J. Quanr. Specrrosc. Rodiar. Transfer Vol. 57, No. 2, pp. 145-155,
1997
Published by Elsevier Science Ltd. Printed in Great Britain
0022-4073/97 $17.00 + 0.00
Pergamon
PII: SOO22-4073(%)00129-X
LINE MIXING
SUM RULES FOR THE ANALYSIS OF
MULTIPLET SPECTRA
A. S. PINE
Optical
Technology
Division,
National
Institute of Standards
MD 20899, U.S.A.
and Technology,
Gaithersburg,
{Received I2 June 1996)
Abstract-In
the impact approximation, the spectrum of transitions overlapped by collisional
broadening is given by a sum over Lorentzian absorption and dispersion profiles about each
transition. The dispersion components arise from the interference among lines coupled by the
collisions, and their magnitudes and signs are given by the line mixing parameters. The line
mixing parameters, as well as the transition intensities, widths and shifts, are obtained from
inversion of the relaxation matrix and may be highly non-linear in pressure when the coupled
lines are strongly overlapped. For any overlap, however, the line mixing parameters summed
over all coupled lines is zero and the total integrated intensities, widths and shifts are
conserved. These sum rules may be used as constraints in least-squares fitting of multiplet
spectra to reduce the number of free parameters and their correlation. Also for the fitting of
spectra at atmospheric pressures, a prescription is given for incorporating Doppler broadening
and Dicke narrowing. Published by Elsevier Science Ltd
1. INTRODUCTION
When pressure-broadened spectral lines overlap, the resulting profile may deviate significantly from
a linear superposition of Lorentzian line shapes if the lines are collisionally coupled. Various aspects
of the phenomenological theory of overlapping lines within the impact approximation were
developed by Baranger,’ Kolb and Griem,* and Fano,’ describing the interference among the
transitions in Liouville or “line” space coupled by a “relaxation” matrix. The interference between
lines leads to a merging of the spectral components and a reduction of their effective widths and
is variously known as rotational or collisional narrowing, coherence transfer and line mixing.
Ben-Reuven4 and Gordon’ presented semiclassical scattering formulations for the relaxation matrix
elements.
Gordon and McGinnis6 calculated line coupling scattering cross sections for a model CO/He
potential for comparison to their measurements of the CO infrared fundamental at He pressures
high enough to blend the P and R branch lines. Later, first- and higher-order approaches to line
mixing theory were developed by Rosenkranz,’ Lam,* and Smith’ for modeling the important
atmospheric microwave spectrum of O2 which exhibits magnetic-dipole-allowed spin splittings in
the ground ‘C, electronic state.
More recently, strong line mixing has been observed in the isotropic Raman Q branches of N2,‘O-2?
C0,23-26and CO 2.27m29
In most of these Raman studies, scattering cross sections were not available,
so the relaxation matrix was modeled empirically using energy gap or energy-corrected-sudden
(ECS) scaling laws to fit the broadening coefficients measued at sufficiently low pressures to
minimize overlap. The broadening coefficient of a line was assumed to be equal to the sum of all
of the rotationally inelastic (AJ # 0) state-to-state collision rates from the J level of the transition.
This sum rule is reasonably well justified for scalar Raman scattering, which does not depend on
the orientation of the molecule, but shows minor deviations due to vibrational relaxation, The
utility of the energy gap or ECS law is then judged on a comparison with, not a fit to, the higher
pressure spectra. Very dramatic line mixing effects are exhibited by the 1388 cm-’ vI Q branch of
145
146
A. S. Pine
C&27.28
where the AB is so small that the lines are overlapped at the Doppler limit and subsequently
broaden at a rate -30 times less than a single isolated P or R branch line in the infrared.
