NBSIR 80-2175

Analysis of the Shape of the

Far-Infrared Spectra of H2 “H2

and H2 -He Collisions

E. Richard Cohen

Rockwell International Science Center

Thousand Oaks, CA 91360

and

George Birnbaum

National Measurement Laboratory

U.S. Department of Commerce

National Bureau of Standards

Washington, DC 20234

r

April 1981

Prepared for the Jet Propulsion Laboratory, California institute of Technology
4
MATIOH4L *UBS4t

U» BTANDA&Dt

UBKAtT

ANALYSIS OF THE SHAPE OF THE

FAR-INFRARED SPECTRA OF H 2-H 2 AND

H2-He COLLISIONS

NBSIR 80-2175

•

- *

JUN 1 0 1981

noi acc.- i/c.

OiLioO

, u:; j

Wl

E. Richard Cohen C.9-

Rockwell International Science Center

Thousand Oaks, CA 91360

and

George Blrnbaum

National Measurement Laboratory

U.S. Department of Commerce

National Bureau of Standards

Washington, DC 20234

April 1981

Prepared for the Jet Propulsion Laboratory, California Institute of Technology.

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS. Ernest Ambler. Director
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Abstract

The collision-induced far-infrared spectra due to H2 “H2 and H2 ~He collisions have been previously measured in the range 20 to 900 cm"^ at 77.4,

195 and 292 K.

These spectra are fitted with a semi-empirical line

shape and the parameters in the shape function are evaluated. The accuracy

of these fittings is discussed. The zeroth and first spectral moments for

the isotropic overlap translational spectrum due to H2 “He collisions are obtained and give a value for the range parameter in the induced-dipole

function in good agreement with that given by an ab initio calculation.
3

1 . INTRODUCTION

Collision-induced absorption in the region 20 - 900 cm"^ has been

measured in normal H2 (

11H2 ) at 77.4 K and equilibrium H2 (eH2 ) and eH2 “He

mixtures at 77.4, 195 and 292 K (ref. 1, hereinafter called I). The density

of gas in this experiment was sufficiently low so that the collisions were

mainly bimolecular. A semi-empirical line shape for such spectra has been

developed and found to represent well the absorption in H2 at 77.4 K from

20 to 500 cm"^ in reference 2 (hereinafter called II). In this work we fit

the measurements in I with the object of obtaining an analytical representation for these spectra for use in analyzing the far-infrared emission

from planetary atmospheres containing H2 and He.^ Some of the problems in

fitting these spectra are discussed.

Although the detailed analysis of these spectra is outside the scope

of this work, we obtain the zeroth and first spectral moments and from

these we obtain the range parameter in the induced dipole model for the

isotropic overlap component of H2 ~He.

A.
4

2. THEORY OF SPECTRAL SHAPE

A semi-empirical approach to obtaining the shape of collision-induced

lines based on assuming a reasonable and convenient form for the correlation

function has been developed in I and reviewed in reference 4, The model

correlation function satisfies certain mathematical and physical properties

and its parameters are related to the moments of the spectrum. Although

there is no guarantee that the resulting shape will fit the data, when it

does it provides a very simple way of representing and analyzing such data.

The expression for the absorption coefficient of the translationalrotational spectrum due to quadrupole-induced and anisotropic overlapinduced dipoles in two-body encounters is given by

f

. 27t^n^ , -pfiuj. ^ ^ JJ'

(2J+1)(J2J' ;00)'

t ^ E ( 1 )

Here n is the number density of the single component gas, p = (kT)"^, J is

the angular momentum quantum number and ^jji is the transition angular frequency,

u)jj, = 27Tc(\)j, -Vj) . In practice, the density is frequently given in amagats

,

p, in which case n = pno, where no is the number density at STP. The rotational energy levels (expressed in wavenumbers) are given by^

Vj = 59.3392J(J+1) - 0.04599J^(J+1)^ + 0 . 000052J^(J+1)^ cm"^ . (2)

The Boltzmann factor Pj is given by

Pj = (2J+l)gje

-PEJ

(3)

where Ej = hcVj is the energy of rotational state J and the statistical factors due to nuclear spin are

gj = 1 , J even ; gj

= 3 , J odd

The quantity (2J+l)Pj for various energy levels is presented in Table 1.
5

Table 1

Population statistics for equilibrium hydrogen.

