Neutron Scattering

1 ik, r 2rnn fexp(iklr -i"'I)
h 2 4zIr-r'l
(3 .17) is nothing else but a mathematical formulation of the well-known Huygens principle
for wave propagation.
The approximation (3 .17) assumes that the incident plane wave is only scattered once from
the potential V(r') . For a stronger potential and larger sample, multiple scattering processes
can occur . Again, this can be deduced from the integral equation (3 .16) by further iteration .
For simplification we introduce a new version of equation (3 .16) by writing the integral over
the "Greens function" as operator G :
V -V/ +GVV (3 .18)
The so-called first Born approximation, which gives the kinematical scattering theory is
obtained by substituting the wave function yr on the right hand side by yi
y/ ' =V° +GVyr ° (3 .l9)
This first approximation can be represented by a simple diagram as a sum of an incident plane
wave and a wave scattered once from the potential V.
The second approximation is obtained by substituting the solution of the first approximation
(3 .19) on the right hand side of equation (3 .18) :
V2 = Vf' +GVV1
=V° +GVyi +GVGVyi (3 .20)
Or in a diagrammatic form :
3-8