Absorption and Dispersion in the O2 Microwave Spectrum at Atmospheric Pressures Earl W

Home , A band (NATO), Spectrum

Absorption and dispersion in the O2 microwave spectrum at atmospheric pressures Earl W. Smith National Bureau of Standards. Boulder, Colorado 80302 (Received 8 January 1981; accepted 24 February 1981)

Calculations are performed for absorption and phase dispersion at various frequencies within the 60 GHz band of 0, from low pressures where the spectral lines are isolated, to atmospheric pressures where they merge to form a continuum band. A perturbation theory proposed by Rosenkranz was tested and found to be valid for pressures up to 100 kPa (1 atm). The “line coupling coefficients”, which describe the transfer of excitation from one radiating state to another, are also studied and various methods for evaluating these coefficients are analyzed and compared with experimental data. It is found that dispersion measurements are extremely sensitive to these coefficients and an experimental procedure for systematically measuring them is outlined; it is shown that such measurements can provide a very sensitive test for theoretical calculations of inelastic transition amplitudes.

I. INTRODUCTION shape expression itself can pose serious problems. For example, the 60 GHz spectrum of 0, studied in this The overlap of spectral lines at high pressures pro- paper requires the calculation of least 44 spectral vides an interesting means for studying inelastic rates. at To illustrate this we first note that the linewidth for a lines; thus, taking careful account of all line coupling collision broadened spectral line is the sum of two terms, terms, the line shape calculation requires inversion an elastic term due to collisions which reorient the of a 44x44 matrix for each value of pressure and fre- radiating dipole, changing the azimuthal quantum number quency of interest. Using a computer, this is expensive but not difficult; however, with such a numerical analy- and an inelastic term due to collisions which trans- M, sis, one loses sight of the effect of the individual in- fer excitation out of the states involved in the radiative elastic rates. In an attempt to avoid this difficulty, transition, e. g., Sec. V of Ref. 1. At high pressures, Rosenkranz’ has employed perturbation theory to derive when the individual spectral lines merge to form a band, a relatively simple line shape expression which provides inelastic collisions which transfer excitation from one physical insight into the effect of the inelastic rates and radiating state to another (called “line coupling transi- is numerically accurate for most calculations. A sec- tions’’) are no longer effective in broadening since they ond major goal of the present paper is to make certain only shift intensity from one part of the band to another; improvements in the Rosenkranz perturbation approach see Gordon,’ p. 454. The half-width w of the band will and to confirm the numerical accuracy of this approach thus be smaller than the half-widths of the isolated by comparison with brute force numerical calculations lines; the observed width Pw (in cm-” or MHz) will of course be larger simply because the pressure P is employing a matrix inversion; specific calculations will be performed for the 60 GHz band of 0,. larger. These line coupling transitions will be studied in terms of their effect on absorption and dispersion The absorption and dispersion of microwaves due to (the frequency dependent part of the index of refraction). oxygen at atmospheric pressures are of interest in It will be shown that the dispersion is extremely sensi- their own right for a variety of communications and tive to these inelastic transitions and that, in some cases, remote sensing problems. Thus, there exist accurate it permits a direct measurement of the inelastic rate absorption and dispersion measurements over a wide coupling two radiating states. Such measurements could range of pressures, thereby making this microwave provide a useful addition to the usual measurements of spectrum an attractive test case for microwave line decay rates,, etc., since the latter are usually given by broadening theories. It is hoped that the present analy- a sum of inelastic transitions whereas the former are sis will help to clarify the interpretation of some of these often determined by a single state to state transition experimental data. probability. Direct measurements of slate to state transition rates can provide a very stringent test for It. GENERAL THEORY theoretical calculations. Thus, a major goal of the present paper is to demonstrate this type of analysis A. Properties of the O2 microwave spectrum by studying the inelastic transition rates recently cal- The microwave spectrum of 0, is due to a series of culated by Lam314for the 60 GHz band of 0, and com- magnetic dipole transitions within the 3C, electronic paring them with experimental measurements per - ground state. The nuclear rotation N couples to the formed on this band. 5*8 electronic spin S = 1 to produce a total angular momentum In order to study the effect of inelastic transitions on J which may take the values J=N, N* 1. This triplet absorption and dispersion, it is first necessary to have of rotational states is split by spin-spin and spin-orbit a line shape theory which is numerically accurate and, interactions, resulting in the energy level structure at the same time, analytically simple enough to permit shown in Fig. 1. Within each triplet (labeled by N) there a clear determination of the role of each inelastic rate. are two allowed microwave transitions J = N- J = N + 1, For a band composed of many spectral lines, the line denoted by vi, and J=N-J=N-l, denoted by v;.

6658 J. Chem. Phys. 74(12), 15 June 1981 Earl W. Smith: O2 microwave spectrum 6659

J about 1/5 of the normal separation between lines (see Table I). These doublets merge at pressures around 5 kPa (0.05 atm) and all lines begin to merge with their neighbors at pressures around 30 kPa (0.3 atm), finally producing an unstructured absorption band at 100 kPa (1 atm). This band extends from 50 to 70 GHz with a 4 1120GHz collision broadened line, the vi line, at 120 GHz (see Fig. 2). At pressures above 0.1 atm it is no longer possible to describe the spectrum as a simple superposition of overlapping but noninterfering Lorentzian lines because the line coupling transitions have already begun to produce observable effects (discussed in detail in Sec. 1111. $ 776 GHz

B. The line shape expression and notation In this paper we will use the impact theory” to describe the collision broadening of the spectrum. The impact theory assumes that (1) the radiating or absorb- ing molecule is perturbed by a sequence of binary col- 1431 GHz lisions with other particles and (2) the duration of these collisions is short enough that, if a collision occurs, it can be completed during all times of interest for spectral line broadening. For broad absorption bands at high pressures, it is worth reviewing the validity of these assumptions. The assumption of binary collisions will break down at high pressures when the particle density n becomes comparable with the “interaction volume” rb3, where b is the “range” of the interaction, e. g., FIG. Rough energy level diagram showing the multiplet 1. for a Lennard-Jones 6-12 potential, b would be the structure of 02. The resonant magnetic dipole transitions are denoted by N* (the 1- line at 118 GHz is somewhat separated order of the internuclear distance at the point of maxi- from the rest of the band). For N 5 5, the state J-N - 1 has a mum attractive interaction. For strong interactions lower energy than J =N + 1 and for N > 5 the converse is true. between permanent dipoles, b may be somewhat larger than the position of the attractive well; hence, thevalidity of the bindary collision approximation should be re- considered for pressures above MPa (10 atm). There These transitions, referred to as the “resonant transi- 1 is no problem with the present work since the highest tions,” lie in the range 50-70 GHz except for =119 vi pressure considered is 100 kPa (1 atm). The second GHz; a list of transition frequencies and line strengths assumption, often called the completed collision as- is given in Table I (obtained from Ref. 8). sumption, means that all collisions will be described There are also some magnetic dipole transitions J=N+l-J=N+l, J=N-l-J=N-l, and J=N-J=N (referred to as O;, O;, and ON, respectively) which TABLE I. Partial list of spectral lines occur at zero frequency and are usually called “non- and their frequencies; a more complete resonant transitions. ” At atmospheric pressures these list is given in Table I of Ref. 8. nonresonant lines make a small contribution (1%-10%) to the absorption and dispersion in the 60 GHz region; Line (GHz) hence, they will be included in our analysis. The vi 17* 63.5685 and vi lines are occasionally referred to simply as N’ 15‘ 62.9980 3- and N- lines in cases where there can be no confusion 62.4863 Doublet 13* 62.4112 1 with the 0; and 0; lines. I 11’ 61.8002 The resonant transitions produce a series of isolated 9’ 61.1506 7+ Lorentzian spectral lines whose widths have been mea- ‘O’ 4348 Doublet sured and compared with theoretical calculations many 5- 60.3061 t times. The most recent ~ork~*~*’has produced very 5+ 59.5910 7- 59.1642 satisfactory agreement between theory and experiment 3+ 58*4466 Doublet so that one feels confident that these linewidths are known 9- 58.3239 1 accurately to within about 5% (theoretical uncertainties 11- 57.6125 may be slightly larger due to the lack of an accurate po- 13- 56.9682 15- tential surface). 56*3634 Doublet I+ 56.2648 t There are four pairs of lines (3-, 13+), (7+, 5-), (Y, 1 7- 55.7838 9-), and (15-, 1’) that have a frequency spacing which is

