Neutron Scattering

scattering is observed with the average scattering length b . In addition there exists ineoherent
scattering as a result ofthe isotope disorder . It is not able of interference and isotropie .
We now consider spin dependent scattering from one isotope with the nuclear spinj. Both the
nuclear spin as well as the neutron spin are statistically distributed. There exist two compound
spin states :
(i) neutron and nuclear spin are parallel : then I=j + 1/z. This spin state has the multiplicity
of 2j + 2 . Its scattering length is b+ .
(ii) neutron and nuclear spin are antiparallel I = j - 1/2 the multiplicity is 2j and the
corresponding scattering length equals b - . The a priory probabilities for the compound
spin states are given by the number of possibilities for their realization divided by the
total number.
+_ 2j+2 _ j+1
p
2j + 2 + 2j 2j + 1
p - = j
2j+1
(2 .30)
The corresponding average scattering lengths become
b = p + b+ + p- b- (2 .31)
b2 = p + b +2 + p - b -2
We now consider the proton as an example . Here j= 1/2, b + =1 .04 . 10 -12 cm,
b- = -4 .74 - 10-12cm, p+= 1/4, p = 1/4. Inserting into Eq .[2 .31] we fmd b = -0 .375 -
10-12 cm
and b 2 = 6 .49 . 10-24cm2 . These values lead to a coherent cross section
'7,.h = 4z . b 2 =1 .77 .10-24cm2 . For the incoherent cross section, we obtain
6;,, = 4z - (b 2 - b2 ) = 79 .8 .10 -24 cm 2 . This value is the largest incoherent cross sections of all
isotopes and makes the hydrogen atom the prime incoherent scatterer which can be exploited
for hydrogen containing materials .
2- 17