Neutron Scattering

With this definition the pair correlation function can be written as time-dependent
density-density correlation function :
(5 G(r , t) = N f d 3 r' (p(r' - r, 0)p(e , t)> .
.47)
With this form of the dynamic pair correlation function the dynamical structure factor
can be-analogously to equation (5.1l)-written as the double Fourier transform of the
correlator of the particle density :
Scoh(Q, (5 W) = f~ dtexp(-iwt) f d3r d 3r' exp (iQ - r)
00 J
(p(r' -?', 0)p(r , t» . .48)
We now define the density operator in reciprocal space as the Fourier transform of (5 .46) :
and obtain for the dynamic structure factor
pQ (t) - E exp (iQ - r i (t)) (5 .49)
Scoh (Q, W)
1
27fhN f c dtexp(-iwt) (PQ(0)p_Q(t)) .
o0
(5 .50)
Correspondingly, the intermediate scattering function is
Scoh(Q,t) = 17 (PQ(0)P-Q(t))
(5 .51)
which after insertion of (5 .49) turns out to be equivalent to (5 .42) .
Analogously, one can define a self correlation function by setting i = j in the preceding
equations leading to
G9 (r, t) = N
~- J d3r' (6 (?- - e + ri (0 )) 6 (r' - ri(t»Î (5 .52)
as the equivalent of (5 .45) .
The pair correlation function has some general properties :
1 . For spatially homogeneous systems the integrand in (5 .47) is independent of r' which
can be arbitrarily set to the origin 0 :
G(r, t) = N
(p(-r, 0>0, t)> = 1
(p(0, 0)p(r, t) ) .
(5 .53)
5- 17