Neutron Scattering

with the macroscopic scattering cross-section dl/M in units of [cml ], the sample volume
V, the number N of atoms in the sample and the coherent scattering length b; ofthe atom i at
the position r i . One can consider the resultant diffraction pattern as a coherent superposition
of spherecal waves, emanating from single atoms with an amplitude determined by the
Fia .
14 .1 : Two pencils ofrays ofthe scatteringprobe in two particles of different size
coherent scattering length . In the region of small angle scattering the relation Q < 27c / a is
always fulfilled with the lattice constant a.
according to
e -I
_>
length density p(r) =p and volume V . From Eq.(14 .2) one gets
Then the sum in Eq.(14 .1) can be approximated
Jd3rPUe
r
; QI
_-
by an integral of the coherent scattering length density p(r) = b ; / S2 (atomic volume S2) and
the phase factor . In this approximation one get the basic relationshipfor SANS :
dl LQ)=
V
~Q_ z
d3r pr~e -
(14 .2)
In a ferst example we consider an homogeneous sample wich the constant coherent scattering
_d~ =F2 fd je'-
M V ~ ~
3-
L=(27c)s PZ g(Q)
V
Samples or apertures in front of the Sample are typically of cm size . Diffraction at those large
heterogeneities occurs within angles which are too Small to bc resolved ; such diffraction can
be approximated by a Delta fonction . Diffraction caused by the mean coherent scattering
14-2