1.2.1 Definition of Parameters

There are several parameters used to describe the absorption of light by gas molecules. We start with the definition of the absorption cross-section for a single gas molecule, where the molecule is viewed as having an effective area, σ, for the capture or absorption of a photon from a beam of intensity, I(x,y), as shown in Figure 1.1.

Figure 1.1 Description of the parameters used in the Beer–Lambert law.

Considering a cross-sectional area of dA = dxdy , the probability of photon capture or absorption by the molecule is σ/dA. Over a short length, dz, in the propagation direction, the beam will encounter N_gdAdz gas molecules and the rate of photon absorption or change in beam intensity will therefore be:

$\mathrm{d}I\left(x,y\right)=-I\left(x,y\right)\left(\frac{\sigma }{\mathrm{d}A}\right)\left(N_g\mathrm{d}A\mathrm{d}z\right)=-I\left(x,y\right)\sigma N_g\mathrm{d}z$(1.1)

where N_g is the number density of the target gas molecules (number per unit volume) and we assume that this density is uniform in the (x,y) transverse plane and along beam direction, z.

The power in the beam is P =  ∬ I(x, y)dxdy and hence integration of (1.1) leads to the Beer–Lambert law for the transmitted power, P_out, after a path length, l, over which the beam and gas interact:

where P_in is the incident beam power.

As shown in Figure 1.2, the absorption cross-section, σ, is dependent on the optical frequency, ν, gas pressure, p, and temperature, T, and may be written as σ(ν, p, T) = S(T)ϕ(ν, p, T), where ϕ(ν, p, T) describes the lineshape function and S is the line strength per molecule defined by:

$S\left(T\right)={\int }_{-\infty }^{\infty }\sigma \left(\nu ,p,T\right)\mathrm{d}\nu$(1.3)

Note that with the above definitions, ϕ(ν, p, T) is normalised so that ${\int }_{-\infty }^{\infty }\varphi \left(\nu ,p,T\right)\mathrm{d}\nu =1$.

Figure 1.2 Typical absorption cross-section of a gas molecule and the associated parameters.

The number density, N_g, in (1.2) may be expressed in terms of the mole fraction or partial pressure of the target gas using the ideal gas law and Dalton’s law:

$p=\left(\frac{n}{V}\right)\mathit{RT}=\left(\frac{n_1+n_2+\dots +n_i}{V}\right)\mathit{RT}=p_1+p_2+\dots +p_i$(1.4)

where n is the total number of moles and n_i and p_i are the number of moles and the partial pressure of each of the constituent gases, respectively.

Hence we can write for the number density of the target gas:

$N_g=N_A\left(\frac{n_g}{V}\right)=\frac{p{X}_g}{k_{B}T}=\frac{p_g}{k_{B}T}$(1.5)

where N_A is Avogadro’s number, k_B = R/N_A is the Boltzmann constant, p is the total gas pressure, n_g is the number of moles, p_g the partial pressure and X_g = n_g/n is the mole fraction of the target gas.

Hence the Beer–Lambert law may be written as:

where S^′(T) = S(T)/(k_BT).

Another commonly used parameter is the absorption coefficient, α, where the absorption cross-section or line strength for a single molecule is multiplied by N₀ as follows:

where N₀ = p₀/(k_BT₀) ≅ 2.5 × 10²⁵m⁻³ is the molecular density at normal temperature and pressure (NTP, T₀ = 293.15 K, p₀ = 1 atm pressure or 101.325 kPa).

The Beer–Lambert law is then:

where the target gas concentration is given by C = N_g/N₀ = (T₀/T)(pX_g/p₀) = (T₀/T)(p_g/p₀), with T₀ = 293.15 K and p₀ = 1atm pressure.

Equations (1.2), (1.6) and (1.8) represent useful equivalent forms of the Beer–Lambert law for calculating the gas absorption.

1.2.2 Absorption Lineshape Functions

The absorption lineshape function, ϕ(ν, p, T), for gases is determined by three line-broadening mechanisms, namely, natural (or lifetime) broadening, Doppler broadening and collisional (or pressure) broadening [Reference Demtroder1, Reference Fox2].

