chapter  8
Spectroscopic Structure Factor
Pages 42

One important application of diffusion is to spectroscopic measurement of what is called the structure factor that is detected in the time domain or the frequency domain. The problem is best understood in the context of gas phase spectroscopy in an infrared or microwave regime. Consider an optically active atom in the surroundings of a buffer gas composed of optically neutral atoms. The density of optically active atoms is so low that we can consider them one at a time. If an active atom emits a radiation of wave vector

k at a time t1, the phase of the emitted radiation is contained in a factor:

φ = − •

exp ( ( )),1ik r t (8.1)

where ( )1r t is the instantaneous position of the emitter. The measured struc-

ture factor (for t1 < t2) given by Dattagupta (1987) is

S k t t e e ik r t ik r t

(8.2)

where the angular brackets <….> indicate a statistical average over the randomness of the position vectors. Now,

∫= + ′ υ ′( ) ( ) ( ),2 1 1

r t r t dt t t

(8.3)

and υ( )t is the velocity vector. Hence

S k t t e i dt k t t

(8.4)

If υ( )t is a stationary stochastic process (see Chapters 2 through 5), time t1 can

be chosen to be the origin of time, in which case the structure factor becomes

( ; ) . ( . ( ))

0S k t e i dt k t t

(8.5)

Interestingly, the experimentally measurable structure factor is identical to the characteristic functional of a stochastic process [cf. Equation (3.31)] if we take

( )k t to be independent of t. By fixing the direction of observation of the

emitted radiation, the relevant velocity is the component along that direction, denoted by (t). Hence, Equation (8.5) simplifies to:

( , ) . ( )

0S k t e ik t dt

(8.6)

8.2.1 Weak Collision Model: Gaussian Process

We first consider a “heavy” active atom in the midst of lighter buffer gas atoms. The active atom is kicked around randomly due to thermal fluctuations but because of the mass mismatch, the effects of collisions are weak so that the post-and pre-collision velocities differ only marginally. Consequently, the underlying stochastic process, i.e., velocity ν(t) can be characterized by a linear damping term in the equation of motion according to the Langevin equation (Chapter 4). In such a weak collision model, the resultant stochastic process is a Gaussian and hence the expressions derived earlier for the characteristics functional can be directly utilized here. Thus, from Equation (3.33),

∫ ∫∫= < υ > − < υ υ >( , ) exp[ ( ) 2 ( ) ( ) ].1 1 0

1 2S k t ik dt t k

dt dt t t t tt

(8.7)

However, because ν(t) is a stationary process,

< υ > = < υ >( ) ,1t

< υ υ > = < υ υ − > >( ) ( ) (0) ( ) , .1 2 2 1 2 1t t t t t t (8.8)

Furthermore, for an optically active atom in thermal equilibrium with a buffer gas, its velocity is governed by a Maxwellian distribution. Additionally, Doob’s theorem for a one-dimensional Gaussian process applies (cf. Section 3.3.1). Therefore, from Equations (3.34) and (4.28),

< υ > =

< υ υ − > = −λ −

(0) ( ) .2 1 | |2 1t t K T m

In this context, λ has the interpretation of the mean rate of collisions, i.e., λ−1 is the mean free time of collisions. Thus

(8.10)

Alternatively, we may avoid introducing the velocity in Equation (8.3) altogether and work directly in the position space, i.e., with Equation (8.2) which yields

S k t e eik x t k x t (8.11)

for a Gaussian process, in which x(t) follows the Langevin equation. Substitution of Equation (4.44) into Equation (8.11) directly leads to Equation (8.10). The structure factor in the frequency domain, also called the spectral line shape, is obtained from the one-sided Fourier transform of Equation (8.10):

∫ω = pi − ω − λ λ − + −λ ∞

( , ) 1

Re exp ( 1 ) , 2

S k dt i t k K T m

t eB t (8.12)

which can be developed as a continued fraction:

ω = pi

ω +

ω + λ + ω + λ +

( , ) 1

Re 1

/ 2 /

2 ...