Neutron Scattering

In addition it is often useful to define the intermediate scattering function which denotes
the time dependent parts of definitions (5 .38) and (5 .39) before Fourier transform :
Sinc(Q,t) - N (exp(-iQ Ei (o))exp(iQ ri(t))) (5 .41)
Scoh(Q,t) - N
E(exp(-iQ - r i (0» exp(iQ- r j (t») .
(5 .42)
2,7
If one compares this result with the definition of the structure factor (5.8-5 .11) one
recognizes that Sc oh(Q, w) is
time of a dynamical pair correlation function G(r, t) :
G(r, t) = C2r/ I d3Q
3
exp (-
in an analogous way the Fourier transform in space and
E `exp (-iQ - r i(0» exp (iQ - r j (t)) > . (5 .43)
N
The derivation of the relation between the coherent dynamical structure factor Swh(Q, W)
and the generalized pair correlation function requires a strict quantum mechanical calculation
.
This problem results from the fact that the coordinate vector operators commute
only at identical times . Therefore, in all algebraic manipulations the order of r i (0) and
ri(o) must not be interchanged .
To begin, one writes the operator exp (-iQ - ri (0» as the Fourier transform of the delta
function :
exp (-iQ - ri (0» = J
d 3 r'Ö (r - ri (0» exp(-iQ . r') . (5 .44)
Using this expression equation (5 .43) can be rewritten as
3
G(r, t) = (1) 1 ~- (Id 3 r 'b (r i - ri
27r N i, j
(0))
J
d3Q exp (-iQ . r - iQ . r + iQ . r j (t))
= (27r)3ö (r + r' - ri (t) )
N E J d 3 r' (6 (r - r' +r(0)) Ö (r' - rj (t)) ) (5 .45)
1,9
without changing the order of the operators at different times .
Now the particle density operator is
particle position operators :
introduced as a sum over delta functions at the
p (r, t) - Z- 6 (r - Ei (t» .
(5 .46)
5-1 6