1.2.2.2 Doppler Broadening

The random thermal motion of gas molecules means that the optical frequency ‘seen’ by a molecule is shifted due to the Doppler effect. The frequency shift will be positive or negative depending on whether the z-component of the molecular velocity, ν_z, is in the opposite or in the same direction as the beam propagation along the z-axis (the velocity component perpendicular to the beam will have no effect). The frequency shift is given by:

$\frac{\nu -{\nu }_0}{{\nu }_0}=v_z/c$(1.9)

where ν₀ is the centre frequency of the transition.

The number of gas molecules with a velocity component between ν_z and ν_z + dν_z is given by the one-dimensional Maxwell–Boltzmann distribution:

$N\left(v_z\right)\mathrm{d}{v}_z=N_T{\left(\frac{m}{2\pi k_{B}T}\right)}^{\frac{1}{2}}exp\left\{-\frac{m{v}_z^2}{2k_{B}T}\right\}\mathrm{d}{v}_z$(1.10)

where N_T is the total number of molecules and m is the molecular mass.

Substituting v_z = (c/ν₀)(ν − ν₀) and dv_z = (c/ν₀)dν from (1.9) into (1.10) and, since the absorbed power in the range ν to ν + dν will be proportional to the number of molecules in the range ν_z to ν_z + dν_z, we obtain a normalised Gaussian lineshape for Doppler broadening as:

${\varphi }_G\left(\nu ,T\right)=\frac{1}{{\gamma }_D\sqrt{\pi }}exp\left\{-{\left(\frac{\nu -{\nu }_0}{{\gamma }_D}\right)}^2\right\}$(1.11)

with the 1/e half-linewidth,

${\gamma }_D={\nu }_0\sqrt{\frac{2k_{B}T}{m{c}^2}}\cong \frac{1.29\times 10^2}{{\lambda }_0}\sqrt{\frac{T}{M}}$(1.12)

where M is the molecular weight in atomic mass units. The half-width-half-maximum linewidth is ${\gamma }_{\mathit{\text{HWHM}}}={\gamma }_D\sqrt{ln\left(2\right)}$. Typically, for CO₂ at an absorption wavelength of ~2 µm, with M = 44 and T ~300 K, γ_HWHM ~ 0.14 GHz.

1.2.2.3 Collisional Broadening

Gas molecules are subject to random collisions and, as the molecules approach each other during collisions, their energy levels will be perturbed, each level by differing amounts. Hence the frequency of photon absorption during a collision will differ from the unperturbed situation, resulting in collisional broadening of the absorption line. From a simple analysis, we would expect the line broadening to be dependent on the collision rate or inversely on the mean time, τ_c, between collisions. In turn, the collision rate will be proportional to the collision cross-section, σ_c, the total number density of molecules present, N_D = p/k_BT (from the ideal gas law) and the average molecular speed, $\bar{v}=\sqrt{8k_{B}T/\mathit{\pi m}}$ (from the Maxwell–Boltzmann distribution). Hence:

$\frac{1}{{\tau }_c}~{\sigma }_c{N}_D\bar{v}={\sigma }_{c}p\sqrt{\frac{8}{\mathit{\pi m}{k}_{B}T}}$(1.13)

Collisional (or pressure) broadening produces a normalised Lorentzian lineshape of the form:

${\varphi }_L\left(\nu \right)=\left(\frac{1}{\pi {\gamma }_c}\right){\left\{1+{\left(\frac{\nu -{\nu }_0}{{\gamma }_c}\right)}^2\right\}}^{-1}$(1.14)

where, from (1.13), the half-linewidth will have a pressure and temperature dependence of the form:

${\gamma }_c={\gamma }_0\frac{p}{p_0}{\left(\frac{T_0}{T}\right)}^n$(1.15)

where γ₀ is the half-linewidth at NTP and the temperature index, n, is ½ in the simple model outlined above, but in practice may differ from this value.

