Ferromagnetism (F.Lüttner)
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Ferromagnetism Characteristics experiments and its „origin“ (by Florian Lüttner) Contents • General expressions for describing magnetism and ferromagnetism • Different kinds of magnetism - short overview • What is ferromagnetism? - the phenomenon and ist characteristics • Ferromagnetism ferrimagnetism and antiferromagnetism • Observation of magnetic structures • „Origin of ferromagnetism“ - Singlet and tripplet states for a two electron system - Spin Hamiltonian - Heisenbergmodel - Meanfield approximation - curie-temperature General expressions Magnetization: ππππππ‘ππ ππππππ‘ π= π’πππ‘ π£πππ’ππ • Vectorfield that expresses the density of permanent or induced magnetic moments π π π π = ππ π= π = ππ’ππππ ππ ππππππ‘ππ ππππππ‘π ππ π πππππ π = π£πππ’ππ ππ π‘βπ π πππππ π = ππππππ‘ππ ππππππ‘ π = ππ’ππππ ππππ ππ‘π¦ ππ ππππππ‘ππ ππππππ‘π Magnetic susceptibility: Dimensionless proportionality constant between the degree of magnetization and the applied magnetic field π πΆπΊπ χ = π΅ π0 π ππΌ χ = π΅ π΅ = ππππππ πππππ ππππππ‘ππ πππππ πππ‘πππ ππ‘π¦ π = ππππππ‘ππ§ππ‘πππ π0 = ππππππ‘ππ πππππππππππ‘π¦ Magnetic moment: Determines the angular momentum (or torque) a magnet will experience in an external applied magnetic field π =π×π΅ π = ππππππππ π‘ππππ’π π΅ = ππ₯π‘πππππ ππππππ‘ππ πππππ π = ππππππ‘ππ ππππππ‘ π is the vector that relates the aligning torque on an object from an externally applied magnetic field to the field vector itself. With an unknown sample or object you can measure the torque by an applied known external magnetic field and get the magnetic moment. Gyromagnetic ratio: Ratio of its magnetic moment to its angular momentum (classical body) The gyromagnetic ratio can be written as π πΎ= 2ππ Therefore the magnetic moment is π = πΎπΏ πΏ = ππππ’πππ ππππππ‘π’π ππ = πππ π ππ π‘βπ ππππ π ππππ ππππ¦ π = πβππππ ππ π‘βπ ππππ π ππππ ππππ¦ For an isolated electron (quantummechanical) An electron has a spin ο no classical rotation (quantummechanical phenomenon) so ππ πΎπ ≠ 2ππ But with a correction factor or electron g-factor ππ −π πΎπ = π 2ππ π πΎπ = ππ΅ = Bohr magneton ππ ππ΅ β ππ = 2 1 + πΌ= 1 137 πΌ +β― 2π ππππ − π π‘ππ’ππ‘π’ππ ππππ π‘πππ‘ For this ππ is ππ = 2.0023193043617 And therefore πΎπ is calculated as πΎπ = 1.760859708 ∗ 1011 πππ π ∗π Electron spin: Follows the same mathematical laws as quantized angular momenta, but • No classical rotation ο quantummechanical phenomenon • Spin as an intrinsic form of angular momentum for elementary particles, hadrons and nuclei with a definite magnitude and a direction (up and down) • Spin quantum number only takes half-integer values (0, ½, 1, 3/2, 2, …) • Only spindirection can be changed in spin up or spin down but not ist value • The calculation of the magnetic moment of an electron needs an electron g factor different from 1 (like in classical cases) π π = 2 (n is any non − negative integer) Spin angular momentum π= β π (π + 1) 2π Spin angular momentum π= So S also can be calculated with π = β π (π + 1) 2π π 2 β π= π(π + 2) 4π For this the electron (fermion) spin quantum number and the angular momentum can be calculated with π=1 for this and in this case 1 π = 2 β π= 3 4π Short overview • Paramagnetism -χ<0 - magnetization is different from 0 as long as an external applied magnetic field exists - atoms, molecules and lattice defects with an odd number of electrons (the total spin of a system must not be zero) - free atoms and ions with a partly filled inner shell - metals - few compounds with an even number of electrons (molecular oxygen) • Diamagnetism -χ>0 - magnetization is different from 0 as long as an external applied magnetic field exists - the inner magnetization is in the opposite direction to the applied external field (want to vanish the field) - Bismut - Carbon - superconducter (crowds the magnetic field lines of the applied field out of the superconducter) • Ferromagnetism - spontanious magnetic moment (saturation moment) - electron spins and magnetic moment have to be arranged in a regular manner What is ferromagnetism? Ferromagnetic orders • Spontanious magnetization ο saturation magnetization • Vanishes not ο temperature under the Curie-temperature (π < ππ ) • π > ππ the order of electronspins vanishes after the external magnetic field is turned off (Paramagnetism) • From internal interaction of the magnetic moments of the spins (exchange field) Characteristics (hysteresis) Materials with different forms of magnetism Ferromagnetism, ferrimagnetism and antiferromagnetism • Use the language appropriate of solids (magnetic ions localized at lattice points) ferromagnetism • Here we have in general non vanishing vector moments of the magnetic ions below a temperature ππ (ordered spin orientation) • Same direction of magnetic moments (spins) ο add up to a net magnetization density ο ferromagnetism • Microscopic magnetic ordering is revealed by the existance of a macroscopic bulk magnetization density • Not all magnetic moments have to be the same Antiferromagnetism • Magnetic ordering of the individual local moments vavanishes to zero • No spontanious magnetization • Microscopic magnetic ordering can not be revealed by a macroscopic bulk magnetization density • Local moments orientated like two interpenetrating sublattices with the same structure ferrimagnetism • Here we have even a non vanishing vector moments of the magnetic ions • Not the same direction of magnetic moments (spins) at all • Exchange coupling between nearest neighbours may favor antiparallel orientation • Neighbour has not the same value of magnetic moment • Leaving a net magnetic moment for the solid as a whole • Some more complex arrangements are possible • Describtion not with the above three types • Specifiing the ordering in terms of spin density At any point π along any direction π§ the density π π§ (π) is described by 1 π π§ π = π↑ π − π↓ (π) 2 π↑ π πππ π↓ π πππ π‘βπ ππππ‘ππππ’π‘ππππ ππ π‘βπ π‘π€π π ππππππ’πππ‘ππππ π‘π π‘βπ πππππ‘πππππ ππππ ππ‘π¦ (ππ π§ ππππππ‘πππ) therefore πππ π§ (π) ≠ 0 for ferrimagnets and ferromagnets and πππ π§ π = 0 for antiferromagnets Observation of magnetic structure • Can be observed by the scattering pattern of neutron scattering • Neutrons have a magnetic moment ο couples to the elctronic spin • also nonmagnetic Bragg reflection of neutrons by ionic nuclei ο additional peaks in the elastic neutron scattering cross section • Over ππ the magnetic scattering peaks are vanished • Crosssection for neutron electron scattering of the same order of magnitude as for neutron nuclei interaction • Determination of the distribution direction and the order of the magnetic moments • Detemine the magnetic structure of antiferromagnets Origin of ferromagnetism • Magnetic effects in a material by the pauli priciple (even with no spin dependent terms in the Hamiltonian) • consider Two-electron system (with spin independet Hamiltonian) • Ψ stationary state is the product of a purely orbital stationary state • Ψ(π1 , π2 ) satisfies orbital Schrödinger equation π»Ψ = − β π»1 2 + π»2 2 Ψ + π π1 , π2 Ψ 2π π»Ψ = πΈΨ • Four spin states with both electrons in levels of definite π π§ |↑↑ , |↑↓ , |↓↑ , |↓↓ • Choose linearcombinations of these states to get definite values of the total spin π (and its component ππ§ ) Singlet state Triplet states 1 Combination of two spin − 2 particles can only carry a total spin of 1 or 0 (occupy triplet or singlet state) Any combination (like |↑↑ ) can be calculated by π 1 , π1 π 2 , π2 = |π 2 , π2 ⊗ |π 2 , π2 1 1 Span a 4-dimensional room (π 1/2 = 2 for spin − 2 particles) Calculated with Clebsch-Gordon coefficients π π π |π , π = 2 πΆπ11π π 1 , π1 π 2 , π2 2π π1 +π2 =π Substituting in the four basis states 1 1 , + 2 2 1 1 , + 2 2 1 1 , − 2 2 1 1 , − 2 2 1 1 , + 2 2 1 1 , − 2 2 1 1 , + 2 2 1 1 , − 2 2 for ↑↑ for ↑↓ for ↓↑ for ↓↓ Therefore you get three states with total spin angular momentum 1 and one state with total spin angular