The same energy gap or ECS law methodology has been applied to infrared Q branches of linear
molecules such as C02,2s35 N20,36,37HCCH,38 and HCN. 39.40
However, the sum rule equating the
broadening of one line with the total inelastic coupling to other lines within the Q branch is not
valid in this case where reorientations and degenerate vibrational levels are present;5.9 so that an
empirical coupling factor has been introduced to fit the pressure-blended spectra. Consideration
of angular momentum coupling within the 10s or ECS framework4’43 has shown that the effective
coupling is about 50%, which agrees well with the observations on C02,30-35.43
but not adequately
with the greater and lesser decouplings seen in HCCH and HCN respectively.384
For polyatomic molecules that are not linear, we have no simple models for the relaxation matrix
elements, but even more fine structure splittings available, providing opportunity for line mixing
at modest pressures. Consequently, it has proved difficult to fit blended spectra of some simple
molecules at atmospheric pressures. In particular, we noted that additive Voigt or Rautian line
shape fits to the CH, v3 band Q branch” were unstable above w I3 kPa and that linear
extrapolations from lower pressure yielded significant deviations from the measurements. Though
the deviations were characteristic of line mixing, we were unable to gauge the contributions of
quasielastic (AJ = 0) collisions among the various tetrahedral components relative to the inelastic
(AJ # 0) collisions seen in linear molecules. However, the J manifolds in the P and R branches
of a spherical top are well separated so that quasielastic line coupling can be isolated at atmospheric
pressures.
have recently reported simultaneous multispectrum fits to the CH4 v3 P(J) and
Benner et a145,46
R(J) manifolds, recorded with a Fourier transform interferometer, demonstrating the presence of
line mixing using the Rosenkranz first-order approximation.7 In the accompanying paper:’ we
present tunable laser measurements of these P(J) and R(J) manifolds with somewhat higher
resolution, sensitivity and precision, indicating the presence of higher-order line mixing. We also
impose symmetry-based collisional selection rules and zero sum rules on the mixing coefficients.
The sum rules on the mixing coefficients, as well as on the integrated line intensities, the line shifts
and widths, follow mathematically from the impact line mixing formulationms as will be shown
in this note. They are quite general, valid to all mixing orders, and are independent of the nature
of the relaxation matrix, the transition or energy level quantum numbers or the tensor properties
of the probe. Most importantly, they provide physical constraints on the non-linear least-squares
fitting of real data, reducing the number of independent parameters, their mutual correlation and
uncertainties.
We will also present some illustrative examples involving coupled doublets and simulated J
manifolds for methane as well as the modifications necessary to account for Doppler broadening
and Dicke narrowing.
2. LINE
Within the impact approximation,
can be written 1I-9
MIXING
SUM
RULES
the absorption coefficient for a spectrum of overlapping lines
IC(O) = (N&r) Im 1 j~~({rnl[ol i WI.”
00
-
iWl-‘l~hwn
.
(1)
I
Here N, is the density of active molecules, the indices m and n represent transitions in Liouville
or “line” spaceim3and imply all of the relevant quantum labels of the initial and final radiative states
coupled by a transition moment p, p” is the fractional population in the lower level of transition
n. The matrices between the double-bracketed state vectors are the unit matrix, 1, a diagonal
matrix, 00 such that ((rnja In>> = o.&%,,~, where e$, are the unperturbed transition frequencies in
the absence of collisions, and W is the “relaxation” or scattering rate matrix whose elements are
given by collisional scattering cross sections.‘-9 The real and imaginary parts of the diagonal
elements of W are the broadening, rl, and shift, Si, respectively, of each transition in the absence
of overlap. The off-diagonal elements of W are minus the line coupling or coherence transfer rates.
Line mixing sum rules
147
The reverse rates are related by detailed balance, W,, = W,,,,p,/p,,. In principle, the double sum
in Eq. (1) is over all transitions in the spectrum, but in practice can be restricted to isolated regions
of overlapping lines.