J. Chem. Phys., Vol. 74, No. 12, 15 June 1981 6660 Earl W. Smith: O2 microwave spectrum

loot1 I I I I I I I 113 lines“; one should therefore be aware of the possibility of encountering nonimpact effects in very broad molecu- lar bands. Within the framework of the impact theory, the expressions for the absorption coefficient k(v) and the index of refraction n(v)are given by’,

2 (2nv),N Re n(v)- 1 = - - 3ckT

where AN(v)= - ReZ(v) is called the phase dispersion’ -1 3k A and the complex line shape function Z(v) is given by

i z(V) = c dj(j 1 [ (V - VO)- iPW]-’l k)dk pk . (4 ) jk In Eq. (4), Ij) and I k) denote “doubled state vectors” which are given in terms of ordinary 0, states INSJM) and the Wigner 3j symbol [e.g., Eqs. (18)and (19) of Ref. 131

INSJ, N’SJ’, KQ) =- (- I)J’+M’+K MM’ 40 60 80 100 120 140 Frequency in GHz

FIG. 2. Absorption and dispersion as a function of frequency, illustrating the difference between a pure Van Vleck-Weisskopf (VVW) calculation (dashed curve) and a calculation which ac- dj counts for line coupling effects (solid curve). Throughout this The in Eq. (4) denote the reduced matrix elements paper, a VVW calculation refers to a simple superposition of [e. g., Eqs. (21) and (22) of Ref. 131 Lorentzian lines whose linewidths (Table 11) are determined by low pressure measurements; this sum of Lorentzian lines includes the “resonant lines” at the frequencies vj> 0, ‘honreson- ant lines” for which vj=O, and “negative resonant lines” for which vj< 0 (discussed in Sec. IIB).

by an S matrix which represents the effect of a com- where the d, are the components of the magnetic dipole plete collision. To show that this assumption limits moment d and pk denote the density matrix elements the impact theory to the line center, we recall that the (statistical weights) which are well approximated by14 intensity Z(v) (for either emission or absorption) at a frequency separation Av = (v - vo)from the line center vo is given by the Fourier transform of a dipole cor- relation function3’’’ C(t)=(d(t) d(0)):

Z(v)= JoaC(t)exp(2niAvt)dt . where E,/k =2.0685 N(N+l)is the energy of the Nth multiplet in K. The frequency v in Eq. (4) is diagonal, Since the exponential oscillates rapidly for large values (,jI vik) = vbja, and vo is also a diagonal matrix (j 1 vol k) of t, the integral is determined mainly by t,<1/(2nAv). = vj 6j,, where vj is the energy difference (divided by If the average duration of a collision is 7, then there h so as to give a frequency) between the two states NJ will be a region in the line wings Av> 1/(2717) where the and N‘J’ which are combined to form the doubled state intensity is determined entirely by times t< 7, which I j) [cf. Eq. (511. Notice that the only states I j) and are so short that most collisions cannot be regarded as lk) required by Eq. (4) are those which correspond to completed and the S matrix does not adequately describe allowed dipole transitions (i. e., those for which dj and their effect. For 0, at 300 K, T= 5x s; hence, the d, are nonzero); hence, the only frequencies vj which impact theory can be used for Av<300 GHz, which is appear are those corresponding to allowed spectral quite adequate for all Av of interest for the 60 GHz lines. Consequently, the vectors I ,j) and lk) are often band. However, for some infrared bands such as those referred to as states in “line space.” The matrix of HC1, the line spacing is on the order of 1/(2117) and (,j I zul k ), whose form will be given in Sec. 111, there- nonimpact effects have been seen between the spectral fore represents the coupling between the spectral lines

J. Chem. Phys., Vol. 74, No. 12, 15 June 1981 Earl W. Smith: O2 microwave spectrum 666 1

j and k which was discussed above; the diagonal elements taining some classical approximations which can be wjj give the linewidth and shift for the spectral line j. improved upon. In this section we will extend the perturbation approach using the more common quantum In our calculations of the 0, microwave spectrum mechanical notation and will avoid some of the classical we have included three groups of spectral lines. The approximations retained by Rosenkranz. first group contains the 44 resonant transitions j, for NS43, the second group contains the 44 nonresonant The main notational differences from Rosenkranz the- line's at u =O denoted by 0% , and the third group con- ory are found by comparing our d, and p, [Eqs. (6) and ' tains 44 "negative resonance lines" which act like a (71, respectively] with Roeenkranz [Eqs. (4)-(7)]. Our group of spectral lines located at negative frequencies d, are reduced matrix elements of the dipole operator - j,. The negative resonance lines occur because the [see Lam, Eqs. (65)-(70)] which are proportional to line profile is given by a Fourier transform [Eq. (l)] the Rosenkranz di multiplied by d2N+1 and our'p, are which contains terms like exp[i(u - u$] as well as reduced matrix elements of the density matrix which are exp [i(u+ u,)t]; the former give rise to the normal reso- equal to his 9, divided by (2N+1). Our matrix w,, will nant transitions (i I vo 1j) = v, > 0 in Eq. (4), and the latter therefore equal his w,,, multiplied by d(2N' +1)/(2N+l). produce the negative resonance lines (jI uo I j ) = uj < 0 in Eq. (41, which contribute to I(u) as though they were As noted above, the basic idea behind the Rosenkranz spectral lines located at negative frequencies. Since the approach is to invert the matrix (u - uo)- iPw by using bandwidth is the order of 20 GHz, these three groups of perturbation theory. Specifically, one expresses the spectral lines are well separated from one another; eigenvalues and eigenvectors of the matrix uo +iPw by hence, the nonresonant lines and the negative resonance a perturbation expansion in powers of Pwj, /ujk, where lines contribute to the observed band at 60 GHz only by ujk= uj - uk is the difference between the frequencies of means of their far line wings. This contribution is two different spectral lines. It is then possible to di- usually on the order of 1%or less; however, when the agonalize the matrix vo + iPw by means of a transforma- dispersion due to j, lines is very small, namely, near tion x-'(vO+~PW)X so that band center (see Fig, 2), the contribution from the O;, and - 4 lines can be quite important (see the discussion of the 5' dispersion in Sec. II1.D). where AI are the eigenvalues of uo +iPw and Xz, is a Since these three groups of lines are well separated matrix whose columns are the normalized eigenvectors. from one another, it is possible to neglect any line The line shape [Eq. (4)] is then given by coupling between groups (recall the discussion of the importance of line coupling terms in Sec. I). This has been verified by numerical calculations which show that line coupling to either the o*, or jN produces changes smaller than 0.001%. Neglecting line coupling between which is quite easy to evaluate since it does not involve the three groups of lines makes the matrix (u - uo)-iPw a matrix inversion. in Eq. (4) block diagonal and each of the three blocks, The validity of this approach requires that Pw,,/ corresponding to the groups Oi, and is then 4, - 4, ujk be small for all lines j and k. The size of these a 44x 44 matrix. The matrix inversion is thus greatly terms will be discussed further in Sec. I11 but for the simplified but still very time consuming, being more than moment we note that there is an immediate problem be- 100 times slower than the perturbation calculation dis- cause uj =O for all of the nonresonant lines; hence, cussed in the following section. wjk/uj,will diverge when both j and k are nonresonant We mention in passing that the group of nonresonant lines. Rosenkranz used an idea from Gordon's clas- lines 0, corresponding to J = N - J = N transitions has not sical theory to merge all nonresonant lines into one and been included because its transition dipole strengths are to replace the coupling to this nonresonant line by a an order of magnitude smaller than those for the o', classical expression [Ref. 7, Eq. (lo)]. While this transitions (Lam4, p. 104). approximation is not at all serious (the nonresonant lines produce at most a 10% effect), it is also unneces- sary if one makes a minor change in the mathematical C. Perturbation theory procedure. Our perturbation expansion is derived in the Rosenkranz' has developed a perturbation theory ap- Appendix and the results are summarized in Eqs. (A17)- proach to the matrix inversion in Eq. (4) which provides (A21), where C j#k is to be interpreted such that, when- a very useful physical insight into the role played by the ever k corresponds to a nonresonant line v, =0, the sum line coupling coefficients w,,. His work is an expanded over j includes only the resonant lines u,>O and the quantum mechanical version of an earlier classical theory "negative resonant" lines uj

