Neutron Scattering

leads to the eigenvalue problem
~q -
i -v
7Il
Il
z
-v
2 2
z
z
- v
q
The eigenvalues and eigenvectors of this 3x3 matrix are
lation . Including normalization yields
l~1 = 0 ; 1~ 2 / 3 = 2 - 3 v
1 1 1
21 = 3
(1,1,1) ; q2 = 3 q3 = 3
(1 , e * , E) (17 .31)
with phasefactors e = exp(23) and e* complex conjugated . Initial conditions are equal popu-
1 2
Pl (t) = 3
+ exp(- 3 vt) (17 .32)
3 2
P2 (t) =P3 (t) = Z (1- Pl(t» (17 .33)
For t=oc all sites are indeed equally populated . The corresponding average proton density
distribution represents the jump geometry . The density distribution is also dynamically stable,
since jumps into and out off the site are in equilibrium .
Omitting the index at p the first term of the self correlation funetion is for r 1
GS (r, t)
= b(r)P(t) + -
(b(r - ('12) + b(r - r13)) (1 - P(t))
(17 .34)
The FT of G, (r, t) with respect to space yields the intermediate scattering funetion
IS (Q, t) = P(t) + 3 (1- P("))--I(Q)
(17 .35)
Using abbreviations r12 = r2 -r1 the structure factor is
A(Q) = eo .s(Qr12 ) + eos(Qr 23 ) + eo .s(Qr31 ) (17 .36)
FT with respect to time yields the scattering funetion of a single crystal .
orientation of the methyl group, rzj , with respect to the scattering vector Q
It depends on the
Szn,(Q, ~1J )
_
(3
+ gA(Q))a(~) + (3 - 9A(Q)) - w2 + (ZV)2
3
1,11
(17 .37)
In general samples are polycrystals . Powder averaging yields
(3 + 2,jo(Qd»s(~,j 3 ) +
(3 - 2,jo(Qd» 3
--2 +(2v)2
s
(17 .38)
with the Bessel funetion .7 o(Qd) = Si n Qd
Qd
'
17- 1 2