The N2-broadened water vapor absorption line shape and infrared continuum absorption—I. Theoretical development

.[ Quant. Spectrosc. gadiat. Trans[er Vol. 28, No. 2, pp. 81-101, 1982

0022-.4073182/0600~1-21503.00/0 Pergamon Press Ltd.

Printed in Great Britain,

THE N2-BROADENED WATER VAPOR ABSORPTION LINE SHAPE AND INFRARED CONTINUUM ABSORPTION'I'mI. THEORETICAL DEVELOPMENT MICHAEL E. THOMAS* The Johns Hopkins University, Applied Physics Laboratory, Laurel, MD 20707, U.S.A. and ROBERT J. NORDSTROM Battelle ColumbusLaboratories, Columbus, OH 43201, U.S.A.

(Received 17 September 1981) Abstract--Attenuation of infrared radiation in the troposphere is dominated by water vapor absorption. Most past work on the spectroscopy of water vapor has been on the analysis of the rotational and vibrational bands. As a result, a thorough listing of line positions, line strengths, and half-widths exists today. Because of the weak continuum absorption in the water vapor windows centered at 10 and 4 ~,m, efforts to measure and model the continuum have not been as successful as efforts in the analysis of the bands. Yet, a precise understanding of continuum absorption is important for long path energy transmission. For this reason, the study of the water vapor windows has had a long and speculativehistory. The purpose of this study is to demonstrate the importance of far wing phenomena in characterizing H20 continuum absorption. A total line shape for water vapor-nitrogen interactions valid under tropospheric conditions is derived. The complexity of such a model is substantial and many approximations are necessary to obtain a tractable theory. This effort represents one of the initial attempts at a far wingline shape solution of continuum absorption. Applicationof the model to existingexperimentaldata is presented in the following article. INTRODUCTION Strong absorption bands of water vapor and carbon dioxide separate spectral regions of high infrared transmittance within the lower atmosphere into several spectral windows. It has long been known that attenuation of radiation within these transmittance windows is greater than predicted from calculations of local Lorentzian absorption line contributions and from possible aerosol contributions. The residual absorption in excess of the predicted amount has been called continuum absorption since it is a slowly varying function of frequency. This continuum absorption represents a depression in the spectral background from the 100% transmittance level, and has been linked to water vapor content along the absorbing path. Continuum absorption in the 8-14/~m region has been the topic of a great deal of study) -5 Investigations of its behavior as a function of total atmospheric pressure, partial water vapor pressure, and temperature have been made. Similar measurements have been performed in the 3.5-4/~m region, 6'7 although these experiments have been more difficult because the continuum absorption in this region is much weaker than the absorption aroung 10/~m. The exact mechanism of water vapor continuum absorption within a transmittance window has not been understood. Originally it was thought to be the result of wings of very strong water vapor lines located in the absorption bands bordering the transmittance window. However, it was shown 2 that self-broadened Lorentzian absorption line wings were not adequate to model the 10/~m continuum absorption caused by pure water vapor samples. Furthermore, absorption in the wings of Lorentzian lines near 10/~m increases with increasing temperature while experiments show the measured absorption actually decreases with increasing temperature, s'9 Based on these inadequacies of the Lorentz absorption line wing theory to model the water vapor continuum, other mechanisms for the absorption were sought. Infrared absorption by tThis work done under ARO Contract DAAG-29-77-C-0010while both authors were with The Ohio State University, ElectroScience Laboratory, Columbus, OH 43212, U.S.A. ~Author to whom correspondence should be sent. 81

82

MICHAEl E. THOMASand ROBER]J. NORDSTROM

water dimers I°'" or by more complex molecular clusters ~2 have been proposed as possible attenuation mechanisms. It is argued that water vapor aggregates could have unresolved absorption bands throughout the i.r. spectral region and although the fractional concentration of aggregates would be small compared to the concentration of H20 molecules, the absorption could still be large enough to be detected in the transmittance windows. Perhaps the most convincing argument in favor of the attenuation being caused by water vapor aggregates is that this mechanism predicts the observed negative temperature dependence. It is argued that since the attenuation is directly proportional to the number of aggregates present, which in turn is proportional to exp(-AHlkT) where AH is the change in enthalpy on aggregate formation and is negative, the resulting attenuation will exhibit a negative temperature dependence. Furthermore, measurements of the temperature dependence of water vapor continuum absorption in the 8-14/~m spectral region are in good agreement with predictions based on hydrogen-hydrogen bond dissociation rates. ~3 Of course, there is always the possibility that both absorption mechanisms are occurring simultaneously. Measurements of the temperature dependence of continuum absorption at 1203 cm -~ by Montgomery~4 showed that, as the temperature of the water vapor sample was increased from room temperature, the absorption reached a minimum. This behavior was interpreted by Montgomery as showing the competitive effects of dimer absorption and far wing absorption. He suggested that initially the measured absorption decreased as the temperature increased because of dimer dissociation. Above 400K, however, with dimers depleted, far wings of the strong absorption lines of water vapor began to dominate and push the absorption up slowly. Figure I demonstrates that the accumulative effects caused by the overlap of absorption line wings can be significant. These spectra of the 4.3/zm band of CO2 are used here to visualize the continuum-type absorption which extends well beyond the band-head. This absorption is clearly caused by the far wings of resonant absorption lines within the band. To model this continuum absorption properly, the complete shape of the absorption line profile for each CO2 line in the absorption and is needed. The spectrum of water vapor does not have the clear spectral regions as in Fig. 1 in which to demonstrate far wing absorption. Nevertheless, the continuum exhibited in the i.r. spectrum of water vapor must be at least partially influenced by far wings of resonant absorption lines of the main bands. Any thorough continuum model must correctly incorporate the nature of far wing absorption. Only then can the importance of absorption mechanisms based on molecular aggregates be established. In this study, we have begun to investigate the importance of far wing absorption for characterizing the water vapor continuum. An i.r. line shape for pressure-broadened water absorption has been derived which is valid under tropospheric conditions. It was found during the study that the traditional assumptions which lead to the Lorentz line shape could not be used when considering the far wings of water vapor absorption lines. The total, normalized line profile was used to predict the absorption at specific frequencies within the 8-14/zm and 3.5-4/~m transmittance windows. The experimental methods and the results are discussed in detail in another paper. ~5 In this section, we shall discuss the development of the i.r. absorption line profile for water vapor molecules.

