The-Cambridge-Handbook-of-Physics-Formulas (1)

main April 22, 2003 15:22 This page intentionally left blank main April 22, 2003 15:22 The Cambridge Handbook of Physics Formulas The Cambridge Handbook of Physics Formulas is a quick-reference aid for students and professionals in the physical sciences and engineering. It contains more than 2 000 of the most useful formulas and equations found in undergraduate physics courses, covering mathematics, dynamics and mechanics, quantum physics, thermodynamics, solid state physics, electromagnetism, optics, and astrophysics. An exhaustive index allows the required formulas to be located swiftly and simply, and the unique tabular format crisply identifies all the variables involved. The Cambridge Handbook of Physics Formulas comprehensively covers the major topics explored in undergraduate physics courses. It is designed to be a compact, portable, reference book suitable for everyday work, problem solving, or exam revision. All students and professionals in physics, applied mathematics, engineering, and other physical sciences will want to have this essential reference book within easy reach. Graham Woan is a senior lecturer in the Department of Physics and Astronomy at the University of Glasgow. Prior to this he taught physics at the University of Cambridge where he also received his degree in Natural Sciences, specialising in physics, and his PhD, in radio astronomy. His research interests range widely with a special focus on low-frequency radio astronomy. His publications span journals as diverse as Astronomy & Astrophysics, Geophysical Research Letters, Advances in Space Science, the Journal of Navigation and Emergency Prehospital Medicine. He was co-developer of the revolutionary CURSOR radio positioning system, which uses existing broadcast transmitters to determine position, and he is the designer of the Glasgow Millennium Sundial. main April 22, 2003 15:22 main April 22, 2003 15:22 The Cambridge Handbook of Physics Formulas 2003 Edition GRAHAM WOAN Department of Physics & Astronomy University of Glasgow    Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge  , UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521573498 © Cambridge University Press 2000 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2000 - - ---- eBook (EBL) --- eBook (EBL) - - ---- hardback --- hardback - - ---- paperback --- paperback Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. main April 22, 2003 15:22 Contents Preface page vii How to use this book 1 1 3 Units, constants, and conversions 1.1 Introduction, 3 • 1.2 SI units, 4 • 1.3 Physical constants, 6 • 1.4 Converting between units, 10 • 1.5 Dimensions, 16 • 1.6 Miscellaneous, 18 2 Mathematics 19 2.1 Notation, 19 • 2.2 Vectors and matrices, 20 • 2.3 Series, summations, and progressions, 27 • 2.4 Complex variables, 30 • 2.5 Trigonometric and hyperbolic formulas, 32 • 2.6 Mensuration, 35 • 2.7 Differentiation, 40 • 2.8 Integration, 44 • 2.9 Special functions and polynomials, 46 • 2.10 Roots of quadratic and cubic equations, 50 • 2.11 Fourier series and transforms, 52 • 2.12 Laplace transforms, 55 • 2.13 Probability and statistics, 57 • 2.14 Numerical methods, 60 3 Dynamics and mechanics 63 3.1 Introduction, 63 • 3.2 Frames of reference, 64 • 3.3 Gravitation, 66 3.4 Particle motion, 68 • 3.5 Rigid body dynamics, 74 • 3.6 Oscillating systems, 78 • 3.7 Generalised dynamics, 79 • 3.8 Elasticity, 80 • 3.9 Fluid • dynamics, 84 4 Quantum physics 89 4.1 Introduction, 89 • 4.2 Quantum definitions, 90 • 4.3 Wave mechanics, 92 • 4.4 Hydrogenic atoms, 95 • 4.5 Angular momentum, 98 • 4.6 Perturbation theory, 102 • 4.7 High energy and nuclear physics, 103 5 Thermodynamics 5.1 Introduction, 105 • 5.2 Classical thermodynamics, 106 • 5.3 Gas laws, 110 • 5.4 Kinetic theory, 112 • 5.5 Statistical thermodynamics, 114 • 5.6 Fluctuations and noise, 116 • 5.7 Radiation processes, 118 105 main 6 April 22, 2003 15:22 Solid state physics 123 6.1 Introduction, 123 • 6.2 Periodic table, 124 • 6.3 Crystalline structure, 126 • 6.4 Lattice dynamics, 129 • 6.5 Electrons in solids, 132 7 Electromagnetism 135 7.1 Introduction, 135 • 7.2 Static fields, 136 • 7.3 Electromagnetic fields (general), 139 • 7.4 Fields associated with media, 142 • 7.5 Force, torque, and energy, 145 • 7.6 LCR circuits, 147 • 7.7 Transmission lines and waveguides, 150 • 7.8 Waves in and out of media, 152 • 7.9 Plasma physics, 156 8 Optics 161 8.1 Introduction, 161 • 8.2 Interference, 162 • 8.3 Fraunhofer diffraction, 164 • 8.4 Fresnel diffraction, 166 • 8.5 Geometrical optics, 168 • 8.6 Polarisation, 170 • 8.7 Coherence (scalar theory), 172 • 8.8 Line radiation, 173 9 Astrophysics 175 9.1 Introduction, 175 • 9.2 Solar system data, 176 • 9.3 Coordinate transformations (astronomical), 177 • 9.4 Observational astrophysics, 179 • 9.5 Stellar evolution, 181 • 9.6 Cosmology, 184 Index 187 main April 22, 2003 15:22 Preface In A Brief History of Time, Stephen Hawking relates that he was warned against including equations in the book because “each equation... would halve the sales.” Despite this dire prediction there is, for a scientific audience, some attraction in doing the exact opposite. The reader should not be misled by this exercise. Although the equations and formulas contained here underpin a good deal of physical science they are useless unless the reader understands them. Learning physics is not about remembering equations, it is about appreciating the natural structures they express. Although its format should help make some topics clearer, this book is not designed to teach new physics; there are many excellent textbooks to help with that. It is intended to be useful rather than pedagogically complete, so that students can use it for revision and for structuring their knowledge once they understand the physics. More advanced users will benefit from having a compact, internally consistent, source of equations that can quickly deliver the relationship they require in a format that avoids the need to sift through pages of rubric. Some difficult decisions have had to be made to achieve this. First, to be short the book only includes ideas that can be expressed succinctly in equations, without resorting to lengthy explanation. A small number of important topics are therefore absent. For example, Liouville’s theorem can be algebraically succinct (˙ = 0) but is meaningless unless ˙ is thoroughly (and carefully) explained. Anyone who already understands what ˙ represents will probably not need reminding that it equals zero. Second, empirical equations with numerical coefficients have been largely omitted, as have topics significantly more advanced than are found at undergraduate level. There are simply too many of these to be sensibly and confidently edited into a short handbook. Third, physical data are largely absent, although a periodic table, tables of physical constants, and data on the solar system are all included. Just a sighting of the marvellous (but dimensionally misnamed) CRC Handbook of Chemistry and Physics should be enough to convince the reader that a good science data book is thick. Inevitably there is personal choice in what should or should not be included, and you may feel that an equation that meets the above criteria is missing. If this is the case, I would be delighted to hear from you so it can be considered for a subsequent edition. Contact details are at the end of this preface. Likewise, if you spot an error or an inconsistency then please let me know and I will post an erratum on the web page. main April 22, 2003 15:22 Acknowledgments This venture is founded on the generosity of colleagues in Glasgow and Cambridge whose inputs have strongly influenced the final product. The expertise of Dave Clarke, Declan Diver, Peter Duffett-Smith, Wolf-Gerrit Früh, Martin Hendry, Rico Ignace, David Ireland, John Simmons, and Harry Ward have been central to its production, as have the linguistic skills of Katie Lowe. I would also like to thank Richard Barrett, Matthew Cartmell, Steve Gull, Martin Hendry, Jim Hough, Darren McDonald, and Ken Riley who all agreed to field-test the book and gave invaluable feedback. My greatest thanks though are to John Shakeshaft who, with remarkable knowledge and skill, worked through the entire manuscript more than once during its production and whose legendary red pen hovered over (or descended upon) every equation in the book. What errors remain are, of course, my own, but I take comfort from the fact that without John they would be much more numerous. Contact information A website containing up-to-date information on this handbook and contact details can be found through the Cambridge University Press web pages at us.cambridge.org (North America) or uk.cambridge.org (United Kingdom), or directly at radio.astro.gla.ac.uk/hbhome.html. Production notes This book was typeset by the author in LATEX 2ε using the CUP Times fonts. The software packages used were WinEdt, MiKTEX, Mayura Draw, Gnuplot, Ghostscript, Ghostview, and Maple V. Comments on the 2002 edition I am grateful to all those who have suggested improvements, in particular Martin Hendry, Wolfgang Jitschin, and Joseph Katz. Although this edition contains only minor revisions to the original its production was also an opportunity to update the physical constants and periodic table entries and to reflect recent developments in cosmology. main April 22, 2003 15:22 How to use this book The format is largely self-explanatory, but a few comments may be helpful. Although it is very tempting to flick through the pages to find what you are looking for, the best starting point is the index. I have tried to make this as extensive as possible, and many equations are indexed more than once. Equations are listed both with their equation number (in square brackets) and the page on which they can be found. The equations themselves are grouped into self-contained and boxed “panels” on the pages. Each panel represents a separate topic, and you will find descriptions of all the variables used at the right-hand side of the panel, usually adjacent to the first equation in which they are used. You should therefore not need to stray outside the panel to understand the notation. Both the panel as a whole and its individual entries may have footnotes, shown below the panel. Be aware of these, as they contain important additional information and conditions relevant to the topic. Although the panels are self-contained they may use concepts defined elsewhere in the handbook. Often these are cross-referenced, but again the index will help you to locate them if necessary. Notations and definitions are uniform over subject areas unless stated otherwise. main April 22, 2003 15:22 main January 23, 2006 16:6 1 Chapter 1 Units, constants, and conversions 1.1 Introduction The determination of physical constants and the definition of the units with which they are measured is a specialised and, to many, hidden branch of science. A quantity with dimensions is one whose value must be expressed relative to one or more standard units. In the spirit of the rest of the book, this section is based around the International System of units (SI). This system uses seven base units1 (the number is somewhat arbitrary), such as the kilogram and the second, and defines their magnitudes in terms of physical laws or, in the case of the kilogram, an object called the “international prototype of the kilogram” kept in Paris. For convenience there are also a number of derived standards, such as the volt, which are defined as set combinations of the basic seven. Most of the physical observables we regard as being in some sense fundamental, such as the charge on an electron, are now known to a relative standard uncertainty,2 ur , of less than 10−7 . The least well determined is the Newtonian constant of gravitation, presently standing at a rather lamentable ur of 1.5 × 10−3 , and the best is the Rydberg constant (ur = 7.6 × 10−12 ). The dimensionless electron g-factor, representing twice the magnetic moment of an electron measured in Bohr magnetons, is now known to a relative uncertainty of only 4.1 × 10−12 . No matter which base units are used, physical quantities are expressed as the product of a numerical value and a unit. These two components have more-or-less equal standing and can be manipulated by following the usual rules of algebra. So, if 1 · eV = 160.218 × 10−21 · J then 1 · J = [1/(160.218 × 10−21 )] · eV. A measurement of energy, U, with joule as the unit has a numerical value of U/ J. The same measurement with electron volt as the unit has a numerical value of U/ eV = (U/ J) · ( J/ eV) and so on. 1 The metre is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second. The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram. The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom. The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 metre apart in vacuum, would produce between these conductors a force equal to 2 × 10−7 newton per metre of length. The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12; its symbol is “mol.” When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles. The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 × 1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian. 2 The relative standard uncertainty in x is defined as the estimated standard deviation in x divided by the modulus of x (x = 0). main January 23, 2006 16:6 4 Units, constants, and conversions 1.2 SI units SI base units physical quantity length mass time interval electric current thermodynamic temperature amount of substance luminous intensity a Or name metrea kilogram second ampere kelvin mole candela symbol m kg s A K mol cd “meter”. SI derived units physical quantity catalytic activity electric capacitance electric charge electric conductance electric potential difference electric resistance energy, work, heat force frequency illuminance inductance luminous flux magnetic flux magnetic flux density plane angle power, radiant flux pressure, stress radiation absorbed dose radiation dose equivalenta radioactive activity solid angle temperatureb a To name katal farad coulomb siemens volt ohm joule newton hertz lux henry lumen weber tesla radian watt pascal gray sievert becquerel steradian degree Celsius symbol kat F C S V Ω J N Hz lx H lm Wb T rad W Pa Gy Sv Bq sr ◦ C equivalent units mol s−1 C V −1 As Ω−1 J C−1 V A−1 Nm m kg s−2 s−1 cd sr m−2 V A−1 s cd sr Vs V s m−2 m m−1 J s−1 N m−2 J kg−1 [ J kg−1 ] s−1 m2 m−2 K distinguish it from the gray, units of J kg−1 should not be used for the sievert in practice. Celsius temperature, TC , is defined from the temperature in kelvin, TK , by TC = TK − 273.15. b The main January 23, 2006 16:6 5 1.2 SI units 1 SI prefixesa factor 1024 1021 1018 1015 1012 109 106 103 102 101 prefix yotta zetta exa peta tera giga mega kilo hecto decab symbol Y Z E P T G M k h da factor 10−24 10−21 10−18 10−15 10−12 10−9 10−6 10−3 10−2 10−1 prefix yocto zepto atto femto pico nano micro milli centi deci symbol y z a f p n µ m c d a The kilogram is the only SI unit with a prefix embedded in its name and symbol. For mass, the unit name “gram” and unit symbol “g” should be used with these prefixes, hence 10−6 kg can be written as 1 mg. Otherwise, any prefix can be applied to any SI unit. b Or “deka”. Recognised non-SI units physical quantity area energy length plane angle pressure time mass volume a These name barn electron volt ångström fermia microna degree arcminute arcsecond bar minute hour day unified atomic mass unit tonnea,b litrec are non-SI names for SI quantities. “metric ton.” c Or “liter”. The symbol “l” should be avoided. b Or symbol b eV Å fm µm bar min h d SI value 10−28 m2 1.602 18 × 10−19 J 10−10 m 10−15 m 10−6 m (π/180) rad (π/10 800) rad (π/648 000) rad 105 N m−2 60 s 3 600 s 86 400 s u t l, L 1.660 54 × 10−27 kg 103 kg 10−3 m3 ◦ main January 23, 2006 16:6 6 Units, constants, and conversions 1.3 Physical constants The following 1998 CODATA recommended values for the fundamental physical constants can also be found on the Web at physics.nist.gov/constants. Detailed background information is available in Reviews of Modern Physics, Vol. 72, No. 2, pp. 351–495, April 2000. The digits in parentheses represent the 1σ uncertainty in the previous two quoted digits. For example, G = (6.673±0.010)×10−11 m3 kg−1 s−2 . It is important to note that the uncertainties for many of the listed quantities are correlated, so that the uncertainty in any expression using them in combination cannot necessarily be computed from the data presented. Suitable covariance values are available in the above references. Summary of physical constants speed of light in vacuuma permeability of vacuumb permittivity of vacuum constant of gravitationc Planck constant h/(2π) elementary charge magnetic flux quantum, h/(2e) electron volt electron mass proton mass proton/electron mass ratio unified atomic mass unit fine-structure constant, µ0 ce2 /(2h) inverse Rydberg constant, me cα2 /(2h) Avogadro constant Faraday constant, NA e molar gas constant Boltzmann constant, R/NA Stefan–Boltzmann constant, π 2 k 4 /(60h̄3 c2 ) Bohr magneton, eh̄/(2me ) a By 2.997 924 58 4π =12.566 370 614 . . . 1/(µ0 c2 ) 0 =8.854 187 817 . . . G 6.673(10) h 6.626 068 76(52) h̄ 1.054 571 596(82) e 1.602 176 462(63) 2.067 833 636(81) Φ0 eV 1.602 176 462(63) 9.109 381 88(72) me 1.672 621 58(13) mp mp /me 1 836.152 667 5(39) u 1.660 538 73(13) α 7.297 352 533(27) 1/α 137.035 999 76(50) 1.097 373 156 854 9(83) R∞ 6.022 141 99(47) NA F 9.648 534 15(39) R 8.314 472(15) k 1.380 650 3(24) ×108 m s−1 ×10−7 H m−1 ×10−7 H m−1 F m−1 ×10−12 F m−1 ×10−11 m3 kg−1 s−2 ×10−34 J s ×10−34 J s ×10−19 C ×10−15 Wb ×10−19 J ×10−31 kg ×10−27 kg σ 5.670 400(40) ×10−8 W m−2 K−4 µB 9.274 008 99(37) ×10−24 J T−1 c µ0 definition, the speed of light is exact. exact, by definition. Alternative units are N A−2 . c The standard acceleration due to gravity, g, is defined as exactly 9.806 65 m s−2 . b Also ×10−27 kg ×10−3 ×107 m−1 ×1023 mol−1 ×104 C mol−1 J mol−1 K−1 ×10−23 J K−1 main January 23, 2006 16:6 7 1.3 Physical constants 1 General constants speed of light in vacuum permeability of vacuum permittivity of vacuum impedance of free space constant of gravitation Planck constant in eV s h/(2π) in eV s Planck mass, (h̄c/G)1/2 Planck length, h̄/(mPl c) = (h̄G/c3 )1/2 Planck time, lPl /c = (h̄G/c5 )1/2 elementary charge magnetic flux quantum, h/(2e) Josephson frequency/voltage ratio Bohr magneton, eh̄/(2me ) in eV T−1 µB /k nuclear magneton, eh̄/(2mp ) in eV T−1 µN /k Zeeman splitting constant 2.997 924 58 4π =12.566 370 614 . . . 1/(µ0 c2 ) 0 =8.854 187 817 . . . µ0 c Z0 =376.730 313 461 . . . G 6.673(10) h 6.626 068 76(52) 4.135 667 27(16) h̄ 1.054 571 596(82) 6.582 118 89(26) mPl 2.176 7(16) lPl 1.616 0(12) tPl 5.390 6(40) e 1.602 176 462(63) 2.067 833 636(81) Φ0 2e/h 4.835 978 98(19) 9.274 008 99(37) µB 5.788 381 749(43) 0.671 713 1(12) 5.050 783 17(20) µN 3.152 451 238(24) 3.658 263 8(64) µB /(hc) 46.686 452 1(19) c µ0 ×108 m s−1 ×10−7 H m−1 ×10−7 H m−1 F m−1 ×10−12 F m−1 Ω Ω ×10−11 m3 kg−1 s−2 ×10−34 J s ×10−15 eV s ×10−34 J s ×10−16 eV s ×10−8 kg ×10−35 m ×10−44 s ×10−19 C ×10−15 Wb ×1014 Hz V −1 ×10−24 J T−1 ×10−5 eV T−1 K T−1 ×10−27 J T−1 ×10−8 eV T−1 ×10−4 K T−1 m−1 T−1 Atomic constantsa fine-structure constant, µ0 ce2 /(2h) inverse Rydberg constant, me cα2 /(2h) R∞ c R∞ hc R∞ hc/e Bohr radiusb , α/(4πR∞ ) a See also the Bohr model on page 95. b Fixed nucleus. α 1/α R∞ a0 7.297 352 533(27) 137.035 999 76(50) 1.097 373 156 854 9(83) 3.289 841 960 368(25) 2.179 871 90(17) 13.605 691 72(53) 5.291 772 083(19) ×10−3 ×107 m−1 ×1015 Hz ×10−18 J eV ×10−11 m main January 23, 2006 16:6 8 Units, constants, and conversions Electron constants electron mass in MeV electron/proton mass ratio electron charge electron specific charge electron molar mass, NA me Compton wavelength, h/(me c) classical electron radius, α2 a0 Thomson cross section, (8π/3)re2 electron magnetic moment in Bohr magnetons, µe /µB in nuclear magnetons, µe /µN electron gyromagnetic ratio, 2|µe |/h̄ electron g-factor, 2µe /µB 9.109 381 88(72) ×10−31 kg 0.510 998 902(21) MeV ×10−4 me /mp 5.446 170 232(12) −e −1.602 176 462(63) ×10−19 C ×1011 C kg−1 −e/me −1.758 820 174(71) Me 5.485 799 110(12) ×10−7 kg mol−1 2.426 310 215(18) ×10−12 m λC re 2.817 940 285(31) ×10−15 m σT 6.652 458 54(15) ×10−29 m2 −9.284 763 62(37) ×10−24 J T−1 µe −1.001 159 652 186 9(41) −1 838.281 966 0(39) 1.760 859 794(71) ×1011 s−1 T−1 γe ge −2.002 319 304 3737(82) me Proton constants proton mass in MeV proton/electron mass ratio proton charge proton specific charge proton molar mass, NA mp proton Compton wavelength, h/(mp c) proton magnetic moment in Bohr magnetons, µp /µB in nuclear magnetons, µp /µN proton gyromagnetic ratio, 2µp /h̄ 1.672 621 58(13) 938.271 998(38) 1 836.152 667 5(39) 1.602 176 462(63) 9.578 834 08(38) 1.007 276 466 88(13) 1.321 409 847(10) 1.410 606 633(58) 1.521 032 203(15) 2.792 847 337(29) 2.675 222 12(11) ×10−27 kg MeV 1.674 927 16(13) 939.565 330(38) mn /me 1 838.683 655 0(40) mn /mp 1.001 378 418 87(58) Mn 1.008 664 915 78(55) 1.319 590 898(10) λC,n −9.662 364 0(23) µn µn /µB −1.041 875 63(25) µn /µN −1.913 042 72(45) 1.832 471 88(44) γn ×10−27 kg MeV mp mp /me e e/mp Mp λC,p µp γp ×10−19 C ×107 C kg−1 ×10−3 kg mol−1 ×10−15 m ×10−26 J T−1 ×10−3 ×108 s−1 T−1 Neutron constants neutron mass in MeV neutron/electron mass ratio neutron/proton mass ratio neutron molar mass, NA mn neutron Compton wavelength, h/(mn c) neutron magnetic moment in Bohr magnetons in nuclear magnetons neutron gyromagnetic ratio, 2|µn |/h̄ mn ×10−3 kg mol−1 ×10−15 m ×10−27 J T−1 ×10−3 ×108 s−1 T−1 main January 23, 2006 16:6 9 1.3 Physical constants 1 Muon and tau constants 1.883 531 09(16) 105.658 356 8(52) 3.167 88(52) mτ 1.777 05(29) mµ /me 206.768 262(30) −e −1.602 176 462(63) −4.490 448 13(22) µµ 4.841 970 85(15) 8.890 597 70(27) −2.002 331 832 0(13) gµ mµ muon mass in MeV tau mass in MeV muon/electron mass ratio muon charge muon magnetic moment in Bohr magnetons, µµ /µB in nuclear magnetons, µµ /µN muon g-factor ×10−28 kg MeV ×10−27 kg ×103 MeV ×10−19 C ×10−26 J T−1 ×10−3 Bulk physical constants Avogadro constant atomic mass constanta in MeV Faraday constant molar gas constant Boltzmann constant, R/NA in eV K−1 molar volume (ideal gas at stp)b Stefan–Boltzmann constant, π 2 k 4 /(60h̄3 c2 ) Wien’s displacement law constant,c b = λm T NA mu F R k Vm σ b 6.022 141 99(47) 1.660 538 73(13) 931.494 013(37) 9.648 534 15(39) 8.314 472(15) 1.380 650 3(24) 8.617 342(15) 22.413 996(39) 5.670 400(40) 2.897 768 6(51) ×1023 mol−1 ×10−27 kg MeV ×104 C mol−1 J mol−1 K−1 ×10−23 J K−1 ×10−5 eV K−1 ×10−3 m3 mol−1 ×10−8 W m−2 K−4 ×10−3 m K mass of 12 C/12. Alternative nomenclature for the unified atomic mass unit, u. temperature and pressure (stp) are T = 273.15 K (0◦ C) and P = 101 325 Pa (1 standard atmosphere). c See also page 121. a= b Standard Mathematical constants pi (π) exponential constant (e) Catalan’s constant Euler’s constanta (γ) Feigenbaum’s constant (α) Feigenbaum’s constant (δ) Gibbs constant golden mean Madelung constantb a See also Equation (2.119). structure. b NaCl 3.141 2.718 0.915 0.577 2.502 4.669 1.851 1.618 1.747 592 281 965 215 907 201 937 033 564 653 828 594 664 875 609 051 988 594 589 459 177 901 095 102 982 749 633 793 045 219 532 892 990 466 894 182 238 235 015 860 822 671 170 848 190 462 360 054 606 283 853 361 204 636 643 287 603 512 902 203 053 586 212 383 471 514 090 873 820 370 834 035 279 352 932 082 218 466 157 370 544 ... ... ... ... ... ... ... ... ... main January 23, 2006 16:6 10 1.4 Units, constants, and conversions Converting between units The following table lists common (and not so common) measures of physical quantities. The numerical values given are the SI equivalent of one unit measure of the non-SI unit. Hence 1 astronomical unit equals 149.597 9 × 109 m. Those entries identified with a “∗ ” in the second column represent exact conversions; so 1 abampere equals exactly 10.0 A. Note that individual entries in this list are not recorded in the index, and that values are “international” unless otherwise stated. There is a separate section on temperature conversions after this table. unit name abampere abcoulomb abfarad abhenry abmho abohm abvolt acre amagat (at stp) ampere hour ångström apostilb arcminute arcsecond are astronomical unit atmosphere (standard) atomic mass unit value in SI units 10.0∗ A 10.0∗ C 1.0∗ ×109 F 1.0∗ ×10−9 H ∗ 1.0 ×109 S ∗ 1.0 ×10−9 Ω ∗ 10.0 ×10−9 V 4.046 856 ×103 m2 44.614 774 mol m−3 ∗ 3.6 ×103 C ∗ ×10−12 m 100.0 ∗ 1.0 lm m−2 290.888 2 ×10−6 rad 4.848 137 ×10−6 rad ∗ 100.0 m2 149.597 9 ×109 m ∗ 101.325 0 ×103 Pa 1.660 540 ×10−27 kg bar barn baromil barrel (UK) barrel (US dry) barrel (US liquid) barrel (US oil) baud bayre biot bolt (US) brewster British thermal unit bushel (UK) bushel (US) butt (UK) cable (US) calorie 100.0∗ 100.0∗ 750.1 163.659 2 115.627 1 119.240 5 158.987 3 1.0∗ 100.0∗ 10.0 36.576∗ 1.0∗ 1.055 056 36.36 872 35.23 907 477.339 4 219.456∗ 4.186 8∗ ×103 Pa ×10−30 m2 ×10−6 m ×10−3 m3 ×10−3 m3 ×10−3 m3 ×10−3 m3 s−1 ×10−3 Pa A m ×10−12 m2 N−1 ×103 J ×10−3 m3 ×10−3 m3 ×10−3 m3 m J continued on next page . . . main January 23, 2006 16:6 11 1.4 Converting between units 1 unit name candle power (spherical) carat (metric) cental centare centimetre of Hg (0 ◦ C) centimetre of H2 O (4 ◦ C) chain (engineers’) chain (US) Chu clusec cord cubit cumec cup (US) curie value in SI units 4π lm 200.0∗ ×10−6 kg 45.359 237 kg 1.0∗ m2 1.333 222 ×103 Pa 98.060 616 Pa 30.48∗ m 20.116 8∗ m 1.899 101 ×103 J 1.333 224 ×10−6 W 3.624 556 m3 457.2∗ ×10−3 m 1.0∗ m3 s−1 236.588 2 ×10−6 m3 37.0∗ ×109 Bq darcy day day (sidereal) debye degree (angle) denier digit dioptre Dobson unit dram (avoirdupois) dyne dyne centimetres 986.923 3 86.4∗ 86.164 09 3.335 641 17.453 29 111.111 1 19.05∗ 1.0∗ 10.0∗ 1.771 845 10.0∗ 100.0∗ ×10−15 m2 ×103 s ×103 s ×10−30 C m ×10−3 rad ×10−9 kg m−1 ×10−3 m m−1 ×10−6 m ×10−3 kg ×10−6 N ×10−9 J electron volt ell em emu of capacitance emu of current emu of electric potential emu of inductance emu of resistance Eötvös unit esu of capacitance esu of current esu of electric potential esu of inductance esu of resistance erg 160.217 7 1.143∗ 4.233 333 1.0∗ 10.0∗ 10.0∗ 1.0∗ 1.0∗ 1.0∗ 1.112 650 333.564 1 299.792 5 898.755 2 898.755 2 100.0∗ ×10−21 J m ×10−3 m ×109 F A ×10−9 V ×10−9 H ×10−9 Ω ×10−9 m s−2 m−1 ×10−12 F ×10−12 A V ×109 H ×109 Ω ×10−9 J faraday fathom fermi Finsen unit firkin (UK) 96.485 3 1.828 804 1.0∗ 10.0∗ 40.914 81 ×103 C m ×10−15 m ×10−6 W m−2 ×10−3 m3 continued on next page . . . main January 23, 2006 16:6 12 Units, constants, and conversions unit name firkin (US) fluid ounce (UK) fluid ounce (US) foot foot (US survey) foot of water (4 ◦ C) footcandle footlambert footpoundal footpounds (force) fresnel funal furlong value in SI units 34.068 71 ×10−3 m3 28.413 08 ×10−6 m3 29.573 53 ×10−6 m3 ∗ 304.8 ×10−3 m 304.800 6 ×10−3 m 2.988 887 ×103 Pa 10.763 91 lx 3.426 259 cd m−2 42.140 11 ×10−3 J 1.355 818 J 1.0∗ ×1012 Hz 1.0∗ ×103 N 201.168∗ m g (standard acceleration) gal gallon (UK) gallon (US liquid) gamma gauss gilbert gill (UK) gill (US) gon grade grain gram gram-rad gray 9.806 65∗ 10.0∗ 4.546 09∗ 3.785 412 1.0∗ 100.0∗ 795.774 7 142.065 4 118.294 1 π/200∗ 15.707 96 64.798 91∗ 1.0∗ 100.0∗ 1.0∗ m s−2 ×10−3 m s−2 ×10−3 m3 ×10−3 m3 ×10−9 T ×10−6 T ×10−3 A turn ×10−6 m3 ×10−6 m3 rad ×10−3 rad ×10−6 kg ×10−3 kg J kg−1 J kg−1 hand hartree hectare hefner hogshead horsepower (boiler) horsepower (electric) horsepower (metric) horsepower (UK) hour hour (sidereal) Hubble time Hubble distance hundredweight (UK long) hundredweight (US short) 101.6∗ 4.359 748 10.0∗ 902 238.669 7 9.809 50 746∗ 735.498 8 745.699 9 3.6∗ 3.590 170 440 130 50.802 35 45.359 24 ×10−3 m ×10−18 J ×103 m2 ×10−3 cd ×10−3 m3 ×103 W W W W ×103 s ×103 s ×1015 s ×1024 m kg kg inch inch of mercury (0 ◦ C) inch of water (4 ◦ C) 25.4∗ 3.386 389 249.074 0 ×10−3 m ×103 Pa Pa jansky 10.0∗ ×10−27 W m−2 Hz−1 continued on next page . . . main January 23, 2006 16:6 13 1.4 Converting between units 1 unit name jar value in SI units 10/9∗ ×10−9 F kayser kilocalorie kilogram-force kilowatt hour knot (international) 100.0∗ 4.186 8∗ 9.806 65∗ 3.6∗ 514.444 4 m−1 ×103 J N ×106 J ×10−3 m s−1 lambert langley langmuir league (nautical, int.) league (nautical, UK) league (statute) light year ligne line line (magnetic flux) link (engineers’) link (US) litre lumen (at 555 nm) 10/π ∗ 41.84∗ 133.322 4 5.556∗ 5.559 552 4.828 032 9.460 73∗ 2.256∗ 2.116 667 10.0∗ 304.8∗ 201.168 0 1.0∗ 1.470 588 ×103 cd m−2 ×103 J m−2 ×10−6 Pa s ×103 m ×103 m ×103 m ×1015 m ×10−3 m ×10−3 m ×10−9 Wb ×10−3 m ×10−3 m ×10−3 m3 ×10−3 W maxwell mho micron mil (length) mil (volume) mile (international) mile (nautical, int.) mile (nautical, UK) mile per hour milliard millibar millimetre of Hg (0 ◦ C) minim (UK) minim (US) minute (angle) minute minute (sidereal) month (lunar) 10.0∗ 1.0∗ 1.0∗ 25.4∗ 1.0∗ 1.609 344∗ 1.852∗ 1.853 184∗ 447.04∗ 1.0∗ 100.0∗ 133.322 4 59.193 90 61.611 51 290.888 2 60.0∗ 59.836 17 2.551 444 ×10−9 Wb S ×10−6 m ×10−6 m ×10−6 m3 ×103 m ×103 m ×103 m ×10−3 m s−1 ×109 m3 Pa Pa ×10−9 m3 ×10−9 m3 ×10−6 rad s s ×106 s nit noggin (UK) 1.0∗ 142.065 4 cd m−2 ×10−6 m3 oersted ounce (avoirdupois) ounce (UK fluid) ounce (US fluid) 1000/(4π)∗ 28.349 52 28.413 07 29.573 53 A m−1 ×10−3 kg ×10−6 m3 ×10−6 m3 pace parsec 762.0∗ 30.856 78 ×10−3 m ×1015 m continued on next page . . . main January 23, 2006 16:6 14 Units, constants, and conversions unit name peck (UK) peck (US) pennyweight (troy) perch phot pica (printers’) pint (UK) pint (US dry) pint (US liquid) point (printers’) poise pole poncelot pottle pound (avoirdupois) poundal pound-force promaxwell psi puncheon (UK) value in SI units 9.092 18∗ ×10−3 m3 8.809 768 ×10−3 m3 1.555 174 ×10−3 kg ∗ 5.029 2 m 10.0∗ ×103 lx 4.217 518 ×10−3 m 568.261 2 ×10−6 m3 550.610 5 ×10−6 m3 473.176 5 ×10−6 m3 351.459 8∗ ×10−6 m 100.0∗ ×10−3 Pa s 5.029 2∗ m 980.665∗ W 2.273 045 ×10−3 m3 453.592 4 ×10−3 kg 138.255 0 ×10−3 N 4.448 222 N 1.0∗ Wb 6.894 757 ×103 Pa 317.974 6 ×10−3 m3 quad quart (UK) quart (US dry) quart (US liquid) quintal (metric) 1.055 056 1.136 522 1.101 221 946.352 9 100.0∗ ×1018 J ×10−3 m3 ×10−3 m3 ×10−6 m3 kg rad rayleigh rem REN reyn rhe rod roentgen rood (UK) rope (UK) rutherford rydberg 10.0∗ 10/(4π) 10.0∗ 1/4 000∗ 689.5 10.0∗ 5.029 2∗ 258.0 1.011 714 6.096∗ 1.0∗ 2.179 874 ×10−3 Gy ×109 s−1 m−2 sr−1 ×10−3 Sv S ×103 Pa s Pa−1 s−1 m ×10−6 C kg−1 ×103 m2 m ×106 Bq ×10−18 J scruple seam second (angle) second (sidereal) shake shed slug square degree statampere statcoulomb 1.295 978 290.949 8 4.848 137 997.269 6 100.0∗ 100.0∗ 14.593 90 (π/180)2∗ 333.564 1 333.564 1 ×10−3 kg ×10−3 m3 ×10−6 rad ×10−3 s ×10−10 s ×10−54 m2 kg sr ×10−12 A ×10−12 C continued on next page . . . main January 23, 2006 16:6 15 1.4 Converting between units 1 unit name statfarad stathenry statmho statohm statvolt stere sthéne stilb stokes stone value in SI units 1.112 650 ×10−12 F 898.755 2 ×109 H 1.112 650 ×10−12 S 898.755 2 ×109 Ω 299.792 5 V 1.0∗ m3 1.0∗ ×103 N 10.0∗ ×103 cd m−2 100.0∗ ×10−6 m2 s−1 6.350 293 kg tablespoon (UK) tablespoon (US) teaspoon (UK) teaspoon (US) tex therm (EEC) therm (US) thermie thou tog ton (of TNT) ton (UK long) ton (US short) tonne (metric ton) torr townsend troy dram troy ounce troy pound tun 14.206 53 14.786 76 4.735 513 4.928 922 1.0∗ 105.506∗ 105.480 4∗ 4.185 407 25.4∗ 100.0∗ 4.184∗ 1.016 047 907.184 7 1.0∗ 133.322 4 1.0∗ 3.887 935 31.103 48 373.241 7 954.678 9 ×10−6 m3 ×10−6 m3 ×10−6 m3 ×10−6 m3 ×10−6 kg m−1 ×106 J ×106 J ×106 J ×10−6 m ×10−3 W−1 m2 K ×109 J ×103 kg kg ×103 kg Pa ×10−21 V m2 ×10−3 kg ×10−3 kg ×10−3 kg ×10−3 m3 XU 100.209 ×10−15 m yard year (365 days) year (sidereal) year (tropical) 914.4∗ 31.536∗ 31.558 15 31.556 93 ×10−3 m ×106 s ×106 s ×106 s Temperature conversions TK temperature in kelvin TC temperature in ◦ Celsius From degrees Celsiusa TK = TC + 273.15 From degrees Fahrenheit TK = TF − 32 + 273.15 1.8 (1.2) TF From degrees Rankine TK = TR 1.8 (1.3) TR temperature in ◦ Rankine a The (1.1) temperature in ◦ Fahrenheit term “centigrade” is not used in SI, to avoid confusion with “10−2 of a degree”. main January 23, 2006 16:6 16 1.5 Units, constants, and conversions Dimensions The following table lists the dimensions of common physical quantities, together with their conventional symbols and the SI units in which they are usually quoted. The dimensional basis used is length (L), mass (M), time (T), electric current (I), temperature (Θ), amount of substance (N), and luminous intensity (J). physical quantity acceleration action angular momentum angular speed area Avogadro constant bending moment Bohr magneton Boltzmann constant bulk modulus capacitance charge (electric) charge density conductance conductivity couple current current density density electric displacement electric field strength electric polarisability electric polarisation electric potential difference energy energy density entropy Faraday constant force frequency gravitational constant Hall coefficient Hamiltonian heat capacity Hubble constant1 illuminance impedance symbol a S L, J ω A, S NA Gb µB k, kB K C q ρ G σ G, T I, i J, j ρ D E α P V E, U u S F F ν, f G RH H C H Ev Z dimensions L T−2 L2 M T−1 L2 M T−1 T−1 L2 N−1 L2 M T−2 L2 I L2 M T−2 Θ−1 L−1 M T−2 L−2 M−1 T4 I2 TI L−3 T I L−2 M−1 T3 I2 L−3 M−1 T3 I2 L2 M T−2 I L−2 I L−3 M L−2 T I L M T−3 I−1 M−1 T4 I2 L−2 T I L2 M T−3 I−1 L2 M T−2 L−1 M T−2 L2 M T−2 Θ−1 T I N−1 L M T−2 T−1 L3 M−1 T−2 L3 T−1 I−1 L2 M T−2 L2 M T−2 Θ−1 T−1 L−2 J L2 M T−3 I−2 SI units m s−2 Js m2 kg s−1 rad s−1 m2 mol−1 Nm J T−1 J K−1 Pa F C C m−3 S S m−1 Nm A A m−2 kg m−3 C m−2 V m−1 C m2 V −1 C m−2 V J J m−3 J K−1 C mol−1 N Hz m3 kg−1 s−2 m3 C−1 J J K−1 s−1 lx Ω continued on next page . . . Hubble constant is almost universally quoted in units of km s−1 Mpc−1 . There are about 3.1 × 1019 kilometres in a megaparsec. 1 The main January 23, 2006 16:6 17 1.5 Dimensions 1 physical quantity impulse inductance irradiance Lagrangian length luminous intensity magnetic dipole moment magnetic field strength magnetic flux magnetic flux density magnetic vector potential magnetisation mass mobility molar gas constant moment of inertia momentum number density permeability permittivity Planck constant power Poynting vector pressure radiant intensity resistance Rydberg constant shear modulus specific heat capacity speed Stefan–Boltzmann constant stress surface tension temperature thermal conductivity time velocity viscosity (dynamic) viscosity (kinematic) volume wavevector weight work Young modulus symbol I L Ee L L, l Iv m, µ H Φ B A M m, M µ R I p n µ h P S p, P Ie R R∞ µ, G c u, v, c σ σ, τ σ, γ T λ t v, u η, µ ν V, v k W W E dimensions L M T−1 L2 M T−2 I−2 M T−3 L2 M T−2 L J L2 I L−1 I L2 M T−2 I−1 M T−2 I−1 L M T−2 I−1 L−1 I M M−1 T2 I L2 M T−2 Θ−1 N−1 L2 M L M T−1 L−3 L M T−2 I−2 L−3 M−1 T4 I2 L2 M T−1 L2 M T−3 M T−3 L−1 M T−2 L2 M T−3 L2 M T−3 I−2 L−1 L−1 M T−2 L2 T−2 Θ−1 L T−1 M T−3 Θ−4 L−1 M T−2 M T−2 Θ L M T−3 Θ−1 T L T−1 L−1 M T−1 L2 T−1 L3 L−1 L M T−2 L2 M T−2 L−1 M T−2 SI units Ns H W m−2 J m cd A m2 A m−1 Wb T Wb m−1 A m−1 kg m2 V −1 s−1 J mol−1 K−1 kg m2 kg m s−1 m−3 H m−1 F m−1 Js W W m−2 Pa W sr−1 Ω m−1 Pa J kg−1 K−1 m s−1 W m−2 K−4 Pa N m−1 K W m−1 K−1 s m s−1 Pa s m2 s−1 m3 m−1 N J Pa main January 23, 2006 16:6 18 1.6 Units, constants, and conversions Miscellaneous Greek alphabet A α alpha N ν nu B β beta Ξ ξ xi Γ γ gamma O o omicron ∆ δ delta Π π pi E epsilon P ρ rho Z ζ zeta Σ σ ς sigma H η eta T τ tau Θ θ theta Υ υ upsilon I ι iota Φ φ K κ kappa X χ chi Λ λ lambda Ψ ψ psi M µ mu Ω ω omega ε ϑ ϕ phi Pi (π) to 1 000 decimal places 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989 e to 1 000 decimal places 2.7182818284 5904523536 0287471352 6624977572 4709369995 9574966967 6277240766 3035354759 4571382178 5251664274 2746639193 2003059921 8174135966 2904357290 0334295260 5956307381 3232862794 3490763233 8298807531 9525101901 1573834187 9307021540 8914993488 4167509244 7614606680 8226480016 8477411853 7423454424 3710753907 7744992069 5517027618 3860626133 1384583000 7520449338 2656029760 6737113200 7093287091 2744374704 7230696977 2093101416 9283681902 5515108657 4637721112 5238978442 5056953696 7707854499 6996794686 4454905987 9316368892 3009879312 7736178215 4249992295 7635148220 8269895193 6680331825 2886939849 6465105820 9392398294 8879332036 2509443117 3012381970 6841614039 7019837679 3206832823 7646480429 5311802328 7825098194 5581530175 6717361332 0698112509 9618188159 3041690351 5988885193 4580727386 6738589422 8792284998 9208680582 5749279610 4841984443 6346324496 8487560233 6248270419 7862320900 2160990235 3043699418 4914631409 3431738143 6405462531 5209618369 0888707016 7683964243 7814059271 4563549061 3031072085 1038375051 0115747704 1718986106 8739696552 1267154688 9570350354 main January 23, 2006 16:6 Chapter 2 Mathematics 2 2.1 Notation Mathematics is, of course, a vast subject, and so here we concentrate on those mathematical methods and relationships that are most often applied in the physical sciences and engineering. Although there is a high degree of consistency in accepted mathematical notation, there is some variation. For example the spherical harmonics, Yl m , can be written Ylm , and there is some freedom with their signs. In general, the conventions chosen here follow common practice as closely as possible, whilst maintaining consistency with the rest of the handbook. In particular: scalars a general vectors a unit vectors â scalar product a·b vector cross-product a× b gradient operator Laplacian operator ∇2 derivative ∇ df etc. dx ∂f etc. ∂x n d f dxn ds derivative of r with respect to t partial derivatives nth derivative closed surface integral ṙ closed loop integral dl L matrix mean value (of x) x binomial coefficient factorial ! unit imaginary (i2 = −1) A or aij n r i exponential constant e modulus (of x) |x| natural logarithm ln log to base 10 log10 S main January 23, 2006 16:6 20 2.2 Mathematics Vectors and matrices Vector algebra Scalar producta b Vector product Product rules Lagrange’s identity Scalar triple product a · b = |a||b|cosθ (2.1) x̂ a× b = |a||b|sinθ n̂ = ax bx ŷ ay by ẑ az bz a× b = −b× a a · (b + c) = (a · b) + (a · c) a× (b + c) = (a× b) + (a× c) (2.4) (2.5) (2.6) (a× b) · (c× d) = (a · c)(b · d) − (a · d)(b · c) (2.7) ax (a× b) · c = bx cx (2.8) az bz cz = (b× c) · a = (c× a) · b Vector a with respect to a nonorthogonal basis {e1 ,e2 ,e3 }c a Also (2.9) (2.10) (2.11) a× (b× c) = (a · c)b − (a · b)c (2.12) b = (c× a)/[(a× b) · c] c = (a× b)/[(a× b) · c] n̂ (in) a (a× b)× c = (a · c)b − (b · c)a a = (b× c)/[(a× b) · c] Reciprocal vectors b (2.3) ay by cy θ (2.2) a·b=b·a = volume of parallelepiped Vector triple product a (2.13) (2.14) (2.15) (a · a) = (b · b) = (c · c) = 1 (2.16) a = (e1 · a)e1 + (e2 · a)e2 + (e3 · a)e3 (2.17) known as the “dot product” or the “inner product.” known as the “cross-product.” n̂ is a unit vector making a right-handed set with a and b. c The prime ( ) denotes a reciprocal vector. b Also c b main January 23, 2006 16:6 21 2.2 Vectors and matrices Common three-dimensional coordinate systems z 2 ρ point P θ r y x φ x = ρcosφ = r sinθ cosφ (2.18) y = ρsinφ = r sinθ sinφ (2.19) z = r cosθ (2.20) coordinate system: coordinates of P : volume element: metric elementsa (h1 ,h2 ,h3 ): rectangular (x,y,z) dx dy dz (1,1,1) ρ = (x2 + y 2 )1/2 (2.21) r = (x2 + y 2 + z 2 )1/2 (2.22) θ = arccos(z/r) (2.23) φ = arctan(y/x) (2.24) spherical polar (r,θ,φ) r2 sinθ dr dθ dφ (1,r,r sinθ) cylindrical polar (ρ,φ,z) ρ dρ dz dφ (1,ρ,1) a In an orthogonal coordinate system (parameterised by coordinates q1 ,q2 ,q3 ), the differential line element dl is obtained from (dl)2 = (h1 dq1 )2 + (h2 dq2 )2 + (h3 dq3 )2 . Gradient Rectangular coordinates ∇f = ∂f ∂f ∂f x̂ + ŷ + ẑ ∂x ∂y ∂z (2.25) Cylindrical coordinates ∇f = 1 ∂f ∂f ∂f ρ̂ + φ̂ + ẑ ∂ρ r ∂φ ∂z (2.26) Spherical polar coordinates ∇f = 1 ∂f 1 ∂f ∂f r̂ + θ̂ + φ̂ ∂r r ∂θ r sinθ ∂φ (2.27) General orthogonal coordinates ∇f = q̂ 1 ∂f q̂ 2 ∂f q̂ 3 ∂f + + h1 ∂q1 h2 ∂q2 h3 ∂q3 (2.28) f ˆ scalar field unit vector ρ distance from the z axis qi hi basis metric elements main January 23, 2006 16:6 22 Mathematics Divergence Rectangular coordinates ∇·A= ∂Ax ∂Ay ∂Az + + ∂x ∂y ∂z (2.29) Cylindrical coordinates ∇·A= 1 ∂(ρAρ ) 1 ∂Aφ ∂Az + + ρ ∂ρ ρ ∂φ ∂z (2.30) Spherical polar coordinates ∇·A= General orthogonal coordinates 1 ∂(Aθ sinθ) 1 ∂Aφ 1 ∂(r2 Ar ) + + r2 ∂r r sinθ ∂θ r sinθ ∂φ (2.31) ∂ ∂ 1 (A1 h2 h3 ) + (A2 h3 h1 ) ∇·A= h1 h2 h3 ∂q1 ∂q2 ∂ (A3 h1 h2 ) (2.32) + ∂q3 A Ai vector field ith component of A ρ distance from the z axis qi basis hi metric elements ˆ unit vector A vector field Ai ith component of A ρ distance from the z axis qi basis hi metric elements Curl Rectangular coordinates Cylindrical coordinates Spherical polar coordinates General orthogonal coordinates x̂ ∇× A = ∂/∂x Ax ρ̂/ρ ∇× A = ∂/∂ρ Aρ ẑ ∂/∂z Az ŷ ∂/∂y Ay φ̂ ∂/∂φ ρAφ r̂/(r2 sinθ) ∇× A = ∂/∂r Ar ẑ/ρ ∂/∂z Az θ̂/(r sinθ) q̂ h 1 1 1 ∇× A = ∂/∂q1 h1 h2 h3 h1 A1 (2.33) ∂/∂θ rAθ q̂ 2 h2 ∂/∂q2 h2 A2 (2.34) φ̂/r ∂/∂φ rAφ sinθ q̂ 3 h3 ∂/∂q3 h3 A3 (2.35) (2.36) Radial formsa r ∇r = r ∇·r =3 (2.37) (2.38) ∇r2 = 2r (2.39) ∇ · (rr) = 4r (2.40) a Note −r r3 1 ∇ · (r/r2 ) = 2 r −2r 2 ∇(1/r ) = 4 r ∇ · (r/r3 ) = 4πδ(r) ∇(1/r) = that the curl of any purely radial function is zero. δ(r) is the Dirac delta function. (2.41) (2.42) (2.43) (2.44) main January 23, 2006 16:6 23 2.2 Vectors and matrices Laplacian (scalar) Rectangular ∂2 f ∂2 f ∂2 f 2 f = + + (2.45) ∇ coordinates ∂x2 ∂y 2 ∂z 2 Cylindrical 1 ∂ ∂f 1 ∂2 f ∂2 f 2 f = + (2.46) ∇ ρ + coordinates ρ ∂ρ ∂ρ ρ2 ∂φ2 ∂z 2 Spherical ∂ ∂2 f 1 ∂ 1 ∂f 1 2 2 ∂f f = ∇ r + sinθ + polar 2 2 2 r ∂r ∂r r sinθ ∂θ ∂θ r2 sin θ ∂φ2 coordinates (2.47) ∂ h2 h3 ∂f h3 h1 ∂f 1 ∂ + ∇2 f = General h1 h2 h3 ∂q1 h1 ∂q1 ∂q2 h2 ∂q2 orthogonal h1 h2 ∂f ∂ coordinates + (2.48) ∂q3 h3 ∂q3 f scalar field ρ distance from the z axis qi basis hi metric elements Differential operator identities ∇(fg) ≡ f∇g + g∇f (2.49) ∇ · (fA) ≡ f∇ · A + A · ∇f (2.50) ∇× (fA) ≡ f∇× A + (∇f)× A (2.51) ∇(A · B) ≡ A× (∇× B) + (A · ∇)B + B× (∇× A) + (B · ∇)A (2.52) ∇ · (A× B) ≡ B · (∇× A) − A · (∇× B) (2.53) ∇× (A× B) ≡ A(∇ · B) − B(∇ · A) + (B · ∇)A − (A · ∇)B (2.54) ∇ · (∇f) ≡ ∇ f ≡ 2 f,g scalar fields A,B vector fields A dV vector field volume element Sc V closed surface volume enclosed S ds L surface surface element loop bounding S dl line element f,g scalar fields (2.55) f ∇× (∇f) ≡ 0 (2.56) ∇ · (∇× A) ≡ 0 (2.57) ∇× (∇× A) ≡ ∇(∇ · A) − ∇2 A (2.58) Vector integral transformations Gauss’s (Divergence) theorem Stokes’s theorem (∇ · A) dV = (∇× A) · ds = Green’s second theorem A · dl S (2.60) L (f∇g) · ds = S V = (2.59) Sc Green’s first theorem A · ds V ∇ · (f∇g) dV (2.61) [f∇2 g + (∇f) · (∇g)] dV (2.62) V [f(∇g) − g(∇f)] · ds = S (f∇2 g − g∇2 f) dV V (2.63) 2 main January 23, 2006 16:6 24 Mathematics Matrix algebraa  Matrix definition a11  a21  A= .  .. a12 a22 .. . am1 am2 C = A + B if Matrix addition C = AB Matrix multiplication ··· ··· ··· ···  a1n a2n   ..  .  (2.64) A m by n matrix aij matrix elements amn cij = aij + bij (2.65) if cij = aik bkj (2.66) (AB)C = A(BC) A(B + C) = AB + AC (2.67) (2.68) Transpose matrixb ãij = aji = Ñ... B̃Ã (AB...N) (2.69) Adjoint matrix (definition 1)c A† = Ã∗ (AB...N)† = N† ...B† A† (2.71) (2.72) Hermitian matrixd H† = H (2.73) (2.70) ãij transpose matrix (sometimes aT ij , or aij ) ∗ † complex conjugate (of each component) adjoint (or Hermitian conjugate) H Hermitian (or self-adjoint) matrix examples:  a11  A = a21 a31  a11  Ã = a12 a13 a12 a13  a22  a23  a32 a33 a21 a31 b31  a22  a32  a23 a33  a11 b11 + a12 b21 + a13 b31  AB = a21 b11 + a22 b21 + a23 b31 a31 b11 + a32 b21 + a33 b31 a Terms  b11  B = b21 b12 b13  b22  b23  b32 b33  a11 + b11  A + B = a21 + b21 a31 + b31 a12 + b12 a13 + b13  a22 + b22  a23 + b23  a32 + b32 a33 + b33 a11 b12 + a12 b22 + a13 b32 a21 b12 + a22 b22 + a23 b32 a31 b12 + a32 b22 + a33 b32 a11 b13 + a12 b23 + a13 b33   a21 b13 + a22 b23 + a23 b33  a31 b13 + a32 b23 + a33 b33 are implicitly summed over repeated suffices; hence aik bkj equals k aik bkj . also Equation (2.85). c Or “Hermitian conjugate matrix.” The term “adjoint” is used in quantum physics for the transpose conjugate of a matrix and in linear algebra for the transpose matrix of its cofactors. These definitions are not compatible, but both are widely used [cf. Equation (2.80)]. d Hermitian matrices must also be square (see next table). b See main January 23, 2006 16:6 25 2.2 Vectors and matrices Square matricesa Trace Determinantb Adjoint matrix (definition 2)c Inverse matrix (detA = 0) trA = aii tr(AB) = tr(BA) (2.74) (2.75) A square matrix aij aii matrix elements implicitly = i aii detA = ijk... a1i a2j a3k ... = (−1)i+1 ai1 Mi1 = ai1 Ci1 (2.76) (2.77) (2.78) tr det Mij trace determinant (or |A|) minor of element aij det(AB...N) = detAdetB...detN (2.79) Cij cofactor of the element aij adj adjA = C̃ij = Cji (2.80) ∼ adjoint (sometimes written Â) transpose 1 unit matrix δjk Kronecker delta (= 1 if i = j, = 0 otherwise) U unitary matrix Hermitian conjugate adjA Cji = detA detA AA−1 = 1 a−1 ij = (AB...N) Orthogonality condition Symmetry Unitary matrix examples:  a11  A = a21 a12 a31 a32 a22 −1 =N −1 (2.81) (2.82) −1 ...B A −1 (2.83) aij aik = δjk i.e., Ã = A (2.84) −1 If A = Ã, If A = −Ã, (2.85) A is symmetric (2.86) A is antisymmetric (2.87) U† = U−1 a13 (2.88)   a23  a33 trA = a11 + a22 + a33 B= b11 b12 b21 b22 † trB = b11 + b22 detA = a11 a22 a33 − a11 a23 a32 − a21 a12 a33 + a21 a13 a32 + a31 a12 a23 − a31 a13 a22 detB = b11 b22 − b12 b21  a22 a33 − a23 a32 1  −1 A = −a21 a33 + a23 a31 detA a21 a32 − a22 a31 b22 −b12 1 −1 B = detB −b21 b11 −a12 a33 + a13 a32 a11 a33 − a13 a31 −a11 a32 + a12 a31 a12 a23 − a13 a22   −a11 a23 + a13 a21  a11 a22 − a12 a21 are implicitly summed over repeated suffices; hence aik bkj equals k aik bkj . b ijk... is defined as the natural extension of Equation (2.443) to n-dimensions (see page 50). Mij is the determinant of the matrix A with the ith row and the jth column deleted. The cofactor Cij = (−1)i+j Mij . c Or “adjugate matrix.” See the footnote to Equation (2.71) for a discussion of the term “adjoint.” a Terms 2 main January 23, 2006 16:6 26 Mathematics Commutators Commutator definition [A,B] = AB − BA = −[B,A] (2.89) [·,·] commutator Adjoint [A,B]† = [B† ,A† ] (2.90) † adjoint Distribution [A + B,C] = [A,C] + [B,C] (2.91) Association [AB,C] = A[B,C] + [A,C]B (2.92) Jacobi identity [A,[B,C]] = [B,[A,C]] − [C,[A,B]] (2.93) (2.94) σi 1 i Pauli spin matrices 2 × 2 unit matrix i2 = −1 δij Kronecker delta Pauli matrices 0 1 1 0 1 0 σ3 = 0 −1 σ1 = Pauli matrices 0 i 1 1= 0 σ2 = −i 0 0 1 Anticommutation σ i σ j + σ j σ i = 2δij 1 (2.95) Cyclic permutation σ i σ j = iσ k (2.96) 2 (σ i ) = 1 (2.97) Rotation matricesa   1 0 0 cosθ sinθ  R1 (θ) = 0 0 −sinθ cosθ   cosθ 0 −sinθ 1 0  R2 (θ) =  0 sinθ 0 cosθ   cosθ sinθ 0 R3 (θ) =  −sinθ cosθ 0 0 0 1 Rotation about x1 Rotation about x2 Rotation about x3 Euler angles  Ri (θ) matrix for rotation about the ith axis θ rotation angle α β γ rotation about x3 rotation about x2 rotation about x3 R rotation matrix (2.98) (2.99) (2.100) cosγ cosβ cosα − sinγ sinα cosγ cosβ sinα + sinγ cosα R(α,β,γ) = −sinγ cosβ cosα − cosγ sinα −sinγ cosβ sinα + cosγ cosα sinβ cosα sinβ sinα a Angles  −cosγ sinβ sinγ sinβ  cosβ (2.101) are in the right-handed sense for rotation of axes, or the left-handed sense for rotation of vectors. i.e., a vector v is given a right-handed rotation of θ about the x3 -axis using R3 (−θ)v → v . Conventionally, x1 ≡ x, x2 ≡ y, and x3 ≡ z. main January 23, 2006 16:6 27 2.3 Series, summations, and progressions 2.3 Series, summations, and progressions Progressions and summations Arithmetic progression Sn = a + (a + d) + (a + 2d) + ··· + [a + (n − 1)d] n = [2a + (n − 1)d] 2 n = (a + l) 2 (2.103) a d number of terms sum of n successive terms first term common difference (2.104) l last term r common ratio (2.102) n Sn Sn = a + ar + ar2 + ··· + arn−1 1 − rn =a 1−r a S∞ = (|r| < 1) 1−r (2.106) Arithmetic mean 1 x a = (x1 + x2 + ··· + xn ) n (2.108) . a arithmetic mean Geometric mean x g = (x1 x2 x3 ...xn )1/n (2.109) . g geometric mean (2.110) . h harmonic mean Geometric progression Harmonic mean 1 1 1 + + ··· + x h = n x1 x2 xn Relative mean magnitudes x a ≥ x g ≥ x n i=1 n i=1 n Summation formulas i=1 n i=1 ∞ i=1 ∞ i=1 ∞ Euler’s constanta a γ 0.577215664... h (2.105) (2.107) −1 if xi > 0 for all i (2.111) n i = (n + 1) 2 (2.112) n i2 = (n + 1)(2n + 1) 6 (2.113) i3 = n2 (n + 1)2 4 (2.114) i4 = n (n + 1)(2n + 1)(3n2 + 3n − 1) 30 (2.115) 1 1 1 (−1)i+1 = 1 − + − + ... = ln2 i 2 3 4 (2.116) 1 1 1 π (−1)i+1 = 1 − + − + ... = 2i − 1 3 5 7 4 (2.117) π2 1 1 1 1 = 1 + + + + ... = 2 i 4 9 16 6 i=1 1 1 1 γ = lim 1 + + + ··· + − lnn n→∞ 2 3 n i dummy integer γ Euler’s constant (2.118) (2.119) 2 main January 23, 2006 16:6 28 Mathematics Power series n(n − 1) 2 n(n − 1)(n − 2) 3 x + x + ··· 2! 3! Binomial seriesa (1 + x)n = 1 + nx + Binomial coefficientb n Binomial theorem (a + b)n = Taylor series (about a)c f(a + x) = f(a) + xf (1) (a) + Taylor series (3-D) f(a + x) = f(a) + (x · ∇)f|a + Maclaurin series f(x) = f(0) + xf (1) (0) + (2.120) n n! Cr ≡ ≡ r r!(n − r)! (2.121) n n n−k k a b k (2.122) k=0 x2 (2) xn−1 (n−1) f (a) + ··· + f (a) + ··· 2! (n − 1)! (2.123) (x · ∇)2 (x · ∇)3 f|a + f|a + ··· 2! 3! (2.124) x2 (2) xn−1 (n−1) f (0) + ··· + f (0) + ··· 2! (n − 1)! (2.125) a If n is a positive integer the series terminates and is valid for all x. Otherwise the (infinite) series is convergent for |x| < 1. b The coefficient of xr in the binomial series. c xf (n) (a) is x times the nth derivative of the function f(x) with respect to x evaluated at a, taken as well behaved around a. (x · ∇)n f|a is its extension to three dimensions. Limits nc xn → 0 as xn /n! → 0 as n → ∞ if n→∞ |x| < 1 (for any fixed c) (2.126) (for any fixed x) (2.127) (1 + x/n)n → ex as xlnx → 0 as x→0 (2.129) sinx → 1 as x x→0 (2.130) If n→∞ f(a) = g(a) = 0 or ∞ then (2.128) f(x) f (1) (a) = (1) x→a g(x) g (a) lim (l’Hôpital’s rule) (2.131) main January 23, 2006 16:6 29 2.3 Series, summations, and progressions Series expansions exp(x) 1+x+ ln(1 + x) x− ln 1+x 1−x x2 x3 + + ··· 2! 3! (2.132) (for all x) (2.133) (−1 < x ≤ 1) x3 x5 x7 2 x + + + + ··· 3 5 7 (2.134) (|x| < 1) x2 x3 x4 + − + ··· 2 3 4 cos(x) 1− x2 x4 x6 + − + ··· 2! 4! 6! (2.135) (for all x) sin(x) x− x3 x5 x7 + − + ··· 3! 5! 7! (2.136) (for all x) tan(x) x+ x3 2x5 17x7 + + ··· 3 15 315 (2.137) (|x| < π/2) sec(x) 1+ x2 5x4 61x6 + + + ··· 2 24 720 (2.138) (|x| < π/2) csc(x) 31x5 1 x 7x3 + + + + ··· x 6 360 15120 (2.139) (|x| < π) cot(x) 1 x x3 2x5 − − − − ··· x 3 45 945 (2.140) (|x| < π) arcsin(x)a x+ (2.141) (|x| < 1) arctan(x)b cosh(x) 1 x3 1 · 3 x5 1 · 3 · 5 x7 + + ··· 2 3 2·4 5 2·4·6 7  x3 x5 x7   x − + − + ···    3 5 7   1 π 1 1 − + 3 − 5 + ···  2 x 3x 5x     1 1 1 π  − − + − + ··· 2 x 3x3 5x5 x2 x4 x6 1 + + + + ··· 2! 4! 6! (|x| ≤ 1) (2.142) (x > 1) (x < −1) (2.143) (for all x) sinh(x) x+ x3 x5 x7 + + + ··· 3! 5! 7! (2.144) (for all x) tanh(x) x− x3 2x5 17x7 + − + ··· 3 15 315 (2.145) (|x| < π/2) a arccos(x) = π/2 − arcsin(x). b arccot(x) = π/2 − arctan(x). Note that arcsin(x) ≡ sin−1 (x) etc. 2 main January 23, 2006 16:6 30 Mathematics Inequalities Triangle inequality |a1 | − |a2 | ≤ |a1 + a2 | ≤ |a1 | + |a2 |; n n ≤ a |ai | i i=1 Cauchy inequality 2.4 (2.148) b1 ≥ b2 ≥ b3 ≥ ... ≥ bn n n n then n ai bi ≥ ai bi and n i=1 b i=1 2 ≤ ai bi i=1 Schwarz inequality (2.147) i=1 a1 ≥ a2 ≥ a3 ≥ ... ≥ an if Chebyshev inequality (2.146) n a2i i=1 n (2.149) (2.150) i=1 b2i (2.151) i=1 2 b b f(x)g(x) dx ≤ [f(x)]2 dx [g(x)]2 dx a a (2.152) a Complex variables Complex numbers z complex variable Cartesian form z = x + iy (2.153) i x,y i2 = −1 real variables Polar form z = reiθ = r(cosθ + isinθ) (2.154) r θ amplitude (real) phase (real) Modulusa |z| = r = (x2 + y 2 )1/2 |z1 · z2 | = |z1 | · |z2 | (2.155) (2.156) |z| modulus of z argz argument of z z∗ complex conjugate of z = reiθ n integer y x arg(z1 z2 ) = argz1 + argz2 θ = argz = arctan Argument Complex conjugate Logarithmb a Or z ∗ = x − iy = re−iθ arg(z ∗ ) = −argz ∗ z · z = |z| 2 lnz = lnr + i(θ + 2πn) “magnitude.” principal value of lnz is given by n = 0 and −π < θ ≤ π. b The (2.157) (2.158) (2.159) (2.160) (2.161) (2.162) main January 23, 2006 16:6 31 2.4 Complex variables Complex analysisa Cauchy– Riemann equationsb Cauchy– Goursat theoremc Cauchy integral formulad Laurent expansione if f(z) = u(x,y) + iv(x,y) ∂u ∂v = then ∂x ∂y ∂u ∂v =− ∂y ∂x f(z) dz = 0 a Closed (2.164) complex variable i2 = −1 x,y real variables f(z) function of z u,v real functions (2.165) c f(z) 1 f(z0 ) = dz 2πi c z − z0 f(z) n! dz f (n) (z0 ) = 2πi c (z − z0 )n+1 f(z) = ∞ an (z − z0 )n (2.166) (2.167) (2.168) (n) nth derivative an Laurent coefficients a−1 residue of f(z) at z0 z dummy variable y c2 n=−∞ f(z ) 1 dz 2πi c (z − z0 )n+1 f(z) dz = 2πi enclosed residues where an = Residue theorem (2.163) z i c1 (2.169) (2.170) c z0 x c contour integrals are taken in the counterclockwise sense, once. condition for f(z) to be analytic at a given point. c If f(z) is analytic within and on a simple closed curve c. Sometimes called “Cauchy’s theorem.” d If f(z) is analytic within and on a simple closed curve c, encircling z . 0 e Of f(z), (analytic) in the annular region between concentric circles, c and c , centred on z . c is any closed curve 1 2 0 in this region encircling z0 . b Necessary 2 main January 23, 2006 16:6 32 2.5 Mathematics Trigonometric and hyperbolic formulas Trigonometric relationships sin(A ± B) = sinAcosB ± cosAsinB (2.171) cos(A ± B) = cosAcosB ∓ sinAsinB (2.172) tanA ± tanB 1 ∓ tanAtanB 1 cosAcosB = [cos(A + B) + cos(A − B)] 2 1 sinAcosB = [sin(A + B) + sin(A − B)] 2 1 sinAsinB = [cos(A − B) − cos(A + B)] 2 (2.173) (2.174) (2.175) 2 cos2 A + sin2 A = 1 (2.177) 1 sec2 A − tan2 A = 1 (2.178) 0 csc A − cot A = 1 (2.179) sin2A = 2sinAcosA (2.180) cos2A = cos A − sin A (2.181) 2 tan2A = 2 2tanA 1 − tan2 A (2.183) cos3A = 4cos3 A − 3cosA (2.184) tan x −6 −4 −2 0 2 4 6 4 6 x (2.185) (2.186) 4 2 (2.187) (2.188) tx co 1 cos A = (1 + cos2A) 2 1 2 sin A = (1 − cos2A) 2 1 cos3 A = (3cosA + cos3A) 4 1 sin3 A = (3sinA − sin3A) 4 2 −2 sec x A−B A+B cos 2 2 A+B A−B sinA − sinB = 2cos sin 2 2 A+B A−B cosA + cosB = 2cos cos 2 2 A+B A−B cosA − cosB = −2sin sin 2 2 −1 (2.182) sin3A = 3sinA − 4sin3 A sinA + sinB = 2sin x 2 sin 2 cos x (2.176) csc x tan(A ± B) = 0 −2 −4 (2.189) −6 −4 −2 0 x (2.190) (2.191) (2.192) 2 main January 23, 2006 16:6 33 2.5 Trigonometric and hyperbolic formulas Hyperbolic relationshipsa sinh(x ± y) = sinhxcoshy ± coshxsinhy (2.193) cosh(x ± y) = coshxcoshy ± sinhxsinhy (2.194) tanh(x ± y) = tanhx ± tanhy 1 ± tanhxtanhy 1 coshxcoshy = [cosh(x + y) + cosh(x − y)] 2 1 sinhxcoshy = [sinh(x + y) + sinh(x − y)] 2 1 sinhxsinhy = [cosh(x + y) − cosh(x − y)] 2 2 (2.195) (2.196) (2.197) (2.198) 4 co cosh2 x − sinh2 x = 1 (2.199) 2 sech2 x + tanh2 x = 1 (2.200) 0 coth2 x − csch2 x = 1 (2.201) −2 sinh2x = 2sinhxcoshx (2.202) cosh2x = cosh2 x + sinh2 x (2.203) sh x tanhx sin hx tanhx tanh2x = 2tanhx 1 + tanh2 x (2.205) cosh3x = 4cosh x − 3coshx (2.206) x−y x+y cosh 2 2 x+y x−y sinhx − sinhy = 2cosh sinh 2 2 x+y x−y coshx + coshy = 2cosh cosh 2 2 x+y x−y coshx − coshy = 2sinh sinh 2 2 sinhx + sinhy = 2sinh 1 cosh2 x = (cosh2x + 1) 2 1 sinh2 x = (cosh2x − 1) 2 1 cosh3 x = (3coshx + cosh3x) 4 1 sinh3 x = (sinh3x − 3sinhx) 4 a These −3 −2 −1 0 1 2 3 x (2.204) sinh3x = 3sinhx + 4sinh3 x 3 −4 (2.207) 4 (2.208) 2 cothx sech x (2.209) 0 cs −2 (2.210) −4 −3 (2.211) (2.212) (2.213) (2.214) can be derived from trigonometric relationships by using the substitutions cosx → coshx and sinx → isinhx. ch x −2 −1 0 x 1 2 3 main January 23, 2006 16:6 34 Mathematics Trigonometric and hyperbolic definitions de Moivre’s theorem (cosx + isinx)n = einx = cosnx + isinnx (2.215) cosx = 1 ix −ix e +e 2 (2.216) coshx = 1 x −x e +e 2 (2.217) sinx = 1 ix −ix e −e 2i (2.218) sinhx = 1 x −x e −e 2 (2.219) tanx = sinx cosx (2.220) tanhx = sinhx coshx (2.221) cosix = coshx (2.222) coshix = cosx (2.223) sinix = isinhx (2.224) sinhix = isinx (2.225) cotx = (tanx)−1 (2.226) cothx = (tanhx)−1 (2.227) secx = (cosx)−1 (2.228) sechx = (coshx)−1 (2.229) cscx = (sinx)−1 (2.230) cschx = (sinhx)−1 (2.231) 1.6 Inverse trigonometric functionsa in cs ar nx arcta x 1 0 (2.234) x 1.6 (2.235) (2.236) π − arcsinx 2 (2.237) 1 in the angle range 0 ≤ θ ≤ π/2. Note that arcsinx ≡ sin−1 x etc. 0 arccsc x a Valid 1 (2.233) 1 x arccotx = arctan arccosx = (2.232) ar cc os x x cot arc x arcsinx = arctan (1 − x2 )1/2 (1 − x2 )1/2 arccosx = arctan x 1 arccscx = arctan (x2 − 1)1/2 arcsecx = arctan (x2 − 1)1/2 1 cx arcse 2 3 x 4 5 main January 23, 2006 16:6 35 2.6 Mensuration Inverse hyperbolic functions arsinhx ≡ sinh−1 x = ln x + (x2 + 1)1/2 (2.238) 2 (2.239) 0 arsi −1 1+x 1 artanhx ≡ tanh−1 x = ln (2.240) 2 1−x x+1 1 −1 arcothx ≡ coth x = ln (2.241) 2 x−1 1 (1 − x2 )1/2 −1 + arsechx ≡ sech x = ln x x (2.242) 1 (1 + x2 )1/2 + arcschx ≡ csch−1 x = ln x x (2.243) |x| < 1 −2 −1 0 x 4 3 arcoth x 0<x≤1 2 x = 0 ar 1 arc se Mensuration Moiré fringesa −1 1 1 dM = − d1 d2 Rotational patternb d dM = 2|sin(θ/2)| (2.244) dM a From b From overlapping linear gratings. identical gratings, spacing d, with a relative rotation θ. Moiré fringe spacing d1,2 grating spacings d (2.245) 1 |x| > 1 sch x ch x 1 x Parallel pattern 2 nh x 0 2.6 x arc x≥1 nh ta ar x arcoshx ≡ cosh−1 x = ln x + (x2 − 1)1/2 1 osh for all x θ common grating spacing relative rotation angle (|θ| ≤ π/2) 2 main January 23, 2006 16:6 36 Mathematics Plane triangles Sine formulaa Cosine formulas a b c = = sinA sinB sinC (2.246) a2 = b2 + c2 − 2bccosA (2.247) b +c −a 2bc a = bcosC + ccosB (2.248) 2 2 cosA = Tangent formula C A−B a−b = cot tan 2 a+b 2 Area 1 = absinC 2 a2 sinB sinC = 2 sinA = [s(s − a)(s − b)(s − c)]1/2 1 where s = (a + b + c) 2 area a The C 2 (2.249) b a A B c (2.250) (2.251) (2.252) (2.253) (2.254) diameter of the circumscribed circle equals a/sinA. Spherical trianglesa Sine formula Cosine formulas Analogue formula sinb sinc sina = = sinA sinB sinC (2.255) cosa = cosbcosc + sinbsinccosA (2.256) cosA = −cosB cosC + sinB sinC cosa (2.257) sinacosB = cosbsinc − sinbcosccosA (2.258) Four-parts formula cosacosC = sinacotb − sinC cotB (2.259) Areab E =A+B +C −π (2.260) a On a unit sphere. called the “spherical excess.” b Also a b C A B c main January 23, 2006 16:6 37 2.6 Mensuration Perimeter, area, and volume Perimeter of circle P = 2πr (2.261) P r perimeter radius Area of circle A = πr2 (2.262) A area Surface area of spherea A = 4πR 2 (2.263) R sphere radius Volume of sphere 4 V = πR 3 3 (2.264) V volume a semi-major axis b E semi-minor axis elliptic integral of the second kind (p. 45) eccentricity (= 1 − b2 /a2 ) P = 4aE(π/2, e) 2 1/2 a + b2 2π 2 (2.265) Area of ellipse A = πab (2.267) Volume of ellipsoidc V = 4π Surface area of cylinder b Perimeter of ellipse (2.266) abc 3 e 2 (2.268) c third semi-axis A = 2πr(h + r) (2.269) h height Volume of cylinder V = πr2 h (2.270) Area of circular coned A = πrl (2.271) l slant height Volume of cone or pyramid V = Ab h/3 (2.272) Ab base area Surface area of torus A = π 2 (r1 + r2 )(r2 − r1 ) (2.273) r1 inner radius r2 outer radius Volume of torus V= Aread of spherical cap, depth d A = 2πRd (2.275) d cap depth (2.276) Ω z α solid angle distance from centre half-angle subtended Volume of spherical cap, depth d π2 2 2 (r − r )(r2 − r1 ) 4 2 1 2 V = πd Solid angle of a circle from a point on its axis, z from centre d R− 3 (2.274) z Ω = 2π 1 − 2 2 1/2 (z + r ) = 2π(1 − cosα) (2.277) (2.278) defined by x2 + y 2 + z 2 = R 2 . approximation is exact when e = 0 and e 0.91, giving a maximum error of 11% at e = 1. c Ellipsoid defined by x2 /a2 + y 2 /b2 + z 2 /c2 = 1. d Curved surface only. a Sphere b The α z r main January 23, 2006 16:6 38 Mathematics Conic sections y y y b equation parametric form foci x x a a a parabola ellipse hyperbola y 2 = 4ax x2 y 2 + =1 a2 b2 x2 y 2 − =1 a2 b2 x = acost y = bsint √ (± a2 − b2 ,0) x = ±acosht y = bsinht √ (± a2 + b2 ,0) x = t2 /(4a) y=t (a,0) eccentricity e=1 directrices x = −a √ √ a2 − b2 e= a a x=± e e= a2 + b2 a x=± a e x Platonic solidsa solid (faces,edges,vertices) tetrahedron (4,6,4) cube (6,12,8) octahedron (8,12,6) dodecahedron (12,30,20) icosahedron (20,30,12) a Of volume surface area √ a3 2 12 √ a 3 a3 6a2 √ a3 2 3 √ 2a2 3 √ a3 (15 + 7 5) 4 √ 5a3 (3 + 5) 12 2 3a2 √ 5(5 + 2 5) √ 5a2 3 circumradius inradius √ a 6 4 √ a 3 2 √ a 6 12 a √ 2 a √ 6 √ a 50 + 22 5 4 5 5 a √ 3+ 4 3 √ a√ 3(1 + 5) 4 a 4 √ 2(5 + 5) a 2 side a. Both regular and irregular polyhedra follow the Euler relation, faces − edges + vertices = 2. main January 23, 2006 16:6 39 2.6 Mensuration Curve measure Length of plane curve Surface of revolution Volume of revolution Radius of curvature b 1+ l= a dy dx b dx y 1+ A = 2π a start point (2.279) b y(x) l end point plane curve length dy dx (2.280) A surface area (2.281) V volume ρ radius of curvature 2 1/2 dx b y 2 dx V =π a 2 1/2 a ρ= 1+ dy dx 2 3/2 d2 y dx2 −1 (2.282) Differential geometrya τ r tangent curve parameterised by r(t) v |ṙ(t)| (2.284) n principal normal (2.285) b binormal |ṙ× r̈| |ṙ|3 (2.286) κ curvature ρ= 1 κ (2.287) ρ radius of curvature λ= ... ṙ · (r̈× r ) |ṙ× r̈|2 (2.288) λ torsion Unit tangent τ̂ = ṙ ṙ = |ṙ| v (2.283) Unit principal normal r̈ −v̇ τ̂ n̂ = |r̈ −v̇ τ̂ | Unit binormal b̂ = τ̂ × n̂ Curvature κ= Radius of curvature Torsion n̂ Frenet’s formulas τ̂˙ = κv n̂ (2.289) ˙ n̂ = −κv τ̂ + λv b̂ (2.290) ˙ b̂ = −λv n̂ (2.291) osculating plane normal plane τ̂ rectifying plane r b̂ origin a For a continuous curve in three dimensions, traced by the position vector r(t). 2 main January 23, 2006 16:6 40 2.7 Mathematics Differentiation Derivatives (general) Power Product Quotient Function of a functiona Leibniz theorem Differentiation under the integral sign General integral Logarithm Exponential Inverse functions a The “chain rule.” d n du (u ) = nun−1 dx dx dv du d (uv) = u +v dx dx dx d u ! 1 du u dv = − dx v v dx v 2 dx d du d [f(u)] = [f(u)] · dx du dx n n d u n dv dn−1 u dn [uv] = v + + ··· 0 dxn 1 dx dxn−1 dxn n k n−k n d v n d v d u + ··· + u + k n−k n dxn k dx dx q d f(x) dx = f(q) (p constant) dq p q d f(x) dx = −f(p) (q constant) dp p v(x) du d dv − f(u) f(t) dt = f(v) dx u(x) dx dx d (logb |ax|) = (xlnb)−1 dx d ax (e ) = aeax dx −1 dy dx = dy dx −3 d2 x d2 y dy = − dy 2 dx2 dx 2 −5 d2 y dy d3 x dy d3 y = 3 − dy 3 dx2 dx dx3 dx (2.292) n power index (2.293) u,v functions of x (2.294) (2.295) f(u) function of u(x) n (2.296) k binomial coefficient (2.297) (2.298) (2.299) (2.300) (2.301) (2.302) (2.303) (2.304) b a log base constant main January 23, 2006 16:6 41 2.7 Differentiation Trigonometric derivativesa d (sinax) = acosax dx (2.305) d (cosax) = −asinax dx (2.306) d (tanax) = asec2 ax dx (2.307) d (cscax) = −acscax · cotax dx (2.308) d (secax) = asecax · tanax dx (2.309) d (cotax) = −acsc2 ax dx (2.310) d (arcsinax) = a(1 − a2 x2 )−1/2 dx (2.311) d (arccosax) = −a(1 − a2 x2 )−1/2 dx (2.312) d (arctanax) = a(1 + a2 x2 )−1 dx (2.313) a 2 2 d (arccscax) = − (a x − 1)−1/2 dx |ax| (2.314) d (arccotax) = −a(a2 x2 + 1)−1 dx (2.316) a 2 2 d (arcsecax) = (a x − 1)−1/2 (2.315) dx |ax| aa is a constant. Hyperbolic derivativesa d (sinhax) = acoshax dx (2.317) d (coshax) = asinhax dx (2.318) d (tanhax) = asech2 ax dx (2.319) d (cschax) = −acschax · cothax dx (2.320) d (sechax) = −asechax · tanhax dx (2.321) d (cothax) = −acsch2 ax dx (2.322) d (arsinhax) = a(a2 x2 + 1)−1/2 dx (2.323) d (arcoshax) = a(a2 x2 − 1)−1/2 dx (2.324) d (artanhax) = a(1 − a2 x2 )−1 dx (2.325) a d (arcschax) = − (1 + a2 x2 )−1/2 (2.326) dx |ax| a d (arsechax) = − (1 − a2 x2 )−1/2 dx |ax| (2.327) aa is a constant. d (arcothax) = a(1 − a2 x2 )−1 dx (2.328) 2 main January 23, 2006 16:6 42 Mathematics Partial derivatives Total differential Reciprocity Chain rule Jacobian Change of variable ∂f ∂f ∂f dx + dy + dz ∂x ∂y ∂z ∂g ∂x ∂y = −1 ∂x y ∂y g ∂g x df = f f(x,y,z) (2.330) g g(x,y) J Jacobian u v w u(x,y,z) v(x,y,z) w(x,y,z) V V volume in (x,y,z) volume in (u,v,w) mapped to by V ∂f ∂f ∂x ∂f ∂y ∂f ∂z = + + (2.331) ∂u ∂x ∂u ∂y ∂u ∂z ∂u ∂x ∂x ∂x ∂u ∂v ∂w ∂(x,y,z) ∂y ∂y ∂y (2.332) = J= ∂(u,v,w) ∂u ∂v ∂w ∂z ∂z ∂z ∂u ∂v ∂w f(x,y,z) dxdy dz = f(u,v,w)J dudv dw V V Euler– Lagrange equation (2.329) if b I= (2.333) F(x,y,y ) dx a then d ∂F = ∂y dx δI = 0 when ∂F ∂y y dy/dx a,b fixed end points (2.334) Stationary pointsa saddle point maximum minimum ∂f ∂f = = 0 at (x0 ,y0 ) ∂x ∂y Additional sufficient conditions Stationary point if (necessary condition) (2.335) 2 2 ∂ f ∂2 f ∂2 f > 2 2 ∂x ∂y ∂x∂y 2 ∂2 f ∂2 f ∂2 f > ∂x2 ∂y 2 ∂x∂y 2 for maximum ∂2 f < 0, ∂x2 and for minimum ∂2 f > 0, ∂x2 and for saddle point 2 ∂ f ∂2 f ∂2 f < 2 2 ∂x ∂y ∂x∂y a Of quartic minimum a function f(x,y) at the point (x0 ,y0 ). Note that at, for example, a quartic minimum (2.336) (2.337) (2.338) ∂2 f ∂x2 2 = ∂∂yf2 = 0. main January 23, 2006 16:6 43 2.7 Differentiation Differential equations Laplace ∇2 f = 0 (2.339) f f(x,y,z) Diffusiona ∂f = D∇2 f ∂t (2.340) D diffusion coefficient Helmholtz ∇2 f + α2 f = 0 (2.341) α constant Wave ∇2 f = (2.342) c wave speed (2.343) l integer (2.344) m integer 1 ∂2 f c2 ∂t2 Associated Legendre d 2 dy (1 − x ) + l(l + 1)y = 0 dx dx d m2 2 dy y=0 (1 − x ) + l(l + 1) − dx dx 1 − x2 Bessel x2 Hermite d2 y dy + 2αy = 0 − 2x dx2 dx Laguerre x d2 y dy + αy = 0 + (1 − x) dx2 dx (2.347) Associated Laguerre x dy d2 y + αy = 0 + (1 + k − x) dx2 dx (2.348) k integer Chebyshev (1 − x2 ) (2.349) n integer Euler (or Cauchy) x2 (2.350) a,b constants Bernoulli dy + p(x)y = q(x)y a dx (2.351) p,q functions of x Airy d2 y = xy dx2 (2.352) Legendre d2 y dy + (x2 − m2 )y = 0 +x 2 dx dx d2 y dy + n2 y = 0 −x dx2 dx d2 y dy + by = f(x) + ax 2 dx dx (2.345) (2.346) a Also known as the “conduction equation.” For thermal conduction, f ≡ T and D, the thermal diffusivity, ≡ κ ≡ λ/(ρcp ), where T is the temperature distribution, λ the thermal conductivity, ρ the density, and cp the specific heat capacity of the material. 2 main January 23, 2006 16:6 44 Mathematics 2.8 Integration Standard formsa xn+1 x dx = n+1 n u dv = [uv] − v du (n = −1) (2.355) (2.357) lnax dx = x(lnax − 1) (2.359) aa x2 1 xlnax dx = lnax − 2 2 1 1 dx = ln(a + bx) a + bx b −1 1 dx = 2 (a + bx) b(a + bx) 1 x dx = ln|x2 ± a2 | x2 ± a2 2 (2.363) (2.365) (2.369) x dx = (x2 ± a2 )1/2 (2.373) 2 (x ± a2 )1/2 ax x 1 − a a2 f (x) dx = lnf(x) f(x) bax dx = x! 1 (2.371) dx = arcsin a (a2 − x2 )1/2 and b are non-zero constants. 1 xn 1 (2.367) dx = ln x(xn + a) an xn + a (2.356) xe dx = e (2.361) dv dx (2.354) u dx dx 1 dx = ln|x| x ax u dx − 1 e dx = eax a ax uv dx = v (2.353) bax alnb (2.358) (2.360) (b > 0) (2.362) 1 1 a + bx dx = − ln x(a + bx) a x (2.364) bx 1 1 dx = arctan 2 2 2 a +b x ab a (2.366) 1 1 x − a ln dx = x2 − a2 2a x + a (2.368) x −1 dx = (x2 ± a2 )n 2(n − 1)(x2 ± a2 )n−1 (2.370) 1 dx = ln|x + (x2 ± a2 )1/2 | (x2 ± a2 )1/2 (2.372) x! 1 1 arcsec dx = a a x(x2 − a2 )1/2 (2.374) main January 23, 2006 16:6 45 2.8 Integration Trigonometric and hyperbolic integrals sinx dx = −cosx (2.375) cosx dx = sinx (2.377) tanx dx = −ln|cosx| (2.379) coshx dx = sinhx (2.378) tanhx dx = ln(coshx) (2.380) x cscx dx = ln tan 2 (2.381) x cschx dx = ln tanh 2 (2.382) secx dx = ln|secx + tanx| (2.383) sechx dx = 2arctan(ex ) (2.384) cotx dx = ln|sinx| (2.385) cothx dx = ln|sinhx| (2.386) sinmx · sinnx dx = sin(m − n)x sin(m + n)x − 2(m − n) 2(m + n) sinmx · cosnx dx = − cosmx · cosnx dx = Error function Complementary error function sin(m − n)x sin(m + n)x + 2(m − n) 2(m + n) Gamma function (2.389) 0 2 π 1/2 (2.390) ∞ exp(−t2 ) dt (2.391) x x πt2 πt2 dt; S(x) = dt sin 2 2 0 0 1/2 π 1+i erf (1 − i)x C(x) + i S(x) = 2 2 x t e dt (x > 0) Ei(x) = −∞ t ∞ Γ(x) = tx−1 e−t dt (x > 0) x cos φ F(φ,k) = 0 1 (1 − k 2 sin2 θ)1/2 0 E(φ,k) = also page 167. (m2 = n2 ) (2.388) x erfc(x) = 1 − erf(x) = 0 Elliptic integrals (trigonometric form) (m2 = n2 ) (2.387) exp(−t2 ) dt π 1/2 Exponential integral 2 erf(x) = C(x) = Fresnel integralsa (m2 = n2 ) cos(m − n)x cos(m + n)x − 2(m − n) 2(m + n) Named integrals a See (2.376) sinhx dx = coshx φ dθ (1 − k 2 sin2 θ)1/2 dθ (first kind) (second kind) (2.392) (2.393) (2.394) (2.395) (2.396) (2.397) 2 main January 23, 2006 16:6 46 Mathematics Definite integrals ∞ e−ax dx = 2 0 ∞ 1 π !1/2 2 a xe−ax dx = 1 2a xn e−ax dx = n! an+1 2 0 ∞ 0 (a > 0) (2.398) (a > 0) (2.399) (a > 0; n = 0,1,2,...) (2.400) 2 b π !1/2 exp(2bx − ax ) dx = exp (a > 0) a a −∞ " ∞ 1 · 3 · 5 · ... · (n − 1)(2a)−(n+1)/2 (π/2)1/2 n > 0 and even n −ax2 x e dx = n > 1 and odd 2 · 4 · 6 · ... · (n − 1)(2a)−(n+1)/2 0 1 p!q! (p,q integers > 0) xp (1 − x)q dx = (p + q + 1)! 0 ∞ ∞ 1 π !1/2 2 cos(ax ) dx = sin(ax2 ) dx = (a > 0) 2 2a 0 0 ∞ 2 ∞ sinx sin x π dx = dx = 2 x x 2 0 0 ∞ 1 π (0 < a < 1) dx = a (1 + x)x sinaπ 0 2.9 ∞ 2 (2.401) (2.402) (2.403) (2.404) (2.405) (2.406) Special functions and polynomials Gamma function Definition Γ(z) = ∞ tz−1 e−t dt [(z) > 0] (2.407) (n = 0,1,2,...) (2.408) 0 n! = Γ(n + 1) = nΓ(n) Relations Stirling’s formulas (for |z|,n 1) 1/2 Γ(1/2) = π z Γ(z + 1) z! = = w w!(z − w)! Γ(w + 1)Γ(z − w + 1) 1 1 + − ··· Γ(z) e−z z z−(1/2) (2π)1/2 1 + 12z 288z 2 (2.409) (2.410) (2.411) n! nn+(1/2) e−n (2π)1/2 (2.412) ln(n!) nlnn − n (2.413) main January 23, 2006 16:6 47 2.9 Special functions and polynomials Bessel functions x !ν (−x2 /4)k Jν (x) = 2 k!Γ(ν + k + 1) Jν (x) ∞ Series expansion Yν (x) = Jν (x)cos(πν) − J−ν (x) sin(πν) Approximations " x ν 1 π 2 2 1/2 sin x − 12 νπ − π4 πx (0 ≤ x ν) (x ν) (0 < x ν) (x ν) Iν (x) = (−i)ν Jν (ix) π Kν (x) = iν+1 [Jν (ix) + iYν (ix)] 2 π !1/2 jν (x) = Jν+ 1 (x) 2 2x Modified Bessel functions Spherical Bessel function (2.415) order (ν ≥ 0) ν 1 Jν (x) Γ(ν+1) 2 2 1/2 cos x − 12 νπ − π4 πx " −Γ(ν) x −ν Yν (x) (2.414) k=0 Bessel function of the first kind Yν (x) Bessel function of the second kind Γ(ν) Gamma function J0 0.5 J1 (2.416) 0 −0.5 (2.417) (2.418) (2.419) −1 0 Iν (x) Y0 Y1 2 4 x 6 8 10 modified Bessel function of the first kind Kν (x) modified Bessel function of the second kind jν (x) (2.420) spherical Bessel function of the first kind [similarly for yν (x)] Legendre polynomialsa d2 Pl (x) dPl (x) + l(l + 1)Pl (x) = 0 − 2x dx2 dx (2.421) Legendre equation (1 − x2 ) Rodrigues’ formula Pl (x) = Recurrence relation (l + 1)Pl+1 (x) = (2l + 1)xPl (x) − lPl−1 (x) 1 Orthogonality −1 Explicit form 1 dl 2 (x − 1)l 2l l! dxl Pl (x)Pl (x) dx = Pl (x) = 2−l l/2 m=0 Expansion of plane wave Kronecker delta l binomial coefficients (2.423) (2.424) l 2l − 2m l−2m (2.425) x m l exp(ikz) = exp(ikr cosθ) ∞ (2l + 1)il jl (kr)Pl (cosθ) = l=0 δll (2.422) 2 δll 2l + 1 (−1)m l Legendre polynomials order (l ≥ 0) Pl m (2.426) k z (2.427) jl wavenumber propagation axis z = r cosθ spherical Bessel function of the first kind (order l) P0 (x) = 1 P2 (x) = (3x2 − 1)/2 P4 (x) = (35x4 − 30x2 + 3)/8 P1 (x) = x P3 (x) = (5x3 − 3x)/2 P5 (x) = (63x5 − 70x3 + 15x)/8 a Of the first kind. 2 main January 23, 2006 16:6 48 Mathematics Associated Legendre functionsa Associated Legendre equation d dP m (x) m2 P m (x) = 0 (1 − x2 ) l + l(l + 1) − dx dx 1 − x2 l (2.428) Plm associated Legendre functions dm Pl (x), 0 ≤ m ≤ l dxm (l − m)! m P (x) Pl−m (x) = (−1)m (l + m)! l Pl Legendre polynomials !! 5!! = 5 · 3 · 1 etc. δll Kronecker delta Plm (x) = (1 − x2 )m/2 From Legendre polynomials Recurrence relations (2.429) (2.430) m (x) = x(2m + 1)Pmm (x) Pm+1 (2.431) Pmm (x) = (−1)m (2m − 1)!!(1 − x2 )m/2 (2.432) m m (l − m + 1)Pl+1 (x) = (2l + 1)xPlm (x) − (l + m)Pl−1 (x) (2.433) 1 Orthogonality −1 Plm (x)Plm (x) dx = (l + m)! 2 δll (l − m)! 2l + 1 (2.434) P00 (x) = 1 P10 (x) = x P11 (x) = −(1 − x2 )1/2 P20 (x) = (3x2 − 1)/2 P21 (x) = −3x(1 − x2 )1/2 P22 (x) = 3(1 − x2 ) a Of the first kind. Plm (x) can be defined with a (−1)m factor in Equation (2.429) as well as Equation (2.430). Legendre polynomials 1 P3 2 P1 2 P2 0.5 associated Legendre functions P2 P0 3 P P4 2 P5 0 1 P00 1 P 0 −0.5 2 0 P1 0 1 P1 −1 −1 −1 −0.5 0 x 0.5 1 −1 0 x 1 main January 23, 2006 16:6 49 2.9 Special functions and polynomials Spherical harmonics 1 ∂ ∂ 1 ∂2 Yl m + l(l + 1)Yl m = 0 sinθ + 2 sinθ ∂θ ∂θ sin θ ∂φ2 (2.435) 1/2 2l + 1 (l − m)! Yl m (θ,φ) = (−1)m Plm (cosθ)eimφ 4π (l + m)! (2.436) Differential equation Definitiona 2π π φ=0 θ=0 f(θ,φ) = Plm associated Legendre functions Y∗ complex conjugate Kronecker delta m Yl (θ,φ)Yl (θ,φ)sinθ dθ dφ = δmm δll (2.437) ∞ l alm Yl m (θ,φ) l=0 m=−l 2π Laplace series spherical harmonics m∗ Orthogonality Yl m δll (2.438) f continuous function ψ continuous function a,b constants π where alm = φ=0 θ=0 Yl m∗ (θ,φ)f(θ,φ)sinθ dθ dφ (2.439) if Solution to Laplace equation ∇2 ψ(r,θ,φ) = 0, ψ(r,θ,φ) = ∞ l then $ # Yl m (θ,φ) · alm rl + blm r−(l+1) l=0 m=−l 1 4π Y00 (θ,φ) = Y1±1 (θ,φ) = ∓ Y2±1 (θ,φ) = ∓ Y30 (θ,φ) = 1 2 Y3±2 (θ,φ) = 1 4 3 sinθ e±iφ 8π 15 sinθ cosθ e±iφ 8π 7 (5cos2 θ − 3)cosθ 4π 105 2 sin θ cosθ e±2iφ 2π (2.440) 3 cosθ 4π 5 3 1 cos2 θ − Y20 (θ,φ) = 4π 2 2 15 sin2 θ e±2iφ Y2±2 (θ,φ) = 32π 1 21 ±1 sinθ(5cos2 θ − 1)e±iφ Y3 (θ,φ) = ∓ 4 4π 1 35 3 ±3iφ ±3 sin θ e Y3 (θ,φ) = ∓ 4 4π Y10 (θ,φ) = a Defined for −l ≤ m ≤ l, using the sign convention of the Condon–Shortley phase. Other sign conventions are possible. 2 main January 23, 2006 16:6 50 Mathematics Delta functions Kronecker delta Threedimensional Levi–Civita symbol (permutation tensor)a " 1 if i = j δij = 0 if i = j δii = 3 (2.442) 123 = 231 = 312 = 1 132 = 213 = 321 = −1 (2.443) δij Kronecker delta i,j,k,... indices (= 1,2 or 3) all other ijk = 0 ijk klm = δil δjm − δim δjl δij ijk = 0 ilm jlm = 2δij (2.444) (2.445) (2.446) ijk ijk = 6 (2.447) b a Dirac delta function (2.441) b " 1 if a < 0 < b δ(x) dx = 0 otherwise f(x)δ(x − x0 ) dx = f(x0 ) (2.449) δ(x − x0 )f(x) = δ(x − x0 )f(x0 ) δ(−x) = δ(x) (2.450) (2.451) δ(ax) = |a|−1 δ(x) (2.452) δ(x) nπ e (a = 0) (n 1) Levi–Civita symbol (see also page 25) δ(x) f(x) Dirac delta function smooth function of x a,b constants (2.448) a −1/2 −n2 x2 ijk (2.453) general symbol ijk... is defined to be +1 for even permutations of the suffices, −1 for odd permutations, and 0 if a suffix is repeated. The sequence (1,2,3,... ,n) is taken to be even. Swapping adjacent suffices an odd (or even) number of times gives an odd (or even) permutation. a The 2.10 Roots of quadratic and cubic equations Quadratic equations (a = 0) Equation ax2 + bx + c = 0 Solutions √ −b ± b2 − 4ac x1,2 = 2a −2c √ = b ± b2 − 4ac Solution combinations x1 + x2 = −b/a x1 x2 = c/a (2.454) x a,b,c variable real constants x1 ,x2 quadratic roots (2.455) (2.456) (2.457) (2.458) main January 23, 2006 16:6 51 2.10 Roots of quadratic and cubic equations Cubic equations Equation ax3 + bx2 + cx + d = 0 (a = 0) 3c b2 − 2 a a 3 9bc 27d 1 2b − 2 + q= 27 a3 a a ! ! 3 2 q p + D= 3 2 p= Intermediate definitions 1 3 If D ≥ 0, also define: !1/3 −q + D1/2 u= (2.463) 2 !1/3 −q − D1/2 (2.464) v= 2 y1 = u + v (2.465) u − v 1/2 −(u + v) y2,3 = ±i 3 (2.466) 2 2 1 real, 2 complex roots (if D = 0: 3 real roots, at least 2 equal) Solutionsa Solution combinations ay n (2.459) xn = yn − b 3a 2 (2.460) (2.461) D discriminant (2.462) If D < 0, also define: −3/2 −q |p| φ = arccos 2 3 1/2 |p| φ y1 = 2 cos 3 3 1/2 |p| φ±π y2,3 = −2 cos 3 3 (2.467) (2.468) (2.469) 3 distinct real roots (2.470) x1 + x2 + x3 = −b/a (2.471) x1 x2 + x1 x3 + x2 x3 = c/a x1 x2 x3 = −d/a (2.472) (2.473) are solutions to the reduced equation y 3 + py + q = 0. x variable a,b,c,d real constants xn cubic roots (n = 1,2,3) main January 23, 2006 16:6 52 Mathematics 2.11 Fourier series and transforms Fourier series nπx nπx ! a0 an cos + + bn sin 2 L L n=1 L 1 nπx dx an = f(x)cos L −L L 1 L nπx dx bn = f(x)sin L −L L ∞ inπx cn exp f(x) = L n=−∞ L −inπx 1 f(x)exp cn = dx 2L −L L L ∞ a2 1 2 1 an + b2n |f(x)|2 dx = 0 + 2L −L 4 2 ∞ (2.474) f(x) = Real form Complex form Parseval’s theorem (2.475) = |cn | periodic function, period 2L an ,bn Fourier coefficients cn complex Fourier coefficient || modulus (2.476) (2.477) (2.478) (2.479) n=1 ∞ f(x) 2 (2.480) n=−∞ Fourier transforma ∞ F(s) = −∞ ∞ Definition 1 f(x) = f(x)e−2πixs dx (2.481) 2πixs (2.482) F(s)e ds f(x) F(s) −∞ ∞ f(x)e−ixs dx 1 ∞ f(x) = F(s)eixs ds 2π −∞ ∞ 1 f(x)e−ixs dx F(s) = √ 2π −∞ ∞ 1 f(x) = √ F(s)eixs ds 2π −∞ F(s) = Definition 2 Definition 3 a All (2.483) −∞ (2.484) (2.485) (2.486) three (and more) definitions are used, but definition 1 is probably the best. function of x Fourier transform of f(x) main January 23, 2006 16:6 53 2.11 Fourier series and transforms Fourier transform theoremsa Convolution f(x) ∗ g(x) = ∞ −∞ f(u)g(x − u) du (2.487) f,g general functions ∗ convolution Convolution rules f ∗g =g ∗f f ∗ (g ∗ h) = (f ∗ g) ∗ h (2.488) (2.489) f Convolution theorem f(x)g(x) F(s) ∗ G(s) (2.490) Autocorrelation f ∗ (x) f(x) = (2.491) f∗ complex conjugate of f Wiener– Khintchine theorem 2 f ∗ (x) f(x) |F(s)| Crosscorrelation f ∗ (x) g(x) = Correlation theorem ∗ h(x) j(x) H(s)J (s) h,j H real functions H(s) h(x) J(s) j(x) Parseval’s relationb Parseval’s theoremc Derivatives ∞ −∞ ∞ −∞ ∞ ∞ −∞ |f(x)| dx = 2 Fourier transform relation correlation (2.492) f ∗ (u − x)g(u) du f(x)g ∗ (x) dx = −∞ f ∗ (u − x)f(u) du g f(x) F(s) G(s) g(x) (2.493) (2.494) J ∞ F(s)G∗ (s) ds (2.495) −∞ ∞ −∞ |F(s)|2 ds (2.496) df(x) (2.497) 2πisF(s) dx df(x) dg(x) d [f(x) ∗ g(x)] = ∗ g(x) = ∗ f(x) dx dx dx (2.498) a Defining the Fourier transform as F(s) = called the “power theorem.” c Also called “Rayleigh’s theorem.” b Also %∞ −2πixs −∞ f(x)e dx. Fourier symmetry relationships f(x) even odd real, even real, odd imaginary, even complex, even complex, odd real, asymmetric imaginary, asymmetric F(s) even odd real, even imaginary, odd imaginary, even complex, even complex, odd complex, Hermitian complex, anti-Hermitian definitions real: f(x) = f ∗ (x) imaginary: f(x) = −f ∗ (x) even: f(x) = f(−x) odd: f(x) = −f(−x) Hermitian: f(x) = f ∗ (−x) anti-Hermitian: f(x) = −f ∗ (−x) 2 main January 23, 2006 16:6 54 Mathematics Fourier transform pairsa ∞ f(x)e−2πisx dx F(s) = f(ax) 1 F(s/a) |a| f(x − a) e−2πias F(s) (2πis)n F(s) (2.502) 1 (2.503) δ(x − a) e−2πias (2.504) e−a|x| xe−a|x| e−x /a2 sinax cosax δ(x − ma) f(x) (2.499) −∞ (a = 0, real) (a real) (2.500) (2.501) n d f(x) dxn δ(x) ∞ 2 m=−∞ " 0 x<0 f(x) = (“step”) 1 x>0 " 1 |x| ≤ a f(x) = (“top hat”) 0 |x| > a   1 − |x| |x| ≤ a a f(x) = (“triangle”)  0 |x| > a a Equation 2a a2 + 4π 2 s2 (a > 0) 8iπas (a > 0) (a2 + 4π 2 s2 )2 √ 2 2 2 a πe−π a s 1 a ! a ! δ s− −δ s+ 2i 2π 2π ! a a ! 1 δ s− +δ s+ 2 2π 2π ∞ ! 1 n δ s− a n=−∞ a (2.505) (2.506) (2.507) (2.508) (2.509) (2.510) 1 i δ(s) − 2 2πs (2.511) sin2πas = 2asinc2as πs (2.512) 1 (1 − cos2πas) = asinc2 as (2.513) 2π 2 as2 (2.499) defines the Fourier transform used for these pairs. Note that sincx ≡ (sinπx)/(πx). main January 23, 2006 16:6 55 2.12 Laplace transforms 2.12 Laplace transforms Laplace transform theorems Definitiona & Convolutionb Inversec ∞ F(s) = L{f(t)} = F(s) · G(s) = L f(t)e−st dt 0 ∞ L{} Laplace transform (2.515) F(s) G(s) L{f(t)} L{g(t)} (2.516) ∗ convolution γ constant n integer > 0 a constant u(t) unit step function (2.514) ' f(t − z)g(z) dz 0 = L{f(t) ∗ g(t)} γ+i∞ 1 f(t) = est F(s) ds 2πi γ−i∞ = residues (for t > 0) & dn f(t) dtn ' n−1 (2.517) (2.518) dr f(t) dtr t=0 (2.519) Transform of derivative L Derivative of transform dn F(s) = L{(−t)n f(t)} dsn (2.520) Substitution F(s − a) = L{eat f(t)} (2.521) (2.522) Translation e−as F(s) = L{u(t − a)f(t − a)} " 0 (t < 0) where u(t) = 1 (t > 0) = sn L{f(t)} − r=0 sn−r−1 (2.523) |e−s0 t f(t)| is finite for sufficiently large t, the Laplace transform exists for s > s0 . known as the “faltung (or folding) theorem.” c Also known as the “Bromwich integral.” γ is chosen so that the singularities in F(s) are left of the integral line. a If b Also 2 main January 23, 2006 16:6 56 Mathematics Laplace transform pairs f(t) =⇒ F(s) = L{f(t)} = ∞ f(t)e−st dt (2.524) 0 δ(t) =⇒ 1 1 =⇒ 1/s (2.525) (s > 0) n! (s > 0, n > −1) sn+1 π 1/2 t =⇒ 4s3 π −1/2 =⇒ t s 1 eat =⇒ (s > a) s−a 1 teat =⇒ (s > a) (s − a)2 s (1 − at)e−at =⇒ (s + a)2 2 t2 e−at =⇒ (s + a)3 a sinat =⇒ 2 (s > 0) s + a2 s cosat =⇒ 2 (s > 0) s + a2 a sinhat =⇒ 2 (s > a) s − a2 s (s > a) coshat =⇒ 2 s − a2 a e−bt sinat =⇒ (s + b)2 + a2 s+b e−bt cosat =⇒ (s + b)2 + a2 tn =⇒ e−at f(t) =⇒ F(s + a) (2.526) (2.527) (2.528) (2.529) (2.530) (2.531) (2.532) (2.533) (2.534) (2.535) (2.536) (2.537) (2.538) (2.539) (2.540) main January 23, 2006 16:6 57 2.13 Probability and statistics 2.13 Probability and statistics 2 Discrete statistics x = Mean 1 N N xi (2.541) (2.542) var[·] unbiased variance (2.543) σ i=1 1 (xi − x )2 N −1 N Variancea var[x] = data series series length mean value xi N · i=1 Standard deviation σ[x] = (var[x]) 1/2 standard deviation 3 N xi − x N skew[x] = (2.544) (N − 1)(N − 2) σ i=1 4 N 1 xi − x kurt[x] −3 (2.545) N σ Skewness Kurtosis N i=1 data series to correlate r correlation coefficient a If x is derived from the data, {x }, the relation is as shown. If x is known independently, then an unbiased i estimate is obtained by dividing the right-hand side by N rather than N − 1. b Also known as “Pearson’s r.” Correlation coefficientb r = i=1 (xi − x )(y − y ) i N N 2 2 i=1 (xi − x ) i=1 (yi − y ) x,y (2.546) Discrete probability distributions distribution pr(x) Binomial n Geometric Poisson mean variance domain np np(1 − p) (x = 0,1,... ,n) (2.547) (1 − p)x−1 p 1/p (1 − p)/p2 (x = 1,2,3,...) (2.548) λx exp(−λ)/x! λ λ (x = 0,1,2,...) (2.549) x n px (1 − p)n−x x binomial coefficient main January 23, 2006 16:6 58 Mathematics Continuous probability distributions distribution pr(x) mean variance domain Uniform 1 b−a a+b 2 (b − a)2 12 (a ≤ x ≤ b) (2.550) Exponential λexp(−λx) 1/λ 1/λ2 (x ≥ 0) (2.551) Normal/ Gaussian −(x − µ)2 1 √ exp 2σ 2 σ 2π µ σ2 (−∞ < x < ∞) (2.552) r 2r (x ≥ 0) (2.553) (x ≥ 0) (2.554) (−∞ < x < ∞) (2.555) Chi-squareda Rayleigh Cauchy/ Lorentzian a With e−x/2 x(r/2)−1 2r/2 Γ(r/2) 2 −x x exp σ2 2σ 2 a π(a2 + x2 ) σ ( π/2 (none) 2σ 2 1 − π! 4 (none) r degrees of freedom. Γ is the gamma function. Multivariate normal distribution Density function # $ exp − 12 (x − µ)C−1 (x − µ)T pr(x) = (2π)k/2 [det(C)]1/2 (2.556) pr k C probability density number of dimensions covariance matrix x µ variable (k dimensional) vector of means T Mean µ = (µ1 ,µ2 ,... ,µk ) (2.557) transpose det determinant µi mean of ith variable Covariance C = σij = xi xj − xi xj (2.558) σij components of C Correlation coefficient r= (2.559) r correlation coefficient (2.560) xi normally distributed deviates yi deviates distributed uniformly between 0 and 1 Box–Muller transformation σij σi σj x1 = (−2lny1 )1/2 cos2πy2 x2 = (−2lny1 )1/2 sin2πy2 (2.561) main January 23, 2006 16:6 59 2.13 Probability and statistics Random walk Onedimensional rms displacement Threedimensional Mean distance rms distance pr(x) = −x 1 exp 2 1/2 2Nl 2 (2πNl ) xrms = N 1/2 l a !3 (2.563) exp(−a2 r2 ) 1/2 3 where a = 2Nl 2 1/2 8 N 1/2 l r = 3π (2.564) rrms = N 1/2 l (2.566) pr(r) = π 1/2 Conditional probability pr(x) = Joint probability Bayes’ theorema pr(y|x) = pr(x|y )pr(y ) dy N displacement after N steps (can be positive or negative) probability density of x %∞ ( −∞ pr(x) dx = 1) number of steps l step length (all equal) xrms root-mean-squared displacement from start point r radial distance from start point probability density of r %∞ ( 0 4πr2 pr(r) dr = 1) (most probable distance)−1 x pr(x) (2.562) Bayesian inference a In 2 pr(r) a (2.565) r mean distance from start point rrms root-mean-squared distance from start point pr(x) probability (density) of x (2.567) pr(x|y ) conditional probability of x given y pr(x,y) = pr(x)pr(y|x) (2.568) pr(x,y) joint probability of x and y pr(x|y) pr(y) pr(x) (2.569) this expression, pr(y|x) is known as the posterior probability, pr(x|y) the likelihood, and pr(y) the prior probability. 2 main January 23, 2006 16:6 60 Mathematics 2.14 Numerical methods Straight-line fittinga {xi },{yi } Data n points (2.570) {wi } (2.571) Model y = mx + c (2.572) Residuals di = yi − mxi − c (2.573) Weighted centre ! 1 wi yi wi xi , (x,y) = wi Weights b Weighted moment D= Intercept y = mx + c (0,c) (x,y) x (2.574) wi (xi − x)2 (2.575) 1 wi (xi − x)yi D 2 1 wi di var[m] D n−2 m= Gradient y c = y − mx 2 1 wi di x2 var[c] + wi D n−2 (2.576) (2.577) (2.578) (2.579) a Least-squares b If fit of data to y = mx + c. Errors on y-values only. the errors on yi are uncorrelated, then wi = 1/var[yi ]. Time series analysisa Discrete convolution M/2 (r s)j = sj−k rk (2.580) k=−(M/2)+1 Bartlett (triangular) window Welch (quadratic) window Hanning window Hamming window a The j − N/2 wj = 1 − N/2 (2.581) 2 j − N/2 N/2 2πj 1 wj = 1 − cos 2 N 2πj wj = 0.54 − 0.46cos N wj = 1 − ri si response function time series M response function duration wj windowing function N length of time series 1 (2.582) w 0.6 (2.583) Hamming Welch 0.8 Bartlett 0.4 Hanning 0.2 0 (2.584) 0 0.2 time series runs from j = 0...(N − 1), and the windowing functions peak at j = N/2. 0.4 0.6 j/N 0.8 1 main January 23, 2006 16:6 61 2.14 Numerical methods Numerical integration h 2 f(x) x x0 Trapezoidal rule x0 Simpson’s rulea aN xN xN x0 xN h h f(x) dx (f0 + 2f1 + 2f2 + ··· 2 + 2fN−1 + fN ) fi (2.585) N = (xN − x0 )/N (subinterval width) fi = f(xi ) number of subintervals h f(x) dx (f0 + 4f1 + 2f2 + 4f3 + ··· 3 + 4fN−1 + fN ) (2.586) must be even. Simpson’s rule is exact for quadratics and cubics. Numerical differentiationa 1 df [−f(x + 2h) + 8f(x + h) − 8f(x − h) + f(x − 2h)] dx 12h 1 ∼ [f(x + h) − f(x − h)] 2h (2.587) (2.588) d2 f 1 [−f(x + 2h) + 16f(x + h) − 30f(x) + 16f(x − h) − f(x − 2h)] 2 dx 12h2 1 ∼ 2 [f(x + h) − 2f(x) + f(x − h)] h d3 f 1 [f(x + 2h) − 2f(x + h) + 2f(x − h) − f(x − 2h)] ∼ dx3 2h3 (2.589) (2.590) (2.591) a Derivatives of f(x) at x. h is a small interval in x. Relations containing “” are O(h4 ); those containing “∼” are O(h2 ). Numerical solutions to f(x) = 0 Secant method xn+1 = xn − xn − xn−1 f(xn ) f(xn ) − f(xn−1 ) (2.592) f xn function of x f(x∞ ) = 0 Newton–Raphson method xn+1 = xn − f(xn ) f (xn ) (2.593) f = df/dx main January 23, 2006 16:6 62 Mathematics Numerical solutions to ordinary differential equationsa if Euler’s method and then if and Runge–Kutta method (fourth-order) then dy = f(x,y) dx h = xn+1 − xn (2.594) (2.595) 2 yn+1 = yn + hf(xn ,yn ) + O(h ) dy = f(x,y) dx h = xn+1 − xn k1 = hf(xn ,yn ) k2 = hf(xn + h/2,yn + k1 /2) k3 = hf(xn + h/2,yn + k2 /2) k4 = hf(xn + h,yn + k3 ) k1 k2 k3 k4 yn+1 = yn + + + + + O(h5 ) 6 3 3 6 (2.596) (2.597) (2.598) (2.599) (2.600) (2.601) (2.602) (2.603) dy differential equations (ODEs) of the form dx = f(x,y). Higher order equations should be reduced to a set of coupled first-order equations and solved in parallel. a Ordinary main January 23, 2006 16:6 Chapter 3 Dynamics and mechanics 3 3.1 Introduction Unusually in physics, there is no pithy phrase that sums up the study of dynamics (the way in which forces produce motion), kinematics (the motion of matter), mechanics (the study of the forces and the motion they produce), and statics (the way forces combine to produce equilibrium). We will take the phrase dynamics and mechanics to encompass all the above, although it clearly does not! To some extent this is because the equations governing the motion of matter include some of our oldest insights into the physical world and are consequentially steeped in tradition. One of the more delightful, or for some annoying, facets of this is the occasional use of arcane vocabulary in the description of motion. The epitome must be what Goldstein1 calls “the jabberwockian sounding statement” the polhode rolls without slipping on the herpolhode lying in the invariable plane, describing “Poinsot’s construction” – a method of visualising the free motion of a spinning rigid body. Despite this, dynamics and mechanics, including fluid mechanics, is arguably the most practically applicable of all the branches of physics. Moreover, and in common with electromagnetism, the study of dynamics and mechanics has spawned a good deal of mathematical apparatus that has found uses in other fields. Most notably, the ideas behind the generalised dynamics of Lagrange and Hamilton lie behind much of quantum mechanics. 1 H. Goldstein, Classical Mechanics, 2nd ed., 1980, Addison-Wesley. main January 23, 2006 16:6 64 3.2 Dynamics and mechanics Frames of reference Galilean transformations r = r + vt t = t (3.1) (3.2) Velocity u = u + v (3.3) Momentum p = p + mv (3.4) Time and positiona Angular momentum J = J + mr × v + v× p t (3.5) Kinetic energy 1 T = T + mu · v + mv 2 2 (3.6) a Frames r,r v position in frames S and S velocity of S in S t,t time in S and S u,u velocity in frames S and S p,p particle momentum in frames S and S m particle mass J ,J angular momentum in frames S and S T ,T kinetic energy in frames S and S γ Lorentz factor v c velocity of S in S speed of light S m S r r vt coincide at t = 0. Lorentz (spacetime) transformationsa −1/2 v2 γ = 1− 2 c (3.7) x = γ(x + vt ); y = y ; x = γ(x − vt) y = y (3.8) (3.9) z = z; z = z Lorentz factor Time and position (3.10) v ! v ! (3.11) t = γ t + 2 x ; t = γ t − 2 x c c Differential dX = (cdt,−dx,−dy,−dz) four-vectorb (3.12) S S x,x t,t X x-position in frames S and S (similarly for y and z) time in frames S and S v x x spacetime four-vector frames S and S coincident at t = 0 in relative motion along x. See page 141 for the transformations of electromagnetic quantities. b Covariant components, using the (1,−1,−1,−1) signature. a For Velocity transformationsa Velocity ux + v ux = ; 1 + ux v/c2 uy ; uy = γ(1 + ux v/c2 ) uz uz = ; γ(1 + ux v/c2 ) a For ux − v 1 − ux v/c2 uy uy = γ(1 − ux v/c2 ) uz uz = γ(1 − ux v/c2 ) ux = γ Lorentz factor = [1 − (v/c)2 ]−1/2 v c velocity of S in S speed of light ui ,ui particle velocity components in frames S and S (3.13) (3.14) (3.15) frames S and S coincident at t = 0 in relative motion along x. S S u v x x main January 23, 2006 16:6 65 3.2 Frames of reference Momentum and energy transformationsa Momentum and energy px = γ(px + vE /c2 ); py = py ; px = γ(px − vE/c2 ) (3.16) py = py (3.17) pz = pz ; pz = pz E = γ(E + vpx ); E = γ(E − vpx ) 2 2 2 E −p c =E −p c 2 2 2 Four-vectorb a For (3.18) (3.19) γ Lorentz factor = [1 − (v/c)2 ]−1/2 v c velocity of S in S speed of light px ,px E,E x components of momentum in S and S (sim. for y and z) energy in S and S (rest) mass total momentum in S momentum four-vector = m20 c4 (3.20) m0 p P = (E/c,−px ,−py ,−pz ) (3.21) P S S v x x frames S and S coincident at t = 0 in relative motion along x. components, using the (1,−1,−1,−1) signature. b Covariant Propagation of lighta ! v ν = γ 1 + cosα ν c Doppler effect (3.22) cosθ + v/c 1 + (v/c)cosθ cosθ − v/c cosθ = 1 − (v/c)cosθ cosθ = Aberration b Relativistic beamingc P (θ) = (3.23) (3.24) sinθ 2γ 2 [1 − (v/c)cosθ]2 (3.25) ν ν α frequency received in S frequency emitted in S arrival angle in S γ Lorentz factor = [1 − (v/c)2 ]−1/2 v velocity of S in S c speed of light θ,θ emission angle of light in S and S S y c α x S S y y v θ c x x P (θ) angular distribution of photons in S frames S and S coincident at t = 0 in relative motion along x. travelling in the opposite sense has a propagation angle of π + θ radians.