Neutron Scattering

9
We note one important exception, the dynamical scattering theory, which considers the
scattering problem close to a Bragg reflection in a crystal . Then multiple beam interferences
are important and the Born approximation ceases to apply . For most practical puiposes,
however, the Born approximation is valid. Under the Born assumption the scattered wave
function becomes
1 e`k~= 'I 2m,
y/ = er-r - - = V (I:') e 'kI dzr' (2 .15)
4z Ir-r~I
z
In order to arrive at a final expression, we have to expand all expressions containing r and r'
around r. Thereby, we consider that r' is a sample coordinate and small compared to r. We
have
1 '
- 1 + O 1
Ir-r I r ( rr
)
(2 .16)
r-r'-
Ir-r'I -- grad (r) = r-r'
Inserting Eq.[2.16] into Eq.[2 .15] gives
1 e'~ - =k=~' 2m
e d'r' V/ e' - - -
'- =` V (r') e'~I
(2 .17)
4z r hz
k
: direction ofthe scattered wave k'
Irl
EI
Eq . [2 .17] may be written as
(k'-
IVI k)
m
e (2 .18)
2Z hz r
fp)
leads to the final expression for the cross section Eq .[2 .8]
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