Neutron Scattering

(A+k2 ) yr = u(Z:) yr
V( u~r)= z " r)
The wave equation is solved by the appropriate Greenfunction
(A2+k2)GL-r') = -4zd(r-r')
'b ~ -=' I
G(r-L' ) = e Ir _ 0
(2 .12a)
(2.12b)
u (r) yr (r) = f dr' S (r - r') u(r') yr(r')
(2 .12c)
where G(r) is the Greenfunction solving the wave equation with the o-function as
inhomogeneity . Using Eq.[2 .12c] the wave Eq.[2 .11] may be formally solved by
v/ (r) = e'=
_ 1 f dr' G (L-r') u ( L' ) V ( r' ) (2 .13)
4; -
In Eq .[2 .13] the first part is the solution of the homogenous equation and the second part the
particular solution of the inhomogeneous one . The integral Eq .[2 .13] may now be solved by
iteration. Starting with the incoming wave vP= eikr as the zero order solution, the v+ 1 order
is obtained from the order vby
e'k -
41 f G(r-L') u(L') y/v(r') d 3 r' (2 .14)
In the Born approximation we consider the first order solution which describes single
scattering processes . All higher order processes are then qualified as multiple scattering
events . The Born approximation is valid for weak potentials . 6'°` 0 1 where a is the size
a z
of the scattering object . For a single nucleus this estimation gives about 10-7 and the Born
approximation is well fulfilled .
2- 1 1