% c Angular distribution of photons from a source, isotropic and stationary in S . π P (θ) dθ = 1. 0 a For b Light Four-vectorsa Covariant and contravariant components x 0 = x0 x1 = −x1 x2 = −x2 x3 = −x3 (3.26) Scalar product xi yi = x0 y0 + x1 y1 + x2 y2 + x3 y3 (3.27) 1 0 1 1 0 x = γ[x + (v/c)x ]; 2 2 x =x ; covariant vector components xi contravariant components xi ,x i four-vector components in frames S and S Lorentz transformations x0 = γ[x + (v/c)x ]; xi x = γ[x0 − (v/c)x1 ] 0 1 x = γ[x − (v/c)x ] 3 x =x 1 3 0 (3.28) v Lorentz factor = [1 − (v/c)2 ]−1/2 velocity of S in S c speed of light γ (3.29) (3.30) frames S and S , coincident at t = 0 in relative motion along the (1) direction. Note that the (1,−1,−1,−1) signature used here is common in special relativity, whereas (−1,1,1,1) is often used in connection with general relativity (page 67). a For 3 main January 23, 2006 16:6 66 Dynamics and mechanics Rotating frames Vector transformation dA dt = S dA dt + ω× A (3.31) S Acceleration v̇ = v̇ + 2ω× v + ω× (ω× r ) (3.32) Coriolis force F cor = −2mω× v (3.33) F cen = −mω× (ω× r ) (3.34) = +mω 2 r ⊥ (3.35) Centrifugal force Motion relative to Earth Foucault’s penduluma a The 3.3 A S S any vector stationary frame rotating frame ω angular velocity of S in S v̇,v̇ accelerations in S and S v r F cor coriolis force m particle mass (3.36) mÿ = Fy − 2mωe ẋsinλ mz̈ = Fz − mg + 2mωe ẋcosλ (3.37) (3.38) Ωf = −ωe sinλ (3.39) ω λ nongravitational force latitude z y x local vertical axis northerly axis easterly axis Ωf pendulum’s rate of turn F cen r ⊥ F cen centrifugal force r ⊥ perpendicular to particle from rotation axis Fi mẍ = Fx + 2mωe (ẏ sinλ − ż cosλ) velocity in S position in S m r ωe y z x λ ωe Earth’s spin rate sign is such as to make the rotation clockwise in the northern hemisphere. Gravitation Newtonian gravitation Newton’s law of gravitation Newtonian field equationsa Fields from an isolated uniform sphere, mass M, r from the centre a The Gm1 m2 r̂ 12 F1= 2 r12 (3.40) g = −∇φ (3.41) ∇2 φ = −∇ · g = 4πGρ (3.42)  GM  − 2 r̂ (r > a) r g(r) =  − GMr r̂ (r < a) 3  a GM  − (r > a) r φ(r) =   GM (r2 − 3a2 ) (r < a) 2a3 gravitational force on a mass m is mg. (3.43) m1,2 masses F 1 force on m1 (= −F 2 ) r 12 ˆ vector from m1 to m2 unit vector G g φ constant of gravitation gravitational field strength gravitational potential ρ mass density r vector from sphere centre M a mass of sphere radius of sphere M (3.44) a r main January 23, 2006 16:6 67 3.3 Gravitation General relativitya Line element Christoffel symbols and covariant differentiation ds2 = gµν dxµ dxν = −dτ2 (3.45) ds dτ gµν invariant interval proper time interval metric tensor 1 Γαβγ = g αδ (gδβ,γ + gδγ,β − gβγ,δ ) 2 φ;γ = φ,γ ≡ ∂φ/∂xγ Aα;γ = Aα,γ + Γαβγ Aβ (3.46) dxµ Γαβγ differential of xµ Christoffel symbols (3.47) (3.48) ,α ;α φ partial diff. w.r.t. xα covariant diff. w.r.t. xα scalar Bα;γ = Bα,γ − Γβαγ Bβ (3.49) Aα Bα contravariant vector covariant vector R αβγδ Riemann tensor vµ tangent vector (= dxµ /dλ) affine parameter (e.g., τ for material particles) R αβγδ = Γαµγ Γµβδ − Γαµδ Γµβγ + Γαβδ,γ − Γαβγ,δ Riemann tensor Geodesic equation (3.50) Bµ;α;β − Bµ;β;α = R γµαβ Bγ (3.51) Rαβγδ = −Rαβδγ ; Rβαγδ = −Rαβγδ Rαβγδ + Rαδβγ + Rαγδβ = 0 (3.52) (3.53) Dv µ =0 Dλ (3.54) where DAµ dAµ ≡ + Γµαβ Aα v β Dλ dλ (3.55) λ Geodesic deviation D2 ξ µ = −R µαβγ v α ξ β v γ Dλ2 (3.56) ξµ geodesic deviation Ricci tensor Rαβ ≡ R σασβ = g σδ Rδασβ = Rβα (3.57) Rαβ Ricci tensor Einstein tensor Gµν = R µν − Gµν Einstein tensor R Ricci scalar (= g µν Rµν ) Einstein’s field equations Gµν = 8πT µν (3.59) T µν p stress-energy tensor pressure (in rest frame) Perfect fluid T µν = (p + ρ)uµ uν + pg µν (3.60) ρ uν density (in rest frame) fluid four-velocity Schwarzschild solution (exterior) 1 µν g R 2 (3.58) −1 2M 2M dr2 ds2 = − 1 − dt2 + 1 − r r + r2 (dθ2 + sin2 θ dφ2 ) (3.61) spherically symmetric mass (see page 183) (r,θ,φ) spherical polar coords. t time M Kerr solution (outside a spinning black hole) ∆ − a2 sin2 θ 2 2Mr sin2 θ dt − 2a dt dφ 2 2 2 2 (r2 + a2 )2 − a2 ∆sin2 θ 2 2 dr + 2 dθ2 sin θ dφ + + 2 ∆ a ∆ angular momentum (along z) ≡ J/M ≡ r2 − 2Mr + a2 2 ≡ r2 + a2 cos2 θ J ds2 = − (3.62) a General relativity conventionally uses the (−1,1,1,1) metric signature and “geometrized units” in which G = 1 and c = 1. Thus, 1kg = 7.425 × 10−28 m etc. Contravariant indices are written as superscripts and covariant indices as subscripts. Note also that ds2 means (ds)2 etc. 3 main January 23, 2006 16:6 68 Dynamics and mechanics 3.4 Particle motion Dynamics definitionsa F force Newtonian force F = mr̈ = ṗ (3.63) m r mass of particle particle position vector Momentum p = mṙ (3.64) p momentum Kinetic energy 1 T = mv 2 2 (3.65) T kinetic energy v particle velocity Angular momentum J = r× p (3.66) J angular momentum Couple (or torque) G = r× F (3.67) G couple Centre of mass (ensemble of N particles) N mi r i R0 = i=1 N i=1 mi (3.68) R0 mi position vector of centre of mass mass of ith particle ri position vector of ith particle a In the Newtonian limit, v c, assuming m is constant. Relativistic dynamicsa Lorentz factor −1/2 v2 γ = 1− 2 c γ v Lorentz factor particle velocity c speed of light p relativistic momentum m0 particle (rest) mass (3.71) F t force on particle time (3.69) Momentum p = γm0 v Force F= Rest energy Er = m0 c2 (3.72) Er particle rest energy Kinetic energy T = m0 c2 (γ − 1) (3.73) T relativistic kinetic energy E = γm0 c2 (3.74) E total energy (= Er + T ) Total energy (3.70) dp dt 2 2 = (p c + m20 c4 )1/2 (3.75) a It is now common to regard mass as a Lorentz invariant property and to drop the term “rest mass.” The symbol m0 is used here to avoid confusion with the idea of “relativistic mass” (= γm0 ) used by some authors. Constant acceleration v = u + at v 2 = u2 + 2as 1 s = ut + at2 2 u+v t s= 2 (3.76) (3.77) (3.78) (3.79) u initial velocity v t s final velocity time distance travelled a acceleration main January 23, 2006 16:6 69 3.4 Particle motion Reduced mass (of two interacting bodies) r m2 m1 centre of mass r2 r1 m1 m2 m1 + m 2 m2 r r1 = m1 + m 2 −m1 r r2 = m1 + m 2 (3.80) µ reduced mass mi interacting masses (3.81) ri position vectors from centre of mass (3.82) r |r| r = r1 − r2 distance between masses Moment of inertia I = µ|r|2 (3.83) I moment of inertia Total angular momentum J = µr× ṙ (3.84) J angular momentum Lagrangian 1 L = µ|ṙ|2 − U(|r|) 2 (3.85) L U Lagrangian potential energy of interaction µ= Reduced mass Distances from centre of mass 3 Ballisticsa Velocity v = v0 cosα x̂ + (v0 sinα − gt) ŷ (3.86) v0 initial velocity v α velocity at t elevation angle v 2 = v02 − 2gy (3.87) g gravitational acceleration (3.88) ˆ t unit vector time h maximum height l range Trajectory gx2 y = xtanα − 2 2v0 cos2 α Maximum height h= v02 sin2 α 2g (3.89) Horizontal range l= v02 sin2α g (3.90) a Ignoring the curvature and rotation of the Earth and frictional losses. g is assumed constant. ŷ v0 h α l x̂ main January 23, 2006 16:6 70 Dynamics and mechanics Rocketry Escape velocitya Specific impulse Exhaust velocity (into a vacuum) 2GM vesc = r Isp = vesc G M escape velocity constant of gravitation mass of central body r Isp central body radius specific impulse u g R effective exhaust velocity acceleration due to gravity molar gas constant γ Tc µ ∆v ratio of heat capacities combustion temperature effective molecular mass of exhaust gas rocket velocity increment Mi Mf M pre-burn rocket mass post-burn rocket mass mass ratio (3.95) N Mi ui number of stages mass ratio for ith burn exhaust velocity of ith burn (3.96) t θ burn time rocket zenith angle ∆vah ∆vhb velocity increment, a to h velocity increment, h to b ra rb radius of inner orbit radius of outer orbit 1/2 (3.91) u g (3.92) 2γRTc u= (γ − 1)µ 1/2 (3.93) Rocket equation (g = 0) Mi ∆v = uln Mf Multistage rocket ∆v = N ≡ ulnM (3.94) ui lnMi i=1 In a constant gravitational field ∆v = ulnM − gtcosθ Hohmann cotangential transferb GM ∆vah = ra GM ∆vhb = rb 1/2 2rb ra + rb 1/2 1− 1/2 2ra ra + rb −1 (3.97) 1/2 transfer ellipse, h a b (3.98) a From the surface of a spherically symmetric, nonrotating body, mass M. between coplanar, circular orbits a and b, via ellipse h with a minimal expenditure of energy. b Transfer main January 23, 2006 16:6 71 3.4 Particle motion Gravitationally bound orbital motiona α GMm ≡− r r Potential energy of interaction U(r) = − Total energy J2 α α E =− + =− r 2mr2 2a (3.100) Virial theorem (1/r potential) E = U /2 = −T U = −2T Orbital equation (Kepler’s 1st law) r0 = 1 + ecosφ , r a(1 − e2 ) r= 1 + ecosφ Rate of sweeping area (Kepler’s 2nd law) J dA = = constant dt 2m Semi-major axis a= r0 α = 2 1−e 2|E| Semi-minor axis b= Eccentricityb Semi-latusrectum Pericentre Apocentre Speed Period (Kepler’s 3rd law) or (3.99) U(r) potential energy G constant of gravitation M central mass m α orbiting mass ( M) GMm (for gravitation) E J total energy (constant) total angular momentum (constant) (3.101) (3.102) T kinetic energy · mean value (3.103) r0 r semi-latus-rectum distance of m from M (3.104) e φ eccentricity phase (true anomaly) (3.105) A area swept out by radius vector (total area = πab) (3.106) a semi-major axis b semi-minor axis 2a J r0 = (3.107) (1 − e2 )1/2 (2m|E|)1/2 1/2 1/2 2EJ 2 b2 e= 1+ = 1 − (3.108) mα2 a2 J 2 b2 = = a(1 − e2 ) (3.109) mα a r0 = a(1 − e) (3.110) rmin = 1+e r0 = a(1 + e) (3.111) rmax = 1−e 2 1 2 − v = GM (3.112) r a 1/2 m m !1/2 = 2πa3/2 P = πα 3 2|E| α (3.113) r0 = m A r0 r φ M ae 2b rmax rmin rmin pericentre distance rmax apocentre distance v orbital speed P orbital period an inverse-square law of attraction between two isolated bodies in the nonrelativistic limit. If m is not M, then the equations are valid with the substitutions m → µ = Mm/(M + m) and M → (M + m) and with r taken as the body separation. The distance of mass m from the centre of mass is then rµ/m (see earlier table on Reduced mass). Other orbital dimensions scale similarly, and the two orbits have the same eccentricity. b Note that if the total energy, E, is < 0 then e < 1 and the orbit is an ellipse (a circle if e = 0). If E = 0, then e = 1 and the orbit is a parabola. If E > 0 then e > 1 and the orbit becomes a hyperbola (see Rutherford scattering on next page). a For 3 main January 23, 2006 16:6 72 Dynamics and mechanics Rutherford scatteringa y trajectory for α < 0 b x scattering centre a χ rmin trajectory for α > 0 Scattering potential energy Scattering angle Closest approach Semi-axis Eccentricity Motion trajectoryb Scattering centrec Rutherford scattering formulad a Nonrelativistic rmin a (α<0) (α>0) α U(r) = − r " < 0 repulsive α > 0 attractive |α| χ tan = 2 2Eb α |α| χ rmin = csc − 2E 2 |α| = a(e ± 1) |α| 2E 1/2 2 2 4E b χ +1 = csc e= α2 2 a= 4E 2 2 y 2 x − 2 =1 α2 b 2 1/2 α 2 x=± + b 4E 2 dσ 1 dN = dΩ n dΩ α !2 4 χ = csc 4E 2 (3.114) (3.115) U(r) potential energy r particle separation α constant scattering angle total energy (> 0) impact parameter (3.116) χ E b (3.117) rmin closest approach (3.118) a e hyperbola semi-axis eccentricity (3.119) (3.120) (3.121) x,y position with respect to hyperbola centre (3.122) dσ dΩ (3.123) (3.124) differential scattering cross section n beam flux density dN number of particles scattered into dΩ Ω solid angle treatment for an inverse-square force law and a fixed scattering centre. Similar scattering results from either an attractive or repulsive force. See also Conic sections on page 38. b The correct branch can be chosen by inspection. c Also the focal points of the hyperbola. d n is the number of particles per second passing through unit area perpendicular to the beam. main January 23, 2006 16:6 73 3.4 Particle motion Inelastic collisionsa m1 m2 v1 m1 v2 Before collision v2 After collision Coefficient of restitution v2 − v1 = (v1 − v2 ) = 1 if perfectly elastic = 0 if perfectly inelastic Loss of kinetic energyb T −T = 1 − 2 T (3.125) (3.126) (3.127) coefficient of restitution vi vi pre-collision velocities post-collision velocities T ,T total KE in zero momentum frame before and after collision mi particle masses (3.128) m1 − m2 (1 + )m2 v1 + v2 m1 + m 2 m1 + m 2 m2 − m1 (1 + )m1 v2 = v2 + v1 m1 + m 2 m1 + m 2 v1 = Final velocities m2 v1 (3.129) (3.130) the line of centres, v1 ,v2 c. zero momentum frame. a Along b In Oblique elastic collisionsa m1 Directions of motion Relative separation angle tanθ1 = θ2 = θ a Collision After collision m1 v2 θ1 v1 v m2 sin2θ m1 − m2 cos2θ   > π/2 θ1 + θ2 = π/2   < π/2 θ angle between centre line and incident velocity θi mi final trajectories sphere masses (3.134) v incident velocity of m1 (3.135) vi final velocities (3.131) (3.132) if m1 < m2 if m1 = m2 if m1 > m2 (m21 + m22 − 2m1 m2 cos2θ)1/2 v m1 + m 2 2m1 v v2 = cosθ m1 + m 2 v1 = Final velocities m2 θ Before collision θ2 m2 (3.133) between two perfectly elastic spheres: m2 initially at rest, velocities c. 3 main January 23, 2006 16:6 74 3.5 Dynamics and mechanics Rigid body dynamics Moment of inertia tensor Moment of inertia tensora % Iij = (r2 δij − xi xj ) dm (y 2 + z 2 ) dm %  I =  − xy dm % − xz dm (3.136) % − xy dm %  − xz dm % 2 %  (x + z 2 ) dm − yz dm  % % 2 − yz dm (x + y 2 ) dm r r 2 = x2 + y 2 + z 2 δij Kronecker delta moment of inertia tensor dm mass element xi position vector of dm Iij components of I I (3.137) tensor with respect to centre of mass ai ,a position vector of centre of mass m mass of body Iij I12 = I12 − ma1 a2 (3.138) I11 = I11 + m(a22 + a23 ) (3.139) Iij = Iij + m(|a| δij − ai aj ) (3.140) Angular momentum J = Iω (3.141) J angular momentum ω angular velocity Rotational kinetic energy 1 1 T = ω · J = Iij ωi ωj 2 2 (3.142) T kinetic energy Parallel axis theorem 2 a I are the moments of inertia of the body. I (i = j) are its products of inertia. The integrals are over the body ii ij volume. Principal axes  0 0 I3 Principal moment of inertia tensor  I1 I =  0 0 Angular momentum J = (I1 ω1 ,I2 ω2 ,I3 ω3 ) (3.144) J angular momentum ωi components of ω along principal axes Rotational kinetic energy 1 T = (I1 ω12 + I2 ω22 + I3 ω32 ) 2 (3.145) T Moment of inertia ellipsoida T = T (ω1 ,ω2 ,ω3 ) ∂T (J is ⊥ ellipsoid surface) Ji = ∂ωi " ≥ I3 generally I1 + I 2 = I3 flat lamina ⊥ to 3-axis Perpendicular axis theorem Symmetries a The 0 I2 0 I1 = I2 = I3 asymmetric top I1 = I2 = I3 I1 = I2 = I3 symmetric top spherical top ellipsoid is defined by the surface of constant T . (3.143) I principal moment of inertia tensor Ii principal moments of inertia kinetic energy (3.146) I3 (3.147) I1 I2 (3.148) lamina (3.149) main January 23, 2006 16:6 75 3.5 Rigid body dynamics Moments of inertiaa Thin rod, length l Solid sphere, radius r Spherical shell, radius r Solid cylinder, radius r, length l Solid cuboid, sides a,b,c I1 = I2 = I3 0 l ml 2 12 (3.150) 2 I1 = I2 = I3 = mr2 5 I1 = m(b2 + c2 )/12 I2 = m(c2 + a2 )/12 (3.156) (3.157) I3 = m(a2 + b2 )/12 (3.158) I3 = Elliptical lamina, semi-axes a,b 3 I2 (3.153) l I1 I2 r I3 (3.155) 2 a I3 I2 b c (3.159) h I3 I 2 I r (3.160) 1 2 Solid ellipsoid, semi-axes a,b,c I3 I1 3 h m r2 + 20 4 3 mr2 10 I1 r (3.154) I1 = I2 = I2 I1 (3.152) 2 I1 = I2 = I3 = mr2 3 m 2 l2 I1 = I2 = r + 4 3 1 I3 = mr2 2 Solid circular cone, base radius r, height hb I3 (3.151) 2 I1 = m(b + c )/5 I2 = m(c2 + a2 )/5 (3.161) (3.162) I3 = m(a2 + b2 )/5 (3.163) I1 = mb2 /4 (3.164) 2 I2 = ma /4 I3 = m(a2 + b2 )/4 I3 a c b I2 I1 I2 b I3 a (3.165) (3.166) I1 I2 2 I1 = I2 = mr /4 I3 = mr2 /2 Disk, radius r c Triangular plate a With m I3 = (a2 + b2 + c2 ) 36 (3.167) (3.168) r I1 I3 a (3.169) b I3 c respect to principal axes for bodies of mass m and uniform density. The radius of gyration is defined as k = (I/m)1/2 . b Origin of axes is at the centre of mass (h/4 above the base). c Around an axis through the centre of mass and perpendicular to the plane of the plate. main January 23, 2006 16:6 76 Dynamics and mechanics Centres of mass Solid hemisphere, radius r d = 3r/8 from sphere centre (3.170) Hemispherical shell, radius r d = r/2 from sphere centre (3.171) Sector of disk, radius r, angle 2θ 2 sinθ d= r 3 θ from disk centre (3.172) Arc of circle, radius r, angle 2θ d=r from circle centre (3.173) Arbitrary triangular lamina, height ha d = h/3 perpendicular from base (3.174) Solid cone or pyramid, height h d = h/4 perpendicular from base (3.175) 3 (2r − h)2 from sphere centre 4 3r − h shell: d = r − h/2 from sphere centre (3.176) solid: d = Spherical cap, height h, sphere radius r Semi-elliptical lamina, height h ah sinθ θ d= 4h 3π (3.177) from base (3.178) is the perpendicular distance between the base and apex of the triangle. Pendulums Simple pendulum Conical pendulum Torsional penduluma l θ02 1 + + ··· P = 2π g 16 P period g gravitational acceleration (3.179) l length θ0 maximum angular displacement (3.180) α cone half-angle 1/2 l cosα g 1/2 lI0 P = 2π C P = 2π Equal double pendulumc a Assuming P 2π l √ (2 ± 2)g α m I0 moment of inertia of bob (3.181) 1 (ma2 + I1 cos2 γ1 P 2π mga 1/2 2 2 + I2 cos γ2 + I3 cos γ3 ) (3.182) m l Compound pendulumb l θ0 C torsional rigidity of wire (see page 81) l I0 a distance of rotation axis from centre of mass m mass of body Ii principal moments of inertia γi angles between rotation axis and principal axes 1/2 (3.183) the bob is supported parallel to a principal rotation axis. b I.e., an arbitrary triaxial rigid body. c For very small oscillations (two eigenmodes). a I1 I3 I2 l m l m main January 23, 2006 16:6 77 3.5 Rigid body dynamics Tops and gyroscopes J herpolhode 3 ω space cone invariable plane J3 polhode Ωp body cone moment of inertia ellipsoid 3 θ support point a 2 gyroscope prolate symmetric top Euler’s equations a Free symmetric topb (I3 < I2 = I1 ) Free asymmetric topc Steady gyroscopic precession G1 = I1 ω̇1 + (I3 − I2 )ω2 ω3 G2 = I2 ω̇2 + (I1 − I3 )ω3 ω1 (3.184) (3.185) G3 = I3 ω̇3 + (I2 − I1 )ω1 ω2 (3.186) I1 − I3 ω3 I1 J Ωs = I1 Ωb = Ω2b = (I1 − I3 )(I2 − I3 ) 2 ω3 I1 I2 Gi external couple (= 0 for free rotation) Ii ωi principal moments of inertia angular velocity of rotation (3.187) Ωb Ωs body frequency space frequency (3.188) J total angular momentum Ωp θ J3 precession angular velocity angle from vertical angular momentum around symmetry axis mass (3.189) Ω2p I1 cosθ − Ωp J3 + mga = 0 (3.190) (slow) (fast) (3.191) " Mga/J3 Ωp J3 /(I1 cosθ) mg m g a gravitational acceleration distance of centre of mass from support point moment of inertia about support point Gyroscopic stability J32 ≥ 4I1 mgacosθ (3.192) Gyroscopic limit (“sleeping top”) J32 I1 mga (3.193) Nutation rate Ωn = J3 /I1 (3.194) Ωn nutation angular velocity Gyroscope released from rest Ωp = (3.195) t time a Components mga (1 − cosΩn t) J3 I1 are with respect to the principal axes, rotating with the body. body frequency is the angular velocity (with respect to principal axes) of ω around the 3-axis. The space frequency is the angular velocity of the 3-axis around J , i.e., the angular velocity at which the body cone moves around the space cone. c J close to 3-axis. If Ω2 < 0, the body tumbles. b b The main January 23, 2006 16:6 78 3.6 Dynamics and mechanics Oscillating systems Free oscillations x oscillating variable t γ ω0 time damping factor (per unit mass) undamped angular frequency A φ amplitude constant phase constant ω angular eigenfrequency Ai amplitude constants (3.202) ∆ an logarithmic decrement nth displacement maximum (3.203) Q quality factor 2 Differential equation dx d x + ω02 x = 0 + 2γ dt2 dt Underdamped solution (γ < ω0 ) Critically damped solution (γ = ω0 ) Overdamped solution (γ > ω0 ) x = Ae−γt cos(ωt + φ) (3.197) where ω = (ω02 − γ 2 )1/2 (3.198) x = e−γt (A1 + A2 t) (3.199) x = e−γt (A1 eqt + A2 e−qt ) (3.200) − ω02 )1/2 (3.201) where q = (γ 2 an 2πγ = an+1 ω π ω0 if Q= 2γ ∆ Logarithmic decrementa ∆ = ln Quality factor (3.196) Q1 a The decrement is usually the ratio of successive displacement maxima but is sometimes taken as the ratio of successive displacement extrema, reducing ∆ by a factor of 2. Logarithms are sometimes taken to base 10, introducing a further factor of log10 e. Forced oscillations Differential equation Steadystate solutiona dx d2 x + ω02 x = F0 eiωf t + 2γ dt2 dt x = Aei(ωf t−φ) , where A = F0 [(ω02 − ωf2 )2 + (2γωf )2 ]−1/2 F0 /(2ω0 ) [(ω0 − ωf )2 + γ 2 ]1/2 2γωf tanφ = 2 ω0 − ωf2 (γ ωf ) x oscillating variable (3.204) t γ time damping factor (per unit mass) (3.205) ω0 undamped angular frequency (3.206) F0 force amplitude (per unit mass) (3.207) ωf A φ forcing angular frequency amplitude phase lag of response behind driving force (3.208) Amplitude resonanceb 2 ωar = ω02 − 2γ 2 (3.209) ωar amplitude resonant forcing angular frequency Velocity resonancec ωvr = ω0 (3.210) ωvr velocity resonant forcing angular frequency Quality factor Q= (3.211) Q quality factor Impedance Z = 2γ + i (3.212) Z impedance (per unit mass) a Excluding ω0 2γ ωf2 − ω02 ωf the free oscillation terms. frequency for maximum displacement. c Forcing frequency for maximum velocity. Note φ = π/2 at this frequency. b Forcing main January 23, 2006 16:6 79 3.7 Generalised dynamics 3.7 Generalised dynamics Lagrangian dynamics Action S q q̇ action (δS = 0 for the motion) generalised coordinates generalised velocities L t Lagrangian time m mass v r U velocity position vector potential energy T kinetic energy m0 γ (rest) mass Lorentz factor (3.217) +e φ A positive charge electric potential magnetic vector potential (3.218) pi generalised momenta t2 L(q, q̇,t) dt S= (3.213) t1 d dt Euler–Lagrange equation ∂L ∂q̇i − ∂L =0 ∂qi (3.214) 1 L = mv 2 − U(r,t) 2 =T −U Lagrangian of particle in external field Relativistic Lagrangian of a charged particle L=− Generalised momenta pi = (3.215) (3.216) 2 m0 c − e(φ − A · v) γ ∂L ∂q̇i Hamiltonian dynamics Hamiltonian H= pi q̇i − L (3.219) L pi q̇i Lagrangian generalised momenta generalised velocities (3.220) H qi Hamiltonian generalised coordinates (3.221) v r particle speed position vector (3.222) U T potential energy kinetic energy m0 c (rest) mass speed of light +e φ positive charge electric potential A vector potential p t particle momentum time i ∂H ; ∂pi Hamilton’s equations q̇i = Hamiltonian of particle in external field 1 H = mv 2 + U(r,t) 2 =T +U Relativistic Hamiltonian of a charged particle ṗi = − ∂H ∂qi H = (m20 c4 + |p − eA|2 c2 )1/2 + eφ ∂f ∂g ∂f ∂g − [f,g] = ∂qi ∂pi ∂pi ∂qi i Poisson brackets [qi ,g] = ∂g , ∂pi ∂g = 0, ∂t ∂S ∂S + H qi , ,t = 0 ∂t ∂qi [H,g] = 0 if Hamilton– Jacobi equation ∂g ∂qi dg =0 dt [pi ,g] = − (3.223) (3.224) (3.225) (3.226) (3.227) f,g arbitrary functions [·,·] Poisson bracket (also see Commutators on page 26) S action 3 main January 23, 2006 16:6 80 3.8 Dynamics and mechanics Elasticity Elasticity definitions (simple)a Stress Strain τ = F/A e = δl/l F (3.228) (3.229) e stress applied force cross-sectional area strain δl l change in length length τ F A Young modulus (Hooke’s law) E = τ/e = constant (3.230) E Young modulus Poisson ratiob σ=− δw/w δl/l (3.231) σ δw Poisson ratio change in width A l w w width apply to a thin wire under longitudinal stress. b Solids obeying Hooke’s law are restricted by thermodynamics to −1 ≤ σ ≤ 1/2, but none are known with σ < 0. Non-Hookean materials can show σ > 1/2. a These Elasticity definitions (general) Stress tensora Strain tensor force i direction area ⊥ j direction 1 ∂uk ∂ul + ekl = 2 ∂xl ∂xk (3.232) τij = τij stress tensor (τij = τji ) ekl strain tensor (ekl = elk ) (3.233) uk xk displacement to xk coordinate system Elastic modulus τij = λijkl ekl (3.234) λijkl elastic modulus Elastic energyb 1 U = λijkl eij ekl 2 (3.235) U potential energy Volume strain (dilatation) δV = e11 + e22 + e33 ev = V (3.236) ev δV volume strain change in volume V volume Shear strain 1 1 ekl = (ekl − ev δkl ) + ev δkl 3 ) *+ , )3 *+ , (3.237) δkl Kronecker delta p hydrostatic pressure pure shear Hydrostatic compression dilatation τij = −pδij a τ are normal stresses, τ (i = j) are torsional stresses. ii ij b As usual, products are implicitly summed over repeated (3.238) indices. main January 23, 2006 16:6 81 3.8 Elasticity Isotropic elastic solids E 2(1 + σ) Eσ λ= (1 + σ)(1 − 2σ) µ= Lamé coefficients Longitudinal modulusa Ml = E(1 − σ) = λ + 2µ (1 + σ)(1 − 2σ) 1 [τii − σ(τjj + τkk )] E σ (ejj + ekk ) τii = Ml eii + 1−σ t = 2µe + λ1tr(e) Shear modulus (rigidity modulus) a In strain in i direction τii e stress in i direction strain tensor (3.246) 9µK µ + 3K Poisson ratio eii (3.245) E 2(1 + σ) τT = µθsh 3K − 2µ σ= 2(3K + µ) longitudinal elastic modulus (3.244) µ= E= Ml (3.243) 2 E =λ+ µ 3(1 − 2σ) 3 1 1 ∂V =− KT V ∂p T −p = Kev Young modulus (3.240) Young modulus Poisson ratio (3.242) K= Bulk modulus (compression modulus) µ,λ Lamé coefficients E σ (3.241) eii = Diagonalised equationsb (3.239) t stress tensor 1 unit matrix tr(·) trace K bulk modulus KT isothermal bulk modulus V volume (3.247) p T pressure temperature (3.248) ev µ volume strain shear modulus (3.249) τT θsh transverse stress shear strain τT (3.250) θsh (3.251) an extended medium. aligned along eigenvectors of the stress and strain tensors. b Axes Torsion Torsional rigidity (for a homogeneous rod) G=C Thin circular cylinder C = 2πa3 µt (3.253) Thick circular cylinder 1 C = µπ(a42 − a41 ) 2 (3.254) Arbitrary thin-walled tube 4A2 µt C= P (3.255) Long flat ribbon 1 C = µwt3 3 (3.256) φ l (3.252) G twisting couple C l φ a torsional rigidity rod length twist angle in length l radius t µ wall thickness shear modulus a1 inner radius a2 outer radius A cross-sectional area perimeter P w cross-sectional width G a φ l A t w t 3 main January 23, 2006 16:6 82 Dynamics and mechanics Bending beamsa E Gb = Rc EI = Rc Bending moment (3.257) Gb E Rc bending moment Young modulus radius of curvature (3.258) ds ξ W area element distance to neutral surface from ds moment of area displacement from horizontal end-weight l x beam length distance along beam w beam weight per unit length 2 ξ ds I y Light beam, horizontal at x = 0, weight at x = l x! 2 W l− x y= 2EI 3 (3.259) 4 d y = w(x) dx4  2 2  π EI/l Fc = 4π 2 EI/l 2   2 π EI/(4l 2 ) Heavy beam Euler strut failure a The (3.260) EI (free ends) (fixed ends) (1 free end) (3.261) Fc critical compression force l strut length ds ξ neutral surface (cross section) x y W free Fc Fc fixed radius of curvature is approximated by 1/Rc d2 y/dx2 . Elastic wave velocitiesa In an infinite isotropic solidb In a fluid vt vl speed of transverse wave speed of longitudinal wave (3.263) µ ρ shear modulus density (3.264) Ml longitudinal modulus ! E(1−σ) = (1+σ)(1−2σ) (3.265) K bulk modulus vl(i) speed of longitudinal wave (displacement i) vt(i) E speed of transverse wave (displacement i) Young modulus σ k t Poisson ratio wavenumber (= 2π/λ) plate thickness (in z, t λ) vφ a torsional wave velocity rod radius ( λ) vt = (µ/ρ)1/2 (3.262) vl = (Ml /ρ)1/2 1/2 2 − 2σ vl = vt 1 − 2σ vl = (K/ρ)1/2 On a thin plate (wave travelling along x, plate thin in z) vl(x) = k z y x In a thin circular rod a Waves E ρ(1 − σ 2 ) 1/2 (3.266) vt(y) = (µ/ρ)1/2 1/2 Et2 vt(z) = k 12ρ(1 − σ 2 ) (3.267) vl = (E/ρ)1/2 (3.269) 1/2 (3.270) vφ = (µ/ρ) 1/2 ka E vt = 2 ρ (3.268) (3.271) that produce “bending” are generally dispersive. Wave (phase) speeds are quoted throughout. waves are also known as shear waves, or S-waves. Longitudinal waves are also known as pressure waves, or P-waves. b Transverse main January 23, 2006 16:6 83 3.8 Elasticity Waves in strings and springsa In a spring vl = (κl/ρl ) 1/2 (3.272) vl κ l speed of longitudinal wave spring constantb spring length ρl mass per unit lengthc On a stretched string vt = (T /ρl )1/2 (3.273) vt T speed of transverse wave tension On a stretched sheet vt = (τ/ρA )1/2 (3.274) τ ρA tension per unit width mass per unit area amplitude assumed wavelength. b In the sense κ = force/extension. c Measured along the axis of the spring. a Wave Propagation of elastic waves Acoustic impedance Z= F force = response velocity u̇ = (E ρ)1/2 E ρ Wave velocity/ impedance relation then Mean energy density (nondispersive waves) 1 U = E k 2 u20 2 1 = ρω 2 u20 2 P = Uv Normal coefficientsa Snell’s lawb if v = (3.275) Z F impedance stress force (3.276) u strain displacement (3.277) E elastic modulus (3.278) ρ v density wave phase velocity (3.279) U energy density (3.280) k ω wavenumber angular frequency (3.281) u0 P maximum displacement mean energy flux (3.282) r reflection coefficient (3.283) t τ transmission coefficient stress (3.284) θi θr angle of incidence angle of reflection 1/2 Z = (E ρ)1/2 = ρv ur τr Z1 − Z2 =− = ui τi Z1 + Z2 2Z1 t= Z1 + Z2 r= sinθi sinθr sinθt = = vi vr vt θt angle of refraction stress and strain amplitudes. Because these reflection and transmission coefficients are usually defined in terms of displacement, u, rather than stress, there are differences between these coefficients and their equivalents defined in electromagnetism [see Equation (7.179) and page 154]. b Angles defined from the normal to the interface. An incident plane pressure wave will generally excite both shear and pressure waves in reflection and transmission. Use the velocity appropriate for the wave type. a For 3 main January 23, 2006 16:6 84 3.9 Dynamics and mechanics Fluid dynamics Ideal fluidsa Continuityb Kelvin circulation ∂ρ + ∇ · (ρv) = 0 ∂t Γ = v · dl = constant = ω · ds (3.285) ρ v t density fluid velocity field time (3.286) Γ dl circulation loop element ds element of surface bounded by loop ω vorticity (= ∇× v) (3.287) S Euler’s equation c ∂v ∇p + (v · ∇)v = − +g ∂t ρ ∂ or (∇× v) = ∇× [v× (∇× v)] ∂t (3.289) pressure gravitational field strength (v · ∇) advective operator p g (3.288) Bernoulli’s equation (incompressible flow) 1 2 ρv + p + ρgz = constant 2 (3.290) z altitude 1 2 γ p v + + gz = constant (3.291) 2 γ −1 ρ 1 = v 2 + cp T + gz (3.292) 2 γ Bernoulli’s equation (compressible adiabatic flow)d ratio of specific heat capacities (cp /cV ) specific heat capacity at constant pressure temperature Hydrostatics ∇p = ρg (3.293) Adiabatic lapse rate (ideal gas) g dT =− dz cp (3.294) cp T a No thermal conductivity or viscosity. generally. c The second form of Euler’s equation applies to incompressible flow only. d Equation (3.292) is true only for an ideal gas. b True Potential flowa v = ∇φ (3.295) v velocity ∇ φ=0 (3.296) φ velocity potential ω = ∇×v = 0 ω vorticity Vorticity condition (3.297) F drag force on moving sphere Drag force on a sphereb 1 2 F = − πρa3 u̇ = − Md u̇ 3 2 a u̇ ρ sphere radius sphere acceleration fluid density Velocity potential 2 (3.298) Md displaced fluid mass a For incompressible fluids. effect of this drag force is to give the sphere an additional effective mass equal to half the mass of fluid displaced. b The main January 23, 2006 16:6 85 3.9 Fluid dynamics Viscous flow (incompressible)a Fluid stress τij = −pδij + η ∂vi ∂vj + ∂xj ∂xi Navier–Stokes equationb ∇p η ∂v + (v · ∇)v = − − ∇× ω + g ∂t ρ ρ ∇p η 2 + ∇ v +g =− ρ ρ Kinematic viscosity ν = η/ρ a I.e., τij p η fluid stress tensor hydrostatic pressure shear viscosity vi δij velocity along i axis Kronecker delta (3.300) v ω fluid velocity field vorticity (3.301) g ρ gravitational acceleration density (3.302) ν kinematic viscosity (3.299) ∇ · v = 0, η = 0. bulk (second) viscosity. b Neglecting Laminar viscous flow Between parallel plates vz (y) = ∂p 1 y(h − y) 2η ∂z Along a circular pipea 1 ∂p vz (r) = (a2 − r2 ) 4η ∂z dV πa4 ∂p = Q= dt 8η ∂z Circulating between concentric rotating cylindersb 4πηa2 a2 Gz = 2 1 22 (ω2 − ω1 ) a2 − a1 Along an annular pipe η flow velocity direction of flow distance from plate shear viscosity p pressure (3.304) r (3.305) a V distance from pipe axis pipe radius volume (3.303) (3.306) vz z y Gz axial couple between cylinders per unit length ωi angular velocity of ith cylinder a1 π ∂p 4 (a22 − a21 )2 4 a2 Q= a − a1 − 8η ∂z 2 ln(a2 /a1 ) Q (3.307) inner radius outer radius volume discharge rate a Poiseuille b Couette flow. flow. Draga On a sphere (Stokes’s law) F = 6πaηv (3.308) F drag force a radius On a disk, broadside to flow F = 16aηv (3.309) v η velocity shear viscosity On a disk, edge on to flow F = 32aηv/3 (3.310) a For Reynolds numbers 1. z h y r a a1 ω1 ω2 a2 3 main January 23, 2006 16:6 86 Dynamics and mechanics Characteristic numbers Re Reynolds number ρ U density characteristic velocity L η characteristic scale-length shear viscosity F g Froude number gravitational acceleration S Strouhal number τ characteristic timescale P cp λ Prandtl number Specific heat capacity at constant pressure thermal conductivity (3.315) M c Mach number sound speed (3.316) Ro Ω Rossby number angular velocity inertial force ρUL = η viscous force (3.311) F= inertial force U2 = Lg gravitational force (3.312) Strouhal numberb S= Uτ evolution scale = L physical scale (3.313) Prandtl number ηcp momentum transport = P= λ heat transport Reynolds number Re = Froude numbera speed U = c sound speed Mach number M= Rossby number Ro = a Sometimes b Sometimes inertial force U = ΩL Coriolis force (3.314) the square root of this expression. L is usually the fluid depth. the reciprocal of this expression. Fluid waves Sound waves K vp = ρ 1/2 = dp dρ 1/2 In an ideal gas (adiabatic conditions)a γRT vp = µ Gravity waves on a liquid surfaceb ω 2 = gk tanhkh  !  1 g 1/2 vg 2 k  (gh)1/2 ω2 = Capillary–gravity waves (h λ) ω 2 = gk + b Amplitude bulk modulus pressure density γp = ρ (3.318) γ R T ratio of heat capacities molar gas constant (absolute) temperature (3.319) µ vg h mean molecular mass group speed of wave liquid depth λ k g wavelength wavenumber gravitational acceleration ω angular frequency σ surface tension (3.317) 1/2 (h λ) (3.320) (h λ) (3.321) σk 3 ρ the waves are isothermal rather than adiabatic then vp = (p/ρ)1/2 . wavelength. c In the limit k 2 gρ/σ. a If wave (phase) speed K p ρ 3 Capillary waves (ripples)c σk ρ vp 1/2 (3.322) main January 23, 2006 16:6 87 3.9 Fluid dynamics Doppler effecta Source at rest, observer moving at u |u| ν = 1 − cosθ ν vp Observer at rest, source moving at u ν = ν a For 1 |u| 1 − cosθ vp (3.323) ν ,ν observed frequency ν emitted frequency vp wave (phase) speed in fluid (3.324) u θ velocity angle between wavevector, k, and u k θ u 3 plane waves in a stationary fluid. Wave speeds Phase speed vp = vg = Group speed ω = νλ k dω dk = vp − λ (3.325) vp phase speed ν ω λ frequency angular frequency (= 2πν) wavelength k wavenumber (= 2π/λ) vg group speed (3.326) dvp dλ (3.327) Shocks Mach wedgea sinθw = Kelvin wedgeb λK = Spherical adiabatic shockc Rankine– Hugoniot shock relationsd vp vb 4πvb2 3g θw = arcsin(1/3) = 19◦ .5 r (3.328) (3.329) (3.330) 2 1/5 Et ρ0 p2 2γM21 − (γ − 1) = p1 γ +1 v1 ρ2 γ +1 = = v2 ρ1 (γ − 1) + 2/M21 (3.331) (3.332) (3.333) θw wedge semi-angle vp vb wave (phase) speed body speed λK characteristic wavelength gravitational acceleration g r E t shock radius energy release time ρ0 density of undisturbed medium 1 2 upstream values downstream values p v T pressure velocity temperature ρ density γ ratio of specific heats M Mach number a Approximating the wake generated by supersonic motion of a body in a nondispersive medium. b For gravity waves, e.g., in the wake of a boat. Note that the wedge semi-angle is independent of v . b c Sedov–Taylor relation. d Solutions for a steady, normal shock, in the frame moving with the shock front. If γ = 5/3 then v /v ≤ 4. 1 2 T2 [2γM21 − (γ − 1)][2 + (γ − 1)M21 ] = T1 (γ + 1)2 M21 (3.334) main January 23, 2006 16:6 88 Dynamics and mechanics Surface tension surface energy area surface tension = length σlv = Definition Laplace’s formulaa ∆p = σlv Capillary constant 2σlv cc = gρ Capillary rise (circular tube) h= 1 1 + R1 R2 (3.335) surface tension (liquid/vapour interface) ∆p pressure difference over surface Ri principal radii of curvature cc ρ g capillary constant liquid density gravitational acceleration rise height (3.336) (3.337) 1/2 2σlv cosθ ρga σlv (3.338) h surface R2 R1 h a θ (3.339) θ contact angle a tube radius σwv wall/vapour surface σwv − σwl tension (3.340) cosθ = Contact angle σwl wall/liquid surface σlv tension a For a spherical bubble in a liquid ∆p = 2σ /R. For a soap bubble (two surfaces) ∆p = 4σ /R. lv lv σwv σwl θ σlv main January 23, 2006 16:6 Chapter 4 Quantum physics 4.1 Introduction Quantum ideas occupy such a pivotal position in physics that different notations and algebras appropriate to each field have been developed. In the spirit of this book, only those formulas that are commonly present in undergraduate courses and that can be simply presented in tabular form are included here. For example, much of the detail of atomic spectroscopy and of specific perturbation analyses has been omitted, as have ideas from the somewhat specialised field of quantum electrodynamics. Traditionally, quantum physics is understood through standard “toy” problems, such as the potential step and the one-dimensional harmonic oscillator, and these are reproduced here. Operators are distinguished from observables using the “hat” notation, so that the momentum observable, px , has the operator pˆx = −ih̄∂/∂x. For clarity, many relations that can be generalised to three dimensions in an obvious way have been stated in their one-dimensional form, and wavefunctions are implicitly taken as normalised functions of space and time unless otherwise stated. With the exception of the last panel, all equations should be taken as nonrelativistic, so that “total energy” is the sum of potential and kinetic energies, excluding the rest mass energy. 4 main January 23, 2006 16:6 90 4.2 Quantum physics Quantum definitions Quantum uncertainty relations h λ p = h̄k p= De Broglie relation Planck–Einstein relation a Dispersion p,p h h̄ particle momentum Planck constant h/(2π) λ de Broglie wavelength k E de Broglie wavevector energy ν ω frequency angular frequency (= 2πν) a,b · observablesb expectation value (∆a)2 dispersion of a (4.6) â [·,·] operator for observable a commutator (see page 26) (4.1) (4.2) E = hν = h̄ω (4.3) (∆a)2 = (a − a )2 (4.4) = a2 − a 2 (4.5) General uncertainty relation 1 (∆a)2 (∆b)2 ≥ i[â, b̂] 4 Momentum–position uncertainty relationc ∆p∆x ≥ h̄ 2 (4.7) x particle position Energy–time uncertainty relation ∆E ∆t ≥ h̄ 2 (4.8) t time Number–phase uncertainty relation ∆n∆φ ≥ 1 2 (4.9) n number of photons φ wave phase 2 a Dispersion in quantum physics corresponds to variance in statistics. b An observable is a directly measurable parameter of a system. c Also known as the “Heisenberg uncertainty relation.” Wavefunctions Probability density Probability density currenta pr(x,t) dx = |ψ(x,t)|2 dx ∂ψ ∗ h̄ ∗ ∂ψ −ψ j(x) = ψ 2im ∂x ∂x # $ h̄ ψ ∗ (r)∇ψ(r) − ψ(r)∇ψ ∗ (r) j= 2im 1 = (ψ ∗ p̂ψ) m (4.11) j,j probability density current h̄ (Planck constant)/(2π) (4.12) x position coordinate p̂ momentum operator m particle mass (4.13) real part of t time (4.14) ∂ψ ∂t (4.15) H Hamiltonian (4.16) V potential energy E total energy ∇·j =− Schrödinger equation Ĥψ = ih̄ Particle stationary statesb − h̄2 ∂2 ψ(x) + V (x)ψ(x) = Eψ(x) 2m ∂x2 particles. In three dimensions, suitable units would be particles m−2 s−1 . Schrödinger equation for a particle, in one dimension. b Time-independent pr probability density ψ wavefunction ∂ (ψψ ∗ ) ∂t Continuity equation a For (4.10) main January 23, 2006 16:6 91 4.2 Quantum definitions Operators Hermitian conjugate operator Position operator Momentum operator pˆnx = Kinetic energy operator h̄2 ∂2 T̂ = − 2m ∂x2 Hamiltonian operator ∗ (âφ) ψ dx = (4.18) h̄n ∂n in ∂xn L̂2 = Lˆx + Lˆy + L̂z Parity operator P̂ ψ(r) = ψ(−r) 2 Expectation value Time dependence Relation to eigenfunctions Ehrenfest’s theorem a Equation arbitrary integer ≥ 1 momentum coordinate T h̄ kinetic energy (Planck constant)/(2π) m particle mass (4.21) H V Hamiltonian potential energy (4.22) Lz (4.23) L angular momentum along z axis (sim. x and y) total angular momentum P̂ parity operator r position vector 2 L̂z = x̂pˆy − ŷ pˆx 2 a = â = 2 (4.24) Ψ∗ âΨ dx - . ∂â i d â = [Ĥ, â] + dt h̄ ∂t if âψn = an ψn and Ψ = then a = |cn |2 an a expectation value of a (4.26) â Ψ x operator for a (spatial) wavefunction (spatial) coordinate (4.27) t h̄ time (Planck constant)/(2π) ψn an n eigenfunctions of â eigenvalues dummy index cn probability amplitudes (4.25) = Ψ|â|Ψ complex conjugate x,y position coordinates px (4.20) h̄ ∂2 + V (x) 2m ∂x2 ∗ n (4.19) Angular momentum operators Expectation valuea (4.17) xˆn = xn Ĥ = − Hermitian conjugate operator ψ,φ normalisable functions â φ∗ âψ dx cn ψn (4.28) d r = p dt (4.29) m r particle mass position vector d p = −∇V dt (4.30) p V momentum potential energy m (4.26) uses the Dirac “bra-ket” notation for integrals involving operators. The presence of vertical bars distinguishes this use of angled brackets from that on the left-hand side of the equations. Note that a and â are taken as equivalent. 4 main January 23, 2006 16:6 92 Quantum physics Dirac notation Matrix elementa n,m eigenvector indices ψn∗ âψm anm = dx = n|â|m (4.31) anm matrix element ψn basis states (4.32) â x operator spatial coordinate Bra vector bra state vector = n| (4.33) ·| bra Ket vector ket state vector = |m (4.34) |· ket Scalar product n|m = Ψ cn wavefunction probability amplitudes if Ψ= ψn∗ ψm dx then a = m a The 4.3 (4.36) cn ψn n Expectation (4.35) c∗n cm anm (4.37) n Dirac bracket, n|â|m , can also be written ψn |â|ψm . Wave mechanics Potential stepa V (x) V0 incident particle i Potential function Wavenumbers h̄2 k 2 = 2mE h̄2 q 2 = 2m(E − V0 ) Probability currentsb a One-dimensional x 0 " 0 V (x) = V0 Amplitude reflection coefficient Amplitude transmission coefficient ii (x < 0) (x ≥ 0) (x < 0) (x > 0) (4.38) V V0 particle potential energy step height h̄ (Planck constant)/(2π) (4.39) (4.40) k,q particle wavenumbers m particle mass E total particle energy r= k−q k+q (4.41) r amplitude reflection coefficient t= 2k k+q (4.42) t amplitude transmission coefficient ji particle flux in zone i jii particle flux in zone ii h̄k (1 − |r|2 ) m h̄q jii = |t|2 m ji = (4.43) (4.44) interaction with an incident particle of total energy E = KE + V . If E < V0 then q is imaginary and |r|2 = 1. 