Neutron Scattering

In real liquids however usually an interaction exists which depends (only) on the distance
of the atoms r12 = r2 - El . We express the deviation from (5 .5) by a pair correlation
function defined by
g(ri2) = P((~P~))
With the pair distribution function n(rl, r2) = N(N - 1)P(El , r2) which expresses the
probability density for any pair of atoms to occupy positions rl
and r2 we obtain
2)
(5 g(L12) =
.7)
n('2r po
because in the lirait of large N we have N(N - 1) ~ N2 .
The qualitative features of the pair correlation function are shown in figure 5 .1 . For r --> 0
one finds g(r) = 0 because two atours cannot be at the saine position .
(5 .6)
Usually, this is
also true for distances r < ro because the atours cannot penetrate each other and have a
"hard core" radius ro .
For r --> oc the lirait is g(r) = 1 because the interactions decay
with distance and the P(rl , r2 ) reverts to its default value 5 .5 . At intermediate distances
g(r) shows a peak because the probability density which is lacking at r < ro must be
compensated .
Its location is usually close to the minimum of the interatomic potential,
i .e . close to the average next neighbour distance rnn .
Using the pair correlation function one can formulate the differential neutron cross section
for a monatomic liquid'
Here (d (F /dÇZ) coh
du
CdZ)
coh
N
= Ibl 2 E exp N - (ri - ri» ~ - (5 .8)
(i ' j= I
denotes the angle dependent coherent scattering cross section and b is
the average scattering length of the liquid atours . Q is the scattering vector, i .e . the
difference between incoming and scattered wave vector of the radiation, Q = k - k' .
average in the expression can be evaluated using the pair correlation function (5 .7) :
do,
()
dQ coh
_
- Ibl 2
+ ~E exp (1Q - (ri - rj
C1 i4j
1 Exactly speaking, this expression is valid only for monisotopic liquids . Otherwise,
there would be an additional incoherent terni like the one discussed for the dynamic
correlation function in section 5 .3 .
only visible as a flat background in the experiment .
the cross section is still correctly represented by (5 .8) .
The
Nevertheless, this terni is just a constant and therefore
5- 2
The Q dependent (coherent) part of