MODERN QUANTUM KINETIC THEORY AND SPECTRAL LINE SHAPES

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LOUIS MONCHICK
MODERN QUANTUM KINETIC THEORY
AND SPECTRAL LINE SHAPES
The modern quantum kinetic theory of spectral line hapes i outlined and a typical calculation of a Raman scattered line hape described. The distingui hing feature of thi calculation i that it was completely
ab initio and therefore con tituted a test of modern quantum kinetic theor ,the tate of the art in computing
molecular-scattering cross sections, and novel method of olving kinetic equation. The computation employed a large assortment of tools: group theory finite-element method. cIa ic method of solving coupled sets of ordinary differential equations, graph method of combining angular momenta, and matrix
methods of solving integral equations. Agreement with experimental re ult wa excellent.
THE PROBLEM
Almo t with the inception of quantum mechanics,
theorists-mostly nuclear and atomic physici t -had
developed the general scattering equations neces ary to
describe collision between fundamental particles. With
almost no exception, however, only a few angular
momentum and spin states were invol ed; as a re ult,
these calculations were not computationally intensive. In
principle, the general methodology was applicable to the
calculation of molecular inela tic and reactive colli ion,
which are important ingredients of reaction rates. tran port properties, and, as in the present case, line broadening. Calculation of these cro s sections, however,
requires inclusion of many rotation, spin, and orbital
angular momentum tates a well as several vibrational
and electronic state. The number of required states
increases linearly and sometime quadratically with the
molecular weight and complexity of the molecule; the
result is that problem involving molecular colli ion
were forced to await the advent of modern computers
before they could be attacked. Even then the door had
been opened only a chink, becau e the peed of machine
had increased linearly with time, but the ize and duration of a computation had increased a the cube of the
number of tate; therefore, despite the advance in
kinetic and scattering theory, only rather approximate
method have been applied to most real problems. The e
methods, often elegant and ingenious and alway pIau ible, nevertheless had limited applicability and omehow
made simplifying a umption about the fundamental
physics of the proce s.
At present, only systems composed of hydrogen (H 2 ) ,
deuterium (D 2) , and helium (He) can be attacked with a
minimum of ad hoc as umptions. As a re ult, the e are
the only calculation that can be compared directly with
experimental results and thus serve as benchmark for
simpler approximations. The calculation presented here
will be confined to a He-D2 sy tern becau e of the availability of extremely accurate experimental data for com246
parison. 1 the a ailability of an extremely good model of
the interaction of helium ith two deuterium atoms, and
the feasibilit of carrying out the calculation in a finite
time and at a rea onable co t. The goal which we did not
quite reach. a an entirely ab initio computation containing a umption only about the rate of convergence
of the numerical procedure u ed, ith no e peri mental
quantitie except Planck' and Boltzmann ' con tant
and the electronic charg and mas of the electron, deuteron , and helium nucleu . A de cribed in the following
ections, practicalit forced u to compromi e for an approximation; judging from the outcome. thi approximation wa quite good. The ne t ection pre ent a preliminary qualitati e de cription of the theor of line shapes,
followed by a brief account of the kinetic theory of line
hape and then a de cription of the calculation.
PRELIM I ARY REMARKS
Spectra, the interaction of light and matter, have been
and till are a major tool for tud ing electronic structure of atom and molecule. Loo ely speaking, this
ubject might be characterized a the tudy of intramolecular force . becau e energ level are determined b a balance bet een kinetic energy and coulombic interaction. More recentl . the realization that
colli ion influence width of pectral line hape ha led
to the u e of pectra to mea ure intermolecular force .
which are manife ted in e eral contra ting mechani m
that affect the width and hape of pectral line . An
i olated atom or a molecule in an excited tate i
meta table and, if it doe not tran fer it energ to omething el e fir t, radiate it extra energy in the form of a
photon. The energ of the photon will not corre pond e actly to the difference in energ Ie el of the excited and
ground tate, t1E = Ef - Ei == liwif. where Ii i Planck'
con tant di ided b 27r and wif i the frequenc of the
transition, but it will be di tributed 0 er a range of ener10hns Hopkins APL Technical
Di~ e5( .
l oilime
n,
limber 3 (/991)
gies. The plot of the intensity of the emitted light versus
the energy of the photon,
called the line shape, is
more or less a Lorentzian
nw,
(1)
where 7 is the e-folding lifetime of the excited state and
withe angular frequency. It can be seen that the width,
~w = IWI /2 - Wil l, at which the Lorentzian is one-half the
maximum, occurs at ~w = 7, which is just what one
would expect from the Heisenberg uncertainty principle.