One can avoid the matrix inversion at each frequency, o, in Eq. (1) by diagonalizing the
non-Hermitian matrix o. + iW with the matrix A such that,‘,2
A-‘.(oo + iW).A = 52,
(2)
where the diagonal matrix R has complex eigenvalues R,, = (0, + iym)S,, and the matrix A has
columns made up of eigenvectors for the corresponding eigenvalues. If we consider the pL, as
components of a vector and the pm as diagonal elements of a diagonal matrix p, then Eq. (1) can
be written,‘.2.*.9
K(O)= Waln) Im 1
m
-
(P~A),(A-'~~.~)~/(w
If we define the real and imaginary parts of the numerator
o,,, - iy,,,) .
(3)
in the sum as,
&, = Re{(~.A),,(A-‘.p.~),},
(44
qrn= Im{(rc.A),(A-‘.p.Cc),},
(4b)
then Eq. (3) becomes a sum over Lorentzian and dispersion terms,
As shown previously,lm6 no approximations have been used in deriving Eqs. (3-5) from Eq. (1)
so they are valid for any overlap. These expressions are reviewed here simply to introduce the
notation for what follows and to indicate the physical origin of the generalized line parameters,
&II, Vnt,w, and ym, to be fit to an experimental spectrum. &,,may be considered the coupled line
strength, I],,,the line mixing parameter, and w, and ymare the renormalized line frequency and half
0
width, respectively. If the off-diagonal W-matrix elements were zero, then A+l,
w,,,+w,,,,
&,,+p~p,,,, and Y]~+O, and K(W) would be the usual sum over Lorentzians with
+Ef,Ym-+Yll,
widths and shifts linear in pressure as are all the W-matrix elements. In general, however, the line
parameters are complicated functions of the W-matrix elements and may be considerably
non-linear with pressure under conditions of strong overlap. Rosenkranz’ and Smith9 have given
linear and quadratic expansions for the parameters in the realm of weak overlap. Their line mixing
coefficients are obtained by perturbation theory from the q,,,above divided by pn, for Rosenkranz’
and &pm for Smith,9 and their W-matrix elements are written explicitly linear in pressure.
We note that the dot products in Eqs. (3) and (4) are (p.A),,, = &p,A,,,, and
(A-‘.p.p), = X,A;ippnpn; so the sum of their product over all lines in the spectrum gives
Here we have reversed the order of summation and noted that ~,,A,,,A,;,I 5 6,” by orthonormality
of the eigenvectors defining the inverse matrix.’ Thus the sum of all the line strengths and i x the
line mixing parameters is purely real and equal to the sum of the unperturbed line strengths. This
is an expression of the sum rule that the total integrated intensities are conserved and the line mixing
parameters sum to zero, independent of the overlap or the magnitude of the line coupling elements.
Similarly, because the trace of a matrix is invariant to diagonalization, the sum of the line shifts,
6, = CD,,,
- oi, and widths are preserved; that is
1 (L + iynl)
m
=
C (Sf + iy,O).
”
(7)
148
A. S. Pine
As stated earlier, these sum rules are exact and may be used to reduce the number of independent
parameters in a fit. Alternatively, they may be used to test the reliability of a fit when all parameters
are unconstrained.
In some cases, usually for symmetry reasons, lines or groups of lines that overlap may not be
coupled. For example, the A, E and F symmetry nuclear spin modifications in tetrahedral molecules
or the singlet-triplet levels in HCCH or HOH will have essentially no collisional cross relaxation.
This leads to block diagonalization of the W-matrix for which the sum rules, Eqs. (6) and (7),
pertain to each symmetry group separately.
