Revised Edition: 2016
ISBN 978-1-283-49456-4
© All rights reserved.
Published by:
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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
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Chapter 5 - Fluorescence Correlation Spectroscopy
Chapter 6 - Rotational Spectroscopy
Chapter 7 - Electromagnetic Spectrum
Chapter 8 - Molecular Vibration
Chapter 9 - Fourier Transform
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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.
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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.
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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.
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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).
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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.
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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
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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
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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
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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
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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
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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
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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.
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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:
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(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
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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.
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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
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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.
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Chapter- 2
Fourier Transform Infrared Spectroscopy
and Infrared Spectroscopy Correlation Table
Fourier transform infrared spectroscopy
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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
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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.
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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
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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.
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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
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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.
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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.
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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.
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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
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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
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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
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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)
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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
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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
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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.
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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
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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.
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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
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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
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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.
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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.
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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.
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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.
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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
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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.
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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.
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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
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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
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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
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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.
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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
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•
•
•
•
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-
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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.
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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:
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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).
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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
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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):
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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
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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.
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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.
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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
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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).
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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)
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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,
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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
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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
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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
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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.
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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.
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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.
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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.
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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)
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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.
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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.
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Types of radiation
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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
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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
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,
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.
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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:
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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.
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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).
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WT
Original function showing oscillation 3 hertz.
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Real and imaginary parts of integrand for Fourier transform at 3 hertz
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Real and imaginary parts of integrand for Fourier transform at 5 hertz
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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
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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
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Uniform continuity and the Riemann–Lebesgue lemma
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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.
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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):
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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
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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
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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.
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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
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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.
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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
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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):
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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
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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.
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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).
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•
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
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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:
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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.
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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.
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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
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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)
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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.
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