Van Hove theory

In what follows, we will quote results of the so-called Van Hove theory for correlation func-tions in the case of neutron scattering. The derivation starts from the general form of the cross section (34). By rewriting the δ-function in an integral representation (Fourier transform) and introducing time-dependent Heisenberg operators, one can show [1, 2, 4] that the cross sec-tion for inelastic or quasielastic scattering takes the following form:

( ) ( )

As discussed for the case of elastic scattering in section 3.4, the cross section separates into a coherent and incoherent part. In ( 0), we have introduced the so-called coherent and incoherent scattering functions S

( )

4^Q,ω

coh and S_inc

( )

^Q,ω . These functions depend solely on the system under investigation and not on the detailed interaction between its constituents and the probe. The strength of this interaction is represented by the coupling constants in front of the scattering functions. The scattering function is given by

( )

^Q, ₂^{1 dte i t d3reiQrG}

( )

^r,t

i. e. is a double Fourier transform of the spatial and temporal pair correlation function:

( )

= ∑∫ δ

(

−

( ) )

⋅δ

(

+ −

^{( )} )

= ∫ ρ

( ) (

ρ +

)

Here, <...> denotes the thermal average. (42) shows that G

( )

r,t can be interpreted as the cor-relater of the particle density. Thus (40) - (42) represent a natural generalisation of the con-cept of a Patterson function discussed in section 2.2. Besides the scattering functions, which depend on scattering vector and energy transfer, it is often useful to introduce the intermediate scattering functions

( )

≡ ∑ − ⋅ ^{ }⋅ ⋅

^{( )}

which are related to the scattering function by a Fourier transform:

( ) ( )

^Q,t

The incoherent scattering function is given by a double Fourier transform

( ) ( ) ( )

r,t

of the self correlation function

( )

= _{∑ ∫} δ

(

−

( ) )

⋅δ

(

+ −

( ) )

In a natural generalisation of the situation for static scattering, the coherent scattering arises from the correlation of a pair of particles, while the incoherent scattering arises from scattering from single particles (see figure 13).

r

_j

(0)

Fig. 13: Illustration of the correlations responsible for coherent scattering (left) and incoherent scattering (right). In the case of coherent scattering, the position of particle j at time 0 is correlated with the position of particle i at time t, while in incoherent scattering the movements of the individual particles are visible.

Let us finally discuss the case of so-called integral scattering, where the experimental conditions are such that the energy change during the scattering process is not resolved, but instead an integration over all energies is being performed. This scattering function for integral scattering S(Q) is given by (here we drop the distinction between coherent and incoherent scattering for simplicity):

)

(47) shows that with integral scattering, the instantaneous correlations are being measured: a snap shot of the sample at time t = 0 is observed.

Correlation functions are rather abstract concepts and we want to illustrate them for the exam-ple of a simexam-ple liquid. The constituents of this liquid (atoms, molecules) are assumed to be spherical particles, which have a strong repulsive interaction potential for short distances and an attractive one for larger distances (see figure 14). The minimum in the interaction potential will give rise to a preferred nearest neighbour distance. Due to the hardcore potentials, the spheres cannot penetrate into each other and produce the excluded region in the pair correlation function for small distances r. This naturally explains the shape of the t = 0 correlation function, depicted in figure 14. The scattering function for integral scattering is obtained via a Fourier transform and is also depicted in figure 14.

t = 0 correlation function integral scattering Interaction potential

Fig. 14: Snap shot (top right), pair interaction potential (top left), t = 0 correlation function (bottom left) and the scattering function for integral scattering (bottom right) for a simple liquid.

Let us now look at the time dependence of the pair correlation and self correlation functions (see figure 15).

Fig. 15: Schematic plot of the pair correlation (solid line) and self correlation functions (dotted line) on the left and the resulting intermediate scattering functions on the right for a simple liquid at different times.

S(Q,t) S(Q,t) S(Q,t)

For t = 0 the self correlation function is given by a δ-function at r = 0. The pair correlation function follows the static correlation function. For intermediate times, the self correlation function broadens to a bell-shaped function due to the diffusion process, while the pair corre-lation function loses its structure. Finally in the long time limit, the self correcorre-lation simply vanishes for a liquid while the pair correlation function assumes a constant value. The inco-herent intermediate scattering function as the Fourier transform of the self correlation function is Q-independent at t = 0, decays for intermediate t with respect to the t = 0 value, where the decay is faster for higher Q, and finally vanishes in the long time limit. The coherent scattering function is given by the static coherent scattering function for t = 0. It decays for intermediate times with respect to the t = 0 value, but the decay is less pronounced at the structure factor maximum. In the long time limit, the coherent scattering function decays to zero for any Q just as the incoherent scattering function.

Let us return to the case of integral scattering from a simple liquid. We make the simplification that the liquid consists of identical spherical and isotropic particles. Then it is easy to show [1, 4] that the intensity is given by

( ) ( )











 +ρ ∫ ⋅

⋅

⇒

Vs

R Q ei R g 3R d 2 1

Q f static~

I

( )

2 S

( )

Q

Q

f ⋅

≡ (48)

i. e. the intensity for static scattering separates into a pre-factor depending on the experimental geometry and the type of radiation, the square of a form factor, which describes scattering from a single particle, and the scattering function (for the static case often referred to as struc-ture factor), which contains information on the particle correlation and is independent of the type of radiation used. The structure factor gives an average snap shot picture g(R) of the sample. As an example, figure 16 shows the structure factor of liquid ³⁶Ar at 85 K. The points are deduced from a neutron scattering experiment, the curve is generated by a molecular dy-namics calculation using a Lennard-Jones potential. The insert shows the pair correlation function g(R) of liquid Ar calculated by a Fourier transform of the data [7].

Fig. 16: Static scattering function for liquid ³⁶Ar at 85 K and resulting pair correlation function (insert) [7].

Im Dokument A 1 Scattering (Seite 21-25)