3. EXAMPLES
3.1. Doublets
To illustrate the sum rules, it is useful to discuss some realistic and representative cases. The
coupled doublet is the simplest case where Eq. (1) can be inverted analytically as has been shown
by Ben-Reuven4 and Henck and Lehmann4* for the inversion doublets in NH3. The results can also
be applied to line shapes of l-type doublets in polar linear mo1ecules,49 A-doublets in open shell
species,” and many other systems where pairs of collisionally coupled lines overlap away from more
remote spectral lines. The examples considered above4@z49
involved balanced doublets with the two
components equal in intensity (cl, = p2, p, = p2) and width (W,, = Wz2, W,2 = W2,) and initially
separated by A = WY- wi with no intrinsic shifts, Im{ W,,,,} = 0. Their expressions and discussions
can be summarized in Fig. l(a), where we show the reduced line parameters of Eq. (5) as a linear
function of pressure as measured by the overlap given by the average width, y. = (W,, + W22)/2,
divided by the initial separation, A. The solid curves correspond to the parameters for line 1
and the dashed for line 2. The &,, and Q,, parameters are scaled to the average intensity,
50 = (/.G, + PL:P2W
The balanced doublet parameters in Fig. l(a) exhibit distinctive behavior in two regions
separated by singularities in the intensity and mixing coefficients. At lower pressures, the widths
and intensities do not vary from their average values, while the magnitude of the dispersion
components start out linearly and then diverge, and the shifts begin quadratically until the lines
coalesce exactly. After coalescence the shifts stop, the dispersion components vanish abruptly, and
the two remaining Lorentizians develop different widths with the broader having a negative
intensity. The symmetry of the plot about 1 for the ym/yoand about 0 for the others indicates the
sum rules or Eqs. (6) and (7). The cancellations arising from the sum rules prevent any singularities
in the spectrum where the rrn and q,,, parameters diverge.
The position of the singularities depends on the magnitude of the off-diagonal W-matrix elements
and occurs for W,2W2, = A2/4 as long as W,, = Wz2 and is independent of the relative line
intensities. In Fig. l(a), we have arbitrarily chosen 1W,,l/ W,, = 0.5, but larger ratios just move the
singularities to the left and smaller to the right. If, however, we unbalance the widths of the doublet,
the singularities are rapidly suppressed as shown in Fig. l(b) where we have chosen W2,/W,, = 0.9.
In this case, all of the parameters vary non-linearly with pressure throughout, but the sum rules,
indicated by the symmetry of the diagram, persist. We note that line mixing averages the positions
of the doublet, but not the widths, contrary to our expectations in an earlier study of differential
broadening of the A-doublets in NO.%
3.2. Multiplets
We now address a case characteristic of a J manifold in a spherical top molecule, pertinent to
the accompanying study,47 where many fine structure components may overlap with restrictive
collisional coupling selection rules. In Fig. 2, we show a low pressure experimental trace of the
R(6) manifold of methane in the triply-degenerate v3C-H stretching band along with a stimulated
pressure-broadened spectrum at PO= 13.33 kPa (100 torr) in an Ar buffer gas. The low pressure
trace has essentially Doppler-limited resolution with negligible instrumental distortion.” The
manifold consists of 6 allowed lines with different tetrahedral symmetry labels and nuclear spin
weights as given in Table 1 along with initial frequencies and estimated W-matrix elements ignoring
any intrinsic shifts. The frequency units in Table 1 are arbitrary grid points, not cm-‘, and the
pressure units are scaled by PO. Since the intermolecular potential is expanded in angular operators
Line mixing sum rules
Balanced
149
Mixing
Doublet
2.0
A = wo2-wo,
1.5
. .
. .
--__
---___
I
al
0.5
a,
5
---_
(t2/#o)-l
1.0
-_
-.
0.0
I
. .
_.
9,/t*
--.-___
&,A-----‘--.
C2/#0)-1
-.
.
:
.\
. .
.
-0.5
.
