Revised Edition: 2016 ISBN 978-1-283-49456-4 © All rights reserved. Published by: Research World 48 West 48 Street, Suite 1116, New York, NY 10036, United States Email: info@wtbooks.com Table of Contents Chapter 1 - Infrared Spectroscopy Chapter 2 - Fourier Transform Infrared Spectroscopy and Infrared Spectroscopy Correlation Table Chapter 3 - Near-infrared Spectroscopy Chapter 4 - Two-dimensional Infrared Spectroscopy and Two-dimensional Correlation Analysis WT Chapter 5 - Fluorescence Correlation Spectroscopy Chapter 6 - Rotational Spectroscopy Chapter 7 - Electromagnetic Spectrum Chapter 8 - Molecular Vibration Chapter 9 - Fourier Transform ________________________WORLD TECHNOLOGIES________________________ Chapter- 1 Infrared Spectroscopy Infrared spectroscopy (IR spectroscopy) is the spectroscopy that deals with the infrared region of the electromagnetic spectrum, that is light with a longer wavelength and lower frequency than visible light. It covers a range of techniques, mostly based on absorption spectroscopy. As with all spectroscopic techniques, it can be used to identify and study chemicals. A common laboratory instrument that uses this technique is a Fourier transform infrared (FTIR) spectrometer. WT The infrared portion of the electromagnetic spectrum is usually divided into three regions; the near-, mid- and far- infrared, named for their relation to the visible spectrum. The higher energy near-IR, approximately 14000–4000 cm−1 (0.8–2.5 μm wavelength) can excite overtone or harmonic vibrations. The mid-infrared, approximately 4000– 400 cm−1 (2.5–25 μm) may be used to study the fundamental vibrations and associated rotational-vibrational structure. The far-infrared, approximately 400–10 cm−1 (25– 1000 μm), lying adjacent to the microwave region, has low energy and may be used for rotational spectroscopy. The names and classifications of these subregions are conventions, and are only loosely based on the relative molecular or electromagnetic properties. Theory Infrared spectroscopy exploits the fact that molecules absorb specific frequencies that are characteristic of their structure. These absorptions are resonant frequencies, i.e. the frequency of the absorbed radiation matches the frequency of the bond or group that vibrates. The energies are determined by the shape of the molecular potential energy surfaces, the masses of the atoms, and the associated vibronic coupling. In particular, in the Born–Oppenheimer and harmonic approximations, i.e. when the molecular Hamiltonian corresponding to the electronic ground state can be approximated by a harmonic oscillator in the neighborhood of the equilibrium molecular geometry, the resonant frequencies are determined by the normal modes corresponding to the molecular electronic ground state potential energy surface. Nevertheless, the resonant frequencies can be in a first approach related to the strength of the bond, and the mass of the atoms at either end of it. Thus, the frequency of the vibrations can be associated with a particular bond type. ________________________WORLD TECHNOLOGIES________________________ Number of vibrational modes In order for a vibrational mode in a molecule to be "IR active," it must be associated with changes in the permanent dipole. A molecule can vibrate in many ways, and each way is called a vibrational mode. For molecules with N atoms in them, linear molecules have 3N – 5 degrees of vibrational modes, whereas nonlinear molecules have 3N – 6 degrees of vibrational modes (also called vibrational degrees of freedom). As an example H2O, a non-linear molecule, will have 3 × 3 – 6 = 3 degrees of vibrational freedom, or modes. Simple diatomic molecules have only one bond and only one vibrational band. If the molecule is symmetrical, e.g. N2, the band is not observed in the IR spectrum, but only in the Raman spectrum. Unsymmetrical diatomic molecules, e.g. CO, absorb in the IR spectrum. More complex molecules have many bonds, and their vibrational spectra are correspondingly more complex, i.e. big molecules have many peaks in their IR spectra. WT The atoms in a CH2 group, commonly found in organic compounds, can vibrate in six different ways: symmetric and antisymmetric stretching, scissoring, rocking, wagging and twisting: Symmetrical stretching Antisymmetrical stretching Scissoring Rocking Wagging Twisting (These figures do not represent the "recoil" of the C atoms, which, though necessarily present to balance the overall movements of the molecule, are much smaller than the movements of the lighter H atoms). ________________________WORLD TECHNOLOGIES________________________ Special effects The simplest and most important IR bands arise from the "normal modes," the simplest distortions of the molecule. In some cases, "overtone bands" are observed. These bands arise from the absorption of a photon that leads to a doubly excited vibrational state. Such bands appear at approximately twice the energy of the normal mode. Some vibrations, socalled 'combination modes," involve more than one normal mode. The phenomenon of Fermi resonance can arise when two modes are similar in energy, Fermi resonance results in an unexpected shift in energy and intensity of the bands. Practical IR spectroscopy The infrared spectrum of a sample is recorded by passing a beam of infrared light through the sample. Examination of the transmitted light reveals how much energy was absorbed at each wavelength. This can be done with a monochromatic beam, which changes in wavelength over time, or by using a Fourier transform instrument to measure all wavelengths at once. From this, a transmittance or absorbance spectrum can be produced, showing at which IR wavelengths the sample absorbs. Analysis of these absorption characteristics reveals details about the molecular structure of the sample. When the frequency of the IR is the same as the vibrational frequency of a bond, absorption occurs. WT This technique works almost exclusively on samples with covalent bonds. Simple spectra are obtained from samples with few IR active bonds and high levels of purity. More complex molecular structures lead to more absorption bands and more complex spectra. The technique has been used for the characterization of very complex mixtures. Sample preparation Gaseous samples require a sample cell with a long pathlength (typically 5–10 cm), to compensate for the diluteness. Liquid samples can be sandwiched between two plates of a salt (commonly sodium chloride, or common salt, although a number of other salts such as potassium bromide or calcium fluoride are also used). The plates are transparent to the infrared light and do not introduce any lines onto the spectra. Solid samples can be prepared in a variety of ways. One common method is to crush the sample with an oily mulling agent (usually Nujol) in a marble or agate mortar, with a pestle. A thin film of the mull is smeared onto salt plates and measured. The second method is to grind a quantity of the sample with a specially purified salt (usually potassium bromide) finely (to remove scattering effects from large crystals). This powder mixture is then pressed in a mechanical press to form a translucent pellet through which the beam of the spectrometer can pass. A third technique is the "cast film" technique, which is used mainly for polymeric materials. The sample is first dissolved in a suitable, non hygroscopic solvent. A drop of this solution is deposited on surface of KBr or NaCl cell. The solution is then evaporated to dryness and the film formed on the cell is ________________________WORLD TECHNOLOGIES________________________ analysed directly. Care is important to ensure that the film is not too thick otherwise light cannot pass through. This technique is suitable for qualitative analysis. The final method is to use microtomy to cut a thin (20–100 µm) film from a solid sample. This is one of the most important ways of analysing failed plastic products for example because the integrity of the solid is preserved. It is important to note that spectra obtained from different sample preparation methods will look slightly different from each other due to differences in the samples' physical states. Comparing to a reference WT Schematics of a two-beam absorption spectrometer. A beam of infrared light is produced, passed through an interferometer (not shown), and then split into two separate beams. One is passed through the sample, the other passed through a reference. The beams are both reflected back towards a detector, however first they pass through a splitter, which quickly alternates which of the two beams enters the detector. The two signals are then compared and a printout is obtained. This "two-beam" setup gives accurate spectra even if the intensity of the light source drifts over time. To take the infrared spectrum of a sample, it is necessary to measure both the sample and a "reference" (or "control"). This is because each measurement is affected by not only the light-absorption properties of the sample, but also the properties of the instrument (for example, what light source is used, what detector is used, etc.). The reference measurement makes it possible to eliminate the instrument influence. Mathematically, the sample transmission spectrum is divided by the reference transmission spectrum. The appropriate "reference" depends on the measurement and its goal. The simplest reference measurement is to simply remove the sample (replacing it by air). However, sometimes a different reference is more useful. For example, if the sample is a dilute ________________________WORLD TECHNOLOGIES________________________ solute dissolved in water in a beaker, then a good reference measurement might be to measure pure water in the same beaker. Then the reference measurement would cancel out not only all the instrumental properties (like what light source is used), but also the light-absorbing and light-reflecting properties of the water and beaker, and the final result would just show the properties of the solute (at least approximately). A common way to compare to a reference is sequentially: First measure the reference, then replace the reference by the sample, then measure the sample. This technique is not perfectly reliable: If the infrared lamp is a bit brighter during the reference measurement, then a bit dimmer during the sample measurement, the measurement will be distorted. More elaborate methods, such as a "two-beam" setup (see figure), can correct for these types of effects to give very accurate results. FTIR WT An interferogram from an FTIR measurement. The horizontal axis is the position of the mirror, and the vertical axis is the amount of light detected. This is the "raw data" which can be Fourier transformed to get the actual spectrum. Fourier transform infrared (FTIR) spectroscopy is a measurement technique that allows one to record infrared spectra. Infrared light is guided through an interferometer and then through the sample (or vice versa). A moving mirror inside the apparatus alters the distribution of infrared light that passes through the interferometer. The signal directly recorded, called an "interferogram", represents light output as a function of mirror position. A data-processing technique called Fourier transform turns this raw data into the desired result (the sample's spectrum): Light output as a function of infrared wavelength (or equivalently, wavenumber). As described above, the sample's spectrum is always compared to a reference. There is an alternate method for taking spectra (the "dispersive" or "scanning monochromator" method), where one wavelength at a time passes through the sample. The ________________________WORLD TECHNOLOGIES________________________ dispersive method is more common in UV-Vis spectroscopy, but is less practical in the infrared than the FTIR method. One reason that FTIR is favored is called "Fellgett's advantage" or the "multiplex advantage": The information at all frequencies is collected simultaneously, improving both speed and signal-to-noise ratio. Another is called "Jacquinot's Throughput Advantage": A dispersive measurement requires detecting much lower light levels than an FTIR measurement. There are other advantages, as well as some disadvantages, but virtually all modern infrared spectrometers are FTIR instruments. Absorption bands WT Wavenumbers listed in cm−1. Uses and applications Infrared spectroscopy is widely used in both research and industry as a simple and reliable technique for measurement, quality control and dynamic measurement. It is also used in forensic analysis in both criminal and civil cases, enabling identification of polymer degradation for example. The instruments are now small, and can be transported, even for use in field trials. With increasing technology in computer filtering and manipulation of the results, samples in solution can now be measured accurately (water produces a broad absorbance across the range of interest, and thus renders the spectra unreadable without this computer treatment). Some instruments will also automatically tell you what substance is being measured from a store of thousands of reference spectra held in storage. By measuring at a specific frequency over time, changes in the character or quantity of a particular bond can be measured. This is especially useful in measuring the degree of polymerization in polymer manufacture. Modern research instruments can take infrared measurements across the whole range of interest as frequently as 32 times a second. This can be done whilst simultaneous measurements are made using other techniques. This makes the observations of chemical reactions and processes quicker and more accurate. Infrared spectroscopy has been highly successful for applications in both organic and inorganic chemistry. Infrared spectroscopy has also been successfully utilized in the field of semiconductor microelectronics: for example, infrared spectroscopy can be applied to semiconductors like silicon, gallium arsenide, gallium nitride, zinc selenide, amorphous silicon, silicon nitride, etc. ________________________WORLD TECHNOLOGIES________________________ Isotope effects The different isotopes in a particular species may give fine detail in infrared spectroscopy. For example, the O–O stretching frequency (in reciprocal centimeters) of oxyhemocyanin is experimentally determined to be 832 and 788 cm−1 for ν(16O–16O) and ν(18O–18O), respectively. By considering the O–O bond as a spring, the wavenumber of absorbance, ν can be calculated: where k is the spring constant for the bond, c is the speed of light, and μ is the reduced mass of the A–B system: WT (mi is the mass of atom i). The reduced masses for respectively. Thus 16 O–16O and 18 O–18O can be approximated as 8 and 9 Where ν is the wavenumber; [wavenumber = frequency/(speed of light)] The effect of isotopes, both on the vibration and the decay dynamics, has been found to be stronger than previously thought. In some systems, such as silicon and germanium, the decay of the anti-symmetric stretch mode of interstitial oxygen involves the symmetric stretch mode with a strong isotope dependence. For example, it was shown that for a natural silicon sample, the lifetime of the anti-symmetric vibration is 11.4 ps. When the isotope of one of the silicon atoms is increased to 29Si, the lifetime increases to 19 ps. In similar manner, when the silicon atom is changed to 30Si, the lifetime becomes 27 ps. Two-dimensional IR Two-dimensional infrared correlation spectroscopy analysis is the application of 2D correlation analysis on infrared spectra. By extending the spectral information of a perturbed sample, spectral analysis is simplified and resolution is enhanced. The 2D synchronous and 2D asynchronous spectra represent a graphical overview of the spectral ________________________WORLD TECHNOLOGIES________________________ changes due to a perturbation (such as a changing concentration or changing temperature) as well as the relationship between the spectral changes at two different wavenumbers. WT Pulse Sequence used to obtain a two-dimensional Fourier transform infrared spectrum. The time period τ1 is usually referred to as the coherence time and the second time period τ2 is known as the waiting time. The excitation frequency is obtained by Fourier transforming along the τ1 axis. Nonlinear two-dimensional infrared spectroscopy is the infrared version of correlation spectroscopy. Nonlinear two-dimensional infrared spectroscopy is a technique that has become available with the development of femtosecond infrared laser pulses. In this experiment, first a set of pump pulses are applied to the sample. This is followed by a waiting time, wherein the system is allowed to relax. The typical waiting time lasts from zero to several picoseconds, and the duration can be controlled with a resolution of tens of femtoseconds. A probe pulse is then applied resulting in the emission of a signal from the sample. The nonlinear two-dimensional infrared spectrum is a two-dimensional correlation plot of the frequency ω1 that was excited by the initial pump pulses and the frequency ω3 excited by the probe pulse after the waiting time. This allows the observation of coupling between different vibrational modes; because of its extremely high time resolution, it can be used to monitor molecular dynamics on a picosecond timescale. It is still a largely unexplored technique and is becoming increasingly popular for fundamental research. As with two-dimensional nuclear magnetic resonance (2DNMR) spectroscopy, this technique spreads the spectrum in two dimensions and allows for the observation of cross peaks that contain information on the coupling between different modes. In contrast to 2DNMR, nonlinear two-dimensional infrared spectroscopy also involves the excitation to overtones. These excitations result in excited state absorption peaks located below the diagonal and cross peaks. In 2DNMR, two distinct techniques, COSY and NOESY, are frequently used. The cross peaks in the first are related to the scalar coupling, while in the ________________________WORLD TECHNOLOGIES________________________ later they are related to the spin transfer between different nuclei. In nonlinear twodimensional infrared spectroscopy, analogs have been drawn to these 2DNMR techniques. Nonlinear two-dimensional infrared spectroscopy with zero waiting time corresponds to COSY, and nonlinear two-dimensional infrared spectroscopy with finite waiting time allowing vibrational population transfer corresponds to NOESY. The COSY variant of nonlinear two-dimensional infrared spectroscopy has been used for determination of the secondary structure content proteins. WT ________________________WORLD TECHNOLOGIES________________________ Chapter- 2 Fourier Transform Infrared Spectroscopy and Infrared Spectroscopy Correlation Table Fourier transform infrared spectroscopy WT Fourier transform infrared spectroscopy (FTIR) is a technique which is used to obtain an infrared spectrum of absorption, emission, photoconductivity or Raman scattering of a solid, liquid or gas. An FTIR spectrometer simultaneously collects spectral data in a wide spectral range. This confers a significant advantage over a dispersive spectrometer which measures intensity over a narrow range of wavelengths at a time. FTIR technique has made dispersive infrared spectrometers all but obsolete (except sometimes in the near infrared) and opened up new applications of infrared spectroscopy. The term Fourier transform infrared spectroscopy originates from the fact that a Fourier transform (a mathematical algorithm) is required to convert the raw data into the actual spectrum. Conceptual introduction ________________________WORLD TECHNOLOGIES________________________ An interferogram from an FTIR spectrometer. The horizontal axis is the position of the mirror, and the vertical axis is the amount of light detected. This is the "raw data" which can be transformed into an actual spectrum. The goal of any absorption spectroscopy (FTIR, ultraviolet-visible ("UV-Vis") spectroscopy, etc.) is to measure how well a sample absorbs light at each wavelength. The most straightforward way to do this, the "dispersive spectroscopy" technique, is to shine a monochromatic light beam at a sample, measure how much of the light is absorbed, and repeat for each different wavelength. (This is how UV-Vis spectrometers work, for example.) Fourier transform spectroscopy is a less intuitive way to obtain the same information. Rather than shining a monochromatic beam of light at the sample, this technique shines a beam containing many different frequencies of light at once, and measures how much of that beam is absorbed by the sample. Next, the beam is modified to contain a different combination of frequencies, giving a second data point. This process is repeated many times. Afterwards, a computer takes all these data and works backwards to infer what the absorption is at each wavelength. WT The beam described above is generated by starting with a broadband light source—one containing the full spectrum of wavelengths to be measured. The light shines into a certain configuration of mirrors, called a Michelson interferometer, that allows some wavelengths to pass through but blocks others (due to wave interference). The beam is modified for each new data point by moving one of the mirrors; this changes the set of wavelengths that pass through. As mentioned, computer processing is required to turn the raw data (light absorption for each mirror position) into the desired result (light absorption for each wavelength). The processing required turns out to be a common algorithm called the Fourier transform (hence the name, "Fourier transform spectroscopy"). The raw data is sometimes called an "interferogram". Michelson interferometer Schematic diagram of a Michelson interferometer, configured for FTIR ________________________WORLD TECHNOLOGIES________________________ In a Michelson interferometer adapted for FTIR, light from the polychromatic infrared source, approximately a black-body radiator, is collimated and directed to a beam splitter. Ideally 50% of the light is reflected towards the fixed mirror and 50% is transmitted towards the moving mirror. Light is reflected from the two mirrors back to the beam splitter and (ideally) 50% of the original light passes into the sample compartment. There, the light is focussed on the sample. On leaving the sample compartment the light is refocused on to the detector. The difference in optical path length between the two arms to the interferometer is known as the retardation. An interferogram is obtained by varying the retardation and recording the signal from the detector for various values of the retardation. The form of the interferogram when no sample is present depends on factors such as the variation of source intensity and splitter efficiency with wavelength. This results in a maximum at zero retardation, when there is constructive interference at all wavelengths, followed by series of "wiggles". The position of zero retardation is determined accurately by finding the point of maximum intensity in the interferogram. When a sample is present the background interferogram is modulated by the presence of absorption bands in the sample. WT There are two principle advantages for a FT spectrometer compared to a scanning (dispersive) spectrometer. 