Volume 144, number 3
CHEMICAL PHYSICS LETTERS
COHERENCE TRANSFER BETWEEN ROTATION-INVERSION
IN THE v3 FUNDAMENTAL OF NH3
26 February 1988
TRANSITIONS,
Steven A. HENCK and Kevin K. LEHMANN
Department of Chemistry, Princeton University, Princeton, NJ 08544, USA
Received 19 November 1987; in final form 28 December 1987
The study of the absorption profile of the uI ‘R 44 and pP SBlines of ammonia at various pressures is reported. The two lines of
the observed inversion doublets broaden as the pressure is increased until they overlap. Once the lines have overlapped the
spectral profile narrows and then begins to broaden again but at a markedly different rate. This effect is due to collisions which
transfer coherence (often called cross relaxation) between the transitions of the given inversion doublets. Modeling of this effect
shows that the rate of coherence transfer is 71% and 75% of the total collisional dephasing rate for the ‘R 4, and pP S8 lines,
respectively.
The study of the absorption profile of the v3 ‘R 44
and pP g8 lines of ammonia at various pressures is
reported. The two lines of the observed inversion
doublets broaden as the pressure is increased until
they overlap. Once the lines have overlapped the
spectral profile narrows and then begins to broaden
again but at a markedly different rate. This effect is
modeled from the density matrix for a four-level
system.
The coherence transfer narrowing of ammonia in
the gas phase is a process in which the oscillation frequency of a molecule’s dipole is changed by a collision and then changed back again by a subsequent
collision before the dipole has had time to dephase
[ 11. The effect of randomly timed collisions is that
the phase of the molecular dipole undergoes a random walk with an average step size given by the
change in angular frequency divided by the collision
rate. In the high pressure limit, the time required to
dephase the molecular dipole is proportional to the
collision rate which leads to a resonance width that
decreases with increasing pressure. In contrast, molecular pressure broadening occurs when collisions
change the oscillator’s frequency without retention
of its phase continuity or do not return the oscillating dipole to its original frequency before dephasing
occurs.
One spectroscopic property of ammonia which
makes it an ideal probe for the investigation of coherence transfer is its inversion doubling of levels.
This tunneling motion converts each rotational level
into a pair of levels, symmetric (s) and antisymmetric (a) with respect to the plane of inversion,
which are very closely spaced compared to rotational
separations. This is the justification for the four-level
model [ 21. Since the splitting of these levels changes
only slightly from pair to pair, half of the collisions
should occur with a molecule allowing a resonant
transition to occur (as++sa). This process should facilitate the transfer of coherence.
Previous work on coherence transfer in ammonia
in the microwave region has been done by Bleaney
and Loubser [ 31. However, it was not until BenReuven [4] that this work was interpreted in the
framework of coherence transfer. In their work, the
transitions from individual J, K lines were blended
together, so they could only determine an average
coherence transfer rate over the thermal distribution
weighted by the microwave transition Hiinl-London
factors which are proportional to K’/J( J+ 1). In the
far infrared, Lightman and Ben-Reuven [ 561 studied the coherence transfer phenomena in R branch
rotation-inversion transitions. This work could not
resolve the individual K transitions for a given J. The
R branch transition intensities preferentially weight
the lower K states which are not efficient coherence
0 009-2614/88/$ 03.50 0 Elsevier Science Publishers B.V.
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281
Volume 144, number 3
CHEMICAL PHYSICS LETTERS
transfer states. Lightman and Ben-Reuven reported
coherence transfer rates for several foreign gases, but
not for self-broadening. In several recent papers by
Broquier et al.’[ 7,8], the coherence transfer between
ammonia inversion doublets in the v4 fun&mental
band caused by collisions with hydrogen and helium
was reported, but again no self-broadening results.
These systems were analyzed by a semi-classical calculation using an ab initio potential energy surface.