This is called the natural width of a spectral line because,
absent any external influences (collisions, fields, measuring apparatus) or motion of the radiating molecule,
this width is what would be observed.
In the real world, a molecule is never isolated. It has
frequent collisions with other molecules, less frequent
ones with the walls, and occasional interactions with
externally imposed fields . Consequently, the state,
particularly the velocity, of an individual radiating molecule is not known exactly, but it is distributed according
to some probability rule, usually taken to be the
Maxwell-Boltzmann distribution law
changes shape. This phenomenon, called Dicke narrowing, has been observed many times since it was first
predicted. 2
What has been neglected up to this point is that every
collision introduces an uncertai.nty ·in the phase of the
radiator, which may be reinterpreted as an uncertainty in
energy, leading to a widening of the line shape. At high
energies and increasing collision frequencies , this effect
becomes more important than . Dicke narrowing and
produces the pressure broadening generally seen at moderate and high pressures. As is. well known, the "drunken
man's walk" is a diffusion mechanism' that measures the
average momentum transferred per collision. This value,
in turn, is a function of the 'average componen't of force
along the initial direction 'of relat.ive motion 'of two
molecules? As an empirical rule of thumb, this process
is determined mainly by elastic collisions and the orientation-averaged intermolecular potential. Line broadening, on the other hand, can be a sensitive function of inelastic and reorientating collisions that can occur only if
the potential has orientation-dependent components. Observation of a process dependent on the presence of
orientation-dependent components is 'what interests
chemical physicists.
THE KINETIC THEORY OF LINE SHAPES
(2)
where
a is the aggregate quantum number or the set of all
the quantum numbers describing all the possible
degrees of freedom ,
Ecx is the energy of this state, including kinetic and
thermal,
k is Boltzmann 's constant,
T is the absolute temperature,
f~O) is the probability of the molecule occupying that
state, and
'J{-I is a normalization constant.
The kinetic energy component of Ecx is mv 2, where m is
mass and v is velocity, which means that the radiating
molecule is seldom at rest relative to an observer, and
therefore every possible value of velocity along the line
of sight shifts the frequency of the radiated line by the
well-known Doppler shift formula. The center of gravity
of each corresponding Lorentzian is shifted as well; thus,
the observed line shape must be the resultant of all possible shifted natural line shapes. The result is the familiar
Doppler line shape, essentially a Gaussian.
Doppler line shapes are seen only at pressures low
enough that the mean time required to complete a radiative transition is reduced to values below the mean time
between collisions. At somewhat higher pressures, radiating molecules will collide and change direction several
times before the radiation process is completed. Consequently, molecules no longer move in straight lines but
execute something like the "drunken man's walk." As a
result, even if a molecule does not change its speed, the
average velocity along a line Of sight, as seen by an observer, decreases, and the line shape narrows and
Johns Hopkins APL Technical Digest, Volllme 12, Number 3 (199 1)
The descriptions in the previous section were confined to an elementary "hand-waving" level and could be
refined; but the resulting theory, although easily visualized, would be rather clumsy and jerry-built. The most
rigorous way to proceed is to solve the Schrodinger
equation or its equivalent, the von Neumann equation,
for N interacting molecules and N photons. The number
of interacting molecules, however, is on the order of
10 23 , which is completely out of the range of feasibility.
The popular approach of direct simulation by Monte
Carlo methods is also infeasible: even though the number of particles is manageable, 100 to 1000, correlations
between gas-phase collisions are negligible, and relevant
scattering events are' few. It is simpler to use single-particle kinetic equations in which the effects of all possible
interactions are accounted for by effective collision cross
sections.
Because radiative transitions are functions of the initial and final states and an electromagnetic field , it is natural to use density matrices, pi, the superscript 1 signifying the description of a single representative molecule.
The macroscopic value of any classical dyna,mic variable, A, is then determined byI
ij
ij
(3)
where A is the quantum operator corresponding to A, and
Ii) and (j I are the generalized Hilbert-space vectors· corresponding to the states i and j. In this representation, p~
and A j i are seen to be the quantum analogs of a ·classical
247
L. Manchick
distribution function and a classical dynamic variable.
Indeed the diagonal element of p I, pt can be identified
as the population den ity of the ith quantum state. The
off-diagonal element, pt, i 7:- j, which vary with time a
exp[ih-I(E i - Ej )t] in the absence of any interaction, are
interference term. In the pre ence of an external field .
they have the added ignificance of being a mea ure of
the polarization induced by the field.
Another characteri tic of inglet-den ity matrice i
that, becau se they describe single, repre entative molecules, all information concerning fluctuations is 10 t.