THEORY The absorption coefficient, k(v), for attenuation caused by absorption lines can be written

k(v) = ~ Sji(v - voi),

(i)

where & is the line strength of the ith absorption line, and j~ (v - VoOis the line shape function of the ith absorption line centered at frequency v0~.It can be assumed that any differences in the line shapes of different water vapor lines can be expressed parametrically within the same overall function. That is, j~(v - vo~)is better written as j ( v - Vo~;fl~) where/3~ represents a set of parameters which represent the ith absorption line. The line shape function is normalized by

~

2250

'

2275

,, ,

LABORATORY

2300

..............

2325

,,lilt1,11,Ill " " , ,

ELECTROSCIENCE

"

2400

I

O0

2425

I

qli! ¸:

2450

!

BACKGROUND CO 2

CO z M O L E C U L E S

TORR

WAVENUMBERS

/ ~iI

2375

BY

M E T E R PATH

ABSORPTION 300

Fig. 1. Far wing absorption by C02 molecules.

2350

,~;'

FAR WING

~ r 2475

,

I, II !!II ~l !:'

! 2500

.

/

! 2525

7 4 0 TORR CO 2

~¢~;f, bt,mi iJhlllill I1~

.~'~,~

I!lllI ,!III1~1~~

; 2550

L

r-

:3"

o

o

,cI

o"

z

84

MICHAEL. E. THOMAS and ROBERT J. NORDSTROM

the requirement

for all i. Any description of the absorption coefficient, k(v), then can be divided into a line strength factor and a line shape factor. The Hamiltonian The starting point for the development of the spectral line profile is the total Hamiltonian describing the molecular system. In this case, the system consists of a number n, of gas-phase, interacting molecules which can absorb i.r. radiation. Also present are other molecules which can interact with the absorbing molecules via collisions. Thus, the Hamiltonian can be written as

&ot(f) = Ha,+ Hb+ Hp+ Ha,&) + H,(t),

(3)

where H,, is the Hamiltonian for the absorbing molecule, Hb the Hamiltonian for the buffer molecules, HP the Hamiltonian for the photon field, HolP(f) the interaction Hamiltonian between the absorbing molecule and photon field, H,(t) the collision Hamiltonian which describes molecule-molecule interactions. Time-dependent terms are indicated explicity. The collision Hamiltonian will be written to account for absorber-absorber molecule interactions (spectral self-broadening) and to account for absorber-buffer molecule interactions (spectral foreign-broadening). Ha, represents the absorbing molecule. The added subscript 1 indicates the specific molecule considered to be doing the absorption while the n, - 1 identical molecules are considered for the moment to be perturbers. The unperturbed Hamiltonian for the absorbing molecule is

where HaR,,Tdescribes the rotational energy of the molecule, H,,,, the vibrational energy of the molecule, H,,ELthe electronic energy of the molecule, Her, the translation energy of the molecule. The first three terms are fundamental to the study of molecular spectroscopy and to the production of spectral lines. The term HqRNwill not be directly included in this analysis. Motion perturbed line profiles such as the Doppler profile can be derived from the term. In the total Hamiltonian, the term Hb is a many-particle Hamiltonian. This includes foreign molecules and molecules identical to the absorber. It can be written as

Here, n, is the number of absorbing molecules, K represents the number of types of foreign molecules, and nk is the number of molecules of the kth type. Each Hamiltonian is structured in the same way as Eq. (4), i.e., the rotational, vibrational, electronic, and translational energies are represented. The free space photon Hamiltonian is

where a:(q) is the photon creation operator and a,(q) is the photon annihilation operator for a photon of polarization u and wave number q corresponding to the angular frequency o = clql, The interaction Hamiltonian between the absorbing molecule and the photon field is

Ha&) = $$P

* A@,t) + A@,t) . PI +

$z Nr, f) * A(r, t),

The N2-broadened water vapor absorption fine shape and infrared continuum absorption--I

85

where p is the particle momentum operator and the photon field magnetic vector potential operator is F27r/ic 11/2 ir . ] A(r, t) = [ - ~ J la~(q),,,(q) expO(q • r - (at)) + a~(q)E*(q) exp(-i(q • r - (at))j~.

(8)

In this expression, E~(q) is a unit vector perpendicular to the free space photon propagation direction and represents the polarization of the electromagnetic radiation. To first order, the interaction Hamiltonian becomes H a w ( t ) = H aob lp

• exp(-ioJt)+

~ exp0o~t), . Ha,v

(9)

where the A. A term has been dropped. The term H~1b represents the collection of operators and factors which multiply the negative exponential after Eqs. (7) and (8) have been combined. This term represents the absorption of radiation by the molecule, t7 Similarly, the term H ~m alp represents the emission of radiation by the molecule. Finally, H c ( t ) represents the intermolecular potential which couples the absorbing molecule to molecules surrounding it. It is the least understood term in the total Hamiltonian, and yet, it is the most important when attempting to understand the nature of resonant line profiles. A general many-body expression can be written as

He(t)= ~ Ha,~,(t)+ ~--Irn=l Hb*al(t)"

(10)

Each term is a binary interaction of the external fields of the molecule. Again, K is the number of different foreign gases and nk is the number of molecules of the kth gas. Figure 2 illustrates the geometry involved in such an interaction. For electrically neutral molecules (no ionic interaction) with permanent electrostatic potentials, a single Hamiltonian in Eq. (10) takes the form TM Hb.k.~(R(t);

r.,k, rl) = (VEl¢~t~o.t~a~+ VMagnetostatic)Rot (11)

-- VDispersive- VInductive "t" VRepulsiv e.

As shown in Fig. 2, R ( t ) is the intermolecular distance between the absorber and perturber. The quantifies rink and rm locate the charged particles within the molecular coordinates of the foreign and absorber molecules respectively. The magnetostatic potential function is significantly smaller than the electrostatic potential and thus is ignored hereafter.

\ zl

PHOTON

/

~

MOLECULE

I

Yl

MOLECULE

2

xI

'

"~ vt

/

/ X2

Fig. 2. Interaction geometry.