These calculations revealed that the coherence transfer rate is more sensitive to the potential than the
pressure broadening rate. However, due to the short
range interactions of hydrogen and helium, they predominantly sample the repulsive wall of the potential. On the contrary, ammonia-ammonia collisions
are dominated by long range dipole-dipole interactions and are thus more likely to cause coherence
transfer.
Optical selection rules helped us to predict lines
where coherence transfer was most likely to occur. A
perpendicular band was chosen because the lines are
closer together (separated by Avinv rather than
Yin”+Vi,,). In order to maximize the collisional dipole matrix element across the inversion J must equal
K. The Hiinl-London factors for a perpendicular
transition predict the strongest lines for J= K are the
‘R and PP lines. Of ammonia’s two perpendicular
bands, v3 and v4, the v3 band was chosen because v4
was perturbed by 215. A final factor used in selecting
lines for study was that there be no other strong transitions nearby. This was necessary so we could treat
the absorption in the region of interest as coming
from one inversion doublet even up to pressures of
several atmospheres. In the ammonia spectra for
J= K lines in the v3band, the only lines which seemed
to be isolated and intense enough were the ‘R 44 and
pP and 8s lines,
We used an F-center laser (FCL) setup shown in
fig. 1. Portions of the light from the FCL are sampled
to check for spectral purity (at the spectrum analyzer) , frequency calibration (with the vernier, germanium etalons and a reference gas cell containing
ammonia at 2 Torr), and intensity (at the reference
detector). The ammonia transitions were assigned
with data from Angst1 et al. [ 91. The remainder of
the light was passed through the sample. The reference and the sample detectors were fast (100 MHz)
liquid-nitrogen-cooled indium antimonide detec282
26 February 1988
IR
Deteeters
Fig. 1. F-center laser setup. The IBM-AT scans the laser and records the data for later processing. The air path lengths are adjusted to minimize the effect of water absorption. The ratio of
the Ge etalons’thicknesses is close to the golden ratio. This makes
the fringe patterns as incommensurate as possible to facilitate the
alignment of scans.
tors. The air path lengths of both reference and sample beams were adjusted to be equal in order to
minimize the effect of atmospheric water absorption
which was strong in this region. The other detectors
were made from lead selenide plates. The outputs of
the lock-in amplifiers were digitized by an IBM AT.
The IBM also scanned the intercavity etalon and advanced the FCL grating. The vernier etalons were
solid germanium with thicknesses such that their ratio was close to the golden ratio, which is the most
irrational number. This guaranteed that the fringe
patterns were as incommensurate as possible. This
facilitated piecing together individual scans which
were two-thirds of a wavenumber long.
The ammonia used was purchased from MG Industries and was purified by freezing the lecture cylinder in liquid nitrogen and pumping out the residual
gas. This was repeated until the vapor pressure was
undetectable to ensure that all of the nitrogen and
hydrogen were removed from the cylinder. When the
cells were filled, the lecture cylinder was placed in a
m-xylenelliquid nitrogen slush. This slush holds the
cylinder at approximately -47 ’C and freezes most
of the water vapor in the cylinder while allowing the
ammonia to have enough vapor pressure to fill the
cell. For the low pressure ‘R 44 lines a 75 cm white
Volume 144. number 3
CHEMICAL PHYSICS LETTERS
26 February 1988
r
Low Pressure
Dota
rR
4.4
21.9
d(a)
_’
A””
Pressure Broadening
MHz/torr
*(I
rR 4.4
OS rR 4.b
-
Fit
00
” “‘. Fit
60.00
20.00
to 4.4
doto
pP 8.8
to 8.8data
100.00
Pressure (tom)
Fig. 2. Half widths at half maximum versus pressure. The slopes
of these lines determine the pressure broadening rates.
cell was used, while for the low pressure pP 8* lines
a 38 cm cell was used. All of the high pressure data
were taken using a 10 cm cell.