7)2/co
‘X\
-\
\
-1.0
\
\
\
\
\
\
\
\
\
-1.5
A)
\
\
\
,
-2.0
0.0
0.2
0.4
0.6
0.8
\
\
;
7?2/60
’
l
il___-_---------------------.
l
I
I
I
I
I
I
I
I
I
I
I
I
I
I
1.0
1.2
62/A
(#,/fo)-1
1.4
1 .6
1.8
2.0
1.6
1.8
2.0
YO/A
Unbalanced
Doublet
Mixing
2.0
w,,/w,,
1.5
=
0.50
,-.
A = oo2-oo,
1.0
0.5
L
B)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
YO/A
Fig. 1. Reduced line parameters for a coupled doublet with (a) balanced widths, W,Z= WI,, and (b)
unbalanced widths, IV,?= 0.9W11. In both cases PU?
= ~1, p? = PI, 1WIZI= OSWl,. A = 07 - o:,
yu = (WI1 + W22)/2,5” = (pip, +&Q/2.
150
A. S. Pine
1.2
CH4/Ar
1
I
I
I
I
I,
v3 R6 Simulation
I,
I
I
I
I,
I,
I
I
I
I
I
I
I
I
I
I
I,,
,
,
,
,
,
,
,
I-
0.8
tu
cu
i
k
0.4
I’ (Ar)
kPa
-
0.0
1 sim-fit
r
set
refit
-0.4
13.33
x3
f
i
:
eta=0
!
::
with eta=0
""""""""""""""""""
3086.0
3086.2
J-
3085.8
3085.6
Wavenumberhm-1
Fig. 2. Experimental Doppler-limited and simulated pressure-broadened spectra of the R(6) manifold of
the VIband of methane. Doppler-limited trace recorded at P = 13 Pa and T = 295 K; simulation for Ar
buffer at P = 13.33 kPa; relative absorbance scales are arbitrary. From left to right, the line symmetries
are E, F, A, F, F, A. Dotted trace is least-squares fit using Z,tt”# = 0 sum rule constraint. Simulation-fit
residuals are shown dotted, displaced below the spectra and multiplied by three for visibility. Residuals
( x 3) also shown for sum-rule fit parameters, but with all )I~ set to zero, and for refit parameters with
all rfmconstrained to zero.
of only Al or AZ symmetry in the Td group, there is zero coupling between lines of different A,
E or F symmetry,” thus, the two A and three F lines mix only among themselves.
The simulated spectrum is calculated using Eqs. (l)-(5) yielding the line parameters of Eq. (5),
also given in Table 1. The simulated spectrum includes a convolution with Dicke-narrowed Doppler
Line mixing
Table
m
rN
c,,,
1
El
2
d,
WI,,,
W,,,,
Wl,,,
W&,,,
I. Line parameters
for simulated
2
F22
3
3
A2
5
1213.16
1241.45
1323.29
13.21
0.
14.61
0.
0.
13.36
151
sum rules
CHI vx R(6) manifold
I
4
F2
3
I
1597.59
0.
-0.27
0.
15.01
WS,,,
in Ar buffer.7
5
Fl
3
6
Al
5
1
1662.40
1719.51
0.
-0.81
0.
-2.31
14.49
0.
0.
-4.20
0.
0.
13.74
W,,,,
w,, $
W.>§
w,,,a
6, II
Y., :
40 $2
1200+
13.1600
13.1576 (II)
13.161 (53)
0.
1200+
41.4517
41.4505 (8)
41.413 (39)
0.232
1
1300+
23.3346
23.3346 (3)
23.702 (18)
5.884
l500f
97.6726
97.6702 (7)
98.972 (33)
II.181
1600+
62.3163
62.3158 (8)
60.935 (32)
-11.410
1700f
19.4656
19.4658 (3)
19.060 (19)
- 5.940
13.3600
13.3595 (3)
13.469 (20)
- 0.008
15.0106
15.0089 (7)
14.703 (36)
0.113
14.4894
14.4900 (9)
14.143 (35)
-0.118
13.7400
13.7408 (4)
13.959 (21)
0.008
13.2100
13.2077 (16)
13.566 (57)
0.
