Neutron Scattering

_du _
mn
\2
A2-~2z r2~
I(k'
IVI
k)I2 (2 .19)
We now also allow for inelastic processes, where the sample undergoes a change of its state
from I Â) -> I Â') . For the cross section we now also have to consider the changes of state as
well as different length of the wave vectors of the incoming and outgoing waves, which lead
to factors k' and k in the cuvent calculation.
(2 d6 k' ~ m n
dÇ2) (,k=, ;~) - k 2z
2
I(k',Z' IVI k,)1 2
.20)
The scattering event must fulfill energy and momentum conservation .
fmally at the double differential cross section
With that we arrive
a26
_ _k'
M aü) k (2z
M)n
h2
\ 2
Pa 1 (k', a' IVI k,a)2 s(hw + (2 .21)
The summation over  is carried out over all possible initial states À of the system with their
appropriate probability P Â. The sum over V is the surr over all final states, the o-function
takes care of the energy conservation, thereby hao is the energy transfer of the neutron to the
system . This double differential cross section will be discussed in detail in the lecture on
correlation functions .
2 .6 Elementary scattering processes
2 .6 .1 The Fermi pseudo potential
The interaction of the neutron with a nucleon occurs under the strong interaction on a length
scale of 1 .5 - 10-15m. For that process, the Born criterium is not fulfilled and for the scattering
process on a single nucleus the Born series would have to be summed up . Fortunately, this is
a problem of nuclear physics and for the pulposes of neutron scattering a phenomenological
approach suffices . Considering that the wavelengths of thermal neutrons are in the order of
10-1°m we realize, that they are much larger than the dimension of a nucleus of about 10-15m.
2- 13