This characteri tic is not a drawback a regard line
shapes, however, becau e they are directly related to the
spectral response function, which in turn is related to p, a
Laplace transform of p ,
(4)
quenc of the line. According to the Heisenberg uncertainty principle. the uncertaint in the radiative lifetime
varie in er el with thi frequency difference; thus, the
duration of a colli ion cannot be considered negligible
far out in the ing of a pectral line. These reservations
can be wai ed in the pre ent ca e, which concerns very
narro ,exactl r onant. i olated line . The second appro imation i Boltzmann' sfoss:ahlansatz, which has
been iolated b man computer experiments designed
to imulate den e ga e and liquid but which, it seems,
can be afel ignored at 10 and moderate pressures. The
third i well ati fied b th experiment considered here.
The rather complex term Tp (2) has been written as the
re ult of operating on a den it matrix p(2) with a "superoperator" or tetradic T: both terms are common usu to write Tp (2) in a
age. The fir t appro imation allo
form that lightl re embl the familiar Boltzmann colli ion operat~r: - .6
(5 )
Here P and it adjoint, p T, are the operator equivalent of
the polarization of the radiating molecule induced b an
electromagnetic field . In the presence of a weak or
moderate- trength electromagnetic field, the density matrix obeys the following equation if the tran lational
degrees of freedom can be treated cla icall/ a
where ( Tp (2») i a horthand notation of the average urn
of all the interaction of the other N - 1 molecules with
the Nth, and p(2) i the density matrix of a pair of
molecules obtained by fixing their quantum tate and
averaging and umming over the tate of the remaining
N - 2 molecule. Thu defined, p(1) i a complex functional of p(I), (Tp (1») i a complex functional of p(2) , and
the reduced kinetic equation is in principle exact even
though fluctuation phenomena are buried deep in ide
(Tp (2»).
In actuality, however, the exact form of ( Tp(2») is not
known , and we have to use a set of approximations that,
although for the mo t part upply a very good de cription
of dilute ga es have not gone unchallenged. The e approximations are a follow : (1) the interaction between
the two molecule is described from knowledge of the
results of completed collision that occur on a time cale
that is much maIler than that of any other proces in the
gas; (2) before a collision, or sets of collisions, the tate
of two molecules are totally uncorrelated; and (3) only
first-order corrections need be retained for deviation
from local equilibrium. The fir t assumption is known to
violate detailed balance in the theory of pectra of overlapping lines and to yield poor descriptions of the far
wings of broad spectral lines because the spectral response function depends directly on the difference of the
radiation field frequency and the natural resonant fre-
248
Here f, which will b di cu ed briefl in the next ection i the familiar t-matri of cattering theory; l i it
adjoint; and fp (2) {" . tp(2l . and p(2) /'; are matri product.
All integration and implied urn are under tood to be
confined to the energ
hell. that i , tho e colli ion
where energy icon erv d. Th
cond and third approximation allo u to replace p(- ) b Po') f ~:.
Here we ha e implicitl a umed that the He concentration i 0 much larger than the D1 oncentration that
the He i al a in quilibrium: thi equilibrium reduces
orne hat bet een Tp and the Boltzmann colli ion operator, but th kinetic equation still retain t-matrice rather than cro
ection a a necessary
component becau e of th po ibility of interference pattern et up after e ry colli ion. The initial wave packet,
e entiall a partiall 10 aliz d plain wave, i broken up
into a et of cattered a e corre ponding to the set of
final tate populated b the colli ion . The vibrational
and electronic energ Ie el u uall are ufficiently well
eparated to allo
catter d a e to eparate in space
before the next colli ion and not mutuall interfere (Fig.
1). early degenerate tate. uch a tho e that occur in
rotation multiplet and lectronic fine tructure. u uall
cannot eparate a well a tho e hown in the figure,
re ulting in interference.
Equation 7 how that, becau e P D2 i not kno n e actly but only implicitly a the solution of a rather complicated equation, it will be a rather complex function of
all the tran lational and internal degree of freedom . A
in the ca e of the Boltzmann equation, p rna be implified lightly becau e of everal con traint that T mu t
ati fy in all tage of approximation. One i that T be
independent of the frame of reference; in particular, it
Johns Hopkins A PL Technical Digest , l 'ollime 12 .
I
limber 3 (1991)
Modern Quantum Kin etic Theory and Spectral Line Shapes
Iq Primary
beam
LI ::;; J ::;; q + L .
Because of the rotational invariance of T , this choice
of expansion has the desirable consequence that all
tetradics of T and other rotationally invariant superoperators are block diagonal in J and M. In mathematical notation,
J ;r= J', M ;r= M' .