Broadening will, however, depend on whether collisions occur between molecules of the same species (self-broadening by the target gas) or cross-broadening from different species. The overall broadening may be approximated as the weighted sum of self-broadening and cross-broadening effects. For example, for the target gas in air:

${\gamma }_c\cong \frac{\left\{p_g{\gamma }_{0s}+\left(p-p_g\right){\gamma }_{0a}\right\}}{p_0}{\left(\frac{T_0}{T}\right)}^n$(1.16)

where γ_0s and γ_0a are the self- and air-broadened half-widths at NTP and p_g is the partial pressure of the target gas.

Note from the above discussion that the Doppler linewidth is dependent only on temperature, whereas pressure broadening depends on pressure and inversely on temperature. At NTP, pressure-broadened linewidths are typically of the order of several GHz and dominate over Doppler broadening. However, at lower pressures and/or higher temperatures where the pressure and Doppler-broadened linewidths are comparable, the lineshape may be described by a convolution of the normalised Gaussian and Lorentzian lineshapes:

${\varphi }_V\left(\nu -{\nu }_0\right)=\left\{\left(\frac{1}{{\gamma }_D\sqrt{\pi }}\right)exp\left[-{\left(\frac{\nu -{\nu }_0}{{\gamma }_D}\right)}^2\right]\right\}\otimes \left\{\left(\frac{1}{\pi {\gamma }_c}\right){\left[1+{\left(\frac{\nu -{\nu }_0}{{\gamma }_c}\right)}^2\right]}^{-1}\right\}$(1.17)

This convolution function is also normalised and may be written as:

${\varphi }_V\left(\nu -{\nu }_0\right)=\frac{1}{{\gamma }_D\sqrt{\pi }}V\left(x,a\right)$(1.18)

where V(x, a) is the Voigt function [Reference Voigt3] defined by the integral:

$V\left(x,a\right)=\frac{a}{\pi }{\int }_{-\infty }^{\infty }\frac{e^{-u^2}}{a^2+{\left(x-u\right)}^2}\mathrm{d}u$(1.19)

with x = (ν − ν₀)⁄γ_D , a = γ_c⁄γ_D.

The Voigt function cannot be evaluated analytically, but various approximations are given in the literature [Reference Drayson4–Reference Huang and Yung7]. Approximations to the Voigt linewidth are discussed by Olivero and Longbothum [Reference Olivero and Longbothum5].

1.4.1 Rotational Lines of Gases

A molecule may be excited into rotational states through the interaction of the oscillatory electric field of the radiation with the electric dipole moment of the molecule. Hence the existence of a pure rotational line spectrum depends on whether the molecule has a permanent dipole moment. Symmetric linear molecules such as CO₂ and highly symmetric spherical molecules such as CH₄ have no permanent electric dipole moment and hence no pure rotational line spectrum in the microwave region. However, as discussed below, rotational line spectra for these molecules may appear in combination with vibrational states.

In describing molecular rotation, it is convenient to consider three principal axes of rotation through the centre of gravity of the molecule. A key factor in determining the rotational energy states and hence the rotational line spectrum is the moment of inertia associated with each axis, which in turn depends on the shape and symmetry of the molecule. For diatomic gases such as CO and linear molecules such as CO₂ (see Figure 1.3a), the moment of inertia for rotation around the bond axis is approximately zero, while rotation around the other two orthogonal axes has the same moment of inertia so these molecules possess only one series of rotational absorption lines. Highly symmetric spherical molecules such as CH₄ have, to a first approximation, the same moment of inertia for all three principal axes so likewise have a single series of rotational lines (although fine structure exists and may be observed at low pressure). Most molecules have three different moments of inertia, e.g. H₂O, SO₂, NO₂, etc., so possess three basic series of rotational line spectra and combination series from rotation about more than one axis.

Figure 1.3 Rotation and vibration of the CO₂ molecule (a) the three axes of rotation, (b) symmetric stretch mode, (c) anti-symmetric stretch mode, (d) bending mode.

