Comments on: Absorption Line Parameter Measurements Using Laser Spectroscopy

A. D. May

University of Toronto, Physics Department, Toronto, M5S 1A7. Received 9 March 1973. It is the purpose of this Letter to point out that the dis­

cussion given by Kucerovsky et al.1 is misleading and that the use of a Voigt profile to analyze the absorption spec­trum of a gas is in general physically incorrect. To begin let us review briefly the theory of spectral line shapes. Perhaps the simplest way to discuss spectral profiles is in terms of temporal correlation functions, the Fourier trans­form of which yields the (power) spectrum. For the Case we wish to consider it is the time dependence of the emit­ting or absorbing dipole that determines this correlation function or the spectrum. Neglecting radiation damping an isolated molecule fixed in space will have a dipole completely periodic in time or a time correlation function of infinite width of the form Φ(t) = exp(-iω0t). Thus it will absorb at a single frequency ω0· There are two rea­sons why the spectrum is spread out in the neighborhood of ω0· First, collisions perturb the internal states of the molecule, i.e., modulate the dipole and shorten the coher­ence time of the dipole to a time of the order of the time between collisions. This effect of collisions leads to the well-known broadening and shifting of spectral lines that increase linearly with density in the low density region. The details of how this occurs depend both on the type of intermolecular force and on the nature of the transition being studied and have been the subject of many theoreti­cal studies. Nevertheless the general result is that the correlation function can be written in the form Φ(t) = exp(-βρt - iω0t), where ρ is the density and B the broad­ening coefficient. This leads to a Lorentzian profile with

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a full width at half-height given by Δv1/2 = Bρ/π. If B contains an imaginary part, a shift in frequency also oc­curs. Physically frequency shifts arise from systematic collision induced phase shifts in the radiating or absorbing dipole. (Note that Kucerovsky et al.1 have not included the possibility of a frequency shift in their analysis.) The second reason why the spectrum is spread out near ω0 is that the absorbing molecules are not fixed in space being bombarded by perturbers but are moving through the in­cident radiation field. If collisions do not affect the inter­nal states, it may easily be shown that the correct correla­tion function2-3 is just Gs(kt) exp(-iωot), where Gs(kt) is the spatial Fourier transform of Gs(r,t), the probability of finding a molecule at r at time t given that it was at the origin at t equal to zero, i.e., Gs(rt) is the self part of the pair correlation function of Van Hove.4 The self part alone appears since we are considering incoherent absorp­tion or emission. Intuitively only this one spatial compo­nent should appear since it is the motion of the molecule in the direction of the wave vector k that modifies the phase of the radiation that the molecules see. For very dilute gases the appropriate form of Gs(rt) leads to the well-known Doppler broadened profile. However at den­sities where the product of the mean free path Λ and k is less than 1 the correct physical description of translational motion is one of diffusion and Gs(rt) takes on the form5

[ 1 / ( 4 Π D t ) 3 / 2 ] exp[-r2 / (4Dt)] so that Gs(kt) becomes exp(-Dk2t). This leads to a Lorentzian profile with a full width at half-height given by Δv1/2 = (k2D)/π, where D is the coefficient of diffusion.6,7 As D varies inversely with the density, the width arising from the translational mo­tion decreases with increasing density, an effect described as Dicke8 narrowing in the literature.

Let us now consider the analysis of Kucerovsky et al. In their treatment of the translational motion they have used without question a model for freely propagating mol­ecules. There is no doubt that in the two cases consid­ered, ir absorption by CH4 and C2H3Cl, the temporal evo­lution of the internal motion is collision dominated. Whether the evolution of the translational motion as it contributes to the spectrum is also collision dominated depends upon whether kλ is less than or greater than 1. For CH4 absorbing at 3.392 μm and with a Lennard-Jones collision diameter of 3.8 Å (Ref. 9) kΛ equals 1 at 100 Torr. Thus in Fig. 6 of Ref. 1 the use of a Voigt pro­file is completely justified only below, say, 10 Torr. It is not justified around 100 Torr and higher and is probably the reason why the analysis fails as it does. Because of Dicke narrowing the absorption profile is not as broad as they have computed so that the extinction is larger than calculated. (Of course a shifting of the line toward the laser frequency would also give a similar discrepancy.) For C2H3CI the longer wavelength (27.972 μm) and the larger collision diameter mean that the Voigt profile can­not be used at pressures above say ½ Torr. The observa­tion of Kucerovsky et al. extends from ½ Torr to 1000 Torr.

Although Gs(r,t) is not known with precision except in the two extreme limits, there do exist models10 that span the intermediate region and that reduce exactly to the known forms at very low and very high densities. It is thus possible to calculate an absorption spectrum in a reasonable manner for any density.

The true situation is even more complicated than im­plied here. It is the assumption that the evolutions of the translational motion and the internal motion are uncorre­c t e d that allows one to write the total correlation func­tions as a product of Gs(kt) and e x p ( - β ρ t - iω0t) or the

spectrum as a convolution of two spectral profiles. Such a correlation can exist if collisions are responsible for both types of temporal evolution. At very low densities (kΛ » 1) where collisions do not affect the translational motion a product of correlation functions is correct, and the Voigt profile should give an accurate description of the profile of an isolated absorption or emission spectral line of a gas. For kλ less than 1 the situation is more complicated.11

Experimentally, however, it appears that the product form is adequate as a first approximation.12 This is for­tunate since it means that experiments such as those of Kucerovsky et al.1 or Eng et al.13 can be used as a tool to study Gs(r,t), an important quantity in nonequilibrium statistical mechanics. Finally it seems worthwhile to mention that the discussion given above is restricted to isolated lines. For overlapping lines a simple summation cannot always be performed if additional effects like mo­tional narrowing14 >15 occur.

References 1. Z. Kucerovsky, E. Brannen, D. G. Rumbold, and W. J. Sar­

gent, Appl. Opt. 12, 226 (1973). 2. To the author's knowledge the proof of this has never been

published, but it can be given in a few lines and is simpler than the closely related proof for coherent scattering as given by Komarov and Fischer.3

3. L. I. Komarov and I. Z. Fischer, Sov. Phys. JETP 16, 1358 (1963).

4. L. Van Hove, Phys. Rev. 95, 249 (1958). 5. G. H. Vineyard, Phys. Rev. 110, 999 (1958). 6. The astute reader will realize that the proof of the Doppler

profile usually given in textbooks, and traceable to Rayleigh,7

is wrong. One cannot take a shortcut by establishing a one-to-one correspondence between molecular velocity and fre­quency shift. Molecules will maintain their Boltzmann dis­tribution of velocities independent of the density, but the spectrum that results from their motion will be Gaussian only at low densities.

7. Lord Rayleigh, Nature 8, 474 (1873). 8. R. H. Dicke, Phys. Rev. 89, 472 (1953). 9. J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular

Theory of Gases and Liquids (Wiley, New York, 1964). 10. M. Nelkin and A. Ghatak, Phys. Rev. 135, A4.1621 (1964). 11. J. I. Gersten and H. M. Foley, J. Opt. Soc. Am. 58, 933

(1968). 12. B. K. Gupta and A. D. May, Can. J. Phys. 50, 1747 (1972). 13. R. S. Eng, A. R. Calawa, T. C. Harman, and P. Kelly, Appl.

Phys. Lett. 21, 303 (1972). 14. P. Dion and A. D. May, Can. J. Phys. 51, 36 (1973). 15. M. Perrot and J. Lascombe, Compt. Rend. 273, 25 (1973).

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