Ph.D. thesis (pdf) - dirac

4.3. Inelastic Scattering Experiments 53
the neutron and r is the position of the nucleus. b is the scattering length of the
nucleus, this quantity depends on the spin state of the nucleus and can hence differ
for different atoms of the same species.
The calculation of the scattering cross section is based on the following assumptions.
(i) That the neutron can always both before and after scattering, be described by
a plane wave. (ii) That the neutron is only scattered once by the sample (iii) The
probability of a transition where the neutron goes from state Q to Q ′ and the sample
goes from λ to λ ′ is given by first order perturbation theory “Fermis Golden rule”,
meaning that it is proportional to the matrix element |〈Q ′ λ ′ |V |Qλ〉| 2 .
The result is the basic expression for the partial differential cross-section
∂ 2 σ
∂Ω∂E = Q out
Q in
1
2π
∑
∫ ∞
b i b j 〈exp(−iQ ·r i (0))exp(iQ ·r j (t))〉exp(−iωt)dt.
i,j
−∞
where the sums over i and j are to be taken over all the atoms in the system.
(4.3.4)
In the case of photons it is the interaction with the electrons that strongly dominates
over the interaction with the nucleus. The dominating term is due to Thomson
scattering which describes the coupling between the electronic current and the electric
photon field 7 . When using photons in the study of the structure and dynamics
on an intermolecular scale it is assumed that the electrons have a fixed position
with respect to the nucleus (the adiabatic approximation). By doing so it becomes
possible to factor out the relative positions of the electrons in a form factor. (The
expression for the cross section is given the next section). The form factor then plays
a role similar to that of the scattering length in neutron scattering. The two major
differences between the scattering length and the form factor is (i) the form factor is
Q-dependent and this leads to an intrinsic decrease of intensity as Q increases. (ii)
the magnitude of the form factor is proportional to the number of electrons meaning
that larger atoms give larger contribution to the scattering.
4.3.2 Coherent and Incoherent Scattering
The scattering length b depends on the spin state of the nucleus interacting with
the neutron beam. Since we consider scattering from a large system b i b j is replaced
by its average value b i b j . Assuming that b i and b j are statistically independent it
7 Considering only Thomson scattering means that we again ignore magnetic interactions.