Y2

86

MICHAEt,E. THOMASand ROBERTJ. NORDSTROM

A brief description of the potential terms is necessary. A multipole expansion of the electrostatic potential is WElectrostatic =

VDo(r~k, r 0 + Voq(r~k, r 0 , Voo(r~k,r 0 + V~(rmk, r 0 + . . . R3(t ) R4(t ) -r R~(t )

(12)

where Voo(rmk, rO is the dipole-dipole interaction term, Voo(r,.k, r0 the dipole-quadrupole interaction term, Voo(r~k,r~) the dipole-octopole interaction term, Voo(r~k, r0 the quadrupolequadrupole interaction term. These potentials can be expressed as

Voo(r,.k, r0 = JZmk~lf(O,,k,f~mk,l~l, rink,r), Voo(r.,k, r0 = ~mkOIgl(~mk, ~1, r~k, r) + OmgP'lg2(~,nk, ~'~1, r~k, r), Voo(r.~k, r0 = ~rakRlhl(~'~mk, ~'~1, rink, 7) + Rmkld, lh2(~-~mk, ~t, rink, r), Voo(r,,,k, rl) = Q~kQiq(lq,,k, ol, r,,k, r). The quantity ~.trnkis the dipole moment of the ruth molecule of the kth type and/z~ is the dipole moment of the absorber. Employing the same convention with the subscripts, 0 and /~ represent the mean quadrupole and octopole moments of the molecules, respectively. The symbols f~mk and f~l represent the parameters necessary to establish the orientation of the multipole moment of both the perturber and absorber molecules in a space-fixed coordinate system. These parameters are arguments of expressions f, g, h, and q, which determine the angular dependence of the electrostatic interaction potentials. The molecules are rotating during the collision. The total rotation during interaction will depend on the rotational energy of the molecule and the duration of the collision. The forces produced by the electrostatic potential will tend to orient the molecules in a minimum potential energy configuration. A weighted average over the rotation is performed in order to account for the ability of the absorber molecule to achieve this preferred orientation. The average is f dO VEI exp(- VEI/kT)p(Oo, ta, fmk, f R~

(13)

(Vel)Rot ="

f dl~ exp(-VEJkT)p(f~o, td, f~k, f~) Here, dO is the differential element of solid angle for all possible orientations and p(l~o, td, fRk, fR) is a pulse function with unit amplitude and a pulse duration which depends explictly on the initial orientation f~o, the collision duration time ta, and the rotational frequencies of the perturbing and absorbing molecules fRk and fR, respectively. For collisions with large impact parameters, the collision duration time is very small and (VE)Rot ~ VEi.

(14)

For collisions with small impact parameters, the electrostatic potentials can be so strong that the interaction can no longer be considered to be a small perturbation to the unperturbed Hamiltonian of the absorber molecule. This occurs during close encounters and during the formation of liquids and solids. The ability of the molecule to rotate is inhibited and the absorber molecule takes a relatively fixed orientation. At this extreme, the average over orientations breaks down and in fact the formulation of the unperturbed Hamiltonian as Eq. (4) is no longer valid. Other long-range electrostatic potentials are of interest in molecular collision dynamics. The most important of these are induction and dispersion potentials, both of which vary as R-6(t). Induction potentials represent the interaction of a permanent electric multipole of one molecule with an induced multipole of another molecule. The resulting potential is attractive. Dispersion potentials arise from random motion of electrons about the nuclei generating instantaneous multipoles within the bounds of the uncertainty principle. A discussion of these potentials is given elsewhere TM and approximate forms are 1~

The Nrbroadened water vapor absorption line shape and infrared continuum absorption--I

Vl.duCtiv~= - [/~tim~ + #2k61] R6(t) ,

87

(15a)

t~e2(r2) VDispersive --

(15b)

~-g( t ) '

where 6 is the mean electric polarizability of the molecule, and e is the charge on an electron. The final term in the collision Hamiltonian as written in Eq. (11) is Vaovm,~v¢.During very close encounters between molecules, repulsive forces such as electron cloud interaction can occur. For this study, such potentials were not explicitly considered. THE T R A N S I T I O N RATE

Following the work of Anderson 2° and Tsao and Curnutte,~ the rate for a radiation absorption transition from a lower molecular energy level to an upper molecular energy level separated by E. - E~ = hto0 is

~= ~ R e { f°

d~"exp(-iAtor)Tr[poH :,~S-~'(r)H~,~S~(z)]},

(16a)

where the time development operator for the collision Hamiltonian at time ~"is So(r) = exp ~h y f He(l"') d~".

(16b)

In Eq. (16), Po is the density matrix generated by the initial states and Ao = to- o0 is the amount of off-resonance between the radiation frequency oJ and energy level spacing. This equation can be written in a compact form as

F

= ~ Re{fo dr exp(-iAtor) 1 X

(a,,.I.M.s,,n,~,..ISc- 1 ('r)H °,pSc('r)lal.fIMtslnqoO}. b

ab

b

(17)

In this expression, gt is the degeneracy of the lower level, pdg, = PvibPRotis the density matrix over vibrational and rotational states, Op is the density matrix of the photon field, n.,. and n..t are the photon numbers when the system is in the upper (final) and lower (initial) states, respectively, and Pd is the density matrix over vibrational and rotational states of the broadening molecule. The a quantum numbers represent all quantum numbers other than J and M necessary to describe the upper and lower levels of the absorbing molecule. The terms s b and S~ represent all quantum numbers necessary to describe the lower and upper levels of the broadening molecule. The summations in Eq. (17) are over states and should be written as

I

at, Jr, Ma

s ~, ql, °'I

where q~is the initial wavenumber state of the photon and cr~is the polarization state of the photon. A similar expansion can be written for the upper, final state. We set l = alJiMI and u = aJ,M.. Since H "b is independent of s~ and s~, Eq. (17) becomes alp

F

|~u

2Re[ ~

[ o ,M, P~ 1 quCru EE o,,(in,,od H°b°,,,1" qlcr! aJ.M~

Eb b

s.s I

x f.o d. e-la~'(us:n.~.lS-~t(.)H::pSc(.)lls~nq~,)]

(18)

88

MICHAEL E. THOMAS and ROBERT J. NORDSTROM

Reducing the sums for nontrivial matrix elements. F

p,~

-~rRe

=

I~u

I

X

~. op(ln,~,.lH ,a,,~,,olunqo.,,)~. p.~b u qlczt qu,Cru

-iAw'r

lfo~dre

~"

(us b n q,~ulS - ic ('r)H aba,pSc('r)[ ls b n..i)].

(19)

The subscripts u and I have been dropped from the S b since the Kronecker- 6 in Eq. (18) requires l = u. By inserting unit operators between the time development operators and the interaction Hamiltonian H~p the second matrix element expression in Eq. (19) becomes

' b t t b' ~ ! (us no,.. S,.- t ('7")u ~s b nq,,.)(u s n,~,..Ig ab.,~,[l i s b n.,~l)

E

u,

3

b

n'qm v I h'

x(r s""n;odsA',')lls %o-0.

(20)

Since S~(~') is independent of the photon field and H ~ does not operate on states of the i ! broadening molecule, s b'= s b', nq~t = nq~, and nq~ = nqo-u. The adiabatic assumption21'22 is now made requiring u = u,' S b = s b, and l= l'. By this approximation, the r.h.s, of Eq. (20) becomes

(usblS:l('r)lusb)(unqo-,,IH'~,llnqo-,XlsbJSA'r)llsb).

