4. Experimental methods
4.3. Infrared spectroscopy
The infrared spectral region is adjacent to the visible spectral region for longer wavelengths.
It can be further classified into three different regions, the Near-Infrared (NIR), Mid-Infrared (MIR) and Far-Mid-Infrared (FIR). While low-energy Far-Mid-Infrared radiation mainly induces rotations of molecules around chemical bonds, MIR and NIR excite vibrations of molecules.
A simple model for the description of molecular vibrations is the harmonic oscillator.
Thereby a two-atomic molecule is represented by two masses m1 and m2 which are connected by a mechanical spring with a force constant k. For a harmonic oscillation the force is proportional to the displacement xaround the resting position of the atom and can be calculated according toHooke’s law.
F(x) = −k·x (4.1)
Integration of Hooke’s law leads to the parabolic harmonic potential V.
V(x) = 1
2 ·k·x2 (4.2)
The frequency of the harmonic oscillation can be derived by solving the second order differential equation
µ· d2x
dt2 =−k·x (4.3)
with the reduced mass
µ= m1·m2
m1+m2 (4.4)
and using a sine function as ansatz.
The vibrational frequency ν is found to be ν= 1
2π
sk
µ (4.5)
4.3. Infrared spectroscopy
˜ ν = ν
c = 1
λ (4.6)
The quantum-mechanical description of the harmonic oscillator leads to the formulation of discrete equidistant energy levels
En= (n+ 1
2)hν (4.7)
and the selection rule for allowed transitions of
∆n =±1 (4.8)
The harmonic potential can only be used as an approximation for low displacements around the resting position, since it e.g. cannot explain the dissociation of a molecule.
For higher displacements, the anharmonicity can be described with theMorsepotential (Figure 4.6).
V(x) =De(1−e−a(x−x0))2 (4.9)
Derepresents the dissociation energy and the constanta is related to the force constant and describes the strength of the bond between the atoms. For x<x0, the repulsion between the atoms strongly increases which is reflected in a steeper increase of the potential barrier of the Morse potential compared to the harmonic potential. For x>x0 theMorsepotential reaches a threshold in form of the dissociation energy leading to the dissociation of the molecule. For the anharmonic oscillator, the discrete energy levels are no longer equidistant to each other and transitions from the ground state to higher energy levels in form of overtone vibrations are allowed. The probability of the appearance of overtone vibrations is strongly reduced though leading to smaller intensities for overtone vibrational bands.
4. Experimental methods
Figure 4.6.: Harmonic potential (left) and Morsepotential (right).
The frequency of the electromagnetic radiation has to match the energy difference of the vibrational states of the molecule in order to excite vibrational transitions. Different forms of vibrations of the molecules exist: symmetric or asymmetric stretching vibrations, bending, rocking, wagging and twisting vibrations. A vibrating molecule is called IR active, when the absorbed radiation induces a change of the dipole moment during the vibration. Homonuclear molecules like e.g. nitrogen (N2) are therefore IR inactive, since they do not have a dipole moment. For carbon dioxide (CO2) the symmetric stretching vibration is IR inactive, while the asymmetric stretching and bending vibrations are IR active. Only for the latter vibrations, a change of the dipole moment is induced.
An infrared spectrum reveals the absorbance of the molecule versus the wavenumber of radiation. A wavenumber is defined as the number of waves that exist over a specified distance and is the reciprocal of the wavelength (equation 4.6). Typically the wavenumber in infrared spectra is given in the unit cm-1.
The first infrared spectroscopic measurements were performed with dispersive spectrome-ters. In the 1960’s the invention of FTIR spectrometers revolutionized the technique, leading to much faster measurements with much better signal-to-noise ratio. Nowadays FTIR spectroscopy is used essentially exclusively due to its significant advantages. Cen-tral part of the FTIR spectrometer is the built-in Michelson interferometer (Figure 4.7).
4.3. Infrared spectroscopy
IR light source
sample
Detector Beam splitter
fixed mirror
movable mirror
Figure 4.7.: Scheme of aMichelsoninterferometer. Polychromatic infrared light is split by a beam splitter and then reflected by a fixed and a movable mirror. Depending on the position of the movable mirror, the recombined beam interferes constructively or destructively and passes the sample for detection.
The infrared light from a silicon carbide globar light source is first split up by a potassium bromide beam splitter. The reflected light beam is reflected by a fixed mirror back to the beam splitter, whereas the transmitted light beam is reflected by a movable mirror back to the beam splitter, where both light beams recombine. Depending on the mirror position, the polychromatic infrared light interferes constructively or destructively.
The recombined light beam is then passed through the sample and focused onto a mercury cadmium telluride (MCT) detector resulting in an interferogram. An interferogram is defined as the dependence of the intensity on the position of the movable mirror.
For a standard static infrared measurement, the movable mirror of the Michelson interferometer moves continuously forward and backward during the acquisition mode. For a single frequency, the interferogram would result in a sinusoidal function with its maxima representing constructive and its minima representing destructive interference. Due to the polychromacity of the infrared light source all wavelengths interfere positively at the so called centerburst of the interferogram, when fixed and movable mirror have the equal distance from the beam splitter, leading to the highest recorded intensity at that position.
Figure 4.8 shows a schematic characteristic interferogram recorded with a polychromatic globar infrared light source. A helium-neon-laser beam (λ = 633 nm) propagating simultaneously through the interferometer serves for calibration and determination
4. Experimental methods
of the movable mirror position. Every zero passage (λ/2) of the recorded sinusoidal interferogram defines a digitalization point of the interferogram. Single channel spectra I(˜ν) of the reference and the measured sample can be obtained via Fouriertransformation of the reference interferogram and sample interferogram according to equation 4.10.
The transmittance spectrum of the sample T is then defined as the quotient of single channel spectrum of the sample I(˜ν)sample and single channel spectrum of the reference I(˜ν)ref erence. The absorption spectrum A(˜ν) can be calculated according to equation 4.11.
Mirror position
Intensity
Figure 4.8.: Typical interferogram for a polychromatic light source (blue), showing the de-pendence of the detected intensity on the position of the movable mirror. The centerburst with highest intensity occurs when the fixed and the movable mirror have the same distance from the beam splitter giving rise to constructive interfer-ence for all frequencies of the polychromatic light. The sinusoidal interferogram of the helium-neon-laser beam (black) defines the digitalization points of the polychromatic light interferogram.
I(˜ν) =
∞
Z
−∞
I(x)D(˜ν)cos(2πνx)dx˜ (4.10)
A(˜ν) = −lg I(˜ν)sample
I(˜ν)ref erence (4.11)