2. Infrared Spectroscopy

Before going on to consider modelling and measurements of absorbing gases in the atmosphere it is important to have some understanding of the processes that result in the absorption taking place. Since the work described herein is directly concerned only with the absorption of infrared radiation by diatomic molecules (hydrogen chloride and hydrogen bromide), this is all that shall be considered in detail although reference shall also be made to the behaviour of other molecular configurations. Standard texts (such as Herzberg [1950] or Bernath [1995]) consider the subject in greater detail.

The absorption of electromagnetic radiation is associated with a change in the energy of the absorber, and the form that this change takes is characteristic of the wavelength of the incident radiation (table 2-1); in the case of the infrared this takes the form of a change in the vibrational and rotational energy of the absorber.

Consider a diatomic molecule to be represented by two point masses separated by a light, rigid rod; the molecule is therefore free only to rotate about its centre of mass. The energies that the molecule (described by Schrödinger equation for a rigid rotator) may possess are given by

(2.1)

where J is the rotational quantum number and is whole. Equation (2.1), therefore, defines a set of energy levels, the spacing of which increases as the square of the quantum number.

It can be shown (Herzberg [1950]) that the angular momentum of the system is given by

(2.2)

By using the classical relation for angular momentum, P = Iw, we may obtain the expression for the angular velocity:

(2.3)

and hence the rotational frequency may be seen to be

(2.4)

which is approximately linear with J.

Classically, radiation of any frequency may be absorbed by a molecule with a permanent dipole moment since there is no restriction on the rotational frequency of the molecule, but quantum theory requires that a molecule may only possess energies given by equation (2.1), and so only finite quanta of energy may be absorbed or emitted. This corresponds to a change between the permitted energy levels which may be described by

s = E''/hc - E'/hc

(2.5)

where s is the wave number of the emitted or absorbed radiation and E’ and E’’ represent the energies of the upper and lower states in question; these are described by

F(J) = E/hc = BJ(J+1)

(2.6)

where the rotational constant

B = h/8p2cI

(2.7)

Using equation (2.6), equation (2.5) may be rewritten as

s = BJ’(J’+1) - BJ’’(J’’+1),

(2.8)

where J’ and J’’ are the rotational quantum numbers in the upper and lower states respectively.

It may be shown (Herzberg [1950]) that the selection rule for J is DJ = ±1; substituting J’=J’’+1 into equation (2.7) we find that

s = 2B(J’’+1)

(2.9)

yielding the result that the spectrum of the rigid rotator consists of a series of lines at frequencies equal to nr (equation (2.4)) and separated by approximately twice the rotational constant.

2.1.2 The Nonrigid Rotator

Now consider the situation where the light rod may be stretched as the molecule rotates. The result of this is that the moment of inertia of the system becomes dependent upon its rotational energy. It may be shown that, to a good approximation, the situation can be handled if equation (2.6) is modified as follows:

F(J) = BJ(J+1) - DJ2(J+1)2

(2.10)

where the rotational constant D is a function of the vibrational frequency w and is defined as

D = 4B3/w2.

(2.11)

The effect of this is to slightly reduce the rate at which the separation of the energy levels increases in the case of the nonrigid rotator compared to the rigid rotator.

2.2 Vibration Spectra of Diatomic Molecules

Consider the situation where the diatomic molecule is represented by two point masses separated by a light, elastic rod. The molecule is not permitted to rotate but the atoms may vibrate in antiphase, alternately compressing and stretching the rod; the system can be likened to a harmonic oscillator.

2.2.1 The Harmonic Oscillator

A harmonic oscillator can be defined as a system where a point mass is acted upon by a force the magnitude of which is proportional to the displacement of the mass from its position of equilibrium, and whose direction is towards that point. The energy levels of the quantum harmonic oscillator are given by

E(u) = hn0(u + ½)

(2.12)

where u is a whole number known as the vibrational quantum number.

2.2.2 The Anharmonic Oscillator

The foregoing argument assumes that the restoring force on the atoms increases with distance ad infinitum, but this is clearly not the case and so the model must be modified.

For the case of a harmonic oscillator, the potential energy function is parabolic and might be represented by

U = f(r-re)2,

(2.13)

where r represents the separation of the nuclei and r_e is their equilibrium separation, but to take account of the fact that the restoring force decreases with increasing r-r_e the potential is better represented by

U = f(r-re)2 - g(r-re)3

(2.14)

where the cubic term (g being much smaller than f) has the effect of lowering the restoring force at greater internuclear separations which has the effect of slightly decreasing the separations of the energy levels at higher values of the vibrational quantum number.

Although the only vibrational transitions of relevance to the work described herein are those that occur between the ground and first excited state, it could be noted that in the case where the frequency of vibration is low or the temperature of the absorber is elevated, the population in the first excited state may become significant and transitions between the first and second vibrational states may occur. The frequencies of these so-called ‘hot bands’ are slightly lower than those of the fundamental but close to them (Thorne [1988]). Since the hydrogen chloride absorption feature of interest is at 2926 cm^-1 and the temperature in the chosen profile never exceeds 270 K, such hot bands need not be considered further.

Summary
Acknowlegements
Contents
Chapters: 1, 2, 3, 4, 5, 6, 7, 8
Appendices
References