(21)

Substituting this result into Eq. (19), the transition rate is 27r

1

~

,b

2 Re

,~,,r = ~z~f,j,O,~g,,q,,,., Pol(unq""lH ~pllnq~')l ~ fo dr e ,a~ qu,tru

x ~ p~(usblS-~I(~')lusb)
(22)

Assuming binary collisions only, the matrix elements of the time-development operator S~(~) can be written as

(lsblS,.(r)lls b) = (ls'~lSc,,olls")".-'(ls

b,ls.,,,,,Its %",. . .

... (Is b"ls,,,b,, IIS b,),k,

(23a)

(us"lS,.'(',')lus")- -( u s ° IS -'.... lus ,~)o,1 (us b, Is~o,,,,us -, ,,, >,, . .. -I . . .
).

bk nk

(23b)

In these expressions, terms similar to Sc,~bk represent quantities averaged over impact parameters and summed over collisional events. The sums over broadening states s b in Eq. (22) can be expanded, viz.

P*~(UsblS~I( T)IUSb)(IsbISA T)tls b> = [ ~ P,a((us'qSc2,'~('r)lus'5('s°laco,,,("')l's'~)),¢o,~.,o]

"°

• ..x [~ p,~,((USh'lS,.2,h,(r)]USb*)(ls°'jS,.,,,b,(r)llS"giRo,,.,o]

nb k

(24)

The N2-broadenedwater vaporabsorptionline shapeand infraredcontinuumabsorption--I

89

The symbol ( )Ro,~,~oindicates the average over impact parameters and velocities and a sum over collision events. For arbitrary A, 1 (A)~,~,to = "~

~r

dv

\2kT]

JRMi n

Ro dRo

(25) '

where V is volume, /x the reduced mass of the binary system, T temperature, and k the Boltzmann constant. Also, because no "> 1, the approximation na - 1 ~ na was used in Eq. (24) It is convenient at this point to define Ca,a(r)

= Z

Calbk(T)-~

Ps a

$a

E $bk

((usalS-E2,o(r)lus°)(ls"lS-,.(~r)lls°>)~.o.,o,

o.g(usNSE2,~(r)lus')(lsNS-,~,(r)lts"~)).o.W

(26)

Then,

P,b(usbIS~'(T)IUS~)(lsblS¢(r)IIsb) sb na nI = Ca,.(r)C,,,b,(r)...

ttk C,,,b~(r)

(27)

= C(r).

With this notation, the transition rate is [2~r 1 ~-~

F = ~ p, ~.Te~, ,~, |--*u

qu,O'u

~, ppl
ql,O'l

X {Re 1 fo~ dr e-ia~" C(r)}).

(28)

Without external perturbations except for photon interactions, He(t) and C(T) = 1. Then, Re ~

dr e-iao'C(r) = 8(Ato),

and the transition rate, as expressed in Eq. (28), takes the familiar form of Fermi's golden rule. The r.h.s, of Eq. (28) is the product of two terms. The first term is fnterpreted as the transition strength, while the second term if an off-resonance amplitude correction to the transition strength. This second term then is the frequency profile of the transit otherwise referred to as the line shape. We write j~(Ato) = 1 Re f o dr exp(-iAtor)C(r),

(29)

where the subscript c refers to the collision broadened line shape. THE CORRELATION FUNCTION The quantity C(r), as defined in Eq. (27), is interpreted as the autocorrelation function between the state of the molecular system at time t = 0 and time t = ~'. Just as an interferogram produced by a Michelson interferometer is the autocorrelation function of the incident electric field and is symmetric about the zero retardation point, so the function C(r) is symmetric about r = 0. Using this symmetry, Eq. (29) can be written jR(AtO) = ~

dr exp(- kor) exp(itoor)C(r)

= Re{~-l(exp(koor)C(r))}. Q S R T Vol. 28, No. 2 - - B

(30)

90

MICHAELE. THOMASand ROBERTJ. NORDSTROM

That is, the line shape function iR(Aco) and the function exp(icoor)C(z) are related by the Fourier transform. To generalize the concept further, let j~(Aco) be the Hilbert transform companion to j~(Aco). We define jc(Aco) = ~:-l(exp(icoo)C(?)) = j~(Aco) + ij~ (Aco).

(31)

and j~(AoJ) represents refractive index effects caused by the collision processes. Since the goal of this study is a more thorough understanding of the water vapor absorption line shape, the refractive index terms will be ignored and the superscript R suppressed. Thus jc(Aco) = Re{~: ~(exp(icoot)C(7))}.

(32)

This equation indicates that a study of the absorption line shape depends on an understanding of the correlation function C(~). An intuitive understanding of the relation between the spectral line shape and the correlation function is shown in Fig. 3. Long correlation times ~- imply that the binary collision perturbation was weak and did not greatly upset the energy levels of the absorber. Thus, appreciable absorption caused by a transition from the lower energy level to the upper energy level will occur at frequencies co close to the unperturbed center frequency coo. In the spectrum, the absorption line is narrow. If, however, the correlation time for the energy levels is short, a strong interaction between absorber and perturber molecule occurred. This can cause the absorber molecule to absorb radiation at frequencies co which are far from coo.The spectral line appears broad. Of course, in an experiment all ranges of correlation times are experienced by the ensemble of absorber molecules. The total line shape is a composite resulting from the different ranges of correlation functions.

The Lorentz line shape It is reasonable, but not entirely correct, to assume that the correlation function is a monotonically decreasing function of positive ?. That is, the molecular system tends to become

Hbmkol(t)

FAR INTERMEDIATE WING ! WING NEAR LINE CENTER R

i.. I I

/i )

I

/

r I

C(T) i

,/

I

I

, r

I

jc ((~)

I I I I =

I

w0

Fig. 3. Comparison among the intermolecular potential, the correlation function, and the spectral line shape.

The N2-broadened water vapor absorption line shape and infrared continuum absorption--I

91

(D

NEAR LINE / CENTER REGION....~

"

Fig. 4. Possible collision trajectories.

less correlated as time proceeds. If this monotonic decay can be written as a simple exponential of the form c(z)

=

exp(-ar)

(33)

From Eq. (32), the line shape becomes jc(zl~o) =

1

Ot

= ~

(34)

which is the familiar Lorentz line shape with a halfwidth a. The Lorentz line shape is an adequate model for the absorption line profile only when the correlation function can be expressed as an exponentially decaying function. It will be shown later that this assumption can be applied to the water vapor line shape only under certain conditions. For the most part, the profile of water vapor absorption lines is decidedly non-Lorentzian.