J

Ej/hc

-1 cm

(2J+l)Pj

Sj

77.4 K 194.7 K 292.4 K

normal hydrogen

77.4 K

0 0 1 0.500 0.191 0.132 0.249

1 118.5 3 0.497 0.717 0.662 0.749

2 354.4 1 0.003 0.070 0.115 0.002

3 705.6 3

- 0.022 0.086 -

4 1168.9 1 - - 0.004 -

5 1740.4 3

- - 0.001 -

para 0.503 0.261 0.251 0.251

ortho 0.497 0.739 0.749 0.749

The induced translation-rotation spectrum of H2 is due to quadrupole

and anisotropic overlap-induced dipoles. The type of contribution to

a given transition is designated by the subscript It is sufficient

for our purpose here to assume that

rcuj-iujj,) (4)

namely that the various contributions to a given transition may be represented by a single shape function, r(u)-u)jj,), with an intensity given by

the sum of the various contributions which are discussed in the Appendix.

The sum which we designate by S is the total strength of the

A A q

quadrupole- induced as well as the quadrupolar overlap-induced dipoles.

The shape function is given by

with

r(u) ) zK^(z)

,^2 2

>

z = -.^2 21 +U)_T^

2 2 ^ . io.k>2

"''3 ” "''2

(5)

= UJ-OJJJ I UJ
6

Here K^Cz) is the modified Bessel function of the second kind with the

properties

,

zK^(z) 1 , z ^ 0 f

K^(z) ^ (7t/2z)^e z >> 1

The time parameter X 2 controls the exponential decay of the wings. When

Ti >> X 2 shape is Lorentzian in the vicinity of the transition frequency

with a half-width given approximately by Au)i = However, when X 2 /X 1 1, as is true for H2 , (5) is never accurately Lorentzian and AuDi differs

appreciably from the half-width that would be obtained by attempting to fit

the data to a Lorentzian, as has been done in the past.®

The Clebsch-Gordan coefficients (J2J’ ;00) are well known; their vanishing for > 2 and their symmetry,

(2J+1)(J2J' ;00)^ = (2J'+1)(J'2J;00)^ , (6)

allow us to write

E

jj’

pj(2J+l) (J2J* ;00) r(uj-uj.rT.

)

with

JJ’

= Ar(u)) + p^rCui-uj^^) +

I

+ P3F(uj+u;33)_

j^p^rcuj-uj^^) + p^rcuj+uj^^)

7

+ if [p3rc»<-u^35) + P5r("^<^35)

A = Z

J

J(J+1) (2J+l)py (2J-1) (2J+3)

(7)

(8)

In (7) the first term represents the translational band (^jji ~ 0) while

the others are the resonance (u)-ujjj.,) and anti-resonance (uu+iUjj,) terms.

This treatment neglects a small contribution to the absorption resulting

from the anisotropy in the polarizability of H2

.

®

For H2 “He collisions, the absorption coefficient becomes

. 2

4ti n

3fic

a% , -p-htu. tu(l-e ^ )

JJ'

S^Vi(uj)

O oh

p (2J+1)(J2J' ;00) S r (U)-UJ )

•J ^ J sJ ( 9 )
7

The strength represents the contributions to the intensity from quadrupolar and anisotropic overlap induction. These components are discussed in

the Appendix. In (9), n^ and n^ are the densities of the components of the

mixture. The term T (in) represents the translational absorption due to

the dipole induced in H2 ~He collisions by isotropic overlap forces.^ By

symmetry such a contribution is impossible for rotational-translational

transitions in H2• Equation (1) with (4) can be fitted with three parameters: two time

constants and and a strength factor Equation (9) can be

ab ab ab"^ fitted with six parameters: T, , and S due to quadrupole and

Iq ab^^ ab ab anisotropic induced dipoles and ~^ 2 ± isotopic overlapinduced dipoles. The actual situation is even more complex, but the present

data do not have sufficient precision to allow the quadrupolar and anisotropic overlap contributions to be resolved into separate components.
8

3 . SPECTRA

The three-parameter least-squares fits of the eH2 “H2 spectra, including the

separate translational and rotational components, are shown in Figures 1 to 3

,

where we have plotted

D(v) = [a(v)/p^] [vtanh(phcv/2) ] ^ (10)

instead of a(v)/p^ to emphasize the translational band at the lower frequencies. The J = 2 0 transition at 354.4 cm"^ (S(0) line) and the J = 3 1 transition at 587.1 cm"^ (S(l) line) are partially resolved in the data.