1. The multiplex or Fellgett's advantage. This arises from the fact that information from all wavelengths is collected simultaneously. It results in a higher Signal-tonoise ratio for a given scan-time or a shorter scan-time for a given resolution. 2. The throughput or Jacquinot's advantage. This results from the fact that, in a dispersive instrument, the monochromator has entrance and exit slits which restrict the amount of light that passes through it. The interferometer throughput is determined only by the diameter of the collimated beam coming from the source. Other minor advantages include less sensitivity to stray light, and "Connes' advantage" (better wavelength accuracy), while a disadvantage is that FTIR cannot use the advanced electronic filtering techniques that often makes its signal-to-noise ratio inferior to that of dispersive measurements. Resolution The interferogram belongs in the length domain. Fourier transform (FT) inverts the dimension, so the FT of the interferogram belongs in the reciprocal length domain, that is the wavenumber domain. The spectral resolution in wavenumbers per cm is equal to the reciprocal of the maximum retardation in cm. Thus a 4 cm−1 resolution will be obtained if the maximum retardation is 0.25 cm; this is typical of the cheaper FTIR instruments. Much higher resolution can be obtained by increasing the maximum retardation. This is not easy as the moving mirror must travel in a near-perfect straight line. The use of corner-cube mirrors in place of the flat mirrors is helpful as an outgoing ray from a corner-cube mirror is parallel to the incoming ray, regardless of the orientation of the mirror about axes perpendicular to the axis of the light beam. Connes measured in 1966 the temperature of the atmosphere of Venus by recording the vibration-rotation spectrum ________________________WORLD TECHNOLOGIES________________________ of Venusian CO2 at 0.1 cm−1 resolution. Michelson himself attempted to resolve the hydrogen Hα emission band in the spectrum of a hydrogen atom into its two components by using his interferometer. p25 A spectrometer with 0.001 cm−1 resolution is now available commercially from Bruker. The throughput advantage is important for highresolution FTIR as the monochromator in a dispersive instrument with the same resolution would have very narrow entrance and exit slits. Beam splitter The beam-splitter can not be made of a common glass, as it is opaque to infrared radiation of wavelengths longer than about 2.5 μm. A thin film, usually of a plastic material, is used instead. However, as any material has a limited range of optical transmittance, several beam-splitters are used interchangeably to cover a wide spectral range. WT Fourier transform The interferogram in practice consists of a set of intensities measured for discrete values of retardation. The difference between successive retardation values is constant. Thus, a discrete Fourier transform is needed. The fast Fourier transform (FFT) algorithm is used. Far-infrared FTIR The first FTIR spectrometers were developed for far-infrared range. The reason for this has to do with the mechanical tolerance needed for good optical performance, which is related to the wavelength of the light being used. For the relatively long wavelengths of the far infrared (~10 μm), tolerances are adequate, whereas for the rock-salt region tolerances have to be better than 1 μm. A typical instrument was the cube interferometer developed at the NPL and marketed by Grubb Parsons. It used a stepper motor to drive the moving mirror, recording the detector response after each step was completed. Mid-infrared FTIR With the advent of cheap microcomputers it became possible to have a computer dedicated to controlling the spectrometer, collecting the data, doing the Fourier transform and presenting the spectrum. This provided the impetus for the development of FTIR spectrometers for the rock-salt region. The problems of manufacturing ultra-high precision optical and mechanical components had to be solved. A wide range of instruments is now available commercially. Although instrument design has become more sophisticated, the basic principles remain the same. Nowadays, the moving mirror of the interferometer moves at a constant velocity, and sampling of the interferogram is triggered by finding zero-crossings in the fringes of a secondary interferometer lit by a helium-neon laser. This confers high wavenumber accuracy on the resulting infrared spectrum and avoids wavenumber calibration errors. ________________________WORLD TECHNOLOGIES________________________ Near-infrared FTIR The near-infrared region spans the wavelength range between the rock-salt region and the start of the visible region at about 750 nm. Overtones of fundamental vibrations can be observed in this region. It is used mainly in industrial applications such as process control. Applications FTIR can be used in all applications where a dispersive spectrometer was used in the past. In addition, the multiplex and throughput advantages have opened up new areas of application. These include: • • • • • GC-IR (gas chromatography-infrared spectrometry). A gas chromatograph can be used to separate the components of a mixture. The fractions containing single components are directed into an FTIR spectrometer, to provide the infrared spectrum of the sample. This technique is complementary to GC-MS (gas chromatography-mass spectrometry). The GC-IR method is particularly useful for identifying isomers, which by their nature have identical masses. The key to the successful use of GC-IR is that the interferogram can be captured in a very short time, typically less than 1 second. FTIR has also been applied to the analysis of liquid chromatography fractions. TG-IR (thermogravimetry-infrared spectrometry) IR spectra of the gases evolved during thermal decompostion are obtained as a function of temperature. Micro-samples. Tiny samples, such as in forensic analysis, can be examined with the aid of an infrared microscope in the sample chamber. An image of the surface can be obtained by scanning. Another example is the use of FTIR to characterize artistic materials in old-master paintings. Emission spectra. Instead of recording the spectrum of light transmitted through the sample, FTIR spectrometer can be used to acquire spectrum of light emitted by the sample. Such emission could be induced by various processes, and the most common ones are luminescence and Raman scattering. Little modification is required to an absorption FTIR spectrometer to record emission spectra and therefore many commercial FTIR spectrometers combine both absorption and emission/Raman modes. Photocurrent spectra. This mode uses a standard, absorption FTIR spectrometer. The studied sample is placed instead of the FTIR detector, and its photocurrent, induced by the spectrometer's broadband source, is used to record the interferrogram, which is then converted into the photoconductivity spectrum of the sample. WT ________________________WORLD TECHNOLOGIES________________________ Infrared spectroscopy correlation table In physical and analytical chemistry, infrared spectroscopy ("IR spectroscopy") is a technique used to identify chemical compounds based on how infrared radiation is absorbed by the compounds' chemical bonds. This is an IR spectroscopy correlation table that lists some general absorption peaks for common types of atomic bonds and functional groups. The absorptions in this range do not apply only to bonds in organic molecules. IR spectroscopy is useful when it comes to analysis of inorganic compounds (such as metal complexes or fluoromanganates) as well. Bond Specific type of bond methyl alkyl methylene methine C═CH2 C═CH C─H Absorption peak 1260 cm−1 1380 cm−1 2870 cm−1 2960 cm−1 1470 cm−1 2850 cm−1 2925 cm−1 2890 cm−1 900 cm−1 2975 cm−1 3080 cm−1 3020 cm−1 900 cm−1 990 cm−1 670–700 cm−1 strong weak medium to strong medium to strong strong medium to strong medium to strong weak strong medium medium medium strong strong 965 cm−1 strong WT Type of bond vinyl monosubstituted alkenes cis-disubstituted alkenes trans-disubstituted alkenes 800–840 cm−1 benzene/sub. benzene 3070 cm−1 700–750 cm−1 monosubstituted benzene 690–710 cm−1 ortho-disub. benzene 750 cm−1 trisubstituted alkenes aromatic Appearance strong strong to medium weak strong strong strong ________________________WORLD TECHNOLOGIES________________________ 750–800 cm−1 meta-disub. benzene 860–900 cm−1 800–860 para-disub. benzene cm−1 alkynes any 3300 cm−1 2720 cm−1 aldehydes any 2820 cm−1 monosub. alkenes 1645 cm−1 1,1-disub. alkenes 1655 cm−1 cis-1,2-disub. alkenes 1660 cm−1 acyclic C─C trans-1,2-disub. 1675 cm−1 alkenes trisub., tetrasub. 1670 cm−1 alkenes 1600 cm−1 conjugated C─C 1650 cm−1 dienes with benzene ring 1625 cm−1 C─C with C═O 1600 cm−1 1640–1680 C═C (both sp2) any cm−1 1450 cm−1 1500 cm−1 any aromatic C═C 1580 cm−1 1600 cm−1 2100–2140 terminal alkynes cm−1 C≡C 2190–2260 disubst. alkynes cm−1 saturated aliph./cyclic 61720 cm−1 membered α,β-unsaturated 1685 cm−1 1685 cm−1 aldehyde/ketone aromatic ketones C═O cyclic 5-membered 1750 cm−1 cyclic 4-membered 1775 cm−1 strong strong strong medium medium medium medium medium WT aldehydes carboxylic 1725 cm−1 saturated carboxylic 1710 cm−1 medium weak strong strong strong strong medium weak to strong (usually 3 or 4) weak very weak (often indisinguishable) influence of conjugation (as with ketones) ________________________WORLD TECHNOLOGIES________________________ acids/derivates acids unsat./aromatic carb. 1680–1690 acids cm−1 esters and lactones anhydrides acyl halides amides carboxylates (salts) amino acid zwitterions carboxylic acids primary amines N─H low concentration high concentration low concentration high concentration any secondary amines any ammonium ions any primary alcohols secondary tertiary C─O 1760 cm−1 1820 cm−1 1800 cm−1 1650 cm−1 1550–1610 cm−1 1550–1610 cm−1 3610–3670 cm−1 3200–3400 cm−1 3500–3560 cm−1 3000 cm−1 3400–3500 cm−1 1560–1640 cm−1 >3000 cm−1 2400–3200 cm−1 1040–1060 cm−1 ~1100 cm−1 1150–1200 cm−1 1200 cm−1 1120 cm−1 1220–1260 cm−1 1250–1300 cm−1 1100–1300 cm−1 influenced by conjugation and ring size (as with ketones) associated amides WT alcohols, phenols O─H 1735 cm −1 phenols ethers any aliphatic aromatic carboxylic acids any esters any broad broad strong strong weak to medium multiple broad peaks strong, broad strong medium two bands (distinct from ketones, which do not ________________________WORLD TECHNOLOGIES________________________ possess a C─O bond) aliphatic amines any C═N any C─N C≡N (nitriles) R─N─C (isocyanides) any R─N═C═S any ordinary often overlapped similar conjugation effects to C═O medium medium WT fluoroalkanes C─X unconjugated conjugated 1020–1220 cm−1 1615–1700 cm−1 2250 cm−1 2230 cm−1 2165–2110 cm−1 2140–1990 cm−1 1000–1100 cm−1 1100–1200 cm−1 540–760 cm−1 500–600 cm−1 500 cm−1 1540 cm−1 1380 cm−1 1520, 1350 cm−1 trifluromethyl chloroalkanes any bromoalkanes any iodoalkanes any N─O nitro compounds aliphatic aromatic two strong, broad bands weak to medium medium to strong medium to strong stronger weaker lower if conjugated ________________________WORLD TECHNOLOGIES________________________ Chapter- 3 Near-infrared Spectroscopy WT Near IR absorption spectrum of dichloromethane showing complicated overlapping overtones of mid IR absorption features. Near-infrared spectroscopy (NIRS) is a spectroscopic method that uses the nearinfrared region of the electromagnetic spectrum (from about 800 nm to 2500 nm). Typical applications include pharmaceutical, medical diagnostics (including blood sugar and oximetry), food and agrochemical quality control, and combustion research, as well as cognitive neuroscience research. ________________________WORLD TECHNOLOGIES________________________ Theory Near-infrared spectroscopy is based on molecular overtone and combination vibrations. Such transitions are forbidden by the selection rules of quantum mechanics. As a result, the molar absorptivity in the near IR region is typically quite small. One advantage is that NIR can typically penetrate much farther into a sample than mid infrared radiation. Nearinfrared spectroscopy is, therefore, not a particularly sensitive technique, but it can be very useful in probing bulk material with little or no sample preparation. The molecular overtone and combination bands seen in the near IR are typically very broad, leading to complex spectra; it can be difficult to assign specific features to specific chemical components. Multivariate (multiple wavelength) calibration techniques (e.g., principal components analysis, partial least squares, or artificial neural networks) are often employed to extract the desired chemical information. Careful development of a set of calibration samples and application of multivariate calibration techniques is essential for near-infrared analytical methods. History WT Near infrared spectrum of liquid ethanol. The discovery of near-infrared energy is ascribed to Herschel in the 19th century, but the first industrial application began in the 1950s. In the first applications, NIRS was used ________________________WORLD TECHNOLOGIES________________________ only as an add-on unit to other optical devices that used other wavelengths such as ultraviolet (UV), visible (Vis), or mid-infrared (MIR) spectrometers. In the 1980s, a single unit, stand-alone NIRS system was made available, but the application of NIRS was focused more on chemical analysis. With the introduction of light-fiber optics in the mid-1980s and the monochromator-detector developments in early-1990s, NIRS became a more powerful tool for scientific research. This optical method can be used in a number of fields of science including physics, physiology, or medicine. It is only in the last few decades that NIRS began to be used as a medical tool for monitoring patients. Instrumentation Instrumentation for near-IR (NIR) spectroscopy is similar to instruments for the UVvisible and mid-IR ranges. There is a source, a detector, and a dispersive element (such as a prism, or, more commonly, a diffraction grating) to allow the intensity at different wavelengths to be recorded. Fourier transform NIR instruments using an interferometer are also common, especially for wavelengths above ~1000 nm. Depending on the sample, the spectrum can be measured in either reflection or transmission. WT Common incandescent or quartz halogen light bulbs are most often used as broadband sources of near-infrared radiation for analytical applications. Light-emitting diodes (LEDs) are also used; they offer greater lifetime and spectral stability and reduced power requirements. The type of detector used depends primarily on the range of wavelengths to be measured. Silicon-based CCDs are suitable for the shorter end of the NIR range, but are not sufficiently sensitive over most of the range (over 1000 nm). InGaAs and PbS devices are more suitable though less sensitive than CCDs. In certain diode array (DA) NIRS instruments, both silicon-based and InGaAs detectors are employed in the same instrument. Such instruments can record both UV-visible and NIR spectra 'simultaneously'. Instruments intended for chemical imaging in the NIR may use a 2D array detector with a acousto-optic tunable filter. Multiple images may be recorded sequentially at different narrow wavelength bands. Many commercial instruments for UV/vis spectroscopy are capable of recording spectra in the NIR range (to perhaps ~900 nm). In the same way, the range of some mid-IR instruments may extend into the NIR. In these instruments, the detector used for the NIR wavelengths is often the same detector used for the instrument's "main" range of interest. ________________________WORLD TECHNOLOGIES________________________ Applications WT NIR sensor for moisture measurement installed on a belt conveyor The primary application of NIRS to the human body uses the fact that the transmission and absorption of NIR light in human body tissues contains information about hemoglobin concentration changes. When a specific area of the brain is activated, the localized blood volume in that area changes quickly. Optical imaging can measure the location and activity of specific regions of the brain by continuously monitoring blood hemoglobin levels through the determination of optical absorption coefficients. Typical applications of NIR spectroscopy include the analysis of foodstuffs, pharmaceuticals, combustion products and a major branch of astronomical spectroscopy. Astronomical spectroscopy Near-infrared spectroscopy is used in astronomy for studying the atmospheres of cool stars where molecules can form. The vibrational and rotational signatures of molecules such as titanium oxide, cyanide, and carbon monoxide can be seen in this wavelength range and can give a clue towards the star's spectral type. It is also used for studying molecules in other astronomical contexts, such as in molecular clouds where new stars are formed. The astronomical phenomenon known as reddening means that near-infrared wavelengths are less affected by dust in the interstellar medium, such that regions inaccessible by optical spectroscopy can be studied in the near-infrared. Since dust and gas are strongly associated, these dusty regions are exactly those where infrared spectroscopy is most useful. The near-infrared spectra of very young stars provide ________________________WORLD TECHNOLOGIES________________________ important information about their ages and masses, which is important for understanding star formation in general. Remote monitoring Techniques have been developed for NIR spectroscopic imaging. These have been used for a wide range of uses, including the remote investigation of plants and soils. Data can be collected from instruments on airplanes or satellites to assess ground cover and soil chemistry. Materials Science Techniques have been developed for NIR spectroscopy of microscopic sample areas for film thickness measurements, research into the optical characteristics of nanoparticles and optical coatings for the telecommunications industry. WT Medical uses Medical applications of NIRS center on the non-invasive measurement of the amount and oxygen content of hemoglobin, as well as the use of exogenous optical tracers in conjunction with flow kinetics. NIRS can be used for non-invasive assessment of brain function through the intact skull in human subjects by detecting changes in blood hemoglobin concentrations associated with neural activity, e.g., in branches