Neutron Scattering

to a volume V is simply a plane wave with wave vector K :
ID'(r) = WE) _ exp(iK - r) . (5 .64)
Using this expression, thé matrix element in thé double differential cross section (5 .27)
can immediately be calculated :
(KIexpiQ-rlr,)
VJv d3rexp(i(Q+ - K)-?')=6(Q+r,-K) . (5 .65)
The resulting delta function expresses the momentum conservation . Only if momentum
is conserved the matrix element is 1-otherwise zero .
In the second step we have to consider energy conservation .
wave vector K is
The energy of the atom with
(5 .66)
where ms , is thé mass of thé scattering atom . For thé évaluation of thé delta function
in (5 .27) we need thé energy différence between thé states K and W .
Because of thé delta
function factor (5 .65) only such states with K' = K - Q have to be considered and for
those thé energy différence is :
(5 E, - E = - (Qz + 2Q ) .
.67)
2m .,
With this result one can calculate the scattering function :
S(Q, w) = E Pub (hw - h (Qz + 2Q - E)~ . (5 .68)
2msc
In thé limit of a large volume V, K becomes a continuons variable .
thé component of K parallel to Q is relevant .
Therefore, (5 .68) can be written as a onedimensional
integral :
In addition only
S(Q, cv) = J
dKP,6
(hw
- 1' (Qz +
2QK))
. (5 .69)
2m sc
The probability of a momentum state K follows from the Boltzmann distribution :
1 1i2K2
P, Z eXp ( 2m-kRT
(5 .70)
with thé state surr being
z z
Z = ~ dr exp _ ~ r,
( 2m kBT)
5- 2 1
_ 2~rme~kB T .
(5 .71)