The J = 4 2 transition at 814.5 cm"^ (S(2) line) can be seen only as a distortion of the high frequency wing of the S(l) line at dry ice and room

temperatures. Contributions to the absorption due to the molecular hexadecapole moment of H2 and to double transitions, where both molecules

change rotational state, are too small to make significant contributions to

the spectrum and were omitted in the analysis. There is a trend at all

temperatures for the fitted spectrum to exceed somewhat the experimental

spectrum in the high frequency wing of the S(l) line. The values of the

parameters and are given in Table 2.

Table 2. Values of the

experimental H

equilibrium

parameters

2 spectra.

in (1), (4) and (5) used

(n) signifies normal

in fitting the

and (e) signifies

T

aa aa 33

1 U q "^iq ^2q

(K)

°6

(ka"^) (lO’^^ s) (lO'^^ s)

77.4 (n) 106.3 ±0.8 10.26 ±0.14 3.91 ±0.25

77.4 (e) 102.8 ±0.6 10.23 ±0.13 4.20 ±0.25

195 (e) 129.5 ±0.5 5.95 ±0.07 2.59 ±0.13

292 (e) 149.7 ±2.4 4.48 ±0.22 2.12 ±0.47

A similar least-squares fit to the normal hydrogen spectrum (not shown)

observed at 77.4 K was made. The agreement of the results at 77.4 K between

nH2 (para-ortho distribution, 1:2.98) and eH2 (para-ortho distribution, 1:1)
9

f

Q.Ol- X 2_ieBeujB/(A)
-195K

10

O

CO0)

!-i

CCJ 3

a*

CO

I 4J

CO

cfl 0) E

o

0) 4

-

1 OJs

nj

J-4 cdcu

I0) CJ .

5

-

4 V ^ O

H H

LO

(Ti

rQT3 iH 1)C

C4M-I X Qj 0) T3

CN

OJD

60

•H

P>4

P p

8-01. X ^_ieBeuiB/(A) q
11

8-01- X 2-*e6BiuB/(A)
12

confirms that the normal population (equilibrated at 292 K) was preserved

at the lower temperature (77.4 K) and that the equilibrium population

(equilibrated at 77.4 K) was indeed achieved in the measurement. The three

percent difference in the intensity factor S in the two measurements at

77.4 K confirms that the gas densities were determined consistently to

within an error which could not have greatly exceeded one percent.

Other adjustments to the data have been made in which we have allowed

the time constants to be different for the translational and rotational bands.

However, this introduces very little change in the adequacy of the adjustment.

In general, the differences in the two values of Xi and I 2 , taking into account

the statistical correlation which exists between the values, is not statistically significant.

We believe the accuracy of the fitted parameters to be of the order of

5%, although relative values and the temperature dependence may be more re- liable. Up to this point our analysis is independent of any models of the

intermolecular potential. We can evaluate the integrals Ig(T), I'(a/p,T) and

I"(a/p,T) introduced in the Appendix only if we know the correct form of the

(quantum mechanical) distribution function g(x) as well as the parameters a

and e of the Lennard-Jones model, or the equivalent parameters of whatever

more accurate potential model is used. However, for our purpose here we present only empirical fits. From the data in Table 2, we can express the

Si Si strength by the expression

= 142(T/T_)^‘^^ KA^

q 0

with Tq = 273.15 K. Similarly we find

= 4.85(T/T_)"°'^^^ • lO'^^ s Iq 0

= 2.17(T/T_)“^‘^^^ • 10 “^^

s 2q 0

Theory leads one to expect^ the product X 1 I 2 T to be roughly independent

of temperature, varying only as Ig(T)/Iio(T) assuming that the quadrupoleinduced dipole predominates. For a classical Lennard-Jones potential, an analysis of the asymptotic behavior of the integrals shows that this ratio

varies from a T^ dependence at low temperatures (s >> kT) to T"^/® at high

temperatures. The observed variation from the above fittings is
13

4. THE REDUCED H^-He SPECTRA

The experimental and fitted spectra at 77, 195 and 292 K, are shown in

Figures 5 to 7 for the reduced H2 "He spectra, where we have plotted

' ^Pa/Ph'^aa^'’^ '

The spectral function D(v) is defined by (10) and the subscript m refers to

the mixture whose density is given by ~ P]^ • Also shown are the

separate isotropic overlap and quadrupole-induced components. The fitted

parameters are given in Table 3. It appears that the discrepancies between

the fitted and measured reduced spectra for H2 ~He may be attributed to the

inaccuracy in the experimental results. As explained in I, this arises in

the case of H2 ~He mixtures with too high an H2 concentration which yields

as a small difference between two large numbers.