1/|q| is then a measure of the tunnelling depth. b Particle flux with the sign of increasing x. main January 23, 2006 16:6 93 4.3 Wave mechanics Potential wella V (x) incident particle i −a ii a iii 0 Potential function " 0 V (x) = −V0 Wavenumbers h̄2 k 2 = 2mE h̄2 q 2 = 2m(E + V0 ) Amplitude reflection coefficient Amplitude transmission coefficient (|x| > a) (|x| ≤ a) (4.45) (4.46) (4.47) (4.48) 2kqe−2ika 2kq cos2qa − i(q 2 + k 2 )sin2qa (4.49) t= h̄k (1 − |r|2 ) m h̄k jiii = |t|2 m Ramsauer effectc En = −V0 + a One-dimensional −V0 ie−2ika (q 2 − k 2 )sin2qa r= 2kq cos2qa − i(q 2 + k 2 )sin2qa Probability currentsb Bound states (V0 < E < 0)d (|x| > a) (|x| < a) ji = (4.50) (4.51) n2 h̄2 π 2 8ma2 " |k|/q tanqa = −q/|k| q 2 − |k|2 = 2mV0 /h̄2 (4.52) even parity odd parity x V particle potential energy V0 h̄ 2a well depth (Planck constant)/(2π) well width k,q particle wavenumbers m E particle mass total particle energy r amplitude reflection coefficient t amplitude transmission coefficient ji jiii particle flux in zone i particle flux in zone iii n integer > 0 En Ramsauer energy (4.53) (4.54) interaction with an incident particle of total energy E = KE + V > 0. flux in the sense of increasing x. c Incident energy for which 2qa = nπ, |r| = 0, and |t| = 1. d When E < 0, k is purely imaginary. |k| and q are obtained by solving these implicit equations. b Particle 4 main January 23, 2006 16:6 94 Quantum physics Barrier tunnellinga V (x) V0 incident particle i ii −a " 0 V (x) = V0 Potential function Wavenumber and tunnelling constant Amplitude reflection coefficient Amplitude transmission coefficient 0 iii a (|x| > a) (|x| ≤ a) (4.55) h̄2 k 2 = 2mE (|x| > a) 2 2 h̄ κ = 2m(V0 − E) (|x| < a) r= t= x (4.56) (4.57) −ie−2ika (k 2 + κ2 )sinh2κa 2kκcosh2κa − i(k 2 − κ2 )sinh2κa V V0 h̄ particle potential energy well depth (Planck constant)/(2π) 2a k κ barrier width incident wavenumber tunnelling constant m E particle mass total energy (< V0 ) r amplitude reflection coefficient t amplitude transmission coefficient |t|2 tunnelling probability ji particle flux in zone i jiii particle flux in zone iii (4.58) 2kκe−2ika (4.59) 2kκcosh2κa − i(k 2 − κ2 )sinh2κa 4k 2 κ2 (4.60) (k 2 + κ2 )2 sinh2 2κa + 4k 2 κ2 16k 2 κ2 exp(−4κa) (|t|2 1) 2 (k + κ2 )2 (4.61) |t|2 = Tunnelling probability Probability currentsb a By h̄k (1 − |r|2 ) m h̄k jiii = |t|2 m ji = (4.62) (4.63) a particle of total energy E = KE + V , through a one-dimensional rectangular potential barrier height V0 > E. flux in the sense of increasing x. b Particle Particle in a rectangular boxa Eigenfunctions Energy levels Ψlmn = 8 abc h2 Elmn = 8M 1/2 sin mπy nπz lπx sin sin a b c (4.64) l 2 m2 n2 + + a2 b2 c2 (4.65) Ψlmn a,b,c l,m,n eigenfunctions box dimensions integers ≥ 1 Elmn h energy Planck constant particle mass M density of states (per unit volume) a Spinless particle in a rectangular box bounded by the planes x = 0, y = 0, z = 0, x = a, y = b, and z = c. The potential is zero inside and infinite outside the box. Density of states ρ(E) dE = 4π (2M 3 E)1/2 dE h3 ρ(E) (4.66) a b x z y c main January 23, 2006 16:6 95 4.4 Hydrogenic atoms Harmonic oscillator Schrödinger equation Energy levelsa h̄2 ∂2 ψn 1 + mω 2 x2 ψn = En ψn − 2m ∂x2 2 En = n + 1 h̄ω 2 (4.68) 2 Eigenfunctions Hermite polynomials aE 0 4.4 (4.67) h̄ (Planck constant)/(2π) m ψn mass nth eigenfunction x n ω displacement integer ≥ 0 angular frequency En total energy in nth state Hn Hermite polynomials y dummy variable 2 Hn (x/a)exp[−x /(2a )] (n!2n aπ 1/2 )1/2 1/2 h̄ where a = mω ψn = (4.69) H0 (y) = 1, H1 (y) = 2y, H2 (y) = 4y 2 − 2 Hn+1 (y) = 2yHn (y) − 2nHn−1 (y) (4.70) is the zero-point energy of the oscillator. Hydrogenic atoms Bohr modela Quantisation condition µrn2 Ω = nh̄ Bohr radius a0 = 0 h2 α = 52.9pm 2 πme e 4πR∞ Orbit radius rn = n2 me a0 Z µ Total energy En = − Fine structure constant α= Hartree energy EH = Rydberg constant R∞ = Rydberg’s formulab µe4 Z 2 µ Z2 = −R hc ∞ me n2 820 h2 n2 rn Ω nth orbit radius orbital angular speed n principal quantum number (> 0) (4.72) a0 µ −e Bohr radius reduced mass ( me ) electronic charge (4.73) Z h h̄ atomic number Planck constant h/(2π) (4.74) En 0 me total energy of nth orbit permittivity of free space electron mass α fine structure constant µ0 permeability of free space (4.71) e2 1 µ0 ce2 = 2h 4π0 h̄c 137 (4.75) h̄2 4.36 × 10−18 J me a20 (4.76) EH Hartree energy (4.77) R∞ Rydberg constant c speed of light me e4 EH me cα2 = 3 2 = 2h 8h 0 c 2hc 1 1 µ 1 = R∞ Z 2 2 − 2 λmn me n m (4.78) λmn photon wavelength m integer > n the Bohr model is strictly a two-body problem, the equations use reduced mass, µ = me mnuc /(me +mnuc ) me , where mnuc is the nuclear mass, throughout. The orbit radius is therefore the electron–nucleus distance. b Wavelength of the spectral line corresponding to electron transitions between orbits m and n. a Because 4 main January 23, 2006 16:6 96 Quantum physics Hydrogenlike atoms – Schrödinger solutiona Schrödinger equation − h̄2 2 Ze2 ∇ Ψnlm − Ψnlm = En Ψnlm 2µ 4π0 r with µ = me mnuc me + mnuc (4.79) Eigenfunctions 1/2 3/2 (n − l − 1)! 2 m xl e−x/2 L2l+1 Ψnlm (r,θ,φ) = n−l−1 (x)Yl (θ,φ) 2n(n + l)! an with a = me a0 , µ Z x= 2r , an and L2l+1 n−l−1 (x) = k=0 µe4 Z 2 820 h2 n2 total energy 0 permittivity of free space h me h̄ Planck constant mass of electron h/2π µ mnuc Ψnlm reduced mass ( me ) mass of nucleus eigenfunctions (4.85) Ze −e charge of nucleus electronic charge n = 1,2,3,... (4.86) Lqp l = 0,1,2,... ,(n − 1) m = 0,±1,±2,... ,±l ∆n = 0 (4.87) (4.88) (4.89) a r associated Laguerre polynomialsc classical orbit radius, n = 1 electron–nucleus separation Yl m spherical harmonics ∆l = ±1 ∆m = 0 or (4.90) (4.91) a0 0 h Bohr radius = πm e2 En = − Radial expectation values a r = [3n2 − l(l + 1)] 2 a2 n2 r2 = [5n2 + 1 − 3l(l + 1)] 2 1 1/r = 2 an 2 2 1/r = (2l + 1)n3 a2 a For (4.81) ±1 a−3/2 −r/a e π 1/2 a−3/2 r −r/2a e Ψ210 = cosθ 4(2π)1/2 a r2 −r/3a a−3/2 r + 2 Ψ300 = 27 − 18 e a a2 81(3π)1/2 r ! r −r/3a a−3/2 6− e sinθ e±iφ Ψ31±1 = ∓ 1/2 a a 81π a−3/2 r2 −r/3a Ψ32±1 = ∓ e sinθ cosθ e±iφ 81π 1/2 a2 Ψ100 = (l + n)!(−x)k (2l + 1 + k)!(n − l − 1 − k)!k! En Total energy Allowed quantum numbers and selection rulesb n−l−1 (4.80) (4.82) (4.83) (4.84) r ! −r/2a a−3/2 2 − e a 4(2π)1/2 a−3/2 r Ψ21±1 = ∓ 1/2 e−r/2a sinθ e±iφ 8π a r ! r −r/3a 21/2 a−3/2 6 − e Ψ310 = cosθ a a 81π 1/2 Ψ200 = a−3/2 r2 −r/3a e (3cos2 θ − 1) 81(6π)1/2 a2 a−3/2 r2 −r/3a 2 ±2iφ Ψ32±2 = e sin θ e 162π 1/2 a2 Ψ320 = a single bound electron in a perfect nuclear Coulomb potential (nonrelativistic and spin-free). dipole transitions between orbitals. c The sign and indexing definitions for this function vary. This form is appropriate to Equation (4.80). b For 2 e main January 23, 2006 16:6 97 4.4 Hydrogenic atoms Orbital angular dependence z (s)2 −0.4 (px )2 0.2 −0.4 −0.2 (py )2 −0.2 y 0.2 0.2 x −0.2 (pz )2 (dx2 −y2 )2 −0.4 (dxz )2 4 (dz 2 )2 (dyz )2 0 0 s orbital (l = 0) s = Y00 = constant p orbitals (l = 1) −1 1 (Y − Y1−1 ) ∝ cosφsinθ 21/2 1 i py = 1/2 (Y11 + Y1−1 ) ∝ sinφsinθ 2 pz = Y10 ∝ cosθ px = dx2 −y2 = d orbitals (l = 2) a See (dxy )2 1 21/2 (Y22 + Y2−2 ) ∝ sin2 θ cos2φ −1 dxz = 1/2 (Y21 − Y2−1 ) ∝ sinθ cosθ cosφ 2 dz 2 = Y20 ∝ (3cos2 θ − 1) i dyz = 1/2 (Y21 + Y2−1 ) ∝ sinθ cosθ sinφ 2 −i dxy = 1/2 (Y22 − Y2−2 ) ∝ sin2 θ sin2φ 2 page 49 for the definition of spherical harmonics. (4.92) Ylm spherical harmonicsa (4.93) (4.94) θ,φ spherical polar coordinates (4.95) (4.96) z (4.97) (4.98) (4.99) (4.100) θ x y φ main January 23, 2006 16:6 98 4.5 Quantum physics Angular momentum Orbital angular momentum Angular momentum operators Ladder operators L̂ = r× p̂ ∂ h̄ ∂ L̂z = x −y i ∂y ∂x h̄ ∂ = i ∂φ 2 2 2 2 L̂ = Lˆx + Lˆy + L̂z 1 ∂ ∂ 1 ∂2 2 = −h̄ sinθ + 2 sinθ ∂θ ∂θ sin θ ∂φ2 L (4.102) p r (4.103) xyz Cartesian coordinates (4.104) rθφ spherical polar coordinates (4.105) Lˆ± = Lˆx ± iLˆy ∂ ∂ ±iφ ± = h̄e icotθ ∂φ ∂θ Lˆ± Y ml = h̄[l(l + 1) − ml (ml ± 1)]1/2 Y ml ±1 (4.107) L̂2 Yl ml = l(l + 1)h̄2 Yl ml (4.109) l l Eigenfunctions and eigenvalues (4.101) ml (l ≥ 0) ml (4.106) (4.108) L̂z Yl = ml h̄Yl (|ml | ≤ l) ml ˆ L̂z [L± Y (θ,φ)] = (ml ± 1)h̄Lˆ± Y ml (θ,φ) (4.110) l-multiplicity = (2l + 1) (4.112) l l h̄ angular momentum linear momentum position vector (Planck constant)/(2π) Lˆ± ladder operators m Yl l spherical harmonics l,ml integers (4.111) Angular momentum commutation relationsa Conservation of angular momentumb [Ĥ, L̂z ] = 0 [L̂z ,x] = ih̄y (4.114) [L̂z ,y] = −ih̄x (4.115) [L̂z ,z] = 0 (4.116) [L̂z , pˆx ] = ih̄pˆy (4.117) [L̂z , pˆy ] = −ih̄pˆx (4.118) [L̂z , pˆz ] = 0 (4.119) [L̂2 , Lˆx ] = [L̂2 , Lˆy ] = [L̂2 , L̂z ] = 0 a The b For (4.113) L p H Lˆ± angular momentum momentum Hamiltonian ladder operators [Lˆx , Lˆy ] = ih̄L̂z [L̂z , Lˆx ] = ih̄Lˆy (4.120) [Lˆy , L̂z ] = ih̄Lˆx [Lˆ+ , L̂z ] = −h̄Lˆ+ (4.122) [Lˆ− , L̂z ] = h̄Lˆ− [Lˆ+ , Lˆ− ] = 2h̄L̂z (4.124) [L̂2 , Lˆ± ] = 0 (4.126) (4.121) (4.123) (4.125) (4.127) commutation of a and b is defined as [a,b] = ab − ba (see page 26). Similar expressions hold for S and J. motion under a central force. main January 23, 2006 16:6 99 4.5 Angular momentum Clebsch–Gordan coefficientsa +1 +3/2 j,−mj |l1 ,−m1 ;l2 ,−m2 = (−1)l1 +l2 −j j,mj |l1 ,m1 ;l2 ,m2 1 1 × 1/2 3/2 +1/2 0 +1/2 +1/2 1 1 +1 +1/2 1 3/2 1/2 0 mj +1/2 −1/2 1/2 1/2 +1 −1/2 1/3 2/3 l1 × l 2 j j ... −1/2 +1/2 1/2 −1/2 0 +1/2 2/3 −1/3 m1 m2 coefficients m1 m2 j,mj |l1 ,m1 ;l2 ,m2 +2 +5/2 . . . . . . 3/2 × 1/2 2 2 × 1/2 5/2 +3/2 . . . +1 +3/2 +1/2 1 2 +2 +1/2 1 5/2 3/2 1 +3/2 −1/2 1/4 3/4 +2 −1/2 1/5 4/5 0 +1/2 +1/2 +1/2 3/4 −1/4 2 +1 +1/2 4/5 −1/5 5/2 3/2 1 +1/2 −1/2 1/2 1/2 +1 −1/2 2/5 3/5 −1/2 +1/2 1/2 −1/2 0 +1/2 3/5 −2/5 +2 +5/2 1/2 × 1/2 1×1 2 +1 +1 +1 1 2 1 +1 0 1/2 1/2 0 +1 1/2 −1/2 +1 −1 0 0 −1 +1 +3 2×1 3 +2 +2 +1 1 3 2 +2 0 1/3 2/3 +1 +1 2/3 −1/3 3/2 × 1 0 2 1 0 1/6 1/2 1/3 2/3 0 −1/3 1/6 −1/2 1/3 +1 3 2 1 +2 −1 1/15 1/3 3/5 +1 0 8/15 1/6 −3/10 0 +1 6/15 −1/2 1/10 +1 −1 0 0 −1 +1 5/2 +3/2 +3/2 +1 1 5/2 3/2 +3/2 0 2/5 3/5 +1/2 +1/2 +1 3/5 −2/5 5/2 3/2 1/2 +3/2 −1 1/10 2/5 1/2 1/2 0 3/5 1/15 −1/3 +3 −1/2 +1 3/10 −8/15 1/6 3/2 × 3/2 3 +2 +3/2 +3/2 1 3 2 +3/2 +1/2 1/2 1/2 +1 +1/2 +3/2 1/2 −1/2 3 2 1 +3/2 −1/2 1/5 1/2 3/10 +1/2 +1/2 3/5 0 −2/5 0 −1/2 +3/2 1/5 −1/2 3/10 3 2 1 0 0 +3/2 −3/2 1/20 1/4 9/20 1/4 3 2 1 +1/2 −1/2 9/20 1/4 −1/20 −1/4 1/5 1/2 3/10 −1/2 +1/2 9/20 −1/4 −1/20 1/4 3/5 0 −2/5 −3/2 +3/2 1/20 −1/4 9/20 −1/4 1/5 −1/2 3/10 +7/2 7/2 +5/2 +2 +3/2 1 7/2 5/2 +2 +1/2 3/7 4/7 +3/2 +1 +3/2 4/7 −3/7 7/2 5/2 3/2 +2 −1/2 1/7 16/35 2/5 +2 +1 +1/2 4/7 1/35 −2/5 +1/2 4 3 2 0 +3/2 2/7 −18/35 1/5 7/2 5/2 3/2 1/2 3/14 1/2 2/7 +2 −3/2 1/35 6/35 2/5 2/5 4/7 0 −3/7 +1 +1 −1/2 12/35 5/14 0 −3/10 3/14 −1/2 2/7 4 3 2 1 0 +1/2 18/35 −3/35 −1/5 1/5 +2 −1 1/14 3/10 3/7 1/5 −1 +3/2 4/35 −27/70 2/5 −1/10 +1 0 3/7 1/5 −1/14 −3/10 0 +1 3/7 −1/5 −1/14 3/10 0 −1 +2 1/14 −3/10 3/7 −1/5 4 3 2 1 0 +2 −2 1/70 1/10 2/7 2/5 1/5 +1 −1 8/35 2/5 1/14 −1/10 −1/5 0 0 18/35 0 −2/7 0 1/5 −1 +1 8/35 −2/5 1/14 1/10 −1/5 −2 +2 1/70 −1/10 2/7 −2/5 1/5 2 × 3/2 +4 4 +3 +2 +2 1 4 3 +2 +1 1/2 1/2 +1 +2 1/2 −1/2 +2 0 +1 +1 0 +2 2×2 a Or “Wigner coefficients,” using the Condon–Shortley sign ( convention. Note that a square root is assumed over all coefficient digits, so that “−3/10” corresponds to − 3/10. Also for clarity, only values of mj ≥ 0 are listed here. The coefficients for mj < 0 can be obtained from the symmetry relation j,−mj |l1 ,−m1 ;l2 ,−m2 = (−1)l1 +l2 −j j,mj |l1 ,m1 ;l2 ,m2 . 4 main January 23, 2006 16:6 100 Quantum physics Angular momentum additiona J =L+S Jˆz = L̂z + Sˆz (4.128) ·S Jˆ2 = L̂2 + Sˆ2 + 2L/ Jˆz ψj,mj = mj h̄ψj,mj (4.130) Jˆ2 ψj,mj = j(j + 1)h̄2 ψj,mj (4.132) j-multiplicity = (2l + 1)(2s + 1) (4.133) Mutually commuting sets {L2 ,S 2 ,J 2 ,Jz ,L · S} (4.134) {L2 ,S 2 ,Lz ,Sz ,Jz } (4.135) Clebsch– Gordan coefficientsb |j,mj = Total angular momentum J ,J total angular momentum L,L orbital angular momentum S,S spin angular momentum ψ eigenfunctions (4.129) (4.131) j,mj |l,ml ;s,ms |l,ml |s,ms ml ,ms ms +ml =mj (4.136) mj magnetic quantum number |mj | ≤ j j (l + s) ≥ j ≥ |l − s| {} set of mutually commuting observables |· eigenstates ·|· Clebsch–Gordan coefficients spin and orbital angular momenta as examples, eigenstates |s,ms and |l,ml . “Wigner coefficients.” Assuming no L–S interaction. a Summing b Or Magnetic moments Bohr magneton eh̄ µB = 2me Gyromagnetic ratioa γ= Electron orbital gyromagnetic ratio Spin magnetic moment of an electronb Landé g-factorc a Or (4.137) orbital magnetic moment (4.138) orbital angular momentum −µB h̄ −e = 2me γe = µB Bohr magneton −e h̄ me electronic charge (Planck constant)/(2π) electron mass γ gyromagnetic ratio γe electron gyromagnetic ratio (4.139) (4.140) µe,z = −ge µB ms (4.141) h̄ (4.142) = ±ge γe 2 ge eh̄ =± (4.143) 4me ( µJ = gJ J(J + 1)µB (4.144) µJ,z = −gJ µB mJ (4.145) J(J + 1) + S(S + 1) − L(L + 1) gJ = 1 + 2J(J + 1) (4.146) µe,z z component of spin magnetic moment ge electron g-factor ( 2.002) ms spin quantum number (±1/2) µJ total magnetic moment µJ,z z component of µJ mJ magnetic quantum number J,L,S total, orbital, and spin quantum numbers gJ Landé g-factor “magnetogyric ratio.” b The electron g-factor equals exactly 2 in Dirac theory. The modification g = 2 + α/π + ..., where α is the fine e structure constant, comes from quantum electrodynamics. c Relating the spin + orbital angular momenta of an electron to its total magnetic moment, assuming g = 2. e main January 23, 2006 16:6 101 4.5 Angular momentum Quantum paramagnetism 1 0.8 0.6 0.4 0.2 0 −10 −5 B∞ (x) = L(x) B4 (x) B1 (x) B1/2 (x) = tanhx −0.2 −0.4 −0.6 −0.8 −1 5 x 10 (2J + 1)x x 2J + 1 1 coth coth BJ (x) = − 2J 2J 2J 2J Brillouin function  J +1x BJ (x) 3J L(x) BJ (x) Brillouin function (x 1) (J 1) B1/2 (x) = tanhx Mean magnetisationa M for isolated spins (J = 1/2) a Of (4.147) (4.148) (4.149) µB B M = nµB JgJ BJ JgJ (4.150) kT µB B M 1/2 = nµB tanh kT (4.151) J total angular momentum quantum number L(x) Langevin function = cothx − 1/x (see page 144) M n gJ mean magnetisation number density of atoms Landé g-factor µB B k Bohr magneton magnetic flux density Boltzmann constant T M temperature mean magnetisation for J = 1/2 (and gJ = 2) 1/2 an ensemble of atoms in thermal equilibrium at temperature T , each with total angular momentum quantum number J. 4 main January 23, 2006 16:6 102 4.6 Quantum physics Perturbation theory Time-independent perturbation theory Unperturbed states Ĥ0 ψn = En ψn Ĥ0 ψn En unperturbed Hamiltonian eigenfunctions of Ĥ0 eigenvalues of Ĥ0 n integer ≥ 0 Ĥ Ĥ perturbed Hamiltonian perturbation ( Ĥ0 ) (4.154) Ek || perturbed eigenvalue ( Ek ) Dirac bracket (4.155) ψk perturbed eigenfunction ( ψk ) (4.152) (ψn nondegenerate) Perturbed Hamiltonian Ĥ = Ĥ0 + Ĥ Perturbed eigenvaluesa Ek = Ek + ψk |Ĥ |ψk |ψk |Ĥ |ψn |2 + ... + Ek − En (4.153) n=k Perturbed eigenfunctionsb ψk = ψk + ψk |Ĥ |ψn n=k Ek − En ψn + ... a To second order. b To first order. Time-dependent perturbation theory Ĥ0 unperturbed Hamiltonian ψn En n eigenfunctions of Ĥ0 eigenvalues of Ĥ0 integer ≥ 0 (4.157) Ĥ Ĥ (t) t perturbed Hamiltonian perturbation ( Ĥ0 ) time (4.158) Ψ wavefunction (4.159) ψ0 h̄ initial state (Planck constant)/(2π) cn probability amplitudes Γi→f transition probability per unit time from state i to state f Unperturbed stationary states Ĥ0 ψn = En ψn (4.156) Perturbed Hamiltonian Ĥ(t) = Ĥ0 + Ĥ (t) Schrödinger equation [Ĥ0 + Ĥ (t)]Ψ(t) = ih̄ ∂Ψ(t) ∂t Ψ(t = 0) = ψ0 Ψ(t) = cn (t)ψn exp(−iEn t/h̄) Perturbed n wavewhere functiona −i t cn = ψn |Ĥ (t )|ψ0 exp[i(En − E0 )t /h̄] dt h̄ 0 Fermi’s golden rule 2π Γi→f = |ψf |Ĥ |ψi |2 ρ(Ef ) h̄ (4.160) (4.161) (4.162) ρ(Ef ) density of final states a To first order. main January 23, 2006 16:6 103 4.7 High energy and nuclear physics 4.7 High energy and nuclear physics Nuclear decay Nuclear decay law N(t) = N(0)e−λt Half-life and mean life T1/2 = N(t) number of nuclei remaining after time t (4.163) ln2 λ T = 1/λ t time (4.164) λ decay constant (4.165) T1/2 half-life T mean lifetime Successive decays 1 → 2 → 3 (species 3 stable) N1 (t) = N1 (0)e−λ1 t (4.166) −λ1 t −λ2 t −e N1 (0)λ1 (e N2 (t) = N2 (0)e−λ2 t + λ2 − λ1 N1 N2 N3 ) (4.167) λ1 e−λ2 t − λ2 e−λ1 t λ1 −λ2 t N3 (t) = N3 (0) + N2 (0)(1 − e ) + N1 (0) 1 + λ2 λ2 − λ1 (4.168) population of species 1 population of species 2 population of species 3 decay constant 1 → 2 decay constant 2 → 3 v 3 = a(R − x) v velocity of α particle Geiger’s lawa (4.169) Geiger–Nuttall rule x a R distance from source constant range logλ = b + clogR (4.170) b, c constants for each series α, β, and γ N A B number of neutrons mass number (= N + Z) semi-empirical binding energy Z av as number of protons volume term (∼ 15.8MeV) surface term (∼ 18.0MeV) ac aa ap Coulomb term (∼ 0.72MeV) asymmetry term (∼ 23.5MeV) pairing term (∼ 33.5MeV) a For α particles in air (empirical). Nuclear binding energy Liquid drop modela B = av A − as A2/3 − ac  −3/4  +ap A δ(A) −ap A−3/4   0 Semi-empirical mass formula a Coefficient Z2 (N − Z)2 + δ(A) − aa 1/3 A A Z, N both even Z, N both odd otherwise M(Z,A) = ZMH + Nmn − B values are empirical and approximate. (4.171) (4.172) (4.173) M(Z,A) atomic mass MH mass of hydrogen atom mn neutron mass 4 main January 23, 2006 16:6 104 Quantum physics Nuclear collisions Breit–Wigner formulaa π Γab Γc g k 2 (E − E0 )2 + Γ2 /4 2J + 1 g= (2sa + 1)(2sb + 1) σ(E) = Total width Γ = Γab + Γc Resonance lifetime τ= Born scattering formulab h̄ Γ σ(E) cross-section for a + b → c (4.174) k g incoming wavenumber spin factor (4.175) E E0 Γ total energy (PE + KE) resonant energy width of resonant state R (4.176) Γab partial width into a + b Γc partial width into c τ resonance lifetime (4.177) J 2 dσ 2µ ∞ sinKr 2 = V (r)r dr dΩ h̄2 0 Kr (4.178) Mott scattering formulac α 2χ Acos lntan α !2 dσ χ χ h̄v 2 = csc4 + sec4 + dΩ 4E 2 2 sin2 2χ cos χ2 (4.179) ! 2 2 dσ 4 − 3sin χ α (A = −1, α vh̄) (4.180) dΩ 2E sin4 χ total angular momentum quantum number of R sa,b spins of a and b dσ dΩ differential collision cross-section µ reduced mass K = |kin − kout | (see footnote) r radial distance V (r) potential energy of interaction h̄ (Planck constant)/2π α/r scattering potential energy χ scattering angle v A closing velocity = 2 for spin-zero particles, = −1 for spin-half particles the reaction a + b ↔ R → c in the centre of mass frame. a central field. The Born approximation holds when the potential energy of scattering, V , is much less than the total kinetic energy. K is the magnitude of the change in the particle’s wavevector due to scattering. c For identical particles undergoing Coulomb scattering in the centre of mass frame. Nonidentical particles obey the Rutherford scattering formula (page 72). a For b For Relativistic wave equationsa Klein–Gordon equation (massive, spin zero particles) Weyl equations (massless, spin 1/2 particles) Dirac equation (massive, spin 1/2 particles) a Written (∇2 − m2 )ψ = ∂2 ψ ∂t2 ∂ψ ∂ψ ∂ψ ∂ψ = ± σx + σy + σz ∂t ∂x ∂y ∂z (iγ µ ∂µ − m)ψ = 0 ∂ ∂ ∂ ∂ , , , where ∂µ = ∂t ∂x ∂y ∂z (γ 0 )2 = 14 ; in natural units, with c = h̄ = 1. ψ wavefunction (4.181) (4.182) (4.183) (4.184) (γ 1 )2 = (γ 2 )2 = (γ 3 )2 = −14 (4.185) m particle mass t time ψ spinor wavefunction σ i Pauli spin matrices (see page 26) i i2 = −1 γ µ Diracmatrices: 1 0 γ0 = 2 0 −12 0 σi γi = −σ i 0 1n n × n unit matrix main January 23, 2006 16:6 Chapter 5 Thermodynamics 5.1 Introduction The term thermodynamics is used here loosely and includes classical thermodynamics, statistical thermodynamics, thermal physics, and radiation processes. Notation in these subjects can be confusing and the conventions used here are those found in the majority of modern treatments. In particular: • The internal energy of a system is defined in terms of the heat supplied to the system plus the work done on the system, that is, dU = d Q + d W . • The lowercase symbol p is used for pressure. Probability density functions are denoted by pr(x) and microstate probabilities by pi . • With the exception of specific intensity, quantities are taken as specific if they refer to unit mass and are distinguished from the extensive equivalent by using lowercase. Hence specific volume, v, equals V /m, where V is the volume of gas and m its mass. Also, the specific heat capacity of a gas at constant pressure is cp = Cp /m, where Cp is the heat capacity of mass m of gas. Molar values take a subscript “m” (e.g., Vm for molar volume) and remain in upper case. • The component held constant during a partial differentiation is shown after a vertical bar; is the partial differential of volume with respect to pressure, holding temperature hence ∂V ∂p T constant. The thermal properties of solids are dealt with more explicitly in the section on solid state physics (page 123). Note that in solid state literature specific heat capacity is often taken to mean heat capacity per unit volume. 5 main January 23, 2006 16:6 106 5.2 Thermodynamics Classical thermodynamics Thermodynamic laws Thermodynamic temperaturea T ∝ lim(pV ) Kelvin temperature scale T /K = 273.16 (5.1) p→0 lim(pV )T p→0 (5.2) lim(pV )tr T V p thermodynamic temperature volume of a fixed mass of gas gas pressure K tr kelvin unit temperature of the triple point of water p→0 dU change in internal energy First lawb dU = d Q + d W Entropyc dS = d Qrev d Q ≥ T T (5.3) d W work done on system d Q heat supplied to system S experimental entropy (5.4) T temperature reversible change a As determined with a gas thermometer. The idea of temperature is associated with the zeroth law of thermodynamics: If two systems are in thermal equilibrium with a third, they are also in thermal equilibrium with each other. b The d notation represents a differential change in a quantity that is not a function of state of the system. c Associated with the second law of thermodynamics: No process is possible with the sole effect of completely converting heat into work (Kelvin statement). rev Thermodynamic worka Hydrostatic pressure d W = −p dV (5.5) Surface tension d W = γ dA (5.6) p (hydrostatic) pressure dV volume change d W work done on the system γ surface tension dA change in area Electric field d W = E · dp (5.7) E electric field dp induced electric dipole moment Magnetic field d W = B · dm (5.8) B magnetic flux density dm induced magnetic dipole moment Electric current d W = ∆φ dq (5.9) ∆φ potential difference a The dq charge moved sources of electric and magnetic fields are taken as being outside the thermodynamic system on which they are working. main January 23, 2006 16:6 107 5.2 Classical thermodynamics Cycle efficiencies (thermodynamic)a Heat engine η= work extracted Th − Tl ≤ heat input Th (5.10) Refrigerator η= Tl heat extracted ≤ work done Th − Tl (5.11) Heat pump η= b Otto cycle a The Th heat supplied ≤ work done Th − Tl γ−1 V2 work extracted =1− η= heat input V1 η Th Tl efficiency higher temperature lower temperature V1 V2 compression ratio γ ratio of heat capacities (assumed constant) (5.12) (5.13) equalities are for reversible cycles, such as Carnot cycles, operating between temperatures Th and Tl . reversible “petrol” (heat) engine. b Idealised Heat capacities Constant volume d Q ∂U ∂S CV = = =T dT V ∂T V ∂T V Constant pressure Cp = d Q ∂H ∂S = =T dT p ∂T p ∂T p Difference in heat capacities Cp − CV = = Ratio of heat capacities γ= ∂U ∂V +p ∂V T ∂T p V T βp2 κT Cp κT = CV κS CV heat capacity, V constant Q T heat temperature V U S volume internal energy entropy Cp p H heat capacity, p constant pressure enthalpy βp κT isobaric expansivity isothermal compressibility γ κS ratio of heat capacities adiabatic compressibility βp V isobaric expansivity volume T temperature κT isothermal compressibility p pressure (5.21) κS adiabatic compressibility (5.22) KT isothermal bulk modulus (5.23) KS (5.14) (5.15) (5.16) (5.17) (5.18) Thermodynamic coefficients Isobaric expansivitya βp = 1 ∂V V ∂T p Adiabatic compressibility 1 ∂V κT = − V ∂p T 1 ∂V κS = − V ∂p S Isothermal bulk modulus KT = Isothermal compressibility Adiabatic bulk modulus a Also 1 ∂p = −V κT ∂V T 1 ∂p = −V KS = κS ∂V S (5.19) (5.20) adiabatic bulk modulus called “cubic expansivity” or “volume expansivity.” The linear expansivity is αp = βp /3. 5 main January 23, 2006 16:6 108 Thermodynamics Expansion processes ∂T T 2 ∂(p/T ) =− ∂V U CV ∂T V 1 ∂p =− T −p CV ∂T V ∂T T 2 ∂(V /T ) µ= = p ∂p H Cp ∂T 1 ∂V = T −V Cp ∂T p η= Joule expansiona Joule–Kelvin expansionb a Expansion b Expansion η T p Joule coefficient temperature pressure (5.25) U CV internal energy heat capacity, V constant (5.26) µ V H Joule–Kelvin coefficient volume enthalpy Cp heat capacity, p constant (5.24) (5.27) with no change in internal energy. with no change in enthalpy. Also known as a “Joule–Thomson expansion” or “throttling” process. Thermodynamic potentialsa dU = T dS − pdV + µdN Internal energy (5.28) H = U + pV (5.29) dH = T dS + V dp + µdN (5.30) F =U −TS (5.31) dF = −S dT − pdV + µdN (5.32) G = U − T S + pV (5.33) = F + pV = H − T S dG = −S dT + V dp + µdN (5.34) (5.35) Grand potential Φ = F − µN dΦ = −S dT − pdV − N dµ (5.36) (5.37) Gibbs–Duhem relation −S dT + V dp − N dµ = 0 (5.38) A = U − T0 S + p0 V (5.39) dA = (T − T0 )dS − (p − p0 )dV (5.40) Enthalpy Helmholtz free energyb Gibbs free energy Availability c U T S internal energy temperature entropy µ N chemical potential number of particles H p enthalpy pressure V volume F Helmholtz free energy G Gibbs free energy Φ grand potential A availability T0 temperature of surroundings pressure of surroundings p0 a dN=0 for a closed system. b Sometimes called the “work function.” c Sometimes called the “thermodynamic potential.” main January 23, 2006 16:6 109 5.2 Classical thermodynamics Maxwell’s relations Maxwell 1 ∂T ∂p =− ∂V S ∂S V Maxwell 2 ∂V ∂T = ∂p S ∂S p Maxwell 3 Maxwell 4 = = ∂2 U ∂S∂V ∂2 H ∂p∂S U internal energy T V temperature volume (5.42) H S p enthalpy entropy pressure (5.43) F Helmholtz free energy (5.44) G Gibbs free energy (5.41) ∂p ∂S ∂2 F = = ∂T V ∂V T ∂T ∂V ∂V ∂S ∂2 G = =− ∂T p ∂p T ∂p∂T Gibbs–Helmholtz equations ∂(F/T ) V ∂T ∂(F/V ) G = −V 2 T ∂V 2 ∂(G/T ) H = −T p ∂T U = −T 2 F U G Helmholtz free energy internal energy Gibbs free energy (5.46) H T enthalpy temperature (5.47) p V pressure volume (5.45) Phase transitions Heat absorbed Clausius–Clapeyron equationa Coexistence curveb L = T (S2 − S1 ) dp S2 − S1 = dT V2 − V1 L = T (V2 − V1 ) −L p(T ) ∝ exp RT Ehrenfest’s equationc dp βp2 − βp1 = dT κT 2 − κT 1 1 Cp2 − Cp1 = V T βp2 − βp1 Gibbs’s phase rule P+F=C+2 L (latent) heat absorbed (1 → 2) (5.48) T S temperature of phase change entropy (5.49) p V pressure volume (5.50) 1,2 phase states (5.51) R molar gas constant (5.52) βp isobaric expansivity (5.53) κT Cp isothermal compressibility heat capacity (p constant) P number of phases in equilibrium (5.54) F number of degrees of freedom C number of components a Phase boundary gradient for a first-order transition. Equation (5.50) is sometimes called the “Clapeyron equation.” b For V V , e.g., if phase 1 is a liquid and phase 2 a vapour. 2 1 c For a second-order phase transition. 5 main January 23, 2006 16:6 110 5.3 Thermodynamics Gas laws Ideal gas Joule’s law U = U(T ) (5.55) U T internal energy temperature Boyle’s law pV |T = constant (5.56) p pressure V volume Equation of state (Ideal gas law) pV = nRT (5.57) n R number of moles molar gas constant pV γ = constant T V (γ−1) = constant (5.58) (5.59) γ ratio of heat capacities (Cp /CV ) Adiabatic equations γ (1−γ) = constant 1 (p2 V2 − p1 V1 ) ∆W = γ −1 T p nRT γ −1 (5.60) ∆W work done on system (5.61) Internal energy U= Reversible isothermal expansion ∆Q = nRT ln(V2 /V1 ) (5.63) ∆Q heat supplied to system Joule expansiona ∆S = nR ln(V2 /V1 ) (5.64) ∆S (5.62) 1,2 initial and final states change in entropy of the system a Since ∆Q = 0 for a Joule expansion, ∆S is due entirely to irreversibility. Because entropy is a function of state it has the same value as for the reversible isothermal expansion, where ∆S = ∆Q/T . Virial expansion Virial expansion Boyle temperature B2 (T ) pV = RT 1 + V B3 (T ) + + ··· V2 (5.65) B2 (TB ) = 0 (5.66) p pressure V R T volume molar gas constant temperature Bi virial coefficients TB Boyle temperature main January 23, 2006 16:6 111 5.3 Gas laws Van der Waals gas Equation of state p+ a (Vm − b) = RT Vm2 (5.67) T temperature a,b van der Waals’ constants Tc = 8a/(27Rb) (5.68) 2 pc = a/(27b ) Vmc = 3b Critical point Reduced equation of state pr + p pressure Vm molar volume R molar gas constant 3 (3Vr − 1) = 8Tr Vr2 Tc critical temperature (5.69) (5.70) pc critical pressure Vmc critical molar volume (5.71) pr Vr = p/pc = Vm /Vmc Tr = T /Tc Dieterici gas Equation of state p= −a RT exp Vm − b RT Vm (5.72) T temperature a ,b Dieterici’s constants Tc = a /(4Rb ) (5.73) 2 2 pc = a /(4b e ) Critical point (5.74) Vmc = 2b Reduced equation of state pr = (5.75) 2 Tr exp 2 − 2Vr − 1 Vr Tr Van der Waals gas 1.4 1.1 1.2 0.8 pr pr Tr = 1.2 1.0 1 0.9 0.6 0.4 0.2 0 0.8 0 1 2 3 Vr 4 5 5 p pressure Vm molar volume R molar gas constant (5.76) 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 Tc critical temperature pc critical pressure Vmc critical molar volume e = 2.71828... pr = p/pc Vr Tr = Vm /Vmc = T /Tc Dieterici gas Tr = 1.2 1.1 1.0 0.9 0.8 0 1 2 3 Vr 4 5 main January 23, 2006 16:6 112 5.4 Thermodynamics Kinetic theory Monatomic gas 1 p = nmc2 3 Pressure (5.77) Equation of state of an ideal gas pV = NkT Internal energy N 3 U = NkT = mc2 2 2 (5.79) 3 CV = Nk 2 (5.80) Heat capacities Entropy (Sackur– Tetrode equation)a a For (5.78) 5 Cp = CV + Nk = Nk 2 Cp 5 γ= = CV 3 S = Nk ln the uncondensed gas. The factor mkT 2πh̄2 mkT 2πh̄2 (5.81) pressure number density = N/V particle mass c2 mean squared particle velocity V k N volume Boltzmann constant number of particles T temperature U internal energy CV Cp γ heat capacity, constant V heat capacity, constant p ratio of heat capacities S entropy h̄ e = (Planck constant)/(2π) = 2.71828... (5.82) 3/2 5/2 V e !3/2 p n m N (5.83) is the quantum concentration of the particles, nQ . Their thermal de −1/3 Broglie wavelength, λT , approximately equals nQ . Maxwell–Boltzmann distributiona Particle speed distribution Particle energy distribution rms speed Most probable speed a Probability −E 2E exp kT π 1/2 (kT )3/2 1/2 1/2 pr(E) dE = Mean speed pr m probability density particle mass k T Boltzmann constant temperature c particle speed E particle kinetic energy (= mc2 /2) (5.86) c mean speed (5.87) crms root mean squared speed (5.88) ĉ m !3/2 −mc2 pr(c) dc = exp 4πc2 dc 2πkT 2kT (5.84) 8kT πm 1/2 1/2 3kT 3π = c crms = m 8 1/2 2kT π !1/2 ĉ = = c m 4 c = density functions normalised so that %∞ 0 pr(x) dx = 1. dE (5.85) most probable speed main January 23, 2006 16:6 113 5.4 Kinetic theory Transport properties l d n mean free path molecular diameter particle number density pr probability x linear distance J c molecular flux mean molecular speed D diffusion coefficient (5.95) H λ T heat flux per unit area thermal conductivity temperature (5.96) ρ cV density specific heat capacity, V constant η dynamic viscosity (5.97) x displacement of sphere in x direction after time t (5.98) k t a Boltzmann constant time interval sphere radius dM dt mass flow rate pipe radius pipe length Mean free patha l= √ 1 2πd2 n (5.89) Survival equationb pr(x) = exp(−x/l) (5.90) Flux through a planec 1 J = nc 4 (5.91) Self-diffusion (Fick’s law of diffusion)d J = −D∇n (5.92) 2 where D lc 3 (5.93) Thermal conductivityd H = −λ∇T 1 ∂T ∇2 T = D ∂t (5.94) for monatomic gas Viscosity d Brownian motion (of a sphere) 5 λ ρlc cV 4 1 η ρlc 2 x2 = Free molecular flow (Knudsen flow)e kT t 3πηa 4Rp3 dM = dt 3L 2πm k 1/2 p1 1/2 T1 − p2 1/2 T2 Rp L m particle mass p pressure a For a perfect gas of hard, spherical particles with a Maxwell–Boltzmann speed distribution. b Probability of travelling distance x without a collision. c From the side where the number density is n, assuming an isotropic velocity distribution. Also known as “collision number.” d Simplistic kinetic theory yields numerical coefficients of 1/3 for D, λ and η. e Through a pipe from end 1 to end 2, assuming R l (i.e., at very low pressure). p (5.99) Gas equipartition k T energy per quadratic degree of freedom Boltzmann constant temperature CV Cp N heat capacity, V constant heat capacity, p constant number of molecules f n R number of degrees of freedom number of moles molar gas constant γ ratio of heat capacities Eq Classical equipartitiona 1 Eq = kT 2 Ideal gas heat capacities 1 1 CV = fNk = fnR 2 2 f Cp = Nk 1 + 2 2 Cp =1+ γ= CV f a System in thermal equilibrium at temperature T . (5.100) (5.101) (5.102) (5.103) 5 main January 23, 2006 16:6 114 5.5 Thermodynamics Statistical thermodynamics Statistical entropy Boltzmann formulaa Gibbs entropyb S = k lnW k lng(E) S = −k pi lnpi entropy Boltzmann constant number of accessible microstates (5.105) S k W (5.106) g(E) density of microstates with energy E i sum over microstates (5.104) i pi probability that the system is in microstate i N two-level systems W= N! (N − n)!n! (5.107) N n number of systems number in upper state N harmonic oscillators W= (Q + N − 1)! Q!(N − 1)! (5.108) Q total number of energy quanta available a Sometimes b Sometimes called “configurational entropy.” Equation (5.105) is true only for large systems. called “canonical entropy.” Ensemble probabilities Microcanonical ensemblea Partition functionb pi 1 pi = W Z= (5.109) −βEi e (5.110) i Canonical ensemble (Boltzmann distribution)c pi = Grand partition function Ξ= Grand canonical ensemble (Gibbs distribution)d pi = a Energy 1 −βEi e Z −β(Ei −µNi ) e (5.111) (5.112) i 1 −β(Ei −µNi ) e Ξ (5.113) fixed. called “sum over states.” c Temperature fixed. d Temperature fixed. Exchange of both heat and particles with a reservoir. b Also W Z probability that the system is in microstate i number of accessible microstates partition function β Ei sum over microstates = 1/(kT ) energy of microstate i k T Boltzmann constant temperature Ξ µ grand partition function chemical potential Ni number of particles in microstate i i main January 23, 2006 16:6 115 5.5 Statistical thermodynamics Macroscopic thermodynamic variables Helmholtz free energy F = −kT lnZ (5.114) Grand potential Φ = −kT lnΞ Internal energy U =F +TS =− Entropy S =− ∂ lnZ ∂β V ,N ∂F ∂(kT lnZ) = V ,N ∂T V ,N ∂T ∂F ∂(kT lnZ) p=− = T ,N ∂V T ,N ∂V ∂(kT lnZ) ∂F µ= =− V ,T ∂N V ,T ∂N Pressure Chemical potential F k T Helmholtz free energy Boltzmann constant temperature Z partition function (5.115) Φ Ξ grand potential grand partition function (5.116) U β internal energy = 1/(kT ) S entropy N number of particles (5.118) p pressure (5.119) µ chemical potential (5.117) 5 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 Bose–Einstein distribution 1.2 (µ = 0) Fermi–Dirac distribution 1 5 0.6 5 10 β =1 0.4 10 0.2 50 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0 0.2 0.4 0.6 0.8 i Bose–Einstein distributiona Fermi–Dirac distributionb fi = fi = 1 (5.121) eβ(i −µ) + 1 2 Bose condensation temperature Tc = h̄ 2m 2 6π n g 2/3 2/3 n 2πh̄2 mk gζ(3/2) bosons. fi ≥ 0. fermions. 0 ≤ fi ≤ 1. c For noninteracting particles. At low temperatures, µ . F b For 2 fi mean occupation number of ith state (5.120) eβ(i −µ) − 1 F = 1 1.2 1.4 1.6 1.8 i 1 Fermi energyc a For (µ = 1) 50 0.8 β =1 fi fi Identical particles (5.122) β = 1/(kT ) i µ energy quantum for ith state chemical potential F h̄ Fermi energy (Planck constant)/(2π) n m g particle number density particle mass spin degeneracy (= 2s + 1) ζ Riemann zeta function ζ(3/2) 2.612 Tc Bose condensation temperature (5.123) main January 23, 2006 16:6 116 Thermodynamics Population densitiesa −(χmj − χlj ) nmj gmj = exp Boltzmann nlj glj kT excitation −hν g mj lm equation exp = glj kT −χij ! gij exp Zj (T ) = kT Partition i function nij −χij ! gij exp = Nj Zj (T ) kT Saha equation (general) gij h3 χIj − χij ! (2πme kT )−3/2 exp nij = n0,j+1 ne g0,j+1 2 kT Saha equation (ion populations) Nj Zj (T ) h3 χIj ! (2πme kT )−3/2 exp = ne Nj+1 Zj+1 (T ) 2 kT nij (5.124) gij number density of atoms in excitation level i of ionisation state j (j = 0 if not ionised) level degeneracy (5.125) χij excitation energy relative to the ground state (5.126) νij h k photon transition frequency Planck constant Boltzmann constant T temperature Zj partition function for ionisation state j (5.128) Nj total number density in ionisation state j (5.129) ne me χIj electron number density electron mass ionisation energy of atom in ionisation state j (5.127) a All equations apply only under conditions of local thermodynamic equilibrium (LTE). In atoms with no magnetic splitting, the degeneracy of a level with total angular momentum quantum number J is gij = 2J + 1. 5.6 Fluctuations and noise Thermodynamic fluctuationsa Pressure fluctuations pr(x) ∝ exp[S(x)/k] −A(x) ∝ exp kT −1 2 ∂ A(x) var[x] = kT ∂x2 kT 2 ∂T var[T ] = kT = ∂S V CV ∂V var[V ] = −kT = κT V kT ∂p T ∂S var[S] = kT = kCp ∂T p KS kT ∂p var[p] = −kT = ∂V S V Density fluctuations var[n] = Fluctuation probability General variance Temperature fluctuations Volume fluctuations Entropy fluctuations a In n2 n2 κT kT var[V ] = V2 V (5.130) (5.131) (5.132) pr probability density x S A unconstrained variable entropy availability var[·] mean square deviation k Boltzmann constant T temperature V volume CV heat capacity, V constant (5.134) p κT pressure isothermal compressibility (5.135) Cp heat capacity, p constant (5.136) KS adiabatic bulk modulus (5.137) n number density (5.133) part of a large system, whose mean temperature is fixed. Quantum effects are assumed negligible. main January 23, 2006 16:6 117 5.6 Fluctuations and noise Noise Nyquist’s noise theorem dw = kT · β(eβ − 1)−1 dν = kTN dν kT dν (hν kT ) Johnson (thermal) noise voltagea vrms = (4kTN R∆ν)1/2 Shot noise (electrical) Irms = (2eI0 ∆ν)1/2 Noise figureb fdB = 10log10 1 + Relative power a Thermal b Noise G = 10log10 P2 P1 TN T0 w exchangeable noise power k T Boltzmann constant temperature TN β ν noise temperature = hν/(kT ) frequency (5.141) h vrms R Planck constant rms noise voltage resistance (5.142) ∆ν Irms −e bandwidth rms noise current electronic charge (5.143) I0 fdB T0 mean current noise figure (decibels) ambient temperature (usually taken as 290 K) G decibel gain of P2 over P1 (5.138) (5.139) (5.140) (5.144) P1 , P2 power levels voltage over an open-circuit resistance. figure can also be defined as f = 1 + TN /T0 , when it is also called “noise factor.” 5 main January 23, 2006 16:6 118 5.7 Thermodynamics Radiation processes Radiometrya Ω radiant energy radiance (generally a function of position and direction) angle between dir. of dΩ and normal to dA solid angle (5.146) A t area time (5.147) Φe radiant flux We radiant energy density dV differential volume of propagation medium Qe Le Radiant energyb Qe = Le cosθ dA dΩ dt J (5.145) θ Radiant flux (“radiant power”) Radiant energy densityc Φe = ∂Qe ∂t W Le cosθ dA dΩ = We = ∂Qe ∂V Jm−3 (5.148) Me = ∂Φe ∂A Wm−2 (5.149) Radiant exitanced = Me radiant exitance Le cosθ dΩ (5.150) z Irradiancee ∂Φe Ee = ∂A = −2 (5.151) Le cosθ dΩ (5.152) Wm (normal) x θ dΩ dA φ Radiant intensity ∂Φe Ie = ∂Ω = a Radiometry (5.153) Le cosθ dA (5.154) 1 ∂ 2 Φe cosθ dAdΩ 1 ∂Ie = cosθ ∂A Le = Radiance Wsr−1 Wm−2 sr−1 Ee Ie irradiance radiant intensity (5.155) (5.156) is concerned with the treatment of light as energy. called “total energy.” Note that we assume opaque radiant surfaces, so that 0 ≤ θ ≤ π/2. c The instantaneous amount of radiant energy contained in a unit volume of propagation medium. d Power per unit area leaving a surface. For a perfectly diffusing surface, M = πL . e e e Power per unit area incident on a surface. b Sometimes y main January 23, 2006 16:6 119 5.7 Radiation processes Photometrya Luminous energy (“total light”) Qv = Luminous flux ∂Qv Φv = ∂t Lv cosθ dA dΩ dt = Luminous densityb Luminous exitancec Illuminance (“illumination”)d Luminous intensitye Luminance (“photometric brightness”) (5.158) Lv cosθ dA dΩ (5.159) ∂Qv ∂V lmsm−3 Mv = ∂Φv ∂A lx Ev = Iv = = Ω solid angle A t Φv area time luminous flux Wv luminous density V volume (5.162) Lv cosθ dΩ (5.164) cd (5.165) Lv cosθ dA (5.166) 1 ∂ 2 Φv cosθ dAdΩ 1 ∂Iv = cosθ ∂A Lv = Luminous efficacy Luminous efficiency V (λ) = K(λ) Kmax z (normal) (5.163) Φv Lv Iv K= = = Φe Le Ie a Photometry angle between dir. of dΩ and normal to dA (5.161) lmm−2 ∂Φv ∂Ω θ Mv luminous exitance Lv cosθ dΩ ∂Φv ∂A = (5.160) (lmm−2 ) luminous energy luminance (generally a function of position and direction) (5.157) lumen (lm) Wv = = lms Qv Lv cdm−2 x θ dA φ y Ev Iv illuminance luminous intensity K Le luminous efficacy radiance Φe Ie V radiant flux radiant intensity luminous efficiency (5.167) (5.168) lmW−1 (5.169) (5.170) λ wavelength Kmax spectral maximum of K(λ) is concerned with the treatment of light as seen by the human eye. instantaneous amount of luminous energy contained in a unit volume of propagating medium. c Luminous emitted flux per unit area. d Luminous incident flux per unit area. The derived SI unit is the lux (lx). 1lx = 1lmm−2 . e The SI unit of luminous intensity is the candela (cd). 1cd = 1lmsr−1 . b The dΩ 5 main January 23, 2006 16:6 120 Thermodynamics Radiative transfera Flux density (through a plane) z Fν = (normal) Wm−2 Hz−1 Iν (θ,φ)cosθ dΩ (5.171) θ dΩ x φ Mean intensityb Jν = 1 4π Iν (θ,φ) dΩ Wm−2 Hz−1 (5.172) Fν Iν Jν Spectral energy densityc 1 uν = c Specific emission coefficient ν jν = ρ Iν (θ,φ) dΩ −3 Jm −1 Hz (5.173) uν Ω θ jν Wkg−1 Hz−1 sr−1 (5.174) ν ρ Gas linear absorption coefficient (αν 1) αν = nσν = Opacityd κν = Optical depth τν = αν ρ 1 lν m−1 kg−1 m2 κν ρ ds Transfer equatione 1 dIν = −κν Iν + jν ρ ds dIν = −αν Iν + ν or ds Kirchhoff’s lawf Sν ≡ Emission from a homogeneous medium Iν = Sν (1 − e−τν ) a The jν ν = κν αν y flux density specific intensity (Wm−2 Hz−1 sr−1 ) mean intensity spectral energy density solid angle angle between normal and direction of Ω specific emission coefficient emission coefficient (Wm−3 Hz−1 sr−1 ) density αν linear absorption coefficient (5.175) n σν lν particle number density particle cross section mean free path (5.176) κν opacity τν ds optical depth, or optical thickness line element Sν source function (5.177) (5.178) (5.179) (5.180) (5.181) definitions of these quantities vary in the literature. Those presented here are common in meteorology and astrophysics. Note particularly that the ambiguous term specific is taken to mean “per unit frequency interval” in the case of specific intensity and “per unit mass per unit frequency interval” in the case of specific emission coefficient. b In radio astronomy, flux density is usually taken as S = 4πJ . ν c Assuming a refractive index of 1. d Or “mass absorption coefficient.” e Or “Schwarzschild’s equation.” f Under conditions of local thermal equilibrium (LTE), the source function, S , equals the Planck function, B (T ) ν ν [see Equation (5.182)]. main January 23, 2006 16:6 121 5.7 Radiation processes Blackbody radiation 1050 105 1010 K 109 K νm (T ) = c/λm (T ) 108 K 1 107 K 10−5 106 K 105 K 10−10 104 K 103 K 10−15 100K 10−20 106 brightness (Bλ /Wm−2 m−1 sr−1 ) brightness (Bν /Wm−2 Hz−1 sr−1 ) 1010 1040 1010 K 1030 1020 107 K 106 K 105 K 1010 104 K 1 103 K 100K 10−10 2.7K 2.7K 108 1010 1012 1014 1016 1018 10−20 10−14 10−12 10−10 10−8 10−6 10−4 10−2 1022 1020 −1 hν 2hν 3 exp − 1 c2 kT dν Bλ (T ) = Bν (T ) dλ −1 hc 2hc2 = 5 exp −1 λ λkT Spectral energy density Rayleigh–Jeans law (hν kT ) Wien’s law (hν kT ) Wien’s displacement law Stefan– Boltzmann lawb 4π Bν (T ) c 4π uλ (T ) = Bλ (T ) c uν (T ) = Bν (5.183) Bλ (5.184) h surface brightness per unit frequency (Wm−2 Hz−1 sr−1 ) surface brightness per unit wavelength (Wm−2 m−1 sr−1 ) Planck constant Jm−3 Hz−1 (5.185) c k speed of light Boltzmann constant Jm−3 m−1 (5.186) T temperature uν,λ spectral energy density 2kT 2 2kT ν = 2 c2 λ −hν 2hν 3 Bν (T ) = 2 exp c kT " 5.1 × 10−3 mK for Bν λm T = 2.9 × 10−3 mK for Bλ ∞ M =π Bν (T ) dν Bν (T ) = 0 5 4 = 102 (5.182) Bν (T ) = Planck functiona 1 wavelength (λ/m) frequency (ν/Hz) 2π k T 4 = σT 4 15c2 h3 Energy density 4 u(T ) = σT 4 c Greybody M = σT 4 = (1 − A)σT 4 a With λm (T ) 109 K 108 K Jm−3 Wm−2 (5.187) (5.188) λm wavelength of maximum brightness M σ exitance Stefan–Boltzmann constant ( 5.67 × 10−8 Wm−2 K−4 ) (5.192) u energy density (5.193) A mean emissivity albedo (5.189) (5.190) (5.191) respect to the projected area of the surface. Surface brightness is also known simply as “brightness.” “Specific intensity” is used for reception. b Sometimes “Stefan’s law.” Exitance is the total radiated energy from unit area of the body per unit time. 5 main January 23, 2006 16:6 main January 23, 2006 16:6 Chapter 6 Solid state physics 6.1 Introduction This section covers a few selected topics in solid state physics. There is no attempt to do more than scratch the surface of this vast field, although the basics of many undergraduate texts on the subject are covered. In addition a period table of elements, together with some of their physical properties, is displayed on the next two pages. 