with a relative error of < 1%. The above series converges very slowly for y > 10 and for this range of values, wD can be evaluated, with a relative error < 1%, from the truncated series

where v(y) = i/lny. The Lorentz equivalent width can be computed from

Table 4.1 The coefficients b and c used to evaluate the Ladenburg-Reiche function L(x) for x < 7 using eqn (4-45). (Vardavas 1993)

n |
K |
Cn |

1 |
2.2499997 |
0.56249985 |

2 |
1.2656208 |
0.21093573 |

3 |
0.3163866 |
0.03954289 |

4 |
0.0444479 |
0.00443319 |

5 |
0.0039444 |
0.00031761 |

6 |
0.0002100 |
0.00001109 |

where x(a,y) = y/(2a^/Tr) and L(x) is the Ladenburg-Reiche function given by L(x) = xe~K[J0(ix) - iJ1(ix)], (4.45)

which can be computed rapidly, with a relative error < 1%, for x < 7 from

6 2n

using series expansions for the Bessel functions J0 and Ji (see Abramowitz and Stegun 1965). The coefficients b and c are given in Table 4.1. For x > 7, the simple expression wl = 7T3\fay [l — exp(—i/i/7r/4a)]2 (4.47)

can be used with a relative error of < 1%. Near x = 7, the relative error rises to just 2% for both of the above expressions.

The Voigt equivalent width can be evaluated from the expression

with c0 = 1.12, ci = 1.22, b0 = 0.47, and bx = 0.98. The error is generally < 1% with a maximum value of about 5%. The equivalent width wV can be set equal

H Cl

FlG. 4.5. Examples of heteronuclear (HCl) and homonuclear (O2) diatomic molecules. The heteronuclear molecules have an electric dipole moment.

to wD within 1% error for a < 10~6 while for a > 1.5 the equivalent width wV can be set equal to wL within a 2% error. In the limit of y ^ 1, both the Doppler and Lorentz equivalent widths vary linearly with the absorber amount

whereas for y ^ 1, the Lorentz equivalent width varies as the square root of the absorber amount wL(a,y) = n~^/ay. (4.53)

For large y, the Doppler equivalent width varies as the square root of the logarithm of the absorber amount

4.3 Rotational lines and bands

Let us consider a simple model of a rotating diatomic molecule composed of solid spheres and joined by a rigid bond. We first note that for the molecule to interact with the radiation field it must have a permanent electric dipole moment (Fig. 4.5), and this requires that the molecule is made up of heteronuclear atoms, such as HCl, and not homonuclear diatomic molecules, such as O2, which have no dipole moment and thus there are no vibration-rotation transitions and so they do not absorb or emit in the infra-red. In the rigid-rotator model, the molecule can rotate about an axis perpendicular to the bond axis going through the centre of gravity of the system, as shown in Fig. 4.6 The moment of inertia, I, is then given by

where ro is the distance between the atoms and the reduced mass, m, is given by m\m2 m = -—

FlG. 4.6. Rigid-rotating diatomic molecule of radius ro.

Typical values for ro, indicative of the length of the molecule, is about 1 A. Solution of the Schrodinger equation results in quantized rotational energy levels h2

where J is the rotational quantum number and h is Planck's constant. If we introduce the rotational constant, B, given by h

then the allowed rotational energy levels are given by

EJ = hcBJ (J +1), J = 0,1, 2, 3, •••, (4.59) with energy difference between rotational energy levels

or, in terms of wavenumber

AwJ;J +i = 2B( J +1) J = 0,1, 2, 3, •••, (4.61)

with rotational transitions allowed according to the selection rule AJ = ±1, where here +1 corresponds to excitation and -1 to de-excitation. We thus have rotational energy levels according to the series 2B, 6B, 12B,... and so we have spectral lines with line centre at 2B, 4B, 6B,..., as shown in Fig. 4.7 We thus have an infinite number of rotational levels, however, the populations of molecules that have been excited by collisions to such energy levels above the ground state (J = 0) are given by the Boltzmann distribution, if we have thermodynamic equilibrium,

no where n0 corresponds to the population of the ground state. A typical value for B is 2 cm-1 and so for Earth-type temperatures, say T = 300 K, n1/n0 ^ 0.98.

co J

FlG. 4.7. Allowed rotational levels for the rigid-rotator model and corresponding spectral line series.

