Download Component Manual for the Neutron Ray

This means that the integrated oscillations (around 1) of Scoh (q) are directly related to
the density of the material ρ. In practice, the function S(q) is often known on a restricted
range q ∈ [0, qmax ], due to either limitations in the sample molecular dynamics simulation,
or the measurement itself. In first approximation we consider that Eq. (8.52) can be
applied in this range, i.e. we neglect the large q contributions provided S(q) − 1 converges
faster than 1/q 2 . This is usually true after 2-3 oscillations of S(q) in liquids. Then, in
isotropic liquid-like materials, Eq. (8.52) provides a normalisation sum-rule for S.
8.7.3
Theoretical side - scattering in the sample
The Eq. 8.44 controls the scattering in the whole sample volume. Its implementation in
a propagative Monte Carlo neutron code such as McStas can be summarised as follows:
1. Compute the propagation path length in the material by geometrical intersections
between the neutron trajectory and the sample volume.
2. Evaluate the total cross section from the integration of the scattering law over the
accessible dynamical range (Section 8.7.3).
3. Use the total cross section to determine the probability of interaction for each neutron
along the path length, and select a scattering position.
4. Weight neutron interaction with the absorption probability and select the type of
interaction (coherent or incoherent).
5. Select the wave vector and energy transfer from the dynamic structure factor S(q, ω)
used as a probability distribution (Section 8.7.3). Apply the detailed balance.
6. Check whether selection rules can be solved (Section 8.7.3). If they cannot, repeat
(5).
This procedure is iterated until the neutron leaves the sample. We shall now detail the
key steps of this implementation.
Evaluating the cross sections and interaction probability
Following Sears [31], the total scattering cross section for incoming neutrons with initial
energy Ei is
σs (Ei ) =
ZZ
d2 σ
Nσ
dΩdEf =
dΩdEf
4π
ZZ
kf
S(q, ω)dΩdEf
ki
(8.53)
where the integration runs over the entire space and all final neutron energies. As the
dynamic structure factor is defined in the q, ω space, the integration requires a variable
change. Using the momentum conservation law and the solid angle relation Ω = 2π(1 −
cosθ), were θ is the solid angle opening, we draw:
ZZ
σS(q, ω)q
σs (Ei ) = N
dqdω.
(8.54)
2ki2
98
Risø–R–1538(rev.ed.)(EN)