1.4.2 Vibrational Lines of Gases

In a simple model, the bond between the atoms of a molecule may be viewed as a spring obeying Hooke’s law, and the vibrations of a molecule as a mass–spring system forming a simple harmonic oscillator. According to the rules of quantum mechanics, the allowed energy states of the oscillator are quantised according to E_n = (v + ½)hf₀ where f₀ is the classical oscillation frequency, v = 0, 1, 2, …, and changes in the vibrational state are restricted to: Δv =  ±1. However, the real behaviour of molecular bonds differs from the simple Hooke’s law model and the vibrations are better described as an anharmonic oscillator, as shown in Figure 1.4, with quantised energy states, E_n = (v + ½)hf_a{1 − x_a(v + ½)}, where f_a is a characteristic frequency and x_a is the anharmonicity factor. The spacing between the energy states is no longer uniform and the allowed transitions are governed by the selection rule, Δv =  ±1,  ±2,  ±3, … However, the probability of large jumps is small and at normal temperatures the populations of states with v ≥ 1 is small, so the main vibrational transitions of interest are the fundamental, first and second overtones, corresponding to ν changing from 0 → 1, 0 → 2, 0 → 3, respectively, with diminishing probability and line strength. The fundamental transition 0 → 1 is typically in the mid-IR region for gases, with the near-IR region corresponding to first and second overtones giving absorption lines which are two or more orders of magnitude weaker than the fundamental. Note that at high temperature, transitions from the v = 1 state (known as a vibrational hot-band) may become important.

Figure 1.4 The energy levels of the anharmonic oscillator.

The above discussion applies in principle to each vibration mode of a molecule. In general, a molecule containing N atoms can undergo 3N – 6 modes of vibration (3N – 5 for linear molecules). This includes N – 1 bond-stretching motions, with the others being bending motions. Of course, a vibration mode must produce a change in the dipole moment of the molecule for it to interact with the exciting radiation; for example, the symmetric stretching mode of the linear CO₂ molecule (see Figure 1.3b) does not give rise to vibrational absorption lines, whereas the anti-symmetric stretching mode (Figure 1.3c) does. Animations of the vibration modes of various molecules, such as methane, may be viewed on the web, see, for example, reference [Reference Greeves13].

As well as pure vibrational modes, a molecule may be excited into a superposition of vibration modes giving vibrational energy levels corresponding to combination and difference bands. For example if ν₂ represents the (doubly degenerate) vibration mode of CH₄ at ~ 6.5 µm and ν₃ represents the (triply degenerate) asymmetric stretching mode ~3.3 µm, then the first overtone, 2ν₃, gives rise to near-IR absorption lines around 1.66 µm and the combination, ν₂ + 2ν₃, gives lines around 1.33 µm (note that IR-inactive vibration modes may appear in combination with other modes). Table 1.1 provides a list of the vibration modes and absorption regions (λ  >1 µm) for some common gases.

1.4.3 Rovibrational Lines

In general, molecules may simultaneously undergo both vibration and rotation, giving rise to rovibrational energy states. This includes molecules such as CO₂, CH₄, etc., which have no permanent dipole moment and hence no pure rotational spectra, but undergo dipole changes during vibration. As noted earlier, the energy separation of the vibrational states of a gas molecule is typically in the mid-IR region, whereas for rotational states the energy separation is two or three orders of magnitude smaller, in the microwave region. The large difference in these energies means that, to a first approximation, the total energy of a combined rovibrational state is simply the sum of the separate rotational and vibrational energies. Associated with each vibrational absorption line there is a series of closely spaced rotational lines. For linear molecules where the vibration mode produces dipole changes parallel to the major axis of rotational symmetry (e.g. the anti-symmetric stretching mode of CO₂ in Figure 1.3c), transitions between vibrational states must be accompanied by a change in the rotational state, ΔJ =  ±1, but for a vibration mode producing perpendicular dipole changes (e.g. the bending mode of N₂O), ΔJ = 0,  ±1 is allowed. For the vibration modes of non-linear molecules, the selection rules are more complex, but ΔJ = 0,  ±1 is often allowed (e.g. the asymmetric stretching mode of CH₄). Rovibrational lines corresponding to ΔJ = −1, 0, +1 are described as the P, Q, R branches, respectively. Figure 1.5 illustrates the above principles showing the origin of typical rovibrational absorption lines in the near-IR with the P, Q and R branches.

Figure 1.5 Characteristics of typical rovibrational absorption lines in the near-IR showing the P, Q and R branches.