We see that the population of the first excited state is not very different from that of the ground state. Since the line intensity of a transition, k, is linearly proportional to the population of the lower state of the transition, we see that the line intensity for rotational lines decreases slowly with J, as shown in Fig. 4.8, lower curve. The lower curve gives the shape of the envelope of the variation of line intensity of the rotational lines with lower level J, relative to the intensity

FlG. 4.9. The three projections of the angular momentum vector for J =1.

of the line with the ground level (J = 0) as its lower level. We notice that by J =18 the intensity has become insignificant. Thus, we have a band of molecular rotational lines with a characteristic envelope that describes the distribution of the strength of each line. Here, the first line, corresponding to J = 0 ^ J =1 transitions, is centred at 2B cm-1.

Atomic or molecular energy levels that correspond to more than one allowed quantum state are termed degenerate. In the case of rotational energy levels, there is a law of quantum mechanics that allows the angular momentum vector, T, along the axis of rotation, to have components, along some specified reference direction, whose magnitude is an integral (including zero) multiple of h. The magnitude of the angular momentum vector, P, is related to the energy of level

where _

so that Pi = V2Ti. We can obtain three projections of the vector P of magnitude a/2 along the vertical axis that have magnitudes -1, 0 and +1, as shown in Fig. 4.9. Thus, we say that the rotational level J = 1 is three-fold degenerate, i.e. three allowed quantum states with the same energy EJ. Generally, a level J has (2 J + 1)-fold degeneracy and this increases the population of molecules in each rotational level J by the factor 2 J +1, so that

The resultant band envelope is shown in Fig. 4.8 (upper curve), which exhibits a peak in nJ/n0 at J = 7, and hence in line intensity, for the transition J =7 ^

J = 8, corresponding to the rotational spectral line centred at 14B cm-1. It can be shown that the band peak occurs for the transition whose lower level, J, is closest to the value V/T/2MB - 1/2, where T is in Kelvin and B is in cm 1.

4.4 Vibrational lines and bands

The diatomic molecular rigid rotator is an idealization that does not take into account the fact that the atoms are not rigidly bound and that the distance between the atoms is not fixed. A diatomic molecule can, to a first approximation, be regarded as a harmonic vibrator-rotator. If we consider only the vibrational energy transitions then the allowable vibrational levels are given by v + ^jhuj0c v = 0,1,2, • • •, (4.66)

where is the wave number of the fundamental vibration and v is the vibrational quantum number. The selection rule for the harmonic vibrator is Av = ±1. Thus, all vibrational transitions correspond to the same energy difference and hence to a single spectral line with wave number wo (Fig. 4.10). If the molecule is either compressed or stretched, the energy required to do this increases as the radius is decreased or increased from its equilibrium value, just like a spring. As in the case of the simple rotator, a vibrating molecule is radiatively (infra-red) active if it has a residual electric dipole moment. This is not possible for homonuclear diatomic molecules and some vibration modes of linear polyatomic molecules, such as CO2, as we will discuss later.

We saw that the lowest energy level of a rotator was zero whereas for the harmonic vibrator it is ^huja. This is a consequence of the fact that the atoms that constitute the molecule can never be completely at rest with respect to each other, a result that is also expected from the Heisenberg Uncertainty Principle.

In the case of the diatomic harmonic vibrator model, there is a restoring force that increases as the separation of the atoms increases. In reality, if the atoms are at a sufficient distance apart, the bond between them will weaken and the molecule will dissociate into its constituent atoms. We expect that for small amplitudes of vibration the molecule will behave like a harmonic oscillator. At sufficiently large internuclear distance the molecule will proceed towards dissociation as show in Fig. 4.11. To account for molecular dissociation, the anharmonic vibrator model is introduced where vibrational transitions become progressively smaller as the bond is stretched, so that the energy difference between vibrational levels decreases as the vibrational quantum number, v, increases. To a first approximation the wave number of the energy levels is given by