Table 3. Values of the parameters in (9) (4) and (5) used in

fitting the reduced eH^-He spectra.

Isotropic Overlap Quadrupolar Plus

Anisotropic Overlap

T

1 ab

^li

ab

^2i q

ab

"iq

ab

" 2 ,

(K) (kIS (lO'^^ s) (lO"^"^ s) (lo"^^ s) (10‘^^ s)

77.4 34.47 3 . 64 4.66 6.93 7.02 13.44

195 81.34 1.99 4.22 19.14 4.13 2.38

292 119 .

1

1.91 2.52 21.0 4.20 2.86

The rapid decrease in ^t low frequencies at 195 and 292 K might

be attributed to the inter-collisional interference effect.^ However, an analysis of the data shows that these dips are probably not statistically

significant. It should be noted that dividing a(v) by vtanh(phcu/2) to

obtain D(v) greatly amplifies any errors in the low frequency data.

On comparing the H2 spectra (Figures 1 to 3) with the reduced H2 ~He

spectra (Figures 4 to 6) we note that translational band intensity of the
-He:

292K

14
15

v/cm"'

Figure

5.

eH2

~He,

195

K.

Six-parameter

least-squares

defined

by

(10)

and

(11)*
-He:

77.4K

16

r—" ' I r 1 I I I O

CO

0

+J •H

<4-1

CO

CUu

nJ

;3cr

CO

1 4-1 COCO

(U

E

u

^40) 4J

CUs

cO •

cOeu rH

I ^

X

•H T3

C/3 ficO O

rH <r ^

- T3

<U CU 33 fl

I *H

CM M-l 33 (U

(U T3

sO

(UV43

00 •H

g.OI. X ^_iBBeiuB/(A)'’®a
17

latter is much greater than that of the former. This is due to the isotropic overlap induction in H2 ~He collisions. On the other hand, the S(0)

and S(l) lines in H2 “He have much less intensity than in H2

. This results

from the much smaller polarizability of He compared with H2 which greatly

reduces the quadrupole- induced dipole component. Finally, the S(0) and

S(l) lines in H2 ~He at a given temperature are broader than those in H2

. Such a result may be understood if anisotropic overlap makes a significant

contribution to the intensity of these lines. The width of an overlap

component should be considerably greater than that of a quadrupolar-induced

line because of the shorter range and consequently shorter collision duration

of the former interaction.

Undoubtedly the inaccuracy in the data contributed to our difficulty

in obtaining a unique set of parameters from the fitting. It was surprising, nevertheless, to find that these fittings showed so much flexibility

and that a wide range of parameter sets could approximate the experimental

curves with equal accuracy. For example, the overall rms fractional error

in the fit at 195 K is only 8 percent. However, the adjustment is statistically loose and the least squares fit shows very strong correlation

between the time constants and the strength parameter. One can adjust for

changes in the former by making compensating changes in the latter. Thus,

the quadrupolar and anisotropic-overlap strength parameter S shows a standard

error of 25 percent, T 1 X 2 a standard error of 50 percent and I 2 /X 1 a standard

error of 200 percent. The isotropic overlap parameters are much more

Apart from the imprecision of the data, we believe that a six-parameter

fit to a rather featureless spectrum such as H2 “He, particularly at 195 and

292 K, provides too many degrees of freedom unless the parameters are restricted by theory. Attempts to reduce the number of variables by invoking a relation in II between the product I 1 I 2 and molecular parameters was frustrated in part because of the unknown admixture of quadrupole and

anisotropic overlap interactions.

3!^ 3b

Since as explained above, the parameters S , and not well

defined, there is little to be gained by trying to determine the temperature

precisely defined: = 8.4 ±1.1

li 2i

± 5.1.
18

dependence of can be fittedthe quadrupolar parameters. The isotropic parameters,

with reasonable accuracy to yield

= 112(T/T KA^

I o tf = 1.74CT/T s II o

if! - 3.4(T/T s 2x o

however

,
19

5. SPECTRAL MOMENT ANALYSIS

We can obtain immediately from the parameters of the fitted spectrum

the experimental invariants

,

-2 a = n laf

aa f aa

(uj)du) ( 12 )

- ^ - aa f

^aa

J u)tan

(u))du)

u)tanh(pfiw/2) (13)

Thus, following the derivation of (31b) and (35) in II we obtain

aa aa / 6kT ^ Of = Y I —r- +

q

*q )

C-i to /

(14)

Iq 2q

for quadrupole and anisotropic overlap-induced dipoles. From (1), (4)

and (13) we find that

aa 7T „aa

(15)

The quantity y (6kT/I) is the contribution to a from the rotational

q aa / aa ai^ transitions AJi^ = ±2, AJo = 0, while v / is the translational

^ q / Iq 2q

contribution.

