A. One-Dimensional Measurements: Raman Line Shape and Free Induction Decays
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Vibrational Dephasing in Liquids
387
Fourier transform of CFID :
Iiso
1
D
0
CFID
e
i
0
d
2
0
Although experiments are most directly expressed in terms of the
oscillator coordinate q , the quantity of physical relevance is the instantaneous oscillator frequency ω t , or, expressed more conveniently, the
instantaneous deviation from the mean frequency, υω t D ω t
hωi. This
frequency deviation is the direct consequence of time-dependent forces
exerted on the oscillator by its environment. At a semiclassical level, the
FID correlation function is related to the time-dependent frequency by
CFID
D
exp
υω t dt
i
3
0
Unfortunately, the conversion from υω t to CFID t involves a signiﬁcant
averaging and loss of information.
This loss of information is easier to see after applying a cumulant
approximation (61). This approximation is equivalent to assuming that the
distribution of υω is always Gaussian and simpliﬁes Equation (3) to
CFID
t0
dt0
D exp
0
0
Cω t00 dt00
Cω t D hυω t υω 0 i
4
5
where
indicates operator multiplication. The FID correlation function
is derived from the frequency correlation function Cω t by averCFID
aging in time.
If we also assume that the vibrational frequency is modulated by
a single process, υω t is fully characterized by two parameters: ω , the
rms magnitude of υω, and ω , the correlation time of υω t . In the KuboAnderson model (61–63), the frequency correlation decay is taken to be
exponential:
Cω t D 2ω e
t/
ω
6
and the FID correlation function becomes
CFID
D exp
2ω
2
ω
e
/
ω
1C
ω
7
In the limit ω ω − 1, called the fast modulation or homogeneous limit,
the FID correlation function reduces to
CFID
De
/T2
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8
388
Berg
where T2 D 2ω ω , whereas in the limit ω
lation or inhomogeneous limit, it becomes
CFID
De
2
/22ω
ω
× 1, called the slow modu9
In each limit, the experimental decay is characterized by a single parameter,
whereas the underlying frequency correlation function is characterized by
at least two parameters. This problem exists for other models as well. In
general, the problem of inverting CFID
to ﬁnd Cω t is underdetermined
and requires model-dependent assumptions to complete an analysis. This
problem reﬂects the loss of information caused by the time averaging in
Equation (4).
If we restrict ourselves to the Kubo-Anderson model, the problem
does not seem entirely intractable. The shape of the decay distinguishes the
limits: the fast limit has an exponential decay; the slow limit has a Gaussian
decay. In the intermediate regime [Equation (7)], the details of the decay
shape can be used to ﬁnd both ω and ω .
However, these results are highly model dependent (64–66). For
example, a model based on a random distribution of dipole-dipole interactions gives an exponential decay in the slow modulation limit (64).
As another example, a nonexponential frequency correlation decay gives
different intermediate results than the Kubo-Anderson model (67,68).
The situation becomes even more complicated when multiple
dephasing processes are considered. Figure 2c shows FID curves from two
distinctly different modulation processes. In case #1 the are two processes
modulating the vibrational frequency: one fast and one slow. In case
#2, there is a single process at an intermediate rate. Real experimental
data contain both random noise and systematic errors from tails of
other transitions, weak overtones, hot bands, and isotope lines and from
background emissions from the sample. In the face of these factors,
the subtle differences between CFID
in case #1 and in case #2 are
almost impossible to distinguish, despite the signiﬁcant differences in the
underlying physics.
In the fast modulation limit [Equation (8)], the loss of information is
fundamental. The lifetime of the frequency perturbations is short compared
to their magnitude, and the uncertainty principle precludes a full characterization of υω t by any experimental technique. However, in the slow
modulation limit and in the intermediate regimes, the loss of information in
the FID is not fundamental. The next section shows that the Raman echo
contains additional information about the rate of the frequency ﬂuctuations
that is not present in the FID. By using a combination of Raman echo and
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Vibrational Dephasing in Liquids
389
FID measurements, a much more robust and more nearly unique model of
Cω t can be extracted.
B. A Two-Dimensional Measurement: The Raman Echo
Loring and Mukamel pointed out that additional information can be obtained
by a higher order spectroscopy, the Raman echo (26). The corresponding
observed correlation function is [see Equation (3)]
CRE
1,
3
D
1C 3
exp i
1
υω t dt
i
υω t dt
.
10
0
1
The Raman echo correlation function has two time variables and provides
direct information on the rate of frequency modulation (26,69,70). In
essence, the echo tests whether υω t differs between two different time
intervals, whereas the FID only examines υω t during a single time
interval [Equation (3)]. If υω t changes slowly, i.e., ω × 1 , 3 , the two
integrals in Equation (10) cancel, with complete cancellation at 1 D 3 .
More speciﬁcally,
CRE
1, 3
D CFID j
3j
1
11
in the slow modulation limit. On the other hand, if υω t randomizes
quickly, the change in sign in the integrals is unimportant, no cancellation
occurs, and
CRE
1, 3
D CFID
1
C
12
3
in the fast modulation limit. Thus, a comparison of the Raman echo to the
FID provides qualitative and model-independent discrimination between
fast and slow modulation dephasing.
For more complex systems with intermediate modulation or multiple
modulation time scales, a cumulant approximation is useful [see Equation
(4)] (34):
CRE
1, 3
1
D exp
2 2
0
t0
0
dt00 C0ω t00
1C 3
dt0
0
.
3
dt0 C 2
dt0
0
13
The echo and FID decays derive from the same frequency correlation function, but the echo dissects Cω t with a more complex and informative
operator.
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390
Berg
Figure 1 Two-dimensional contour plots of the log of the Raman-echo correlation
function, ln CRE 1 , 3 , showing the effect of changing the rate of solvent-induced
perturbations. All cases give a Raman line with the same FWHM 5 cm 1 and
FIDs with similar decay times but give very different Raman echo results. (a) Fast
modulation; (b) intermediate modulation (ω D 3.32 cm 1 , ω D 1.60 ps); (c) slow
modulation. Calculations are based on a single Kubo-Anderson process [Equations
(7)–(9)].
This idea is illustrated in Fig. 1, which shows the behavior of the
Raman echo signal for fast, intermediate, and slow modulation dephasing.
In each of the cases shown, the Raman linewidth is the same, and the FID
decays times are very similar. Nonetheless, the Raman echo decays are
qualitatively different, depending on the rate of frequency modulation.
The additional information in the Raman echo is even more important
when multiple dephasing mechanisms might be operating simultaneously.
Copyright © 2001 by Taylor & Francis Group, LLC