Internuclear distance (Angstroms)

FiG. 4.10. Potential curve and energy levels of the diatomic harmonic vibrator. All transitions involve the same energy difference, and hence correspond to the same spectral line.

where we have introduced a first-order correction of anharmonicity through the constant xe, and ue is the harmonic oscillator frequency or equilibrium frequency. The constant xe is always positive and small so that the separation of the energy levels of the anharmonic oscillator decreases slowly with increasing v, as shown in Fig. 4.11. When the energy of vibration is equal to or greater than the dissociation energy, De, the molecule breaks up into its constituent atoms. The selection rules for the anharmonic oscillator are Av = ±1, ±2, ±3, •••. Although larger energy transitions are now allowed, unlike the rigid rotator the populations of the first and second excited states are very small compared with the ground-state population. For vibration, the energy separation of the energy levels is of the order of 1000 cm-1. Thus, assuming thermodynamic equilibrium and using the Boltzmann distribution for the populations of the excited states relative to the ground state, we find that at atmospheric temperatures the population of the first excited level is typically 1% of the ground state. Thus, the three most significant vibrational transitions originate from the ground state with jumps to the first three excited states. Absorption of radiation for these transitions

FlG. 4.11. Potential curve and energy levels of the diatomic anharmonic vibrator. Values for the dissociation energy and internuclear distance are representative for the molecule HCl. The equilibrium oscillator frequency is = 2990.6 cm"1, req = 1.274 Â, and De = 12we. We note that the spacing between energy levels must decrease for dissociation to occur for v=28.

becomes rapidly weaker with increasing vibrational quantum number. We note that weaker transitions may become more significant as the temperature rises to 1000 K.

The corresponding vibrational spectral lines for each transition occur at the following energies:

For example, the molecule HCl has we = 2990.6 cm"1 and xe = 0.0174, resulting in:

strong weak negligible.

FlG. 4.12. Vibrational spectral lines of the anharmonic vibrator. The lines are centred near multiples of the equilibrium vibrational frequency we. Only the spectral lines corresponding to the fundamental frequency of oscillation and the first two overtones are significant.

Thus, for the anharmonic vibrator we observe three spectral lines, of different strengths, at wave numbers close to we, 2ue, and 3ue, as shown in Fig. 4.12

The molecule can undergo both vibration and rotation and so can absorb photons of a larger variety in frequency than the simple vibrator or rigid rotator. The rotational and vibrational states of a diatomic molecule are coupled even though to a first approximation the combined rotational-vibrational energy of the molecule can be determined from the sum of its rotational and vibrational components. This is a good approximation because the energies of rotation and vibration are so different, typically of order 10 cm-1 and 1000 cm-1, respectively. Thus, the molecule performs at least 100 rotations for each vibration. In this, the Born-Oppenheimer approximation, the energy of each state is simply the sum of the allowed rotational and vibrational energies. We note that a more general formulation would include allowed energies of electronic transitions, whose energies are coupled to the rotational and vibrational states, and that can have energies corresponding to visible and ultraviolet radiation.

The allowed energy levels of the anharmonic vibrator-rotator, to a first approximation, are given by cov = BJ{J+l)-DJ2{J+lf + + + we.xe I' = 0,1, 2, • • •,

The extra rotational term, which removes the assumption of a rigid rotator and involves the rotational constant, D, arises from the centrifugal force of rotation that increases the internuclear distance between the atoms, and hence affects both rotation and vibration. This is because, in reality, we cannot have a rigid rotator that is also vibrating. However, the effect of the centrifugal force correction is very small as the constant D is much smaller than the rotational constant B. To a very good approximation we can write

C0V = BJ(J + 1) + + M UJe - + ^ weie = 0, 1, 2, • • • , (4.69)

Internuclear distance

Internuclear distance

FlG. 4.13. Permitted transitions (A J = ±1) between rotational levels corresponding to the vibrational ground level v = 0 and the first vibrational excited state v = 1.