(9)
This is shown schematically in Figure 2, specifically,
E E
E
hi! qL
1M
x
foo
KJU,J/ qLv; j',Jt q'L'v ')
o
-1M
( ' ) ,2d '
x P)'/ij'q'L'
V
V
v .
Figure 1. A wave packet approaches a target, in this example
(top), a diatomic molecule . A short time after the collision (bottom) , the primary beam , slightly attenuated, is on the verge of colliding with another molecule. Elastically scattered waves are radiating outward with the same relative velocity as the primary beam
and have moved the same distance from the target. Inelastically
scattered waves have moved away at a slower speed and are
well separated from the elastically scattered waves.
should be rotationally invariant. It is then natural to
expand PD2 in an infinite series of density matrices
minimally affected by the orientation of the gas relative
to an observer. These matrices are chosen to be the irreducible representations of the rotation group, U]0qLeV) ,
which have the property that all matrices with a given
value of J and M transform under rotations into linear
sums of the other matrices with the same value of J and
M, and no others. 7 Then P can be expanded as
(8)
The eigenfunctions, U, are really operators and must couple rotation eigenfunctions of the translational momentum , Y;ev), to the rotational angular momentum of the
states before and after the radiative transition; q is the
magnitude of the pseudo-angular momentum resultant of
Vi m) and (j ! m! I, and J is the resultant of adding the
momenta Land q. By the usual quantization rules, the allowed values of q and J are restricted to
Johns Hopkins APL Technical Digest, Volume 12, Number 3 (1991)
(10)
As shown in Equation 10, other desirable dividends of
this elementary application of group theory are as follows: (1) the kernel and the expansion coefficient under
the integral sign are now scalars; (2) the integration over
velocity orientations has been buried in the kernel; and
(3) the kernel itself is a symmetric function of the dynamic variables before the collision, which are represented by the primed quantities, and the postcollision variables, which are represented by the unprimed quantities.
The last feature now allows us to employ any of the battery of methods developed for integral equations with
symmetric kernels. 8
Tangent to the blocks diagonal in J (Fig. 2) are two
parallel diagonal rows that result from the coupling of
the J, J ± 1 blocks by the drift terms in the quantum
Boltzmann equation. This comes about because, although the collision terms are independent of the placement of an observer, the presence of an electromagnetic
field defines a preferred direction that enters into the drift
terms. At the higher densities, the collision terms-represented by the diagonal blocks-dominate, and the yg
component of the density matrix is effectively decoupled
from the higher-order nonspherical terms. The line shape
is then determined by the pressure-broadening cross section and assumes the familiar Lorentzian shape. At
progressively lower densities, however, the drift terms
become more important, and more and more nonspherical terms are coupled by the kinetic equation, affecting
the spherical term to at least second order. The most important of these are the Y~' ±l tensorial components. The
matrix element of the collision operator sandwiched by
249
L. Monchick
W; • • •• W tv W 1'·
• ••
W ~ W l'
• • ••
W ~ W; ••••
x
X
L=O
X
X
W~
X
W; X
X
X
X
L=1
X
X
X
X
X
X
X
X
X
X
L=2
X
X
X
X
X
-
X
X
X
X
Figure 2. The matrix of the collision kernel KL( Wi, Wi) , where
the velocity distribution function has been expanded in spherical
harmonics Y~ (f') , and Wi is the postcollision and W; the precollision values of the (normalized) velocity evaluated at the pivot
points of a suitably chosen Gaussian integration algorithm. Each
X corresponds to a nonvanishing matrix element.
these components is the momentum transfer cross section, long familiar to the kinetic theory of diffusion. The
line width, which is dominated by the relaxation term at
high densities , decreases as the density is decreased until
the diffusion terms, which vary inversely with the density, assume the dominant role and the line width increase
again, eventually approaching the Doppler line hape at
very low pressures.
THE COMPUTATION
Several years ago, scientists at the National Institute
of Standards and Technology' began extremely accurate
measurements of Raman scattering of D2 in everal
gases, including He. These experiments measured the
polarized Stokes-Raman Q-branch (.t::.J = 0) scattering of
frequencies corresponding to the l' = 0
1 vibrational
transition of D2 at densities ranging from a fraction of an
Amagat to 8 Amagats. An Amagat is a unit of density
and volume defined as the ratio of the number of
molecules per cubic centimeter to the number of
molecules per cubic centimeter in a perfect gas at 1 atmosphere of pressure and at 0° C. In coherent antiStokes-Raman scattering a laser beam is formed with a
multiple of some frequency approximating the separation of two quantum levels of a chosen molecule. The
sample is irradiated with this beam and a probing beam
with a different tunable frequency. Spectral transitions
are now induced; thi s is called scattering because photons are taken out of the primary beam. If the transition is
to a higher level, this is called anti-Stokes scattering. If
the initial and final rotational states are the same, thi is
called a Q-branch transition.
