Neutron Scattering

Thereby we now also consider the spin states ofthe nuclei before and after scattering s and s' .
The initial state probabilities are given by Ps .
d6_ -
P exp (iQ (R -R,, » (s' b;, b s > (2 .25)
dÇ2 SS' il'
We commence with spin independent interactions and assume that spatial coordinates and
scattering lengths are not correlated - that means different isotopes are distributed randomly .
Then Eq .[2 .25] may be evaluated to
_d6
M ,
(bâ b,)
exp (iQ (R, -2R, .»
(2 .26)
In order to evaluate Eq.[2 .26] further, we introduce two scattering lengths averages - the
mean square average and the mean scattering length .
They are given by
N
b2=1 ~b2 ; b =l ~,br
N ,
(2 .27)
With these defmitions, the average product of b, and bt' becomes
(br br~ = (1 _ Srr, ) b2 +611 , b 2
(2 .28)
Finally, introducing Eq .[2 .28] into the expression for the cross section Eq.[2 .26] we obtain
d6 -1 [(1 _ Srr')
-2 + srr , -2] exp iQ ( Rj - Rj' )
dÇ2
i,,
(2 .29)
-2
-)+ b2 b2 2.iQ~Ë exp, -R I ,)
il ,
Obviously, the cross section contains two contributions .
A coherent one where we have
constructive interference of the neutron waves eminating from the different nuclei . This
2- 16