Neutron Scattering

(T2 is related to the mean square displacement . The theory yields the generally valid relation
(r2(t)) = 3a 2 (t) = 411
x
r 2G"(r, t) dr (17 .4)
o
In the gaussian approximation one uses this relation also for any other translational or rotational
motion despite they have time dependences (T(t) different to that of a rare gas . The
problem is thus reduced to determine the mean square displacement of a dynamical process.
The justification of the gaussian approximation is that it works .
17.2 Translation
The simplest translation is that of a rare gas . Hydrogen on interstitial sites in a metal is often
treated as a lattice gas . Self diffusion close to the melting point via vacancies has a similar
character. Atomic liquids represent the simplest example for diffusion . But the most common
liquids are made up by molecules which show additional rotational degrees of freedom.
17.2 .1 Macroscopic diffusion
Diffusion of a monoatomic liquid obeys macroscopically Fick's law
On, (r '
t) - - Dp 2 ra(r, t) (17 .5)
at
withthe number density n (r, t) - G s (r, t) and the diffusion constant D . For isotropie diffusion
2 a2 2 a
ar 2 r ar
in spherical coordinates . G s (r, t) from (17 .2) is a solution of 17 .5 if
(17 .6)
a 2 (t) = 2D 1 t 1 +c (17 .7)
For long times (small energy transfer Ah,w) c can be neglected and Fourier-transformation
(FT) in space and time yields the scattering function
S(Q , w) - w 2 (DQ2)2
a
(17 .8)
Both, half width and scattering intensity of this Lorentzian allow a determination of the diffusion
coefficient D via
I = 2h:DQ2
1
S(Q~ 0) _ ~rriDCQ 2 . (17 .9)
17-3