of Cognitive psychology as a partial replacement for fMRI techniques. NIRS can be used on infants, where fMRI cannot (at least in the United States), and NIRS is much more portable than fMRI machines, even wireless instrumentation is available, which enables investigations in freely moving subjects). However, NIRS cannot fully replace fMRI because it can only be used to scan cortical tissue, where fMRI can be used to measure activation throughout the brain. The application in functional mapping of the human cortex is called optical topography (OT), near infrared imaging (NIRI) or functional NIRS (fNIRS). The term optical tomography is used for three-dimensional NIRS. The terms NIRS, NIRI and OT are often used interchangeably, but they have some distinctions. The most important difference between NIRS and OT/NIRI is that OT/NIRI is used mainly to detect changes in optical properties of tissue simultaneously from multiple measurement points and display the results in the form of a map or image over a specific area, whereas NIRS provides quantitative data in absolute terms on up to a few specific points. The latter is also used to investigate other tissues such as, e.g., muscle, breast and tumors. By employing several wavelengths and time resolved (frequency or time domain) and/or spatially resolved methods blood flow, volume and oxygenation can be quantified. These measurements are a form of oximetry. Applications of oximetry by NIRS methods include the detection of illnesses which affect the blood circulation (e.g., peripheral vascular disease), the detection and assessment of breast tumors, and the optimization of ________________________WORLD TECHNOLOGIES________________________ training in sports medicine. These techniques can also be used for industry or agro processes in order to predict particle size/density. The use of NIRS in conjunction with a bolus injection of indocyanine green (ICG) has been used to measure cerebral blood flow and cerebral metabolic rate of oxygen consumption in neonatal models. NIRS is starting to be used in pediatric critical care, to help deal with cardiac surgery post-op. Indeed, NIRS is able to measure venous oxygen saturation (SVO2), which is determined by the cardiac output, as well as other parameters (FiO2, hemoglobin, oxygen uptake). Therefore, following the NIRS gives critical care physicians a notion of the cardiac output. NIRS is liked by patients, because it is non-invasive, is painless, and uses non-ionizing radiation. The instrumental development of NIRS/NIRI/OT has proceeded tremendously during the last years and, in particular, in terms of quantification, imaging and miniaturization. WT Particle measurement NIR is often used in particle sizing in a range of different fields, including studying pharmaceutical and agricultural powders. Industrial uses As opposed to NIRS used in optical topography, general NIRS used in chemical assays does not provide imaging by mapping. For example, a clinical carbon dioxide analyzer requires reference techniques and calibration routines to be able to get accurate CO2 content change. In this case, calibration is performed by adjusting the zero control of the sample being tested after purposefully supplying 0% CO2 or another known amount of CO2 in the sample. Normal compressed gas from distributors contains about 95% O2 and 5% CO2, which can also be used to adjust %CO2 meter reading to be exactly 5% at initial calibration. ________________________WORLD TECHNOLOGIES________________________ Chapter- 4 Two-dimensional Infrared Spectroscopy and Two-dimensional Correlation Analysis Two-dimensional infrared spectroscopy WT Pulse Sequence used to obtain a two-dimensional Fourier transform infrared spectrum. The time period τ1 is usually referred to as the coherence time and the second time period τ2 is known as the waiting time. The excitation frequency is obtained by Fourier transforming along the τ1 axis. Two-dimensional infrared spectroscopy (2DIR) is a nonlinear infrared spectroscopy technique that has the ability to correlate vibrational modes in condensed-phase systems. This technique provides information beyond linear infrared spectra, by spreading the vibrational information along multiple axes, yielding a frequency correlation spectrum. A frequency correlation spectrum can offer structural information such as vibrational mode coupling, anharmonicities, along with chemical dynamics such as energy transfer rates and molecular dynamics with femtosecond time resolution. 2DIR experiments have only become possible with the development of ultrafast lasers and the ability to generate femtosecond infrared pulses. ________________________WORLD TECHNOLOGIES________________________ Systems studied Among the many systems studied with infrared spectroscopy are water, metal carbonyls, short polypeptides, proteins, and DNA oligomers. Experimental Approaches There are two main approaches to two-dimensional spectroscopy, the Fourier-transform method, in which the data is collected in the time-domain and then Fourier-transformed to obtain a frequency-frequency 2D correlation spectrum, and the frequency domain approach in which all the data is collected directly in the frequency domain. Time domain WT The time-domain approach consists of applying two pump pulses. The first pulse creates a coherence between the vibrational modes of the molecule and the second pulse creates a population, effectively storing information in the molecules. After a determined waiting time, ranging from a zero to a few hundred picoseconds, an interaction with a third pulse again creates a coherence, which, due to an oscillating dipole, radiates an infrared signal. The radiated signal is heterodyned with a reference pulse in order to retrieve frequency and phase information; the signal is usually collected in the frequency domain using a spectrometer yielding detection frequency ω3. A two-dimensional Fourier-transform along ω1 then yields a (ω1, ω3) correlation spectrum. Frequency domain Similarly, in the frequency-domain approach, a narrowband pump pulse is applied and, after a certain waiting time, then a broadband pulse probes the system. A 2DIR correlation spectrum is obtained by plotting the probe frequency spectrum at each pump frequency. ________________________WORLD TECHNOLOGIES________________________ Spectral Interpretation WT Schematic picture of a 2DIR spectrum. The blue peaks on the diagonal corresponds to bleaching of the ground state. The red peaks corresponds to absorption of the excited states. The smaller cross peaks arise due to coupling between the two states. The linear absorption spectrum is indicated above the 2DIR spectrum. The two peaks here reveal no information on coupling between the two states. After the waiting time in the experiment it is possible to reach doubly excited states. This results in the appearance of an overtone peak. The anharmonicity of a vibration can be read from the spectra as the distance between the diagonal peak and the overtone peak. One obvious advantage of 2DIR spectra over normal linear absorption spectra is that they reveal the coupling between different states. This for example allows for the determination of the angle between the involved transition dipoles. The true power of 2DIR spectroscopy is that it allows following dynamical processes as chemical exchange, vibrational population transfer, and molecular reorientation on the sub-picosecond time scale. It has successfully been used to study hydrogen bond forming and breaking and to determine the transition state geometry of a structural rearrangement in an iron carbonyl compound. ________________________WORLD TECHNOLOGIES________________________ Two-dimensional correlation analysis Two dimensional correlation analysis is a mathematical technique that is used to study changes in measured signals. As mostly spectroscopic signals are discussed, sometime also two dimensional correlation spectroscopy is used and refers to the same technique. In 2D correlation analysis, a sample is subjected to an external perturbation while all other parameters of the system are kept at the same value. This perturbation can be a systematic and controlled change in temperature, pressure, pH, chemical composition of the system, or even time after a catalyst was added to a chemical mixture. As a result of the controlled change (the perturbation), the system will undergo variations which are measured by a chemical or physical detection method. The measured signals or spectra will shown systematic variations that are processed with 2D correlation analysis for interpretation. WT When one considers spectra that consist of few bands, it is quite obvious to determine which bands are subject to a changing intensity. Such a changing intensity can be caused for example by chemical reactions. However, the interpretation of the measured signal becomes more tricky when spectra are complex and bands are heavily overlapping. Two dimensional correlation analysis allows one to determine at which positions in such a measured signal there is a systematic change in a peak, either continuous rising or drop in intensity. 2D correlation analysis results in two complementary signals, which referred to as the 2D synchronous and 2D asynchronous spectrum. These signals allow amongst others 1. to determine the events that are occurring at the same time (in phase) and those events that are occurring at different times (out of phase) 2. to determine the sequence of spectral changes 3. to identify various inter- and intramolecular interactions 4. band assignments of reacting groups 5. to detect correlations between spectra of different techniques, for example near infrared spectroscopy (NIR) and Raman spectroscopy History 2D correlation analysis originated from 2D NMR spectroscopy. Isao Noda developed perturbation based 2D spectroscopy in the 1980s. This technique required sinusoidal perturbations to the chemical system under investigation. This specific type of the applied perturbation severely limited its possible applications. Following research done by several groups of scientists, perturbation based 2D spectroscopy could be developed to a more extended and generalized broader base. Since the development of generalized 2D correlation analysis in 1993 based on Fourier transformation of the data, 2D correlation analysis gained widespread use. Alternative techniques that were simpler to calculate, for example the disrelation spectrum, were also developed simultaneously. Because of its computational efficiency and simplicity, the Hilbert transform is nowadays used for the calculation of the 2D spectra. To date, 2D correlation analysis is not only used for the ________________________WORLD TECHNOLOGIES________________________ interpretation of many types of spectroscopic data (including XRF, UV/VIS spectroscopy, fluorescence, infrared, and Raman spectra), although its application is not limited to spectroscopy. Properties of 2D correlation analysis WT Demo dataset consisting of signals at specific intervals (1 out of 3 signals on a total of 15 signals is shown for clarity), peaks at 10 and 20 are rising in intensity whereas the peaks at 30 and 40 have a decreasing intensity 2D correlation analysis is frequently used for its main advantage: increasing the spectral resolution by spreading overlapping peaks over two dimensions and as a result simplification of the interpretation of one dimensional spectra that are otherwise visually indistinguishable from each other. Further advantages are its ease of application and the possibility to make the distinction between band shifts and band overlap. Each type of spectral event, band shifting, overlapping bands of which the intensity changes in the opposite direction, band broadening, baseline change, etc. has a particular 2D pattern. ________________________WORLD TECHNOLOGIES________________________ Presence of 2D spectra WT Schematic presence of a 2D correlation spectrum with peak positions represented by dots. Region A is the main diagonal containing autopeaks, off-diagonal regions B contain cross-peaks. 2D synchronous and asynchronous spectra are basically 3D-datasets and are generally represented by contour plots. X- and y-axes are identical to the x-axis of the original dataset, whereas the different contours represent the magnitude of correlation between the spectral intensities. The 2D synchronous spectrum is symmetric relative to the main diagonal. The main diagonal thus contains positive peaks. As the peaks at (x,y) in de 2D synchronous spectrum are a measure for the correlation between the intensity changes at x and y in the original data, these main diagonal peaks are also called autopeaks and the main diagonal signal is referred to as autocorrelation signal. The off-diagonal crosspeaks can be either positive or negative. On the other hand the asynchronous spectrum is asymmetric and never has peaks on the main diagonal. ________________________WORLD TECHNOLOGIES________________________ Generally contour plots of 2D spectra are oriented with rising axes from left to right and top to down. Other orientations are possible, but interpretation has to be adapted accordingly. Calculation of 2D spectra Suppose the original dataset D contains the n spectra in rows. The signals of the original dataset are generally preprocessed. The original spectra are compared to a reference spectrum. By subtracting a reference spectrum, often the average spectrum of the dataset, so called dynamic spectra are calculated which form the corresponding dynamic dataset E. The presence and interprentation may be dependent on the choice of reference spectrum. The equations below are valid for equally spaced measurements of the perturbation. WT Calculation of the synchronous spectrum A 2D synchronous spectrum expresses the similarity between spectral of the data in the original dataset. In generalized 2D correlation spectroscopy this is mathematically expressed as covariance (or correlation). where: • • • • Φ is the 2D synchronous spectrum ν1 en ν1 are two spectral channels yν is the vector composed of the signal intensities in E in column ν n the number of signals in the original dataset Calculation of the asynchronous spectrum Orthogonal spectra to the dynamic dataset E are obtained with the Hilbert-transform: where: • • • • • Ψ is the 2D synchronous spectrum ν1 en ν1 are two spectral channels yν is the vector composed of the signal intensities in E in column ν n the number of signals in the original dataset N the Noda-Hilbert transform matrix ________________________WORLD TECHNOLOGIES________________________ The values of N, Nj, k are determined as follows: • 0 if j = k if j ≠ k • where: • • j the row number k the column number Interpretation Interpretation of two-dimensional correlation spectra can be considered to consist of several stages. WT Detection of peaks of which the intensity changes in the original dataset Autocorrelation signal on the main diagonal of the synchronous 2D spectrum of the figure below (arbitrary axis units) As real measurement signals contain a certain level of noise, the derived 2D spectra are influenced and degraded with substantial higher amounts of noise. Hence, interpretation begins with studying the autocorrelation spectrum on the main diagonal of the 2D synchronous spectrum. In the 2D synchronous main diagonal signal on the right 4 peaks ________________________WORLD TECHNOLOGIES________________________ are visible at 10, 20, 30, and 40. This indicates that in the original dataset 4 peaks of changing intensity are present. The intensity of peaks on the autocorrelation spectrum are directly proportional to the relative importance of the intensity change in the original spectra. Hence, if an intense band is present at position x, it is very likely that a true intensity change is occurring and the peak is not due to noise. Additional techniques help to filter the peaks that can be seen in the 2D synchronous and asynchronous spectra. Determining the direction of intensity change WT Example of a two-dimensional correlation spectrum. Open circles in this simplified view represent positive peaks, while discs represent negative peaks ________________________WORLD TECHNOLOGIES________________________ It is not always possible to unequivocally determine the direction of intensity change, such as is for example the case for highly overlapping signals next to each other and of which the intensity changes in the opposite direction. This is where the off diagonal peaks in the synchronous 2D spectrum are used for: 1. if there is a positive cross-peak at (x, y) in the synchronous 2D spectrum, the intensity of the signals at x and y changes in the same direction 2. if there is a negative cross-peak at (x, y) in the synchronous 2D spectrum, the intensity of the signals at x and y changes in the opposite direction As can be seen in the 2D synchronous spectrum on the right, the intensity changes of the peaks at 10 and 30 are related and the intensity of the peak at 10 and 30 changes in the opposite direction (negative cross-peak at (10,30)). The same is true for the peaks at 20 and 40. WT Determining the sequence of events Most importantly, with the sequential order rules, also referred to as Noda's rules, the sequence of the intensity changes can be determined. By carefully interpreting the signs of the 2D synchronous and asynchronous cross peaks with the following rules, the sequence of spectral events during the experiment can be determined: 1. if the intensities of the bands at x and y in the dataset are changing in the same direction, the synchronous 2D cross peak at (x,y) is positive 2. if the intensities of the bands at x and y in the dataset are changing in the opposite direction, the synchronous 2D cross peak at (x,y) is negative 3. if the change at x mainly precedes the change in the band at y, the asynchronous 2D cross peak at (x,y) is positive 4. if the change at x mainly follows the change in the band at y, the asynchronous 2D cross peak at (x,y) is negative 5. if the asynchronous 2D cross peak at (x,y) is negative, the interpretation of rule 1 and 2 for the synchronous 2D peak at (x,y) has to be reversed where x and y are the positions on the x-xaxis of two bands in the original data that are subject to intensity changes. Following the rules above. It can be derived that the changes at 10 and 30 occur simultaneously and the changes in intensity at 20 and 40 occur simultaneously as well. Because of the positive asynchronous cross-peak at (10, 20), the changes at 10 and 30 (predominantly) occur before the intensity changes at 20 and 40. ________________________WORLD TECHNOLOGIES________________________ Chapter- 5 Fluorescence Correlation Spectroscopy Fluorescence correlation spectroscopy (FCS) is a correlation analysis of fluctuation of the fluorescence intensity. The analysis provides parameters of the physics under the fluctuations. One of the interesting applications of this is an analysis of the concentration fluctuations of fluorescent particles (molecules) in solution. In this application, the fluorescence emitted from a very tiny space in solution containing a small number of fluorescent particles (molecules) is observed. The fluorescence intensity is fluctuating due to Brownian motion of the particles. In other words, the number of the particles in the sub-space defined by the optical system is randomly changing around the average number. The analysis gives the average number of fluorescent particles and average diffusion time, when the particle is passing through the space. Eventually, both the concentration and size of the particle (molecule) are determined. Since the method is observing a small number of molecule in a very tiny spot, it is a very sensitive analytical tool. Both parameters are important in biochemical research, biophysics, and chemistry. In contrast to other methods, such as HPLC analysis, FCS has no physical separation process and has a good spatial resolution determined by the optics. These are of great advantage. Moreover, the method enables us to observe fluorescence-tagged molecules in the biochemical pathway in intact living cells. This opens a new area, "in situ or in vivo biochemistry": tracing the biochemical pathway in intact cells and organs. WT Commonly, FCS is employed in the context of optical microscopy, in particular confocal or two-photon microscopy. In these techniques light is focused on a sample and the measured fluorescence intensity fluctuations (due to diffusion, physical or chemical reactions, aggregation, etc.) are analyzed using the temporal autocorrelation. Because the measured property is essentially related to the magnitude and/or the amount of fluctuations, there is an optimum measurement regime at the level when individual species enter or exit the observation volume (or turn on and off in the volume). When too many entities are measured at the same time the overall fluctuations are small in comparison