J. Chem. Phys., Vol. 74, No. 12, 15 June 1981 6662 Earl W. Smith: 0, microwave spectrum

He viz11- + lm vZr where, using Eqs. (A201, (A21), and some straightfor- and ward but tedious algebra,

(12)

I Equation (10)is identical to Rosenkranz [Eq. (A2)]and possible to measure a pressure shift since all lines his Gkkin Eq. (A6) agrees with our Eq. (12)to first or- overlap; nonetheless, it is safe to neglect any linear der in P. Next using [Ben-Reuven, l3 Eq. (41)l pressure shift since it would have been detected ai the lower pressures (line center frequencies can be mea- (13) sured with very high accuracy at low pressures’). Our theory contains a quadratic pressure shift 6vk which cannot be ruled out on the basis of low pressure measure- (14) ments; hence, it will be retained. Comparing Eq. (17) with Rosenkranz [Eq. (2)] it is clear that our formal results are identical to first order in pressure; our Eq. (17)also contains two second order contributions gkkand 6vk. We must emphasize, however, that our matrices dk, pk, and wkj differ from those used by Rosenkranz by the factors (2N+1) discussed earlier.

D. Line coup1 ing coefficients- Rosen kranz prescription When Rosenkranz developed his theory for the 0, microwave spectrum, there were no calculations of line coupling coefficients wjkavailable and even the half-widths were only known to within an order of magnitude. He therefore adopted an approximate prescription for evaluating wjkwhich provides order of magnitude accuracy. However, as more accurate measurements became available, attempts to fit the growing body of experimental data by using the Rosenkranz prescription for the wjkbegan to produce results that were inconsis- tent. The central equation in the Rosenkranz prescription for calculating wjkis Cjwjk=O [Ref. 7, Eq. (1411. ,This where 6vk is a second order shift defined by result is obtained by assuming that wjk is proportional to a cross section [Ref. 7, Eq. (813 which would result (19) in this conservation of probability summation rule. Al- though one often speaks of an “optical cross section,” In deriving Eqs. (17)and (18)we have assumed that it should be emphasized that this is not a true cross wkk is real, thereby neglecting any first order pressure section in the usual probabilistic sense. The actual shifts. This is consistent with experimental observa- quantum mechanical expression for wjk is obtained tions at low pressures.5 At high pressures it is not from Lam3 [Eqs. (301, (34), (511, and(48)I in the form

J. Chem. Phys., Vol. 74, No. 12, 15 June 1981 Earl W. Smith: O2 microwave spectrum 6663 where j represents the line N, J, - NbJ, , k represents serious problem for Lam's half -width calculations since the line N,, J,, - Nb. Jbl, { * - *IAvdenotes an average over half -widths are determined by diagonal S matrix ele- perturbers, and I Ma Ji Mi)and I Ni JL M;)are the initial ments which are strongly affected by diagonal phase states for the collisional transitions described by the shift matrix elements q,, for which w,, =O; nonetheless, s matrices. In a semiclassical formulation [e. g., the problem was observable even there (see Sec. HI. A Lam,' Eqs. (26)-(28)], is an average over im- of Ref. 9). For inelastic processes involving off diagonal pact parameters and velocities. Equation (19) is similar S matrix elements it is much more serious, producing to, but not equal to, an inelastic transition rate and in errors on the order of a factor of 2 or 3. We have particular it does not satisfy wik=O. By detailed nu- therefore used a crude semiemperical correction for merical calculations with Lam's3 wfkwe find that the this effect, l7 multiplying Lam's line coupling rates wfk sum of off diagonal wjkis an order of magnitude smaller by exp [- (6?Ejkr,,, /CZ)~/~+ (6 +O. 5Ejk)/kT],where r, than wkk;thus, one may expect Rosenkranz's prescrip- =3.67 A, (~=19,and 6=156 cm-' are the exponential tion to be accurate only to within an order of magnitude. - 6 potential parameters for the Oz-O, interaction, Z In fact, Rosenkranz's wjk are all within an order of mag- is the average velocity which is 6.25~lo4 cm/s-' at nitude of those calculated by Lam with most being a 300 K, Ejk is the energy spacing B, [Nj(Nj- 1) -N~(N~ factor of 2 or 3 off. Liebe et also used the Rosen- - l)]between the rotational levels Nj and Nk , and Be kranz prescription together with a set of measured half - =43.1 GHz or 1.44 cm-'. widths and again the calculated line coupling coefficients Lam's other main approximation is the assumption differed by factors of 2 or 3 from Lam's calculations and in addition the spectrum was extremely sensitive to that the molecules can make inelastic transitions without any effect on their translational energy. This can- the nonresonant linewidth, producing a value smaller not be rigorously true since conservation of energy de- than 1/2 of the value calculated by Lam. mands that any increase or decrease in rotational energy E. Line coupling coefficients-Lam's calculations be balanced by a corresponding decrease or increase in translational kinetic energy. However, this approxima- Lam3 has employed a semiclassical theory developed tion, very common in semiclassical line broadening by Dillon' to calculate linewidths and line coupling co- theories, is justified if the inelastic transitions involve efficients. His calculated linewidths agree closely with energy changes AE,, which are small compared with the measurements by Liebe et a1.' and with independent kT. The argument is that small changes in translational calculations using a slightly different type of semiclas- energy cause small changes in the trajectory which sical theory. ' While this agreement for half -widths produce small changes in the phase integrals q,, [Eq. gives some confidence in the accuracy of the theory, (20)] whose effect on the S matrix can be shown to be it is nonetheless clear that Lam has employed two ap- negligible. Of course, there will always be some small proximations which are not valid for transitions be- translational velocities v such that mv2/2 is comparable tween the multiplets, i. e., AN#O transitions. These to AEab; however, it is argued that these low velocities are (1) the infinite order sudden approximation for the make a very small contribution to the velocity average S matrix and (2) the assumption that the target molecule which is weighted by v3exp(-mv2/2kT), so that even can make an upward inelastic transition without taking large errors in S (e. g., factors of 2) have no effect. energy from the translational motion. It will therefore This argument is sufficient for calculating half -widths be necessary to modify his wjkin order to correct these since one encounters only diagonal S matrix elements, approximations. and it is sufficient for inelastic transitions to states To infinite order sudden approximation assumes that of lower energy. However, inelastic transitions to one may neglect the exponential factors exp(iw,,t) which states of higher energy cannot take place at all unless appear in the phase integrals the kinetic energy exceeds the inelastic energy increase AEab. While this threshold behavior only occurs for kinetic energies mv2/2 5 E,, :< kT. which make a small gab = exp(iw,b t)Va,(t) dt (20) 1.1 contribution to the velocity average, the error incurred that govern the S matrix S =exp(-ig) [see Lam,3 Eqs. in replacing Sac= 0 by a finite number is quite large and (54) and (57)]. The argument for this approximation is results in sizable errors (10'10-20'10) even after the velocity average, i. e., in place of the velocity average that, with the collision centered at t =0, the interaction V(t)vanishes when t>T, where T denotes the collision duration time; hence, exp(iw,,t)s 1 if wab ~<1,where JOm v3S( $) e-mvzJzkTdv=($) 1-ES(E)e-EJkTdE w,, is 2s times the energy difference (in frequency units) between the states a and b. This approximation has been =I, (21) studied in great detail (e. g., Goldflam et al. )le and it is we should have known to produce inelastic rates and half-widths which are too large when wab T> 1. For 0,, the average col- (3)JAI ES(E)e-EJ kT dE lision duration is TE~X sec and the frequency differences w,, range from 2~x431GHzg 2.7X 10" s-' for N=l-N=3 to 2~x2155GHz~1.4~10'~ s-'forN=ll = (5)e-AEl kT (E +AE)S(E+AE)e'E/kTdE I0 -N=13. Thus, we see that this approximation is not well satisfied for any of the AN #O transitions and it ES(E)~-EI kT dE =e-AEl kT1 . (22) breaks down rather badly for larger N. This was not a

