Neutron Scattering

Starting point is the exact atomic structure of the Sample . In this section a crystal with
some density modulations is chosen (Fig. 6 .2 top row) . The potential V(r) of such a System can
be written as a product of the undisturbed infinite crystal lattice potential Vatt (r) and the
modulation Vmod(r) . As it was shown in previous sections in Born approximation the scatternd
intensity
I(Q) -IA(Q)IZ _ fV(r)exp(iQ .r)d'r2 = If Viatt(Y)Vod (E )exp(iQ - r)d 3r 2
= IF{Vatt(Y)V_d(Y)1(Q)IZ
of the sample can be calculated by performing a Fourier transformation F{ V(r) 1 of V(J . The
convolution theorem for Fourier transformations can be used to modify equation (6 .1) . This
theorem states that the convolution © of two Fourier transformed functions f=F{gi 1 gives the
same result as the Fourier transformation of the product of both functions gi . Thus,
F{g , 'gz }= F{g , }©F[92) = .f 0f2 = ff (q)f, (q - Q)d s q
(6 .2)
where the integral is the definition of the convolution .
It is also known from previous sections that for an infinite periodic crystal the lattice
potential Vatt (r) can be written as a surr of delta-functions weighted with the scattering length
and located at the position of the atoms . The Fourier transformation of V,att (r) also yields delta
functions : the Bragg peaks at the reciprocal lattice positions . In contrast, the Fourier
transformation of Vmod(r) is usually a `smooth' function which is strongly decreasing for large
IQI " The result of the convolution is depicted in the second row of Figure 6 .2.
By doing a small angle scattering experiment only wave vector transfers Q with a mean
value close to 0 are considered. All other values are omitted. The result of the magnification
around Q=0 is shown in the third row of Figure 6 .2 . In good approximation it is identical to a
convolution of just a single delta-function at 1=0 with the Fourier transformation of V,od(r) . In
real space (Fourier backtransformation) a single delta peak corresponds to an infinite sample
with homogeneous potential which turns out to be the averaged value of Vatt(r) . The fourth
row of Figure 6 .2 proofs that no information about the atomic structure is necessary to explain
a Small angle scattering pattern.
6 . 3