with the selection rules being Av = ±1, ±2, • • • and AJ = ±1. These rules correspond to simultaneous vibrational and rotational energy changes. In Fig. 4.13 are shown the allowed vibrational-rotational transitions. For the v = 0 ^ v = 1 vibrational transition we can write the energy of the rotational transitions as where m > 0 corresponds to A J = +1, and m < 0 corresponds to A J = — 1. The vibration-rotation spectrum consists of equally spaced (2B cm-1) rotational lines on each side of the vibration-rotation band centre, w1. The idealized envelope or contour of rotational line strength about the band centre is shown in Fig. 4.14. Lines at wave numbers below the band centre correspond to m < 0 and are referred to as the P branch of the band while lines at wave numbers above the band centre correspond to m > 0 and are referred to as the R branch. No transitions are allowed for m = 0. The separation of the P and R maxima is given approximately by 2.36a/BT where B is the rotational constant in cm 1 and T is the temperature in K. Thus for HCl with B = 10.44 cm-1 the separation is 132 cm-1 at T = 300 K. As the temperature increases the P and R peaks move to higher m values and so more rotational lines contribute to the band absorption. The first and second overtone bands have the same rotational structure but the line intensities are greatly reduced.

FlG. 4.14. The idealized envelope of line strength exhibiting the P and R branches of a molecular vibrational-rotational infra-red absorption band.

In reality, the P and R branches are not symmetric due to the interaction of vibration and rotation (breakdown of the Born-Oppenheimer approximation) that results in the rotational constant, B, decreasing with increasing vibrational quantum number, v. The result is that the spacing of the rotational lines decreases with increasing m. Furthermore, since in the Boltzmann distribution of the populations of the rotational levels, the rotational quantum number, J, of the lower level of an absorption transition appears in the exponential term, then line intensities are higher for the R branch (originating from lower J values) than for the P branch.

There is an increase in the complexity of the band envelope and vibrational-rotational structure when we consider the polyatomic molecules CO2, H2O, CH4, NH3 and O3 that play an important role in the terrestrial greenhouse effect. This band complexity results from the many different mechanisms that affect the selection rules and rotational line intensities that result in deviations from the idealized band envelope of a diatomic molecule shown in Fig. 4.14 (see Herzberg 1945 and 1950, Banwell and McCash 1994).

As a relatively simple example of a polyatomic molecule, let us consider CO2 that is a linear molecule that can have three modes of vibration. The most important to atmospheric thermal infra-red absorption are the bending mode and the asymmetric stretch mode, while the symmetric stretch mode does not have an electric dipole and so is infra-red inactive. The vibrational quantum numbers of the symmetric stretch, bending and asymmetric stretch modes are represented by vi,V2, v3, respectively. In general, there can be transitions between the vibrational levels of the different modes, and hence vibrational levels, in the simplest way, are represented by viv2v3. Thus, the ground state of the molecule is denoted by 000 and the first excited state of the bending mode is denoted by 010. The most important vibration-rotation band for atmospheric infra-red absorption is the fundamental band of the bending mode that is centred at = 667.3 cm"1 = 15 ¡m and corresponds to the transition 000 — 010. As we have seen, the energy separation between the vibrational levels decreases slowly with vi-brational quantum number and so transitions between consecutive subordinate levels result in bands also centred near 15 ¡m. In fact, there are at least 10 such important bands, termed hot bands.

An important characteristic of the rotational lines of the fundamental band of CO2 is that alternate lines are missing, so that the spacing between the lines is 4B and not the expected 2B. Many of its hot bands also have this characteristic that arises due to nuclear spin effects in symmetric molecules (see Herzberg 1945). The CO2 molecule can lose its symmetry due to isotopic effects. The fundamental bands of the symmetric isotopes O16C12O16 and O16C13O16 have alternate lines missing, whilst those of the asymmetric Isotopes O18C12O16 and O17C12O18 do not have alternate lines missing. The fundamental band, the hot bands and the isotopic bands of CO2 with band centres near 15 ¡m overlap each other over the wave number interval between approximately 550 and 800 cm"1. This overlap needs to be taken into account in the calculation of atmospheric transmission in this spectral interval. Generally, other molecules, such as H2O, also have bands that absorb in this spectral interval and so care must be taken to include also this overlap.

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