1.4.4 Examples of Gas Absorption Spectra

An extremely useful source of information on spectral line parameters of atmospheric gases and pollutants is the HITRAN database [Reference Gordon, Rothman and Hill14, Reference Rothman15] which also includes an extensive list of references. The HITRAN Application Programming Interface (HAPI) [Reference Kochanov, Gordon and Rothman16, Reference Kochanov, Gordon and Rothman17] and HITRAN on the Web Reference Rothman[18] may be used for the modelling of absorption spectra, including plotting absorption coefficients, and absorption and transmission functions for a variety of important gases and mixtures, with various adjustable parameters such as temperature, pressure, lineshape (Voigt, Lorentz or Doppler), path length, wavenumber range, etc. Outputs are provided in graphical or tabular form which may be downloaded for processing by EXCEL or other programmes.

Figures 1.6–1.12 show some of the stronger near-IR absorption lines for several common and important gases plotted using HITRAN on the Web Reference Rothman[18] with a concentration length product of Cl = 1 cm in (1.8). For comparison, the mid-IR absorption lines for CO, CO₂ and CH₄ with a much smaller value of Cl = 0.01 cm are shown in Figure 8.1 in Chapter 8, clearly showing the relatively weak strength of the near-IR lines. Note that for our purposes we have plotted the spectra in terms of wavelength (µm) rather than the commonly used wavenumber (cm^–1) and hence the order of the P, Q and R branches is reversed. The transmittance (T) on the vertical axis corresponds to the ratio, P_out/P_in, as defined by (1.2), (1.6) or (1.8). Note that the vertical scale differs between graphs to clearly show the line depths. All the examples of the near-IR spectra shown in Figures 1.6–1.12 are for the most abundant isotopologue of the pure gas at a pressure of 1 atm, temperature of 296 K and an absorption path length of 1 cm (hence Cl = 1 cm) with the assumption of a Voigt profile for the lineshape function.

Figure 1.6 Near-IR absorption lines for CO plotted using HITRAN on the Web Reference Rothman[18].

Figure 1.7 Near-IR absorption lines for CO₂: (a) 1.6 μm region (b) 2 μm region plotted using HITRAN on the Web Reference Rothman[18].

Figure 1.8 Near-IR absorption lines for C₂H₂ plotted using HITRAN on the Web Reference Rothman[18].

Figure 1.9 Near-IR absorption lines for CH₄ showing the P, Q and R branches plotted using HITRAN on the Web Reference Rothman[18].

Figure 1.10 Near-IR absorption lines for H₂O in the 1.4μm region plotted using HITRAN on the Web Reference Rothman[18].

Figure 1.11 Near-IR absorption lines for NH₃ plotted using HITRAN on the Web Reference Rothman[18].

Figure 1.12 Near-IR absorption lines for H₂S: (a) 1.6 μm region, (b) 1.9 μm region plotted using HITRAN on the Web Reference Rothman[18].

The absorption coefficient defined by (1.8) at line centre (in units of cm^–1) for a particular line may be easily obtained from these figures as approximately the line depth when it is small or more accurately as α₀ = −ln (T_min) where T_min is the line centre transmittance. The optical attenuation at line centre (in dB cm^–1) is 4.34α₀. Table 1.2 provides a list of the line centre absorption coefficient and attenuation for some absorption lines calculated from the figures, which will be useful later in discussing the sensitivity of near-IR and fibre optic systems. Similar data for a range of other gases may be obtained from HITRAN, including gases such as O₂ which have no rotational or vibrational spectra but have electronic bands (O₂ has a spectral band centred around 762 nm).

Note the characteristic shape of the intensity envelope of the P and R branches as clearly shown in Figures 1.6–1.8. This arises from the multiplication of two functions governing the population of the rotational levels, namely, the exponentially decreasing function describing the population of the J-th state following Boltzmann statistics and the linearly increasing degeneracy of the J-th state by the factor (2J + 1). An additional factor determining the population of rotational levels is the influence of nuclear spin Reference Banwell and McCash[10]. As a result, for linear molecules possessing a centre of symmetry such as CO₂, alternate rotational levels in the P and R branches have zero intensity. For C₂H₂, the rotational lines alternate in strength due to the nuclear spin of hydrogen atoms (see Figure 1.8).