This set of experimental conditions included line
shapes dominated by the Dicke narrowing as well as
--+
250
those characterized by high-pre ure broadening. At the
ame time, Robert Blackmore (a postdoctoral fellow at
APL) . Sheldon Green (a cienti t at ASA), and I undertook a calculation of the e effect in D2 immersed in He,
making a minimal number of extraneous physical assumption . Thi calculation required (1) ascertaining the
intermolecular force between D} and He, (2) solving the
scattering equation for all e ent relevant to line-shape
broadening, and (3 ) in erting the re ulting cross sections
into the appropriate kinetic theory and solving for the
pectral re pon e function , which produces the line
shape.
A glance at the colli ion kernels describing classic
structurele particle how that they are not simple.
The quantum ver ion i e en more complex. For as simple a sy tem a D}-He, a man a eight angular momenta are coupled in the final complex formula for the collision kernel, K j(j;Jf qLl' ; /J;.q'L'v '). The first step is
the computation of matri element of the D _-He potential energy for a mall et of rotational and ibrational
state; the econd tep i the olution of the scattering
equations for the t-matrice : the third step is the insertion
of these t-matrice into colli ion kernel; and the fourth
tep i the olution of the quantum kinetic equation.
Some ear ago, a molecular orbital calculation was
carried out for the electronic energy of two hydrogenlike atom and a helium atom clamped in space at several
configuration .9 For thi clamp d potential energy surface, V( lr I. IRI. I' . R). hich i kno n a the MeyerHariharan-Kutzelnigg potential. Irl i the distance between D2 and He. IRI i the D - D internuclear distance,
and I' . Ria mea ure of the relati e orientation of D2
and He. The potential energ urface i al 0 valid for two
D atom and one He atom clamped in pace. The accuracy of thi urface i borne out b many comparisons of
experiment and prediction ba ed on it. At each D 2- He
eparation and relati e orientation. the potential energy
urface wa calculated at fi e D - D eparations. It is now
convenient to replace the R-dependent potential energy
urface, V. b the ibrational matri elements of V, that
i , b a et of integral of - and iched by the initial and
final vibrational tate. The method chosen to solve the
wa e equation for the relati e nuclear motion was, by interpolation in the ariable R. to evaluate V at the pivot
points of a uitable Hermite polynomial. This procedure
i imilar in pirit to the p eudo pectral methods decribed by Ku and Ro enberg in thi i sue. The differential equation in R could then be converted to a matrix
equation which wa then olved with the u ual normalization and boundary condition. When the ibrational
energy eigenvalue calculated at large D 2- He eparation
were compared with pectro copic value, accuracie of
=0.2% were indicated for the vibrational wa e functions , X,.(IRI ). Most of the calculation of ibrationalrotational matrix element
f
dR [x ,.,(IRI)Y{,>(R) ] * -
(II'I IRI
r·
R)x,.(IRI)Y~,(R )
,
(11 )
Johns Hopkins
PL Technical
D;~ esr .
l'oilime 1'2,
limber 3 (!99!)
Modern Quantum Kinetic Theory and Spectral Line Shapes
were carried out with rigid rotor rotational wave functions, Y~/R) , with rotational energies corrected for centrifugal stretching. A final set also included centrifugal
stretching in the vibrational wave function. The vibrational and rotational quantum numbers included were v =
0, 1 and} = 0, 1, 2, 3, 4, 5, 6, 7. These calculations were
carried out in overnight runs on an IBM AT, and the potential energy matrices were then sent via BITNET to New
York for step two.
Because this is a quantum mechanical system, the
cattering information is obtained not from trajectories
but fro m solving the Schrodinger equation with the solutions having the asymptotic form
laborious but straightforward. In line broadening, the
coding involved summing products of two t-matrices and
terms representing all the possible couplings of the magnitudes of six angular momenta. No chains of interior
matrix multiplications could be optimized other than the
sums over magnetic quantum numbers that already had
been carried out analytically and were now buried inside
the coupling coefficients, which are known in the trade
as 6-J symbols. The code was optimized somewhat by
evaluating the 6-J symbols recursively, but even then a
typical calculation took between thirty minutes and two
hours on the Ahrndahl 5890-200E.