momentum 0 • Total wave function change sign under simultanious ninterchange of the spin (π 1 , π 2 ) and orbital parts (π1 , π2 ) • Ψ is the product of its spin and orbital parts • Therefore if Ψ do not change sign under interchange of π1 πππ π2 ο solution is symmetric ο must describe states with π = 0 • Ψ change sign under interchange of π1 πππ π2 ο antisymmetric solution ο must describe states with π = 1 • If πΈπ and πΈπ‘ are the lowest eigenvalues of the spin-independent orbital schrödinger equation associated to the symmetric and antisymmetric solution of it then the groundstate will only have spin 0 or spin 1 πΈπ < πΈπ‘ → π = 0 πΈπ > πΈπ‘ → π = 1 • This one can require only by an examination of the spin-independent Schrödinger equation • For two electron problem the ground-state wave function have to be symmetric • Only for two electron system • Have to estimate πΈπ − πΈπ‘ that can be generalized to the anologous problem of a N-atom solid Spin Hamiltonian • A way to express dependence of a two-electron configuration spin on the singlettriplet energy splitting • Important for the analyzationof energies of the spin configuration of real insulating solids • protons with great distances ο two independent hydrogen atoms described by the ground state (fourfold degenerated with two possible orientations of each electron spin) • Bring the protons closer to each other (molecule) ο splitting of the fourfold degeneracy (due to atominteractions) • This interaction is small compared with other excitation energies of a two-electron system • Simplifying by ignoring higher states over fourfold degenerated • Molecule as a simple four-state system • Represent general state of the molecule by linear combination of the forur lowest states • Convenient to have an operator Spin Hamiltonian • Same eigenvalues as in the original Hamiltonian within the four state manifold • Eigenfunctions give the spin of each state • Each individual spin operator satisfies ππ Therefore the total spin satisfies 2 1 1 = +1 2 2 3 ππ 2 = 4 πΊ2 = πΊ1 + πΊ2 2 3 2 πΊ = + 2πΊ1 β πΊ2 2 πΊ2 has the eigenvalues π(π + 1) and with the equation above for πΊ2 we get the eigenvalues 3 1 of the operator πΊ1 β πΊ2 as − 4 for the singlet state and 4 for each triplet state Therefore π» π πππ is 1 π πππ β = (πΈπ + 3πΈπ‘ ) − (πΈπ − πΈπ‘ )πΊ1 β πΊ2 4 1 Redefining the zero of energy and omit the constant 4 (πΈπ + 3πΈπ‘ ) which is common to all four states we get β π πππ = −π½πΊ1 β πΊ2 Heisenberg model • β π πππ is the scalar product of the spin operators πΊ1 πππ πΊ2 ο parallel spins for π½ > 0 (leads to ferromagnets) and antiparallel spins for π½ < 0 (leads to ferrimagnets and antiferromagnets) • Coupling in the spin Hamiltonian do not depend on the spatial direction with respect to π 1 − π 2 but on the ralative orientation of the two spins • β π πππ is Isotropic but we need unisotropic terms to describe ferromagnetism • Include terms that braeak rotational symmetrie in spin space • Spin Hamiltonian only for insulating materials (Hubbard model for metals) • Therefore N widely seperated ions with a small overlap of the electron wavefunction • Only interaction between nearest neighbours ππ πππ ππ • Ground state will be 2π + 1 π − ππππ gegenerated • Spin Hamiltonian describes the splitting of this ground state when ions are some closer together that the splittings are small compared with any other excitation energies • The eigenvalues of the ππ πππ ππ gives the split levels • For many cases of interest β π πππ is in the form of the two-spin case summed over all pairs of ions β π πππ = π½ππ πΊπ β πΊπ π,π • π½ππ is the exchange coupling constant (exchange energy) • Exchange interaction between localized spins only from coulomb repulsion and from pauli principle • For angular momentums depending on orbital as well as on spin parts the Hamiltonian depends on the absolute spin oriantations as well as on the relative ones Mean field approximation • Early analysis of the ferromagnetic effects by P. Weiss