J. Chem. Phys., Vol. 74, No. 12, 15 June 1981 6664 Earl W. Smith: O2 microwave spectrum

The latter constitutes an approximation in which we Pressure in kPa have used S(E + AE)s S(E)and (E + AE)s E within the integral. Based on earlier calculations, these approximations should be good for AE<

We will therefore correct Lam’s line coupling coef - 1 ficients wjkfor this effect by simply multiplying them by \ exp(- E,,/kT) with El, = Ej -E,, where E, and E, are the energies of .the multiplets in which the lines vj and vk are found. This is done whenever Ej>E,; the wjk are not changed by this effect when Ej E, , (24) =wj,(Lam)Cms , Ej

Lam’s method of calculating xijk becomes somewhat time consuming and expensive for large N so he performed explicit calculations only for N511 and approximated the values for 135 N533 by various extrapolation procedures. The functional form of these extrapolations was given [Eqs. (113) and (114) of Ref. 31 but unfortunately none of the extrapolation coefficients were published. We have therefore performed a similar extrapolation for N> 11 but we find that uncertainties in this extrapolation procedure can produce uncertainties in 2vjk ranging from 10% 58.4366 GHz for small N to 50% for values of N around 33. For the 10 MHz 4- frequencies v: and vi (Figs. 3 and 4) the contribution from N> 11 lines becomes important for pressures above 26 kPa (200 Torr) and for vi1 (Fig. 5) these lines are dominant at all pressures. These uncertainties for I 1 I I 1 I 1 1 N> 11 are expected to produce some disparity between 200 400 600 800 our calculations and Lam’s results. In Figs. 3-5, Pressure in torr we have plotted experimental data5 together with Lam’s FIG. 3. Absorption at 58.4366 GHz as a function of pressure results3 and our calculations. Our results include cal- at a temperature of 300 K. Comparison of calculations with culations with Lam’s wJkcoefficients denoted in the experimental data5 and with Lam’s calculations. ’ figures by “Lam’s us, ” calculations with the corrected

J. Chem. Phys., Vol. 74, No. 12, 15 June 1981 Earl W. Smith: O2 microwave spectrum 6663

Pressure in kPa is not known well enough to permit a serious analysis 40 80 of such a small effect. One can only conclude that the I I influence of the line coupling coefficients is clearly observable but their effect on the absorption coefficient 54.77 GHz is too small to distinguish between the different methods v;,+ 100 MHz of calculation. 20 vvw / B. Line coupling effects in absosrption and dispersion In the previous section it was noted that, for P< 100 E Y kPa (1 atm), the absorption is not very sensitive to the & 15 line coupling coefficients. The reason for this may be V seen in Eq. (17) where, to first order in pressure, the c .- numerator is proportional to the sum wkk+ (v - vk)yk . c .-0 The terms w,, result in a series of simple Lorentzian c 0 profiles (a Van Vleck-Weisskopf calculation) while the z IO W “interference coefficients” yk contain the effect of the line coupling coefficients wjk[see Eq. (15)]. The half- widths w,, are the order of 200 MHz/Pa (1.5-2 MHz/ Torr) whereas, from Table 11, we-see that y, is typical- ly the order of 200 Pa or 1.5X lom4Torr-’. The correction term (v - vk)yk is thus comparable to wkkonly when (v - vk) is on the order of lo4 MHz, i. e., these correction terms are important only in the far line I/ wings where the Lorentzian has fallen off to lo-’ of its ov I I I I I I I I 0 200 400 600 800 value at the line center. It is thus not surprising that the line coupling coefficients do not have a big effect Pressure in torr on absorption. FIG. 5. Absorption at 54.77 GHz as a function of pressure at Dispersion, on the other hand, is extremely sensitive a temperature of 300 K. Comparison with experimental data5 and with Lam’s calculations. to the line coupling coefficients as seen from Eq. (18) in which the numerator is, to first order in pressure, proportional to (v - vk)+ (PwkR)(Pyk).The terms (u - v,) again give the dispersion for a series of simple Lorentz- given in Eq. (23) denoted by “modified and cal- wjk w,” ian lines while (Pw,,)(Py,) represents the lowest order culations using wjk=O for j # k denoted by “ VVw” to correction due to line coupling effects. In this case the provide a comparison with a simple Van Vleck- normal dispersion, proportional to (v - v,), vanishes at Weisskopf calculation and thereby illustrate the magni- line center so the addition of the correction term tude of the effect of the line coupling coefficients. (Pwkk)(Pyk)is very important. In fact, so much informa- In spite of the uncertainties for N> 11, our calculation is obtained from the dispersion calculations that it tions using Lam’s wjkagree to within 4% with the results is desirable, for the sake of clarity, to discuss each obtained by Lam, thereby providing a check on our com- frequency separately. puter program using matrix inversion. Calculations We note in passing that the computer programs for performed with the second order perturbation theory dispersion calculations, both by matrix inversion and discussed in Sec. IIC’agreed with our matrix inversion perturbation theory, have been checked by detailed com- calculations to better than 1%so they were not plotted. parisons with the Rosenkranz results7 and with those For the absorption coefficient, our perturbation calcula- obtained by Liebe et al.’; the latter authors supplied a tions performed with only first order terms agreed with copy of their program to facilitate this comparison. the second order results to better than 1%for all pressures up to 100 kPa (1 atm). This is somewhat better than one would predict for a perturbation expansion in C. Dispersion at 61.141 GHz powers of since all such terms are about 0.1 Pwj,/v,, Our calculations of dispersion for this frequency are or smaller for P=100 kPa (1 atm); however, there is compared with the measurements by Liebe et al. in considerable cancellation among the higher order terms Fig. 6. The frequency of 61.141 GHz lies 10 MHz to the which reduces the actual error. The perturbation cal- red of the 9’ line; hence, the dispersion is dominated culation breaks down rapidly for pressures the order of by this line for pressures below 6.7 kPa (50 Torr). The 5 atm or larger. half-width for the 9’ line is 230 MHz/Pa or 1.74 MHz/ From Figs. 3-5 it is clear that our corrections to Torr and y9+ = -4.7~10-3/Pa = - 3.5x lO-’/Torr so the Lam’s wjk[Eqs. (21)-(23)] slightly improve the agree- first order correction due to line coupling (Pwkk)(Pyk) ment with experimental data for u’g and vi1 while the will be comparable to or larger than the frequency de- agreement for the vi line is slightly worse. These tuning (v - v,) =10 MHz for pressures greater than 53 changes, while observable, are not at all significant kPa (400 Torr). Unfortunately, the neighboring lines because the effect is very small and the 0,-0, potential also make a sizable contribution at these pressures so