to the total signal and may not be resolvable – in the other direction, if the individual fluctuation-events are too sparse in time, one measurement may take prohibitively too long. FCS is in a way the fluorescent counterpart to dynamic light scattering, which uses coherent light scattering, instead of (incoherent) fluorescence. When an appropriate model is known, FCS can be used to obtain quantitative information such as • diffusion coefficients ________________________WORLD TECHNOLOGIES________________________ • • • • hydrodynamic radii average concentrations kinetic chemical reaction rates singlet-triplet dynamics Because fluorescent markers come in a variety of colors and can be specifically bound to a particular molecule (e.g. proteins, polymers, metal-complexes, etc.), it is possible to study the behavior of individual molecules (in rapid succession in composite solutions). With the development of sensitive detectors such as avalanche photodiodes the detection of the fluorescence signal coming from individual molecules in highly dilute samples has become practical. With this emerged the possibility to conduct FCS experiments in a wide variety of specimens, ranging from materials science to biology. The advent of engineered cells with genetically tagged proteins (like green fluorescent protein) has made FCS a common tool for studying molecular dynamics in living cells. History WT Signal-correlation techniques were first experimentally applied to fluorescence in 1972 by Magde, Elson, and Webb, who are therefore commonly credited as the "inventors" of FCS. The technique was further developed in a group of papers by these and other authors soon after, establishing the theoretical foundations and types of applications. Beginning in 1993, a number of improvements in the measurement techniques—notably using confocal microscopy, and then two-photon microscopy—to better define the measurement volume and reject background—greatly improved the signal-to-noise ratio and allowed single molecule sensitivity. Since then, there has been a renewed interest in FCS, and as of August 2007 there have been over 3,000 papers using FCS found in Web of Science. In addition, there has been a flurry of activity extending FCS in various ways, for instance to laser scanning and spinning-disk confocal microscopy (from a stationary, single point measurement), in using cross-correlation (FCCS) between two fluorescent channels instead of autocorrelation, and in using Förster Resonance Energy Transfer (FRET) instead of fluorescence. Typical FCS setup The typical FCS setup consists of a laser line (wavelengths ranging typically from 405– 633 nm (cw), and from 690–1100 nm (pulsed)), which is reflected into a microscope objective by a dichroic mirror. The laser beam is focused in the sample, which contains fluorescent particles (molecules) in such high dilution, that only a few are within the focal spot (usually 1–100 molecules in one fL). When the particles cross the focal volume, they fluoresce. This light is collected by the same objective and, because it is red-shifted with respect to the excitation light it passes the dichroic mirror reaching a detector, typically a photomultiplier tube or avalanche photodiode detector. The resulting electronic signal can be stored either directly as an intensity versus time trace to be analyzed at a later point, or computed to generate the autocorrelation directly (which requires special acquisition cards). The FCS curve by itself only represents a time- ________________________WORLD TECHNOLOGIES________________________ spectrum. Conclusions on physical phenomena have to be extracted from there with appropriate models. The parameters of interest are found after fitting the autocorrelation curve to modeled functional forms. The measurement volume The measurement volume is a convolution of illumination (excitation) and detection geometries, which result from the optical elements involved. The resulting volume is described mathematically by the point spread function (or PSF), it is essentially the image of a point source. The PSF is often described as an ellipsoid (with unsharp boundaries) of few hundred nanometers in focus diameter, and almost one micrometre along the optical axis. The shape varies significantly (and has a large impact on the resulting FCS curves) depending on the quality of the optical elements (it is crucial to avoid astigmatism and to check the real shape of the PSF on the instrument). In the case of confocal microscopy, and for small pinholes (around one Airy unit), the PSF is well approximated by Gaussians: WT where I0 is the peak intensity, r and z are radial and axial position, and ωxy and ωz are the radial and axial radii, and ωz > ωxy. This Gaussian form is assumed in deriving the functional form of the autocorrelation. Typically ωxy is 200–300 nm, and ωz is 2–6 times larger. One common way of calibrating the measurement volume parameters is to perform FCS on a species with known diffusion coefficient and concentration (see below). Diffusion coefficients for common fluorophores in water are given in a later section. The Gaussian approximation works to varying degrees depending on the optical details, and corrections can sometimes be applied to offset the errors in approximation. Autocorrelation function The (temporal) autocorrelation function is the correlation of a time series with itself shifted by time τ, as a function of τ: where is the deviation from the mean intensity. The normalization (denominator) here is the most commonly used for FCS, because then the correlation at τ = 0, G(0), is related to the average number of particles in the measurement volume. ________________________WORLD TECHNOLOGIES________________________ Interpreting the autocorrelation function To extract quantities of interest, the autocorrelation data can be fitted, typically using a nonlinear least squares algorithm. The fit's functional form depends on the type of dynamics (and the optical geometry in question). Normal diffusion The fluorescent particles used in FCS are small and thus experience thermal motions in solution. The simplest FCS experiment is thus normal 3D diffusion, for which the autocorrelation is: WT where a = ωz / ωxy is the ratio of axial to radial e − 2 radii of the measurement volume, and τD is the characteristic residence time. This form was derived assuming a Gaussian , measurement volume. Typically, the fit would have three free parameters—G(0), and τD--from which the diffusion coefficient and fluorophore concentration can be obtained. With the normalization used in the previous section, G(0) gives the mean number of diffusers in the volume <N>, or equivalently—with knowledge of the observation volume size—the mean concentration: where the effective volume is found from integrating the Gaussian form of the measurement volume and is given by: τD gives the diffusion coefficient: Anomalous diffusion If the diffusing particles are hindered by obstacles or pushed by a force (molecular motors, flow, etc.) the dynamics is often not sufficiently well-described by the normal diffusion model, where the mean squared displacement (MSD) grows linearly with time. Instead the diffusion may be better described as anomalous diffusion, where the temporal dependenc of the MSD is non-linear as in the power-law: ________________________WORLD TECHNOLOGIES________________________ where Da is an anomalous diffusion coefficient. "Anomalous diffusion" commonly refers only to this very generic model, and not the many other possibilities that might be described as anomalous. Also, a power law is, in a strict sense, the expected form only for a narrow range of rigorously defined systems, for instance when the distribution of obstacles is fractal. Nonetheless a power law can be a useful approximation for a wider range of systems. The FCS autocorrelation function for anomalous diffusion is: where the anomalous exponent α is the same as above, and becomes a free parameter in the fitting. WT Using FCS, the anomalous exponent has been shown to be an indication of the degree of molecular crowding (it is less than one and smaller for greater degrees of crowding). Polydisperse diffusion If there are diffusing particles with different sizes (diffusion coefficients), it is common to fit to a function that is the sum of single component forms: where the sum is over the number different sizes of particle, indexed by i, and αi gives the weighting, which is related to the quantum yield and concentration of each type. This introduces new parameters, which makes the fitting more difficult as a higher dimensional space must be searched. Nonlinear least square fitting typically becomes unstable with even a small number of τD,is. A more robust fitting scheme, especially useful for polydisperse samples, is the Maximum Entropy Method. Diffusion with flow With diffusion together with a uniform flow with velocity v in the lateral direction, the autocorrelation is: where τv = ωxy / v is the average residence time if there is only a flow (no diffusion). ________________________WORLD TECHNOLOGIES________________________ Chemical relaxation A wide range of possible FCS experiments involve chemical reactions that continually fluctuate from equilibrium because of thermal motions (and then "relax"). In contrast to diffusion, which is also a relaxation process, the fluctuations cause changes between states of different energies. One very simple system showing chemical relaxation would be a stationary binding site in the measurement volume, where particles only produce signal when bound (e.g. by FRET, or if the diffusion time is much faster than the sampling interval). In this case the autocorrelation is: where WT is the relaxation time and depends on the reaction kinetics (on and off rates), and: is related to the equilibrium constant K. Most systems with chemical relaxation also show measureable diffusion as well, and the autocorrelation function will depend on the details of the system. If the diffusion and chemical reaction are decoupled, the combined autocorrelation is the product of the chemical and diffusive autocorrelations. Triplet state correction The autocorrelations above assume that the fluctuations are not due to changes in the fluorescent properties of the particles. However, for the majority of (bio)organic fluorophores—e.g. green fluorescent protein, rhodamine, Cy3 and Alexa Fluor dyes— some fraction of illuminated particles are excited to a triplet state (or other non-radiative decaying states) and then do not emit photons for a characteristic relaxation time τF. Typically τF is on the order of microseconds, which is usually smaller than the dynamics of interest (e.g. τD) but large enough to be measured. A multiplicative term is added to the autocorrelation account for the triplet state. For normal diffusion: where is the fraction of particles that have entered the triplet state and is the corresponding triplet state relaxation time. If the dynamics of interest are much slower ________________________WORLD TECHNOLOGIES________________________ than the triplet state relaxation, the short time component of the autocorrelation can simply be truncated and the triplet term is unnecessary. Common fluorescent probes The fluorescent species used in FCS is typically a biomolecule of interest that has been tagged with a fluorophore (using immunohistochemistry for instance), or is a naked fluorophore that is used to probe some environment of interest (e.g. the cytoskeleton of a cell). The following table gives diffusion coefficients of some common fluorophores in water at room temperature, and their excitation wavelengths. Fluorescent dye Rhodamine 6G Rhodamine 110 Tetramethyl rhodamine Cy3 Cy5 carboxyfluorescein Alexa-488 Atto655-maleimide Atto655-carboxylicacid 2′, 7′-difluorofluorescein (Oregon Green488) (×10−10 m2 s−1) Excitation wavelength (nm) 2.8, 3.0, 4.14 ± 0.05 @ 25.00 °C 514 2.7 488 2.6 543 2.8 543 2.5, 3.7 ± 0.15 @ 25.00 °C 633 3.2 488 1.96,4.35 @ 22.5±0.5 °C 488 4.07 ± 0.1 @ 25.00 °C 663 4.26 ± 0.08 @ 25.00 °C 663 WT 4.11 ± 0.06 @ 25.00 °C 498 Variations of FCS FCS almost always refers to the single point, single channel, temporal autocorrelation measurement, although the term "fluorescence correlation spectroscopy" out of its historical scientific context implies no such restriction. FCS has been extended in a number of variations by different researchers, with each extension generating another name (usually an acronym). Fluorescence cross-correlation spectroscopy (FCCS) FCS is sometimes used to study molecular interactions using differences in diffusion times (e.g. the product of an association reaction will be larger and thus have larger diffusion times than the reactants individually); however, FCS is relatively insensitive to molecular mass as can be seen from the following equation relating molecular mass to the diffusion time of globular particles (e.g. proteins): ________________________WORLD TECHNOLOGIES________________________ where is the viscosity of the sample and is the molecular mass of the fluorescent species. In practice, the diffusion times need to be sufficiently different—a factor of at least 1.6--which means the molecular masses must differ by a factor of 4. Dual color fluorescence cross-correlation spectroscopy (FCCS) measures interactions by crosscorrelating two or more fluorescent channels (one channel for each reactant), which distinguishes interactions more sensitively than FCS, particularly when the mass change in the reaction is small. Brightness analysis methods (N&B, PCH, FIDA, Cumulant Analysis) Fluorescence cross correlation spectroscopy overcomes the weak dependence of diffusion rate on molecular mass by looking at multicolor coincidence. What about homointeractions? The solution lies in brightness analysis. These methods use the heterogeneity in the intensity distribution of fluorescence to measure the molecular brightness of different species in a sample. Since dimers will contain twice the number of fluorescent labels as monomers, their molecular brightness will be approximately double that of monomers. As a result, the relative brightness is sensitive a measure of oligomerization. The average molecular brightness ( ) is related to the variance (σ2) and the average intensity ( ) as follows: WT Here fi and εi are the fractional intensity and molecular brigthness, respectively, of species i. Two- and three- photon FCS excitation Several advantages in both spatial resolution and minimizing photodamage/photobleaching in organic and/or biological samples are obtained by two-photon or three-photon excitation FCS. FRET-FCS Another FCS based approach to studying molecular interactions uses fluorescence resonance energy transfer (FRET) instead of fluorescence, and is called FRET-FCS. With FRET, there are two types of probes, as with FCCS; however, there is only one channel and light is only detected when the two probes are very close—close enough to ensure an interaction. The FRET signal is weaker than with fluorescence, but has the advantage that there is only signal during a reaction (aside from autofluorescence). Image correlation spectroscopy (ICS) When the motion is slow (in biology, for example, diffusion in a membrane), getting adequate statistics from a single-point FCS experiment may take a prohibitively long ________________________WORLD TECHNOLOGIES________________________ time. More data can be obtained by performing the experiment in multiple spatial points in parallel, using a laser scanning confocal microscope. This approach has been called Image Correlation Spectroscopy (ICS). The measurements can then be averaged together. Another variation of ICS performs a spatial autocorrelation on images, which gives information about the concentration of particles. The correlation is then averaged in time. A natural extension of the temporal and spatial correlation versions is spatio-temporal ICS (STICS). In STICS there is no explicit averaging in space or time (only the averaging inherent in correlation). In systems with non-isotropic motion (e.g. directed flow, asymmetric diffusion), STICS can extract the directional information. A variation that is closely related to STICS (by the Fourier transform) is k-space Image Correlation Spectroscopy (kICS). There are cross-correlation versions of ICS as well. WT Scanning FCS variations Some variations of FCS are only applicable to serial scanning laser microscopes. Image Correlation Spectroscopy and its variations all were implemented on a scanning confocal or scanning two photon microscope, but transfer to other microscopes, like a spinning disk confocal microscope. Raster ICS (RICS), and position sensitive FCS (PSFCS) incorporate the time delay between parts of the image scan into the analysis. Also, low dimensional scans (e.g. a circular ring)—only possible on a scanning system—can access time scales between single point and full image measurements. Scanning path has also been made to adaptively follow particles. Spinning disk FCS, and spatial mapping Any of the image correlation spectroscopy methods can also be performed on a spinning disk confocal microscope, which in practice can obtain faster imaging speeds compared to a laser scanning confocal microscope. This approach has recently been applied to diffusion in a spatially varying complex environment, producing a pixel resolution map of a diffusion coefficient. The spatial mapping of diffusion with FCS has subsequently been extended to the TIRF system. Spatial mapping of dynamics using correlation techniques had been applied before, but only at sparse points or at coarse resolution. Total internal reflection FCS Total internal reflection fluorescence (TIRF) is a microscopy approach that is only sensitive to a thin layer near the surface of a coverslip, which greatly minimizes background fluorscence. FCS has been extended to that type of microscope, and is called TIR-FCS. Because the fluorescence intensity in TIRF falls off exponentially with distance from the coverslip (instead of as a Gaussian with a confocal), the autocorrelation function is different. ________________________WORLD TECHNOLOGIES________________________ Other fluorescent dynamical approaches There are two main non-correlation alternatives to FCS that are widely used to study the dynamics of fluorescent species. Fluorescence recovery after photobleaching (FRAP) In FRAP, a region is briefly exposed to intense light, irrecoverably photobleaching fluorophores, and the fluorescence recovery due to diffusion of nearby (non-bleached) fluorophores is imaged. A primary advantage of FRAP over FCS is the ease of interpreting qualitative experiments common in cell biology. Differences between cell lines, or regions of a cell, or before and after application of drug, can often be characterized by simple inspection of movies. FCS experiments require a level of processing and are more sensitive to potentially confounding influences like: rotational diffusion, vibrations, photobleaching, dependence on illumination and fluorescence color, inadequate statistics, etc. It is much easier to change the measurement volume in FRAP, which allows greater control. In practice, the volumes are typically larger than in FCS. While FRAP experiments are typically more qualitative, some researchers are studying FRAP quantitatively and including binding dynamics. A disadvantage of FRAP in cell biology is the free radical perturbation of the cell caused by the photobleaching. It is also less versatile, as it cannot measure concentration or rotational diffusion, or colocalization. FRAP requires a significantly higher concentration of fluorophores than FCS. WT Particle tracking In particle tracking, the trajectories of a set of particles are measured, typically by applying particle tracking algorithms to movies. Particle tracking has the advantage that all the dynamical information is maintained in the measurement, unlike FCS where correlation averages the dynamics to a single smooth curve. The advantage is apparent in systems showing complex diffusion, where directly computing the mean squared displacement allows straightforward comparison to normal or power law diffusion. To apply particle tracking, the particles have to be distinguishable and thus at lower concentration than required of FCS. Also, particle tracking is more sensitive to noise, which can sometimes affect the results unpredictably. ________________________WORLD TECHNOLOGIES________________________ Chapter- 6 Rotational Spectroscopy WT Part of the rotational-vibrational spectrum of carbon monoxide (CO) gas (from FTIR), showing the presence of P- and R- branches. Frequency is on the x-axis, and transmittance on the y-axis. Rotational spectroscopy or microwave spectroscopy studies the absorption and emission of electromagnetic radiation (typically in the microwave region of the electromagnetic spectrum) by molecules associated with a corresponding change in the rotational