with the mean field theory • This theory fails to predict spinwaves at low and high temperatures but a good approximation at ππ Suppose we focus our attention on a particular Bravais lattice site R in the Heisenberg Hamiltonian β π πππ = π½ππ πΊπ β πΊπ π,π • With π± πΉ − πΉ´ = π± πΉ´ − πΉ > 0 (ferromagnet) • Represents for each ion the total angular momentum with spin and orbital part usual to take these fictitious spins to be parallel to the magnetic moment of the ion 1 β=− πΊ πΉ β πΊ πΉ´ π± πΉ − πΉ´ − πππ΅ π» πΊπ (πΉ) 2 ´ πΉπΉ πΉ • And isolating from β π πππ those terms containing πΊ πΉ π± πΉ − πΉ´ πΊ πΉ´ + πππ΅ π» ββ = −πΊ πΉ β πΉ≠πΉ´ • Has the form of an energy of a spin in an effective external field π»πππ 1 =π»+ πππ΅ π± πΉ − πΉ´ πΊ πΉ´ πΉ´ • But this is an operator depending on a complicated way on the detailed spin configuration of all the other spin at different sites from R Mean field approximation • Replace π»πππ with its termal equilibrium mean value (replace all spin values by its termal equilibrium) • In ferromagnets all spins have the same mean value • In terms of the total magnetization density it is π½ π΄ πΊ(πΉ) = π΅ πππ© Replace each spin in π»πππ = π» + 1 πππ΅ π± πΉ − πΉ´ πΊ πΉ´ πΉ´ by its mean value πΊ(πΉ) = π½ π΄ π΅ πππ© We arrive at the effective field π»πππ = π» + λπ π½ with λ = π΅ π±π πππ© π and π½0 = π π±(πΉ) Every magnetic atom experiences not only the magnetization of its nearest neighbour but an average magnetization of all the other magnetic atoms With a spontanious magnetization at it is usual in ferromagnetics with no applied field we can assume π»πππ as π»πππ = λπ Curie temperature 1 • It is known that π ∝ π • Since π»πππ is given by π»πππ = λπ • without an applied field the magnetization is given by π = π0 λπ π • With π0 the magnetization density with no applied field • Seperate in a pair of equations like π π = π0 x π π π = π₯ λ • Put both equations as graphs in a plot π • Important are the intersections of π0 x and λ π₯ π • If and only if the slope of the straight line λ is less than that one of π0 x at the origin, π0 ´ (0) at a nonzero value of x • π0 ´ (0) can be calculated in terms of zero field susceptibility χ0 , calculated in the absence of interactions, for ππ0 χ0 = ππ» π»=0 π0 ´ (0) χ0 = π • From the analysis of a set of identical ions of angular momentum J one can get curies law π (πππ΅ )2 π½(π + 1) χ= π 3 ππ΅ π with ππ΅ π β« πππ΅ π» • Compare curies law with χ0 = π0 ´ (0) π (can read off the value of π0 ´ (0) ) • Critical temperature is given by ππ = π πππ΅ π 3ππ΅ 2 π π+1 λ π π+1 ππ = π½0 3ππ΅ • Not exactly • More precise with the Isingmodel • Simplification of the Heisenbergmodel only with spins on one axes (z-axes) and only spins with the value π = ±1 • π½ππ ≠ 0 only for neighboring ions 1 β=− π½ππ ππ β ππ − π―π ππ 2 π,π π=1 Sources Books • Ashcroft/Mermin, Solid State Physics Ch. 31, 32, 33 • Charles Kittel, Introductions to Solid State Physics (eight edition) Ch.12, 13 Weblinks • • • • • • • http://de.wikipedia.org/wiki/Ferromagnetismus http://en.wikipedia.org/wiki/Gyromagnetic_ratio http://de.wikipedia.org/wiki/Diamagnetismus http://de.wikipedia.org/wiki/Paramagnetismus http://en.wikipedia.org/wiki/Magnetic_susceptibility http://en.wikipedia.org/wiki/Magnetization http://de.wikipedia.org/wiki/Elektronenspin • http://www.google.de/imgres?imgurl=http%3A%2F%2Fupload.wikimedia.org%2Fwi kipedia%2Fcommons%2Fthumb%2F0%2F04%2FPermeability_by_Zureks.svg%2F220 pxPermeability_by_Zureks.svg.png&imgrefurl=http%3A%2F%2Fde.wikipedia.org%2Fwi ki%2FParamagnetismus&h=170&w=220&tbnid=4qdQ9MfddgwCgM%3A&zoom=1&d ocid=19Wq_TNMEP5fmM&ei=FSc4U4OmGsfOtQbikIHoCQ&tbm=isch&iact=rc&dur= 1325&page=1&start=0&ndsp=15&ved=0CFsQrQMwAQ • http://hyperphysics.phy-astr.gsu.edu/hbase/solids/imgsol/hystcurves.gif • http://hyperphysics.phy-astr.gsu.edu/hbase/solids/imgsol/hyloop.gif • http://de.wikipedia.org/wiki/Ising-Modell • http://de.wikipedia.org/wiki/Heisenberg-Modell_%28Quantenmechanik%29 Thank you for your attention!!