In order to get the rate at which the lines pressure
broaden, the data taken at low pressures where the
two lines did not overlap were fit to the sum of two
Lorentzians which shared the same baseline. The parameters for the fits were the center frequencies, the
heights, the half widths at half maximums (hwhm),
and a baseline slope and intercept. From the slope of
a plot of the hwhms versus pressure, we could determine the pressure broadening rate (fig. 2). The
presure broadening rate for the ‘R 44 lines is 21.9
(0.93) MHz/Torn and for the pP 8, lines it is 24.6
(0.79) MHZ/TOE These rates compare satisfactorily with the microwave pressure broadening rate of
24.4 and 24.7 MHz/Torr, respectively [ lo].
The data, taken at high pressures where the two
lines are overlapped, was lit using the following
model (fig. 3). We write the dipole moment as
~=/&&%c+Pbd
+cac.)
and the time derivatives of the density matrices as
bae= iw,,p,, - Rpac + rpbd
and
&
=iwbdPbd
-fhd
+rPac
,
where w, is the frequency of the transition between
Fig. 3. Four-level model system. w., connects levels a and c. wbd
connects levels b and d. a and b are the symmetric (s) and antisymmetric (a) inversion levels of a given rotational level in the
ground vibrational state. c and d are the symmetric and antisymmetric inversion levels in the ug fundamental.
x and y; R is the rate at which coherence leaves the
transition (21~ times the low pressure broadening
rate); and r is the rate at which coherence is transferred into the transition from the coherence between the other states in our four-level system (i.e.
the rate at which coherence is transferred from between a, c to between b, d and vice versa).
These equations, but with R equal to r, were written by Anderson [ 111. However, R will be greater
than r because of the following processes: (a) inelastic (J or K changing) collisions; (b) molecular
reorientational collisions (Am) which am modulate
the molecular radiation; (c) collisions which transfer
the coherence to pador& which do not radiate (Only
the ground or excited state changes inversion symmetry); and (d) pure dephasing with elastic and inversion changing collisions.
The Fourier transform of the time autocorrelation
function of the dipole matrix element is proportional
to the linear absorption spectrum [ 121. The above
model can be integrated analytically. For a coherence transfer rate less than the separation between
the two transitions (i.e. r < Aw) the spectrum is
+pR-r(n-p-w)
R2+(o-6+/3)2
where ~=~(o~~+o~),Ao=~(o~~-o~),
o=2nu
is the angular frequency, and /3= [ ( Ao)’ -r2] “2.
This spectrum is the sum of two Lorentzian lines
with hwhm equal to the rate at which coherence is
lost from the transitions (R). As the rate at which
283
Volume 144. number 3
CHEMICAL PHYSICS LETTERS
the coherence is transferred between the transitions
goes to zero (i.e. lim,,, I(w)), +Aw and we have
two Lorentzians centered at w,, and tit,& As the rate
at which the coherence is transferred into the transition approaches the frequency separation of the
transitions (i.e. limr+J( w)), p--*0 and the centers
collapse.
For a coherence transfer rate greater than the separation between the two transitions (i.e. r > Ao) the
spectrum is
(P-WR+P)
+ (R+p)2+(W-c3)2
>’
where p= [r* - (AU)*] I’*, This spectrum is the difference of two Lorentzians with the same center frequency but with different hwhm. The first term is a
narrow line; the second is a weaker but broader
Lorentzian curve. As the coherence transfer rate becomes much larger than the frequency separation between the isolated lines (i.e. r 3 Aw), P-r. In this
limit the amplitude of the second term goes to zero,
while the first Lorentzian has a hwhm equal to R-r,
which should be the sum of the rates for the processes discussed above.
In fitting the high pressure data we constrained cue,
Ao and R (27~times the pressure broadening rate
from the fits to the isolated lines times the pressure),
and allowed r, the height, and the baseline to vary.