For the reduced H2 ~He spectrum, (12) to (15) apply, where aa is replaced

by ab for the quadrupole and anisotropic overlap-induced contribution and

n^ is replaced by n^ and n^ . There is, in addition, the isotropic overlap

contribution for which

and

ab ab

I, . li 2i

271 ab

3kTc ^i

ab ab

a , = a + a

.

ab q 1 ab

( 16 )

(17)

Table 4 gives the various values of y computed from (15) and the

values of S given in Table 2. Table 5 gives the values of (6kT/I) and

(tita)”^- We see from Tables 4 and 5 that the translational contribution to

a is between 15 and 17 percent depending on the temperature.
20

is due to collisions,

a a Table 4. Values of y defined by (13). y

ab ^

Y- is due to isotropic overlap induction in H^'He collisions

^ 3. 1!) and Yq is due to quadrupolar and anisotropic overlap induction in H^-He collisions.

T

(K)

aa

,.-58 5

10 s cm

ab

,.-58 5

10 s cm

ab

,.-58 5

10 s cm

77.4 (n) 3.012 — —

77.4 (e) 2.913 1.954 0.3928

195 (e) 1.458 1.834 0.4312

292 (e) 1.123 1.786 0.3150

aa Table 5. Values of a , (6kT)/I which is proportional to the rotational

contribution and

(

aa aa\ -1

which is proportional to the

translational contribution to aaa in collision.

T (6kT)/I

(K) (10^® s'2)

aaa

q

-1 5 . s cm )

77.4 (n) 0.1360 0.02493 4.847

77.4 (e) 0.1360 0.02327 4.640

195 (e) 0.3421 0.06489 5.934

292 (e) 0.5138 0.1053 6.953

33 33

In I, and were evaluated from data in the region 20 to 900

cm"^ and from data from 900 to 1600 cm ^ obtained elsewhere. It was necessary, however, to adjust the data of the latter experiment to agree

with the results in I by forcing a fit in the overlapping region 800 to 900

cm"^. The present values of y^^ are within 0.2 percent of those obtained

a a ^ in I. The present a at 77 K is less than that in I by 4 percent and at

195 K and 292 K it is greater than the values given in I by about 1.3

percent

.
21

As is evident from the spectral fits for the reduced H2 -He spectrum

(Figures 4-6), the isotropic overlap contribution is dominant. Table 4

3. Id shows that y. the isotropic overlap contribution, is on the average 5 times

^

’ ab greater than that due to quadrupole and anisotropic overlap induction y

ab ab ab v ab ^ A comparison of Of. and Of shows that a. Sot 1 q 1 q

To proceed one step further with our analysis, we can immediately

obtain the range of the isotropic overlap-induced dipole, from^

2 kT ab ab

P = — T, . . ov m li 2x (18)

Here m is the reduced mass and and are integrals involving the radial

distribution function defined in II. Since (p^^/a)^ << 1, only an approximate evaluation of Ip,/I„ is necessary and I„ « I is a valid first approxi- K i2 K

mation.

o Using a = 2.77 A and m = i.34 amu = 2.226 • g for the reduced

o mass of H2 “He, we find from the data given in Table 6 that p = 0.288 A at

o o o 77 K, 0.323 A at 194 K and 0.299 A at 292 K. The average is 0.303 A

o which is in satisfactory agreement with the value = 0.324 A given by

an ab initio calculation.^^

Table 6. ^ . ab ab)-l ab Values of (i., t. , Of

, Iq 2q q _ collisions, where

. ab ab^-1 , sb ^ tt (t.,.I».) and a. for H„-He

li 2i X 2

is proportional to the translational contribution of Of . . ab

^ , ab abv-1 ab

T (t, l_ ) Of Iq 2q q

.,-28 - 2 . rin "31 -1 5 . (K) (10 s ) (10 s cm )

. ab ab.-l

(10^® s'^)

ab

Of

,

X .,--31 -1 5.