In the final stage, the kernels were evaluated at the
pivot points of a Laguerre polynomial of order N = 30,
converting the integrals over v in Equation 10 into a
30 x 30 matrix multiplying a vector of order 30. As is
well known, this conversion is equivalent to approximating the density-matrix coefficients with an expansion in
Laguerre polynomials truncated at the 30th term. In principle, the density matrix could have been expanded
directly in Laguerre polynomials, and each of the matrix
elements could have been evaluated individually as in
the older texts on kinetic theory. The present procedure
avoids that step by doing it automatically. Remembering
that, with the last approximation, J = L, Equation 6 could
then be mapped onto a matrix equation
I
\-')'/11 '
(12)
The usual practice in molecular-scattering calculations is
to expand a prototype wave function in the set of all possible internal states- and a few closed channels as welland a large number of partial waves representing the
resultant of the rotational and orbital angular momenta of
the trajectories of relative motion. This representation is
chosen because scattering equations are rotationally invariant in the relative motion frame of reference and become block diagonal if the expansion set is taken to be
the set of total angular momentum eigenfunctions. The
required number is established by making a few test calculations to determine the rate of convergence. The timeindependent Schrodinger equation is then solved at a single total energy for all possible independent boundary
conditions, and the solutions are then combined into
wave functions having the asymptotic form of Equation
12. The requisite scattering information is determined by
the asymptotic value of the wave function at the completion of the collision and succinctly represented by the
contracted set of data composed of the t-matrices. The tmatrices represent the contracted set. These data are calculated automatically by a software package called MOLSCAT,lO which requires as input only the potential energy
matrices calculated in step one. The scattering information was destined for a Q (il) = 0) branch v = 0 - 1 Raman line, requiring calculations for v = 0 and for v = 1.
The required calculations were performed on a NASA IBM
360 during several overnight runs, and the results were
returned to APL on tape.
In step three, the t-matrices were recoupled to form
the collision kernels. This step was necessary because
the scattering matrices and the collision kernels are rotationally invariant in different spaces. Development of a
formula for the collision kernels was not a straightforward task,7 but rather depended on choosing a momentum space transformation that is not at all obvious. One
final approximation was introduced that set q = 0 and
reduced the size of the computation. The subsequent
coding of the resultant formula for the computer was
Johns Hopkins APL Technical Digest, Volume 12, Number 3 (J99 J )
A ·x = y .
Matrix A is composed of 30 x 30 blocks clustered along
the diagonal with the blocks ordered by increasing
values of J. As noted in the previous section, the blocks
are coupled by two parallels to the diagonal (cf. Fig. 2),
which become increasingly important at lower densities.
Preliminary calculations indicated that including Jblocks up to J = 5 was sufficient to represent the spherical part of the density matrix near the Dicke narrowing
minimum (Fig. 3). The final computation then proceed-
300~----~------~------~----~------~
c
o
°u
c
..2 200
c
o
's
.0
-c
U5
'5 100
C
-0
o
2
Q5
ro
"S
u
4
z::::::~ ~
>
0
3
OJ
(5
~
Lmax = 1
-100L-----~------~------~----~------~
o
2
8
4
6
Normalized molecular velocity
10
Figure 3. The deviation of the L = 0 component of the density
matrix, p , from a Maxwell-Boltzmann distribution as a function of
a normalized velocity and of the maximum value of L when p is
expanded in a series of spherical harmonics, Y~ (iJ)_
251
L. Monchick
80
ed by inverting the 180 x 180 sparse matrix at a series of
densities and frequencies . This relat ively mall calculation could be done overnight on a 386 Pc.
c
0
~
RESULTS
60
c
.2
Q)
The experimental re ults to be reproduced were obtained by fir t forming a harply focu ed, almo t monochromatic laser beam corresponding to the fl.i = 0, fl.1' = 0
- 1 transition of D 2 . The beam entered a cell containing
a 90% He: 10% D} mixture and traversed it about forty
times. The value that we have called "experimental "
have been corrected for backward cattering and interferences that occur each time a ray traversing the cell cro es another ray. The mea ured line hape were then attributed to Raman cattering in the forward direction ,
and the line width and hifts were calculated fro m
phenomenological formula fit to the mea urements.
In the calculations reported here, the frequency grid
was fine enough to allow evaluation of the widths and
shifts directly from the numerical re ults. The highdensity widths and shifts vary almost linearly with density. The asymptotic density coefficient of these quantities are shown in Table 1. 11 Computation incorporating
centrifugal stretching and vibrationally inelastic effect
do not cause any ignificant improvement. Con idering
the availability of only five points with which to evaluate
vibrational wave functions and the fact that width and
shift coefficients are evaluated by the asymptotic variation , the agreement with experimental result i about a
good as can be expected.