J. Chem. Phys., Vol. 74, No. 12, 15 June 1981 6666 Earl W. Smith: O2 microwave spectrum

TABLE 11. -ist o hz I-widths wkk, interference coefficients y,, and second order coefficients G, and dvk for the first 20 rotational manifolds.

wM (MHz/Torr) yk (Torr-') Gk(Torr-') dv, (MHz/Torr*) N Minus Plus Minus Plus Minus Plus Minus Plus 1 2.14 2.21 - 7.77~ + 3.06 x - 1.24~lo-'' - 8.91 x 10-8 -6.75~io+ 2. lox 3 1.99 1.96 -2.88X10-4 4. 50 - 5.25~lo-' - 1.36 x lo-? -2.71X10-~ 4.74 5 1.89 1.86 -2.74 3.35 - 7.04 -5. oox - 3.85 4.39 7 1.82 1.79 -2.34 2.15 -4.08 +3.94 - 3.26 3. 06 9 1.76 1.74 -8.36~10-~ 9.80x10-6 +7.48 8.02 - 2.16 1.86 11 1.71 1.70 +7.53 - 1.38~ 5.66 5.69 -1.14 8.15X 13 1.67 1.66 1.92lX1O4 -2.31 3.67 1.16 -5.23X10-6 3.76 15 1.64 1.63 2.74 -2.95 3.02~io-$ - 8. i4x io-$ -2.43 1.64 17 1.60 1.60 3.31 - 3.43 -1.97x10-8 -3.23X lo-' - 9. O~Xio-? 3. 65~10-~ 19 1.57 1.57 3.73 - 3.81 -4.09 -4.34 +7.98X lo-'' -4.53 21 1.54 1.54 4.06 -4.11 -5.47 - 6.26 5.78~lo-' - 9.29 23 1.51 1. 51 4.33 - 4.31 -6.00 -6.17 9.50~ - 1.13x 10-6 25 1.49 1.49 4.53 - 4.48 - 9.19 - 9.05 1.13~10-~ -1.25 27 1.46 1.46 4.65 - 4.59 - 7.79 - 7.60 1.20 - 1.29 29 1.44 1.44 4.77 - 4.68 - I. 02 x io-? - 9.87 1.22 -1.29 31 1.41 1.41 4.86 -4.75 -8. 72x lo-' -8.34 1.20 -1.25 33 1.39 1.39 4.93 - 4.80 - 1.O~X 10-~ - i.04X 1.16 -1.19 35 1.36 1.36 4.98 - 4.84 - 9.66 x - 9.15 x lo-' 1.11 - 1.13 37 1.33 1.33 5.09 -4.94 -8.52XlO-' -8.04X10-8 1.21 -1.21 39 1.30 1.30 7.19 - 6.95 - 1.95~lo-' -i.82~10-~ 1.78~lo+ - 1.72~

the effect of line coupling at (v - vi) = - 10 MHz is much nant and y, is then negative. smaller than it would be at line center = v'). Never- (v It is thus clear that the correction for detailed balance theless, it is easy to see from Fig. 6 that our correc- [Eq. (2511 is extremely important due to the subtraction tions to Lam's wjkare basically correct and also very of upward from downward rates in the calculation of yk; important in terms of achieving agreement with the in the case of yg+ it even produces a change of sign. The experimental data. At this frequency, the interference coefficients yk [Eq. (1511 calculated with Lam's wjk actually have the wrong sign and produce an increase in Pressure in kPa dispersion above the VVW calculation whereas the experimental data lie below the VVW curve. Our correc- 40 80 I I I I I I tions to Lam's wjkproduce a sign change in the ya which lowers the dispersion below the VVW curve and produces reasonably good agreement with experiment. The reason for this rather dramatic effect, namely, the sign change, can be seen by looking at the interference coefficient yv, which is the most important of the yk for this frequency. From Eq. (15) we see that yk is given by a sum of terms which, due to the l/ujk factors, is dominated by the lines closest in frequency to vg+:

?I9+~~2~7+,/~*7+)+2(Wll+,/~,ll+)+. * * , (26) where we have used (d, /d,) g (dll+/d,) g 1, which is correct to about 10%. Both w7+, and wll+% are negative as is v,,,; hence, the first term is negative while the second is positive and they tend to cancel out one another to some extent. This cancellation is typical for all yk because the contribution from vi> vk lines has a sign opposite to that for vj < v,. Lam's calculation gives < wll+% and, since < the second term w7+, I v,,,+ I vs7+, wo dominates and y, is positive. Our correction for the -2.0 - infinite order sudden approximation [Eq. (24)] reduces I 1 I I I I I both and wll+, slightly but our correction in Eq. (251, for the fact that upward rates such as wllr9+should be reduced by exp(- AE/kT) relative to downward rates wLkll+, reduces wll+, whereas w7+, is unchanged. The latter correction causes the first term to become domi-

J. Chem. Phys., Vol. 74, No. 12, 15 June 1981 Earl W. Smith: O2 microwave spectrum 6667

Pressure in kPa y5* provides 99% of the contribution due to line coupling 0 12 effects and at 26 kPa (200 Torr) it still represents over I8,II 75% of this effect. Thus, for Ps13 kPa (Ps100Torr), 59.591 GHz the difference between the experimental data and the 5+ Line Center VVW calculation provides a direct measurement of y5+, whose accuracy is limited in this case by the f 10% un- certainty in the experimental data. From this comparison it is clear that our modified wjaproduce a y5+ coefficient which is in excellent agreement with experimen- L? -0.05 tal data whereas the unmodified wjkas calculated by a, Lam produce a y5+ which is a factor of 2 too large. This analysis may be carried one step further by noting that [cf. Eq. (1511

y5+ 2[(0.789 wSl5+/1144.4) - (0.853 w~+,~+/843.8)] +. . .