The resulting values of r are found in table 1. The
values of R from the low pressure data are 137
MHzlTorr for the ‘R q4 lines and 155 MHz/Torr for
the pP 8, lines. If we take the ratio of the average value
of r to R we find that for the ‘R 44 lines 71% and
for the pP 8* lines 75% of the coherence is transferred
between the levels of the given inversion doublet
upon collision. This can be seen graphically in fig. 4
where the fit to the ‘R 4, data taken at 1241 Torr is
plotted with the expected spectrum if no coherence
transfer had occurred (i.e. r = 0). These results can
be compared with the results of Ben-Reuven’s analysis of the high pressure microwave absorption [4]
where the effective R is 144 MHz/Torr and the effective r is 94 MHz/Torr or 65% of R. As expected
the self-narrowing of ammonia (75Oh) is larger than
the narrowing by hydrogen [ 6,7] (300/o) and by he284
26 February 1988
Table 1
r from the fits to the ‘R 44 and pP 8s lines at various pressures.
2u on the order of 1 MHz/Torr for all tits
Pressure
(Torr)
r ‘R 4+,
(MHz/Torr)
206
311
413
517
621
124
776
827
931
1034
1138
1241
102
106
99
r PP 88
( MHdTorr)
111
116
97
97
92
119
92
131
124
114
114
93
lium [ 71 ( < 10%) because most ammonia-ammonia collisions are long range dipole-dipole collisions
while hydrogen and helium collisions are shorter
range hard-sphere-type collisions.
Self-broadening of ammonia is a very fast process
and is believed to be dominated by the long range
dipole-dipole interactions. For such interactions,
pure dephasing collisions are typically assumed to be
Predicted
Spectral
Line
With and Without
Coherence
’
--spectral
3501.50
Llw?
“““‘No
3504.50
Frequency
Transfer
Coher.
Trans.
3507.50
(cm-
1)
Fig. 4. Graphic demonstration of coherence transfer. The solid
line is the observed ‘R 44 absorption at 1241 Torr. The calculated
resonance shape is presented with no coherence transfer (r = 0,
dashed line) and with the best tit coherence transfer rate (r =93
MHz/Torr, open circles). The wings of the experimental curve
could not be extended due to the wings of the other transitions.
Yohune 144. number 3
CHEMICAL PHYSICS LETTERS
negligible [ 131. Molecular reorientation collisions,
due to the Am = & 1, 0 selection rule, are important
only for very low J, Therefore, the most important
collisions for self-broadening would be expected to
be inelastic, K, J, or inversion symmetry changing
collisions. However, due to the dipole selection rules,
AK should equal zero. We expect that for the S8level
the rate of J changing collisions to be much slower
than for the 44 level. This is because the only dipoleallowed transitions (to 9s and 5, respectively) have
a smaller transition dipole moment for the S6 level
than for the 4, level, and because the energy separation is much larger (199 versus 99 cm-‘). Furthermore, the 4, level is near the peak of the
Boltzmann distribution, so collisional encounters
with molecules that have possible near-resonant AJ
changes are frequent, while the S8 level has a small
Boltzmann factor and possible resonant collisions
should be fewer. In our experiment, inversion symmetry changing collisions produce coherence transfer narrowing while K and J changing collisions will
not. We expected that the difference between coherence transfer and pressure broadening rates to reflect
the rate of J changing collisions. Thus, the similar
coherence transfer rates of the 44 and 8s transitions
that we observed are unexpected.
In the high pressure microwave experiments [ 31,
pure J changing collisions would not be expected to
dephase the microwave absorption since the new J
value would have a very similar inversion frequency.
Thus, the similarity of our results for the 44 and f&
lines with the microwave results of Ben-Reuven [ 41
appear to imply that pure dephasing processes dom-
26 February 1988
inate the loss of coherence for ammonia-ammonia
collisions. Confirmation of this interpretation will
require coherence transfer calculations on a realistic
ammonia-ammonia potential.
The authors wish to thank the Research Corporation and the Camille and Henry Dreyfus Foundation for their support. We would also like to thank
H. Rabitz for useful discussions and M. Broquier, A.
Picard-Bersellini, H. Aroui and G.D. Billing for
making their results available to us prior to
publication.
References
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285