(10 s cm )

77.4 0.0106 0.576 0.0590 3.810

195 0.1017 1.914 0.1191 8.458

292 0.0833 1.880 0.2078 12.89
22

6 , Conclusions

The far infrared absorption spectra of H2~H2 and H2“He collisions have

been fitted with a serai-empirical line shape. For H2~H2 the results appear

to be satisfactory; for H2-He the fit is less good. In H2-H2 collisions

only a single quadrupolar component was needed to fit the experimental data.

The strength and the time constants of this line shape due to unresolved

pure quadrupole plus quadrupolar overlap components have been fitted by an empirical power law temperature dependence which will allow the data to be

conveniently interpolated for temperatures in the range 70 - 300 K. However,

data at additional temperatures would be very useful in establishing the

SiSi accuracy of these empirical power laws. The strength, , can be resolved

into quadrupole and overlap contributions if one uses independent information on the H2 quadrupole moment and intermolecular potential.

The H2“He spectra were determined less accurately since the observed

spectrum of the mixture is the sum of the H2“H2 and the H2-He contributions.

In H2“He the dominant interaction is isotropic, and although the quadrupolar

component is easily seen, the accuracy with which it has been separated

from the H2“H2 quadrupolar absorption is poor. Therefore, only the isotropic

overlap parameters have been fitted to an interpolating temperature dependence.

Data extending to higher frequencies (up to 2000 cra"^)^^ may improve the

accuracy with which the parameters of the quadrupolar component can be

determined

.

7 . Acknowledgments

The authors thank Dr. G. S. Orton and Mr. V. G. Kunde for helpful discussions . This report was prepared for the Jet Propulsion Laboratory, California

Institute of Technology, under contract NAS 7

- 100 , sponsored by the PLANETARY

ATMOSPHERES PROGRAM OFFICE, OFFICE of SPACE SCIENCE, NATIONAL AERONAUTICS

and SPACE ADMINISTRATION.
23

APPENDIX

The R-dependence of the induced-dipole in H2 may be obtained from the

work of Poll and Van Kranendonk^ and is given by

M(R) = 2 V3 2| + M (R)

R

+ P^(R) (Al)

Here ^|3aQR~'^ is the quadrupole-induced dipole, a = |- (ajj + 2a|) is the

average polarizability and R is the intermolecular separation. The quadrupole

moment, 0, is defined by 0 = /p,(.) [:

2 1 , 2 ^2 . z - y (x + y ) 47tr^dr where p^(r)

is the charge density in the molecule. The induced dipoles PsCR) and PiCR) are due to an anisotropic overlap interaction and are assumed to be represented by

(A2)

M3(R) =

MjCR) =

(A3)

where |J° is the magnitude at R = a, p is a range parameter and a is a molecular diameter. The induced dipoles PsCR) and Pi(R) arise from terms

with different rotational symmetry such that the former can interfere with

the quadrupole-induced dipole whereas the latter cannot.^ The factor of 2

in (Al) is due to the induction of a dipole in molecule a by molecule b and

vice versa. The strength parameter S is given by

.00 2

S = / g(R) |p(R)| 47tR^dR , ‘'0

(A4)

where g(R) is the radial distribution function.

With the dipole given by (Al) we find

0 aa ^ /30^ 2^j3aQ[i„ "1 0 2 3 0 2 \ + I'(CT/p2) + a (P2)^I”(^/P3) + cr'^CP^) I”(cr/p^)j . (A5)

a

The integrals I are defined by

= / g(x)x ^47Tx^dx

*^0 (A6)
24

I'(cr/p) = J'g(x)x ,

I” (a/p) = f g(x)e ^^*^^^47ix^dx

•'o

If the overlap is sufficiently small, only the first two terms in (A5) will

be significant, namely, the contribution due to the quadrupole- induced

dipole and the term resulting from the interference between this dipole and

MsCR)-

In H2 “He collisions involving the quadrupole and anisotropic-induced

dipoles, the strength factor is

, /3a?0^\ , ^ 2V3ct,e ab , Ig + I

^ ^ab ab

where 0^ is the quadrupole moment of H2 , 0(^ is the polarizability of He,

a , is the molecular diameter for H2 -He collisions and the factor of 2 is

not present because 0^ = 0. In collisions between two different particles,

there is, in addition to the induced dipoles presented previously which

produce rotational-translational transitions, an induced isotropic overlap

dipole which produces only translational transitions. This is given by

ab 3 , 0^^2_„. , . S. = a , (p.) I’ (a , /p ) 1 ab 1 ab ov

(AlO)

when the isotropic overlap dipole is assumed to have the form

(a , -R)/p

,T^^ 0 ab "^ov

p (R) = p.e ^ov 1

(A7)