14.6100
14.6126 (15)
14.878 (42)
0.001
2.0000
1.9993 (3)
2.0137 (9)
0.
2.999
3.0008 (4)
3.0206 (14)
-0.088
5.0001
4.9999 (I)
5.0343 (23)
0.051
2.9990
2.9987 (I)
3.0206 (14)
-1.120
0.0000
0.
0.
0.
0.0016 (4)
0.0161
0.0159 (2)
0.
-0.112
0.0144 (3)
0.1060
0.1060 (2)
0.
0.566
0.1060 (2)
0.2099
0.2102 (2)
0.
7.185
0.2102 (2)
3.0011
3.0010 (2)
3.0206 (14)
I .208
-0.2260
-0.2260
0.
6.663
-0.2260
(3)
(3)
4.9999
5.0001 (I)
5.0343 (23)
-0.051
-0.1060
-0.1060
0.
0.566
-0.1060
(2)
(2)
tFrequency
units in grid points, not cm-‘; P,(Ar) = 13.33 kPa = 100 torr.
$Value calculated
from inverting W-matrix using Eqs. (24).
@Value retrieved from fit with Z,,q,, = 0 for each A, E or F symmetry block; uncertainties
in parentheses are lu standard
errors from fit in terms of last digits.
TValue retrieved from fit with Q, = 0 and t., cc c,,; uncertainties
in parentheses are la standard errors from fit in terms of
last digits.
/I% shift at 66.66 kPa relative to Doppler width of 18.97 points.
tt% change in coefficient between 6.67 and 66.66 kPa indicating non-linearity.
ISMixing parameters
from fit with no sum rule constraint.
distribution, as discussed in the next section, since the Doppler width seen in the low pressure
(P = 13 Pa) spectrum is comparable to the line broadening. We then fit the simulated spectrum
with only the line mixing parameter sum rule constraint, X,,,Q,,= 0, for each A, E or F symmetry
block. This fit reproduces the spectrum within the “noise” level given by round-off errors of 511O-4
due to an intermediate storage of the spectrum with a 4 digit format. The retrieved parameters
generally agree with the input parameters to within this round-off error as given in Table 1. The
fitted trace is shown dotted, superimposed on the solid curve simulation in Fig. 2, and the residuals,
multiplied by 3, are displaced below the trace for visibility. There are no deviations visible on the
scale of this figure. In order to gauge the effect of line mixing, we then set q,??= 0 but use the other
retrieved parameters from the line mixing fit. This results in the residuals, again x 3, shown in the
next lower trace. Finally we refit the trace with the q,,, constrained to zero, so that the other
parameters can adjust to minimize the deviations. In this case, we also constrained the relative
intensities to the spin weights L,,~,to stabilize the fit (particularly at higher pressures). The residuals
x 3 from this case are shown on the bottom trace of Fig. 2. The large systematic residuals seen
in the lower two traces are characteristic of our experimental results presented in the following
paper4’ when mixing is ignored, and indicate the necessity of including line mixing even at modest
pressures encountered in the upper atmosphere.
Table 1 also compares the retrieved parameters for the fits with q,,,= 0 with the original values.
Here, the deviations are generally much larger than the least-squares lo- standard errors given in
parentheses because of systematic model errors. The non-linearities in the parameters due to
higher-order mixing beyond the Rosenkranz approximation are indicated by the percent change
in the coefficients at a pressure of 66.66 kPa (500 torr), corresponding to highest pressure
152
A. S. Pine
measured.47 Since the intrinsic shifts were arbitrarily set to zero, the shifts due to line mixing are
measured relative to the Doppler width (FWHM), and are a maximum of N 11% for the two F
lines 4 and 5 which attract each other. The non-linearities in the widths are very small, ~0.1%.
The intensities are only N 1% non-linear at worst, and the mixing parameters for the close F lines
are non-linear by -7%. The non-linearities in the shifts and mixing parameters are large
enough to invalidate the use of the Rosenkranz approximation for pressures close to an
atmosphere.