Collision dynamics For this study, it was decided that the best approach for establishing an absorption line profile was to classify the correlation functions into three general categories corresponding to short, intermediate, and long correlation times. By a previous discussion, this classification Table 1. Characteristics of collision dynamics. R > Ra R a > R > Rb

Very little effect on absorbing molecule Interruption broadening near line center phenomena i.

Brief duration of collision

2.

Time between collisions is long compared to

3.

Binary collisions

4.

Long correlation time

d u r a t i o n of collisions

Rb > R > R c

Intermediate r e g i o n - - d l f f i c u l t to model

Rc > R

Statistical b r o a d e n l n g - - f a r wing

1.

i. 2. 3.

Many body problems

Binary collisions 2R D u r a t i o n of collision = - -v Short correlation time

MICHAELE. THOMASand ROBERTJ. NORDSTROM

92

divides the absorption line profile into three regimes corresponding to far wing, intermediate, and near line center domains. Figure 4 illustrates the division of the collision dynamics implied by this approach. Trajectories labeled 1 and 2 represent possible collisional paths. Table 1 lists the general characteristics of collisions in each region. These conditions must be applied to the correlation function in order to extract the proper form of the function to be applied in each region. This division implies jc(A¢o) = Re{~:-1 (exp(iwo'r)CFw(r))+ ~ l(exp(itoor)Clw(r))

+ J;-l(exp(iwor)CNLc(,r))} =

j~Fw(Ato)+ jaw(Ato) + j,.NLc(Aw)

(35)

The term jc~w(&o) describes the far wing collision line shape and is derived from correlation functions with short correlation times. The contributions to the near-line-center shape, jcNLc(Ao~), come from all collisional processes, but are dominated by correlation functions with long correlation times. The intermediate wing region jdw(&O) represents a transition region where correlation functions with both long and short correlation times compete equally. Rather than solving for the line shape function in this region, boundary conditions that the total line shape function and the derivative of the function must be continuous across the domains will be applied. Solutions to the near-line-center problem and the far wing problem can be formulated. The intermediate-wing region, however, is not solvable by direct methods. The collision Hamiltonians for H20-H20 and H20-broadener binary interactions become

/ kl'~f +/,/'101gl + ~101g2 H~., = \R-'J~-i

R4(t)

) + " " --Rot

_ 2tx~6~l+ (l/2)61e2(r~) R6(t)

+" ' "

_/l~,QEg + . . . ) 2~t~62+ (l/2)dle2(r~) Hba, - \ ~ -RotR6(t)

(36a)

(36b)

The terms have been described earlier and the subscripts 1 and 2 represent water vapor and broadener molecule, respectively. The short collision times characteristic of interruption broadening in Table l allow the approximation of Eq. (14) to be made for the near line center problem. Retaining only the leading terms in each Hamiltonian above is equivalent to assuming that the HEO-H20 interaction is a dipole-dipole interaction, while the H20-N2 interaction is a dipole-quadrupole interaction. The limiting case of weak collisions can be modeled with the Hamiltonians Ha,~ = ~ - ~

(37a)

_ ~tQ2 Hb., - R - ~ g.

(37b)

Strong collisions which contribute to the far wings of absorption lines may or may not require evaluation of the rotational average in Eqs. (36a) and (36b). The importance of the rotational average can be evaluated by examining the collision time and the rotational time for the molecules. Choosing the broadening molecules to be N2, an estimate for the collision duration can be made, based on the average thermal velocities and impact parameters of strong collisions for both H:O--H20 and H20-N2 binary collisions. The results show the collision duration to be

10-12 sec

H20-H20

2Ro td = ~ 10 13-10-~2sec H20-N 2.

The N2-broadened water vapor absorption line shape and infrared continuum absorption--I

93

Using a rigid rotor picture of the molecules, typical rotation periods for the N2 molecule are between 10-1° and 10-u see, while for the H20 molecule the rotation periods are between 10 -II and 10-13 sec. These values can be derived from a purely classical view of the molecule using angular momentum values which are derived from representative values of the I quantum number (J ~ 5-20). Since the collision times for H20--N2 are for the most part short compared to the rotational period of the N2 molecule, a "snapshot" view of the interaction can be adopted. That is the N2 broadener molecule does not rotate significantly during the interaction and therefore the rotational average is not performed. Similar results are derived when considering H20-O2 collisions. However, for H20-H20 collisions, the colliding molecule can rotate many times during the collision, making the rotational average important. For far wing absorption line shapes, the binary interaction Hamiltonians become 2

Ha,~=(R--~(t)f>Rot-2p'~fq+(1/2,~'e2(r~) R6(t)

(38a)

~102 Hb=, R--~.

(38b)

=

The average over rotation as defined in Eq. (13) can be performed by recognizing the analogy between this and orientation polarizability calculations.23 2

(39) \R

(t)

/Rot

where L(x) is the Langevin function and x = 2/~2/[kTR3(t)], where T is temperature. Figure 5 shows a plot of this function. If x is small, L(x) .~-x/3, and the rotational average becomes

2

2. 4 1,

(40,

which is similar to results given by Eisenberg and Kauzmann. 19The next term in the expansion of L(x) is of order x 3. This produces a term in the Hamiltonian or order R-12(t). Terms of this order become important during very close collisions, but their influence drops rapidly as R(t) increases. For this study, such terms are not being considered. Thus, from Eq. (38a), the self-broadening influence in the far wings of water vapor absorption lines is controlled to first order by a R-6(t) potential. This differs from the self-broadened nature of the near-line-center Hamiltonian which varied as R-3(t) for large R(t). The foreign-broadening potentials for near-line-center and far wing cases both vary as R-4(t).

1,0

/ 0.8

I

Ia

// I

-Ix I

0.6

/ I

c

0.4

/

ii

,j

0.2 0 0

_ I

L

I

2 x =

I

l

3 4 2 utz kTR3 ( t )

I

5

Fig. 5. Langevin functions.

6

94

MICHAELE. THOMASand ROBERTJ. NORDSTROM

Interruption broadening Table 1 lists the conditions for interruption or impact broadening. These conditions can be applied to a general correlation function Ca,b(r) as defined in Eq. (26). Because interaction time is brief under this assumption, the correlation function can be re-examined after a small change in time.

+ dr) =

ps ((us lSd, (r + dr)lusb)(ls"l&.,b(r

+

dr)Its b))Ro.V,,o

(41)

The time-development operator Sc,~b(r + dr) can be written as a time development operator which evolves the system to time r operated on by a time-development operator which evolves the system from time z to r + dr. That is

Sc.,b(r + dr) = Sc.,b(r-) r + dr)S~,s(r)

(42)

With this, Eq. (41) becomes Ca,b(r +

dr) = ~

p,b((usblS~,b(r ~ r + dr)lusb)(ls~lSc,,,b(T-.> r + dr)llsb>)Ro,~.,o

x <
o.

