Scattering and Correlation Functions

The experimentally determined scattering function is given explicitly by

^ e x p t (Q ,^ ) _ 6 x p ( —h w / 6 ' i n c ( Q , 0 7 ^ (w ) T B ( 2 .3 3 )

where k^ is the Boltzmann constant, B is the background scattering, and

T h eo retica l Background 42

tion, which depends upon details of the system under investigation. The scat­ tering function is convoluted with the instrum ental resolution function 7^(w) (the convolution is denoted by (g>). The detailed balance factor accounts for the asymmetry in the distribution of (energy) states in the system: the scattering function is a symmetric function,

S{Q_,uj) = S { —Q j —oj) (2.34)

i.e. the a priori probability th at a neutron with an energy of huj will effect a transition between two states of the system separated by the same energy is independent of the direction of the transition. However, the probability th at the system will initially be in the higher energy state will be lower by a factor

exp {—huj/kbT) than the probability that it will be in the lower energy state.

Van Hove [42] first suggested describing the scattering function S { Q ,uj)

in terms of the correlation function G{r,t). The correlation function is related to the scattering function by a double Fourier transform ation involving r and t .

In this respect it is useful to define a third function, the interm ediate scattering function /((J,^ ), which represents the Fourier transform ation of G {r,t) with respect to r. The full set of functions is then

I{Q_,t) =

J

S[Q^,u)) = 2 ^

J

The correlation function relevant to incoherent quasielastic neutron scattering is the self pair-correlation function

GsiL-it) = < ^(l ~ E o { t ) R o W ) ^ (2.37)

which describes the conditional probability of a particle being found at r at tim e t if the same particle was at the origin ^ at tim e i = 0. G s { r , t ) must

T h eo retica l Background 43

therefore satisfy the normalization condition

j

The brackets < > in Equation 2.37 denote the therm al average, i.e. the average over all initial states of the system at a given tem perature.

In the case of incoherent scattering, the interm ediate scattering function can be generalized as

^ ex p (iQ .n (i)) e x p ( - : Q .n (0)) >

(2.39) where the sum runs over all particles i and N is the total number of particles. 7inc(Q, t) therefore represents the average over all scatterers z, and decomposes into two parts

/inc(Q, t) = 7inc(0, oo) + 7 ^ ( 0 , 0 (2.40)

which, upon Fourier transformation, become

5'inc(0,w) = 7inc(Q,oo)6(w) + (2.41)

The interm ediate scattering function evaluated at infinite tim e clearly repre­ sents the elastic part of the total scattering upon which the tim e dependent, inelastic scattering is superimposed.

The vector r{t) can be written as

r{t) = r " \ t ) + r ~ ‘(<) + r ‘’^“ ‘’(«) (2.42)

where the superscripts vib, rot and trans represent those parts of r due to vibrational, rotational and translational motion, respectively. The separation

T h eo retica l B ackground 44

of the individual contributions assumes that the different types of motion are independent of each other.

The interm ediate scattering function then becomes

Iinc(Q,t) = X i r ^ ( Q , t ) X (2.43)

and Fourier tranform ation of Equation 2.43 results in

Sinc(Q,u>) = Sy^(Q,u,) 0 ® S l ^ r i Q , ^ ) (2.44)

The total scattering function is the convolution of the vibrational, rotational and translational scattering functions.

In the quasielastic region (|&j| < 2 meV), the vibrational part can be treated as small amplitude, harmonic motion and is then given by the Debye- Waller factor

5T„c = exp ( - Q ^ < >) (2.45)

where < > is the mean square displacement of the oscillating particle. As it is independent of the energy transfer, the Debye-Waller factor reduces to a factor in Equation 2.44.

The exact expressions of the rotational and translational scattering fac­ tors depends upon the nature of the motion. To describe these parts of the scattering function requires a dynamic model, from which the functions are derived (see also Appendix A). In general, if there is no correlation between the motions of distinct particles, the scattering functions take the form

5 i ; f ‘' “ “ (Q,a;) = Ao'°‘' ‘™“(Q) 5(o;) +

T h eo retica l B ackground 45

where Aq{Q) is the elastic incoherent structure factor (EISF) and the Ai{Q)

are the amplitudes of the Lorentzian functions Ciiuo). ^(o;) is D irac’s delta function in lo. The number of terms i in the summ ation depends upon the

dynamic model. The EISF and the amplitudes of the Lorentzian function convey information on the geometry of the motion whereas the Lorentzian functions describe the temporal aspects of the dynamics. They represent the Fourier transformation of exponential functions describing the motion in the tim e-dom ain (see Appendix A). The half width at half maximum (HWHM) of the Lorentian functions are related to the correlation times r of the motion. Exponential decay of the dynamics arises when the motion of a single nucleus is not correlated with the motion of other, structurally independent

nuclei in the system.

Comparison with Equation 2.40 reveals th a t the EISF is equivalent to the interm ediate scattering function evaluated at infinite time. It can be w ritten as

/el ('/n)

=

im r h m

where I denotes intensity and the superscripts el and qe denote the elastic and quasielastic parts thereof. From this description it follows th at the EISF represents the fraction of elastic scattering with respect to the total intensity (within the experimental energy window). The im portance of the EISF is th at it gives direct information on the nature of the observed motion, both in terms of geometrical features of the motion [via its Q-dependence) and in term s of its tem perature dependence. The correlation times of the relevant dynamic processes can be deduced from the widths of the Lorentzian functions Ci {uj ).

T h eo retica l Background_________ ^

tion w ritten as

= J j exp (— - rg]) f (H, f : Ho, 0) f (Ho) cfH cfHo

(2.48) where P (H ,t : Oo,0) is the conditional probability of finding a particle at a position given by H at a time t if the same particle was at the position Qq at

tim e t = 0. P(ilo) is the distribution of all initial positions. The functions P ( n , t : rio ,0) and P(rio} are determined from a dynamic model and some

examples of derivations are given in Appendix A. The models used in this work are discussed in the appropriate chapters in greater detail.

The Fourier transformation of /(Ç , t) with respect to after any addi­ tional averaging required by experimental considerations, gives the required scattering function 5'(Q, w).

The interm ediate scattering function (Equation 2.48) contains the scalar product of the scattering vector Q and the position vector r. To properly ac­ count for the relative orientations of these vectors, the interm ediate scattering function must be averaged over all possible orientations of the position vector r. The resulting scattering function therefore differs, depending upon whether the sample is polycrystalline, semi-oriented or oriented. Averaging procedures are discussed in Appendix A.