Neutron Scattering

17 .3 Rotation
Molecules represent - in first approximation rigid - ensembles of atoms and allow rotation as
new degree of freedom . In the simplest approach the environment is represented as a potential
which determines the single particle excitations . The potential mustshow at least the symmetry
of the molecule . - A classical motion is fully characterized by the motion of a single proton .
17.3 .1 Jump rotation : methyl group in a 3-fold potential
Often the rotational potential is rather strong and forces the molecule to stay most time in
an equilibrium orientations . The dynamics consists in this case of jumps between equivalent
orientations . We call the atomic positions r j , the average time between two jumps T and neglect
the jump time itself. The self correlation function G, (r, t) is the conditional probability of
finding an atom at time t at site r if it was at time t=0 at site r=0 .
GS (r, t) = ~- ,Vpj (t) 6 (r - rj )
j-i
(17 .26)
pj (t) is the occupation probability of site j at time t . The sum averages over all possible starting
conditions = sites of the atour. For uncorrelatedjumps the occupation probabilities obey a finite
system of coupled differential equations, the so-called rate equations
dtp~ (t) -
CN
pa (t) - pj (t)7
A-1
(17 .27)
The first terra describes the all possible jumps into a site, the second the jumps out of this
site . For simplicity it is assumed that all sites show the saure population and that jump times
between any two sites are identical . The considered atom is in the sample:
A simple example is the methyl group . Here N=3 and proton position are ri =(0,0,0)d, r2
the rate
=(1,0,0)d, r3 = (2, 1-3 , 0)d with the proton proton distance d=1 .76 . With v = T
equation for site 1 is
and m.m . with cyclic permutation . The ansatz (p = (p l , p2, p3))
n,
pj = 1 (17 .28)
j=1
d _ v v
dtpl -vpl + 2p2 + 2p3 (17 .29)
p = gexp( ,~ t) (17 .30)
17-11