6 Periodic table (overleaf) Data for the periodic table of elements are taken from Pure Appl. Chem., 71, 1593–1607 (1999), from the 16th edition of Kaye and Laby Tables of Physical and Chemical Constants (Longman, 1995) and from the 74th edition of the CRC Handbook of Chemistry and Physics (CRC Press, 1993). Note that melting and boiling points have been converted to kelvins by adding 273.15 to the Celsius values listed in Kaye and Laby. The standard atomic masses reflect the relative isotopic abundances in samples found naturally on Earth, and the number of significant figures reflect the variations between samples. Elements with atomic masses shown in square brackets have no stable nuclides, and the values reflect the mass numbers of the longest-lived isotopes. Crystallographic data are based on the most common forms of the elements (the α-form, unless stated otherwise) stable under standard conditions. Densities are for the solid state. For full details and footnotes for each element, the reader is advised to consult the original texts. Elements 110, 111, 112 and 114 are known to exist but their names are not yet permanent. main January 23, 2006 16:6 124 Solid state physics 6.2 Periodic table 1 name Hydrogen 1.007 94 1 2 H 1 89 (β) 378 HEX 1.632 13.80 20.28 2 Lithium 6.941 Beryllium 9.012 182 3 Li 4 19 K Rb 55 Cs 87 300 38 boiling point (K) 56 3 4 5 6 7 8 9 Titanium 47.867 Vanadium 50.941 5 Chromium 51.996 1 Manganese 54.938 049 Iron 55.845 Cobalt 58.933 200 Cr 25 Mn 21 Sc [Ca]3d1 559 2 992 HEX 1 757 1 813 Sr 39 Y [Sr]4d1 608 4 475 HEX 1 653 1 798 57 – 71 Fr 502 Zr [Sr]4d2 72 41 Hf Nb [Kr]4d4 5s1 [Ar]3d5 4s1 388 2 943 Molybdenum 95.94 42 Mo [Kr]4d5 5s1 330 10 222 315 BCC 4 973 2 896 4 913 Tantalum 180.947 9 73 24 302 7 194 BCC 3 673 2 180 Niobium 92.906 38 323 8 578 1.593 BCC 4 673 2 750 Hafnium 178.49 V [Ca]3d3 Ta [Yb]5d3 Tungsten 183.84 74 W [Yb]5d4 13 276 319 16 670 330 19 254 316 BCC HEX 1.581 BCC 2 503 4 873 3 293 5 833 3 695 5 823 2 173 Actinides Rutherfordium [261] Ra 89 – 103 104 [Rn]7s2 5 000 BCC 923 973 Zirconium 91.224 40 23 295 6 090 1.587 BCC 3 563 2 193 [Yb]5d2 Radium [226] 88 Ti [Ca]3d2 365 6 507 1.571 HEX 3 613 2 123 Lanthanides Ba 22 331 4 508 1.592 HEX 3 103 1 943 Yttrium 88.905 85 Barium 137.327 614 3 594 BCC 943.2 1 001 [Rn]7s1 c/a (angle in RHL, c/a b/a in ORC & MCL) Scandium 44.955 910 [Xe]6s2 Francium [223] 7 Ca Strontium 87.62 571 2 583 FCC 963.1 1 050 [Xe]6s1 1 900 BCC 301.6 20 [Kr]5s2 Caesium 132.905 45 6 Calcium 40.078 532 1 530 FCC 1 033 1 113 [Kr]5s1 1 533 BCC 312.4 lattice constant, a (fm) melting point (K) 321 1.624 1 363 [Ar]4s2 Rubidium 85.467 8 37 295 1.587 3 563 [Ne]3s2 429 1 738 HEX 1 153 923 [Ar]4s1 862 BCC 336.5 4 508 HEX 1 943 crystal type 229 1.568 2 745 12 Mg Potassium 39.098 3 5 density (kgm−3 ) Be Na [Ne]3s1 symbol Ti 22 [Ca]3d2 Magnesium 24.305 0 966 BCC 370.8 4 electron configuration Sodium 22.989 770 11 Titanium 47.867 [He]2s2 [He]2s1 533 (β) 351 1 846 HEX BCC 453.65 1 613 1 560 3 relative atomic mass (u) atomic number 1s1 [Ca]3d5 7 473 FCC 1 523 Tc [Sr]4d5 Fe [Ca]3d6 891 7 873 BCC 2 333 1 813 Technetium [98] 43 26 Co [Ca]3d7 287 8 800 () 251 HEX 1.623 3 133 1 768 3 203 Ruthenium 101.07 44 27 Ru [Kr]4d7 5s1 Rhodium 102.905 50 45 Rh [Kr]4d8 5s1 11 496 274 12 360 270 12 420 380 HEX 1.604 HEX 1.582 FCC 2 433 4 533 2 603 4 423 2 236 3 973 Rhenium 186.207 75 Re [Yb]5d5 Osmium 190.23 76 Os [Yb]5d6 Iridium 192.217 77 Ir [Yb]5d7 21 023 276 22 580 273 22 550 384 HEX 1.615 HEX 1.606 FCC 3 459 5 873 3 303 5 273 2 720 4 703 Dubnium [262] Seaborgium [263] Bohrium [264] Hassium [265] Meitnerium [268] Rf 105 Db 106 Sg 107 Bh 108 Hs 109 Mt [Ra]5f 14 6d2 [Ra]5f 14 6d3 ? [Ra]5f 14 6d4 ? [Ra]5f 14 6d5 ? [Ra]5f 14 6d6 ? [Ra]5f 14 6d7 ? Lanthanum 138.905 5 Cerium 140.116 Promethium [145] Samarium 150.36 515 1 773 Lanthanides 57 La [Ba]5d1 6 174 HEX 1 193 89 Ce [Ba]4f 1 5d1 59 Ac [Ra]6d1 Thorium 232.038 1 90 Th [Ra]6d2 Pr [Ba]4f 3 377 6 711 (γ) 516 6 779 3.23 FCC HEX 3 733 1 073 3 693 1 204 Actinium [227] Actinides 58 Praseodymium Neodymium 144.24 140.907 65 Pa [Rn]5f 2 6d1 7s2 Nd [Ba]4f 4 367 7 000 3.222 HEX 3 783 1 289 Protactinium 231.035 88 91 60 [Ba]4f 5 366 7 220 3.225 HEX 3 343 1 415 Uranium 238.028 9 92 61 Pm U [Rn]5f 3 6d1 7s2 Np [Rn]5f 4 6d1 7s2 Sm [Ba]4f 6 365 7 536 3.19 HEX 3 573 1 443 Neptunium [237] 93 62 363 7.221 2 063 Plutonium [244] 94 Pu [Rn]5f 6 7s2 285 20 450 666 19 816 618 10 060 531 11 725 508 15 370 392 19 050 1.736 ORC 0.733 MCL 1.773 FCC TET 0.825 ORC FCC 2.056 0.709 0.780 1 323 3 473 2 023 5 063 1 843 4 273 1 405.3 4 403 913 4 173 913 3 503 main January 23, 2006 16:6 125 6.2 Periodic table 18 Helium 4.002 602 He 2 BCC CUB DIA FCC HEX MCL ORC RHL TET (t-pt) 1s2 body-centred cubic simple cubic diamond face-centred cubic hexagonal monoclinic orthorhombic rhombohedral tetragonal triple point 13 14 15 16 17 Boron 10.811 Carbon 12.0107 Nitrogen 14.006 74 Oxygen 15.999 4 Fluorine 18.998 403 2 5 B [Be]2p1 2 466 RHL 2 348 13 Al [Mg]3p1 11 12 Copper 63.546 Zinc 65.39 Ni [Ca]3d8 8 907 FCC 1 728 Cu [Ar]3d10 4s1 30 Pd [Kr]4d10 Silver 107.868 2 47 Ag [Pd]5s1 Cadmium 112.411 48 Platinum 195.078 Pt [Xe]4f 14 5d9 6s1 Gold 196.966 55 79 Au [Xe]4f 14 5d10 6s1 Cd [Pd]5s2 11 995 389 10 500 409 8 647 FCC FCC HEX 1 828 3 233 1 235 2 433 594.2 78 Zn [Ca]3d10 Gallium 69.723 31 49 In [Cd]5p1 Ge [Zn]4p2 452 5 323 DIA 2 473 1211 15 Sn [Cd]5p2 Hg [Yb]5d10 Thallium 204.383 3 81 Tl [Hg]6p1 Lead 207.2 82 Pb [Hg]6p2 P [Mg]3p3 33 Unununium [272] Ununbium [285] 63 Eu [Ba]4f 7 5 248 BCC 1 095 Gadolinium 157.25 64 458 7 870 HEX 1 873 1 587 Americium [243] 95 Am [Ra]5f 7 Gd [Ba]4f 7 5d1 Terbium 158.925 34 65 363 8 267 1.591 HEX 3 533 1 633 Curium [247] 96 Cm [Rn]5f 7 6d1 7s2 Tb [Ba]4f 9 97 Bk [Ra]5f 9 13 670 347 13 510 350 14 780 HEX 3.24 HEX 3.24 HEX 1 449 2 873 1 618 3 383 1 323 114 Dysprosium 162.50 66 Dy [Ba]4f 10 361 8 531 1.580 HEX 3 493 1 683 Berkelium [247] F [Be]2p5 10 S Chlorine 35.452 7 17 As Selenium 78.96 34 Cl [Mg]3p5 331 2 086 1 046 2 030 2.340 ORC ORC 1.229 550 388.47 717.82 172 Se 35 Sb [Cd]5p3 Tellurium 127.60 52 18 451 6 247 57◦ 7 HEX 1 860 723 Bismuth 208.980 38 Bi [Hg]6p3 Te 624 1 656 FCC 239.1 83.81 Br Polonium [209] 84 Po [Hg]6p4 I [Cd]5p5 446 4 953 1.33 ORC 1 263 386.7 87.30 Krypton 83.80 36 Kr 581 119.9 Xenon 131.29 54 Xe [Cd]5p6 727 3 560 FCC 457 161.3 635 1.347 0.659 Astatine [210] 85 532 [Zn]4p6 Iodine 126.904 47 53 Ar 1.324 0.718 [Zn]4p5 [Cd]5p4 27.07 [Mg]3p6 Bromine 79.904 [Zn]4p4 446 Argon 39.948 668 3 000 413 4 808 (γ) 436 3 120 1.308 FCC 54◦ 7 HEX 1.135 ORC 0.672 958 265.90 332.0 115.8 (t-pt) 493 Antimony 121.760 83 16 Ne [Be]2p6 At [Hg]6p5 165.0 Radon [222] 86 Rn [Hg]6p6 337 440 1 233 573 623 202 211 Ununquadium [289] 110 Uun 111 Uuu 112 Uub Europium 151.964 9 1.320 3.162 [Zn]4p3 51 Sulfur 32.066 21 450 392 19 281 408 13 546 300 11 871 346 11 343 495 9 803 475 9 400 FCC FCC RHL 70◦ 32 HEX 1.598 FCC RHL 57◦ 14 CUB 2 041 4 093 1 337.3 3 123 234.32 629.9 577 1743 600.7 2 023 544.59 1 833 527 Ununnilium [271] O [Mg]3p4 Arsenic 74.921 60 566 5 776 RHL 3103 883 Tin 118.710 50 8 [Be]2p4 Phosphorus 30.973 761 543 1 820 ORC 3 533 317.3 Germanium 72.61 1.001 1.695 Indium 114.818 Si [Mg]3p2 32 N [Be]2p3 356 1.631 4.22 Neon 20.179 7 550 1 442 357 1 035 (β) 405 1 460 (γ) 683 1 140 1.32 FCC MCL HEX 1.631 CUB 0.61 (t-pt) 63 77.35 54.36 90.19 53.55 85.05 24.56 Silicon 28.085 5 14 7 325 7 285 (β) 583 6 692 298 7 290 1.886 TET 1.521 TET 0.546 RHL 1 043 429.75 2 343 505.08 2 893 903.8 Mercury 200.59 80 Ga [Zn]4p1 361 7 135 266 5 905 352 8 933 FCC HEX 1.856 ORC 3 263 1 357.8 2 833 692.68 1 183 302.9 Palladium 106.42 46 29 [Be]2p2 2 698 405 2 329 FCC DIA 933.47 2 793 1 683 10 28 C 6 1017 2 266 65◦ 7 DIA 4 273 4 763 Aluminium 26.981 538 Nickel 58.693 4 120 HEX 3-5 98 Cf [Ra]5f 10 342 15 100 3.24 HEX 1 173 Holmium 164.930 32 67 Ho [Ba]4f 11 359 8 797 1.573 HEX 2 833 1 743 Californium [251] Uuq Es [Ra]5f 11 338 3.24 HEX 1 133 68 Er [Ba]4f 12 358 9 044 1.570 HEX 2 973 1 803 Einsteinium [252] 99 Erbium 167.26 Thulium 168.934 21 Ytterbium 173.04 69 Tm 70 [Ba]4f 13 356 9 325 1.570 HEX 3 133 1 823 Yb [Ba]4f 14 Lutetium 174.967 71 Lu [Yb]5d1 354 6 966 (β) 549 9 842 1.570 FCC HEX 2 223 1 097 1 473 1 933 351 1.583 3 663 Fermium [257] Mendelevium [258] Nobelium [259] Lawrencium [262] 100 Fm 101 Md 102 No 103 Lr [Ra]5f 12 [Ra]5f 13 [Ra]5f 14 [Ra]5f 14 7p1 1 803 1 103 1 103 1 903 6 main January 23, 2006 16:6 126 Solid state physics 6.3 Crystalline structure Bravais lattices Volume of primitive cell Reciprocal primitive base vectorsa V = (a× b) · c (6.1) a∗ = 2πb× c/[(a× b) · c] b∗ = 2πc×a/[(a×b) · c] (6.2) (6.3) c∗ = 2πa× b/[(a× b) · c] (6.4) a · a∗ = b · b∗ = c · c∗ = 2π (6.5) ∗ Lattice vector Reciprocal lattice vector ∗ a,b,c V primitive base vectors volume of primitive cell a∗ ,b∗ ,c∗ reciprocal primitive base vectors a · b = a · c = 0 (etc.) (6.6) Ruvw = ua + vb + wc (6.7) Ruvw u,v,w lattice vector [uvw] integers G hkl = ha∗ + kb∗ + lc∗ (6.8) G hkl reciprocal lattice vector [hkl] exp(iG hkl · Ruvw ) = 1 (6.9) i i2 = −1 Weiss zone equationb hu + kv + lw = 0 (6.10) (hkl) Miller indices of planec Interplanar spacing (general) dhkl = 2π Ghkl (6.11) dhkl distance between (hkl) planes Interplanar spacing (orthogonal basis) h2 k 2 l 2 1 = 2+ 2+ 2 2 b c dhkl a (6.12) a Note that this is 2π times the usual definition of a “reciprocal vector” (see page 20). for lattice vector [uvw] to be parallel to lattice plane (hkl) in an arbitrary Bravais lattice. c Miller indices are defined so that G hkl is the shortest reciprocal lattice vector normal to the (hkl) planes. b Condition Weber symbols Converting [uvw] to [UV T W ] 1 U = (2u − v) 3 1 V = (2v − u) 3 1 T = − (u + v) 3 W =w (6.13) (6.14) (6.15) u = (U − T ) v = (V − T ) (6.17) (6.18) w=W (6.19) Zone lawa hU + kV + iT + lW = 0 (6.20) trigonal and hexagonal systems. Weber indices zone axis indices (hkil) Miller–Bravais indices Weber symbol zone axis symbol (6.16) Converting [UV T W ] to [uvw] a For U,V ,T ,W u,v,w [UV T W ] [uvw] main January 23, 2006 16:6 127 6.3 Crystalline structure Cubic lattices lattice lattice parameter volume of conventional cell lattice points per cell 1st nearest neighboursa 1st n.n. distance 2nd nearest neighbours 2nd n.n. distance packing fractionb reciprocal latticec primitive base vectorsd primitive (P) a a3 1 6 a 12 √ a 2 π/6 P a1 = ax̂ a2 = aŷ a3 = aẑ body-centred (I) a a3 2 8 √ a 3/2 6 a √ 3π/8 F a1 = a2 (ŷ + ẑ − x̂) a2 = a2 (ẑ + x̂ − ŷ) a3 = a2 (x̂ + ŷ − ẑ) face-centred (F) a a3 4 12 √ a/ 2 6 a √ 2π/6 I a1 = a2 (ŷ + ẑ) a2 = a2 (ẑ + x̂) a3 = a2 (x̂ + ŷ) a Or “coordination number.” √ close-packed spheres. The maximum possible packing fraction for spheres is 2π/6. c The lattice parameters for the reciprocal lattices of P, I, and F are 2π/a, 4π/a, and 4π/a respectively. d x̂, ŷ, and ẑ are unit vectors. b For Crystal systemsa unit cellb a = b = c; α = β = γ = 90◦ latticesc system symmetry triclinic none monoclinic one diad [010] a = b = c; α = γ = 90◦ , β = 90◦ P, C orthorhombic three orthogonal diads a = b = c; α = β = γ = 90◦ P, C, I, F tetragonal one tetrad [001] a = b = c; α = β = γ = 90◦ P, I trigonald one triad [111] a = b = c; α = β = γ < 120◦ = 90◦ P, R hexagonal one hexad [001] a = b = c; α = β = 90◦ , γ = 120◦ P cubic four triads 111 a = b = c; α = β = γ = 90◦ P, F, I a The P symbol “=” implies that equality is not required by the symmetry, but neither is it forbidden. cell axes are a, b, and c with α, β, and γ the angles between b : c, c : a, and a : b respectively. c The lattice types are primitive (P), body-centred (I), all face-centred (F), side-centred (C), and rhombohedral primitive (R). d A primitive hexagonal unit cell, with a triad [001], is generally preferred over this rhombohedral unit cell. b The 6 main January 23, 2006 16:6 128 Solid state physics Dislocations and cracks Edge dislocation Screw dislocation Screw dislocation energy per unit lengthb Critical crack lengthc (6.21) l̂ · b = b U dislocation energy per unit length (6.23) µ R r0 shear modulus outer cutoff for r inner cutoff for r (6.24) L α (6.25) E σ critical crack length surface energy per unit area Young modulus Poisson ratio (6.22) 2 R µb ln 4π r0 ∼ µb2 U= L= l̂ unit vector line of dislocation b,b Burgers vectora l̂ l̂ · b = 0 4αE π(1 − σ 2 )p20 p0 applied widening stress Burgers vector is a Bravais lattice vector characterising the total relative slip were the dislocation to travel throughout the crystal. b Or “tension.” The energy per unit length of an edge dislocation is also ∼ µb2 . c For a crack cavity (long ⊥ L) within an isotropic medium. Under uniform stress p , 0 cracks ≥ L will grow and smaller cracks will shrink. b b l̂ r L a The Crystal diffraction Laue equations Bragg’s lawa a(cosα1 − cosα2 ) = hλ b(cosβ1 − cosβ2 ) = kλ (6.26) (6.27) c(cosγ1 − cosγ2 ) = lλ (6.28) 2kin .G + |G|2 = 0 Atomic form factor f(G) = Structure factorb S(G) = input wavevector reciprocal lattice vector (6.30) f(G) r ρ(r) atomic form factor position vector atomic electron density fj (G)e−iG·d j (6.31) S(G) n dj structure factor number of atoms in basis position of jth atom within basis K change in wavevector (= kout − kin ) I(K) N scattered intensity number of lattice points illuminated intensity at temperature T intensity from a lattice with no motion mean-squared thermal displacement of atoms vol n kin G α2 ,β2 ,γ2 e−iG·r ρ(r) d3 r j=1 Scattered intensityc I(K) ∝ N 2 |S(K)|2 Debye– Waller factord 1 IT = I0 exp − u2 |G|2 3 a Alternatively, (6.29) h,k,l λ lattice parameters angles between lattice base vectors and input wavevector angles between lattice base vectors and output wavevector integers (Laue indices) wavelength a,b,c α1 ,β1 ,γ1 (6.32) IT I0 (6.33) u2 see Equation (8.32). summation is over the atoms in the basis, i.e., the atomic motif repeating with the Bravais lattice. c The Bragg condition makes K a reciprocal lattice vector, with |k | = |k out |. in d Effect of thermal vibrations. b The main January 23, 2006 16:6 129 6.4 Lattice dynamics 6.4 Lattice dynamics Phonon dispersion relationsa m m1 m m2 (> m1 ) 2a a (2α/µ)1/2 2(α/m)1/2 ω (2α/m1 )1/2 ω (2α/m2 )1/2 − π a 0 k π a − π 2a monatomic chain ka α sin2 m 2 ! α 1/2 ω a! sinc vp = = a k m λ ka α !1/2 ∂ω =a vg = cos ∂k m 2 Diatomic linear chainc Identical masses, alternating spring constants a Along ω2 = 1 α 4 ±α 2 − sin2 (ka) µ µ m1 m2 ω α m phonon angular frequency spring constantb atomic mass (6.36) vp vg λ phase speed (sincx ≡ sinπx πx ) group speed phonon wavelength (6.37) k a mi wavenumber (= 2π/λ) atomic separation atomic masses (m2 > m1 ) µ reduced mass [= m1 m2 /(m1 + m2 )] αi alternating spring constants (6.34) (6.35) 1/2 α1 + α2 1 2 ± (α1 + α22 + 2α1 α2 coska)1/2 m m (6.38) " 0, 2(α1 + α2 )/m if k = 0 = (6.39) 2α1 /m, 2α2 /m if k = π/a ω2 = π 2a k diatomic chain ω2 = 4 Monatomic linear chain 0 m α1 a m α2 α1 infinite linear atomic chains, considering simple harmonic nearest-neighbour interactions only. The shaded region of the dispersion relation is outside the first Brillouin zone of the reciprocal lattice. b In the sense α = restoring force/relative displacement. c Note that the repeat distance for this chain is 2a, so that the first Brillouin zone extends to |k| < π/(2a). The optic and acoustic branches are the + and − solutions respectively. 6 main January 23, 2006 16:6 130 Solid state physics Debye theory E h̄ ω mean energy in a mode at ω (Planck constant)/(2π) phonon angular frequency kB T Boltzmann constant temperature Mean energy per phonon modea h̄ω 1 E = h̄ω + 2 exp[h̄ω/(kB T )] − 1 Debye frequency ωD = vs (6π 2 N/V )1/3 3 1 2 where = + vs3 vl3 vt3 (6.41) (6.42) vs vl vt effective sound speed longitudinal phase speed transverse phase speed Debye temperature θD = h̄ωD /kB (6.43) N V θD number of atoms in crystal crystal volume Debye temperature Phonon density of states g(ω) dω = 3V ω 2 dω 2π 2 vs3 (for 0 < ω < ωD , g = 0 otherwise) Debye heat capacity CV = 9NkB Dulong and Petit’s law 3NkB Debye T 3 law Internal thermal energyb a Or T3 3 θD θD /T 0 (T θD ) T3 12π 4 NkB 3 5 θD U(T ) = where x4 ex dx (ex − 1)2 (T θD ) (6.40) (6.44) (6.45) (6.46) (6.47) g(ω) density of states at ω CV heat capacity, V constant U thermal phonon energy within crystal D(x) Debye function 3NkB CV 0 1 T /θD 2 ωD h̄ω 3 dω ≡ 3NkB T D(θD /T ) exp[h̄ω/(kB T )] − 1 0 3 x y3 D(x) = 3 dy x 0 ey − 1 9N 3 ωD any simple harmonic oscillator in thermal equilibrium at temperature T . zero-point energy. b Neglecting ωD Debye (angular) frequency (6.48) (6.49) main January 23, 2006 16:6 131 6.4 Lattice dynamics Lattice forces (simple) Van der Waals interactiona 3 α2p h̄ω φ(r) = − 4 (4π0 )2 r6 (6.50) B A + r6 r12 σ !12 σ !6 − = 4 r r φ(r) = − Lennard–Jones 6-12 potential (molecular crystals) σ = (B/A)1/6 ; De Boer parameter Coulomb interaction (ionic crystals) r αp particle separation particle polarisability (6.51) h̄ 0 (Planck constant)/(2π) permittivity of free space (6.52) ω angular frequency of polarised orbital (6.53) A,B constants ,σ Lennard–Jones parameters = A2 /(4B) 1/6 at φmin Λ= r= 2 σ h σ(m)1/2 φ(r) two-particle potential energy (6.54) Λ h m de Boer parameter Planck constant particle mass UC lattice Coulomb energy per ion pair e2 4π0 r0 Madelung constant electronic charge nearest neighbour separation a London’s formula for fluctuating dipole interactions, neglecting the propagation time between particles. UC = −αM (6.55) αM −e r0 6 Lattice thermal expansion and conduction Grüneisen parametera Linear expansivityb Thermal conductivity of a phonon gas Umklapp mean free pathc a Strictly, γ=− ∂ lnω ∂ lnV (6.56) 1 ∂p γCV = 3KT ∂T V 3KT V (6.57) 1 CV vs l 3 V lu ∝ exp(θu /T ) α= λ= γ ω V Grüneisen parameter normal mode frequency volume α linear expansivity KT isothermal bulk modulus p pressure T CV λ temperature lattice heat capacity, constant V thermal conductivity (6.58) vs l effective sound speed phonon mean free path (6.59) lu θu umklapp mean free path umklapp temperature (∼ θD /2) the Grüneisen parameter is the mean of γ over all normal modes, weighted by the mode’s contribution to CV . b Or “coefficient of thermal expansion,” for an isotropically expanding crystal. c Mean free path determined solely by “umklapp processes” – the scattering of phonons outside the first Brillouin zone. main January 23, 2006 16:6 132 6.5 Solid state physics Electrons in solids Free electron transport properties Current density Mean electron drift velocity J = −nev d vd = − J n −e current density free electron number density electronic charge vd τ me mean electron drift velocity mean time between collisions (relaxation time) electronic mass (6.62) E σ0 applied electric field d.c. conductivity (J = σE) (6.63) ω a.c. angular frequency σ(ω) a.c. conductivity (6.60) eτ E me (6.61) 2 d.c. electrical conductivity σ0 = a.c. electrical conductivitya σ(ω) = ne τ me σ0 1 − iωτ 1 CV 2 c τ (6.64) 3 V π 2 nkB2 τT = (T TF ) 3me (6.65) λ= Thermal conductivity Wiedemann– Franz lawb Hall coefficient Hall voltage (rectangular strip) λ =L= σT c π 2 kB2 3e2 Ey 1 RH = − = ne Jx Bz VH = RH Bz Ix w (6.66) (6.67) (6.68) CV V total electron heat capacity, V constant volume c2 kB T mean square electron speed Boltzmann constant temperature TF Fermi temperature L λ Lorenz constant ( 2.45 × 10−8 WΩ K−2 ) Jx thermal conductivity RH Hall coefficient Ey Hall electric field Jx applied current density Bz magnetic flux density VH Hall voltage Ix w w Ey Bz VH + applied current (= Jx × cross-sectional area) strip thickness in z an electric field varying as e−iωt . for an arbitrary band structure. c The charge on an electron is −e, where e is the elementary charge (approximately +1.6 × 10−19 C). The Hall coefficient is therefore a negative number when the dominant charge carriers are electrons. a For b Holds main January 23, 2006 16:6 133 6.5 Electrons in solids Fermi gas Electron density of statesa 3/2 2me V g(E) = 2 E 1/2 2π h̄2 3 nV g(EF ) = 2 EF E (6.69) (6.70) electron energy (> 0) g(E) density of states V “gas” volume me electronic mass h̄ (Planck constant)/(2π) kF Fermi wavenumber n number of electrons per unit volume Fermi wavenumber kF = (3π 2 n)1/3 (6.71) Fermi velocity vF = h̄kF /me (6.72) vF Fermi velocity Fermi energy (T = 0) EF = h̄2 kF2 h̄2 = (3π 2 n)2/3 2me 2me (6.73) EF Fermi energy Fermi temperature TF = EF kB (6.74) TF kB Fermi temperature Boltzmann constant π2 g(EF )kB2 T 3 π 2 kB2 = T 2EF Electron heat capacityb (T TF ) CV e = Total kinetic energy (T = 0) 3 U0 = nV EF 5 Pauli paramagnetism M = χHP H 3n = µ0 µ2B H 2EF Landau diamagnetism 1 χHL = − χHP 3 (6.75) (6.76) (6.77) CV e heat capacity per electron T temperature U0 total kinetic energy χHP Pauli magnetic susceptibility a The (6.78) H M magnetic field strength magnetisation (6.79) µ0 µB permeability of free space Bohr magneton (6.80) χHL Landau magnetic susceptibility density of states is often quoted per unit volume in real space (i.e., g(E)/V here). (6.75) holds for any density of states. b Equation Thermoelectricity Thermopower a Peltier effect Kelvin relation a Or J E = + ST ∇T σ (6.81) H = ΠJ − λ∇T (6.82) Π = T ST (6.83) E J σ electrochemical fieldb current density electrical conductivity ST T thermopower temperature H heat flux per unit area Π Peltier coefficient λ thermal conductivity “absolute thermoelectric power.” electrochemical field is the gradient of (µ/e) − φ, where µ is the chemical potential, −e the electronic charge, and φ the electrical potential. b The 6 main January 23, 2006 16:6 134 Solid state physics Band theory and semiconductors Bloch’s theorem Electron velocity Effective mass tensor Scalar effective massa Ψ(r + R) = exp(ik · R)Ψ(r) 1 v b (k) = ∇k Eb (k) h̄ (6.84) (6.85) ∂2 Eb (k) ∂ki ∂kj 2 −1 2 ∂ Eb (k) ∗ m = h̄ ∂k 2 (6.87) m∗ k scalar effective mass = |k| µ vd E particle mobility mean drift velocity applied electric field −e D T electronic charge diffusion coefficient temperature J ne,h µe,h current density electron, hole, number densities electron, hole, mobilities kB Eg Boltzmann constant band gap m∗e,h electron, hole, effective masses (6.88) Net current density J = (ne µe + nh µh )eE (6.89) Semiconductor equation ne nh = (kB T )3 ∗ ∗ 3/2 −Eg /(kB T ) (me mh ) e 2(πh̄2 )3 (6.90) Lh = (Dh τh ) 1/2 h̄ b effective mass tensor components of k eD |v d | = |E| kB T Le = (De τe )1/2 position vector electron velocity (for wavevector k) (Planck constant)/2π band index mij ki µ= p-n junction r vb (6.86) Mobility eV I = I0 exp −1 kB T De Dh + I0 = en2i A Le Na Lh Nd electron eigenstate Bloch wavevector lattice vector Eb (k) energy band −1 mij = h̄2 Ψ k R (6.91) (6.92) (6.93) (6.94) I current I0 V saturation current bias voltage (+ for forward) ni A De,h intrinsic carrier concentration area of junction electron, hole, diffusion coefficients electron, hole, diffusion lengths Le,h electron, hole, recombination times Na,d acceptor, donor, concentrations a Valid for regions of k-space in which E (k) can be taken as independent of the direction of k. b τe,h main January 23, 2006 16:6 Chapter 7 Electromagnetism 7.1 Introduction The electromagnetic force is central to nearly every physical process around us and is a major component of classical physics. In fact, the development of electromagnetic theory in the nineteenth century gave us much mathematical machinery that we now apply quite generally in other fields, including potential theory, vector calculus, and the ideas of divergence and curl. It is therefore not surprising that this section deals with a large array of physical quantities and their relationships. As usual, SI units are assumed throughout. In the past electromagnetism has suffered from the use of a variety of systems of units, including the cgs system in both its electrostatic (esu) and electromagnetic (emu) forms. The fog has now all but cleared, but some specialised areas of research still cling to these historical measures. Readers are advised to consult the section on unit conversion if they come across such exotica in the literature. Equations cast in the rationalised units of SI can be readily converted to the once common Gaussian (unrationalised) units by using the following symbol transformations: 7 Equation conversion: SI to Gaussian units 0 → 1/(4π) µ0 → 4π/c2 B → B/c χE → 4πχE χH → 4πχH H → cH/(4π) M → cM D → D/(4π) A → A/c The quantities ρ, J , E, φ, σ, P, r , and µr are all unchanged. main January 23, 2006 16:6 136 7.2 Electromagnetism Static fields Electrostatics Electrostatic potential E = −∇φ (7.1) Potential differencea φa − φb = Poisson’s Equation (free space) ∇2 φ = − b E · dl = − a a E · dl b (7.2) ρ 0 (7.3) q (7.4) 4π0 |r − r | q(r − r ) E(r) = (7.5) 4π0 |r − r |3 ρ(r )(r − r ) 1 dτ (7.6) E(r) = 4π0 |r − r |3 E φ electric field electrostatic potential φa potential at a φb dl potential at b line element ρ charge density 0 permittivity of free space q point charge φ(r) = Point charge at r Field from a charge distribution (free space) a Between dτ dτ volume element r position vector of dτ volume r r points a and b along a path l. - Magnetostaticsa Magnetic scalar potential B = −µ0 ∇φm φm in terms of the solid angle of a generating current loop φm = Biot–Savart law (the field from a line current) µ0 I B(r) = 4π Ampère’s law (differential form) ∇× B = µ0 J Ampère’s law (integral form) a In free space. (7.7) IΩ 4π Ω loop solid angle I current dl line element in the direction of the current r position vector of dl (7.10) J µ0 current density permeability of free space (7.11) Itot total current through loop (7.8) line dl× (r − r ) |r − r |3 (7.9) B · dl = µ0 Itot φm magnetic scalar potential B magnetic flux density dl s W r I r - main January 23, 2006 16:6 137 7.2 Static fields Capacitancea Of sphere, radius a C = 4π0 r a (7.12) Of circular disk, radius a C = 80 r a (7.13) Of two spheres, radius a, in contact C = 8π0 r aln2 (7.14) Of circular solid cylinder, radius a, length l C [8 + 4.1(l/a)0.76 ]0 r a (7.15) Of nearly spherical surface, area S C 3.139 × 10−11 r S 1/2 (7.16) Of cube, side a C 7.283 × 10−11 r a (7.17) Between concentric spheres, radii a < b C = 4π0 r ab(b − a)−1 (7.18) Between coaxial cylinders, radii a < b 2π0 r per unit length ln(b/a) π0 r per unit length C= arcosh(d/a) π0 r (d a) ln(2d/a) Between parallel cylinders, separation 2d, radii a Between parallel, coaxial circular disks, separation d, radii a a For C= C 0 r πa2 + 0 r a[ln(16πa/d) − 1] d (7.19) (7.20) (7.21) (7.22) conductors, in an embedding medium of relative permittivity r . 7 Inductancea Of N-turn solenoid (straight or toroidal), length l, area A ( l 2 ) L = µ0 N 2 A/l Of coaxial cylindrical tubes, radii a, b (a < b) L= Of parallel wires, radii a, separation 2d Of wire of radius a bent in a loop of radius b a a For µ0 b ln 2π a (7.23) per unit length µ0 2d ln per unit length, (2d a) π a 8b L µ0 b ln − 2 a L currents confined to the surfaces of perfect conductors in free space. (7.24) (7.25) (7.26) main January 23, 2006 16:6 138 Electromagnetism Electric fieldsa Uniformly charged sphere, radius a, charge q Uniformly charged disk, radius a, charge q (on axis, z) Line charge, charge density λ per unit length Electric dipole, moment p (spherical polar coordinates, θ angle between p and r) Charge sheet, surface density σ a For   q r (r < a) 3 E(r) = 4πq0 a (7.27)  r (r ≥ a) 4π0 r3 1 1 q −√ E(z) = z 2π0 a2 |z| z 2 + a2 (7.28) E(r) = λ r 2π0 r2 r pcosθ Er = 2π0 r3 psinθ Eθ = 4π0 r3 σ E= 20 (7.30) − (7.31) θ - K + p (7.32) r = 1 in the surrounding medium. Magnetic fieldsa Uniform infinite solenoid, current I, n turns per unit length Uniform cylinder of current I, radius a " µ0 nI inside (axial) B= 0 outside " µ0 Ir/(2πa2 ) r < a B(r) = r≥a µ0 I/(2πr) Magnetic dipole, moment m (θ angle between m and r) mcosθ Br = µ0 2πr3 µ0 msinθ Bθ = 4πr3 Circular current loop of N turns, radius a, along axis, z B(z) = The axis, z, of a straight solenoid, n turns per unit length, current I a For (7.29) Baxis = (7.33) (7.34) (7.35) θ (7.36) a2 µ0 NI 2 (a2 + z 2 )3/2 (7.38) + Yα1 - z ⊗ Image charges image point image charge b from a conducting plane −b −q b from a conducting sphere, radius a 2 a /b −qa/b −b −q(r − 1)/(r + 1) b +2q/(r + 1) b from a plane dielectric boundary: seen from the dielectric α2 µr = 1 in the surrounding medium. seen from free space K- m (7.37) µ0 nI (cosα1 − cosα2 ) 2 Real charge, +q, at a distance: r main January 23, 2006 16:6 139 7.3 Electromagnetic fields (general) 7.3 Electromagnetic fields (general) Field relationships Conservation of charge ∇·J =− Magnetic vector potential B = ∇× A Electric field from potentials E =− Coulomb gauge condition ∇·A=0 Lorenz gauge condition ∇·A+ Potential field equationsa Expression for φ in terms of ρa Expression for A in terms of J a ∂ρ ∂t ∂A − ∇φ ∂t (7.39) J ρ t current density charge density time (7.40) A vector potential (7.41) φ electrical potential c speed of light (7.42) 1 ∂φ =0 c2 ∂t 1 ∂2 φ ρ − ∇2 φ = c2 ∂t2 0 1 ∂2 A − ∇2 A = µ0 J c2 ∂t2 ρ(r ,t − |r − r |/c) 1 dτ φ(r,t) = 4π0 |r − r | (7.43) dτ (7.44) r r (7.45) µ0 4π J (r ,t − |r − r |/c) dτ |r − r | - dτ volume element (7.46) r position vector of dτ (7.47) µ0 permeability of free space volume A(r,t) = volume a Assumes 7 the Lorenz gauge. Liénard–Wiechert potentialsa Electrical potential of a moving point charge φ= q 4π0 (|r| − v · r/c) (7.48) q charge r vector from charge to point of observation particle velocity q : v r j v Magnetic vector potential of a moving point charge A= µ0 qv 4π(|r| − v · r/c) (7.49) free space. The right-hand sides of these equations are evaluated at retarded times, i.e., at t = t − |r |/c, where r is the vector from the charge to the observation point at time t . a In main January 23, 2006 16:6 140 Electromagnetism Maxwell’s equations Differential form: ∇·E = ρ 0 Integral form: 1 E · ds = ρ dτ 0 (7.50) closed surface volume ∇·B =0 (7.51) B · ds = 0 (7.52) (7.53) closed surface ∂B ∇× E = − ∂t ∂E ∇× B = µ0 J + µ0 0 ∂t E · dl = − (7.54) dΦ dt (7.55) loop B · dl = µ0 I + µ0 0 (7.56) loop E Equation (7.51) is “Gauss’s law” Equation (7.55) is “Faraday’s law” electric field B J ρ magnetic flux density current density charge density surface ds surface element dτ dl Φ volume element line element % linked magnetic flux (= B · ds) % linked current (= J · ds) time I t ∂E · ds (7.57) ∂t Maxwell’s equations (using D and H) Differential form: ∇ · D = ρfree Integral form: D · ds = ρfree dτ (7.58) closed surface ∇·B =0 volume B · ds = 0 (7.60) (7.59) (7.61) closed surface ∂B ∇× E = − ∂t ∂D ∇× H = J free + ∂t D E · dl = − (7.62) dΦ dt loop H · dl = Ifree + (7.64) displacement field ρfree free charge density (in the sense of ρ = ρinduced + ρfree ) B magnetic flux density H magnetic field strength J free free current density (in the sense of J = J induced + J free ) loop E ds dτ (7.63) ∂D · ds ∂t surface electric field surface element volume element dl line element % Φ linked magnetic flux (= B · ds) % Ifree linked free current (= J free · ds) t time (7.65) main January 23, 2006 16:6 141 7.3 Electromagnetic fields (general) Relativistic electrodynamics Lorentz transformation of electric and magnetic fields Lorentz transformation of current and charge densities Lorentz transformation of potential fields Four-vector fields a ⊥ perpendicular to v J ρ current density charge density φ electric potential A magnetic vector potential J current density four-vector (7.66) E B E ⊥ = γ(E + v× B)⊥ B = B (7.67) (7.68) γ B ⊥ = γ(B − v× E/c2 )⊥ (7.69) ρ = γ(ρ − vJ /c2 ) J⊥ = J⊥ (7.70) (7.71) J = γ(J − vρ) (7.72) φ = γ(φ − vA ) (7.73) A⊥ = A⊥ A = γ(A − vφ/c2 ) (7.74) (7.75) J = (ρc,J ) ∼ φ A= ,A ∼ c 1 ∂2 2 = 2 2 ,−∇2 c ∂t (7.76) 2 A = µ0 J ∼ a Other electric field magnetic flux density measured in frame moving at relative velocity v Lorentz factor = [1 − (v/c)2 ]−1/2 parallel to v E = E ∼ (7.77) ∼ potential four-vector A ∼ (7.78) 2 D’Alembertian operator (7.79) sign conventions are common here. See page 65 for a general definition of four-vectors. 7 main January 23, 2006 16:6 142 7.4 Electromagnetism Fields associated with media Polarisation Definition of electric dipole moment Generalised electric dipole moment Electric dipole potential Dipole moment per unit volume (polarisation)a Induced volume charge density p = qa (7.80) p= r ρ dτ (7.81) volume φ(r) = p ·r 4π0 r3 (7.82) ±q a p end charges charge separation vector (from − to +) dipole moment ρ dτ charge density volume element r φ r vector to dτ dipole potential vector from dipole 0 permittivity of free space P = np (7.83) P n polarisation number of dipoles per unit volume ∇ · P = −ρ ind (7.84) ρ ind volume charge density Induced surface charge density σind = P · ŝ (7.85) Definition of electric displacement D = 0 E + P Definition of electric susceptibility Definition of relative permittivityb Atomic polarisabilityc Depolarising fields Clausius–Mossotti equationd a Assuming σind surface charge density ŝ unit normal to surface (7.86) D E electric displacement electric field P = 0 χE E (7.87) χE electrical susceptibility (may be a tensor) r = 1 + χE (7.88) D = 0 r E = E (7.89) (7.90) r relative permittivity permittivity p = αE loc (7.91) α polarisability E loc local electric field Ed depolarising field Nd depolarising factor =1/3 (sphere) =1 (thin slab ⊥ to P) Nd P 0 (7.92) nα r − 1 = 30 r + 2 (7.93) Ed = − − p - + =0 (thin slab to P) =1/2 (long circular cylinder, axis ⊥ to P) dipoles are parallel. The equivalent of Equation (7.112) holds for a hot gas of electric dipoles. permittivity as defined here is for a linear isotropic medium. c The polarisability of a conducting sphere radius a is α = 4π a3 . The definition p = α E 0 0 loc is also used. d With the substitution η 2 = [cf. Equation (7.195) with µ = 1] this is also known as the “Lorentz–Lorenz formula.” r r b Relative main January 23, 2006 16:6 143 7.4 Fields associated with media Magnetisation Definition of magnetic dipole moment dm = I ds Generalised magnetic dipole moment 1 m= 2 Magnetic dipole (scalar) potential φm (r) = Dipole moment per unit volume (magnetisation)a Induced volume current density (7.94) r × J dτ (7.95) volume µ0 m · r 4πr3 (7.96) dτ r φm r µ0 permeability of free space M n magnetisation number of dipoles per unit volume J ind = ∇× M (7.98) J ind volume current density (i.e., A m−2 ) j ind surface current density (i.e., A m−1 ) unit normal to surface Definition of magnetic field strength, H B = µ0 (H + M ) (7.100) M = χH H χB B = µ0 χH χB = 1 + χH (7.101) Definition of magnetic susceptibility (7.99) ŝ B H magnetic flux density magnetic field strength χH magnetic susceptibility. χB is also used (both may be tensors) µr µ relative permeability permeability (7.102) (7.103) B = µ0 µr H = µH (7.104) (7.105) µr = 1 + χH 1 = 1 − χB (7.106) dm, ds out 6 ⊗ in volume element vector to dτ magnetic scalar potential vector from dipole (7.97) j ind = M ×ŝ a Assuming m J dipole moment loop current loop area (right-hand sense with respect to loop current) dipole moment current density M = nm Induced surface current density Definition of relative permeabilityb dm I ds (7.107) all the dipoles are parallel. See Equation (7.112) for a classical paramagnetic gas and page 101 for the quantum generalisation. b Relative permeability as defined here is for a linear isotropic medium. 7 main January 23, 2006 16:6 144 Electromagnetism Paramagnetism and diamagnetism Z B magnetic moment mean squared orbital radius (of all electrons) atomic number magnetic flux density me −e J electron mass electronic charge total angular momentum g Landé g-factor (=2 for spin, =1 for orbital momentum) m r2 Diamagnetic moment of an atom Intrinsic electron magnetic momenta Langevin function m=− m− e2 Zr2 B 6me (7.108) e gJ 2me (7.109) L(x) = cothx − x/3 1 x < 1) (x ∼ (7.110) (7.111) Classical gas paramagnetism (|J | h̄) m0 B M = nm0 L kT Curie’s law χH = µ0 nm20 3kT (7.113) Curie–Weiss law χH = µ0 nm20 3k(T − Tc ) (7.114) a See (7.112) L(x) Langevin function M m0 n T apparent magnetisation magnitude of magnetic dipole moment dipole number density temperature k χH Boltzmann constant magnetic susceptibility µ0 Tc permeability of free space Curie temperature also page 100. Boundary conditions for E, D, B, and H a Parallel component of the electric field Perpendicular component of the magnetic flux density Electric displacementb E continuous (7.115) B⊥ continuous (7.116) ŝ · (D 2 − D 1 ) = σ (7.117) component parallel to interface ⊥ component perpendicular to interface D 1,2 electrical displacements in media 1 & 2 unit normal to surface, directed 1 → 2 surface density of free charge magnetic field strengths in media 1 & 2 surface current per unit width ŝ σ Magnetic field strengthc a At H 1,2 ŝ× (H 2 − H 1 ) = j s the plane surface between two uniform media. σ = 0, then D⊥ is continuous. c If j = 0 then H is continuous. s b If (7.118) js 2 1 ŝ 6 main January 23, 2006 16:6 145 7.5 Force, torque, and energy 7.5 Force, torque, and energy Electromagnetic force and torque Force between two static charges: Coulomb’s law Force between two current-carrying elements Force on a current-carrying element in a magnetic field Force on a charge (Lorentz force) F2= q1 q2 r̂ 2 12 4π0 r12 (7.119) µ0 I1 I2 [dl 2× (dl 1× r̂ 12 )] dF 2 = 2 4πr12 (7.120) dF = I dl× B (7.121) F2 q1,2 r 12 force on q2 charges vector from 1 to 2 ˆ 0 unit vector permittivity of free space line elements currents flowing along dl 1 and dl 2 force on dl 2 dl 1,2 I1,2 dF 2 µ0 permeability of free space dl F I line element force current flowing along dl B magnetic flux density F = q(E + v× B) (7.122) E v electric field charge velocity Force on an electric dipolea F = (p · ∇)E (7.123) p electric dipole moment Force on a magnetic dipoleb F = (m · ∇)B (7.124) m magnetic dipole moment Torque on an electric dipole G = p× E (7.125) G torque Torque on a magnetic dipole G = m× B (7.126) Torque on a current loop G = IL dl L r line-element (of loop) position vector of dl L IL current around loop loop r× (dl L× B) (7.127) simplifies to ∇(p · E) if p is intrinsic, ∇(pE/2) if p is induced by E and the medium is isotropic. b F simplifies to ∇(m · B) if m is intrinsic, ∇(mB/2) if m is induced by B and the medium is isotropic. aF F2 - q1 r 12 dl 1 q2 * r 12 W dlj 2 7 main January 23, 2006 16:6 146 Electromagnetism Electromagnetic energy Electromagnetic field energy density (in free space) 1 B2 1 u = 0 E 2 + 2 2 µ0 Energy density in media 1 u = (D · E + B · H) 2 (7.129) Energy flow (Poynting) vector N = E×H (7.130) Mean flux density at a distance r from a short oscillating dipole N = Total mean power from oscillating dipolea W= ω 4 p20 sin2 θ r 32π 2 0 c3 r3 ω 4 p20 /2 6π0 c3 1 2 u E B energy density electric field magnetic flux density 0 µ0 permittivity of free space permeability of free space D H c electric displacement magnetic field strength speed of light N energy flow rate per unit area ⊥ to the flow direction p0 r amplitude of dipole moment vector from dipole (wavelength) θ ω angle between p and r oscillation frequency (7.132) W total mean radiated power (7.133) Utot total energy dτ volume element r position vector of dτ (7.128) (7.131) Self-energy of a charge distribution Utot = Energy of an assembly of capacitorsb Utot = 1 Cij Vi Vj 2 i j Energy of an assembly of inductorsc Utot = 1 Lij Ii Ij 2 i j Intrinsic dipole in an electric field Udip = −p · E (7.136) Intrinsic dipole in a magnetic field Udip = −m · B (7.137) Hamiltonian of a charged particle in an EM fieldd a Sometimes φ(r)ρ(r) dτ φ ρ electrical potential charge density (7.134) Vi Cij potential of ith capacitor mutual capacitance between capacitors i and j (7.135) Lij mutual inductance between inductors i and j volume H= |pm − qA|2 + qφ 2m called “Larmor’s formula.” b C is the self-capacitance of the ith body. Note that C = C . ii ij ji c L is the self-inductance of the ith body. Note that L = L . ii ij ji d Newtonian limit, i.e., velocity c. (7.138) Udip energy of dipole p electric dipole moment m magnetic dipole moment H pm Hamiltonian particle momentum q m A particle charge particle mass magnetic vector potential main January 23, 2006 16:6 147 7.6 LCR circuits 7.6 LCR circuits LCR definitions Current Ohm’s law I= dQ dt V = IR Ohm’s law (field form) J = σE Resistivity ρ= 1 RA = σ l Q V C= Current through capacitor I =C Self-inductance L= Voltage across inductor V = −L Mutual inductance Φ1 = L21 L12 = I2 Linked magnetic flux through a coil R V I J resistance potential difference over R current through R current density (7.141) E σ ρ electric field conductivity resistivity (7.142) A area of face (I is normal to face) l C V length capacitance potential difference across C I t current through C time Φ I total linked flux current through inductor V potential difference over L Φ1 L12 total flux from loop 2 linked by loop 1 mutual inductance I2 current through loop 2 k coupling coefficient between L1 and L2 (≤ 1) Φ N linked flux number of turns around φ flux through area of turns (7.144) Φ I Φ = Nφ charge (7.143) dV dt |L12 | = k current Q (7.140) Capacitance Coefficient of coupling I (7.139) (7.145) dI dt ( (7.146) L1 L2 (7.147) (7.148) (7.149) φ 7 /l A 7 main January 23, 2006 16:6 148 Electromagnetism Resonant LCR circuits Phase resonant frequencya ω02 = series " 1/LC 1/LC − R 2 /L2 (series) (parallel) (7.150) Tuningb δω 1 R = = ω0 Q ω0 L Quality factor Q = 2π a At ω0 (7.151) stored energy energy lost per cycle L C resonant angular frequency inductance capacitance R L C parallel R resistance δω half-power bandwidth Q quality factor (7.152) which the impedance is purely real. the capacitor is purely reactive. If L and R are parallel, then 1/Q = ω0 L/R. b Assuming Energy in capacitors, inductors, and resistors U C stored energy capacitance Q V L charge potential difference inductance Φ I linked magnetic flux current (7.155) W R power dissipated resistance (7.156) τ r σ relaxation time relative permittivity conductivity Energy stored in a capacitor 1 Q2 1 1 U = CV 2 = QV = 2 2 2 C (7.153) Energy stored in an inductor 1 Φ2 1 1 U = LI 2 = ΦI = 2 2 2 L (7.154) Power dissipated in a resistora (Joule’s law) W = IV = I 2 R = Relaxation time τ= a This V2 R 0 r σ is d.c., or instantaneous a.c., power. Electrical impedance Impedances in series Z tot = Impedances in parallel Z tot = (7.157) Zn n −1 Z −1 n (7.158) n i ωC Impedance of capacitance ZC = − Impedance of inductance Z L = iωL Impedance: Z Inductance: L Conductance: G = 1/R Admittance: Y = 1/Z Capacitance: C Resistance: R = Re[Z] Reactance: X = Im[Z] Susceptance: S = 1/X (7.159) (7.160) main January 23, 2006 16:6 149 7.6 LCR circuits Kirchhoff’s laws Current law Ii = 0 (7.161) Ii currents impinging on node Vi = 0 (7.162) Vi potential differences around loop node Voltage law loop Transformersa I2 Z1 y : V1 I1 V2 z Z2 9 N1 N2 n turns ratio N1 N2 V1 number of primary turns number of secondary turns primary voltage V2 I1 secondary voltage primary current I2 Z out Z in secondary current output impedance input impedance Z1 Z2 source impedance load impedance Turns ratio n = N2 /N1 (7.163) Transformation of voltage and current V2 = nV1 I2 = I1 /n (7.164) (7.165) Output impedance (seen by Z 2 ) Z out = n2 Z 1 (7.166) Input impedance (seen by Z 1 ) Z in = Z 2 /n2 (7.167) a Ideal, with a coupling constant of 1 between loss-free windings. Star–delta transformation 1 1 ‘Star’ Z1 2 Star impedances Delta impedances Z2 Z3 Z12 3 2 Z13 Z23 Zij Zik Zij + Zik + Zjk 1 1 1 Zij = Zi Zj + + Zi Zj Zk Zi = ‘Delta’ 3 (7.168) (7.169) i,j,k node indices (1,2, or 3) Zi impedance on node i Zij impedance connecting nodes i and j 7 main January 23, 2006 16:6 150 7.7 Electromagnetism Transmission lines and waveguides Transmission line relations Loss-free transmission line equations Wave equation for a lossless transmission line Characteristic impedance of lossless line ∂I ∂V = −L ∂x ∂t ∂I ∂V = −C ∂x ∂t 1 ∂2 V ∂2 V = 2 LC ∂x2 ∂t 1 ∂2 I ∂2 I = LC ∂x2 ∂t2 L Zc = C Zc = Wave speed along a lossless line 1 vp = vg = √ LC Input impedance of a terminated lossless line Z in = Zc R + iωL G + iωC Z t coskl − iZc sinkl Zc coskl − iZ t sinkl 2 = Zc /Z t if l = λ/4 r= Line voltage standing wave ratio vswr = (7.171) L C inductance per unit length capacitance per unit length (7.172) x t distance along line time (7.173) Zc characteristic impedance (7.174) ω resistance per unit length of conductor conductance per unit length of insulator angular frequency vp vg phase speed group speed R Characteristic impedance of lossy line Reflection coefficient from a terminated line I potential difference across line current in line V (7.170) Zt −Zc Zt +Zc 1 + |r| 1 − |r| G (7.175) (7.176) Z in (complex) input impedance Z t (complex) terminating impedance k wavenumber (= 2π/λ) (7.177) (7.178) (7.179) l distance from termination r (complex) voltage reflection coefficient (7.180) Transmission line impedancesa Coaxial line Zc = Open wire feeder Zc = b 60 µ b ln √ ln r a 4π 2 a (7.181) µ l 120 l ln √ ln 2 π r r r (7.182) Paired strip Zc = µ d 377 d √ w r w (7.183) Microstrip line 377 Zc √ r [(w/h) + 2] (7.184) a For lossless lines. Zc a characteristic impedance (Ω) radius of inner conductor b µ radius of outer conductor permittivity (= 0 r ) permeability (= µ0 µr ) r l radius of wires distance between wires ( r) d strip separation w strip width ( d) h height above earth plane ( w) main January 23, 2006 16:6 151 7.7 Transmission lines and waveguides Waveguidesa Waveguide equation kg2 = ω 2 m2 π 2 n2 π 2 − 2 − 2 c2 a b Guide cutoff frequency νc = c Phase velocity above cutoff vp = ( Group velocity above cutoff m !2 2a c kg ω a wavenumber in guide angular frequency guide height b m,n c guide width mode indices with respect to a and b (integers) speed of light (7.186) νc ωc cutoff frequency 2πνc (7.187) vp ν phase velocity frequency (7.188) vg group velocity ZTM wave impedance for transverse magnetic modes wave impedance for transverse electric modes impedance of free space ( (= µ0 /0 ) (7.185) n !2 + 2b 1 − (νc /ν)2 vg = c2 /vp = c 1 − (νc /ν)2 Wave impedancesb 1 − (νc ZTE = Z0 / 1 − (νc /ν)2 ZTM = Z0 /ν)2 (7.189) (7.190) ZTE Z0 Field solutions for TEmn modesc ikg c2 ∂Bz ωc2 ∂x ikg c2 ∂Bz By = 2 ωc ∂y nπy mπx cos Bz =B0 cos a b Bx = iωc2 ∂Bz ωc2 ∂y −iωc2 ∂Bz Ey = ωc2 ∂x Ez =0 Ex = Field solutions for TMmn modes ikg c2 ∂Ez ωc2 ∂x ikg c2 ∂Ez Ey = 2 ωc ∂y nπy mπx sin Ez =E0 sin a b Ex = a Equations (7.191) c −iω ∂Ez ωc2 ∂y iω ∂Ez By = 2 ωc ∂x Bz =0 Bx = 7 b x a z y (7.192) are for lossless waveguides with rectangular cross sections and no dielectric. ratio of the electric field to the magnetic field strength in the xy plane. c Both TE and TM modes propagate in the z direction with a further factor of exp[i(k z − ωt)] on all components. g B0 and E0 are the amplitudes of the z components of magnetic flux density and electric field respectively. b The main January 23, 2006 16:6 152 7.8 Electromagnetism Waves in and out of media Waves in lossless media ∂2 E ∇ E = µ 2 ∂t (7.193) ∂2 B ∂t2 (7.194) 2 Electric field Magnetic field ∇2 B = µ Refractive index √ η = r µr Impedance of free space Wave impedance electric field permeability (= µ0 µr ) permittivity (= 0 r ) B t magnetic flux density time v η c wave phase speed refractive index speed of light Z0 impedance of free space Z wave impedance H magnetic field strength (7.195) c 1 v= √ = µ η µ0 Z0 = 376.7 Ω 0 E µr Z = = Z0 H r Wave speed E µ (7.196) (7.197) (7.198) Radiation pressurea Radiation momentum density G= Isotropic radiation 1 pn = u(1 + R) 3 (7.200) Specular reflection pn = u(1 + R)cos2 θi pt = u(1 − R)sinθi cosθi (7.201) (7.202) From an extended sourceb 1+R pn = c From a point source,c luminosity L a On N c2 (7.199) Iν (θ,φ)cos2 θ dΩ dν (7.203) pn = L(1 + R) 4πr2 c G momentum density N c pn Poynting vector speed of light normal pressure u incident radiation energy density R (power) reflectance coefficient pt tangential pressure θi angle of incidence Iν ν Ω specific intensity frequency solid angle θ angle between dΩ and normal to plane L source luminosity (i.e., radiant power) distance from source (7.204) r an opaque surface. spherical polar coordinates. See page 120 for the meaning of specific intensity. c Normal to the plane. b In u θi z θ dΩ (normal) x φ y main January 23, 2006 16:6 153 7.8 Waves in and out of media Antennas z Spherical polar geometry: Field from a short (l λ) electric dipole in free spacea Radiation resistance of a short electric dipole in free space [ṗ] [p] 1 + Er = cosθ 2π0 r2 c r3 [p̈] [ṗ] [p] 1 + + sinθ Eθ = 4π0 rc2 r2 c r3 µ0 [p̈] [ṗ] + 2 sinθ Bφ = 4π rc r 2 2πZ0 l ω2 l 2 = R= 6π0 c3 3 λ 2 l 789 ohm λ r 6 θ U p 6 * φ /x r distance from dipole θ angle between r and p [p] c retarded dipole moment [p] = p(t − r/c) speed of light (7.208) l ω λ dipole length ( λ) angular frequency wavelength (7.209) Z0 impedance of free space ΩA Pn beam solid angle normalised antenna power pattern Pn (0,0) = 1 dΩ differential solid angle (7.211) G antenna gain Ae effective area ΩM main lobe solid angle (7.205) (7.206) (7.207) Beam solid angle ΩA = Pn (θ,φ) dΩ y - (7.210) 4π 4π ΩA Forward power gain G(0) = Antenna effective area Ae = λ2 ΩA (7.212) Power gain of a short dipole 3 G(θ) = sin2 θ 2 (7.213) Beam efficiency efficiency = ΩM ΩA (7.214) antenna temperature Tb (θ,φ)Pn (θ,φ) dΩ (7.215) T sky brightness b 4π temperature a All field components propagate with a further phase factor equal to expi(kr − ωt), where k = 2π/λ. b The brightness temperature of a source of specific intensity I is T = λ2 I /(2k ). ν ν B b Antenna temperatureb 1 TA = ΩA TA 7 main January 23, 2006 16:6 154 Electromagnetism Reflection, refraction, and transmissiona parallel incidence perpendicular incidence Ei Er K Bi θi / θr w Br > θt Ei ηi Er θi / Bi θr w ηt Et * U Br > θt Bt = Bt θi = θr (7.216) Snell’s lawb ηi sinθi = ηt sinθt (7.217) Brewster’s law tanθB = ηt /ηi (7.218) (ηi − ηt )2 (ηi + ηt )2 4ηi ηt T= (ηi + ηt )2 R +T =1 refractive index on incident side ηt refractive index on transmitted side θi θr θt angle of incidence angle of reflection angle of refraction θB Brewster’s angle of incidence for plane-polarised reflection (r = 0) sin(θi − θt ) sin(θi + θt ) 2cosθi sinθt t⊥ = sin(θi + θt ) 2 R⊥ = r⊥ ηt cosθt 2 t T⊥ = ηi cosθi ⊥ ηi − ηt ηi + ηt 2ηi t= ηi + ηt t−r =1 r= (7.227) (7.228) (7.229) ⊥ R electric field parallel to the plane of incidence (power) reflectance coefficient r electric field perpendicular to the plane of incidence amplitude reflection coefficient T (power) transmittance coefficient t amplitude transmission coefficient a For ηi r⊥ = − ηt cosθt 2 t (7.222) T = ηi cosθi Coefficients for normal incidencec R= electric field magnetic flux density Et Law of reflection Fresnel equations of reflection and refraction sin2θi − sin2θt r = (7.219) sin2θi + sin2θt 4cosθi sinθt t = (7.220) sin2θi + sin2θt (7.221) R = r2 E B (7.223) (7.224) (7.225) (7.226) (7.230) (7.231) (7.232) the plane boundary between lossless dielectric media. All coefficients refer to the electric field component and whether it is parallel or perpendicular to the plane of incidence. Perpendicular components are out of the paper. b The incident wave suffers total internal reflection if ηi sinθ > 1. i ηt c I.e., θ = 0. Use the diagram labelled “perpendicular incidence” for correct phases. i main January 23, 2006 16:6 155 7.8 Waves in and out of media Propagation in conducting mediaa Electrical conductivity (B = 0) σ ne τc electrical conductivity electron number density electron relaxation time µ B electron mobility magnetic flux density (7.234) me −e η electron mass electronic charge refractive index (7.235) 0 ν δ permittivity of free space frequency skin depth 2 σ = ne eµ = ne e τc me Refractive index of an ohmic conductorb σ η = (1 + i) 4πν0 Skin depth in an ohmic conductor δ = (µ0 σπν)−1/2 (7.233) 1/2 µ0 permeability of free space a relative permeability, µr , of 1. b Taking the wave to have an e−iωt time dependence, and the low-frequency limit (σ 2πν ). 