A repre entative et of line hape at a given value of
irot (i.e., the rotational quantum number) i hown in Figure 4. Line width a a function of den it and irot are
shown in Figure 5 to 7 for both computed and experimental re ult . The difference between the two is due
en
c
0
a. 40
en
~
~
u
Q)
a.
en 20
o
2
4
6
8
Relative frequency (cm- 1)
Figure 4. The spectral response function for jrot = 0 evaluated
at a set of densities ranging from 0.3 to 6 Amagats . At 6 Amagats,
the spectral response function is almost Lorentzian . (An Amagat
is a unit of volume and density defined as the rati o of the number
of molecules per cubic centimeter to th e number of molecules per
cubic centimeter in a perfect gas at 1 atmosphere of pressure and
0
at 0 C ' jrot is the rotational quantum number. )
in part to difference in th numerical methods used for
extracting half-width from the raw experimental data
and computed line- hap data. Detailed comparison of
calculated and experim ntal line hape how few discernible difference . I} All in all. the re ult were very
gratif ing.
The achie ement of thi tud can be ummarized as
fo llo w :
1. The accu rac of th Me er-Hariharan-Kutzelnigg
He-H} potential energ urface \ a demon trated once
again .
Table 1. Comparison of theoretical and experimental width and shift parameters for the Q(J ) lines of D2 in
He at 298 K.
Width
hift
J
Experimenta
Theory
Y~
Yi1n
Ydep
0
1
2
3
4
5
2.35
1.20
1.73
1.62
1.40
1.20
2.33
1.11
l.66
1.54
1.29
1.14
0.73
0.18
0.40
0.34
0.22
0.16
1.00
0.24
0.54
0.45
0.29
0.23
0.60
0.69
0.72
0.75
0.7
o. 2
E perirnentb
Theory
6.1
6.5
6.
6.9
6.
7.2
7.7
7.7
7.9
8.0
8.2
Note: Theoretical value include centrifugal di tortion in the potential matrix element but not in the rotational energy
levels; vibrational inelasticity i ignored. The theoretical width ha e been decompo ed into th dephasing part. Ydep'
and the inelastic contributions within " = 0 and " = 1,
and y~, re pecti el. LI alue are in 10-3 em-II magal.
NM = not measured .
aReprinted, with penni ion, from Ref. l.
bG. J. Rosasco and W. Hurst, personal communication, 1989.
In
252
Jollns H opkins APL Tecllnical D ir:es1. l oillme 12 , SlImber J (/991)
Modern Quantum Kinetic Theory and Spectral Line Shapes
0.025
, - -- - - - - - r - - - - - . - -- --
- . - - - - - - - -,---------,
0 .020 , - - - - - - , -- - - - . - ----,-------,- - - - ,
0 .020
0.015
~ 0 . 015
E
I'
E
-S.
-S.
~
I
~
IO.010
~
I
0.010
~
I
0.005
0.005
0.0000
2
4
6
Density (Amagats)
8
10
0.0000
Figure 5. The half-width at half-maximum (H WHM ) of the spectral response curve for the Stokes-Raman Q-branch scattering of
the v = 0 - 1 transition of 02 immersed in helium: jrot = 0 (jrot is
the rotational quantum number). The squares are experimental
data. The solid black line is the curve pred icted by the calculation
described here. The circles are the results of a slightly amplified
calculation ; the other symbols and colored lines are the results of
more approximate theories.
0.015
, - - - - - - , - - - - - . --
--,-------r----,
0.010 •
•
0.005
0 .000
L-..-_ _- ' -_ __
o
2
---'-_ _ _.l.....-_ _- ' -_ __
4
6
8
-'
10
Density (Amagats)
Figure 6. The half-width at half-maximum (H WHM ) of the spectral response curve for the Stokes-Raman Q-branch scattering of
the v = 0 - 1 transition of 02 immersed in helium: jrot = 1 (jrot is
the rotational quantum number). The notation is the same as in
Figure 5.
2
4
6
8
10
Density (Amagats)
Figure 7. The half-width at half-maximum (HWHM ) of the spectral response curve for the Stokes-Raman Q-branch scattering of
the v = 0 - 1 transition of 02 immersed in helium: jrot = 2 (jrot is
the rotational quantum number) . The notation is the same as in
Figure 5.
3. Generalized quantum kinetic equations can be
solved by collision kernel methods, otherwise known as
discrete ordinate methods, that had been developed for
neutron diffusion and planetary escape processes, effecting a significant improvement in computational speed
and accuracy. A drawback is that the results are not represented by a simple formula.
4. In principle, extensions to higher-order tensor
polarizations can be made by using sparse matrix and
perturbation methods .
5. In each of the preceding conclusions, the qualifying phrase " in principle" means "provided enough computational power is in hand."
