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P
where z = 12 is the number of nearest neighbours and sq = nn cos(q · rnn ), where in turn
rnn is the lattice positions of the nearest neighbours.
This dispersion relation may be modified with a small effort, since it is given as a separate
c-function attatched to the component.
To calculate
need
R to 3transform the q sum into an integral over the Brillouin
P dσ/dΩ we −3
zone by q → N Vc (2π)
BZ d q. The κ sum can now be removed by expanding the q
integral to infinity. All in all, the partial differential cross section reads
Z
d2 σ ′
1 1
~κ2
2 kf 1
nq + ∓
δ(ω ± ωq )δ(κ ± q)d3 q
(κ, ω) = N b
dΩdEf
ki 2M
~ωq
2 2
2 2
1 1
2 kf ~ κ
= Nb
nκ + ±
δ(~ω ± d1 (κ)).
(8.40)
ki 2M ~ωq
2 2
8.6.2
The algorithm
All neutrons, which hit the sample volume, are scattered into a particular range of solid
angle, ∆Ω, like many other components. One of the difficult things in scattering from a
dispersion is to take care to fulfill the dispersion criteria and to find the correct weight
transformation.
In Phonon simple, the following steps are taken:
1. If the sample is hit, calculate the total path length inside the sample, otherwise leave
the neutron ray unchanged.
2. Choose a scattering point inside the sample
3. Choose a direction for the final wave vector, k̂f within ∆Ω.
4. Calculate possible values of kf so that the dispersion relation is fulfilled for the
corresponding value of kf . (There is always at least one possible kf value [43].)
5. Choose one of the calculated kf values.
6. Propagate the neutron to the scattering point and adjust the neutron velocity according to kf .
7. Calculate and apply the correct weight factor correction, see below.
8.6.3
The weight transformation
Before making the weight transformation, we need to calculate the probability for scattering along one certain direction Ω from one phonon mode. To do this, we must integrate out
the delta functions in the cross section (8.40). We here use that ~ωq = ~2 (ki2 − kf2 )/(2mN ),
R
κ = ki − kf k̂f , and the integration rule δ(f (x)) = (df /dx)(0)−1 . Now, we reach
′
Z
~2 κ2
d2 σ ′
1 1
dσ
2 kf
=
nκ + ±
.
(8.41)
dEf = N b
dΩ j
dΩdEf
ki 2M d1 (κj )J(kf,j )
2 2
where the Jacobian reads
J =1−
Risø–R–1538(rev.ed.)(EN)
mN ∂
(d1 (κ)) .
kf ~2 ∂kf
(8.42)
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