0 a Assuming Electron scattering processesa σR ω Rayleigh cross section radiation angular frequency particle polarisability permittivity of free space Thomson scattering cross sectionc 2 e2 8π σT = (7.237) 3 4π0 me c2 8π = re2 6.652 × 10−29 m2 3 (7.238) α 0 σT me re Thomson cross section electron (rest) mass classical electron radius c speed of light Inverse Compton scatteringd 2 v 4 Ptot = σT curad γ 2 2 3 c Ptot electron energy loss rate urad radiation energy density γ Lorentz factor = [1 − (v/c)2 ]−1/2 Rayleigh scattering cross sectionb σR = Compton scatteringe λ me 4 2 ω α 6π0 c4 (7.236) (7.239) v h (1 − cosθ) me c λ me c2 hν = 1 − cosθ + (1/ε) θ θ φ cotφ = (1 + ε)tan 2 λ − λ = (7.240) (7.241) (7.242) electron speed λ,λ incident & scattered wavelengths ν,ν θ h me c ε incident & scattered frequencies photon scattering angle electron Compton wavelength = hν/(me c2 ) σKN Klein–Nishina cross section ' & 1 1 4 2(ε + 1) ln(2ε + 1) + + − 1− σKN = ε ε2 2 ε 2(2ε + 1)2 σT (ε 1) πr2 1 e ln2ε + (ε 1) ε 2 πre2 Klein–Nishina cross section (for a free electron) a For Rutherford scattering see page 72. by bound electrons. c Scattering from free electrons, ε 1. d Electron energy loss rate due to photon scattering in the Thomson limit (γhν m c2 ). e e From an electron at rest. b Scattering (7.243) (7.244) (7.245) 7 main January 23, 2006 16:6 156 Electromagnetism Cherenkov radiation cone semi-angle θ Cherenkov cone angle c sinθ = ηv c (vacuum) speed of light η(ω) refractive index (7.246) particle velocity v Radiated powera e2 µ0 v 4π Ptot = ωc 7.9 c2 ω dω v 2 η 2 (ω) 1− Ptot total radiated power −e electronic charge (7.247) µ0 ω ωc 0 where η(ω) ≥ a From c v for 0 < ω < ωc free space permeability angular frequency cutoff frequency a point charge, e, travelling at speed v through a medium of refractive index η(ω). Plasma physics Warm plasmas Landau length e2 4π0 kB Te 1.67 × 10−5 Te−1 lL = Electron Debye length 0 kB Te λDe = ne e2 m m Debye number 4 NDe = πne λ3De 3 (7.249) 0 kB Te permittivity of free space Boltzmann constant electron temperature (K) (7.250) λDe electron Debye length φ effective potential q r point charge distance from q (7.253) NDe electron Debye number τe 5 Relaxation times (B = 0)b Characteristic electron thermal speedc a Effective 3/2 Ti τi = 2.09 × 10 ne lnΛ 7 2kB Te vte = me s mi mp (7.254) 1/2 s (7.255) 1/2 1/2 5.51 × 103 Te electron number density (m−3 ) (7.252) 3/2 Te τe = 3.44 × 10 ne lnΛ ne (7.251) q exp(−21/2 r/λDe ) 4π0 r φ(r) = Landau length electronic charge 1/2 69(Te /ne )1/2 Debye screeninga lL −e (7.248) (7.256) ms−1 (7.257) electron relaxation time τi ion relaxation time Ti ion temperature (K) lnΛ Coulomb logarithm (typically 10 to 20) B magnetic flux density vte electron thermal speed me electron mass (Yukawa) potential from a point charge q immersed in a plasma. < T . The Spitzer times for electrons and singly ionised ions with Maxwellian speed distributions, Ti ∼ e conductivity can be calculated from Equation (7.233). c Defined so that the Maxwellian velocity distribution ∝ exp(−v 2 /v 2 ). There are other definitions (see Maxwell– te Boltzmann distribution on page 112). b Collision main January 23, 2006 16:6 157 7.9 Plasma physics Electromagnetic propagation in cold plasmasa (2πνp )2 = Plasma frequency ne e2 = ωp2 0 m e 1/2 νp 8.98ne Hz (7.258) νp ωp plasma frequency plasma angular frequency (7.259) ne me electron number density (m−3 ) electron mass −e 0 electronic charge permittivity of free space η ν k refractive index frequency wavenumber (= 2π/λ) $1/2 # η = 1 − (νp /ν)2 (7.260) Plasma dispersion relation (B = 0) c2 k 2 = ω 2 − ωp2 (7.261) ω c angular frequency (= 2π/ν) speed of light Plasma phase velocity (B = 0) vφ = c/η (7.262) vφ phase velocity Plasma group velocity (B = 0) vg = cη vφ vg = c2 (7.263) (7.264) vg group velocity (7.265) νC cyclotron frequency ωC cyclotron angular frequency νCe electron νC Plasma refractive index (B = 0) Cyclotron (Larmor, or gyro-) frequency Larmor (cyclotron, or gyro-) radius qB = ωC m νCe 28 × 109 B Hz νCp 15.2 × 106 B Hz 2πνC = m v⊥ = v⊥ rL = ωC qB rLe = 5.69 × 10−12 rLp = 10.4 × 10−9 v⊥ ! m B! v⊥ m B Mixed propagation modesb X(1 − X) , η2 = 1 − (1 − X) − 12 Y 2 sin2 θB ± S (7.266) (7.267) νCp proton νC q particle charge B magnetic flux density (T) (7.268) m particle mass (γm if relativistic) (7.269) rL rLe Larmor radius electron rL (7.270) rLp proton rL v⊥ speed ⊥ to B (ms−1 ) θB angle between wavefront normal (k̂) and B (7.271) where X = (ωp /ω)2 , Y = ωCe /ω, 1 and S 2 = Y 4 sin4 θB + Y 2 (1 − X)2 cos2 θB 4 Faraday rotationc µ0 e3 2 ∆ψ = 2 2 λ ne B · dl 8π m c ) *+ e , line 2.63×10−13 = Rλ2 a I.e., (7.272) (7.273) ∆ψ rotation angle λ dl R wavelength (= 2π/k) line element in direction of wave propagation rotation measure plasmas in which electromagnetic force terms dominate over thermal pressure terms. Also taking µr = 1. a collisionless electron plasma. The ordinary and extraordinary modes are the + and − roots of S 2 when θB = π/2. When θB = 0, these roots are the right and left circularly polarised modes respectively, using the optical convention for handedness. c In a tenuous plasma, SI units throughout. ∆ψ is taken positive if B is directed towards the observer. b In 7 main January 23, 2006 16:6 158 Electromagnetism Magnetohydrodynamicsa Sound speed γp vs = ρ 1/2 2γkB T = mp 1/2 (7.274) 166T 1/2 ms−1 vA = Alfvén speed B (µ0 ρ)1/2 (7.276) 2.18 × 10 16 −1/2 Bne ms −1 2µ0 p 4µ0 ne kB T 2vs2 = = 2 B2 B2 γvA Plasma beta β= Direct electrical conductivity σd = Hall electrical conductivity σH = Generalised Ohm’s law (7.275) (7.277) sound (wave) speed ratio of heat capacities hydrostatic pressure ρ kB plasma mass density Boltzmann constant T mp vA temperature (K) proton mass Alfvén speed B µ0 ne magnetic flux density (T) permeability of free space electron number density (m−3 ) β plasma beta (ratio of hydrostatic to magnetic pressure) −e σd electronic charge direct conductivity σ conductivity (B = 0) σH Hall conductivity J E v current density electric field plasma velocity field B̂ = B/|B| µ0 η (7.278) n2e e2 σ 2 ne e2 + σ 2 B 2 (7.279) σB σd ne e (7.280) J = σd (E + v× B) + σH B̂× (E + v× B) (7.281) Resistive MHD equations (single-fluid model)b ∂B = ∇× (v× B) + η∇2 B ∂t ∇p 1 ∂v + (v · ∇)v = − + (∇× B)× B + ν∇2 v ∂t ρ µ0 ρ 1 + ν∇(∇ · v) + g 3 Shear Alfvénic ω = kvA cosθB dispersion relationc Magnetosonic 2 2 4 ω 2 k 2 (vs2 + vA ) − ω 4 = vs2 vA k cos2 θB dispersion d relation vs γ p (7.282) (7.283) ν g permeability of free space magnetic diffusivity [= 1/(µ0 σ)] kinematic viscosity gravitational field strength (7.284) ω k angular frequency (= 2πν) wavevector (k = 2π/λ) θB angle between k and B (7.285) a warm, fully ionised, electrically neutral p+ /e− plasma, µr = 1. Relativistic and displacement current effects are assumed to be negligible and all oscillations are taken as being well below all resonance frequencies. b Neglecting bulk (second) viscosity. c Nonresistive, inviscid flow. d Nonresistive, inviscid flow. The greater and lesser solutions for ω 2 are the fast and slow magnetosonic waves respectively. a For main January 23, 2006 16:6 159 7.9 Plasma physics Synchrotron radiation Power radiated by a single electrona ... averaged over pitch angles v !2 2 sin θ c v !2 2 1.59 × 10−14 B 2 γ 2 sin θ c Ptot = 2σT cumag γ 2 v !2 4 Ptot = σT cumag γ 2 3 c 1.06 × 10−14 B 2 γ 2 c W Single electron emission spectrumb Characteristic frequency eB 3 sinθ νch = γ 2 2 2πme 4.2 × 1010 γ 2 B sinθ Spectral function W (7.287) (7.288) v !2 31/2 e3 B sinθ P (ν) = F(ν/νch ) 4π0 cme 2.34 × 10−25 B sinθF(ν/νch ) (7.286) Hz (7.289) (7.290) expression also holds for cyclotron radiation (v c). total radiated power per unit frequency interval. umag magnetic energy density = B 2 /(2µ0 ) v electron velocity (∼ c) γ Lorentz factor = [1 − (v/c)2 ]−1/2 pitch angle (angle between v and B) magnetic flux density speed of light θ B c P (ν) emission spectrum ν frequency νch characteristic frequency −e 0 me electronic charge free space permittivity electronic (rest) mass (7.292) F spectral function (7.293) K5/3 modified Bessel fn. of the 2nd kind, order 5/3 WHz−1 (7.291) 1 ∞ F(x) = x K5/3 (y)dy "x 2.15x1/3 (x 1) 1/2 −x (x 1) 1.25x e Ptot total radiated power σT Thomson cross section (7.294) (7.295) F(x) 0.5 0 1 x 2 3 4 a This b I.e., 7 main January 23, 2006 16:6 160 Electromagnetism Bremsstrahlunga Single electron and ionb dW ω 2 1 2 ωb Z 2 e6 2 ωb = K + K1 dω γv 24π 4 30 c3 m2e γ 2 v 4 γ 2 0 γv Z 2 e6 24π 4 30 c3 m2e b2 v 2 (7.296) (ωb γv) (7.297) Thermal bremsstrahlung radiation (v c; Maxwellian distribution) −hν dP −51 2 −1/2 = 6.8 × 10 Z T ni ne g(ν,T )exp Wm−3 Hz−1 dV dν kT  16 3 −2 −2 5 2 <  0.28[ln(4.4 × 10 T ν Z ) − 0.76] (hν kT ∼ 10 kZ ) 10 −1 5 2 < where g(ν,T ) 0.55ln(2.1 × 10 T ν ) (hν 10 kZ ∼ kT )   10 −1 −1/2 (hν kT ) (2.1 × 10 T ν ) dP 1.7 × 10−40 Z 2 T 1/2 ni ne dV ω angular frequency (= 2πν) Ze −e 0 ionic charge electronic charge permittivity of free space c me speed of light electronic mass b collision parameterc Wm−3 P electron velocity modified Bessel functions of order i (see page 47) Lorentz factor = [1 − (v/c)2 ]−1/2 power radiated V ν volume frequency (Hz) v Ki γ (7.299) (7.300) W T ni energy radiated electron temperature (K) ion number density (m−3 ) ne electron number density (m−3 ) k h g Boltzmann constant Planck constant Gaunt factor a Classical treatment. The ions are at rest, and all frequencies are above the plasma frequency. spectrum is approximately flat at low frequencies and drops exponentially at frequencies c Distance of closest approach. b The (7.298) > ∼ γv/b. main January 23, 2006 16:6 Chapter 8 Optics 8.1 Introduction Any attempt to unify the notations and terminology of optics is doomed to failure. This is partly due to the long and illustrious history of the subject (a pedigree shared only with mechanics), which has allowed a variety of approaches to develop, and partly due to the disparate fields of physics to which its basic principles have been applied. Optical ideas find their way into most wave-based branches of physics, from quantum mechanics to radio propagation. Nowhere is the lack of convention more apparent than in the study of polarisation, and so a cautionary note follows. The conventions used here can be taken largely from context, but the reader should be aware that alternative sign and handedness conventions do exist and are widely used. In particular we will take a circularly polarised wave as being right-handed if, for an observer looking towards the source, the electric field vector in a plane perpendicular to the line of sight rotates clockwise. This convention is often used in optics textbooks and has the conceptual advantage that the electric field orientation describes a right-hand corkscrew in space, with the direction of energy flow defining the screw direction. It is however opposite to the system widely used in radio engineering, where the handedness of a helical antenna generating or receiving the wave defines the handedness and is also in the opposite sense to the wave’s own angular momentum vector. 8 main January 23, 2006 16:6 162 Optics 8.2 Interference Newton’s ringsa nth dark ring rn2 = nRλ0 nth bright ring rn2 = a Viewed (8.1) 1 n+ Rλ0 (8.2) 2 rn n R radius of nth ring integer (≥ 0) lens radius of curvature λ0 wavelength in external medium R rn in reflection. Dielectric layersa 1 R η1 η1 N × { ηηab RN 1 a single layer η2 multilayer η3 1−R Quarter-wave condition Single-layer reflectanceb Dependence of R on layer thickness, m η3 m λ0 a= η2 4 1 − RN (8.3)  2 η1 η3 − η22     η η + η2 1 3 2 R=  η − η 2  3   1 η1 + η3 (m odd) (8.4) (m even) (−1)m (η1 − η2 )(η2 − η3 ) > 0 (8.5) min if (−1) (η1 − η2 )(η2 − η3 ) < 0 (8.6) R = 0 if η2 = (η1 η3 ) 1/2 and m odd film thickness m thickness integer (m ≥ 0) η2 λ0 R film refractive index free-space wavelength power reflectance coefficient entry-side refractive index exit-side refractive index η1 η3 max if m a (8.7) RN multilayer reflectance N number of layer pairs Multilayer η1 − η3 (ηa /ηb ) ηa refractive index of top RN = (8.8) layer reflectancec η1 + η3 (ηa /ηb )2N ηb refractive index of bottom layer a For normal incidence, assuming the quarter-wave condition. The media are also assumed lossless, with µ = 1. r b See page 154 for the definition of R. c For a stack of N layer pairs, giving an overall refractive index sequence η η ,η η ...η η η (see right-hand diagram). a b 3 1 a b a Each layer in the stack meets the quarter-wave condition with m = 1. 2N 2 main January 23, 2006 16:6 163 8.2 Interference Fabry-Perot etalona ∝1 θ eiφ e2iφ η e3iφ θ h η η incident rays Incremental phase differenceb φ = 2k0 hη cosθ η sinθ = 2k0 hη 1 − η 2 1/2 = 2πn for a maximum Coefficient of finesse Finesse 4R (1 − R)2 π F = F 1/2 2 λ0 = Q ηh F= I0 (1 − R)2 1 + R 2 − 2R cosφ I0 = 1 + F sin2 (φ/2) = I0 A(θ) I(θ) = Transmitted intensity Fringe intensity profile incremental phase difference free-space wavenumber (= 2π/λ0 ) h θ θ cavity width fringe inclination (usually 1) internal angle of refraction η η n cavity refractive index external refractive index fringe order (integer) F R coefficient of finesse interface power reflectance (8.13) F λ0 finesse free-space wavelength (8.14) Q cavity quality factor I transmitted intensity I0 A incident intensity Airy function (8.10) (8.11) (8.12) (8.15) (8.16) (8.17) ∆φ = 2arcsin(F −1/2 ) (8.18) −1/2 (8.19) 2F Chromatic resolving power λ0 R 1/2 πn = nF δλ 1−R 2Fhη (θ 1) λ0 Free spectral rangec δλf = Fδλ c δνf = 2η h a Neglecting φ k0 (8.9) (8.20) ∆φ phase difference at half intensity point δλ minimum resolvable wavelength difference (8.21) (8.22) (8.23) δλf wavelength free spectral range δνf frequency free spectral range any effects due to surface coatings on the etalon. See also Lasers on page 174. adjacent rays. Highest order fringes are near the centre of the pattern. c At near-normal incidence (θ 0), the orders of two spectral components separated by < δλ will not overlap. f b Between 8 main January 23, 2006 16:6 164 8.3 Optics Fraunhofer diffraction Gratingsa coherent plane waves Young’s double slitsb I(s) = I0 cos2 N equally spaced narrow slits I(s) = I0 kDs 2 (8.24) sin(Nkds/2) N sin(kds/2) ∞ 2 (8.25) I(s) diffracted intensity I0 peak intensity θ diffraction angle s D λ = sinθ slit separation wavelength N k d number of slits wavenumber (= 2π/λ) slit spacing n diffraction order (8.26) δ Dirac delta function (8.27) θn angle of diffracted maximum nλ d (8.28) θi angle of incident illumination Reflection grating nλ sinθn − sinθi = d (8.29) Chromatic resolving power λ = Nn δλ (8.30) Grating dispersion n ∂θ = ∂λ dcosθ (8.31) Bragg’s lawc 2asinθn = nλ (8.32) Infinite grating I(s) = I0 Normal incidence sinθn = Oblique incidence sinθn + sinθi = a Unless δ s− n=−∞ nλ d nλ d stated otherwise, the illumination is normal to the grating. narrow slits separated by D. c The condition is for Bragg reflection, with θ = θ . n i b Two D ' N d θi θn θi θn δλ diffraction peak width a atomic plane spacing θn a main January 23, 2006 16:6 165 8.3 Fraunhofer diffraction Aperture diffraction y f( coherent plane-wave illumination, normal to the xy plane x, y) x sy sx z General 1-D aperturea ψ(s) ∝ ∞ f(x)e−iksx dx (8.33) ψ I diffracted wavefunction diffracted intensity (8.34) θ s diffraction angle = sinθ −∞ I(s) ∝ ψψ ∗ (s) aperture amplitude transmission function x,y distance across aperture k wavenumber (= 2π/λ) f General 2-D aperture in (x,y) plane (small angles) Broad 1-D slitb ψ(sx ,sy ) ∝ f(x,y)e−ik(sx x+sy y) dxdy (8.35) ∞ sx sy deflection xz plane deflection ⊥ xz plane peak intensity slit width (in x) wavelength sin2 (kas/2) (kas/2)2 (8.36) ≡ I0 sinc2 (as/λ) (8.37) I0 a λ (8.38) In nth sidelobe intensity (8.39) a b aperture width in x aperture width in y (8.40) J1 D first-order Bessel function aperture diameter λ wavelength I(s) = I0 Sidelobe intensity 2 2 1 In = I0 π (2n + 1)2 Rectangular aperture (small angles) I(sx ,sy ) = I0 sinc2 Circular aperturec I(s) = I0 First minimumd s = 1.22 λ D (8.41) First subsid. maximum s = 1.64 λ D (8.42) Weak 1-D phase object f(x) = exp[iφ(x)] 1 + iφ(x) Fraunhofer limite L asx bsy sinc2 λ λ 2J1 (kDs/2) kDs/2 (∆x)2 λ (n > 0) 2 (8.43) φ(x) phase distribution i i2 = −1 L (8.44) distance of aperture from observation point aperture size ∆x Fraunhofer integral. b Note that sincx = (sinπx)/(πx). c The central maximum is known as the “Airy disk.” d The “Rayleigh resolution criterion” states that two point sources of equal intensity can just be resolved with diffraction-limited optics if separated in angle by 1.22λ/D. e Plane-wave illumination. a The 8 main January 23, 2006 16:6 166 8.4 Optics Fresnel diffraction Kirchhoff’s diffraction formulaa y dS ρ dA x ŝ source θ r ψ0 S P r P (source at infinity) Source at infinity i ψP = − ψ0 λ K(θ) eikr dA r (8.45) plane where: Obliquity factor (cardioid) Source at finite distanceb 1 K(θ) = (1 + cosθ) 2 ψP = − iE0 λ z (8.46) eik(ρ+r) [cos(ŝ · r̂) − cos(ŝ · ρ̂)] dS 2ρr ψP λ k complex amplitude at P wavelength wavenumber (= 2π/λ) ψ0 θ r incident amplitude obliquity angle distance of dA from P ( λ) dA area element on incident wavefront K dS ˆ obliquity factor element of closed surface unit vector s r vector normal to dS vector from P to dS closed surface ρ vector from source to dS E0 amplitude (see footnote) a Also known as the “Fresnel–Kirchhoff formula.” Diffraction by an obstacle coincident with the integration surface can be approximated by omitting that part of the surface from the integral. b The source amplitude at ρ is ψ(ρ) = E eikρ /ρ. The integral is taken over a surface enclosing the point P . 0 (8.47) Fresnel zones y source z1 z2 Effective aperture distancea 1 1 1 = + z z1 z2 (8.48) Half-period zone radius yn = (nλz)1/2 (8.49) Axial zeros (circular aperture) zm = a I.e., R2 2mλ (8.50) observer z z1 z2 effective distance source–aperture distance aperture–observer distance n λ half-period zone number wavelength yn zm nth half-period zone radius distance of mth zero from aperture R aperture radius the aperture–observer distance to be employed when the source is not at infinity. main January 23, 2006 16:6 167 8.4 Fresnel diffraction Cornu spiral √ 0.8 0.6 √ Cornu Spiral 3 3 2 √ ∞ Edge diffraction 2.5 5 S(w) 0.2 2 2 1 2 w 0 1.5 − 21 −0.2 C(w) −2 −0.4 −1 √ − 5 1 √ − 3 −∞ 0.5 intensity CS(w) + 1 (1 + i)2 2 1 0.4 −0.6 −0.8 −0.8 √ − 2 −0.6 −0.4 −0.2 0 0 0.2 0.4 0.6 0.8 −4 −2 0 2 4 w Fresnel integralsa w πt2 dt C(w) = cos 2 0 w πt2 S(w) = dt sin 2 0 Cornu spiral CS(w) = C(w) + iS(w) 1 CS(±∞) = ± (1 + i) 2 ψ0 1 [CS(w) + (1 + i)] 2 21/2 1/2 2 where w = y λz ψP = Edge diffraction Diffraction from a long slitb Diffraction from a rectangular aperture a See b Slit ψ0 ψP = 1/2 [CS(w2 ) − CS(w1 )] 2 1/2 2 where wi = yi λz ψ0 ψP = [CS(v2 ) − CS(v1 )] × 2 [CS(w2 ) − CS(w1 )] 1/2 2 where vi = xi λz 1/2 2 and wi = yi λz also Equation (2.393) on page 45. long in x. (8.51) C S Fresnel cosine integral Fresnel sine integral (8.52) (8.53) (8.54) (8.55) (8.56) CS Cornu spiral v,w length along spiral ψP complex amplitude at P ψ0 λ z unobstructed amplitude wavelength distance of P from aperture plane [see (8.48)] position of edge y (8.57) coherent plane waves 8 (8.58) y1 (8.59) z (8.60) (8.61) (8.62) y2 xi yi positions of slit sides positions of slit top/bottom P main January 23, 2006 16:6 168 8.5 Optics Geometrical optics Lenses and mirrorsa object v r2 x1 f f object u x2 f image r1 lens r u v f MT v R u mirror sign convention + − centred to right centred to left real object virtual object real image virtual image converging lens/ diverging lens/ concave mirror convex mirror erect image inverted image (8.63) L η optical path length refractive index 1 1 1 + = u v f (8.64) dl u v ray path element object distance image distance f focal length Newton’s lens formula x1 x2 = f 2 (8.65) x1 =v −f x2 =u−f Lensmaker’s formula 1 1 1 1 + = (η − 1) − u v r1 r2 ri radii of curvature of lens surfaces Mirror formulac 2 1 1 1 + =− = u v R f (8.67) R mirror radius of curvature Dioptre number D= 1 f (8.68) D dioptre number (f in metres) Focal ratiod n= f d (8.69) n d focal ratio lens or mirror diameter Transverse linear magnification MT = − (8.70) MT transverse magnification Longitudinal linear magnification ML = −MT2 (8.71) ML longitudinal magnification Fermat’s principleb Gauss’s lens formula a Formulas L= η dl is stationary m−1 v u (8.66) assume “Gaussian optics,” i.e., all lenses are thin and all angles small. Light enters from the left. stationary optical path length has, to first order, a length identical to that of adjacent paths. c The mirror is concave if R < 0, convex if R > 0. d Or “f-number,” written f/2 if n = 2 etc. bA main January 23, 2006 16:6 169 8.5 Geometrical optics Prisms (dispersing) α δ θi θt prism Transmission angle sinθt =(η 2 − sin2 θi )1/2 sinα − sinθi cosα (8.72) θi θt α angle of incidence angle of transmission apex angle η refractive index δ angle of deviation Deviation δ = θi + θt − α Minimum deviation condition sinθi = sinθt = η sin Refractive index η= sin[(δm + α)/2] sin(α/2) (8.75) δm minimum deviation Angular dispersiona D= 2sin(α/2) dη dδ = dλ cos[(δm + α)/2] dλ (8.76) D λ dispersion wavelength a At (8.73) α 2 (8.74) minimum deviation. Optical fibres L θm cladding, ηc < ηf Acceptance angle Numerical aperture sinθm = 1 2 (η − ηc2 )1/2 η0 f N = η0 sinθm ηf −1 ηc fibre, ηf (8.77) (8.78) θm η0 ηf maximum angle of incidence exterior refractive index fibre refractive index ηc cladding refractive index N numerical aperture ∆t temporal dispersion L fibre length c speed of light a Of a pulse with a given wavelength, caused by the range of incident angles up to θ . Sometimes called “intermodal m dispersion” or “modal dispersion.” Multimode dispersiona ∆t ηf = L c (8.79) 8 main January 23, 2006 16:6 170 8.6 Optics Polarisation Elliptical polarisationa Elliptical polarisation y iδ i(kz−ωt) E = (E0x ,E0y e )e (8.80) Polarisation angleb 2E0x E0y tan2α = 2 cosδ 2 E0x − E0y (8.81) Ellipticityc e= a−b a (8.82) E k z electric field wavevector propagation axis ωt angular frequency × time E0y E0x E0x x amplitude of E E0y y amplitude of E δ relative phase of Ey with respect to Ex α polarisation angle e a b x α b ellipticity semi-major axis semi-minor axis a θ I(θ) transmitted intensity I(θ) = I0 cos θ (8.83) I0 incident intensity Malus’s law θ polariser–analyser angle a See the introduction (page 161) for a discussion of sign and handedness conventions. b Angle between ellipse major axis and x axis. Sometimes the polarisation angle is defined as π/2 − α. c This is one of several definitions for ellipticity. d Transmission through skewed polarisers for unpolarised incident light. 2 d Jones vectors and matrices Normalised electric fielda Example vectors: Jones matrix Ex ; |E| = 1 Ey 1 1 1 Ex = E45 = √ 0 2 1 1 1 1 1 Er = √ El = √ 2 −i 2 i E= E t = AE i (8.84) E electric field Ex Ey x component of E y component of E E45 45◦ to x axis Er right-hand circular (8.85) El left-hand circular Et Ei A transmitted vector incident vector Jones matrix Example matrices: Linear polariser x Linear polariser at 45◦ Right circular polariser λ/4 plate (fast x) a Known 1 0 0 0 1 1 1 2 1 1 1 1 i 2 −i 1 1 0 eiπ/4 0 i as the “normalised Jones vector.” Linear polariser y Linear polariser at −45◦ Left circular polariser λ/4 plate (fast ⊥ x) 0 0 0 1 1 1 −1 2 −1 1 1 1 −i 2 i 1 1 0 eiπ/4 0 −i main January 23, 2006 16:6 171 8.6 Polarisation Stokes parametersa E0y y V χ pI x α Q E0x 2α 2χ U 2b Poincaré sphere 2a Electric fields Ex = E0x ei(kz−ωt) Ey = E0y ei(kz−ωt+δ) Axial ratiob tanχ = ±r = ± Degree of polarisation (8.89) − Ey2 = pI cos2χcos2α (8.90) (8.91) U = 2Ex Ey cosδ = pI cos2χsin2α (8.92) (8.93) V = 2Ex Ey sinδ = pI sin2χ (8.94) (8.95) p= left circular linear x linear 45◦ to x unpolarised a Using (8.88) I = Ex2 + Ey2 Q = Ex2 Stokes parameters b a (8.86) (8.87) (Q2 + U 2 + V 2 )1/2 ≤1 I Q/I 0 1 0 0 U/I V /I 0 −1 0 0 1 0 0 0 k wavevector ωt δ angular frequency × time relative phase of Ey with respect to Ex χ (see diagram) r axial ratio Ex Ey electric field component x electric field component y E0x field amplitude in x direction E0y field amplitude in y direction α polarisation angle p · degree of polarisation mean over time (8.96) Q/I U/I right circular 0 0 linear y −1 0 linear −45◦ to x 0 −1 V /I 1 0 0 the convention that right-handed circular polarisation corresponds to a clockwise rotation of the electric field in a given plane when looking towards the source. The propagation direction in the diagram is out of the plane. The parameters I, Q, U, and V are sometimes denoted s0 , s1 , s2 , and s3 , and other nomenclatures exist. There is no generally accepted definition – often the parameters are scaled to be dimensionless, with s0 = 1, or to represent power flux through a plane ⊥ the beam, i.e., I = (Ex2 + Ey2 )/Z0 etc., where Z0 is the impedance of free space. b The axial ratio is positive for right-handed polarisation and negative for left-handed polarisation using our definitions. 8 main January 23, 2006 16:6 172 Optics 8.7 Coherence (scalar theory) Mutual coherence function Complex degree of coherence Γ12 (τ) = ψ 1 (t)ψ ∗2 (t + τ) ψ 1 (t)ψ ∗2 (t + τ) [|ψ 1 |2 |ψ 2 |2 ]1/2 Γ12 (τ) = [Γ11 (0)Γ22 (0)]1/2 γ 12 (τ) = Combined intensitya Itot = I1 + I2 + 2(I1 I2 )1/2 [γ 12 (τ)] Fringe visibility V (τ) = if |γ 12 (τ)| is a constant: if I1 = I2 : Complex degree of temporal coherenceb Coherence time and length Complex degree of spatial coherencec (8.98) (8.99) (8.100) 2(I1 I2 )1/2 |γ 12 (τ)| I1 + I2 Imax − Imin V= Imax + Imin V (τ) = |γ 12 (τ)| ψ 1 (t)ψ ∗1 (t + τ) |ψ 1 (t)2 | % I(ω)e−iωτ dω = % I(ω) dω γ(τ) = ∆τc = 1 ∆lc ∼ c ∆ν ψ 1 ψ ∗2 [|ψ 1 |2 |ψ 2 |2 ]1/2 % I(ŝ)eikD·ŝ dΩ = % I(ŝ) dΩ γ(D) = Intensity correlationd I1 I2 = 1 + γ 2 (D) [I1 2 I2 2 ]1/2 Speckle intensity distributione pr(I) = Speckle size (coherence width) λ ∆wc α a From (8.97) 1 −I/I e I Γij τ ψi mutual coherence function temporal interval (complex) wave disturbance at spatial point i t time · γ ij mean over time complex degree of coherence complex conjugate ∗ Itot combined intensity Ii intensity of disturbance at point i real part of (8.101) (8.102) Imax max. combined intensity Imin min. combined intensity (8.103) (8.104) (8.105) γ(τ) degree of temporal coherence I(ω) specific intensity ω radiation angular frequency c speed of light ∆τc coherence time (8.106) ∆lc coherence length ∆ν spectral bandwidth γ(D) degree of spatial coherence (8.107) D (8.108) I(ŝ) specific intensity of distant extended source in direction ŝ (8.109) (8.110) spatial separation of points 1 and 2 dΩ differential solid angle ŝ unit vector in the direction of dΩ k wavenumber pr probability density ∆wc characteristic speckle size (8.111) λ α wavelength source angular size as seen from the screen interfering the disturbances at points 1 and 2 with a relative delay τ. “autocorrelation function.” c Between two points on a wavefront, separated by D. The integral is over the entire extended source. d For wave disturbances that have a Gaussian probability distribution in amplitude. This is “Gaussian light” such as from a thermal source. e Also for Gaussian light. b Or main January 23, 2006 16:6 173 8.8 Line radiation 8.8 Line radiation Spectral line broadening Natural broadeninga (2πτ)−1 I(ω) = (2τ)−2 + (ω − ω0 )2 Natural half-width ∆ω = Collision broadening I(ω) = Collision and pressure half-widthc −1/2 1 2 πmkT ∆ω = = pπd τc 16 1 2τ (8.112) I(ω) normalised intensityb τ lifetime of excited state ω angular frequency (= 2πν) (8.113) ∆ω half-width at half-power ω0 centre frequency p d mean time between collisions pressure effective atomic diameter m k T gas particle mass Boltzmann constant temperature c speed of light τc (πτc )−1 (τc )−2 + (ω − ω0 )2 (8.114) (8.115) 1/2 mc2 mc2 (ω − ω0 )2 exp − I(ω) = 2kT 2kT ω02 π ω02 (8.116) 1/2 2kT ln2 ∆ω = ω0 (8.117) mc2 Doppler broadening Doppler half-width I(ω) ∆ω ω0 a The transition probability per unit time for the state is = 1/τ. In the classical limit of a damped oscillator, the e-folding time of the electric field is 2τ. Both the % natural and collision profiles described here are Lorentzian. b The intensity spectra are normalised so that I(ω) dω = 1, assuming ∆ω/ω 1. 0 c The pressure-broadening relation combines Equations (5.78), (5.86) and (5.89) and assumes an otherwise perfect gas of finite-sized atoms. More accurate expressions are considerably more complicated. Einstein coefficientsa Rij Absorption R12 = B12 Iν n1 (8.118) Spontaneous emission R21 = A21 n2 (8.119) Stimulated emission = B21 Iν n2 R21 (8.120) Coefficient ratios A21 2hν 3 g1 = 2 B12 c g2 B21 g1 = B12 g2 a Note case (8.121) (8.122) transition rate, level i → j (m−3 s−1 ) Bij Einstein B coefficients Iν specific intensity of radiation field A21 Einstein A coefficient ni number density of atoms in quantum level i (m−3 ) h Planck constant ν c gi frequency speed of light degeneracy of ith level that the coefficients can also be defined in terms of spectral energy density, uν = 4πIν /c rather than Iν . In this 3 g 1 . See also Population densities on page 116. = 8πhν c3 g A21 B12 2 8 main January 23, 2006 16:6 174 Optics Lasersa R1 R2 r1 L Cavity stability condition L 0≤ 1− r1 Longitudinal cavity modesb νn = L 1− r2 ≤ 1 (8.123) c n 2L (8.124) 2πL(R1 R2 )1/4 λ[1 − (R1 R2 )1/2 ] 4πL λ(1 − R1 R2 ) νn ∆νc = = 1/(2πτc ) Q Q= Cavity Q Cavity line width ∆ν 2πh(∆νc )2 = νn P Threshold lasing condition R1 R2 exp[2(α − β)L] > 1 a Also r1,2 radii of curvature of end-mirrors L distance between mirror centres νn n c mode frequency integer speed of light (8.125) Q quality factor R1,2 mirror (power) reflectances (8.126) λ (8.127) ∆νc cavity line width (FWHP) τc cavity photon lifetime gl Nu gl Nu − gu Nl (8.128) Schawlow– Townes line width light out r2 (8.129) ∆ν P wavelength line width (FWHP) laser power gu,l degeneracy of upper/lower levels Nu,l number density of upper/lower levels α β gain per unit length of medium loss per unit length of medium see the Fabry-Perot etalon on page 163. Note that “cavity” refers to the empty cavity, with no lasing medium present. b The mode spacing equals the cavity free spectral range. main January 23, 2006 16:6 Chapter 9 Astrophysics 9.1 Introduction Many of the formulas associated with astronomy and astrophysics are either too specialised for a general work such as this or are common to other fields and can therefore be found elsewhere in this book. The following section includes many of the relationships that fall into neither of these categories, including equations to convert between various astronomical coordinate systems and some basic formulas associated with cosmology. Exceptionally, this section also includes data on the Sun, Earth, Moon, and planets. Observational astrophysics remains a largely inexact science, and parameters of these (and other) bodies are often used as approximate base units in measurements. For example, the masses of stars and galaxies are frequently quoted as multiples of the mass of the Sun (1M = 1.989 × 1030 kg), extra-solar system planets in terms of the mass of Jupiter, and so on. Astronomers seem to find it particularly difficult to drop arcane units and conventions, resulting in a profusion of measures and nomenclatures throughout the subject. However, the convention of using suitable astronomical objects in this way is both useful and widely accepted. 9 main January 23, 2006 16:6 176 9.2 Astrophysics Solar system data Solar data equatorial radius mass polar moment of inertia bolometric luminosity effective surface temperature solar constanta absolute magnitude apparent magnitude a Bolometric R M I L T = = = = = MV mV = = 6.960 × 108 m 1.9891 × 1030 kg 5.7 × 1046 kgm2 3.826 × 1026 W 5770K 1.368 × 103 Wm−2 +4.83; Mbol −26.74; mbol = = = 109.1R⊕ 3.32946 × 105 M⊕ 7.09 × 108 I⊕ = = +4.75 −26.82 flux at a distance of 1 astronomical unit (AU). Earth data equatorial radius flatteninga mass polar moment of inertia orbital semi-major axisb mean orbital velocity equatorial surface gravity polar surface gravity rotational angular velocity af R⊕ f M⊕ I⊕ 1AU = = = = = ge gp ωe = = = = 9.166 × 10−3 R 6.37814 × 106 m 0.00335364 = 1/298.183 = 3.0035 × 10−6 M 5.9742 × 1024 kg = 1.41 × 10−9 I 8.037 × 1037 kgm2 11 = 214.9R 1.495979 × 10 m 2.979 × 104 ms−1 (includes rotation) 9.780327ms−2 9.832186ms−2 7.292115 × 10−5 rads−1 equals (R⊕ − Rpolar )/R⊕ . The mean radius of the Earth is 6.3710 × 106 m. the Sun. b About Moon data equatorial radius mass mean orbital radiusa mean orbital velocity orbital period (sidereal) equatorial surface gravity a About Rm Mm am = = = 1.7374 × 106 m 7.3483 × 1022 kg 3.84400 × 108 m 1.03 × 103 ms−1 27.32166d 1.62ms−2 = = = 0.27240R⊕ 1.230 × 10−2 M⊕ 60.27R⊕ = 0.166ge the Earth. Planetary dataa Mercury Venusb Earth Mars Jupiter Saturn Uranusb Neptune Plutob a Using M/M⊕ 0.055 274 0.815 00 1 0.107 45 317.85 95.159 14.500 17.204 0.00251 R/R⊕ 0.382 51 0.948 83 1 0.532 60 11.209 9.449 1 4.007 3 3.882 6 0.187 36 T (d) 58.646 243.018 0.997 27 1.025 96 0.413 54 0.444 01 0.718 33 0.671 25 6.387 2 P (yr) 0.240 85 0.615 228 1.000 04 1.880 93 11.861 3 29.628 2 84.746 6 166.344 248.348 a(AU) 0.387 10 0.723 35 1.000 00 1.523 71 5.202 53 9.575 60 19.293 4 30.245 9 39.509 0 M R mass equatorial radius T rotational period P a M⊕ orbital period mean distance 5.9742 × 1024 kg R⊕ 6.37814 × 106 m 1d 1yr 86400s 3.15569 × 107 s 1AU 1.495979 × 1011 m the osculating orbital elements for 1998. Note that P is the instantaneous orbital period, calculated from the planet’s daily motion. The radii of gas giants are taken at 1 atmosphere pressure. b Retrograde rotation. main January 23, 2006 16:6 177 9.3 Coordinate transformations (astronomical) 9.3 Coordinate transformations (astronomical) Time in astronomy Julian day numbera JD =D − 32075 + 1461 ∗ (Y + 4800 + (M − 14)/12)/4 JD D Y Julian day number day of month number calendar year, e.g., 1963 (9.1) M ∗ calendar month (Jan=1) integer multiply MJD = JD − 2400000.5 (9.2) / MJD integer divide modified Julian day number W = (JD + 1) (9.3) W day of week (0=Sunday, 1=Monday, ... ) LCT UTC local civil time coordinated universal time TZC DSC T time zone correction daylight saving correction Julian centuries between 12h UTC 1 Jan 2000 and 0h UTC D/M/Y + 367 ∗ (M − 2 − (M − 14)/12 ∗ 12)/12 − 3 ∗ ((Y + 4900 + (M − 14)/12)/100)/4 Modified Julian day number Day of week mod 7 Local civil time LCT = UTC + TZC + DSC Julian centuries T= Greenwich sidereal time Local sidereal time JD − 2451545.5 36525 (9.4) (9.5) GMST =6h 41m 50s .54841 + 8640184s .812866T + 0s .093104T 2 − 0s .0000062T 3 LST = GMST + λ◦ 15◦ (9.6) GMST Greenwich mean sidereal time at 0h UTC D/M/Y (for later times use 1s = 1.002738 sidereal seconds) (9.7) LST λ◦ local sidereal time geographic longitude, degrees east of Greenwich a For the Julian day starting at noon on the calendar day in question. The routine is designed around integer arithmetic with “truncation towards zero” (so that −5/3 = −1) and is valid for dates from the onset of the Gregorian calendar, 15 October 1582. JD represents the number of days since Greenwich mean noon 1 Jan 4713 BC. For reference, noon, 1 Jan 2000 = JD2451545 and was a Saturday (W = 6). Horizon coordinatesa Hour angle H = LST − α Equatorial to horizon sina = sinδ sinφ + cosδ cosφcosH −cosδ sinH tanA ≡ sinδ cosφ − sinφcosδ cosH Horizon to equatorial sinδ = sinasinφ + cosacosφcosA −cosasinA tanH ≡ sinacosφ − sinφcosacosA a Conversions (9.8) (9.9) (9.10) (9.11) (9.12) LST local sidereal time H (local) hour angle α δ a right ascension declination altitude A φ azimuth (E from N) observer’s latitude + + − + A, H − − − + between horizon or alt–azimuth coordinates, (a,A), and celestial equatorial coordinates, (δ,α). There are a number of conventions for defining azimuth. For example, it is sometimes taken as the angle west from south rather than east from north. The quadrants for A and H can be obtained from the signs of the numerators and denominators in Equations (9.10) and (9.12) (see diagram). 9 main January 23, 2006 16:6 178 Astrophysics Ecliptic coordinatesa ε = 23◦ 26 21 .45 − 46 .815T Obliquity of the ecliptic ε mean ecliptic obliquity T Julian centuries since J2000.0b (9.14) α δ right ascension declination (9.15) λ β ecliptic longitude ecliptic latitude − 0 .0006T 2 + 0 .00181T 3 Equatorial to ecliptic sinβ = sinδ cosε − cosδ sinεsinα sinαcosε + tanδ sinε tanλ ≡ cosα Ecliptic to equatorial sinδ = sinβ cosε + cosβ sinεsinλ sinλcosε − tanβ sinε tanα ≡ cosλ (9.13) + + (9.16) − + (9.17) − − − + λ, α a Conversions between ecliptic, (β,λ), and celestial equatorial, (δ,α), coordinates. β is positive above the ecliptic and λ increases eastwards. The quadrants for λ and α can be obtained from the signs of the numerators and denominators in Equations (9.15) and (9.17) (see diagram). b See Equation (9.5). Galactic coordinatesa αg = 192◦ 15 δg = 27◦ 24 (9.18) (9.19) lg = 33◦ (9.20) Equatorial to galactic sinb = cosδ cosδg cos(α − αg ) + sinδ sinδg tanδ cosδg − cos(α − αg )sinδg tan(l − lg ) ≡ sin(α − αg ) (9.21) Galactic to equatorial sinδ = cosbcosδg sin(l − lg ) + sinbsinδg cos(l − lg ) tan(α − αg ) ≡ tanbcosδg − sinδg sin(l − lg ) Galactic frame αg δg lg ascending node of galactic plane on equator δ α b declination right ascension galactic latitude (9.22) (9.23) (9.24) right ascension of north galactic pole declination of north galactic pole l galactic longitude between galactic, (b,l), and celestial equatorial, (δ,α), coordinates. The galactic frame is defined at epoch B1950.0. The quadrants of l and α can be obtained from the signs of the numerators and denominators in Equations (9.22) and (9.24). a Conversions Precession of equinoxesa In right ascension α α0 + (3 .075 + 1 .336sinα0 tanδ0 )N In declination δ δ0 + (20 .043cosα0 )N a Right s s α right ascension of date (9.25) α0 N right ascension at J2000.0 number of years since J2000.0 (9.26) δ δ0 declination of date declination at J2000.0 ascension in hours, minutes, and seconds; declination in degrees, arcminutes, and arcseconds. These equations are valid for several hundred years each side of J2000.0. main January 23, 2006 16:6 179 9.4 Observational astrophysics 9.4 Observational astrophysics Astronomical magnitudes Apparent magnitude m1 − m2 = −2.5log10 F1 F2 (9.27) Distance modulusa m − M = 5log10 D − 5 = −5log10 p − 5 (9.28) (9.29) Luminosity– magnitude relation Mbol = 4.75 − 2.5log10 mi Fi apparent magnitude of object i energy flux from object i M m−M absolute magnitude distance modulus D p distance to object (parsec) annual parallax (arcsec) (9.30) L 3.04 × 10(28−0.4Mbol ) Mbol L bolometric absolute magnitude luminosity (W) (9.31) L solar luminosity (3.826 × 1026 W) Fbol 2.559 × 10−(8+0.4mbol ) (9.32) Fbol mbol bolometric flux (Wm−2 ) bolometric apparent magnitude Bolometric correction BC = mbol − mV (9.33) = Mbol − MV (9.34) BC mV bolometric correction V -band apparent magnitude MV V -band absolute magnitude Colour indexb B − V = mB − m V (9.35) U − B = mU − m B (9.36) B −V U −B observed B − V colour index observed U − B colour index Colour excessc E = (B − V ) − (B − V )0 (9.37) Flux– magnitude relation L L E B − V colour excess (B − V )0 intrinsic B − V colour index a Neglecting extinction. the UBV magnitude system. The bands are centred around 365 nm (U), 440 nm (B), and 550 nm (V ). c The U − B colour excess is defined similarly. b Using Photometric wavelengths Mean wavelength Isophotal wavelength Effective wavelength % λR(λ) dλ λ0 = % R(λ) dλ % F(λ)R(λ) dλ F(λi ) = % R(λ) dλ % λF(λ)R(λ) dλ λeff = % F(λ)R(λ) dλ mean wavelength wavelength (9.38) λ0 λ (9.39) R system spectral response F(λ) flux density of source (in terms of wavelength) λi (9.40) isophotal wavelength λeff effective wavelength 9 main January 23, 2006 16:6 180 Astrophysics Planetary bodies Bode’s law a 4 + 3 × 2n DAU = 10 (9.41) Roche limit 1/3 100M 9πρ > 2.46R (if densities equal) 0 ∼ > R∼ 1 1 1 = − S P P⊕ (9.42) (9.43) DAU n planetary orbital radius (AU) index: Mercury = −∞, Venus = 0, Earth = 1, Mars = 2, Ceres = 3, Jupiter= 4, ... R satellite orbital radius M ρ central mass satellite density R0 central body radius S synodic period P planetary orbital period P⊕ Earth’s orbital period a Also known as the “Titius–Bode rule.” Note that the asteroid Ceres is counted as a planet in this scheme. The relationship breaks down for Neptune and Pluto. b Of a planet. Synodic periodb (9.44) Distance indicators Hubble law v = H0 d (9.45) Annual parallax Dpc = p−1 Cepheid variablesa L 1.15log10 Pd + 2.47 (9.47) L MV −2.76log10 Pd − 1.40 (9.48) (9.46) log10 Tully–Fisher relationb MI −7.68log10 Einstein rings θ2 = 2vrot sini − 2.58 (9.49) 4GM c2 ds − dl ds dl Sunyaev– Zel’dovich effectc ∆T = −2 T ... for a homogeneous sphere 4Rne kTe σT ∆T =− T me c2 ne kTe σT dl me c2 (9.50) (9.51) (9.52) v H0 cosmological recession velocity Hubble parameter (present epoch) d (proper) distance Dpc distance (parsec) p annual parallax (±p arcsec from mean) L mean cepheid luminosity L Solar luminosity Pd pulsation period (days) MV absolute visual magnitude MI I-band absolute magnitude vrot observed maximum rotation velocity (kms−1 ) i galactic inclination (90◦ when edge-on) θ ring angular radius M ds dl lens mass distance from observer to source distance from observer to lens T dl apparent CMBR temperature path element through cloud R ne k cloud radius electron number density Boltzmann constant Te σT me electron temperature Thomson cross section electron mass c speed of light relation for classical Cepheids. Uncertainty in MV is ±0.27 (Madore & Freedman, 1991, Publications of the Astronomical Society of the Pacific, 103, 933). b Galaxy rotation velocity–magnitude relation in the infrared I waveband, centred at 0.90µm. The coefficients depend on waveband and galaxy type (see Giovanelli et al., 1997, The Astronomical Journal, 113, 1). c Scattering of the cosmic microwave background radiation (CMBR) by a cloud of electrons, seen as a temperature decrement, ∆T , in the Rayleigh–Jeans limit (λ 1mm). a Period–luminosity main January 23, 2006 16:6 181 9.5 Stellar evolution 9.5 Stellar evolution Evolutionary timescales Free-fall timescalea Kelvin–Helmholtz timescale a For 3π τff = 32Gρ0 1/2 (9.53) −Ug L GM 2 R0 L (9.54) τKH = (9.55) τff G ρ0 free-fall timescale constant of gravitation initial mass density τKH Kelvin–Helmholtz timescale Ug gravitational potential energy M R0 L body’s mass body’s initial radius body’s luminosity the gravitational collapse of a uniform sphere. Star formation Jeans lengtha π dp λJ = Gρ dρ Jeans mass π MJ = ρλ3J 6 Eddington limiting luminosityb 1/2 (9.56) (9.57) 4πGMmp c LE = σT M 1.26 × 1031 M (9.58) W (9.59) λJ G Jeans length constant of gravitation ρ p cloud mass density pressure MJ (spherical) Jeans mass LE M Eddington luminosity stellar mass M solar mass mp proton mass c speed of light σT Thomson cross section that (dp/dρ)1/2 is the sound speed in the cloud. b Assuming the opacity is mostly from Thomson scattering. a Note Stellar theorya r radial distance Mr ρ mass interior to r mass density (9.61) p G pressure constant of gravitation (9.62) Lr luminosity interior to r power generated per unit mass T temperature (9.63) σ κ Stefan–Boltzmann constant mean opacity (9.64) γ ratio of heat capacities, cp /cV Conservation of mass dMr = 4πρr2 dr (9.60) Hydrostatic equilibrium dp −GρMr = dr r2 Energy release dLr = 4πρr2 dr Radiative transport −3 κ ρ Lr dT = dr 16σ T 3 4πr2 Convective transport dT γ − 1 T dp = dr γ p dr a For stars in static equilibrium with adiabatic convection. Note that ρ is a function of r. κ and are functions of temperature and composition. 