Finally, in the last line of Table 1, we give the mixing parameters from a totally unconstrained
fit, and they change sensibly only for the first two lines of E and F symmetry which are partially
blended even at the Doppler limit. Thus, correlation among the parameters and the finite noise
level from round-off errors results in a slight departure from the line mixing sum rules, which
fractionally is not very significant. The lone E transition does acquire a false mixing coefficient
which affects its shift slightly. Real data, however, may have a worse signal-to-noise ratio, may
contain systematic errors due to baseline uncertainties,
non-linearities in the pressure
measurements, in the frequency scale, and in the detector/amplifier/divider/analog-to-digital
converter chain, may have weak extraneous (isotopic, hot band, forbidden, etc.) lines buried in the
background, and in fact, real spectra may not even obey the physical model completely. Then the
extra constraints afforded by the sum rules may help to stabilize the fits and provide better estimates
for the model parameters.47
4. DOPPLER
BROADENING
AND
DICKE
NARROWING
As seen in Fig. 2, Doppler broadening is a significant fraction of the total line widths at
atmospheric pressures for infrared and Raman spectra and must be considered in the line shape
analysis. This is accomplished by a convolution of the Doppler spectral distribution, D(o), with
the line mixing spectrum, K(W), given in Eq. (5), such that
s
+a
a(0) =
IC(O - w’)D(o’)
do’,
-cc
(8)
where the integral of D(o) is normalized to unity. For a thermal equilibrium velocity distribution,
ignoring the velocity-changing collisions, the Doppler profile is Gaussian,
D(w’) = (l/Jxa)exp(
- e/*/a*),
(9)
where CJ= o,(2ke T/MC*)“* is the Doppler half width at e-’ intensity, ke is the Boltzmann constant
and A4 is the mass of the active molecule. Then
s
+a:
a(w) = (Na/K3’*)1
m -m
[Ly + q,,(x - t)exp( - t’) dt/a[(x - t)’ + y’].
(10)
Here we have defined some dimensionless variables scaled to the Doppler width,
t
E
d/o,
x = (w - o,)/a,
y = y&,
(11)
and taken the sum outside the integral since the Doppler width depends on the transition frequency.
Equation (10) may be rewritten in terms of the real and imaginary parts of the complex probability
function5*
s
+m
w(x, Y) = (i/n)
exp( - t’) dt/[x - t + iy];
-co
(12)
Line mixing sum rules
153
a(m) = (Na/,/n) 1 [L Re{w(x, y)} + rlmX
Im{w(x, y)}]/a.
m
(13)
For an isolated line, this is just the usual Voigt profile, s2 here it is generalized for line mixing at
any overlap. A similar expression was given by Strow and Gentry30 in the first-order Rosenkranz
limit. Despite this generalization, Eq. (13) is still restricted to low pressures where the Gaussian
Doppler distribution is valid.