quantum number of the molecule. The use of microwaves in spectroscopy essentially became possible due to the development of microwave technology for RADAR during World War II. Rotational spectroscopy is practical only in the gas phase where the rotational motion is quantized. In solids or liquids the rotational motion is usually quenched due to collisions. Rotational spectrum from a molecule (to first order) requires that the molecule have a dipole moment and that there be a difference between its center of charge and its center of mass, or equivalently a separation between two unlike charges. It is this dipole moment that enables the electric field of the light (microwave) to exert a torque on the molecule, causing it to rotate more quickly (in excitation) or slowly (in de-excitation). Diatomic molecules such as dioxygen (O2), dihydrogen (H2), etc. do not have a dipole ________________________WORLD TECHNOLOGIES________________________ moment and, hence, no purely rotational spectrum. However, electronic excitations can lead to asymmetric charge distributions and thus provide a net dipole moment to the molecule. Under such circumstances, these molecules will exhibit a rotational spectrum. Among the diatomic molecules, carbon monoxide (CO) has one of the simplest rotational spectra. As for tri-atomic molecules, hydrogen cyanide (HC≡N) has a simple rotational spectrum for a linear molecule and hydrogen isocyanide (HN=C:) for a non-linear molecule. As the number of atoms increases, the spectrum becomes more complex, as lines, due to different transitions, start overlapping. Understanding the rotational spectrum In quantum mechanics the free rotation of a molecule is quantized, that is the rotational energy and the angular momentum can take only certain fixed values; what these values are is related simply to the moment of inertia, I, of the molecule. In general, for any molecule, there are three moments of inertia: IA, IB and IC about three mutually orthogonal axes A, B, and C with the origin at the center of mass of the system. A linear molecule is a special case in this regard. These molecules are cylindrically symmetric, and one of the moments of inertia (IA, which is the moment of inertia for a rotation taking ). place along the axis of the molecule) is negligible (i.e., WT The general convention is to define the axes such that the axis A has the smallest moment of inertia (and, hence, the highest rotational frequency) and other axes such that . Sometimes the axis A may be associated with the symmetric axis of the molecule, if any. If such is the case, then IA need not be the smallest moment of inertia. To avoid confusion, we will stick with the former convention for the rest. The particular pattern of energy levels (and, hence, of transitions in the rotational spectrum) for a molecule is determined by its symmetry. A convenient way to look at the molecules is to divide them into four different classes (based on the symmetry of their structure). These are linear molecules (or linear rotors), symmetric tops (or symmetric rotors), spherical tops (or spherical rotors), and asymmetric tops Linear molecules As mentioned earlier, for a linear molecule . For most of the purposes, IA is taken to be zero. For a linear molecule, the separation of lines in the rotational spectrum can be related directly to the moment of inertia of the molecule, and, for a molecule of known atomic masses, can be used to determine the bond lengths (structure) directly. For diatomic molecules, this process is trivial, and can be made from a single measurement of the rotational spectrum. For linear molecules with more atoms, rather more work is required, and it is necessary to measure molecules in which more than one isotope of each atom have been substituted (effectively this gives rise to a set of simultaneous equations that can be solved for the bond lengths). ________________________WORLD TECHNOLOGIES________________________ Examples of linear molecules: dioxygen (O=O), carbon monoxide (O≡C*), hydroxy radical (OH), carbon dioxide (O=C=O), hydrogen cyanide (HC≡N), carbonyl sulfide (O=C=S), chloroethyne (HC≡CCl), acetylene (HC≡CH) Symmetric tops A symmetric top is a molecule in which two moments of inertia are the same. As a matter of historical convenience, spectroscopists divide molecules into two classes of symmetric tops, Oblate symmetric tops (saucer or disc shaped) with IA = IB < IC and Prolate symmetric tops (rugby football, or cigar shaped) with IA < IB = IC. The spectra look rather different, and are instantly recognizable. As for linear molecules, the structure of symmetric tops (bond lengths and bond angles) can be deduced from their spectra. Examples of symmetric tops: WT Oblate: benzene (C6H6), cyclobutadiene (C4H4), ammonia (NH3) Prolate: chloromethane (CH3Cl), propyne (CH3C≡CH) Spherical tops A spherical top molecule can be considered as a special case of symmetric tops with equal moment of inertia about all three axes (IA = IB = IC). Examples of spherical tops: phosphorus tetramer (P4), carbon tetrachloride (CCl4), nitrogen tetrahydride (NH4), ammonium ion (NH4+), sulfur hexafluoride (SF6) Asymmetric tops A molecule is termed an asymmetric top if all three moments of inertia are different. Most of the larger molecules are asymmetric tops, even when they have a high degree of symmetry. In general, for such molecules, a simple interpretation of the spectrum is not normally possible. Sometimes asymmetric tops have spectra that are similar to those of a linear molecule or a symmetric top, in which case the molecular structure must also be similar to that of a linear molecule or a symmetric top. For the most general case, however, all that can be done is to fit the spectra to three different moments of inertia. If the molecular formula is known, then educated guesses can be made of the possible structure, and, from this guessed structure, the moments of inertia can be calculated. If the calculated moments of inertia agree well with the measured moments of inertia, then the structure can be said to have been determined. For this approach to determining molecular structure, isotopic substitution is invaluable. Examples of asymmetric tops: anthracene (C14H10), water (H2O), nitrogen dioxide (NO2) ________________________WORLD TECHNOLOGIES________________________ Structure of rotational spectra Linear molecules WT An energy level diagram showing some of the transitions involved in the IR vibrationrotation spectrum of a linear molecule: P branch (where ΔJ = − 1), Q branch (not always allowed, ΔJ = 0) and R branch (ΔJ = 1) These molecules have two degenerate modes of rotation (IB = IC, IA = 0). Since we cannot distinguish between the two modes, we need only one rotational quantum number (J) to describe the rotational motion of the molecule. The rotational energy levels (F(J)) of the molecule based on rigid rotor model can be expressed as, ________________________WORLD TECHNOLOGIES________________________ where is the rotational constant of the molecule and is related to the moment of inertia of the molecule IB = IC by, Selection rules dictate that during emission or absorption the rotational quantum number has to change by unity; i.e., . Thus, the locations of the lines in a rotational spectrum will be given by WT denotes the lower energy level and denotes higher energy level involved in where the transition. The height of the lines is determined by the distribution of the molecules in the different levels and the probability of transition between two energy levels. We observe that, for a rigid rotor, the transition lines are equally spaced in the wavenumber space. However, this is not always the case, except for the rigid rotor model. For non-rigid rotor model, we need to consider changes in the moment of inertia of the molecule. Two primary reasons for this are Centrifugal distortion When a molecule rotates, the centrifugal force pulls the atoms apart. As a result, the moment of inertia of the molecule increases, thus decreasing . To account for this a centrifugal distortion correction term is added to the rotational energy levels of the molecule. where is the centrifugal distortion constant. Therefore, the line spacing for the rotational mode changes to, Effect of vibration on rotation A molecule is always in vibration. As the molecule vibrates, its moment of inertia changes. Further, there is a fictitious force, Coriolis coupling, between the vibrational motion of the nuclei in the rotating (non-inertial) frame. However, as long as the ________________________WORLD TECHNOLOGIES________________________ vibrational quantum number does not change (i.e., the molecule is in only one state of vibration), the effect of vibration on rotation is not important, because the time for vibration is much shorter than the time required for rotation. The Coriolis coupling is often negligible, too, if one is interested in low vibrational and rotational quantum numbers only. Symmetric top The rotational motion of a symmetric top molecule can be described by two independent rotational quantum numbers (since two axes have equal moments of inertia, the rotational motion about these axes requires only one rotational quantum number for complete description). Instead of defining the two rotational quantum numbers for two independent axes, we associate one of the quantum number (J) with the total angular momentum of the molecule and the other quantum number (K) with the angular momentum of the axis that has different moment of inertia (i.e., axis C for oblate symmetric top and axis A for prolate symmetric tops). The rotational energy F(J,K) of such a molecule, based on rigid rotor assumptions can be expressed in terms of the two previously defined rotational quantum numbers as follows where WT for a prolate symmetric top molecule or and for an oblate molecule. Selection rule for these molecules provide the guidelines for possible transitions. Therefore, . This is so because K is associated with the axis about which the molecule is symmetric and, hence, has no net dipole moment in that direction. Thus, there is no interaction of this mode with the light particles (photons). This gives the transition wavenumbers of which is the same as in the case of a linear molecule. In case of non-rigid rotors, the first order centrifugal distortion correction is given by ________________________WORLD TECHNOLOGIES________________________ The suffixes on the centrifugal distortion constant D indicate the rotational mode involved and are not a function of the rotational quantum number. The location of the transition lines on a spectrum is given by Spherical top Unlike other molecules, spherical top molecules have no net dipole moment, and, hence, they do not exhibit a pure rotational spectrum. WT Asymmetric top The spectrum for these molecules usually involves many lines due to three different rotational modes and their combinations. The following analysis is valid for the general case and collapses to the various special cases described above in the appropriate limit. From the moments of inertia one can define an asymmetry parameter κ as , which varies from -1 for a prolate symmetric top to 1 for an oblate symmetric top. One can define a scaled rotational Hamiltonian dependent on J and κ. The (symmetric) matrix representation of this Hamiltonian is banded, zero everywhere but the main diagonal and the second subdiagonal. The Hamiltonian can be formulated in six different settings, dependent on the mapping of the principal axes to lab axes and handedness. For the most asymmetric, right-handed representation, the diagonal elements are, for Hk,k(κ) = κk2 and the second off-diagonal elements (independent of κ) are . Diagonalising H yields a set of 2J + 1 scaled rotational energy levels Ek(κ). The rotational energy levels of the asymmetric rotor for total angular momentum J are then given by ________________________WORLD TECHNOLOGIES________________________ Hyperfine interaction In addition to the main structure that is observed in microwave spectra due to the rotational motion of the molecules, a whole host of further interactions are responsible for small details in the spectra, and the study of these details provides a very deep understanding of molecular quantum mechanics. The main interactions responsible for small changes in the spectra (additional splittings and shifts of lines) are due to magnetic and electrostatic interactions in the molecule. The particular strength of such interactions differs in different molecules, but, in general, the order of these effects (in decreasing significance) is: 1. electron spin - electron spin interaction (this occurs in molecules with two or more unpaired electrons, and is a magnetic-dipole / magnetic-dipole interaction) 2. electron spin - molecular rotation (the rotation of a molecule corresponds to a magnetic dipole, which interacts with the magnetic dipole moment of the electron) 3. electron spin - nuclear spin interaction (the interaction between the magnetic dipole moment of the electron and the magnetic dipole moment of the nuclei (if present)). 4. electric field gradient - nuclear electric quadrupole interaction (the interaction between the electric field gradient of the electron cloud of the molecule and the electric quadrupole moments of nuclei (if present)). 5. nuclear spin - nuclear spin interaction (nuclear magnetic moments interacting with one another). WT These interactions give rise to the characteristic energy levels that are probed in "magnetic resonance" spectroscopy such as NMR and ESR, where they represent the "zero field splittings," which are always present. ________________________WORLD TECHNOLOGIES________________________ Experimental determination of the spectrum WT Part of the rotational-vibrational spectrum of methane (CH4) gas (from FTIR), showing the presence of P-, Q- and R- branches (purple, top) and a simulation in PGOPHER (black, bottom). Frequency is on the x-axis, and transmittance on the y-axis. Fourier transform infrared (FTIR) spectroscopy can be used to experimentally study rotational spectra. Typical spectra at these wavelengths involve rovibrational excitation, i.e., excitation of both a vibrational and a rotational mode of a molecule. In the past, microwave spectra were determined using a simple arrangement in which low-pressure gas was introduced to a section of waveguide between a microwave source (of variable frequency) and a microwave detector. The spectrum was obtained by sweeping the frequency of the source while detecting the intensity of the transmitted radiation. This experimental arrangement has a major difficulty related to the propagation of microwave radiation through waveguides. The physical size of the waveguide restricts the frequency of the radiation that can be transmitted through it. For a given waveguide size (such as X-band), there is a cutoff frequency, and microwave radiation with smaller frequencies (longer wavelengths) cannot be propagated through the waveguide. In addition, as the frequency is increased, additional modes of propagation become possible, which correspond to different velocities of the radiation propagating down the waveguide (this can be envisaged as the radiation bouncing down the guide, at different angles of reflection). the net result of these considerations is that each size of waveguide is useful only over a rather narrow range of frequencies and must be physically swapped out for a different size of waveguide once this frequency range is exceeded. ________________________WORLD TECHNOLOGIES________________________ From 1980 onward, microwave spectra have often been obtained using Fourier Transform Microwave Spectroscopy - a technique developed by W. H. Flygare at the University of Illinois. Within the last two years, a further development of Fourier Transform Microwave Spectroscopy has occurred, which may well introduce a new renaissance into microwave spectroscopy. This is the use of "Chirped Pulses" to provide an electromagnetic wave that has as its Fourier Transform a very wide range of microwave frequencies. Applications Microwave spectroscopy is commonly used in physical chemistry to determine the structure of small molecules (such as ozone, methanol, or water) with high precision. Other common techniques for determining molecular structure, such as X-ray crystallography do not work very well for some of these molecules (especially the gases) and are not as precise. However, microwave spectroscopy is not useful for determining the structures of large molecules such as proteins. WT Modern microwave spectrometers have very high resolution. When hyperfine structure can be observed, the technique can also provide information on the electronic structures of molecules. Microwave spectroscopy is one of the principal means by which the constituents of the universe are determined from the earth. It is particularly useful for detecting molecules in the interstellar medium (ISM). One of the early surprises in interstellar chemistry came with the discovery of the existence in the ISM of long-chain carbon molecules. It was in attempting to research such molecules in the laboratory that Harry Kroto was led to the laboratory of Rick Smalley and Robert Curl, where it was possible to vaporize carbon under enormous energy conditions. This collaborative experiment led to the discovery of C60, buckminsterfullerene, which led to the award of the 1996 Nobel prize in chemistry to Kroto, Smalley and Curl. ________________________WORLD TECHNOLOGIES________________________ Chapter- 7 Electromagnetic Spectrum WT Although some radiations are marked as "N" for "no" in the diagram, some waves do in fact penetrate the atmosphere, although extremely minimally compared to the other radiations. The electromagnetic spectrum is the range of all possible frequencies of electromagnetic radiation. The "electromagnetic spectrum" of an object is the characteristic distribution of electromagnetic radiation emitted or absorbed by that particular object. The electromagnetic spectrum extends from low frequencies used for modern radio to gamma radiation at the short-wavelength end, covering wavelengths from thousands of kilometers down to a fraction of the size of an atom. The long wavelength limit is the size of the universe itself, while it is thought that the short wavelength limit is in the vicinity of the Planck length, although in principle the spectrum is infinite and continuous. ________________________WORLD TECHNOLOGIES________________________ WT Legend γ= Gamma rays MIR= Mid infrared HF= High freq. HX= Hard X-rays FIR= Far infrared MF= Medium freq. SX= Soft X-rays Radio waves LF= Low freq. EUV= Extreme ultraviolet EHF= Extremely high freq. VLF= Very low freq. NUV= Near ultraviolet SHF= Super high freq. VF/ULF= Voice freq. Visible light UHF= Ultra high freq. SLF= Super low freq. NIR= Near Infrared VHF= Very high freq. ELF= Extremely low freq. Freq=Frequency Range of the spectrum EM waves are typically described by any of the following three physical properties: the frequency f, wavelength λ, or photon energy E. Frequencies range from 2.4×1023 Hz (1 GeV gamma rays) down to the local plasma frequency of the ionized interstellar medium (~1 kHz). Wavelength is inversely proportional to the wave frequency, so gamma rays have very short wavelengths that are fractions of the size of atoms, whereas wavelengths can be as long as the universe. Photon energy is directly proportional to the wave frequency, so gamma rays have the highest energy (around a billion electron volts) ________________________WORLD TECHNOLOGIES________________________ and radio waves have very low energy (around femto electron volts). These relations are illustrated by the following equations: where: • • c = 299,792,458 m/s is the speed of light in vacuum and h = 6.62606896(33)×10−34 J s = 4.13566733(10)×10−15 eV s is Planck's constant. Whenever electromagnetic waves exist in a medium with matter, their wavelength is decreased. Wavelengths of electromagnetic radiation, no matter what medium they are traveling through, are usually quoted in terms of the vacuum wavelength, although this is not always explicitly stated. WT Generally, EM radiation is classified by wavelength into radio wave, microwave, infrared, the visible region we perceive as light, ultraviolet, X-rays and gamma rays. The behavior of EM radiation depends on its wavelength. When EM radiation interacts with single atoms and molecules, its behavior also depends on the amount of energy per quantum (photon) it carries. Spectroscopy can detect a much wider region of the EM spectrum than the visible range of 400 nm to 700 nm. A common laboratory spectroscope can detect wavelengths from 2 nm to 2500 nm. Detailed information about the physical properties of objects, gases, or even stars can be obtained from this type of device. Spectroscopes are widely used in astrophysics. For example, many hydrogen atoms emit a radio wave photon which