(43)

To obtain separate averages, it was assumed that events at time r and time r + d r are independent. 21 Within the limits of the impact approximation, the second average is independent of the sum over quantum numbers of the broadening molecule and in fact is equal to the correlation function C,,b(r) at time r. Thus, Eq. (43) can be written

C,,b(r + dr) = C,,b(r)c~(r -->z + dr),

(44)

where the function ~b(r ~ r + dr) simplifies the notation of the remaining terms in Eq. (43), and is dependent only on dr. A solution to Eq. (44) can be written

C~,b(r) = C~,b(O) exp(-7~,bz),

(45)

Ya,b = < 1 - - ~ psb(UsblS~,b(r"->r +dr)lusb)(lsb[S~,,b(r~r + dr)llsO)Ro~>.

(46)

where 3'~,b can be shown to be

The initial condition C~,b(O)= 1 can be applied, and the total correlation function for a binary mixture can be written by Eq. (27) as na nb C(r) = C~,~(r)C~,b(r)

= exp(-n~y~z) exp(-nbT~,br).

(47)

Since 7~1~and 7~,b are quantities averaged by the process described in Eq. (25), the I/V factor can be brought out explicitly to write C(r) = exp[-(p,%,, +

pbTa,b)r]

(48)

where pa and Pb are the molecular number densities of the absorber and perturber gases respectively. By using this correlation function in Eq. (30), the interruption broadened line shape is 1

J~NLC(At°)= -7T ( A ~

a

R

"+"0~1)2+

(otR)2

(49)

The N2-broadenedwatervaporabsorptionline shapeand infraredcontinuumabsorption--I

95

where aR is the real part of Pa3%, + PbY~,b and a t is the imaginary part of the same function. Thus, the function which describes the near line center profile is the Lorentz line shape. The term a ~ represents a line shift caused by the impact approximation collisions, while a R represents the halfwidth of the absorption line. The derivation of Eq. (49) for the long correlation times implied by the impact approximation did not explicitly depend on the functional forms of the self-broadened and foreign-broadened interaction Hamiltonians written in Eqs. (37a) and (37b). Instead, these interaction potentials influence the nature of % : and %1~ respectively within the derived line shape. The pressure dependence of the line width and line shift terms in Eq. (49) are already apparent since the molecular number densities p, and Pb are proportional to pressure by the ideal gas law. a R

= Alpo + A2Pb

I

ot = BIpa + B2pb

(50)

where A~, A2, Bl, and 82 are constants related to 7~a and 7~1b,P~ is the partial pressure of the absorber and Pb is the partial pressure of the broadener. The temperature dependences of the line width and line shift are more difficult to derive. Here, the explicit forms for the interaction Hamiltonians in Eqs. (37a) and (37b) become important. A second order perturbation expansion of the % : and %1b2°'2t yields a temperature dependence of the form 1 a ~ :~Ty"

(51)

However, this is not always observed.24'25 The approach which has been adopted here is the phase shift method,26'27which consists of only phase modulation of the eigenstates, but to all orders. This approach leads to a temperature dependence for the linewidth of 1 ~y

(52)

for the dipole-dipole interactions modeling self-broadening and 1

= -y~

(53)

for dipole--quadrupole interactions modeling foreign-broadening. Statistical broadening

Statistical broadening is characterized by short correlation functions caused by strong binary interactions. These interactions contribute to the far wing absorption of a spectral line profile. No perturbation method can be applied. However, the phase shift contributions are solvable. In addition, corrections will be included to account for energy level shifts. The time-development operator Sca~(¢) can be written Sca,b(z) = exp(-iB(r))

(54)

where

] ft tO+¢ "q(T) = ~

o

dt Halb(t)

(55)

96

MICHAEL E. THOMASand ROBERTJ. NORDSTROM

and

Ho,b(t) = V/(R(t)) mj.

(56)

The parameter m~ is the power of the exponent in the Hamiltonians of Eqs. (38a) and (38b). For H20-H20 interactions mj = 6, while for H20-N2 or H20-O2, m~ = 4. The correlation function initially written in Eq. (26) can now be expressed

C'~b('r) = ~ O,~((USbIexp(i~(v))IUS")(IsbIexp(--i*I(T))Ilsh))Ro,~,,o] "~

(57)

This can be written

C,,~b(~')"b =

I1 +Pb~ P,o((us~lexp(i'o(~'))lus")(lSbln~exp(-iTl(~'))[Isb)-1)k°'~'~ 1 "b

(58)

where ( )~,~,to ~ l/v( )ko,~,~and Pb = ndV. The fact that nb is very large permits the correlation function to be written2s

C:~b(~') = e x p [ - Pb ~ p~b(1--(USb[ exp(i~](r))lUSb)(lsb I exp(-i~?(z))llsb))b~,o.to]

(59)

The matrix elements in Eq. (59) can be manipulated to yield the form

(usblei'[usb)(ls~'le-i'lls b) = e"n.-.,, + ei'd~t + ei',fl. + ~ ,

(60)

where

7. = (us'l~lus'), ~,

= (ls"l~lts %

iq. = °==---~-.~ ~ (-i)" , (us bIwlu,s b~)..-(u'"-"s~("-%lus% l(n-1)s b(n-l) and

n,-- ~2 ~' (-i)° "! Y ~'(l,bl~l,,~b)" • (r" I'sb("-%llsb) l(n-l)sb(n--I) The prime on the summation sign indicates that all the indices can not be equal. The parameters l~u,i represent level shifts; they have no closed form expression. Let, /b(r) = e i'"II~ + e in'llu + ll~[~t.

(61)

Now the correlation function becomes Ca,b(r) ' "~ = exp(-Pb ~ P,b((1 - e'('~-n'))sbRo,,~,to+fb('r))~,~.Q)

(62)

The phase shift term (1 - ..,~-,0\,/Ro,~,to.... .~.. be solved. The major contribution to the phase term

The N2-broadened water vapor absorption line shape and infrared continuum absorption--I

97

comes in the region to < Ro/V29where v is the velocity of the colliding molecule. The sum over to implied by the bracket in Eq. (62) as defined in Eq. (25) is converted to an integral which gives its major contribution between -Ro/v and Ro/v. In this region ~r~x ~

(xs"l Vslxsb)r(ms

-

2

1) (22_,.0,

mj-even

(63)

where x represents the subscripts l and u. The integral over to gives (1 -

dto(1 - e i<~-~))

eit~-~l))Ro,v, fo =

or

(' - °"'~"-'~'~'

-

/2Ro

(1 - e""-"b)'

(64)