9 main January 23, 2006 16:6 182 Astrophysics Stellar fusion processesa PP i chain PP ii chain p + p → 21 H + e+ + νe 2 + 3 1 H + p → 2 He + γ 3 3 4 + 2 He + 2 He → 2 He + 2p + PP iii chain p + p → 12 H + e+ + νe 2 + 3 1 H + p → 2 He + γ 3 4 7 2 He + 2 He → 4 Be + γ 7 − 7 4 Be + e → 3 Li + νe + + p + p+ → 12 H + e+ + νe + + 2 + 3 1 H + p → 2 He + γ 3 4 7 2 He + 2 He → 4 Be + γ 7 + 8 4 Be + p → 5 B + γ 8 8 + 5 B → 4 Be + e + νe 8 4 4 Be → 2 2 He 7 + 4 3 Li + p → 2 2 He CNO cycle triple-α process 12 + 13 6C + p → 7N + γ 13 13 + 7 N → 6 C + e + νe 13 + 14 6C + p → 7N + γ 14 + 15 7N + p → 8O + γ 15 15 + 8 O → 7 N + e + νe 4 4 8 4 Be + γ 2 He + 2 He 8 4 126 C∗ 4 Be + 2 He 12 ∗ 12 6C → 6C + γ γ photon + p e+ proton positron e− νe electron electron neutrino 15 + 12 4 7 N + p → 6 C + 2 He a All species are taken as fully ionised. Pulsars ω̇ ∝ −ω n Braking index n=2− P P̈ Ṗ 2 Characteristic agea 1 P T= n − 1 Ṗ Magnetic dipole radiation L= µ0 |m̈|2 sin2 θ 6πc3 2πR 6 Bp2 ω 4 sin2 θ = 3c3 µ0 Dispersion measure (9.65) ω rotational angular velocity (9.66) P n rotational period (= 2π/ω) braking index T L characteristic age luminosity µ0 c permeability of free space speed of light (9.67) (9.68) (9.69) D ne dl DM = D dl ne path length to pulsar path element electron number density (9.71) τ ∆τ pulse arrival time difference in pulse arrival time (9.72) νi me observing frequencies electron mass (9.70) 0 b Dispersion −e2 dτ = 2 DM dν 4π 0 me cν 3 1 1 e2 − ∆τ = 2 DM 8π 0 me c ν12 ν22 pulsar magnetic dipole moment pulsar radius magnetic flux density at magnetic pole θ angle between magnetic and rotational axes DM dispersion measure m R Bp n = 1 and that the pulsar has already slowed significantly. Usually n is assumed to be 3 (magnetic dipole radiation), giving T = P /(2Ṗ ). b The pulse arrives first at the higher observing frequency. a Assuming main January 23, 2006 16:6 183 9.5 Stellar evolution Compact objects and black holes rs G M Schwarzschild radius constant of gravitation mass of body Schwarzschild radius rs = 2GM M 3 km c2 M (9.73) Gravitational redshift 1/2 ν∞ 2GM = 1− νr rc2 (9.74) r ν∞ νr distance from mass centre frequency at infinity frequency at r Gravitational wave radiationa Lg = (9.75) mi a Lg orbiting masses mass separation gravitational luminosity Rate of change of orbital period Ṗ = − P orbital period Neutron star degeneracy pressure (nonrelativistic) (3π 2 )2/3 h̄2 p= 5 mn p h̄ pressure (Planck constant)/(2π) mn ρ neutron mass density Relativisticb p= (9.78) u energy density Chandrasekhar massc Maximum black hole angular momentum MCh 1.46M (9.79) MCh Chandrasekhar mass GM 2 c (9.80) c speed of light M solar mass 32 G4 m21 m22 (m1 + m2 ) 5 c5 a5 G5/3 m1 m2 P −5/3 96 (4π 2 )4/3 5 5 c (m1 + m2 )1/3 h̄c(3π 2 )1/3 4 Jm = ρ mn ρ mn 5/3 4/3 Black hole evaporation time τe ∼ M3 × 1066 M3 Black hole temperature T= M h̄c3 10−7 8πGMk M 2 = u 3 1 = u 3 yr (9.76) (9.77) (9.81) K (9.82) Jm maximum angular momentum τe evaporation time T temperature k Boltzmann constant a From two bodies, m and m , in circular orbits about their centre of mass. Note that the frequency of the radiation 1 2 is twice the orbital frequency. b Particle velocities ∼ c. c Upper limit to mass of a white dwarf. 9 main January 23, 2006 16:6 184 9.6 Astrophysics Cosmology Cosmological model parameters Hubble law vr = Hd (9.83) Ṙ(t) R(t) H(z) =H0 [Ωm0 (1 + z)3 + ΩΛ0 H(t) = Hubble parametera (9.84) + (1 − Ωm0 − ΩΛ0 )(1 + z)2 ]1/2 (9.85) Redshift Robertson– Walker metricb Friedmann equationsc Critical density λobs − λem R0 −1 = λem R(tem ) dr2 ds2 =c2 dt2 − R 2 (t) 1 − kr2 2 2 + r (dθ + sin θ dφ ) p ! ΛR 4π GR ρ + 3 2 + 3 c 3 2 8π ΛR Ṙ 2 = GρR 2 − kc2 + 3 3 ρcrit = 3H 2 8πG ρ 8πGρ = ρcrit 3H 2 Λ ΩΛ = 3H 2 kc2 Ωk = − 2 2 R H Ωm + ΩΛ + Ωk = 1 Ωm = Density parameters Deceleration parameter 2 R̈ = − q0 = − R0 R̈0 Ωm0 − ΩΛ0 = 2 Ṙ02 radial velocity Hubble parameter d proper distance 0 R t present epoch cosmic scale factor cosmic time z redshift λobs observed wavelength (9.86) z= 2 vr H λem tem emitted wavelength epoch of emission ds c interval speed of light (9.87) r,θ,φ comoving spherical polar coordinates (9.88) k G curvature parameter constant of gravitation (9.89) p Λ pressure cosmological constant (9.90) ρ (mass) density ρcrit critical density (9.91) (9.92) (9.93) Ωm matter density parameter ΩΛ Ωk lambda density parameter curvature density parameter q0 deceleration parameter (9.94) (9.95) < H < 80kms−1 Mpc−1 ≡ 100hkms−1 Mpc−1 , where called the Hubble “constant.” At the present epoch, 60 ∼ 0∼ h is a dimensionless scaling parameter. The Hubble time is tH = 1/H0 . Equation (9.85) assumes a matter dominated universe and mass conservation. b For a homogeneous, isotropic universe, using the (−1,1,1,1) metric signature. r is scaled so that k = 0,±1. Note that ds2 ≡ (ds)2 etc. c Λ = 0 in a Friedmann universe. Note that the cosmological constant is sometimes defined as equalling the value used here divided by c2 . a Often main January 23, 2006 16:6 185 9.6 Cosmology Cosmological distance measures Look-back time Proper distance Luminosity distancea Flux density– redshift relation Angular diameter distanced tlb (z) = t0 − t(z) r dp = R0 0 (9.96) dr = cR0 (1 − kr2 )1/2 z dL = dp (1 + z) = c(1 + z) 0 F(ν) = L(ν ) 4πd2L (z) t0 t dt R(t) dz H(z) where ν = (1 + z)ν da = dL (1 + z)−2 tlb (z)light travel time from an object at redshift z t0 present cosmic time t(z) cosmic time at z dp R c proper distance cosmic scale factor speed of light 0 dL present epoch luminosity distance (9.98) z H F redshift Hubble parameterb spectral flux density (9.99) ν frequency L(ν) spectral luminosityc (9.97) (9.100) da angular diameter distance k curvature parameter a flat universe (k = 0). The apparent flux density of a source varies as d−2 L . b See Equation (9.85). c Defined as the output power of the body per unit frequency interval. d True for all k. The angular diameter of a source varies as d−1 . a a Assuming Cosmological modelsa 2c [1 − (1 + z)−1/2 ] H0 H(z) = H0 (1 + z)3/2 q0 = 1/2 2 t(z) = 3H(z) ρ = (6πGt2 )−1 dp = Einstein – de Sitter model (Ωk = 0, Λ = 0, p = 0 and Ωm0 = 1) R(t) = R0 (t/t0 ) Concordance model (Ωk = 0, Λ = 3(1 − Ωm0 )H02 , p = 0 and Ωm0 < 1) a Currently popular. c dp = H0 0 z 2/3 (9.101) (9.102) (9.103) (9.105) −1/2 H(z) = H0 [Ωm0 (1 + z)3 + (1 − Ωm0 )] q0 = 3Ωm0 /2 − 1 (1 − Ωm0 )1/2 2 −1/2 t(z) = (1 − Ωm0 ) arsinh 3H0 (1 + z)3/2 H 0 z (9.104) (9.106) Ωm0 dz 1/2 [(1 + z )3 − 1 + Ω−1 m0 ] dp (9.107) speed of light deceleration parameter t(z) time at redshift z c q R Ωm0 (9.108) (9.109) (9.110) proper distance Hubble parameter present epoch redshift cosmic scale factor present mass density parameter G constant of gravitation ρ mass density 9 main January 23, 2006 16:6 main January 23, 2006 16:6 Index Section headings are shown in boldface and panel labels in small caps. Equation numbers are contained within square brackets. A aberration (relativistic) [3.24], 65 absolute magnitude [9.29], 179 absorption (Einstein coefficient) [8.118], 173 absorption coefficient (linear) [5.175], 120 accelerated point charge bremsstrahlung, 160 Liénard–Wiechert potentials, 139 oscillating [7.132], 146 synchrotron, 159 acceleration constant, 68 dimensions, 16 due to gravity (value on Earth), 176 in a rotating frame [3.32], 66 acceptance angle (optical fibre) [8.77], 169 acoustic branch (phonon) [6.37], 129 acoustic impedance [3.276], 83 action (definition) [3.213], 79 action (dimensions), 16 addition of velocities Galilean [3.3], 64 relativistic [3.15], 64 adiabatic bulk modulus [5.23], 107 compressibility [5.21], 107 expansion (ideal gas) [5.58], 110 lapse rate [3.294], 84 adjoint matrix definition 1 [2.71], 24 definition 2 [2.80], 25 adjugate matrix [2.80], 25 admittance (definition), 148 advective operator [3.289], 84 Airy disk [8.40], 165 function [8.17], 163 resolution criterion [8.41], 165 Airy’s differential equation [2.352], 43 albedo [5.193], 121 Alfvén speed [7.277], 158 Alfvén waves [7.284], 158 alt-azimuth coordinates, 177 alternating tensor (ijk ) [2.443], 50 altitude coordinate [9.9], 177 Ampère’s law [7.10], 136 ampere (SI definition), 3 ampere (unit), 4 analogue formula [2.258], 36 angle aberration [3.24], 65 acceptance [8.77], 169 beam solid [7.210], 153 Brewster’s [7.218], 154 Compton scattering [7.240], 155 contact (surface tension) [3.340], 88 deviation [8.73], 169 Euler [2.101], 26 Faraday rotation [7.273], 157 hour (coordinate) [9.8], 177 Kelvin wedge [3.330], 87 Mach wedge [3.328], 87 polarisation [8.81], 170 principal range (inverse trig.), 34 refraction, 154 rotation, 26 Rutherford scattering [3.116], 72 separation [3.133], 73 spherical excess [2.260], 36 units, 4, 5 ångström (unit), 5 I main January 23, 2006 16:6 188 angular diameter distance [9.100], 185 Angular momentum, 98 angular momentum conservation [4.113], 98 definition [3.66], 68 dimensions, 16 eigenvalues [4.109] [4.109], 98 ladder operators [4.108], 98 operators and other operators [4.23], 91 definitions [4.105], 98 rigid body [3.141], 74 Angular momentum addition, 100 Angular momentum commutation relations, 98 angular speed (dimensions), 16 anomaly (true) [3.104], 71 antenna beam efficiency [7.214], 153 effective area [7.212], 153 power gain [7.211], 153 temperature [7.215], 153 Antennas, 153 anticommutation [2.95], 26 antihermitian symmetry, 53 antisymmetric matrix [2.87], 25 Aperture diffraction, 165 aperture function [8.34], 165 apocentre (of an orbit) [3.111], 71 apparent magnitude [9.27], 179 Appleton-Hartree formula [7.271], 157 arc length [2.279], 39 arccosx from arctan [2.233], 34 series expansion [2.141], 29 arcoshx (definition) [2.239], 35 arccotx (from arctan) [2.236], 34 arcothx (definition) [2.241], 35 arccscx (from arctan) [2.234], 34 arcschx (definition) [2.243], 35 arcminute (unit), 5 arcsecx (from arctan) [2.235], 34 arsechx (definition) [2.242], 35 arcsecond (unit), 5 arcsinx from arctan [2.232], 34 series expansion [2.141], 29 arsinhx (definition) [2.238], 35 arctanx (series expansion) [2.142], 29 Index artanhx (definition) [2.240], 35 area of circle [2.262], 37 of cone [2.271], 37 of cylinder [2.269], 37 of ellipse [2.267], 37 of plane triangle [2.254], 36 of sphere [2.263], 37 of spherical cap [2.275], 37 of torus [2.273], 37 area (dimensions), 16 argument (of a complex number) [2.157], 30 arithmetic mean [2.108], 27 arithmetic progression [2.104], 27 associated Laguerre equation [2.348], 43 associated Laguerre polynomials, 96 associated Legendre equation and polynomial solutions [2.428], 48 differential equation [2.344], 43 Associated Legendre functions, 48 astronomical constants, 176 Astronomical magnitudes, 179 Astrophysics, 175–185 asymmetric top [3.189], 77 atomic form factor [6.30], 128 mass unit, 6, 9 numbers of elements, 124 polarisability [7.91], 142 weights of elements, 124 Atomic constants, 7 atto, 5 autocorrelation (Fourier) [2.491], 53 autocorrelation function [8.104], 172 availability and fluctuation probability [5.131], 116 definition [5.40], 108 Avogadro constant, 6, 9 Avogadro constant (dimensions), 16 azimuth coordinate [9.10], 177 B Ballistics, 69 band index [6.85], 134 Band theory and semiconductors, 134 bandwidth and coherence time [8.106], 172 main January 23, 2006 16:6 Index and Johnson noise [5.141], 117 Doppler [8.117], 173 natural [8.113], 173 of a diffraction grating [8.30], 164 of an LCR circuit [7.151], 148 of laser cavity [8.127], 174 Schawlow-Townes [8.128], 174 bar (unit), 5 barn (unit), 5 Barrier tunnelling, 94 Bartlett window [2.581], 60 base vectors (crystallographic), 126 basis vectors [2.17], 20 Bayes’ theorem [2.569], 59 Bayesian inference, 59 bcc structure, 127 beam bowing under its own weight [3.260], 82 beam efficiency [7.214], 153 beam solid angle [7.210], 153 beam with end-weight [3.259], 82 beaming (relativistic) [3.25], 65 becquerel (unit), 4 Bending beams, 82 bending moment (dimensions), 16 bending moment [3.258], 82 bending waves [3.268], 82 Bernoulli’s differential equation [2.351], 43 Bernoulli’s equation compressible flow [3.292], 84 incompressible flow [3.290], 84 Bessel equation [2.345], 43 Bessel functions, 47 beta (in plasmas) [7.278], 158 binomial coefficient [2.121], 28 distribution [2.547], 57 series [2.120], 28 theorem [2.122], 28 binormal [2.285], 39 Biot–Savart law [7.9], 136 Biot-Fourier equation [5.95], 113 black hole evaporation time [9.81], 183 Kerr solution [3.62], 67 maximum angular momentum [9.80], 183 Schwarzschild radius [9.73], 183 189 Schwarzschild solution [3.61], 67 temperature [9.82], 183 blackbody energy density [5.192], 121 spectral energy density [5.186], 121 spectrum [5.184], 121 Blackbody radiation, 121 Bloch’s theorem [6.84], 134 Bode’s law [9.41], 180 body cone, 77 body frequency [3.187], 77 body-centred cubic structure, 127 Bohr energy [4.74], 95 magneton (equation) [4.137], 100 magneton (value), 6, 7 quantisation [4.71], 95 radius (equation) [4.72], 95 radius (value), 7 Bohr magneton (dimensions), 16 Bohr model, 95 boiling points of elements, 124 bolometric correction [9.34], 179 Boltzmann constant, 6, 9 constant (dimensions), 16 distribution [5.111], 114 entropy [5.105], 114 excitation equation [5.125], 116 Born collision formula [4.178], 104 Bose condensation [5.123], 115 Bose–Einstein distribution [5.120], 115 boson statistics [5.120], 115 Boundary conditions for E, D, B, and H, 144 box (particle in a) [4.64], 94 Box Muller transformation [2.561], 58 Boyle temperature [5.66], 110 Boyle’s law [5.56], 110 bra vector [4.33], 92 bra-ket notation, 91, 92 Bragg’s reflection law in crystals [6.29], 128 in optics [8.32], 164 braking index (pulsar) [9.66], 182 Bravais lattices, 126 Breit-Wigner formula [4.174], 104 Bremsstrahlung, 160 bremsstrahlung I main January 23, 2006 16:6 190 single electron and ion [7.297], 160 thermal [7.300], 160 Brewster’s law [7.218], 154 brightness (blackbody) [5.184], 121 Brillouin function [4.147], 101 Bromwich integral [2.518], 55 Brownian motion [5.98], 113 bubbles [3.337], 88 bulk modulus adiabatic [5.23], 107 general [3.245], 81 isothermal [5.22], 107 bulk modulus (dimensions), 16 Bulk physical constants, 9 Burgers vector [6.21], 128 C calculus of variations [2.334], 42 candela, 119 candela (SI definition), 3 candela (unit), 4 canonical ensemble [5.111], 114 entropy [5.106], 114 equations [3.220], 79 momenta [3.218], 79 cap, see spherical cap Capacitance, 137 capacitance current through [7.144], 147 definition [7.143], 147 dimensions, 16 energy [7.153], 148 energy of an assembly [7.134], 146 impedance [7.159], 148 mutual [7.134], 146 capacitance of cube [7.17], 137 cylinder [7.15], 137 cylinders (adjacent) [7.21], 137 cylinders (coaxial) [7.19], 137 disk [7.13], 137 disks (coaxial) [7.22], 137 nearly spherical surface [7.16], 137 sphere [7.12], 137 spheres (adjacent) [7.14], 137 spheres (concentric) [7.18], 137 capacitor, see capacitance capillary Index constant [3.338], 88 contact angle [3.340], 88 rise [3.339], 88 waves [3.321], 86 capillary-gravity waves [3.322], 86 cardioid [8.46], 166 Carnot cycles, 107 Cartesian coordinates, 21 Catalan’s constant (value), 9 Cauchy differential equation [2.350], 43 distribution [2.555], 58 inequality [2.151], 30 integral formula [2.167], 31 Cauchy-Goursat theorem [2.165], 31 Cauchy-Riemann conditions [2.164], 31 cavity modes (laser) [8.124], 174 Celsius (unit), 4 Celsius conversion [1.1], 15 centi, 5 centigrade (avoidance of), 15 centre of mass circular arc [3.173], 76 cone [3.175], 76 definition [3.68], 68 disk sector [3.172], 76 hemisphere [3.170], 76 hemispherical shell [3.171], 76 pyramid [3.175], 76 semi-ellipse [3.178], 76 spherical cap [3.177], 76 triangular lamina [3.174], 76 Centres of mass, 76 centrifugal force [3.35], 66 centripetal acceleration [3.32], 66 cepheid variables [9.48], 180 Cerenkov, see Cherenkov chain rule function of a function [2.295], 40 partial derivatives [2.331], 42 Chandrasekhar mass [9.79], 183 change of variable [2.333], 42 Characteristic numbers, 86 charge conservation [7.39], 139 dimensions, 16 elementary, 6, 7 force between two [7.119], 145 Hamiltonian [7.138], 146 main January 23, 2006 16:6 Index to mass ratio of electron, 8 charge density dimensions, 16 free [7.57], 140 induced [7.84], 142 Lorentz transformation, 141 charge distribution electric field from [7.6], 136 energy of [7.133], 146 charge-sheet (electric field) [7.32], 138 Chebyshev equation [2.349], 43 Chebyshev inequality [2.150], 30 chemical potential definition [5.28], 108 from partition function [5.119], 115 Cherenkov cone angle [7.246], 156 Cherenkov radiation, 156 χE (electric susceptibility) [7.87], 142 χH , χB (magnetic susceptibility) [7.103], 143 chi-squared (χ2 ) distribution [2.553], 58 Christoffel symbols [3.49], 67 circle (arc of) centre of mass [3.173], 76 area [2.262], 37 perimeter [2.261], 37 circular aperture Fraunhofer diffraction [8.40], 165 Fresnel diffraction [8.50], 166 circular polarisation, 170 circulation [3.287], 84 civil time [9.4], 177 Clapeyron equation [5.50], 109 classical electron radius, 8 Classical thermodynamics, 106 Clausius–Mossotti equation [7.93], 142 Clausius-Clapeyron equation [5.49], 109 Clebsch–Gordan coefficients, 99 Clebsch–Gordan coefficients (spin-orbit) [4.136], 100 close-packed spheres, 127 closure density (of the universe) [9.90], 184 CNO cycle, 182 coaxial cable capacitance [7.19], 137 inductance [7.24], 137 coaxial transmission line [7.181], 150 coefficient of 191 coupling [7.148], 147 finesse [8.12], 163 reflectance [7.227], 154 reflection [7.230], 154 restitution [3.127], 73 transmission [7.232], 154 transmittance [7.229], 154 coexistence curve [5.51], 109 coherence length [8.106], 172 mutual [8.97], 172 temporal [8.105], 172 time [8.106], 172 width [8.111], 172 Coherence (scalar theory), 172 cold plasmas, 157 collision broadening [8.114], 173 elastic, 73 inelastic, 73 number [5.91], 113 time (electron drift) [6.61], 132 colour excess [9.37], 179 colour index [9.36], 179 Common three-dimensional coordinate systems, 21 commutator (in uncertainty relation) [4.6], 90 Commutators, 26 Compact objects and black holes, 183 complementary error function [2.391], 45 Complex analysis, 31 complex conjugate [2.159], 30 Complex numbers, 30 complex numbers argument [2.157], 30 cartesian form [2.153], 30 conjugate [2.159], 30 logarithm [2.162], 30 modulus [2.155], 30 polar form [2.154], 30 Complex variables, 30 compound pendulum [3.182], 76 compressibility adiabatic [5.21], 107 isothermal [5.20], 107 compression modulus, see bulk modulus compression ratio [5.13], 107 Compton I main January 23, 2006 16:6 192 scattering [7.240], 155 wavelength (value), 8 wavelength [7.240], 155 Concordance model, 185 conditional probability [2.567], 59 conductance (definition), 148 conductance (dimensions), 16 conduction equation (and transport) [5.96], 113 conduction equation [2.340], 43 conductivity and resistivity [7.142], 147 dimensions, 16 direct [7.279], 158 electrical, of a plasma [7.233], 155 free electron a.c. [6.63], 132 free electron d.c. [6.62], 132 Hall [7.280], 158 conductor refractive index [7.234], 155 cone centre of mass [3.175], 76 moment of inertia [3.160], 75 surface area [2.271], 37 volume [2.272], 37 configurational entropy [5.105], 114 Conic sections, 38 conical pendulum [3.180], 76 conservation of angular momentum [4.113], 98 charge [7.39], 139 mass [3.285], 84 Constant acceleration, 68 constant of gravitation, 7 contact angle (surface tension) [3.340], 88 continuity equation (quantum physics) [4.14], 90 continuity in fluids [3.285], 84 Continuous probability distributions, 58 contravariant components in general relativity, 67 in special relativity [3.26], 65 convection (in a star) [9.64], 181 convergence and limits, 28 Conversion factors, 10 Converting between units, 10 convolution definition [2.487], 53 Index derivative [2.498], 53 discrete [2.580], 60 Laplace transform [2.516], 55 rules [2.489], 53 theorem [2.490], 53 coordinate systems, 21 coordinate transformations astronomical, 177 Galilean, 64 relativistic, 64 rotating frames [3.31], 66 Coordinate transformations (astronomical), 177 coordinates (generalised ) [3.213], 79 coordination number (cubic lattices), 127 Coriolis force [3.33], 66 Cornu spiral, 167 Cornu spiral and Fresnel integrals [8.54], 167 correlation coefficient multinormal [2.559], 58 Pearson’s r [2.546], 57 correlation intensity [8.109], 172 correlation theorem [2.494], 53 cosx and Euler’s formula [2.216], 34 series expansion [2.135], 29 cosec, see csc cschx [2.231], 34 coshx definition [2.217], 34 series expansion [2.143], 29 cosine formula planar triangles [2.249], 36 spherical triangles [2.257], 36 cosmic scale factor [9.87], 184 cosmological constant [9.89], 184 Cosmological distance measures, 185 Cosmological model parameters, 184 Cosmological models, 185 Cosmology, 184 cos−1 x, see arccosx cotx definition [2.226], 34 series expansion [2.140], 29 cothx [2.227], 34 Couette flow [3.306], 85 coulomb (unit), 4 Coulomb gauge condition [7.42], 139 main January 23, 2006 16:6 Index Coulomb logarithm [7.254], 156 Coulomb’s law [7.119], 145 couple definition [3.67], 68 dimensions, 16 electromagnetic, 145 for Couette flow [3.306], 85 on a current-loop [7.127], 145 on a magnetic dipole [7.126], 145 on a rigid body, 77 on an electric dipole [7.125], 145 twisting [3.252], 81 coupling coefficient [7.148], 147 covariance [2.558], 58 covariant components [3.26], 65 cracks (critical length) [6.25], 128 critical damping [3.199], 78 critical density (of the universe) [9.90], 184 critical frequency (synchrotron) [7.293], 159 critical point Dieterici gas [5.75], 111 van der Waals gas [5.70], 111 cross section absorption [5.175], 120 cross-correlation [2.493], 53 cross-product [2.2], 20 cross-section Breit-Wigner [4.174], 104 Mott scattering [4.180], 104 Rayleigh scattering [7.236], 155 Rutherford scattering [3.124], 72 Thomson scattering [7.238], 155 Crystal diffraction, 128 Crystal systems, 127 Crystalline structure, 126 cscx definition [2.230], 34 series expansion [2.139], 29 cschx [2.231], 34 cube electrical capacitance [7.17], 137 mensuration, 38 Cubic equations, 51 cubic expansivity [5.19], 107 Cubic lattices, 127 cubic system (crystallographic), 127 Curie temperature [7.114], 144 193 Curie’s law [7.113], 144 Curie–Weiss law [7.114], 144 Curl, 22 curl cylindrical coordinates [2.34], 22 general coordinates [2.36], 22 of curl [2.57], 23 rectangular coordinates [2.33], 22 spherical coordinates [2.35], 22 current dimensions, 16 electric [7.139], 147 law (Kirchhoff’s) [7.161], 149 magnetic flux density from [7.11], 136 probability density [4.13], 90 thermodynamic work [5.9], 106 transformation [7.165], 149 current density dimensions, 16 four-vector [7.76], 141 free [7.63], 140 free electron [6.60], 132 hole [6.89], 134 Lorentz transformation, 141 magnetic flux density [7.10], 136 curvature in differential geomtry [2.286], 39 parameter (cosmic) [9.87], 184 radius of and curvature [2.287], 39 plane curve [2.282], 39 curve length (plane curve) [2.279], 39 Curve measure, 39 Cycle efficiencies (thermodynamic), 107 cyclic permutation [2.97], 26 cyclotron frequency [7.265], 157 cylinder area [2.269], 37 capacitance [7.15], 137 moment of inertia [3.155], 75 torsional rigidity [3.253], 81 volume [2.270], 37 cylinders (adjacent) capacitance [7.21], 137 inductance [7.25], 137 cylinders (coaxial) capacitance [7.19], 137 inductance [7.24], 137 I main January 23, 2006 16:6 194 cylindrical polar coordinates, 21 D d orbitals [4.100], 97 D’Alembertian [7.78], 141 damped harmonic oscillator [3.196], 78 damping profile [8.112], 173 day (unit), 5 day of week [9.3], 177 daylight saving time [9.4], 177 de Boer parameter [6.54], 131 de Broglie relation [4.2], 90 de Broglie wavelength (thermal) [5.83], 112 de Moivre’s theorem [2.214], 34 Debye T 3 law [6.47], 130 frequency [6.41], 130 function [6.49], 130 heat capacity [6.45], 130 length [7.251], 156 number [7.253], 156 screening [7.252], 156 temperature [6.43], 130 Debye theory, 130 Debye-Waller factor [6.33], 128 deca, 5 decay constant [4.163], 103 decay law [4.163], 103 deceleration parameter [9.95], 184 deci, 5 decibel [5.144], 117 declination coordinate [9.11], 177 decrement (oscillating systems) [3.202], 78 Definite integrals, 46 degeneracy pressure [9.77], 183 degree (unit), 5 degree Celsius (unit), 4 degree kelvin [5.2], 106 degree of freedom (and equipartition), 113 degree of mutual coherence [8.99], 172 degree of polarisation [8.96], 171 degree of temporal coherence, 172 deka, 5 del operator, 21 del-squared operator, 23 del-squared operator [2.55], 23 Delta functions, 50 Index delta–star transformation, 149 densities of elements, 124 density (dimensions), 16 density of states electron [6.70], 133 particle [4.66], 94 phonon [6.44], 130 density parameters [9.94], 184 depolarising factors [7.92], 142 Derivatives (general), 40 determinant [2.79], 25 deviation (of a prism) [8.73], 169 diamagnetic moment (electron) [7.108], 144 diamagnetic susceptibility (Landau) [6.80], 133 Diamagnetism, 144 Dielectric layers, 162 Dieterici gas, 111 Dieterici gas law [5.72], 111 Differential equations, 43 differential equations (numerical solutions), 62 Differential geometry, 39 Differential operator identities, 23 differential scattering cross-section [3.124], 72 Differentiation, 40 differentiation hyperbolic functions, 41 numerical, 61 of a function of a function [2.295], 40 of a log [2.300], 40 of a power [2.292], 40 of a product [2.293], 40 of a quotient [2.294], 40 of exponential [2.301], 40 of integral [2.299], 40 of inverse functions [2.304], 40 trigonometric functions, 41 under integral sign [2.298], 40 diffraction from N slits [8.25], 164 1 slit [8.37], 165 2 slits [8.24], 164 circular aperture [8.40], 165 crystals, 128 infinite grating [8.26], 164 main January 23, 2006 16:6 Index rectangular aperture [8.39], 165 diffraction grating finite [8.25], 164 general, 164 infinite [8.26], 164 diffusion coefficient (semiconductor) [6.88], 134 diffusion equation differential equation [2.340], 43 Fick’s first law [5.93], 113 diffusion length (semiconductor) [6.94], 134 diffusivity (magnetic) [7.282], 158 dilatation (volume strain) [3.236], 80 Dimensions, 16 diode (semiconductor) [6.92], 134 dioptre number [8.68], 168 dipole antenna power flux [7.131], 146 gain [7.213], 153 total [7.132], 146 electric field [7.31], 138 energy of electric [7.136], 146 magnetic [7.137], 146 field from magnetic [7.36], 138 moment (dimensions), 17 moment of electric [7.80], 142 magnetic [7.94], 143 potential electric [7.82], 142 magnetic [7.95], 143 radiation field [7.207], 153 magnetic [9.69], 182 radiation resistance [7.209], 153 dipole moment per unit volume electric [7.83], 142 magnetic [7.97], 143 Dirac bracket, 92 Dirac delta function [2.448], 50 Dirac equation [4.183], 104 Dirac matrices [4.185], 104 Dirac notation, 92 direct conductivity [7.279], 158 directrix (of conic section), 38 195 disc, see disk discrete convolution, 60 Discrete probability distributions, 57 Discrete statistics, 57 disk Airy [8.40], 165 capacitance [7.13], 137 centre of mass of sector [3.172], 76 coaxial capacitance [7.22], 137 drag in a fluid, 85 electric field [7.28], 138 moment of inertia [3.168], 75 Dislocations and cracks, 128 dispersion diffraction grating [8.31], 164 in a plasma [7.261], 157 in fluid waves, 86 in quantum physics [4.5], 90 in waveguides [7.188], 151 intermodal (optical fibre) [8.79], 169 measure [9.70], 182 of a prism [8.76], 169 phonon (alternating springs) [6.39], 129 phonon (diatomic chain) [6.37], 129 phonon (monatomic chain) [6.34], 129 pulsar [9.72], 182 displacement, D [7.86], 142 Distance indicators, 180 Divergence, 22 divergence cylindrical coordinates [2.30], 22 general coordinates [2.32], 22 rectangular coordinates [2.29], 22 spherical coordinates [2.31], 22 theorem [2.59], 23 dodecahedron, 38 Doppler beaming [3.25], 65 effect (non-relativistic), 87 effect (relativistic) [3.22], 65 line broadening [8.116], 173 width [8.117], 173 Doppler effect, 87 dot product [2.1], 20 double factorial, 48 double pendulum [3.183], 76 Drag, 85 I main January 23, 2006 16:6 196 drag on a disk to flow [3.310], 85 on a disk ⊥ to flow [3.309], 85 on a sphere [3.308], 85 drift velocity (electron) [6.61], 132 Dulong and Petit’s law [6.46], 130 Dynamics and Mechanics, 63–88 Dynamics definitions, 68 E e (exponential constant), 9 e to 1 000 decimal places, 18 Earth (motion relative to) [3.38], 66 Earth data, 176 eccentricity of conic section, 38 of orbit [3.108], 71 of scattering hyperbola [3.120], 72 Ecliptic coordinates, 178 ecliptic latitude [9.14], 178 ecliptic longitude [9.15], 178 Eddington limit [9.59], 181 edge dislocation [6.21], 128 effective area (antenna) [7.212], 153 distance (Fresnel diffraction) [8.48], 166 mass (in solids) [6.86], 134 wavelength [9.40], 179 efficiency heat engine [5.10], 107 heat pump [5.12], 107 Otto cycle [5.13], 107 refrigerator [5.11], 107 Ehrenfest’s equations [5.53], 109 Ehrenfest’s theorem [4.30], 91 eigenfunctions (quantum) [4.28], 91 Einstein A coefficient [8.119], 173 B coefficients [8.118], 173 diffusion equation [5.98], 113 field equation [3.59], 67 lens (rings) [9.50], 180 tensor [3.58], 67 Einstein - de Sitter model, 185 Einstein coefficients, 173 elastic collisions, 73 media (isotropic), 81 Index modulus (longitudinal) [3.241], 81 modulus [3.234], 80 potential energy [3.235], 80 elastic scattering, 72 Elastic wave velocities, 82 Elasticity, 80 Elasticity definitions (general), 80 Elasticity definitions (simple), 80 electric current [7.139], 147 electric dipole, see dipole electric displacement (dimensions), 16 electric displacement, D [7.86], 142 electric field around objects, 138 energy density [7.128], 146 static, 136 thermodynamic work [5.7], 106 wave equation [7.193], 152 electric field from A and φ [7.41], 139 charge distribution [7.6], 136 charge-sheet [7.32], 138 dipole [7.31], 138 disk [7.28], 138 line charge [7.29], 138 point charge [7.5], 136 sphere [7.27], 138 waveguide [7.190], 151 wire [7.29], 138 electric field strength (dimensions), 16 Electric fields, 138 electric polarisability (dimensions), 16 electric polarisation (dimensions), 16 electric potential from a charge density [7.46], 139 Lorentz transformation [7.75], 141 of a moving charge [7.48], 139 short dipole [7.82], 142 electric potential difference (dimensions), 16 electric susceptibility, χE [7.87], 142 electrical conductivity, see conductivity Electrical impedance, 148 electrical permittivity, , r [7.90], 142 electromagnet (magnetic flux density) [7.38], 138 electromagnetic boundary conditions, 144 constants, 7 main January 23, 2006 16:6 Index fields, 139 wave speed [7.196], 152 waves in media, 152 electromagnetic coupling constant, see fine structure constant Electromagnetic energy, 146 Electromagnetic fields (general), 139 Electromagnetic force and torque, 145 Electromagnetic propagation in cold plasmas, 157 Electromagnetism, 135–160 electron charge, 6, 7 density of states [6.70], 133 diamagnetic moment [7.108], 144 drift velocity [6.61], 132 g-factor [4.143], 100 gyromagnetic ratio (value), 8 gyromagnetic ratio [4.140], 100 heat capacity [6.76], 133 intrinsic magnetic moment [7.109], 144 mass, 6 radius (equation) [7.238], 155 radius (value), 8 scattering cross-section [7.238], 155 spin magnetic moment [4.143], 100 thermal velocity [7.257], 156 velocity in conductors [6.85], 134 Electron constants, 8 Electron scattering processes, 155 electron volt (unit), 5 electron volt (value), 6 Electrons in solids, 132 electrostatic potential [7.1], 136 Electrostatics, 136 elementary charge, 6, 7 elements (periodic table of), 124 ellipse, 38 (semi) centre of mass [3.178], 76 area [2.267], 37 moment of inertia [3.166], 75 perimeter [2.266], 37 semi-latus-rectum [3.109], 71 semi-major axis [3.106], 71 semi-minor axis [3.107], 71 ellipsoid moment of inertia of solid [3.163], 75 197 the moment of inertia [3.147], 74 volume [2.268], 37 elliptic integrals [2.397], 45 elliptical orbit [3.104], 71 Elliptical polarisation, 170 elliptical polarisation [8.80], 170 ellipticity [8.82], 170 E = mc2 [3.72], 68 emission coefficient [5.174], 120 emission spectrum [7.291], 159 emissivity [5.193], 121 energy density blackbody [5.192], 121 dimensions, 16 elastic wave [3.281], 83 electromagnetic [7.128], 146 radiant [5.148], 118 spectral [5.173], 120 dimensions, 16 dissipated in resistor [7.155], 148 distribution (Maxwellian) [5.85], 112 elastic [3.235], 80 electromagnetic, 146 equipartition [5.100], 113 Fermi [5.122], 115 first law of thermodynamics [5.3], 106 Galilean transformation [3.6], 64 kinetic , see kinetic energy Lorentz transformation [3.19], 65 loss after collision [3.128], 73 mass relation [3.20], 65 of capacitive assembly [7.134], 146 of capacitor [7.153], 148 of charge distribution [7.133], 146 of electric dipole [7.136], 146 of inductive assembly [7.135], 146 of inductor [7.154], 148 of magnetic dipole [7.137], 146 of orbit [3.100], 71 potential , see potential energy relativistic rest [3.72], 68 rotational kinetic rigid body [3.142], 74 w.r.t. principal axes [3.145], 74 thermodynamic work, 106 Energy in capacitors, inductors, and resistors, 148 I main January 23, 2006 16:6 198 energy-time uncertainty relation [4.8], 90 Ensemble probabilities, 114 enthalpy definition [5.30], 108 Joule-Kelvin expansion [5.27], 108 entropy Boltzmann formula [5.105], 114 change in Joule expansion [5.64], 110 experimental [5.4], 106 fluctuations [5.135], 116 from partition function [5.117], 115 Gibbs formula [5.106], 114 of a monatomic gas [5.83], 112 entropy (dimensions), 16 , r (electrical permittivity) [7.90], 142 Equation conversion: SI to Gaussian units, 135 equation of state Dieterici gas [5.72], 111 ideal gas [5.57], 110 monatomic gas [5.78], 112 van der Waals gas [5.67], 111 equipartition theorem [5.100], 113 error function [2.390], 45 errors, 60 escape velocity [3.91], 70 estimator kurtosis [2.545], 57 mean [2.541], 57 skewness [2.544], 57 standard deviation [2.543], 57 variance [2.542], 57 Euler angles [2.101], 26 constant expression [2.119], 27 value, 9 differential equation [2.350], 43 formula [2.216], 34 relation, 38 strut [3.261], 82 Euler’s equation (fluids) [3.289], 84 Euler’s equations (rigid bodies) [3.186], 77 Euler’s method (for ordinary differential equations) [2.596], 62 Euler-Lagrange equation and Lagrangians [3.214], 79 Index calculus of variations [2.334], 42 even functions, 53 Evolutionary timescales, 181 exa, 5 exhaust velocity (of a rocket) [3.93], 70 exitance blackbody [5.191], 121 luminous [5.162], 119 radiant [5.150], 118 exp(x) [2.132], 29 expansion coefficient [5.19], 107 Expansion processes, 108 expansivity [5.19], 107 Expectation value, 91 expectation value Dirac notation [4.37], 92 from a wavefunction [4.25], 91 explosions [3.331], 87 exponential distribution [2.551], 58 integral [2.394], 45 series expansion [2.132], 29 exponential constant (e), 9 extraordinary modes [7.271], 157 extrema [2.335], 42 F f-number [8.69], 168 Fabry-Perot etalon chromatic resolving power [8.21], 163 free spectral range [8.23], 163 fringe width [8.19], 163 transmitted intensity [8.17], 163 Fabry-Perot etalon, 163 face-centred cubic structure, 127 factorial [2.409], 46 factorial (double), 48 Fahrenheit conversion [1.2], 15 faltung theorem [2.516], 55 farad (unit), 4 Faraday constant, 6, 9 Faraday constant (dimensions), 16 Faraday rotation [7.273], 157 Faraday’s law [7.55], 140 fcc structure, 127 Feigenbaum’s constants, 9 femto, 5 Fermat’s principle [8.63], 168 Fermi main January 23, 2006 16:6 Index energy [6.73], 133 temperature [6.74], 133 velocity [6.72], 133 wavenumber [6.71], 133 fermi (unit), 5 Fermi energy [5.122], 115 Fermi gas, 133 Fermi’s golden rule [4.162], 102 Fermi–Dirac distribution [5.121], 115 fermion statistics [5.121], 115 fibre optic acceptance angle [8.77], 169 dispersion [8.79], 169 numerical aperture [8.78], 169 Fick’s first law [5.92], 113 Fick’s second law [5.95], 113 field equations (gravitational) [3.42], 66 Field relationships, 139 fields depolarising [7.92], 142 electrochemical [6.81], 133 electromagnetic, 139 gravitational, 66 static E and B, 136 velocity [3.285], 84 Fields associated with media, 142 film reflectance [8.4], 162 fine-structure constant expression [4.75], 95 value, 6, 7 finesse (coefficient of) [8.12], 163 finesse (Fabry-Perot etalon) [8.14], 163 first law of thermodynamics [5.3], 106 fitting straight-lines, 60 fluctuating dipole interaction [6.50], 131 fluctuation of density [5.137], 116 of entropy [5.135], 116 of pressure [5.136], 116 of temperature [5.133], 116 of volume [5.134], 116 probability (thermodynamic) [5.131], 116 variance (general) [5.132], 116 Fluctuations and noise, 116 Fluid dynamics, 84 fluid stress [3.299], 85 Fluid waves, 86 flux density [5.171], 120 199 flux density–redshift relation [9.99], 185 flux linked [7.149], 147 flux of molecules through a plane [5.91], 113 flux–magnitude relation [9.32], 179 focal length [8.64], 168 focus (of conic section), 38 force and acoustic impedance [3.276], 83 and stress [3.228], 80 between two charges [7.119], 145 between two currents [7.120], 145 between two masses [3.40], 66 central [4.113], 98 centrifugal [3.35], 66 Coriolis [3.33], 66 critical compression [3.261], 82 definition [3.63], 68 dimensions, 16 electromagnetic, 145 Newtonian [3.63], 68 on charge in a field [7.122], 145 current in a field [7.121], 145 electric dipole [7.123], 145 magnetic dipole [7.124], 145 sphere (potential flow) [3.298], 84 sphere (viscous drag) [3.308], 85 relativistic [3.71], 68 unit, 4 Force, torque, and energy, 145 Forced oscillations, 78 form factor [6.30], 128 formula (the) [2.455], 50 Foucault’s pendulum [3.39], 66 four-parts formula [2.259], 36 four-scalar product [3.27], 65 four-vector electromagnetic [7.79], 141 momentum [3.21], 65 spacetime [3.12], 64 Four-vectors, 65 Fourier series complex form [2.478], 52 real form [2.476], 52 Fourier series, 52 Fourier series and transforms, 52 Fourier symmetry relationships, 53 Fourier transform I main January 23, 2006 16:6 200 cosine [2.509], 54 definition [2.482], 52 derivatives and inverse [2.502], 54 general [2.498], 53 Gaussian [2.507], 54 Lorentzian [2.505], 54 shah function [2.510], 54 shift theorem [2.501], 54 similarity theorem [2.500], 54 sine [2.508], 54 step [2.511], 54 top hat [2.512], 54 triangle function [2.513], 54 Fourier transform, 52 Fourier transform pairs, 54 Fourier transform theorems, 53 Fourier’s law [5.94], 113 Frames of reference, 64 Fraunhofer diffraction, 164 Fraunhofer integral [8.34], 165 Fraunhofer limit [8.44], 165 free charge density [7.57], 140 free current density [7.63], 140 Free electron transport properties, 132 free energy [5.32], 108 free molecular flow [5.99], 113 Free oscillations, 78 free space impedance [7.197], 152 free spectral range Fabry Perot etalon [8.23], 163 laser cavity [8.124], 174 free-fall timescale [9.53], 181 Frenet’s formulas [2.291], 39 frequency (dimensions), 16 Fresnel diffraction Cornu spiral [8.54], 167 edge [8.56], 167 long slit [8.58], 167 rectangular aperture [8.62], 167 Fresnel diffraction, 166 Fresnel Equations, 154 Fresnel half-period zones [8.49], 166 Fresnel integrals and the Cornu spiral [8.52], 167 definition [2.392], 45 in diffraction [8.54], 167 Fresnel zones, 166 Fresnel-Kirchhoff formula Index plane waves [8.45], 166 spherical waves [8.47], 166 Friedmann equations [9.89], 184 fringe visibility [8.101], 172 fringes (Moiré), 35 Froude number [3.312], 86 G g-factor electron, 8 Landé [4.146], 100 muon, 9 gain in decibels [5.144], 117 galactic coordinates [9.20], 178 latitude [9.21], 178 longitude [9.22], 178 Galactic coordinates, 178 Galilean transformation of angular momentum [3.5], 64 of kinetic energy [3.6], 64 of momentum [3.4], 64 of time and position [3.2], 64 of velocity [3.3], 64 Galilean transformations, 64 Gamma function, 46 gamma function and other integrals [2.395], 45 definition [2.407], 46 gas adiabatic expansion [5.58], 110 adiabatic lapse rate [3.294], 84 constant, 6, 9, 86, 110 Dieterici, 111 Doppler broadened [8.116], 173 flow [3.292], 84 giant (astronomical data), 176 ideal equation of state [5.57], 110 ideal heat capacities, 113 ideal, or perfect, 110 internal energy (ideal) [5.62], 110 isothermal expansion [5.63], 110 linear absorption coefficient [5.175], 120 molecular flow [5.99], 113 monatomic, 112 paramagnetism [7.112], 144 pressure broadened [8.115], 173 speed of sound [3.318], 86 main January 23, 2006 16:6 Index temperature scale [5.1], 106 Van der Waals, 111 Gas equipartition, 113 Gas laws, 110 gauge condition Coulomb [7.42], 139 Lorenz [7.43], 139 Gaunt factor [7.299], 160 Gauss’s law [7.51], 140 lens formula [8.64], 168 theorem [2.59], 23 Gaussian electromagnetism, 135 Fourier transform of [2.507], 54 integral [2.398], 46 light [8.110], 172 optics, 168 probability distribution k-dimensional [2.556], 58 1-dimensional [2.552], 58 Geiger’s law [4.169], 103 Geiger-Nuttall rule [4.170], 103 General constants, 7 General relativity, 67 generalised coordinates [3.213], 79 Generalised dynamics, 79 generalised momentum [3.218], 79 geodesic deviation [3.56], 67 geodesic equation [3.54], 67 geometric distribution [2.548], 57 mean [2.109], 27 progression [2.107], 27 Geometrical optics, 168 Gibbs constant (value), 9 distribution [5.113], 114 entropy [5.106], 114 free energy [5.35], 108 Gibbs’s phase rule [5.54], 109 Gibbs–Helmholtz equations, 109 Gibbs-Duhem relation [5.38], 108 giga, 5 golden mean (value), 9 golden rule (Fermi’s) [4.162], 102 Gradient, 21 gradient cylindrical coordinates [2.26], 21 201 general coordinates [2.28], 21 rectangular coordinates [2.25], 21 spherical coordinates [2.27], 21 gram (use in SI), 5 grand canonical ensemble [5.113], 114 grand partition function [5.112], 114 grand potential definition [5.37], 108 from grand partition function [5.115], 115 grating dispersion [8.31], 164 formula [8.27], 164 resolving power [8.30], 164 Gratings, 164 Gravitation, 66 gravitation field from a sphere [3.44], 66 general relativity, 67 Newton’s law [3.40], 66 Newtonian, 71 Newtonian field equations [3.42], 66 gravitational collapse [9.53], 181 constant, 6, 7, 16 lens [9.50], 180 potential [3.42], 66 redshift [9.74], 183 wave radiation [9.75], 183 Gravitationally bound orbital motion, 71 gravity and motion on Earth [3.38], 66 waves (on a fluid surface) [3.320], 86 gray (unit), 4 Greek alphabet, 18 Green’s first theorem [2.62], 23 Green’s second theorem [2.63], 23 Greenwich sidereal time [9.6], 177 Gregory’s series [2.141], 29 greybody [5.193], 121 group speed (wave) [3.327], 87 Grüneisen parameter [6.56], 131 gyro-frequency [7.265], 157 gyro-radius [7.268], 157 gyromagnetic ratio definition [4.138], 100 electron [4.140], 100 I main January 23, 2006 16:6 202 proton (value), 8 gyroscopes, 77 gyroscopic limit [3.193], 77 nutation [3.194], 77 precession [3.191], 77 stability [3.192], 77 H H (magnetic field strength) [7.100], 143 half-life (nuclear decay) [4.164], 103 half-period zones (Fresnel) [8.49], 166 Hall coefficient (dimensions), 16 conductivity [7.280], 158 effect and coefficient [6.67], 132 voltage [6.68], 132 Hamilton’s equations [3.220], 79 Hamilton’s principal function [3.213], 79 Hamilton-Jacobi equation [3.227], 79 Hamiltonian charged particle (Newtonian) [7.138], 146 charged particle [3.223], 79 definition [3.219], 79 of a particle [3.222], 79 quantum mechanical [4.21], 91 Hamiltonian (dimensions), 16 Hamiltonian dynamics, 79 Hamming window [2.584], 60 Hanbury Brown and Twiss interferometry, 172 Hanning window [2.583], 60 harmonic mean [2.110], 27 Harmonic oscillator, 95 harmonic oscillator damped [3.196], 78 energy levels [4.68], 95 entropy [5.108], 114 forced [3.204], 78 mean energy [6.40], 130 Hartree energy [4.76], 95 Heat capacities, 107 heat capacity (dimensions), 16 heat capacity in solids Debye [6.45], 130 free electron [6.76], 133 heat capacity of a gas Cp − CV [5.17], 107 Index constant pressure [5.15], 107 constant volume [5.14], 107 for f degrees of freedom, 113 ratio (γ) [5.18], 107 heat conduction/diffusion equation differential equation [2.340], 43 Fick’s second law [5.96], 113 heat engine efficiency [5.10], 107 heat pump efficiency [5.12], 107 heavy beam [3.260], 82 hectare, 12 hecto, 5 Heisenberg uncertainty relation [4.7], 90 Helmholtz equation [2.341], 43 Helmholtz free energy definition [5.32], 108 from partition function [5.114], 115 hemisphere (centre of mass) [3.170], 76 hemispherical shell (centre of mass) [3.171], 76 henry (unit), 4 Hermite equation [2.346], 43 Hermite polynomials [4.70], 95 Hermitian conjugate operator [4.17], 91 matrix [2.73], 24 symmetry, 53 Heron’s formula [2.253], 36 herpolhode, 63, 77 hertz (unit), 4 Hertzian dipole [7.207], 153 hexagonal system (crystallographic), 127 High energy and nuclear physics, 103 Hohmann cotangential transfer [3.98], 70 hole current density [6.89], 134 Hooke’s law [3.230], 80 l’Hôpital’s rule [2.131], 28 Horizon coordinates, 177 hour (unit), 5 hour angle [9.8], 177 Hubble constant (dimensions), 16 Hubble constant [9.85], 184 Hubble law as a distance indicator [9.45], 180 in cosmology [9.83], 184 hydrogen atom eigenfunctions [4.80], 96 energy [4.81], 96 main January 23, 2006 16:6 Index Schrödinger equation [4.79], 96 Hydrogenic atoms, 95 Hydrogenlike atoms – Schrödinger solution, 96 hydrostatic compression [3.238], 80 condition [3.293], 84 equilibrium (of a star) [9.61], 181 hyperbola, 38 Hyperbolic derivatives, 41 hyperbolic motion, 72 Hyperbolic relationships, 33 I I (Stokes parameter) [8.89], 171 icosahedron, 38 Ideal fluids, 84 Ideal gas, 110 ideal gas adiabatic equations [5.58], 110 internal energy [5.62], 110 isothermal reversible expansion [5.63], 110 law [5.57], 110 speed of sound [3.318], 86 Identical particles, 115 illuminance (definition) [5.164], 119 illuminance (dimensions), 16 Image charges, 138 impedance acoustic [3.276], 83 dimensions, 17 electrical, 148 transformation [7.166], 149 impedance of capacitor [7.159], 148 coaxial transmission line [7.181], 150 electromagnetic wave [7.198], 152 forced harmonic oscillator [3.212], 78 free space definition [7.197], 152 value, 7 inductor [7.160], 148 lossless transmission line [7.174], 150 lossy transmission line [7.175], 150 microstrip line [7.184], 150 open-wire transmission line [7.182], 150 203 paired strip transmission line [7.183], 150 terminated transmission line [7.178], 150 waveguide TE modes [7.189], 151 TM modes [7.188], 151 impedances in parallel [7.158], 148 in series [7.157], 148 impulse (dimensions), 17 impulse (specific) [3.92], 70 incompressible flow, 84, 85 indefinite integrals, 44 induced charge density [7.84], 142 Inductance, 137 inductance dimensions, 17 energy [7.154], 148 energy of an assembly [7.135], 146 impedance [7.160], 148 mutual definition [7.147], 147 energy [7.135], 146 self [7.145], 147 voltage across [7.146], 147 inductance of cylinders (coaxial) [7.24], 137 solenoid [7.23], 137 wire loop [7.26], 137 wires (parallel) [7.25], 137 induction equation (MHD) [7.282], 158 inductor, see inductance Inelastic collisions, 73 Inequalities, 30 inertia tensor [3.136], 74 inner product [2.1], 20 Integration, 44 integration (numerical), 61 integration by parts [2.354], 44 intensity correlation [8.109], 172 luminous [5.166], 119 of interfering beams [8.100], 172 radiant [5.154], 118 specific [5.171], 120 Interference, 162 interference and coherence [8.100], 172 intermodal dispersion (optical fibre) [8.79], I main January 23, 2006 16:6 204 169 internal energy definition [5.28], 108 from partition function [5.116], 115 ideal gas [5.62], 110 Joule’s law [5.55], 110 monatomic gas [5.79], 112 interval (in general relativity) [3.45], 67 invariable plane, 63, 77 inverse Compton scattering [7.239], 155 Inverse hyperbolic functions, 35 inverse Laplace transform [2.518], 55 inverse matrix [2.83], 25 inverse square law [3.99], 71 Inverse trigonometric functions, 34 ionic bonding [6.55], 131 irradiance (definition) [5.152], 118 irradiance (dimensions), 17 isobaric expansivity [5.19], 107 isophotal wavelength [9.39], 179 isothermal bulk modulus [5.22], 107 isothermal compressibility [5.20], 107 Isotropic elastic solids, 81 J Jacobi identity [2.93], 26 Jacobian definition [2.332], 42 in change of variable [2.333], 42 Jeans length [9.56], 181 Jeans mass [9.57], 181 Johnson noise [5.141], 117 joint probability [2.568], 59 Jones matrix [8.85], 170 Jones vectors definition [8.84], 170 examples [8.84], 170 Jones vectors and matrices, 170 Josephson frequency-voltage ratio, 7 joule (unit), 4 Joule expansion (and Joule coefficient) [5.25], 108 Joule expansion (entropy change) [5.64], 110 Joule’s law (of internal energy) [5.55], 110 Joule’s law (of power dissipation) [7.155], 148 Joule-Kelvin coefficient [5.27], 108 Index Julian centuries [9.5], 177 Julian day number [9.1], 177 Jupiter data, 176 K katal (unit), 4 Kelvin circulation theorem [3.287], 84 relation [6.83], 133 temperature conversion, 15 temperature scale [5.2], 106 wedge [3.330], 87 kelvin (SI definition), 3 kelvin (unit), 4 Kelvin-Helmholtz timescale [9.55], 181 Kepler’s laws, 71 Kepler’s problem, 71 Kerr solution (in general relativity) [3.62], 67 ket vector [4.34], 92 kilo, 5 kilogram (SI definition), 3 kilogram (unit), 4 kinematic viscosity [3.302], 85 kinematics, 63 kinetic energy definition [3.65], 68 for a rotating body [3.142], 74 Galilean transformation [3.6], 64 in the virial theorem [3.102], 71 loss after collision [3.128], 73 of a particle [3.216], 79 of monatomic gas [5.79], 112 operator (quantum) [4.20], 91 relativistic [3.73], 68 w.r.t. principal axes [3.145], 74 Kinetic theory, 112 Kirchhoff’s (radiation) law [5.180], 120 Kirchhoff’s diffraction formula, 166 Kirchhoff’s laws, 149 Klein–Nishina cross section [7.243], 155 Klein-Gordon equation [4.181], 104 Knudsen flow [5.99], 113 Kronecker delta [2.442], 50 kurtosis estimator [2.545], 57 L ladder operators (angular momentum) [4.108], 98 main January 23, 2006 16:6 Index Lagrange’s identity [2.7], 20 Lagrangian (dimensions), 17 Lagrangian dynamics, 79 Lagrangian of charged particle [3.217], 79 particle [3.216], 79 two mutually attracting bodies [3.85], 69 Laguerre equation [2.347], 43 Laguerre polynomials (associated), 96 Lamé coefficients [3.240], 81 Laminar viscous flow, 85 Landé g-factor [4.146], 100 Landau diamagnetic susceptibility [6.80], 133 Landau length [7.249], 156 Langevin function (from Brillouin fn) [4.147], 101 Langevin function [7.111], 144 Laplace equation definition [2.339], 43 solution in spherical harmonics [2.440], 49 Laplace series [2.439], 49 Laplace transform convolution [2.516], 55 definition [2.514], 55 derivative of transform [2.520], 55 inverse [2.518], 55 of derivative [2.519], 55 substitution [2.521], 55 translation [2.523], 55 Laplace transform pairs, 56 Laplace transform theorems, 55 Laplace transforms, 55 Laplace’s formula (surface tension) [3.337], 88 Laplacian cylindrical coordinates [2.46], 23 general coordinates [2.48], 23 rectangular coordinates [2.45], 23 spherical coordinates [2.47], 23 Laplacian (scalar), 23 lapse rate (adiabatic) [3.294], 84 Larmor frequency [7.265], 157 Larmor radius [7.268], 157 Larmor’s formula [7.132], 146 laser cavity Q [8.126], 174 205 cavity line width [8.127], 174 cavity modes [8.124], 174 cavity stability [8.123], 174 threshold condition [8.129], 174 Lasers, 174 latent heat [5.48], 109 lattice constants of elements, 124 Lattice dynamics, 129 Lattice forces (simple), 131 lattice plane spacing [6.11], 126 Lattice thermal expansion and conduction, 131 lattice vector [6.7], 126 latus-rectum [3.109], 71 Laue equations [6.28], 128 Laurent series [2.168], 31 LCR circuits, 147 LCR definitions, 147 least-squares fitting, 60 Legendre equation and polynomials [2.421], 47 definition [2.343], 43 Legendre polynomials, 47 Leibniz theorem [2.296], 40 length (dimensions), 17 Lennard-Jones 6-12 potential [6.52], 131 lens blooming [8.7], 162 Lenses and mirrors, 168 lensmaker’s formula [8.66], 168 Levi-Civita symbol (3-D) [2.443], 50 l’Hôpital’s rule [2.131], 28 Liénard–Wiechert potentials, 139 light (speed of), 6, 7 Limits, 28 line charge (electric field from) [7.29], 138 line fitting, 60 Line radiation, 173 line shape collisional [8.114], 173 Doppler [8.116], 173 natural [8.112], 173 line width collisional/pressure [8.115], 173 Doppler broadened [8.117], 173 laser cavity [8.127], 174 natural [8.113], 173 Schawlow-Townes [8.128], 174 linear absorption coefficient [5.175], 120 linear expansivity (definition) [5.19], 107 I main January 23, 2006 16:6 206 linear expansivity (of a crystal) [6.57], 131 linear regression, 60 linked flux [7.149], 147 liquid drop model [4.172], 103 litre (unit), 5 local civil time [9.4], 177 local sidereal time [9.7], 177 local thermodynamic equilibrium (LTE), 116, 120 ln(1 + x) (series expansion) [2.133], 29 logarithm of complex numbers [2.162], 30 logarithmic decrement [3.202], 78 London’s formula (interacting dipoles) [6.50], 131 longitudinal elastic modulus [3.241], 81 look-back time [9.96], 185 Lorentz broadening [8.112], 173 contraction [3.8], 64 factor (γ) [3.7], 64 force [7.122], 145 Lorentz (spacetime) transformations, 64 Lorentz factor (dynamical) [3.69], 68 Lorentz transformation in electrodynamics, 141 of four-vectors, 65 of momentum and energy, 65 of time and position, 64 of velocity, 64 Lorentz-Lorenz formula [7.93], 142 Lorentzian distribution [2.555], 58 Lorentzian (Fourier transform of) [2.505], 54 Lorenz constant [6.66], 132 gauge condition [7.43], 139 lumen (unit), 4 luminance [5.168], 119 luminosity distance [9.98], 185 luminosity–magnitude relation [9.31], 179 luminous density [5.160], 119 efficacy [5.169], 119 efficiency [5.170], 119 energy [5.157], 119 exitance [5.162], 119 flux [5.159], 119 Index intensity (dimensions), 17 intensity [5.166], 119 lux (unit), 4 M Mach number [3.315], 86 Mach wedge [3.328], 87 Maclaurin series [2.125], 28 Macroscopic thermodynamic variables, 115 Madelung constant (value), 9 Madelung constant [6.55], 131 magnetic diffusivity [7.282], 158 flux quantum, 6, 7 monopoles (none) [7.52], 140 permeability, µ, µr [7.107], 143 quantum number [4.131], 100 scalar potential [7.7], 136 susceptibility, χH , χB [7.103], 143 vector potential definition [7.40], 139 from J [7.47], 139 of a moving charge [7.49], 139 magnetic dipole, see dipole magnetic field around objects, 138 dimensions, 17 energy density [7.128], 146 Lorentz transformation, 141 static, 136 strength (H) [7.100], 143 thermodynamic work [5.8], 106 wave equation [7.194], 152 Magnetic fields, 138 magnetic flux (dimensions), 17 magnetic flux density (dimensions), 17 magnetic flux density from current [7.11], 136 current density [7.10], 136 dipole [7.36], 138 electromagnet [7.38], 138 line current (Biot–Savart law) [7.9], 136 solenoid (finite) [7.38], 138 solenoid (infinite) [7.33], 138 uniform cylindrical current [7.34], 138 waveguide [7.190], 151 main January 23, 2006 16:6 Index wire [7.34], 138 wire loop [7.37], 138 Magnetic moments, 100 magnetic vector potential (dimensions), 17 Magnetisation, 143 magnetisation definition [7.97], 143 dimensions, 17 isolated spins [4.151], 101 quantum paramagnetic [4.150], 101 magnetogyric ratio [4.138], 100 Magnetohydrodynamics, 158 magnetosonic waves [7.285], 158 Magnetostatics, 136 magnification (longitudinal) [8.71], 168 magnification (transverse) [8.70], 168 magnitude (astronomical) –flux relation [9.32], 179 –luminosity relation [9.31], 179 absolute [9.29], 179 apparent [9.27], 179 major axis [3.106], 71 Malus’s law [8.83], 170 Mars data, 176 mass (dimensions), 17 mass absorption coefficient [5.176], 120 mass ratio (of a rocket) [3.94], 70 Mathematical constants, 9 Mathematics, 19–62 matrices (square), 25 Matrix algebra, 24 matrix element (quantum) [4.32], 92 maxima [2.336], 42 Maxwell’s equations, 140 Maxwell’s equations (using D and H), 140 Maxwell’s relations, 109 Maxwell–Boltzmann distribution, 112 Maxwell-Boltzmann distribution mean speed [5.86], 112 most probable speed [5.88], 112 rms speed [5.87], 112 speed distribution [5.84], 112 mean arithmetic [2.108], 27 geometric [2.109], 27 harmonic [2.110], 27 mean estimator [2.541], 57 mean free path 207 and absorption coefficient [5.175], 120 Maxwell-Boltzmann [5.89], 113 mean intensity [5.172], 120 mean-life (nuclear decay) [4.165], 103 mega, 5 melting points of elements, 124 meniscus [3.339], 88 Mensuration, 35 Mercury data, 176 method of images, 138 metre (SI definition), 3 metre (unit), 4 metric elements and coordinate systems, 21 MHD equations [7.283], 158 micro, 5 microcanonical ensemble [5.109], 114 micron (unit), 5 microstrip line (impedance) [7.184], 150 Miller-Bravais indices [6.20], 126 milli, 5 minima [2.337], 42 minimum deviation (of a prism) [8.74], 169 minor axis [3.107], 71 minute (unit), 5 mirror formula [8.67], 168 Miscellaneous, 18 mobility (dimensions), 17 mobility (in conductors) [6.88], 134 modal dispersion (optical fibre) [8.79], 169 modified Bessel functions [2.419], 47 modified Julian day number [9.2], 177 modulus (of a complex number) [2.155], 30 Moiré fringes, 35 molar gas constant (dimensions), 17 molar volume, 9 mole (SI definition), 3 mole (unit), 4 molecular flow [5.99], 113 moment electric dipole [7.81], 142 magnetic dipole [7.94], 143 magnetic dipole [7.95], 143 moment of area [3.258], 82 moment of inertia I main January 23, 2006 16:6 208 cone [3.160], 75 cylinder [3.155], 75 dimensions, 17 disk [3.168], 75 ellipsoid [3.163], 75 elliptical lamina [3.166], 75 rectangular cuboid [3.158], 75 sphere [3.152], 75 spherical shell [3.153], 75 thin rod [3.150], 75 triangular plate [3.169], 75 two-body system [3.83], 69 moment of inertia ellipsoid [3.147], 74 Moment of inertia tensor, 74 moment of inertia tensor [3.136], 74 Moments of inertia, 75 momentum definition [3.64], 68 dimensions, 17 generalised [3.218], 79 relativistic [3.70], 68 Momentum and energy transformations, 65 Monatomic gas, 112 monatomic gas entropy [5.83], 112 equation of state [5.78], 112 heat capacity [5.82], 112 internal energy [5.79], 112 pressure [5.77], 112 monoclinic system (crystallographic), 127 Moon data, 176 motif [6.31], 128 motion under constant acceleration, 68 Mott scattering formula [4.180], 104 µ, µr (magnetic permeability) [7.107], 143 multilayer films (in optics) [8.8], 162 multimode dispersion (optical fibre) [8.79], 169 multiplicity (quantum) j [4.133], 100 l [4.112], 98 multistage rocket [3.95], 70 Multivariate normal distribution, 58 Muon and tau constants, 9 muon physical constants, 9 mutual capacitance [7.134], 146 inductance (definition) [7.147], 147 Index inductance (energy) [7.135], 146 mutual coherence function [8.97], 172 N nabla, 21 Named integrals, 45 nano, 5 natural broadening profile [8.112], 173 natural line width [8.113], 173 Navier-Stokes equation [3.301], 85 nearest neighbour distances, 127 Neptune data, 176 neutron Compton wavelength, 8 gyromagnetic ratio, 8 magnetic moment, 8 mass, 8 molar mass, 8 Neutron constants, 8 neutron star degeneracy pressure [9.77], 183 newton (unit), 4 Newton’s law of Gravitation [3.40], 66 Newton’s lens formula [8.65], 168 Newton’s rings, 162 Newton’s rings [8.1], 162 Newton-Raphson method [2.593], 61 Newtonian gravitation, 66 noggin, 13 Noise, 117 noise figure [5.143], 117 Johnson [5.141], 117 Nyquist’s theorem [5.140], 117 shot [5.142], 117 temperature [5.140], 117 normal (unit principal) [2.284], 39 normal distribution [2.552], 58 normal plane, 39 Nuclear binding energy, 103 Nuclear collisions, 104 Nuclear decay, 103 nuclear decay law [4.163], 103 nuclear magneton, 7 number density (dimensions), 17 numerical aperture (optical fibre) [8.78], 169 Numerical differentiation, 61 Numerical integration, 61 main January 23, 2006 16:6 209 Index Numerical methods, 60 Numerical solutions to f(x) = 0, 61 Numerical solutions to ordinary differential equations, 62 nutation [3.194], 77 Nyquist’s theorem [5.140], 117 orthorhombic system (crystallographic), 127 Oscillating systems, 78 osculating plane, 39 Otto cycle efficiency [5.13], 107 overdamping [3.201], 78 O p orbitals [4.95], 97 P-waves [3.263], 82 packing fraction (of spheres), 127 paired strip (impedance of) [7.183], 150 parabola, 38 parabolic motion [3.88], 69 parallax (astronomical) [9.46], 180 parallel axis theorem [3.140], 74 parallel impedances [7.158], 148 parallel wire feeder (inductance) [7.25], 137 paramagnetic susceptibility (Pauli) [6.79], 133 paramagnetism (quantum), 101 Paramagnetism and diamagnetism, 144 parity operator [4.24], 91 Parseval’s relation [2.495], 53 Parseval’s theorem integral form [2.496], 53 series form [2.480], 52 Partial derivatives, 42 partial widths (and total width) [4.176], 104 Particle in a rectangular box, 94 Particle motion, 68 partition function atomic [5.126], 116 definition [5.110], 114 macroscopic variables from, 115 pascal (unit), 4 Pauli matrices, 26 Pauli matrices [2.94], 26 Pauli paramagnetic susceptibility [6.79], 133 Pauli spin matrices (and Weyl eqn.) [4.182], 104 Pearson’s r [2.546], 57 Peltier effect [6.82], 133 pendulum compound [3.182], 76 conical [3.180], 76 double [3.183], 76 Oblique elastic collisions, 73 obliquity factor (diffraction) [8.46], 166 obliquity of the ecliptic [9.13], 178 observable (quantum physics) [4.5], 90 Observational astrophysics, 179 octahedron, 38 odd functions, 53 ODEs (numerical solutions), 62 ohm (unit), 4 Ohm’s law (in MHD) [7.281], 158 Ohm’s law [7.140], 147 opacity [5.176], 120 open-wire transmission line [7.182], 150 operator angular momentum and other operators [4.23], 91 definitions [4.105], 98 Hamiltonian [4.21], 91 kinetic energy [4.20], 91 momentum [4.19], 91 parity [4.24], 91 position [4.18], 91 time dependence [4.27], 91 Operators, 91 optic branch (phonon) [6.37], 129 optical coating [8.8], 162 optical depth [5.177], 120 Optical fibres, 169 optical path length [8.63], 168 Optics, 161–174 Orbital angular dependence, 97 Orbital angular momentum, 98 orbital motion, 71 orbital radius (Bohr atom) [4.73], 95 order (in diffraction) [8.26], 164 ordinary modes [7.271], 157 orthogonal matrix [2.85], 25 orthogonality associated Legendre functions [2.434], 48 Legendre polynomials [2.424], 47 P I main January 23, 2006 16:6 210 simple [3.179], 76 torsional [3.181], 76 Pendulums, 76 perfect gas, 110 pericentre (of an orbit) [3.110], 71 perimeter of circle [2.261], 37 of ellipse [2.266], 37 Perimeter, area, and volume, 37 period (of an orbit) [3.113], 71 Periodic table, 124 permeability dimensions, 17 magnetic [7.107], 143 of vacuum, 6, 7 permittivity dimensions, 17 electrical [7.90], 142 of vacuum, 6, 7 permutation tensor (ijk ) [2.443], 50 perpendicular axis theorem [3.148], 74 Perturbation theory, 102 peta, 5 petrol engine efficiency [5.13], 107 phase object (diffraction by weak) [8.43], 165 phase rule (Gibbs’s) [5.54], 109 phase speed (wave) [3.325], 87 Phase transitions, 109 Phonon dispersion relations, 129 phonon modes (mean energy) [6.40], 130 Photometric wavelengths, 179 Photometry, 119 photon energy [4.3], 90 Physical constants, 6 Pi (π) to 1 000 decimal places, 18 Pi (π), 9 pico, 5 pipe (flow of fluid along) [3.305], 85 pipe (twisting of) [3.255], 81 pitch angle, 159 Planck constant, 6, 7 constant (dimensions), 17 function [5.184], 121 length, 7 mass, 7 time, 7 Planck-Einstein relation [4.3], 90 Index plane polarisation, 170 Plane triangles, 36 plane wave expansion [2.427], 47 Planetary bodies, 180 Planetary data, 176 plasma beta [7.278], 158 dispersion relation [7.261], 157 frequency [7.259], 157 group velocity [7.264], 157 phase velocity [7.262], 157 refractive index [7.260], 157 Plasma physics, 156 Platonic solids, 38 Pluto data, 176 p-n junction [6.92], 134 Poincaré sphere, 171 point charge (electric field from) [7.5], 136 Poiseuille flow [3.305], 85 Poisson brackets [3.224], 79 Poisson distribution [2.549], 57 Poisson ratio and elastic constants [3.251], 81 simple definition [3.231], 80 Poisson’s equation [7.3], 136 polarisability [7.91], 142 Polarisation, 170 Polarisation, 142 polarisation (electrical, per unit volume) [7.83], 142 polarisation (of radiation) angle [8.81], 170 axial ratio [8.88], 171 degree of [8.96], 171 elliptical [8.80], 170 ellipticity [8.82], 170 reflection law [7.218], 154 polarisers [8.85], 170 polhode, 63, 77 Population densities, 116 potential chemical [5.28], 108 difference (and work) [5.9], 106 difference (between points) [7.2], 136 electrical [7.46], 139 electrostatic [7.1], 136 energy (elastic) [3.235], 80 energy in Hamiltonian [3.222], 79 main January 23, 2006 16:6 211 Index energy in Lagrangian [3.216], 79 field equations [7.45], 139 four-vector [7.77], 141 grand [5.37], 108 Liénard–Wiechert, 139 Lorentz transformation [7.75], 141 magnetic scalar [7.7], 136 magnetic vector [7.40], 139 Rutherford scattering [3.114], 72 thermodynamic [5.35], 108 velocity [3.296], 84 Potential flow, 84 Potential step, 92 Potential well, 93 power (dimensions), 17 power gain antenna [7.211], 153 short dipole [7.213], 153 Power series, 28 Power theorem [2.495], 53 Poynting vector (dimensions), 17 Poynting vector [7.130], 146 pp (proton-proton) chain, 182 Prandtl number [3.314], 86 precession (gyroscopic) [3.191], 77 Precession of equinoxes, 178 pressure broadening [8.115], 173 critical [5.75], 111 degeneracy [9.77], 183 dimensions, 17 fluctuations [5.136], 116 from partition function [5.118], 115 hydrostatic [3.238], 80 in a monatomic gas [5.77], 112 radiation, 152 thermodynamic work [5.5], 106 waves [3.263], 82 primitive cell [6.1], 126 primitive vectors (and lattice vectors) [6.7], 126 primitive vectors (of cubic lattices), 127 Principal axes, 74 principal moments of inertia [3.143], 74 principal quantum number [4.71], 95 principle of least action [3.213], 79 prism determining refractive index [8.75], 169 deviation [8.73], 169 dispersion [8.76], 169 minimum deviation [8.74], 169 transmission angle [8.72], 169 Prisms (dispersing), 169 probability conditional [2.567], 59 density current [4.13], 90 distributions continuous, 58 discrete, 57 joint [2.568], 59 Probability and statistics, 57 product (derivative of) [2.293], 40 product (integral of) [2.354], 44 product of inertia [3.136], 74 progression (arithmetic) [2.104], 27 progression (geometric) [2.107], 27 Progressions and summations, 27 projectiles, 69 propagation in cold plasmas, 157 Propagation in conducting media, 155 Propagation of elastic waves, 83 Propagation of light, 65 proper distance [9.97], 185 Proton constants, 8 proton mass, 6 proton-proton chain, 182 pulsar braking index [9.66], 182 characteristic age [9.67], 182 dispersion [9.72], 182 magnetic dipole radiation [9.69], 182 Pulsars, 182 pyramid (centre of mass) [3.175], 76 pyramid (volume) [2.272], 37 Q Q, see quality factor Q (Stokes parameter) [8.90], 171 Quadratic equations, 50 quadrature, 61 quadrature (integration), 44 quality factor Fabry-Perot etalon [8.14], 163 forced harmonic oscillator [3.211], 78 free harmonic oscillator [3.203], 78 laser cavity [8.126], 174 I main January 23, 2006 16:6 212 LCR circuits [7.152], 148 quantum concentration [5.83], 112 Quantum definitions, 90 Quantum paramagnetism, 101 Quantum physics, 89–104 Quantum uncertainty relations, 90 quarter-wave condition [8.3], 162 quarter-wave plate [8.85], 170 quartic minimum, 42 R Radial forms, 22 radian (unit), 4 radiance [5.156], 118 radiant energy [5.145], 118 energy density [5.148], 118 exitance [5.150], 118 flux [5.147], 118 intensity (dimensions), 17 intensity [5.154], 118 radiation blackbody [5.184], 121 bremsstrahlung [7.297], 160 Cherenkov [7.247], 156 field of a dipole [7.207], 153 flux from dipole [7.131], 146 resistance [7.209], 153 synchrotron [7.287], 159 Radiation pressure, 152 radiation pressure extended source [7.203], 152 isotropic [7.200], 152 momentum density [7.199], 152 point source [7.204], 152 specular reflection [7.202], 152 Radiation processes, 118 Radiative transfer, 120 radiative transfer equation [5.179], 120 radiative transport (in stars) [9.63], 181 radioactivity, 103 Radiometry, 118 radius of curvature definition [2.282], 39 in bending [3.258], 82 relation to curvature [2.287], 39 radius of gyration (see footnote), 75 Ramsauer effect [4.52], 93 Random walk, 59 Index random walk Brownian motion [5.98], 113 one-dimensional [2.562], 59 three-dimensional [2.564], 59 range (of projectile) [3.90], 69 Rankine conversion [1.3], 15 Rankine-Hugoniot shock relations [3.334], 87 Rayleigh distribution [2.554], 58 resolution criterion [8.41], 165 scattering [7.236], 155 theorem [2.496], 53 Rayleigh-Jeans law [5.187], 121 reactance (definition), 148 reciprocal lattice vector [6.8], 126 matrix [2.83], 25 vectors [2.16], 20 reciprocity [2.330], 42 Recognised non-SI units, 5 rectangular aperture diffraction [8.39], 165 rectangular coordinates, 21 rectangular cuboid moment of inertia [3.158], 75 rectifying plane, 39 recurrence relation associated Legendre functions [2.433], 48 Legendre polynomials [2.423], 47 redshift –flux density relation [9.99], 185 cosmological [9.86], 184 gravitational [9.74], 183 Reduced mass (of two interacting bodies), 69 reduced units (thermodynamics) [5.71], 111 reflectance coefficient and Fresnel equations [7.227], 154 dielectric film [8.4], 162 dielectric multilayer [8.8], 162 reflection coefficient acoustic [3.283], 83 dielectric boundary [7.230], 154 potential barrier [4.58], 94 potential step [4.41], 92 potential well [4.48], 93 main January 23, 2006 16:6 213 Index transmission line [7.179], 150 reflection grating [8.29], 164 reflection law [7.216], 154 Reflection, refraction, and transmission, 154 refraction law (Snell’s) [7.217], 154 refractive index of dielectric medium [7.195], 152 ohmic conductor [7.234], 155 plasma [7.260], 157 refrigerator efficiency [5.11], 107 regression (linear), 60 relativistic beaming [3.25], 65 relativistic doppler effect [3.22], 65 Relativistic dynamics, 68 Relativistic electrodynamics, 141 Relativistic wave equations, 104 relativity (general), 67 relativity (special), 64 relaxation time and electron drift [6.61], 132 in a conductor [7.156], 148 in plasmas, 156 residuals [2.572], 60 Residue theorem [2.170], 31 residues (in complex analysis), 31 resistance and impedance, 148 dimensions, 17 energy dissipated in [7.155], 148 radiation [7.209], 153 resistivity [7.142], 147 resistor, see resistance resolving power chromatic (of an etalon) [8.21], 163 of a diffraction grating [8.30], 164 Rayleigh resolution criterion [8.41], 165 resonance forced oscillator [3.209], 78 resonance lifetime [4.177], 104 resonant frequency (LCR) [7.150], 148 Resonant LCR circuits, 148 restitution (coefficient of) [3.127], 73 retarded time, 139 revolution (volume and surface of), 39 Reynolds number [3.311], 86 ribbon (twisting of) [3.256], 81 Ricci tensor [3.57], 67 Riemann tensor [3.50], 67 right ascension [9.8], 177 rigid body angular momentum [3.141], 74 kinetic energy [3.142], 74 Rigid body dynamics, 74 rigidity modulus [3.249], 81 ripples [3.321], 86 rms (standard deviation) [2.543], 57 Robertson-Walker metric [9.87], 184 Roche limit [9.43], 180 rocket equation [3.94], 70 Rocketry, 70 rod bending, 82 moment of inertia [3.150], 75 stretching [3.230], 80 waves in [3.271], 82 Rodrigues’ formula [2.422], 47 Roots of quadratic and cubic equations, 50 Rossby number [3.316], 86 rot (curl), 22 Rotating frames, 66 Rotation matrices, 26 rotation measure [7.273], 157 Runge Kutta method [2.603], 62 Rutherford scattering, 72 Rutherford scattering formula [3.124], 72 Rydberg constant, 6, 7 and Bohr atom [4.77], 95 dimensions, 17 Rydberg’s formula [4.78], 95 S s orbitals [4.92], 97 S-waves [3.262], 82 Sackur-Tetrode equation [5.83], 112 saddle point [2.338], 42 Saha equation (general) [5.128], 116 Saha equation (ionisation) [5.129], 116 Saturn data, 176 scalar effective mass [6.87], 134 scalar product [2.1], 20 scalar triple product [2.10], 20 scale factor (cosmic) [9.87], 184 scattering angle (Rutherford) [3.116], 72 Born approximation [4.178], 104 Compton [7.240], 155 I main January 23, 2006 16:6 214 crystal [6.32], 128 inverse Compton [7.239], 155 Klein-Nishina [7.243], 155 Mott (identical particles) [4.180], 104 potential (Rutherford) [3.114], 72 processes (electron), 155 Rayleigh [7.236], 155 Rutherford [3.124], 72 Thomson [7.238], 155 scattering cross-section, see cross-section Schawlow-Townes line width [8.128], 174 Schrödinger equation [4.15], 90 Schwarz inequality [2.152], 30 Schwarzschild geometry (in GR) [3.61], 67 Schwarzschild radius [9.73], 183 Schwarzschild’s equation [5.179], 120 screw dislocation [6.22], 128 secx definition [2.228], 34 series expansion [2.138], 29 secant method (of root-finding) [2.592], 61 sechx [2.229], 34 second (SI definition), 3 second (time interval), 4 second moment of area [3.258], 82 Sedov-Taylor shock relation [3.331], 87 selection rules (dipole transition) [4.91], 96 self-diffusion [5.93], 113 self-inductance [7.145], 147 semi-ellipse (centre of mass) [3.178], 76 semi-empirical mass formula [4.173], 103 semi-latus-rectum [3.109], 71 semi-major axis [3.106], 71 semi-minor axis [3.107], 71 semiconductor diode [6.92], 134 semiconductor equation [6.90], 134 Series expansions, 29 series impedances [7.157], 148 Series, summations, and progressions, 27 shah function (Fourier transform of) [2.510], 54 shear modulus [3.249], 81 strain [3.237], 80 viscosity [3.299], 85 waves [3.262], 82 Index shear modulus (dimensions), 17 sheet of charge (electric field) [7.32], 138 shift theorem (Fourier transform) [2.501], 54 shock Rankine-Hugoniot conditions [3.334], 87 spherical [3.331], 87 Shocks, 87 shot noise [5.142], 117 SI base unit definitions, 3 SI base units, 4 SI derived units, 4 SI prefixes, 5 SI units, 4 sidelobes (diffraction by 1-D slit) [8.38], 165 sidereal time [9.7], 177 siemens (unit), 4 sievert (unit), 4 similarity theorem (Fourier transform) [2.500], 54 simple cubic structure, 127 simple harmonic oscillator, see harmonic oscillator simple pendulum [3.179], 76 Simpson’s rule [2.586], 61 sinx and Euler’s formula [2.218], 34 series expansion [2.136], 29 sinc function [2.512], 54 sine formula planar triangles [2.246], 36 spherical triangles [2.255], 36 sinhx definition [2.219], 34 series expansion [2.144], 29 sin−1 x, see arccosx skew-symmetric matrix [2.87], 25 skewness estimator [2.544], 57 skin depth [7.235], 155 slit diffraction (broad slit) [8.37], 165 slit diffraction (Young’s) [8.24], 164 Snell’s law (acoustics) [3.284], 83 Snell’s law (electromagnetism) [7.217], 154 soap bubbles [3.337], 88 solar constant, 176 Solar data, 176 Solar system data, 176 main January 23, 2006 16:6 Index solenoid finite [7.38], 138 infinite [7.33], 138 self inductance [7.23], 137 solid angle (subtended by a circle) [2.278], 37 Solid state physics, 123–134 sound speed (in a plasma) [7.275], 158 sound, speed of [3.317], 86 space cone, 77 space frequency [3.188], 77 space impedance [7.197], 152 spatial coherence [8.108], 172 Special functions and polynomials, 46 special relativity, 64 specific charge on electron, 8 emission coefficient [5.174], 120 heat capacity, see heat capacity definition, 105 dimensions, 17 intensity (blackbody) [5.184], 121 intensity [5.171], 120 specific impulse [3.92], 70 speckle intensity distribution [8.110], 172 speckle size [8.111], 172 spectral energy density blackbody [5.186], 121 definition [5.173], 120 spectral function (synchrotron) [7.295], 159 Spectral line broadening, 173 speed (dimensions), 17 speed distribution (Maxwell-Boltzmann) [5.84], 112 speed of light (equation) [7.196], 152 speed of light (value), 6 speed of sound [3.317], 86 sphere area [2.263], 37 Brownian motion [5.98], 113 capacitance [7.12], 137 capacitance of adjacent [7.14], 137 capacitance of concentric [7.18], 137 close-packed, 127 collisions of, 73 electric field [7.27], 138 geometry on a, 36 gravitation field from a [3.44], 66 215 in a viscous fluid [3.308], 85 in potential flow [3.298], 84 moment of inertia [3.152], 75 Poincaré, 171 polarisability, 142 volume [2.264], 37 spherical Bessel function [2.420], 47 spherical cap area [2.275], 37 centre of mass [3.177], 76 volume [2.276], 37 spherical excess [2.260], 36 Spherical harmonics, 49 spherical harmonics definition [2.436], 49 Laplace equation [2.440], 49 orthogonality [2.437], 49 spherical polar coordinates, 21 spherical shell (moment of inertia) [3.153], 75 spherical surface (capacitance of near) [7.16], 137 Spherical triangles, 36 spin and total angular momentum [4.128], 100 degeneracy, 115 electron magnetic moment [4.141], 100 Pauli matrices, 26 spinning bodies, 77 spinors [4.182], 104 Spitzer conductivity [7.254], 156 spontaneous emission [8.119], 173 spring constant and wave velocity [3.272], 83 Square matrices, 25 standard deviation estimator [2.543], 57 Standard forms, 44 Star formation, 181 Star–delta transformation, 149 Static fields, 136 statics, 63 Stationary points, 42 Statistical entropy, 114 Statistical thermodynamics, 114 Stefan–Boltzmann constant, 9 Stefan–Boltzmann constant (dimensions), 17 I main January 23, 2006 16:6 216 Stefan-Boltzmann constant, 121 Stefan-Boltzmann law [5.191], 121 stellar aberration [3.24], 65 Stellar evolution, 181 Stellar fusion processes, 182 Stellar theory, 181 step function (Fourier transform of) [2.511], 54 steradian (unit), 4 stimulated emission [8.120], 173 Stirling’s formula [2.411], 46 Stokes parameters, 171 Stokes parameters [8.95], 171 Stokes’s law [3.308], 85 Stokes’s theorem [2.60], 23 Straight-line fitting, 60 strain simple [3.229], 80 tensor [3.233], 80 volume [3.236], 80 stress dimensions, 17 in fluids [3.299], 85 simple [3.228], 80 tensor [3.232], 80 stress-energy tensor and field equations [3.59], 67 perfect fluid [3.60], 67 string (waves along a stretched) [3.273], 83 Strouhal number [3.313], 86 structure factor [6.31], 128 sum over states [5.110], 114 Summary of physical constants, 6 summation formulas [2.118], 27 Sun data, 176 Sunyaev-Zel’dovich effect [9.51], 180 surface brightness (blackbody) [5.184], 121 surface of revolution [2.280], 39 Surface tension, 88 surface tension Laplace’s formula [3.337], 88 work done [5.6], 106 surface tension (dimensions), 17 surface waves [3.320], 86 survival equation (for mean free path) [5.90], 113 susceptance (definition), 148 Index susceptibility electric [7.87], 142 Landau diamagnetic [6.80], 133 magnetic [7.103], 143 Pauli paramagnetic [6.79], 133 symmetric matrix [2.86], 25 symmetric top [3.188], 77 Synchrotron radiation, 159 synodic period [9.44], 180 T tanx definition [2.220], 34 series expansion [2.137], 29 tangent [2.283], 39 tangent formula [2.250], 36 tanhx definition [2.221], 34 series expansion [2.145], 29 tan−1 x, see arctanx tau physical constants, 9 Taylor series one-dimensional [2.123], 28 three-dimensional [2.124], 28 telegraphist’s equations [7.171], 150 temperature antenna [7.215], 153 Celsius, 4 dimensions, 17 Kelvin scale [5.2], 106 thermodynamic [5.1], 106 Temperature conversions, 15 temporal coherence [8.105], 172 tensor Einstein [3.58], 67 electric susceptibility [7.87], 142 ijk [2.443], 50 fluid stress [3.299], 85 magnetic susceptibility [7.103], 143 moment of inertia [3.136], 74 Ricci [3.57], 67 Riemann [3.50], 67 strain [3.233], 80 stress [3.232], 80 tera, 5 tesla (unit), 4 tetragonal system (crystallographic), 127 tetrahedron, 38 thermal conductivity main January 23, 2006 16:6 Index diffusion equation [2.340], 43 dimensions, 17 free electron [6.65], 132 phonon gas [6.58], 131 transport property [5.96], 113 thermal de Broglie wavelength [5.83], 112 thermal diffusion [5.93], 113 thermal diffusivity [2.340], 43 thermal noise [5.141], 117 thermal velocity (electron) [7.257], 156 Thermodynamic coefficients, 107 Thermodynamic fluctuations, 116 Thermodynamic laws, 106 Thermodynamic potentials, 108 thermodynamic temperature [5.1], 106 Thermodynamic work, 106 Thermodynamics, 105–121 Thermoelectricity, 133 thermopower [6.81], 133 Thomson cross section, 8 Thomson scattering [7.238], 155 throttling process [5.27], 108 time (dimensions), 17 time dilation [3.11], 64 Time in astronomy, 177 Time series analysis, 60 Time-dependent perturbation theory, 102 Time-independent perturbation theory, 102 timescale free-fall [9.53], 181 Kelvin-Helmholtz [9.55], 181 Titius-Bode rule [9.41], 180 tonne (unit), 5 top asymmetric [3.189], 77 symmetric [3.188], 77 symmetries [3.149], 74 top hat function (Fourier transform of) [2.512], 54 Tops and gyroscopes, 77 torque, see couple Torsion, 81 torsion in a thick cylinder [3.254], 81 in a thin cylinder [3.253], 81 in an arbitrary ribbon [3.256], 81 in an arbitrary tube [3.255], 81 in differential geometry [2.288], 39 217 torsional pendulum [3.181], 76 torsional rigidity [3.252], 81 torus (surface area) [2.273], 37 torus (volume) [2.274], 37 total differential [2.329], 42 total internal reflection [7.217], 154 total width (and partial widths) [4.176], 104 trace [2.75], 25 trajectory (of projectile) [3.88], 69 transfer equation [5.179], 120 Transformers, 149 transmission coefficient Fresnel [7.232], 154 potential barrier [4.59], 94 potential step [4.42], 92 potential well [4.49], 93 transmission grating [8.27], 164 transmission line, 150 coaxial [7.181], 150 equations [7.171], 150 impedance lossless [7.174], 150 lossy [7.175], 150 input impedance [7.178], 150 open-wire [7.182], 150 paired strip [7.183], 150 reflection coefficient [7.179], 150 vswr [7.180], 150 wave speed [7.176], 150 waves [7.173], 150 Transmission line impedances, 150 Transmission line relations, 150 Transmission lines and waveguides, 150 transmittance coefficient [7.229], 154 Transport properties, 113 transpose matrix [2.70], 24 trapezoidal rule [2.585], 61 triangle area [2.254], 36 centre of mass [3.174], 76 inequality [2.147], 30 plane, 36 spherical, 36 triangle function (Fourier transform of) [2.513], 54 triclinic system (crystallographic), 127 trigonal system (crystallographic), 127 Trigonometric and hyperbolic defini- I main January 23, 2006 16:6 218 tions, 34 Trigonometric and hyperbolic formulas, 32 Trigonometric and hyperbolic integrals, 45 Trigonometric derivatives, 41 Trigonometric relationships, 32 triple-α process, 182 true anomaly [3.104], 71 tube, see pipe Tully-Fisher relation [9.49], 180 tunnelling (quantum mechanical), 94 tunnelling probability [4.61], 94 turns ratio (of transformer) [7.163], 149 two-level system (microstates of) [5.107], 114 U U (Stokes parameter) [8.92], 171 UBV magnitude system [9.36], 179 umklapp processes [6.59], 131 uncertainty relation energy-time [4.8], 90 general [4.6], 90 momentum-position [4.7], 90 number-phase [4.9], 90 underdamping [3.198], 78 unified atomic mass unit, 5, 6 uniform distribution [2.550], 58 uniform to normal distribution transformation, 58 unitary matrix [2.88], 25 units (conversion of SI to Gaussian), 135 Units, constants and conversions, 3–18 universal time [9.4], 177 Uranus data, 176 UTC [9.4], 177 V V (Stokes parameter) [8.94], 171 van der Waals equation [5.67], 111 Van der Waals gas, 111 van der Waals interaction [6.50], 131 Van-Cittert Zernicke theorem [8.108], 172 variance estimator [2.542], 57 variations, calculus of [2.334], 42 Vector algebra, 20 Vector integral transformations, 23 vector product [2.2], 20 vector triple product [2.12], 20 Vectors and matrices, 20 Index velocity (dimensions), 17 velocity distribution (Maxwell-Boltzmann) [5.84], 112 velocity potential [3.296], 84 Velocity transformations, 64 Venus data, 176 virial coefficients [5.65], 110 Virial expansion, 110 virial theorem [3.102], 71 vis-viva equation [3.112], 71 viscosity dimensions, 17 from kinetic theory [5.97], 113 kinematic [3.302], 85 shear [3.299], 85 viscous flow between cylinders [3.306], 85 between plates [3.303], 85 through a circular pipe [3.305], 85 through an annular pipe [3.307], 85 Viscous flow (incompressible), 85 volt (unit), 4 voltage across an inductor [7.146], 147 bias [6.92], 134 Hall [6.68], 132 law (Kirchhoff’s) [7.162], 149 standing wave ratio [7.180], 150 thermal noise [5.141], 117 transformation [7.164], 149 volume dimensions, 17 of cone [2.272], 37 of cube, 38 of cylinder [2.270], 37 of dodecahedron, 38 of ellipsoid [2.268], 37 of icosahedron, 38 of octahedron, 38 of parallelepiped [2.10], 20 of pyramid [2.272], 37 of revolution [2.281], 39 of sphere [2.264], 37 of spherical cap [2.276], 37 of tetrahedron, 38 of torus [2.274], 37 volume expansivity [5.19], 107 volume strain [3.236], 80 vorticity and Kelvin circulation [3.287], main January 23, 2006 16:6 219 Index 84 vorticity and potential flow [3.297], 84 vswr [7.180], 150 W wakes [3.330], 87 Warm plasmas, 156 watt (unit), 4 wave equation [2.342], 43 wave impedance acoustic [3.276], 83 electromagnetic [7.198], 152 in a waveguide [7.189], 151 Wave mechanics, 92 Wave speeds, 87 wavefunction and expectation value [4.25], 91 and probability density [4.10], 90 diffracted in 1-D [8.34], 165 hydrogenic atom [4.91], 96 perturbed [4.160], 102 Wavefunctions, 90 waveguide cut-off frequency [7.186], 151 equation [7.185], 151 impedance TE modes [7.189], 151 TM modes [7.188], 151 TEmn modes [7.190], 151 TMmn modes [7.192], 151 velocity group [7.188], 151 phase [7.187], 151 Waveguides, 151 wavelength Compton [7.240], 155 de Broglie [4.2], 90 photometric, 179 redshift [9.86], 184 thermal de Broglie [5.83], 112 waves capillary [3.321], 86 electromagnetic, 152 in a spring [3.272], 83 in a thin rod [3.271], 82 in bulk fluids [3.265], 82 in fluids, 86 in infinite isotropic solids [3.264], 82 magnetosonic [7.285], 158 on a stretched sheet [3.274], 83 on a stretched string [3.273], 83 on a thin plate [3.268], 82 sound [3.317], 86 surface (gravity) [3.320], 86 transverse (shear) Alfvén [7.284], 158 Waves in and out of media, 152 Waves in lossless media, 152 Waves in strings and springs, 83 wavevector (dimensions), 17 weber (unit), 4 Weber symbols, 126 weight (dimensions), 17 Weiss constant [7.114], 144 Weiss zone equation [6.10], 126 Welch window [2.582], 60 Weyl equation [4.182], 104 Wiedemann-Franz law [6.66], 132 Wien’s displacement law [5.189], 121 Wien’s displacement law constant, 9 Wien’s radiation law [5.188], 121 Wiener-Khintchine theorem in Fourier transforms [2.492], 53 in temporal coherence [8.105], 172 Wigner coefficients (spin-orbit) [4.136], 100 Wigner coefficients (table of), 99 windowing Bartlett [2.581], 60 Hamming [2.584], 60 Hanning [2.583], 60 Welch [2.582], 60 wire electric field [7.29], 138 magnetic flux density [7.34], 138 wire loop (inductance) [7.26], 137 wire loop (magnetic flux density) [7.37], 138 wires (inductance of parallel) [7.25], 137 work (dimensions), 17 X X-ray diffraction, 128 Y yocto, 5 yotta, 5 Young modulus and Lamé coefficients [3.240], 81 I main January 23, 2006 16:6 220 and other elastic constants [3.250], 81 Hooke’s law [3.230], 80 Young modulus (dimensions), 17 Young’s slits [8.24], 164 Yukawa potential [7.252], 156 Z Zeeman splitting constant, 7 zepto, 5 zero-point energy [4.68], 95 zetta, 5 zone law [6.20], 126 Index

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