(A8)

(A9)

(All)
25

REFERENCES

1. G. Birnbaum, J. Quant. Spectrosc. Radiat. Transfer 1^, 51 (1978).

2. G. Birnbaum and E. R. Cohen, Can. J. Phys . 593 (1976).

3. R. Hanel, B. Conrath, M. Flasar, V. Kunde, P. Lowman, W. Maguire,

J. Pearl, J. Pirraglia, R. Samuelson, D. Gautier, P. Gierasch,

S. Kumar and C. Ponnampesuma , Science 204 , 972 (1979).

4. G. Birnbaum, Italian Physical Soc. , E. Fermi International School

of Physics, "Intermolecular Spectroscopy and Dynamical Properties

of Dense Systems," edited by J. Van Kranendonk, Italian Physical

Society, 7^ 111 (1980).

5. B. P. Stoicheff, Can. J. Phys. 35 , 730 (1957).

6. For a discussion of collision-induced absorption in H2 and H2 ~rare

gas mixtures see H. Welsh, Inti. Rev. Science 3, 33, Buttersworths

,

London (1972).

7. J. D. Poll and J. Van Kranendonk, Can. J. Phys

.

189 (1961).

8. J. Van Kranendonk and Z . J,. Kiss, Can. J. Phys

.

37, 1187 (1959).

9. J. Van Kranendonk, Can. J. Phys. 1173 (1968).

10. J. W. Mactaggart and J. L. Hunt, Can. J. Phys

.

65 (1969).

11. P. E. S. Wormer and G. Van Dijk, J. Chem. Phys

.

70, 5695 (1979).

12. G. Birnbaum, G. Bachet, E. R. Cohen and P . Dore (to be published)
NBS>114A (H£v. 9-7<)

U.S. DEPT, OF COMM.

BIBLIOGRAPHIC DATA

SHEET

, PUBLICATION OR REPORT NO. NBSIR 80-2175

2. Gov' L Accession No. 3. Recipteni's Accession No.

4. TITLE AND SUBTITLE

Analysis of the Shape of the Far-Infrared Spectra of

and H^-He Collisions

5. Publication Date April 1981

5. Performing Organization Code

7. AUTHOR(S)

E. Richard Cohen and George Birnbaum

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

10. Project/Task/Work Unit No.

11. Contract/Grant No.

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS rsrreer. City. State. ZIP) 13. Type of Report & Period Coverefl

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

I I Document describes a computer program; SF-185, FIPS Software Summary, is attached.

16. ABSTRACT (A 200-word or leaa (actual summary of moat aignidcant information. II document includes a aignilicant bibliography or

literatute survey, mention it here») The collision-induced far-infrared spectra due to H

2 -H2

and H_-He collisions

have been previously measured in the range 20 to 900cm~^ at 77.4, 194.7 and

292.4 K. These spectra are fitted with a semi-empirical line shape and the

parameters in the shape function are evaluated. The accuracy of these fitting is

discussed. The zeroth and first spectral moments for the isotropic overlap translational spectrum due to H^-He collisions are obtained and give a value for the

range parameter in the induced-dipole function in good agreement with that given

by an ab 'initio calculation.

17. KEY WORDS to twelve entries; alphabetical order; capitalize only the lirst letter of the lirst key word unleea a proper name;

separated by semicolons)

Band shape analysis; collision-induced absorption; far infrared; H

2-H

2

collisions;

H

2

~He collisions; planetary atmospheres; quadrupole- induced absorption; spectral

moments

19. SECURITY CLASS 21. NO. OF

(THIS REPORT) PRINTED PAGES

UNCLASSIFIED 25

20. SECURITY CLASS

(THIS PAGE)

22 . Price 5.00

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I "^1 For Official Distribution. Do Not Release to NTIS

I I Order From Sup. of Doc., U.S. Government Printing Office, V/ashington, DC

20402, SD Stock No. SN003-003-

Order From National Technical Information Service (NTIS), Springfield,

VA. 22161 USCOMM-DC
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