Pressure in torr = (zLI~,~+- 1.47 w~+,~+)1.379~ +. . . , FIG. 7. Dispersion" at 59.591 GHz as a function of pressure where wjkis in MHz/Torr and y5+ is in Torr-'. The at a temperature of 300 K. Experimental points, solid and two terms in Eq. (27) provide over 90% of the value of open circles, obtained from two independent unpublished mea- y5+; the sum over the remaining lines contributes less surements. than 10% because the terms are smaller due to the frequency differences in the denominator and the fact that contributions from vj> v5+ have signs opposite to those correction for the infinite order sudden approximation for vj< v5+. This means that y5+ is essentially deter- [Eq. (24)] multiplies both upward and downward rates mined by the difference in line coupling between the 5+ by a factor which decreases as N decreases. This manifold and the two neighboring manifolds. An experi correction thus determines the size of the change due to mental determination of y5+ provides a very sensitive line coupling effects. For the 9' line Cm,'0.65; if this measurement of inelastic transition rates [recall that correction were absent, i. e., Cxosg2.2, our calcula- wjkis proportional to inelastic transition amplitudes tions at 800 Torr would drop to about - 2 in Fig. 6. (1911. Such a change would be clearly observable and the agreement with experimental data would deteriorate As the pressure increases, the interference coef - slightly. We may thus conclude that our Cxos correc- ficients yk for neighboring lines become important and, tion is also approximately correct. Finally, we note for P>40 kPa (300 Torr), the total effect of line coupling that a calculation using first order perturbation theory is greater than the WW contribution, as shown in Fig. agrees with the full matrix inversion calculation to better 8. The dispersion is very small for this frequency be- than 1.5% for pressures less than 106 kPa (800 Torr). cause it lies at the center of the 60 GHz band; see Fig. 2, where the negative contribution from lower frequency lines almost balances out the positive contribution from D. Dispersion at 59.591 GHz higher frequency lines. Due to this strong cancellation, This frequency is at the center of the 5' line; hence, the perturbation calculation is less accurate at this the dispersion due to this line, the k = 5' term in Eq. frequency, being about 8% too high at 80 kPa (600 Torr) (18)) is proportional to Pz(6v5++W~+,~+Y~+). From Table as shown in Fig. 7 and about 15% too high at 107 kPa II it is clear that the second order pressure shift 6v5+ (800 Torr). This was the worst case for the perturba- is negligible compared with the product w5+,$+ y5+; thus, tion calculation and the agreement with the experi- the dispersion due to the 5' line is directly proportional mental data6 is still quite good. to the line coupling coefficient y5+. E. Dispersion at 60.4348 GHz For pressures below 150 Torr, the dispersion at 59.591 GHz is determined by the 5' line and by a simple This frequency lies at the center of the 7+ line which VVW contribution from all neighboring lines, i. e. , for is particularly interesting because it is a member of the k # 5') (v - v,) >> P2(6vk+ wkkyk); hence, the only line (T, 5-) doublet. It should be emphasized that none of the coupling coefficient which influences the dispersion so called doublets in the 60 GHz band arise from any kind below 27 kPa (200 Torr) is y5+. Since the VVW con- of quantum effect such as the splitting of degenerate tribution is determined entirely by line strengths and levels, etc. They all occur simply because the fre- half-widths which are known' to better than 2%, the quencies of two lines happen to lie close to one another. VVW contribution can be calculated with the same ac- It also happens, again by accident, that each of the four curacy and subtracted from experimental measurements doublets (3-, 13'1, (7') 5-), (3') 9-), and (15-, 1') in- to provide an essentially direct measurement of ys+. volves one line from the v> branch and one from the vi branch. Figure 7 shows the comparison between theory and experiment6 for low pressures. The VVW contribution At first glance, it would seem that the relatively is again plotted separately to illustrate the effect due to small frequency separations between the two members line coupling coefficients. For P<13 kPa (P<100 Torr) of a doublet would result in a correspondingly large

J. Chi" Phys., Vol. 74, No. 12, 15 June 1981 6668 Earl W. Smith: O2 microwave spectrum

Pressure in kPa In addition to the above effect on the yk, the relatively 40 80 small frequency spacing between members of a doublet I I I I 1- is also important simply because the lines overlap at ,Lams much lower pressures. This is illustrated very clearly / 'W by the dispersion at the center of the 7+ line where, at / / 10 Torr, the y7+ term in Eq. (18) contributes only 2% of the observed dispersion whereas the wing of the 5- line, i. e., the (v - v;) term in Eq. (18)) provides the rest. Even at 13 kPa (100 Torr), where the y7+ term is an order of magnitude larger, there is a strong overlap effect because the y5- contribution from the neighboring line is almost equal in magnitude to the y7+ term but opposite in sign; the combined effect of these two terms is thus an order of magnitude smaller and again the (v - v5-) contribution represents over 80% of the observed dispersion. That is why the calculations plotted in Fig. 9 show such a small change due to line coupling effects even though the frequency lies exactly at the line center. One should not be misled into think- ing that the interference coefficients themselves are 59.591 GHz 5 +Line Center small simply because the experimental data6 agree so closely with a simple VVW calculation. On the contrary, the interference coefficients for the T and 5- lines are about the same magnitude as any other y, I I I I I I I 1 listed in Table 11; it is simply the strong cancella- 200 400 600 800 tion due to the overlap with the 5- line and the fact that Pressure in torr y7+ and y5- are opposite in sign that drastically reduces the effect of line coupling at this frequency. FIG. 8. Dispersion" at 59.591 GHz as a function of pressure at a temperature of 300 K. Experimental points, solid and open Dispersion at 58.4366 GHz circles, obtained from two independent unpublished measure- F. ments. This frequency lies 10 MHz to the red of the 3' line which is a member of the (3+, 9-) doublet. For this doublet, wg-,>is so small that it contributes only about contribution to the interference coefficients yk, defined 2% to the value of y3+ whereas the w%,% and w~+,~+terms in Eq. (151, because the frequency separation appears contribute 93%. In addition, y9- is less than 0. 2y3+ so in the denominator. However, this does not happen there is no appreciable cancellation from the inter - because the line coupling coefficients wjkare very small ference term of the 9- line [recall that in the (T, 5-) for the doublets. This occurs for two reasons; first of doublet discussed in the previous section there was a all, all doublets except (7+, 5-) involve a rather large very large cancellation effect because I y5- I Iy7+I]. difference in Nand the S matrix elements which couple This means that the effect of line coupling is deter- two such levels will be very small; secondly, the cou- mined entirely by and wl+,%through the y3 term pling betweenthe v> and vi branches is relatively small due to a propensity rulela whereby the S matrix elements (N, J, MIS1 N', J', M' ) are reduced for AJ# AN. Pressure in kPa For example, wjkfor j =vi , k =v> is obtained from Eq. (19) by setting O07A,

(Naj Ja, Nb, Jb; N:, JL, N6, J:) 60.4348 GHz 7+Line Center =(N, N, N, N+1; N', N', N', "-1); I thus, wjkis proportional to (N, N, MalSIN', N', ML) I

X (N, N+1, M,lSINP, "-1, ML)* . ,Lams W Modified W In this product the term (N, N, MalSIN', N', MA) be- / vvw I comes smaller as AN= (N - N') increases and (N, N+1, MbISI N', N' - 1, Mk) is always small because AJ -0.41 = (N - (N' - 1) =AN+2# AN. For these reasons, the +1) 0 20 40 60 80 100 120 coupling between members of a doublet has a much smaller effect on yb than one would initially expect. In Pressure in torr the case of interest here, the coupling between 7' and FIG. 9. Dispersionz1 at 60.4348 GHz as a function of pressure 5- accounts for 23% of the value of y7+. at a temperature of 300 K. Experimental points from Ref. 6.

J. Chem. Phys., Vol. 74, No. 12, 15 June 1981 Earl W. Smith: 0, microwave spectrum 6669

Pressure in k Pa Pressure in kPa 0.2

55.7838 GHz 4t 17- Line Center E Q Q .-C

Modified W -0.21- and LamsW- I I I I I I 0 10 20 30 40 50 Pressure in torr FIG. 10. Dispersion" at 58.4366 GHz as a function of pressure at a temperature of 300 K. Experimental points from Fig. 14 of Ref. 5.

for pressures below 50 Torr; for higher pressures, Pressure in torr y3+ remains dominant, but, due to overlap with neigh- FIG. 12. Dispersion2' at 55.7838 GHz as a function of pres boring lines, the combined effect of the other ya is also sure at a temperature of 300 K. Experimental points from appreciable, contributing about 1/3 as much as y3+at Ref. 6. 107 kPa (800 Torr). From Figs. 10 and 11 it is clear that our calculated our calculated y3+ were four or five times larger but y3+ does not provide very good agreement with the ex- it is hard to see how such a large change in ys could be perimental data. We would obtain good agreement if ' produced by reasonable changes in our correction factors CIos and CDB[Eqs. (24)and (251, respectively] or Prssure in kPo by possible additional errors in Lam's calculations of w%,~+and w3+,%. It is always possible that the experimental data are incorrect but this seems unlikely since the high pressure and low pressure data repre- 58.4366 GHz sent two different runs and both sets of data indicate v;- 10 MHz that our calculations are too low. Consequently, the reason for this discrepancy is not known at the present time.

