Neutron Scattering

where we can arbitrarily set ri = r i (0) because of translation of time invariance . Using
this result the final expression for the double differential cross section is obtained :
k' 1 %
dtexp ( - iwt)
8528w k27rh J-.
(5 PA 1: bi b~ (A exp(-iQ - r j (0» exp(iQ - r i (t» A) . .35)
ij
This equation averages over the scattering length distribution (which may depend on the
spin orientation distribution with respect to the incident neutron's spin) . This produces
coherent and incoherent scattering as explained in lecture 1 . In addition the initial states
of the scattering system are averaged weighted with the probability of their occurrence
PA . The latter is given by the Boltzmann distribution
PA = Z
exp (-EA /kBT) with Z = exp (-EA/k BT) . (5 .36)
We now denote this thermal average by angular brackets ( . . .) while that over the scattering
lengths be written as an overline . . . . Keeping in mind that for equal indices
bibi = Ibil 2 has to be averaged while for unequal indices the scattering lengths itself will
be averaged we end up with the usual separation into incoherent and coherent part :
k' z Ibl
- Ibl z p( J - ~ dtexp(-iwt) .~ ex - iQ . _~(0))ex r_ p(iQ - _Z()) r_ t
27rh
k'
+ Iblz
k 2rh J
~ dtexp( - iwt) E ( A ex p(-iQ
. ri( 0))exp(iQ . r j (t)) A . (5 .37)
z,9
The first term is the incoherent scattering . It involves the coordinate vector operators of
the Same atom at different times . The second, the coherent term correlates also different
atoms at different times . The material dependent parts are now defined as the scattering
functions
Si.,(Q' W) --
Scoh(Q,W)
1 f°°
dtexp(-iwt) E(exp(-iQ - ri(o)) exp(iQ - ri(t))) (5 .38)
27rhN oo i
1 dtexp(-iwt) (exp(-iQ - ri(0» exp(iQ - r j (t)» . (5 .39)
27rhN
1,3
In terms of the scattering functions the double differential cross section can be written as
z
as2a~ _
N ((Iblz - Ibl2) Sin c (Q, W) + Ibl
2
s~h(Q, W)) .
5- 15
(5 .40)