has a wavelength of 21.12 cm. Also, frequencies of 30 Hz and below can be produced by and are important in the study of certain stellar nebulae and frequencies as high as 2.9×1027 Hz have been detected from astrophysical sources. Rationale Electromagnetic radiation interacts with matter in different ways in different parts of the spectrum. The types of interaction can be so different that it seems to be justified to refer to different types of radiation. At the same time, there is a continuum containing all these "different kinds" of electromagnetic radiation. Thus we refer to a spectrum, but divide it up based on the different interactions with matter. ________________________WORLD TECHNOLOGIES________________________ Region of the spectrum Collective oscillation of charge carriers in bulk material (plasma oscillation). An example would be the oscillation of the electrons in an antenna. Radio Microwave through far infrared Near infrared Visible Ultraviolet X-rays Gamma rays High energy gamma rays Main interactions with matter Plasma oscillation, molecular rotation Molecular vibration, plasma oscillation (in metals only) Molecular electron excitation (including pigment molecules found in the human retina), plasma oscillations (in metals only) Excitation of molecular and atomic valence electrons, including ejection of the electrons (photoelectric effect) Excitation and ejection of core atomic electrons, Compton scattering (for low atomic numbers) Energetic ejection of core electrons in heavy elements, Compton scattering (for all atomic numbers), excitation of atomic nuclei, including dissociation of nuclei Creation of particle-antiparticle pairs. At very high energies a single photon can create a shower of high energy particles and antiparticles upon interaction with matter. WT ________________________WORLD TECHNOLOGIES________________________ Types of radiation WT The electromagnetic spectrum While the classification scheme is generally accurate, in reality there is often some overlap between neighboring types of electromagnetic energy. For example, SLF radio waves at 60 Hz may be received and studied by astronomers, or may be ducted along wires as electric power, although the latter is, strictly speaking, not electromagnetic radiation at all. The distinction between X and gamma rays is based on sources: gamma rays are the photons generated from nuclear decay or other nuclear and subnuclear/ particle process, whereas X-rays are generated by electronic transitions involving highly energetic inner atomic electrons. Generally, nuclear transitions are much more energetic than electronic transitions, so usually, gamma-rays are more energetic than X-rays, but ________________________WORLD TECHNOLOGIES________________________ exceptions exist. By analogy to electronic transitions, muonic atom transitions are also said to produce X-rays, even though their energy may exceed 6 megaelectronvolts (0.96 pJ), whereas there are many (77 known to be less than 10 keV (1.6 fJ)) low-energy nuclear transitions (e.g. the 7.6 eV (1.22 aJ) nuclear transition of thorium-229), and despite being one million-fold less energetic than some muonic X-rays, the emitted photons are still called gamma rays due to their nuclear origin. Also, the region of the spectrum of the particular electromagnetic radiation is referenceframe dependent (on account of the Doppler shift for light) so EM radiation which one observer would say is in one region of the spectrum could appear to an observer moving at a substantial fraction of the speed of light with respect to the first to be in another part of the spectrum. For example, consider the cosmic microwave background. It was produced, when matter and radiation decoupled, by the de-excitation of hydrogen atoms to the ground state. These photons were from Lyman series transitions, putting them in the ultraviolet (UV) part of the electromagnetic spectrum. Now this radiation has undergone enough cosmological red shift to put it into the microwave region of the spectrum for observers moving slowly (compared to the speed of light) with respect to the cosmos. However, for particles moving near the speed of light, this radiation will be blue-shifted in their rest frame. The highest energy cosmic ray protons are moving such that, in their rest frame, this radiation is blueshifted to high energy gamma rays which interact with the proton to produce bound quark-antiquark pairs (pions). This is the source of the GZK limit. WT Radio frequency Radio waves generally are utilized by antennas of appropriate size (according to the principle of resonance), with wavelengths ranging from hundreds of meters to about one millimeter. They are used for transmission of data, via modulation. Television, mobile phones, wireless networking and amateur radio all use radio waves. The use of the radio spectrum is regulated by many governments through frequency allocation. Radio waves can be made to carry information by varying a combination of the amplitude, frequency and phase of the wave within a frequency band. When EM radiation impinges upon a conductor, it couples to the conductor, travels along it, and induces an electric current on the surface of that conductor by exciting the electrons of the conducting material. This effect (the skin effect) is used in antennas. EM radiation may also cause certain molecules to absorb energy and thus to heat up, causing thermal effects and sometimes burns. This is exploited in microwave ovens. ________________________WORLD TECHNOLOGIES________________________ Microwaves WT Plot of Earth's atmospheric transmittance (or opacity) to various wavelengths of electromagnetic radiation. The super high frequency (SHF) and extremely high frequency (EHF) of microwaves come next up the frequency scale. Microwaves are waves which are typically short enough to employ tubular metal waveguides of reasonable diameter. Microwave energy is produced with klystron and magnetron tubes, and with solid state diodes such as Gunn and IMPATT devices. Microwaves are absorbed by molecules that have a dipole moment in liquids. In a microwave oven, this effect is used to heat food. Low-intensity microwave radiation is used in Wi-Fi, although this is at intensity levels unable to cause thermal heating. Volumetric heating, as used by microwaves, transfers energy through the material electromagnetically, not as a thermal heat flux. The benefit of this is a more uniform heating and reduced heating time; microwaves can heat material in less than 1% of the time of conventional heating methods. When active, the average microwave oven is powerful enough to cause interference at close range with poorly shielded electromagnetic fields such as those found in mobile medical devices and cheap consumer electronics. Terahertz radiation Terahertz radiation is a region of the spectrum between far infrared and microwaves. Until recently, the range was rarely studied and few sources existed for microwave energy at the high end of the band (sub-millimetre waves or so-called terahertz waves), but applications such as imaging and communications are now appearing. Scientists are also looking to apply terahertz technology in the armed forces, where high frequency waves might be directed at enemy troops to incapacitate their electronic equipment. ________________________WORLD TECHNOLOGIES________________________ Infrared radiation The infrared part of the electromagnetic spectrum covers the range from roughly 300 GHz (1 mm) to 400 THz (750 nm). It can be divided into three parts: • • • Far-infrared, from 300 GHz (1 mm) to 30 THz (10 μm). The lower part of this range may also be called microwaves. This radiation is typically absorbed by socalled rotational modes in gas-phase molecules, by molecular motions in liquids, and by phonons in solids. The water in the Earth's atmosphere absorbs so strongly in this range that it renders the atmosphere effectively opaque. However, there are certain wavelength ranges ("windows") within the opaque range which allow partial transmission, and can be used for astronomy. The wavelength range from approximately 200 μm up to a few mm is often referred to as "sub-millimetre" in astronomy, reserving far infrared for wavelengths below 200 μm. Mid-infrared, from 30 to 120 THz (10 to 2.5 μm). Hot objects (black-body radiators) can radiate strongly in this range. It is absorbed by molecular vibrations, where the different atoms in a molecule vibrate around their equilibrium positions. This range is sometimes called the fingerprint region since the mid-infrared absorption spectrum of a compound is very specific for that compound. Near-infrared, from 120 to 400 THz (2,500 to 750 nm). Physical processes that are relevant for this range are similar to those for visible light. WT Visible radiation (light) Above infrared in frequency comes visible light. This is the range in which the sun and stars similar to it emit most of their radiation. It is probably not a coincidence that the human eye is sensitive to the wavelengths that the sun emits most strongly. Visible light (and near-infrared light) is typically absorbed and emitted by electrons in molecules and atoms that move from one energy level to another. The light we see with our eyes is really a very small portion of the electromagnetic spectrum. A rainbow shows the optical (visible) part of the electromagnetic spectrum; infrared (if you could see it) would be located just beyond the red side of the rainbow with ultraviolet appearing just beyond the violet end. Electromagnetic radiation with a wavelength between 380 nm and 760 nm (790–400 terahertz) is detected by the human eye and perceived as visible light. Other wavelengths, especially near infrared (longer than 760 nm) and ultraviolet (shorter than 380 nm) are also sometimes referred to as light, especially when the visibility to humans is not relevant. If radiation having a frequency in the visible region of the EM spectrum reflects off an object, say, a bowl of fruit, and then strikes our eyes, this results in our visual perception of the scene. Our brain's visual system processes the multitude of reflected frequencies into different shades and hues, and through this not-entirely-understood psychophysical phenomenon, most people perceive a bowl of fruit. ________________________WORLD TECHNOLOGIES________________________ At most wavelengths, however, the information carried by electromagnetic radiation is not directly detected by human senses. Natural sources produce EM radiation across the spectrum, and our technology can also manipulate a broad range of wavelengths. Optical fiber transmits light which, although not suitable for direct viewing, can carry data that can be translated into sound or an image. The coding used in such data is similar to that used with radio waves. Ultraviolet light WT The amount of penetration of UV relative to altitude in Earth's ozone Next in frequency comes ultraviolet (UV). This is radiation whose wavelength is shorter than the violet end of the visible spectrum, and longer than that of an X-ray. Being very energetic, UV can break chemical bonds, making molecules unusually reactive or ionizing them, in general changing their mutual behavior. Sunburn, for example, is caused by the disruptive effects of UV radiation on skin cells, which is the main cause of skin cancer, if the radiation irreparably damages the complex DNA molecules in the cells (UV radiation is a proven mutagen). The Sun emits a large amount of UV radiation, which could quickly turn Earth into a barren desert. However, most of it is absorbed by the atmosphere's ozone layer before reaching the surface. ________________________WORLD TECHNOLOGIES________________________ X-rays After UV come X-rays, which are also ionizing, but due to their higher energies they can also interact with matter by means of the Compton effect. Hard X-rays have shorter wavelengths than soft X-rays. As they can pass through most substances, X-rays can be used to 'see through' objects, most notably diagnostic X-ray images in medicine (a process known as radiography), as well as for high-energy physics and astronomy. Neutron stars and accretion disks around black holes emit X-rays, which enable us to study them. X-rays are given off by stars and are strongly emitted by some types of nebulae. Gamma rays After hard X-rays come gamma rays, which were discovered by Paul Villard in 1900. These are the most energetic photons, having no defined lower limit to their wavelength. They are useful to astronomers in the study of high energy objects or regions, and find a use with physicists thanks to their penetrative ability and their production from radioisotopes. Gamma rays are also used for the irradiation of food and seed for sterilization, and in medicine they are used in radiation cancer therapy and some kinds of diagnostic imaging such as PET scans. The wavelength of gamma rays can be measured with high accuracy by means of Compton scattering. WT Note that there are no precisely defined boundaries between the bands of the electromagnetic spectrum. Radiation of some types have a mixture of the properties of those in two regions of the spectrum. For example, red light resembles infrared radiation in that it can resonate some chemical bonds. ________________________WORLD TECHNOLOGIES________________________ Chapter- 8 Molecular Vibration A molecular vibration ocscurs when atoms in a molecule are in periodic motion while the molecule as a whole has constant translational and rotational motion. The frequency of the periodic motion is known as a vibration frequency. In general, a molecule with N atoms has 3N-6 normal modes of vibration but linear molecules have only 3N-5 normal modes of vibration as rotation about its molecular axis cannot be observed.. A diatomic molecule has one normal mode of vibration. The normal modes of vibration of polyatomic molecules are independent of each other but each normal mode will involve simultaneous vibrations of different parts of the molecule such as different chemical bonds. WT A molecular vibration is excited when the molecule absorbs a quantum of energy, E, corresponding to the vibration's frequency, ν, according to the relation E=hν, where h is Planck's constant. A fundamental vibration is excited when one such quantum of energy is absorbed by the molecule in its ground state. When two quanta are absorbed the first overtone is excited, and so on to higher overtones. To a first approximation, the motion in a normal vibration can be described as a kind of simple harmonic motion. In this approximation, the vibrational energy is a quadratic function (parabola) with respect to the atomic displacements and the first overtone has twice the frequency of the fundamental. In reality, vibrations are anharmonic and the first overtone has a frequency that is slightly lower than twice that of the fundamental. Excitation of the higher overtones involves progressively less and less additional energy and eventually leads to dissociation of the molecule, as the potential energy of the molecule is more like a Morse potential. The vibrational states of a molecule can be probed in a variety of ways. The most direct way is through infrared spectroscopy, as vibrational transitions typically require an amount of energy that corresponds to the infrared region of the spectrum. Raman spectroscopy, which typically uses visible light, can also be used to measure vibration frequencies directly. Vibrational excitation can occur in conjunction with electronic excitation (vibronic transition), giving vibrational fine structure to electronic transitions, particularly with molecules in the gas state. ________________________WORLD TECHNOLOGIES________________________ Simultaneous excitation of a vibration and rotations gives rise to vibration-rotation spectra. Vibrational coordinates The coordinate of a normal vibration is a combination of changes in the positions of atoms in the molecule. When the vibration is excited the coordinate changes sinusoidally with a frequency ν, the frequency of the vibration. Internal coordinates Internal coordinates are of the following types, illustrated with reference to the planar molecule ethylene, • • • • • • WT Stretching: a change in the length of a bond, such as C-H or C-C Bending: a change in the angle between two bonds, such as the HCH angle in a methylene group Rocking: a change in angle between a group of atoms, such as a methylene group and the rest of the molecule. Wagging: a change in angle between the plane of a group of atoms, such as a methylene group and a plane through the rest of the molecule, Twisting: a change in the angle between the planes of two groups of atoms, such as a change in the angle between the two methylene groups. Out-of-plane: a change in the angle between any one of the C-H bonds and the plane defined by the remaining atoms of the ethylene molecule. Another example is in BF3 when the boron atom moves in and out of the plane of the three fluorine atoms. In a rocking, wagging or twisting coordinate the bond lengths within the groups involved do not change. The angles do. Rocking is distinguished from wagging by the fact that the atoms in the group stay in the same plane. In ethene there are 12 internal coordinates: 4 C-H stretching, 1 C-C stretching, 2 H-C-H bending, 2 CH2 rocking, 2 CH2 wagging, 1 twisting. Note that the H-C-C angles cannot be used as internal coordinates as the angles at each carbon atom cannot all increase at the same time. ________________________WORLD TECHNOLOGIES________________________ Vibrations of a Methylene group (-CH2-) in a molecule for illustration The atoms in a CH2 group, commonly found in organic compounds, can vibrate in six different ways: symmetric and antisymmetric stretching, scissoring, rocking, wagging and twisting as shown here: Symmetrical stretching Antisymmetrical stretching Scissoring WT Rocking Wagging Twisting (These figures do not represent the "recoil" of the C atoms, which, though necessarily present to balance the overall movements of the molecule, are much smaller than the movements of the lighter H atoms). Symmetry-adapted coordinates Symmetry-adapted coordinates may be created by applying a projection operator to a set of internal coordinates. The projection operator is constructed with the aid of the character table of the molecular point group. For example, the four(un-normalised) C-H stretching coordinates of the molecule ethene are given by Qs1 = q1 + q2 + q3 + q4 Qs2 = q1 + q2 - q3 - q4 Qs3 = q1 - q2 + q3 - q4 Qs4 = q1 - q2 - q3 + q4 where q1 - q4 are the internal coordinates for stretching of each of the four C-H bonds. Illustrations of symmetry-adapted coordinates for most small molecules can be found in Nakamoto. ________________________WORLD TECHNOLOGIES________________________ Normal coordinates The normal coordinates, denoted as Q, refer to the positions of atoms away from their equilibrium positions, with respect to a normal mode of vibration. Each normal mode is assigned a single normal coordinate, and so the normal coordinate refers to the "progress" along that normal mode at any given time. Formally, normal modes are determined by solving a secular determinant, and then the normal coordinates (over the normal modes) can be expressed as a summation over the cartesian cordinates (over the atom positions). The advantage of working in normal modes is that they diagonalize the matrix governing the molecular vibrations, so each normal mode is an independent molecular vibration, associated with its own spectrum of quantum mechanical states. If the molecule possesses symmetries, it will belong to a point group, and the normal modes will "transform as" an irreducible representation under that group. The normal modes can then be qualitatively determined by applying group theory and projecting the irreducible representation onto the cartesian coordinates. For example, when this treatment is applied to CO2, it is found that the C=O stretches are not independent, but rather there is a O=C=O symmetric stretch and an O=C=O asymmetric stretch. • • WT symmetric stretching: the sum of the two C-O stretching coordinates; the two C-O bond lengths change by the same amount and the carbon atom is stationary. Q = q1 + q2 asymmetric stretching: the difference of the two C-O stretching coordinates; one C-O bond length increases while the other decreases. Q = q1 - q2 When two or more normal coordinates belong to the same irreducible representation of the molecular point group (colloquially, have the same symmetry) there is "mixing" and the coefficients of the combination cannot be determined a priori. For example, in the linear molecule hydrogen cyanide, HCN, The two stretching vibrations are 1. principally C-H stretching with a little C-N stretching; Q1 = q1 + a q2 (a << 1) 2. principally C-N stretching with a little C-H stretching; Q2 = b q1 + q2 (b << 1) The coefficients a and b are found by performing a full normal coordinate