G. Dispersion at 55.7838. 56.9682, and 63.5685 GHz These frequencies correspond to the centers of the 17-, 13-, and 17' lines, respectively. They are grouped together because we have experimental data' only at 400 and 600 Torr which, due to the overlapping of the lines at these pressures, is not sufficient to permit analysis of the individual yn. These data are nonetheless interesting because they provide further confirmation of our method for correcting the line coupling coefficients calculated by Lam (see Figs. 12-14).

IV. CONCLUSIONS In the present paper we have performed calculations of absorption and dispersion [defined in Eq. (3)] at various frequencies in the 60 GHz band of 0, for pres- -0.5 I I I I I I I I I sures ranging from a few Torr, where the spectral 0 200 400 600 800 lines are isolated, to 100 kPa (1 atm), where the lines Pressure in torr have merged to form a continuous absorption band. FIG. 11. Dispersion" at 58.4366 GHz as a function of pressure At the higher pressures, collisions which transfer ex- at a temperature of 300 K. Experimental points from Fig. 14, citation from one radiating state to another, the so solid circles, and Fig. 29, open circles, of Ref. 5. called line coupling transitions, are no longer effective

J. Chem. Phys., Vol. 74, No. 12, 15 June 1981 6670 Earl W. Smith: O2 microwave spectrum

Pressure in kPa the fully classical model of Gordon’ in which the wjk 40 80 were regarded as inelastic transition rates. Actually, I I I I I I I I the wjbare sums over inelastic transition amplitudes (i. e., S matrix elements) which do not satisfy a conser- GHz 56,9682 vation of probability equation 2 jk wjk=O as assumed by 13 Line Center Modified Gordon‘ and Rosenkranz. ’ Consequently, the Rosen- W kranz prescription is incorrect and provides only an order of magnitude estimate of the wjk. Calculations using the Rosenkranz method for absorption and dis- per~ion~*~*~produced results that were in fair agreement with experimental data although clear discre- pancies remained and the results were extremely sensitive to the half-width for the nonresonant lines (e. g., Fig. 4 of Ref. 81, which is contrary to what one would expect. In addition, the best fit to the data required values for the nonresonant linewidth and wjkcoefficients which were in clear disagreement with the values calculated by Lam, differing by factors of 2 to 5. We therefore performed absorption and dispersion calculations using the wjkobtained by Lam. These calculations were not at all sensitive to the nonresonant linewidth as expected; they provided very good agree- Pressure in torr ment with absorption measurements as already noted FIG. 13. Dispersion2’ at 569682 GHz as a function of pressure by Lam3 but they gave extremely poor agreement with at a temperature of 300 K. Experimental points from Ref. 6. dispersion measurements as shown in his Figs. 3-18. It was subsequently determined that the wjkcalcu- as a broadening mechanism. The physics of this problem lated by Lam are in error due to two approximations has been understood for some time’ but a reliable quanti- used in his theory. The first of these is the “infinite tative calculation of the effect has been lacking because order sudden approximation” in which one neglects the (1) a calculation of the line coupling coefficients wjk energy difference between two states when calculating was not practical with older S matrix theories and has the inelastic transition amplitude (i. e. , S matrix) for only recently been attempted using a modern semiclas- transitions from one to the other; this approximation sical approach, 84 and (2) a straightforward calculation is known to overestimate the transition amplitude by of the band shape is itself very time consuming because an amount which increases with the rotational quantum it involves the inversion of a large matrix (about 40 number (e. g., see Ref. 16). Secondly, Lam assumed X 40) for each pressure and frequency of interest. The original motivation for the present work was to first Pressure in k Pa study the validity of the perturbation technique proposed 40 80 by Rosenkranz, which reduces the time required for c I 1 I I I l--l--T- the matrix inversion by about two orders of magnitude, and second to perform calculations with the line cou- 63.5685 GHz pling coefficients obtained by Lam. 3*4 -1 17+ Line Center The perturbation approach was tested by means of comparisons between first and second order perturba- E tion calculations and a complete matrix inversion cal- Q Q culation. It found that the simple first order per- -2 was ._c turbation expressions given in Eqs. (17) and (18) are c accurate to within or 2% for 100 kPa atm) ex- ._0 1% P<, (1 2 cept for the dispersion calculations at the center of the W mQ -3 .- band. At the band center the dispersion passes through n zero (see Fig. 2) due to a strong cancellation in the contributions from neighboring lines. This cancellation requires that the contributions from the individual -4 lines be calculated with very high accuracy and the perturbation theory had some difficulty here as seen in Fig. 8. Perturbation calculations for absorption seemed -5 I I I I I to be accurate to about 500 kPa (5 atm) and at that point 0 200 400 600 800 one may be approaching the limit of validity for a binary Pressure in torr collision theory. FIG. 14. Dispersionz1at 63.5685 GHz as a function of pres- The prescription used by Rosenkranz’ for his calcula- sure at a temperature of 300 K. Experimental points from tion of the line coupling coefficients wjkwas based on Ref. 6.

J. Chem. Phys., Vol. 74, No. 12, 15 June 1981 Earl W. Smith: O2 microwave spectrum 6671 that an upward inelastic transition (i. e., to a state of in Lam’s wjkdue to uncertainties in the intermolecular higher energy) can occur even in the limit of small potential. kinetic energies when actually such a transition is for- bidden for kinetic energies smaller than the energy APPENDIX spacing between the two states; this approximation pre- vents Lam’s wjk from satisfying the detailed balance In this Appendix we will derive our perturbation ex- condition [Eq. (1311. A scheme for correcting these ap- pansion for the eigenvalues A, and eigenvectors I &) of proximations was developed in Sec. II E and calculations the operator uo +iPW introduced in Eq, (4). We will with the modified w,, gave generally very good agree- use a very simple perturbation method (e. g., Merz - ment with all experimental data, an exception being bacher, ’’ WilkinsonZ0)which is slightly complicated by the dispersion for the 3’ line (Figs. 10 and 11). the subspace of zero eigenvalues (i. e., the nonresonant lines). The subspace corresponding to zero eigenvalues It may thus be concluded that absorption is not suf- will be called noand the rest of the space will be Q1 ficiently sensitive to the line coupling coefficients to so that the notation j E noimplies u, = 0 and j E L.2’ means manifest the rather serious errors in the line coupling uj #O. At this point it is convenient to define a set of coefficients wjkcalculated by Lam and the Rosenkranz vectors I c$j) which are orthogonal normalized linear prescription. In the dispersion calculations, on the combinations of Ij )E nodefined by other hand, these errors produce very large effects, sometimes giving the wrong sign as in the case of the 9’ line discussed in Sec. IIIC. i. e., the vectors I @)) are the eigenvectors of the part Perhaps the most important result of the present work of w which exists in no. is the discovery that the dispersion is very sensitive to The perturbation expansions for I &) and A, are de- the line coupling coefficients. In particular, it was fined by found that, near the line center, the low pressure data [P_<20kPa (150 Torr)] were usually governed by a single (A21 interference coefficient y, and this in turn was determined by the difference between the line coupling coef- A, =A:’’ +PA:” +P2A:‘’ +. . . , (A3 ficients wjkfor coupling to the two closest neighboring lines. One of these coefficients will correspond to an where 1 $’)) and Af) are the nth order perturbations and upward inelastic transition and thus be relatively small it is required that (xLo’l xf’) =O for n#O. Substituting whereas the other will correspond to a downward transi- Eqs. (A2) and (A3) into tion and be somewhat larger. This means that a dis- (VO +ipw)1 xk) Xk) (A4 persion measurement will usually be extremely sensitive to one or two inelastic transition rates. A set of and equating like powers of P produces the set of equa- dispersion measurements for a series of spectral lines tions could thus provide a convenient test for calculations of uo I =Ac0)k 1 9 (A5 state to state inelastic transition amplitudes. xio’) x:”)

