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Equation (1.6) can also be rewritten in terms of an autocorrelation function (sometimes known as convolution square function) writing r ′ − r ″ = r
(1.7)
Then
(1.8)
The autocorrelation function has the physical meaning of the overlap between a particle and its ‘ghost particle’ displaced by r (Figure 1.2). This function is the continuous version of the Patterson function familiar from crystallography.
Figure 1.2 Ghost particle construction. The overlap volume (shaded) is the autocorrelation function.
The autocorrelation function for solid geometrical bodies can be calculated analytically. For a sphere of radius R the expression is isotropic and is given by [5–7]
(1.9)
This is a smoothly decaying function of r. The expression for a cylinder is provided in Ref. [8] and can be calculated for other structures, asymptotic expressions for cylinders and discs are given in Eqs. (1.83) and (1.84).
Equation (1.6) can alternatively be written for uncorrelated scatterers as
(1.10)
1.3.2 Isotropic Scattering Systems
For isotropic scattering the scattered intensity will only be a function of the wavenumber q and an orientational average (indicated by <..>Ω) is performed, i.e. Eq. (1.5) becomes
(1.11)
where rjk = rj − rk.
The average over all orientations of rjk can be evaluated as follows
(1.12)
This leads to the Debye equation for scattering from an isotropically averaged ensemble:
(1.13)
Considering a continuous distribution of scattering density, the orientational averaging of Eq. (1.12) has to be performed over Δρ(r) since it is a function of r:
(1.14)
The isotropic average of Eq. (1.14) leads, via Eq. (1.12), to
(1.15)
Here, Dmax is the maximum dimension of a particle (maximum distance from the geometric centre).
In terms of the isotropically averaged autocorrelation function γ(r) this can be written as
(1.16)
Or