These calculations may be considered a validation of
the current state-of-the-art molecular scattering theory
and gas kinetic theory. Although not described here,
these calculations also served as benchmarks for simpler,
more approximate methods. The latter are now being applied to deuterium!hydrogen rotational transitions in the
upper atmospheres of the outer planets (1. Schaefer and
L. Monchick, unpublished data). One immediate application is the estimation of the deuterium!hydrogen concentration in the atmospheres of Jupiter, Neptune, and
Saturn. 13
REFERENCES
Smyth, K. c., Rosasco, G. 1., and Hurst, W. E., " Measurement and Rate Law
Analysis of D2 Q-Branch Line Broadening Coefficients for Collisions with D2,
He, Ar, H2, and CH4," 1. Chem. Phys. 87, 1001-1011 ( 1987).
2 Dicke, R. H., "The Effect of Colli sions upon the Doppler Width of Spectral
Lines," Phys. Rev. 89, 472-473 (1953).
3 Monchick, L. , " The Ehrenfest Theorem and Gas Transport Properties," Physica 78, 64-78 ( 1974).
I
2. Modern molecular-scattering methods have progressed to the point where they are, in principle, applicable to all molecular and kinetic processes.
Johns Hopkins APL Technical Digest, Vo lume 12, Number 3 (1991)
253
L. MOl1chick
4 Hess. S .. "Kinetic Theory of Spectral Line Shape . The Tran ition Between
Doppler Broadeni ng and Colli ional Broadening:' Physica 61. 80-94 ( 197~ ).
5 Monchick, L. . and Humer. L. Woo "Diatomic-Diatomic olecular Collision Integrals for Pressure Broadening and Dicke arrowing: A Generalization of
Hess 's Theory:' 1. Chelll . Phys. 85. 713-71 (19 6).
6Tip. A.. "Tran pon Equation for Dilute Ga es with Imernal Degree~ of Freedom:' PII\'sica 52. 493--:n ( 1971 ).
7 Blackmo;e. R" "Collision Kerneb for the Waldmann-Snider Equation." 1.
Chem . PII\·s. 86. 41 -4197 ( 19 7).
8 Blackmor~. R. 1.. Green. S .. and Monchick. L.. " Dicke arrowing of the Polarized Stoke -Raman Q Branch of the II = 0 - I Transi tion of D, in He." 1.
Chem. PIlI'S. 91. 3846-3 53 (19 9).
9 Meyer, W:. Hariharan. P. C. and Kut zelnigg. V .. " Relined ab illilio Calculations of the Potelllial Energy Surface of He-H2 wi th pecial Empha i on the
Region of the van der V aah Minimum." 1. Chelll. Ph)'.I . 73. 1880- 1897
( 19 0).
10 Hu«'on. J. M" and Green. ,,'.fOLSCAT Complllaliollal Code. Versioll9 , Collaborative Computational Project :-\0. 6. Science and Engineering Research
Council. Cambridge. U.K. (19 6 ).
II Green, "Bla kmore. R" and ~10nchi k. L.. "Comment on Line Widths and
hifh in the SlOke,,-Raman Q Branch of D, in He." 1. Chem. Phys . 91 , 52-55
(19 9).
12 Rosa~ o. G. 1.. Bo\\er~. \\'. 1.. HUN. W. "Smyth. K. C. and May, A. D. ,
" imultaneou FOl'\ard-Backwaru Raman canering Studie of D2:X (X = D2,
He. Ar)." 1. Chem . Pln·s . 9·t 762-:-7633 (1991).
I~ Bezard. B.. Gautier. D.. and ~Ianen. A" "Detectability of HD and NonEquilibrium pecie~ in the Cpper tmo~phere of the Gialll Planets from Their
ubmillimeter pectrum." Aslmll. ASlmphys. 161, 387-411 ( 1986).
THE AUTHOR
a i iting cieJ1li . l. he has remained
at APL. Dr. Monchick ha. worked
in the field of diffu ion-controll d
reaction. molecular colli. ion d)namic . and Iran POI1 propertie of pol a tomic ga e . . Two publication ...
in the la t field made the "tv en ty-two mo t cited" Ii t of article. pubIi hed by APL taff member ( Berl. W. G .. lvhllS Hopkins PL Tech .
D ig. , VoL 7 . 0.3, _21.19 6) .Oneofthe. epaper ' \,,'a alogi\enthe
award of "Ci tation Cia ic" by Cllrrelll COlllellTs.
254
lohllsHopJ..illsAPL
TI'(hllicaIDi~esl . \011l1llE'
12 . SIImber3 (/99JJ

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