Evaluating the real and imaginary parts separately we have

'

4zr{T,~3/,,,

,3./,.,F{__~j)cos(~)mreve n (65)

where vj,, = (usblVjlusb> vj, = F(mj- 1)22-mj

Similarly for the line shift term (2Ro sin(.0u- ~t))~.~ =' _

~jr 3/,.~=4~" (~)3/"'(Vj,_ mj

VJ')3t"~/'JF(-3/mj)"s i n _ -(]2mj31r mj-even -. (66)

The general correlation function from Eq. (62) can now be written,

Separating real and imaginary terms in the exponent,

When a correlation function of this form is substituted into Eq. (32), the expression in the second exponential combines with the OJo to produce a line shift term in the final line shape jc(Ao). It is the amplitude of the correlation function which yields the line profile nb IC~lb(l-)[ = exp[--Abjgbj(z)¢3/mq

(69)

9~

MICHAEI E. THOMAS and ROBERT J. NORDSTROM

where the substitutions hbi = Oh ~ P~ 7j

(70a)

and

gbj(r)=

(70b)

1 + Re[(fb(Z))P~w'h'] Yi'r 3tini

were made. Note that gbi(r) does not depend on pressure but on temperature and the quantum numbers of the interacting molecules• The general correlation function for a binary mixture is then

Cvw('C)= exp[

-

hbigbi{T)'r 3/mj] exp[ - hakgak(Y)T 3link]

(71)

and the far wing line shape is

jc~w(A~o)=l Re{fo~dr exp[- iAo)rlCvw(r)}.

(72)

Power series expansions of the exponentials in Eq. (71) yields a far wing line shape of the form

JcFW(AW)= 1rr Re~'~ '~ A~k~ f~ dz[exp(-iAwr)][- g.k(r)z3/mq'[ gbi(r)r3/"'] ' } [,'=--o¢=--os [ t! Jo -

(73)

Making the substitution 0 = [Atotz, the far wing line shape becomes

jcvw(At°) =~

f--==os! t! IAoJ[, + 3---s+ 3t mk

x Re[fo x dO exp(-i0)[

mj

0

s

(74)

The leading terms in this expression occur for s = 1, t = 0 and for s = 0, t = 1 since the single term where both s and t are zero is zero. These two leading terms have the physical interpretation of being the self-broadened and foreign-broadened contributions to the far wing line shape. If we retain only these leading terms, • 1 Jcvw(hw)= ;{~

h.k

Re[fo~dOexp(-iO)[- gak([-~])]O3/"'] bi

(75)

Evaluation of the integrals requires substitution of Eqs. (61) and (70b). In general, the integral expressions are not tractable. When -gbj and -g,k equal one, no level shifting contribution exists and the integrals become tractable. Therefore, to properly normalize the integral quantities as level shifting contributions, they are divided by

Re[fo~dOexp(-iO)O3/m~]=Re[i-t'+3/mor(1 + 3 ) ]

(76a)

Re[fo~dOexp(-iO)Oa/mJ]=Re[i-~I+3/'0F(1 + 3 ) ]

(76b)

and

The N2-broadened water vapor absorption line shape and infrared continuum absorption--I

99

where appropriate. Therefore, the far wing line shape becomes

jcr,v(A(o) = 1

Aa~(Re[i-tl+31"k)F(1 + 3/mk)])y,.~(mk, A(o)

lao, l'+3m~

+

Ab~[Re(i-(l+31"PF(1 + 3[mj)]) iAtol,+3,m, Yr~i(mj, Ato)

(77)

where mk= 6 for self-broadening and mj = 4 for foreign-broadening by N2 or 02, and the Y-functions are the normalized level shift integrals. The statistical line shape represented by Eq. (77) is similar to the results presented by Fomin and Tvorogov) ° In their development, the term 1

is multiplied by an exponential factor, which represents an asymptotic solution of the Yfunctions in Eq. (77), of the general form

{exp(--Ia~lm-"m)z13}. For self-broadening, this factor in their line shape becomes exp(-lA(o]°56), while for foreignbroadening this factor is exp(- IAtol° .

The p-Iunction In the previous two sections, interruption broadening and statistical broadening were presented. The total line shape is constructed by combining these two resulting line shapes. We have chosen to do this by the application of band-pass and band-stop filter functions to the two derived line shapes. The total line shape is written jc(A I,') = NUcNcL(AV)p(v)+ jcFw(Ap)(1 -- p(v))]

(78)

where we have made a change of variable from frequency to (measured radian/sec) to frequency v (measured in cm-I). The normalization constant N assures that Eq. (2) is satisfied. The function p(v) is a normalized band-pass filter function centered at Vo. Guidelines for choosing a suitable p-function are given in Table 2. Based on these guidelines, the p-function was chosen to be IAvl < 5 cm -~ (79) IA~I ~ 5 cm -~

which is shown in Fig. 6. Table 2. p-Function guidelines. I.

The p-function must be equal to I at V=Vo.

2.

The p-function must be zero for all ~ greater than a cut-off frequency Va'

3.

The p-function must be a monotonically decreasing function in the region ~o C~ CVa"

4.

The p-function must be symmetric about ~ . o

5.

The derivative of the function must be zero at % and at

6.

I t is desirable for the total line shape as expressed by Equation (78) to be easily integrated to evaluate the normalization constant N.

=~a"

100

MICHAELE. THOMASand ROBERTJ. NORDSTROM

I.O

CL

0.5

O

I I

2

3

4

5

6

•'XV ( crn-J ) Fig. 6. p(v) vs Av.