When the mean-free-path for velocity-changing collisions becomes comparable to or less than
the probe wavelength, the Doppler distribution narrows analogously to the effects of line mixing
on overlapped rotational fine structure. This Doppler constriction is known as Dicke narrowing,53
and useful expressions have been given by Galatrys4 and Rautian and SobelmanrP in the weak and
strong collision limits, respectively. Dicke narrowing is by now a commonly observed feature in
high-resolution spectra of isolated lines, with only minor operational distinctions between the two
limits.47,55,56
In the strong collision regime, expected when the buffer gas is heavier than the active
species, the Rautian function for the Doppler profile is”
D(o’) = (l/Jna)
Re{w(x’, z)/[l - Jnzw(x’, z)]},
(14)
where again w(x’, z) is the complex probability function, Eq. (12) with x’ = o’/a and z = /L/g,
and p,,, is a velocity-changing collision rate which may depend in principle on the particular
transition or energy levels. The convolution of the Rautian with Eq. (5) yields
40) = (Na/Jn)c [L Re{r(x,y + z)} + qmIm{r(x, y +
z)}]/o,
(15)
m
where we have defined
r(x, y + z) = w(x, y + z)/[l - Jrrzw(x, y + z)],
(16)
and, again, all the dimensionless variables depend on the transition m. Equation (15) follows from
Eqs. (14) (8) and (5) using Cauchy’s integral formula and the residues for the poles at
w’ = o - w, + iy,,,in the upper half of the complex frequency plane as discussed by Rautian and
Sobelmann55 for the case of isolated lines. In the limit of no velocity-changing collisions,
T(X, y + z)+w(x, y), so Eq. (15) reduces to the Voigt line mixing profile, Eq. (13). Rautian and
SobelmanrP point out that the convolution procedure is only valid when the velocity- and
phase-changing collisions are statistically independent and when the shifts and line widths are
independent of the relative speed of the collision partners.
In the case of diffusional or weak velocity-changing collisions, the convolution of the
Dicke-narrowed Galatry profile with K(O), Eq. (5), using a similar contour integration, yields
do) = (Na/Jn)
1 [h
m
Re{g(x, y,
z)} + q, Im{g(x,
y,
z)}]/o,
(17)
where the complex Galatry function5b56 is
g(x, Y, z) = (l/J4
s
dr exp[ixr - yr + (1 - zr - e-“)/2z2],
(18)
0
and x, y and z have the same definitions as in Eqs. (11) and (14) above.
For the simulated spectrum of Fig. 2, we have performed a direct numerical convolution of the
Dicke-narrowed Rautian distribution, Eq. (14), with the line mixing sum of Eq. (5). Then, we
retrieved the parameters with a least-squares fit using Eq. (15). We employ the Humliceks2
algorithm for computing the complex probability function. The negligible deviations in the sim-fit
residuals verify both Eq. (15) and the non-linear least-squares fitting routine. For this test we used
P,,, = 8.0 points for all transitions (recall Doppler FWHM = 2 In 20 = 18.97 points) and fixed /I”,
at this value in the fits shown in Fig. 2 and Table 1. If we then float the /I,,,in the fit, we find only
A. S. Pine
154
small variations in /3,,,due to parameter correlation, with largest deviations of - - 0.9% for line 5
and - - 0.6% for line 1 and 60.01% for the other line parameters.
We have also generated synthetic spectra for the Dicke-narrowed Galatry profile by direct
numerical convolution of Eq. (5) with Re{g(x’, 0, z)} using the Herbert” continued fraction
algorithm. These spectra could then be fit with the complex Galatry expression, Eq. (17), with
comparable precision to the Rautian fits described above; thus confirming Eq. (17) and the leastsquares fitting procedure.
5. CONCLUSIONS
We have shown that the line mixing parameters for collisionally-coupled multiplets sum to zero
and that the total integrated intensities, broadening and shift parameters are conserved,
independent of the overlap. These line mixing sum rules have been illustrated for coupled doublets
and spherical-top J manifolds, the latter indicating a block diagonal structure among the different
symmetry lines. The sum rules are useful for testing and stabilizing non-linear least-squares fitting
routines, and can reduce the number of free parameters, their correlations and uncertainties in
fitting realistic spectra. Since Doppler broadening and Dicke narrowing are encountered in
atmospheric spectra, we incorporate models of these effects into the formalism for coupled
transitions. An application to measurements of the P and R branches of the v3 band of methane
is given in the following paper.47
Acknowlegemenfs-The author is grateful to T. Gabard for the theoretical justification of the A # E # Fcollisional selection
rules in tetrahedral molecules to J. P. Looney for helpful comments on the extension of the Dicke-narrowing effects to
the Galatry regime. This work was supported by the NASA Upper Atmosphere Research Program.
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