To design such an experiment, one would first estimate a set of half-widths wkkand interference coefficients yk calculated by using Eq. (15) and the “rates” wjkto the two nearest states. The dispersion at the +A:’’/ , (A7 ) center of the uk line would then be given by (yk/wkk) xio’) plus a series of contributions from neighboring lines which are then solved for I xf”) and AF). The zeroth [Eq. (la)]. The (yk/wkk)contribution will generally be order equation [Eq. (A5)] is readily solved by (recall dominant if it is greater than (u - uka)/[(u - u#)2 that uoI 4,) =O since I c$,) is a linear combination of + (Pw,, ,,)2] for the two neighboring lines k’ but even if Ij)E no) this is not the case, the half-widths are generally well A:’) =uk (all k) , ) known and this particular overlap contribution could be (A8 calculated and subtracted out. It would be preferable to avoid contributions from the interference coefficients so from neighboring lines one would want to work at =I@,) (kEQ0). (A9 ) pressures low enough that (Pyk. )(Pwk.), )/[(u - Vk’ +(Pwkek,)‘]is always less than (yk/wm). In this manner The first order equation [Eq. (A6)] may be multiplied it should be possible to design a set of experiments from the left by (k 1 if k E a,, or by (@kl if kE no,to which would measure a set of interference coefficients yield [using Eqs. (A5), (A8), and (A911 yk and thereby provide a good test for calculated in- Ai”=ihlwJk) (k~521), elastic transition amplitudes.

It should be mentioned that the corrected line coupling coefficients calculated in this paper are subject to errors For k~ n1, we obtain the components of I xi’)) by multi- resulting from the approximate nature of the correction plying Eq. (A6) from the left by (jI E nland (@j I using terms in Eqs. (24) and (25) as well as additional errors the notation ujk= vj - uk to obtain

J. Chem. Phys., Vol. 74, No. 12, 15 June 1981 6672 Earl W. Smith: O2 microwave spectrum

to sum over j #k so that ( x:O’I xi”) =O; in fact, the condition (xj0)I xf’) =O for n #O is a general requirement for this type of perturbation theory. Proceeding to the second order equation [Eq. (A7)], we again multiply by (k I if k E 52, or by (@kl if k E 52, and obtain For k~ no, we can determine the components of I xi1)) in &l1by multiplying Eq. (A6) with (jl E 52, but we cannot determine (@jl xi1))or (jl xL1)) for j, kE 52, because (uo - uk) vanishes. We therefore write

(A12) where the ajkwill be determined by the next higher or- We again determine the components of I xL2’) for kc52, der equation. In Eqs. (All) and (A12) we were careful by multiplying Eq. (A7) with (jI E 52, and (@ I E 52,:

I For k t no,multiplying Eq. (A7) by (Gjl provides an ex- in the 52, subspace; neglecting ajkthus represents about pression for the coefficients ajkintroduced in Eq. (A12): a 1% error in dispersion and 0.1% in absorption for the 60 GHz lines. Finally, we neglect the terms Pjk introduced in Eq. (A16) for the same reasons. and multiplying Eq. (A7) by (jl~52, provides With the above approximations, our results may be written in the form

(A161 where again the coefficients Pjk would be determined by the next higher order equation. At this point it is desirable to make some simplifying approximations regarding the nonresonant lines. We first note that for j, k E a,,the diagonal matrix elements (k I wI k)are at least one order of magnitude (A181 greater4than the off diagonal elements (jI wlk). A vec- which are valid for all k provided that we interpret tor I @k) will thus equal I k) plus a small admixture Cjfkas jtnl whenever kg 0,. In order to normalize (10% or less) of other 52, states. Since the nonresonant I xk), we divide by the square root of (recall that lines contribute at most 10% to the dispersion and less (X~~)IX~))=Ofor ~ZO) than 1% to absorption in the vicinity of the 60 GHz lines, we will use 1 @J I k)with a probable error of at most (Xkl Xk) =1 +(Xk1’1 xi1’)+.. . 1%. Our next approximation is to neglect the coefficients ajkdefined in Eq. (A15). This coefficient ap- =1 - c(k/z~Ij)(jl?~/k)/u~,+... . (A19) pears only in Eq. (A121 where it represents the first jCk order correction to the wave function I yk)within the It should be noted that (xkl is the transpose” of I xk), 52, subspace. Using the w matrix elements calculated not the Hermitian adjoint (the matrix u, +iPw is not by Lam (which overestimate ajk), the maximum value Hermitian). The transformation matrix Xj, whose of ajkis about 0. l/atm; thus, even at a pressure of 1 columns are the normalized I xk) eigenvectors is thus atm, cyjk represents only a 10% correction to I xk) with- given by

J. Chem. Phys., Vol. 74, No. 12, 15 June 1981 Earl W. Smith: O2 microwave spectrum 6673

'H. J. Liebe, G. G. Gimmestad, and J. D. Hopponen, IEEE Trans. Antennas Propag. 25, 327 (1977). 'E. W. Smith and M. Giraud, J. Chem. Phys. 71, 4209 (1979). 'OM. Baranger, Phys. Rev. 111, 494 (1958); 112, 855 (1958). "E. W. Boom, D. Frenkel, and J. van der Elsken, J. Chem. Phys. 66, 1826 (1977). "Using Lams3 Eq. (34) and the fact that k(v) and n(v)- 1 are the real and imaginary parts of an analytic function, see for example, R. Loudon, The Quantum Theory of Light (Claren- don, Oxford, 1973), Chaps. 2 and 4. 13A. BenReuven, Phys. Rev. 145, 7 (1966). "J. H. Van Vleck, Phys. Rev. 71, 413 (1947). 15T. A. Dillon and J. T. Godfrey, Phys. Rev. A 5, 599 (1972). I6R. Goldflam, S. Green, and D. J. Kouri, J. Chem. Phys. 67, 4149 (1977). "H. S. W. Massey, Electronic and Ionic Impact Phenomince (Clarendon, Oxford, 1971), Vol. 111, Chap. 17.4, particularly Eqs. (133) and (166). "For the 0, molecule, the spin S is very weakly coupled to the nuclear angular momentum N to form the resultant J. There- fore, a collision which produces a sudden change in N will tend to leave S unchanged, resulting in AS =O or AJ=AN. This is not a rigorous selection rule because S will be able to follow N for slow adiabatic collisions; hence, we observe 'E. W. Smith, M. Giraud, and J. Cooper, J. Chem. Phys. only a propensity for AJ=AN collisions, i. e., the S matrix 65, 1256 (1976). elements for AJ *AN are small but not zero. 'R. G. Gordon, J. Chem. Phys. 46, 448 (1967). i3E. Merzbacher, Quantum Mechanics (Wiley, New York, 1961), 3K. S. Lam, J. Quant. Spectrosc. Radiat. Transfer 17, 351 Chap. 16. (1977). "5. H. Wilkinson, The Algebraic Eigenvalue Problem (Claren- 4K. S. Lam, Ph.D. Thesis, Dept. Of Physics, University of don, Oxford, 1965), pp. 1-7 and 66-70. California, Berkeley, January 1976. "In Figs. 6-14 we have actually plotted AN(v)-Lw(v/2), where 5H. J. Liebe, Dept. of Commerce, Office of Telecommunica- Lw is the dispersion function defined in Eq. (3) and Y is the tions Report 76-65, June 1975. observation frequency, e.g. , v=61.141 GHz in Fig. 6. The 6G. Gimmestad and H. J. Liebe (unpublished data). experimental apparatus of Leibe et al. 5i6i8 is designed to mea- 'P. W. Rosenkranz, IEEE Trans. Antennas Propag. 23, 498 sure this function rather than AN(v) itself in order to sub- (1975). tract out the frequency independent refractivity.

J. Chem. Phys., Vol. 74, No. 12, 15 June 1981