The cut-off at 5 cm -t from the line center was chosen so that the p-function would remain essentially unity within several half-widths of the water vapor line center. This allows interruption broadening to dominate near line center. The stop-band filter function (1- p(v)) allows the statistical line shape to dominate beyond 5 cm -1 (50-100 halfwidths from line center). The region where p(v) and 1 - p(v) are approximately equivalent is the intermediate wing region. This method permits a smooth blending of the interruption broadened and statistical broadened line shapes, and eliminates the necessity for solving or approximating the intermediate wing problem. The exact form of the p-function and the exact choice of cut-off frequency is not very important to the study of the water vapor continuum problem in the i.r. The reason for this is that the continuum absorption is theorized here to be caused by the overlap of the far wings of the strong water vapor absorption lines. The intermediate wing region and near-line-center region enter the problem only indirectly in that they must be known to normalize the total line sphere function properly. Furthermore, the p-function formalism permits the blending of the different pressure and temperature dependences of the interruption broadened and statistical broadened line shapes as will be seen in the following paper) 5 CONCLUDING REMARKS A total line shape, valid for water vapor absorption lines throughout the i.r. has been derived by proper consideration of both impact and statistical collisional processes. Self-broadening is assumed to occur through the electric dipole-dipole interaction, while foreign-broadening is assumed to occur through the electric dipole-quadrupole interaction. Thus, the derived shape is suited for atmospheric (N2 or 02) broadening of H20 absorption lines. Several approximations were made in the process of deriving a proper infrared total line shape for H20 molecules. The following are the major approximations: (!) binary collisions, (2) adiabaticity, (3) retention of only leading terms in the intermolecular potential multipole expansion, (4) the collision Hamiltonian is small compared to the molecular energies, (5) the total line shape can be constructed by normalizing a smooth blending of the near line center and far wing parts. The first four assumptions are standard in many theories. The third and fourth assumptions have the greatest consequence for a realistic theory. Higher order terms in the interaction Hamiltonian can be important in the very far wing. Also, molecular rotational energies, in some cases, are smaller than the high collision energy which generates the far wing. In these cases, the time dependent perturbation scheme developed in this paper breaks down. The last approximation (the p-function concept) permits the practical application of the model. It recognizes that the line shape is a smooth and continuous function and, therefore, attempts to smoothly combine the near-line-center and far wing parts of the total line shape. To do this, the asymptotic solution of the far wing line shape as given by Equation (77) must have the singularity removed at Aw = 0. This is done by simply using the normalized line shape of Varanasi et al., 1° for the phase shift part of the statistical line shape. For large values of Ato, Eq. (77) is obtained, and for Ato = 0 the profile is bounded. Now, the p-function as prescribed by

The N2-broadenedwater vapor absorption line shape and infrared continuum absorption--I

101

Eq. (79) cuts off the statistical line shape around A~o = 0, where it is artificial a n y w a y , and allows the L o r e n t z line shape to dominate as determined b y Eq. (78). In all practical applications thus far, the normalization c o n s t a n t N is essentially 1. This m e a n s the normalized L o r e n t z line shape is unscaled near line center as theory and experiment indicate it should be. This fact in some sense verifies the total line shape construction procedure. The details of this construction are presented in the next paper along with the ability of this line shape to predict c o n t i n u u m absorption in the 8-12 # m and in the 3-4 # m atmospheric transmittance windows. REFERENCES 1. K. Bignell, F. Saiedy, and P. A. Sheppard, L Opt. Soc. Am. 53, 466 (1963). 2. D. E. Burch, Aeronutronic Publication No. U-4784, "Semi-AnnualTechnical Report", Air Force Cambridge Research Laboratories, Contract No. F19628-69-C-0263,Jan. 1970. 3. J. H. McCoy, D. B. Rensch, and R. K. Long, Appl. Opt. g, 1471 (1969). 4. R. J. Nordstrom, M. E. Thomas, J. C. Peterson, E. K. Damon, and R. K. Long, Appl. Opt. 17, 2724 (1978). 5. J. C. Peterson, M. E. Thomas. R. J. Nordstrom, E. K. Damon, and R. K. Long, Appl. Opt. 18, 834 (1979). 6. W. R. Watkins, K. O. White, L. R. Bower, and B. Z. Sojka, "Pressure Dependence of the Water Vapor Continuum Absorption in the 3.5 to 4.0 Micrometer Region", U.S. Army Electronics Research and Development Command, Rep. No. ASL-TR-O017, Sept. 1978. 7. F. S. Mills, "Absorption of Deuterium Fluoride Laser Radiation by the Atmosphere", Ph.D. Dissertation, The Ohio State University, June 1978. 8. J. C. Peterson, "A Study of Water Vapor Absorption at CO2 Laser.Frequencies Using a DifferentialSpectrophone and White Cell", Ph.D. Dissertation, The Ohio State University, June 1978. 9. V. N. Aref'ev and V. I. Dianov-Klokov, Opt. Spectrosc. 42, 488 (1977). 10. P. S. Varanasi, S. Chou, and S. S. Penner, JQSRT 8, 1537 (1968). 11. S. H. Such, J. L. Kassner, Jr., and Y. Yamaguchi, Appl. Opt. lg, 2609 (1979). 12. H. R. Carlon, "Final Report: Infrared Absorption by Water Clusters", Chemical Systems Laboratory, Rep. No. ARCSL-TR-79013, March 1979. 13. R. E. Roberts, J. A. Selby, and L. M. Biberman, Appl. Opt. 15, 2085 (1976). 14. G. P. Montgomery,Jr., Appl. Opt. 17, 2299 (1978). 15. M. E. Thomas and R. J. Nordstrom, "A Synergistic Investigation of the Infrared Water Vapor Continuum: Section IIr', Rep. 784701-7, April 1981, The Ohio State University ElectroScience Laboratory, Department of Electrical Engineering;prepared under Contract DAAG29-77-C-0010for U.S. Army Research Office. 16. G. Baym, Lectures on Quantum Mechanics. Benjamin, San Francisco (1977). 17. M. E. Thomas, "Tropospheric Water Vapor Absorption in the Infrared Window Regions", Rep. 784701-5,Aug. 1979, The Ohio State University ElectroScience Laboratory, Department of Electrical Engineering;prepared under Contract DAAG-29-77-C-0010for U.S. Army Research Office. 18. H. Margenau and N. R. Kestner, Theory of lntermolecular Forces. Pergamon Press, Oxford (1969). 19. D. Eisenberg and W. Kanzmann, The Structure and Properties of Water. Oxford University Press (1969). 20. P. W. Anderson, Phys. Rev. 76, 647 (1949). 21. C. J. Tsao and B. Curnutte, JQSRT 2, 41 (1962). 22. M. Barranger, Atomic and Molecular Processes (Edited by D. R. Bates), Chap. 13. Academic Press, New York (1962). 23. C. Kittel, Introduction to Solid State Physics, 4th Edn. Wiley, New York (1971). 24. E. M. Deutchman and R. F. Calfee, "Two Computer Programs to Produce Theoretical Absorption Spectra of Water Vapor and Carbon Dioxide", ADgl6 369, April 1967. 25. P. Varanasi and F. K. Ko, "Intensity Measurements in the 2 ~m CO2 Bands at Low Temperatures", presented at 33rd Syrup. on Molecular Spectroscopy, The Ohio State University, 1978. 26. M. Mizushima,Phys. Rev. 83, 1061 (1951). 27. H. M. Foley, Phys. Rev. 69, 616 (1946). 28. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, App. Math Series No. 55. 29. C. V. Heer, Statistical Mechanic, Kinetic Theory, and Stochastic Processes, Chap. 11. AcademicPress, New York (1972). 30. V. V. Fomin and S. D. Tvorogov, Appl. Opt. 12, 584 (1973).