analysis by means of the Wilson GF method. Newtonian mechanics Perhaps surprisingly, molecular vibrations can be treated using Newtonian mechanics to calculate the correct vibration frequencies. The basic assumption is that each vibration can be treated as though it corresponds to a spring. In the harmonic approximation the spring obeys Hooke's law: the force required to extend the spring is proportional to the extension. The proportionality constant is known as a force constant, k. The anharmonic oscillator is considered elsewhere. ________________________WORLD TECHNOLOGIES________________________ By Newton’s second law of motion this force is also equal to a reduced mass, μ, times acceleration. Since this is one and the same force the ordinary differential equation follows. The solution to this equation of simple harmonic motion is WT A is the maximum amplitude of the vibration coordinate Q. It remains to define the reduced mass, μ. In general, the reduced mass of a diatomic molecule, AB, is is expressed in terms of the atomic masses, mA and mB, as The use of the reduced mass ensures that the centre of mass of the molecule is not affected by the vibration. In the harmonic approximation the potential energy of the molecule is a quadratic function of the normal coordinate. It follows that the forceconstant is equal to the second derivative of the potential energy. When two or more normal vibrations have the same symmetry a full normal coordinate analysis must be performed. The vibration frequencies,νi are obtained from the eigenvalues,λi, of the matrix product GF. G is a matrix of numbers derived from the masses of the atoms and the geometry of the molecule. F is a matrix derived from forceconstant values. Details concerning the determination of the eigenvalues can be found in. Quantum mechanics In the harmonic approximation the potential energy is a quadratic function of the normal coordinates. Solving the Schrödinger wave equation, the energy states for each normal coordinate are given by ________________________WORLD TECHNOLOGIES________________________ , where n is a quantum number that can take values of 0, 1, 2 ... The difference in energy when n changes by 1 are therefore equal to the energy derived using classical mechanics. Knowing the wave functions, certain selection rules can be formulated. For example, for a harmonic oscillator transitions are allowed only when the quantum number n changes by one, but this does not apply to an anharmonic oscillator; the observation of overtones is only possible because vibrations are anharmonic. Another consequence of anharmonicity is that transitions such as between states n=2 and n=1 have slightly less energy than transitions between the ground state and first excited state. Such a transition gives rise to a hot band. WT Intensities In an infrared spectrum the intensity of an absorption band is proportional to the derivative of the molecular dipole moment with respect to the normal coordinate. The intensity of Raman bands depends on polarizability. ________________________WORLD TECHNOLOGIES________________________ Chapter- 9 Fourier Transform The Fourier transform is a mathematical operation that decomposes a signal into its constituent frequencies. Thus the Fourier transform of a musical chord is a mathematical representation of the amplitudes of the individual notes that make it up. The original signal depends on time, and therefore is called the time domain representation of the signal, whereas the Fourier transform depends on frequency and is called the frequency domain representation of the signal. The term Fourier transform refers both to the frequency domain representation of the signal and the process that transforms the signal to its frequency domain representation. WT In mathematical terms, the Fourier transform transforms one complex-valued function of a real variable into another. In effect, the Fourier transform decomposes a function into oscillatory functions. The Fourier transform and its generalizations are the subject of Fourier analysis. In this specific case, both the time and frequency domains are unbounded linear continua. It is possible to define the Fourier transform of a function of several variables, which is important for instance in the physical study of wave motion and optics. It is also possible to generalize the Fourier transform on discrete structures such as finite groups. The efficient computation of such structures, by fast Fourier transform, is essential for high-speed computing. Definition There are several common conventions for defining the Fourier transform of an integrable function ƒ : R → C (Kaiser 1994). Here we, will use the definition: for every real number ξ. When the independent variable x represents time (with SI unit of seconds), the transform variable ξ represents frequency (in hertz). Under suitable conditions, ƒ can be reconstructed from by the inverse transform: ________________________WORLD TECHNOLOGIES________________________ for every real number x. For other common conventions and notations, including using the angular frequency ω instead of the frequency ξ, see Other conventions and Other notations below. The Fourier transform on Euclidean space is treated separately, in which the variable x often represents position and ξ momentum. Introduction The motivation for the Fourier transform comes from the study of Fourier series. In the study of Fourier series, complicated functions are written as the sum of simple waves mathematically represented by sines and cosines. Due to the properties of sine and cosine it is possible to recover the amount of each wave in the sum by an integral. In many cases it is desirable to use Euler's formula, which states that e2πiθ = cos 2πθ + i sin 2πθ, to write Fourier series in terms of the basic waves e2πiθ. This has the advantage of simplifying many of the formulas involved and providing a formulation for Fourier series that more closely resembles the definition followed here. This passage from sines and cosines to complex exponentials makes it necessary for the Fourier coefficients to be complex valued. The usual interpretation of this complex number is that it gives both the amplitude (or size) of the wave present in the function and the phase (or the initial angle) of the wave. This passage also introduces the need for negative "frequencies". If θ were measured in seconds then the waves e2πiθ and e−2πiθ would both complete one cycle per second, but they represent different frequencies in the Fourier transform. Hence, frequency no longer measures the number of cycles per unit time, but is closely related. WT There is a close connection between the definition of Fourier series and the Fourier transform for functions ƒ which are zero outside of an interval. For such a function we can calculate its Fourier series on any interval that includes the interval where ƒ is not identically zero. The Fourier transform is also defined for such a function. As we increase the length of the interval on which we calculate the Fourier series, then the Fourier series coefficients begin to look like the Fourier transform and the sum of the Fourier series of ƒ begins to look like the inverse Fourier transform. To explain this more precisely, suppose that T is large enough so that the interval [−T/2,T/2] contains the interval on which ƒ is not identically zero. Then the n-th series coefficient cn is given by: Comparing this to the definition of the Fourier transform it follows that since ƒ(x) is zero outside [−T/2,T/2]. Thus the Fourier coefficients are just the values of the Fourier transform sampled on a grid of width 1/T. As T increases the Fourier coefficients more closely represent the Fourier transform of the function. ________________________WORLD TECHNOLOGIES________________________ Under appropriate conditions the sum of the Fourier series of ƒ will equal the function ƒ. In other words ƒ can be written: where the last sum is simply the first sum rewritten using the definitions ξn = n/T, and Δξ = (n + 1)/T − n/T = 1/T. This second sum is a Riemann sum, and so by letting T → ∞ it will converge to the integral for the inverse Fourier transform given in the definition section. Under suitable conditions this argument may be made precise (Stein & Shakarchi 2003). In the study of Fourier series the numbers cn could be thought of as the "amount" of the wave in the Fourier series of ƒ. Similarly, as seen above, the Fourier transform can be thought of as a function that measures how much of each individual frequency is present in our function ƒ, and we can recombine these waves by using an integral (or "continuous sum") to reproduce the original function. WT The following images provide a visual illustration of how the Fourier transform measures whether a frequency is present in a particular function. The function depicted oscillates at 3 hertz (if t measures seconds) and tends quickly to 0. This function was specially chosen to have a real Fourier transform which can easily we must integrate be plotted. The first image contains its graph. In order to calculate −2πi(3t) ƒ(t). The second image shows the plot of the real and imaginary parts of this e function. The real part of the integrand is almost always positive, this is because when ƒ(t) is negative, then the real part of e−2πi(3t) is negative as well. Because they oscillate at the same rate, when ƒ(t) is positive, so is the real part of e−2πi(3t). The result is that when you integrate the real part of the integrand you get a relatively large number (in this case 0.5). On the other hand, when you try to measure a frequency that is not present, as in the case when we look at , the integrand oscillates enough so that the integral is very small. The general situation may be a bit more complicated than this, but this in spirit is how the Fourier transform measures how much of an individual frequency is present in a function ƒ(t). ________________________WORLD TECHNOLOGIES________________________ WT Original function showing oscillation 3 hertz. ________________________WORLD TECHNOLOGIES________________________ WT Real and imaginary parts of integrand for Fourier transform at 3 hertz ________________________WORLD TECHNOLOGIES________________________ WT Real and imaginary parts of integrand for Fourier transform at 5 hertz ________________________WORLD TECHNOLOGIES________________________ WT Fourier transform with 3 and 5 hertz labeled. Properties of the Fourier transform An integrable function is a function ƒ on the real line that is Lebesgue-measurable and satisfies Basic properties Given integrable functions f(x), g(x), and h(x) denote their Fourier transforms by , , and respectively. The Fourier transform has the following basic properties (Pinsky 2002). Linearity ________________________WORLD TECHNOLOGIES________________________ For any complex numbers a and b, if h(x) = aƒ(x) + bg(x), then Translation For any real number x0, if h(x) = ƒ(x − x0), then Modulation For any real number ξ0, if h(x) = e2πixξ0ƒ(x), then Scaling . . The For a non-zero real number a, if h(x) = ƒ(ax), then case a = −1 leads to the time-reversal property, which states: if h(x) = ƒ(−x), then Conjugation If . WT , then In particular, if ƒ is real, then one has the reality condition And if ƒ is purely imaginary, then Duality If Convolution If then , then ________________________WORLD TECHNOLOGIES________________________ Uniform continuity and the Riemann–Lebesgue lemma WT The rectangular function is Lebesgue integrable. The sinc function, which is the Fourier transform of the rectangular function, is bounded and continuous, but not Lebesgue integrable. ________________________WORLD TECHNOLOGIES________________________ The Fourier transform of integrable functions have additional properties that do not always hold. The Fourier transforms of integrable functions ƒ are uniformly continuous and (Katznelson 1976). The Fourier transform of integrable functions also satisfy the Riemann–Lebesgue lemma which states that (Stein & Weiss 1971) The Fourier transform of an integrable function ƒ is bounded and continuous, but need not be integrable – for example, the Fourier transform of the rectangular function, which is a step function (and hence integrable) is the sinc function, which is not Lebesgue integrable, though it does have an improper integral: one has an analog to the alternating harmonic series, which is a convergent sum but not absolutely convergent. WT It is not possible in general to write the inverse transform as a Lebesgue integral. However, when both ƒ and almost every x: are integrable, the following inverse equality holds true for Almost everywhere, ƒ is equal to the continuous function given by the right-hand side. If ƒ is given as continuous function on the line, then equality holds for every x. A consequence of the preceding result is that the Fourier transform is injective on L1(R). The Plancherel theorem and Parseval's theorem Let f(x) and g(x) be integrable, and let and be their Fourier transforms. If f(x) and g(x) are also square-integrable, then we have Parseval's theorem (Rudin 1987, p. 187): where the bar denotes complex conjugation. The Plancherel theorem, which is equivalent to Parseval's theorem, states (Rudin 1987, p. 186): ________________________WORLD TECHNOLOGIES________________________ The Plancherel theorem makes it possible to define the Fourier transform for functions in L2(R), as described in Generalizations below. The Plancherel theorem has the interpretation in the sciences that the Fourier transform preserves the energy of the original quantity. It should be noted that depending on the author either of these theorems might be referred to as the Plancherel theorem or as Parseval's theorem. Poisson summation formula The Poisson summation formula provides a link between the study of Fourier transforms and Fourier Series. Given an integrable function ƒ we can consider the periodic summation of ƒ given by: WT where the summation is taken over the set of all integers k. The Poisson summation formula relates the Fourier series of to the Fourier transform of ƒ. Specifically it states that the Fourier series of is given by: Convolution theorem The Fourier transform translates between convolution and multiplication of functions. If and respectively, ƒ(x) and g(x) are integrable functions with Fourier transforms then the Fourier transform of the convolution is given by the product of the Fourier transforms and (under other conventions for the definition of the Fourier transform a constant factor may appear). This means that if: where ∗ denotes the convolution operation, then: In linear time invariant (LTI) system theory, it is common to interpret g(x) as the impulse response of an LTI system with input ƒ(x) and output h(x), since substituting the unit impulse for ƒ(x) yields h(x) = g(x). In this case, of the system. represents the frequency response ________________________WORLD TECHNOLOGIES________________________ Conversely, if ƒ(x) can be decomposed as the product of two square integrable functions p(x) and q(x), then the Fourier transform of ƒ(x) is given by the convolution of the respective Fourier transforms and . Cross-correlation theorem In an analogous manner, it can be shown that if h(x) is the cross-correlation of ƒ(x) and g(x): then the Fourier transform of h(x) is: WT As a special case, the autocorrelation of function ƒ(x) is: for which Eigenfunctions One important choice of an orthonormal basis for L2(R) is given by the Hermite functions where Hn(x) are the "probabilist's" Hermite polynomials, defined by Hn(x) = (−1)nexp (x2/2) Dn exp (−x2/2). Under this convention for the Fourier transform, we have that In other words, the Hermite functions form a complete orthonormal system of eigenfunctions for the Fourier transform on L2(R) (Pinsky 2002). However, this choice of eigenfunctions is not unique. There are only four different eigenvalues of the Fourier transform (±1 and ±i) and any linear combination of eigenfunctions with the same eigenvalue gives another eigenfunction. As a consequence of this, it is possible to decompose L2(R) as a direct sum of four spaces H0, H1, H2, and H3 where the Fourier transform acts on Hk simply by multiplication by ik. This approach to define the Fourier ________________________WORLD TECHNOLOGIES________________________ transform is due to N. Wiener (Duoandikoetxea 2001). The choice of Hermite functions is convenient because they are exponentially localized in both frequency and time domains, and thus give rise to the fractional Fourier transform used in time-frequency analysis (Boashash 2003). Fourier transform on Euclidean space The Fourier transform can be in any arbitrary number of dimensions n. As with the onedimensional case there are many conventions, for an integrable function ƒ(x) here we takes the definition: WT where x and ξ are n-dimensional vectors, and x · ξ is the dot product of the vectors. The . dot product is sometimes written as All of the basic properties listed above hold for the n-dimensional Fourier transform, as do Plancherel's and Parseval's theorem. When the function is integrable, the Fourier transform is still uniformly continuous and the Riemann–Lebesgue lemma holds. (Stein & Weiss 1971) Uncertainty principle Generally speaking, the more concentrated f(x) is, the more spread out its Fourier transform must be. In particular, the scaling property of the Fourier transform may be seen as saying: if we "squeeze" a function in x, its Fourier transform "stretches out" in ξ. It is not possible to arbitrarily concentrate both a function and its Fourier transform. The trade-off between the compaction of a function and its Fourier transform can be formalized in the form of an Uncertainty Principle by viewing a function and its Fourier transform as conjugate variables with respect to the symplectic form on the time– frequency domain: from the point of view of the linear canonical transformation, the Fourier transform is rotation by 90° in the time–frequency domain, and preserves the symplectic form. Suppose ƒ(x) is an integrable and square-integrable function. Without loss of generality, assume that ƒ(x) is normalized: It follows from the Plancherel theorem that is also normalized. ________________________WORLD TECHNOLOGIES________________________ The spread around x = 0 may be measured by the dispersion about zero (Pinsky 2002) defined by In probability terms, this is the second moment of about zero. The Uncertainty principle states that, if ƒ(x) is absolutely continuous and the functions x·ƒ(x) and ƒ′(x) are square integrable, then (Pinsky 2002). WT The equality is attained only in the case ) where σ > 0 is arbitrary and C1 is such that ƒ is L2– (hence normalized (Pinsky 2002). In other words, where ƒ is a (normalized) Gaussian function, centered at zero. In fact, this inequality implies that: for any in R (Stein & Shakarchi 2003). In quantum mechanics, the momentum and position wave functions are Fourier transform pairs, to within a factor of Planck's constant. With this constant properly taken into account, the inequality above becomes the statement of the Heisenberg uncertainty principle (Stein & Shakarchi 2003). Spherical harmonics Let the set of homogeneous harmonic polynomials of degree k on Rn be denoted by Ak. The set Ak consists of the solid spherical harmonics of degree k. The solid spherical harmonics play a similar role in higher dimensions to the Hermite polynomials in dimension one. Specifically, if f(x) = e−π|x|2P(x) for some P(x) in Ak, then . Let the set Hk be the closure in L2(Rn) of linear combinations of functions of the form f(|x|)P(x) where P(x) is in Ak. The space L2(Rn) is then a direct sum of the spaces Hk and the Fourier transform maps each space Hk to itself and is possible to characterize the action of the Fourier transform on each space Hk (Stein & Weiss 1971). Let ƒ(x) = ƒ0(|x|)P(x) (with P(x) in Ak), then where ________________________WORLD TECHNOLOGIES________________________ Here J(n + 2k − 2)/2 denotes the Bessel function of the first kind with order (n + 2k − 2)/2. When k = 0 this gives a useful formula for the Fourier transform of a radial function (Grafakos 2004). Restriction problems In higher dimensions it becomes interesting to study restriction problems for the Fourier transform. The Fourier transform of an integrable function is continuous and the restriction of this function to any set is defined. But for a square-integrable function the Fourier transform could be a general class of square integrable functions. As such, the restriction of the Fourier transform of an L2(Rn) function cannot be defined on sets of measure 0. It is still an active area of study to understand restriction problems in Lp for 1 < p < 2. Surprisingly, it is possible in some cases to define the restriction of a Fourier transform to a set S, provided S has non-zero curvature. The case when S is the unit sphere in Rn is of particular interest. In this case the Tomas-Stein restriction theorem states that the restriction of the Fourier transform to the unit sphere in Rn is a bounded operator on Lp provided 1 ≤ p ≤ (2n + 2) / (n + 3). WT One notable difference between the Fourier transform in 1 dimension versus higher dimensions concerns the partial sum operator. Consider an increasing collection of measurable sets ER indexed by R ∈ (0,∞): such as balls of radius R centered at the origin, or cubes of side 2R. For a given integrable function ƒ, consider the function ƒR defined by: Suppose in addition that ƒ is in Lp(Rn). For n = 1 and 1 < p < ∞, if one takes ER = (−R, R), then ƒR converges to ƒ in Lp as R tends to infinity, by the boundedness of the Hilbert transform. Naively one may hope the same holds true for n > 1. In the case that ER is taken to be a cube with side length R, then convergence still holds. Another natural candidate is the Euclidean ball ER = {ξ : |ξ| < R}. In order for this partial sum operator to converge, it is necessary that the multiplier for the unit ball be bounded in Lp(Rn). For n ≥ 2 it is a celebrated theorem of Charles Fefferman that the multiplier for the unit ball is never bounded unless p = 2 (Duoandikoetxea 2001). In fact, when p ≠ 2, this shows that not only may ƒR fail to converge to ƒ in Lp, but for some functions ƒ ∈ Lp(Rn), ƒR is not even an element of Lp. ________________________WORLD TECHNOLOGIES________________________ Generalizations Fourier transform on other function spaces It is possible to extend the definition of the Fourier transform to other spaces of functions. Since compactly supported smooth functions are integrable and dense in L2(R), the Plancherel theorem allows us to extend the definition of the Fourier transform to general functions in L2(R) by continuity arguments. Further : L2(R) → L2(R) is a unitary operator (Stein & Weiss 1971, Thm. 2.3). Many of the properties remain the same for the Fourier transform. The Hausdorff–Young inequality can be used to extend the definition of the Fourier transform to include functions in Lp(R) for 1 ≤ p ≤ 2. Unfortunately, further extensions become more technical. The Fourier transform of functions in Lp for the range 2 < p < ∞ requires the study of distributions (Katznelson 1976). In fact, it can be shown that there are functions in Lp with p>2 so that the Fourier transform is not defined as a function (Stein & Weiss 1971). WT Fourier–Stieltjes transform The Fourier transform of a finite Borel measure μ on Rn is given by (Pinsky 2002): This transform continues to enjoy many of the properties of the Fourier transform of integrable functions. One notable difference is that the Riemann–Lebesgue lemma fails for measures (Katznelson 1976). In the case that dμ = ƒ(x) dx, then the formula above reduces to the usual definition for the Fourier transform of ƒ. In the case that μ is the probability distribution associated to a random variable X, the Fourier-Stieltjes transform is closely related to the characteristic function, but the typical conventions in probability theory take eix·ξ instead of e−2πix·ξ (Pinsky 2002). In the case when the distribution has a probability density function this definition reduces to the Fourier transform applied to the probability density function, again with a different choice of constants. The Fourier transform may be used to give a characterization of continuous measures. Bochner's theorem characterizes which functions may arise as the Fourier–Stieltjes transform of a measure (Katznelson 1976). Furthermore, the Dirac delta function is not a function but it is a finite Borel measure. Its Fourier transform is a constant function (whose specific value depends upon the form of the Fourier transform used). Tempered distributions The Fourier transform maps the space of Schwartz functions to itself, and gives a homeomorphism of the space to itself (Stein & Weiss 1971). Because of this it is possible to define the Fourier transform of tempered distributions. These include all the integrable ________________________WORLD TECHNOLOGIES________________________ functions mentioned above, as well as well-behaved functions of polynomial growth and distributions of compact support, and have the added advantage that the Fourier transform of any tempered distribution is again a tempered distribution. The following two facts provide some motivation for the definition of the Fourier transform of a distribution. First let ƒ and g be integrable functions, and let and be their Fourier transforms respectively. Then the Fourier transform obeys the following multiplication formula (Stein & Weiss 1971), Secondly, every integrable function ƒ defines a distribution Tƒ by the relation WT for all Schwartz functions φ. In fact, given a distribution T, we define the Fourier transform by the relation for all Schwartz functions φ. It follows that Distributions can be differentiated and the above mentioned compatibility of the Fourier transform with differentiation and convolution remains true for tempered distributions. Locally compact abelian groups The Fourier transform may be generalized to any locally compact abelian group. A locally compact abelian group is an abelian group which is at the same time a locally compact Hausdorff topological space so that the group operations are continuous. If G is a locally compact abelian group, it has a translation invariant measure μ, called Haar measure. For a locally compact abelian group G it is possible to place a topology on the set of characters so that is also a locally compact abelian group. For a function ƒ in L1(G) it is possible to define the Fourier transform by (Katznelson 1976): ________________________WORLD TECHNOLOGIES________________________ Locally compact Hausdorff space The Fourier transform may be generalized to any locally compact Hausdorff space, which recovers the topology but loses the group structure. Given a locally compact Hausdorff topological space X, the space A=C0(X) of continuous complex-valued functions on X which vanish at infinity is in a natural way a commutative C*-algebra, via pointwise addition, multiplication, complex conjugation, and with norm as the uniform norm. Conversely, the characters of this algebra A, denoted ΦA, are naturally a topological space, and can be identified with evaluation at a point of x, and one has an isometric isomorphism real line, this is exactly the Fourier transform. In the case where X=R is the Non-abelian groups WT The Fourier transform can also be defined for functions on a non-abelian group, provided that the group is compact. Unlike the Fourier transform on an abelian group, which is scalar-valued, the Fourier transform on a non-abelian group is operator-valued (Hewitt & Ross 1971, Chapter 8). The Fourier transform on compact groups is a major tool in representation theory (Knapp 2001) and non-commutative harmonic analysis. Let G be a compact Hausdorff topological group. Let Σ denote the collection of all isomorphism classes of finite-dimensional irreducible unitary representations, along with a definite choice of representation U(σ) on the Hilbert space Hσ of finite dimension dσ for each σ ∈ Σ. If μ is a finite Borel measure on G, then the Fourier–Stieltjes transform of μ is the operator on Hσ defined by where is the complex-conjugate representation of U(σ) acting on Hσ. As in the abelian case, if μ is absolutely continuous with respect to the left-invariant probability measure λ on G, then it is represented as dμ = fdλ for some ƒ ∈ L1(λ). In this case, one identifies the Fourier transform of ƒ with the Fourier–Stieltjes transform of μ. The mapping defines an isomorphism between the Banach space M(G) of finite Borel measures and a closed subspace of the Banach space C∞(Σ) consisting of all sequences E = (Eσ) indexed by Σ of (bounded) linear operators Eσ : Hσ → Hσ for which the norm ________________________WORLD TECHNOLOGIES________________________ is finite. The "convolution theorem" asserts that, furthermore, this isomorphism of Banach spaces is in fact an isomorphism of C* algebras into a subspace of C∞(Σ), in which M(G) is equipped with the product given by convolution of measures and C∞(Σ) the product given by multiplication of operators in each index σ. The Peter-Weyl theorem holds, and a version of the Fourier inversion formula (Plancherel's theorem) follows: if ƒ ∈ L2(G), then WT where the summation is understood as convergent in the L2 sense. The generalization of the Fourier transform to the noncommutative situation has also in part contributed to the development of noncommutative geometry. In this context, a categorical generalization of the Fourier transform to noncommutative groups is Tannaka-Krein duality, which replaces the group of characters with the category of representations. However, this loses the connection with harmonic functions. Alternatives In signal processing terms, a function (of time) is a representation of a signal with perfect time resolution, but no frequency information, while the Fourier transform has perfect frequency resolution, but no time information: the magnitude of the Fourier transform at a point is how much frequency content there is, but location is only given by phase (argument of the Fourier transform at a point), and standing waves are not localized in time – a sine wave continues out to infinity, without decaying. This limits the usefulness of the Fourier transform for analyzing signals that are localized in time, notably transients, or any signal of finite extent. As alternatives to the Fourier transform, in time-frequency analysis, one uses timefrequency transforms or time-frequency distributions to represent signals in a form that has some time information and some frequency information – by the uncertainty principle, there is a trade-off between these. These can be generalizations of the Fourier transform, such as the short-time Fourier transform or fractional Fourier transform, or can use different functions to represent signals, as in wavelet transforms and chirplet transforms, with the wavelet analog of the (continuous) Fourier transform being the continuous wavelet transform. (Boashash 2003). For a variable time and frequency resolution, the De Groot Fourier Transform can be considered. ________________________WORLD TECHNOLOGIES________________________ Applications Analysis of differential equations Fourier transforms and the closely related Laplace transforms are widely used in solving differential equations. The Fourier transform is compatible with differentiation in the following sense: if f(x) is a differentiable function with Fourier transform , then the . This can be used to transform Fourier transform of its derivative is given by differential equations into algebraic equations. Note that this technique only applies to problems whose domain is the whole set of real numbers. By extending the Fourier transform to functions of several variables partial differential equations with domain Rn can also be translated into algebraic equations. WT Fourier transform spectroscopy The Fourier transform is also used in nuclear magnetic resonance (NMR) and in other kinds of spectroscopy, e.g. infrared (FTIR). In NMR an exponentially-shaped free induction decay (FID) signal is acquired in the time domain and Fourier-transformed to a Lorentzian line-shape in the frequency domain. The Fourier transform is also used in magnetic resonance imaging (MRI) and mass spectrometry. Domain and range of the Fourier transform It is often desirable to have the most general domain for the Fourier transform as possible. The definition of Fourier transform as an integral naturally restricts the domain to the space of integrable functions. Unfortunately, there is no simple characterizations of which functions are Fourier transforms of integrable functions (Stein & Weiss 1971). It is possible to extend the domain of the Fourier transform in various ways, as discussed in generalizations above. The following list details some of the more common domains and ranges on which the Fourier transform is defined. • The space of Schwartz functions is closed under the Fourier transform. Schwartz functions are rapidly decaying functions and do not include all functions which are relevant for the Fourier transform. More details may be found in (Stein & Weiss 1971). • The space Lp maps into the space Lq, where 1/p + 1/q = 1 and 1 ≤ p ≤ 2 (Hausdorff–Young inequality). • In particular, the space L2 is closed under the Fourier transform, but here the Fourier transform is no longer defined by integration. • The space L1 of Lebesgue integrable functions maps into C0, the space of continuous functions that tend to zero at infinity – not just into the space of bounded functions (the Riemann–Lebesgue lemma). ________________________WORLD TECHNOLOGIES________________________ • The set of tempered distributions is closed under the Fourier transform. Tempered distributions are also a form of generalization of functions. It is in this generality that one can define the Fourier transform of objects like the Dirac comb. Other notations Other common notations for are these: Though less commonly other notations are used. Denoting the Fourier transform by a capital letter corresponding to the letter of function being transformed (such as f(x) and F(ξ)) is especially common in the sciences and engineering. In electronics, the omega (ω) is often used instead of ξ due to its interpretation as angular frequency, sometimes it is written as F(jω), where j is the imaginary unit, to indicate its relationship with the Laplace transform, and sometimes it is written informally as F(2πf) in order to use ordinary frequency. WT The interpretation of the complex function coordinate form: where: may be aided by expressing it in polar in terms of the two real functions A(ξ) and φ(ξ) is the amplitude and is the phase. Then the inverse transform can be written: which is a recombination of all the frequency components of ƒ(x). Each component is a complex sinusoid of the form e2πixξ whose amplitude is A(ξ) and whose initial phase angle (at x = 0) is φ(ξ). The Fourier transform may be thought of as a mapping on function spaces. This mapping is here denoted and is used to denote the Fourier transform of the function f. This mapping is linear, which means that can also be seen as a linear transformation on the function space and implies that the standard notation in linear algebra of applying a ________________________WORLD TECHNOLOGIES________________________ linear transformation to a vector (here the function f) can be used to write instead of . Since the result of applying the Fourier transform is again a function, we can be interested in the value of this function evaluated at the value ξ for its variable, and this is denoted either as or as . Notice that in the former case, it is implicitly understood that is applied first to f and then the resulting function is evaluated at ξ, not the other way around. In mathematics and various applied sciences it is often necessary to distinguish between a function f and the value of f when its variable equals x, denoted f(x). This means that a formally can be interpreted as the Fourier transform of the values notation like of f at x. Despite this flaw, the previous notation appears frequently, often when a particular function or a function of a particular variable is to be transformed. For example, is sometimes used to express that the Fourier transform of a rectangular function is a sinc function, or WT is used to express the shift property of the Fourier transform. Notice, that the last example is only correct under the assumption that the transformed function is a function of x, not of x0. Other conventions The Fourier transform can also be written in terms of angular frequency: ω = 2πξ whose units are radians per second. The substitution ξ = ω/(2π) into the formulas above produces this convention: Under this convention, the inverse transform becomes: Unlike the convention followed here, when the Fourier transform is defined this way, it is no longer a unitary transformation on L2(Rn). There is also less symmetry between the formulas for the Fourier transform and its inverse. Another convention is to split the factor of (2π)n evenly between the Fourier transform and its inverse, which leads to definitions: ________________________WORLD TECHNOLOGIES________________________ Under this convention, the Fourier transform is again a unitary transformation on L2(Rn). It also restores the symmetry between the Fourier transform and its inverse. Variations of all three conventions can be created by conjugating the complexexponential kernel of both the forward and the reverse transform. The signs must be opposites. Other than that, the choice is (again) a matter of convention. Summary of popular forms of the Fourier transform ordinary frequency unitary ξ (hertz) WT nonunitary angular frequency ω (rad/s) unitary The ordinary-frequency convention is the one most often found in the mathematics literature. In the physics literature, the two angular-frequency conventions are more commonly used. As discussed above, the characteristic function of a random variable is the same as the Fourier–Stieltjes transform of its distribution measure, but in this context it is typical to take a different convention for the constants. Typically characteristic function is defined . As in the case of the "non-unitary angular frequency" convention above, there is no factor of 2π appearing in either of the integral, or in the exponential. Unlike any of the conventions appearing above, this convention takes the opposite sign in the exponential. ________________________WORLD TECHNOLOGIES________________________ Tables of important Fourier transforms The following tables record some closed form Fourier transforms. For functions ƒ(x) , g(x) and h(x) denote their Fourier transforms by , , and respectively. Only the three most common conventions are included. It is sometimes useful to notice that entry 105 gives a relationship between the Fourier transform of a function and the original function, which can be seen as relating the Fourier transform and its inverse. Square-integrable functions The Fourier transforms in this table may be found in (Campbell & Foster 1948), (Erdélyi 1954), or the appendix of (Kammler 2000). Fourier Fourier Fourier transform transform transform unitary, angular non-unitary, Function unitary, ordinary frequency angular frequency frequency 201 202 WT Remarks The rectangular pulse and the normalized sinc function, here defined as sinc(x) = sin(πx)/(πx) Dual of rule 201. The rectangular function is an ideal lowpass filter, and the sinc function is the noncausal impulse response of such a filter. ________________________WORLD TECHNOLOGIES________________________ 203 The function tri(x) is the triangular function 204 Dual of rule 203. 205 206 207 208 The function u(x) is the Heaviside unit step function and a>0. This shows that, for the unitary Fourier transforms, the Gaussian function exp(−αx2) is its own Fourier transform for some choice of α. For this to be integrable we must have Re(α)>0. For a>0. That is, the Fourier transform of a decaying exponential function is a Lorentzian function. The functions Jn (x) are the n-th order Bessel functions of the first kind. The functions Un (x) are the WT ________________________WORLD TECHNOLOGIES________________________ Chebyshev polynomial of the second kind. Hyperbolic secant is its own Fourier transform 209 Distributions The Fourier transforms in this table may be found in (Erdélyi 1954) or the appendix of (Kammler 2000). WT Two-dimensional functions Fourier transform Functions (400 to unitary, ordinary 402) frequency Remarks Fourier transform unitary, angular frequency Fourier transform non-unitary, angular frequency To 400: The variables ξx, ξy, ωx, ωy, νx and νy are real numbers. The integrals are taken over the entire plane. To 401: Both functions are Gaussians, which may not have unit volume. To 402: The function is defined by circ(r)=1 0≤r≤1, and is 0 otherwise. This is the Airy distribution, and is expressed using J1 (the order 1 Bessel function of the first kind). (Stein & Weiss 1971, Thm. IV.3.3) ________________________WORLD TECHNOLOGIES________________________ Formulas for general n-dimensional functions Function Fourier transform unitary, ordinary frequency Fourier transform unitary, angular frequency Fourier transform non-unitary, angular frequency 500 501 502 Remarks WT To 501: The function χ[0,1] is the indicator function of the interval [0,1]. The function Γ(x) is the gamma function. The function Jn/2 + δ is a Bessel function of the first kind, with order n/2+δ. Taking n = 2 and δ = 0 produces 402. (Stein & Weiss 1971, Thm. 4.13) To 502: The formula also holds for all α ≠ −n, −n−1, ... by analytic continuation, but then the function and its Fourier transforms need to be understood as suitably regularized tempered distributions. ________________________WORLD TECHNOLOGIES________________________