Neutron Scattering

The Id rotor : pocket state formalisme
Pocket states are useful basis functions for stronger potentials . They represent a single orientation
of a molecule . Thus =I 123 > means thatproton 1 of the methyl group is at position 1, 2
at 2, 3 at 3 . A rotation is represented by cyclic permutation . We consider the groundstate only .
As outlined for translational tunnelling pocket states are no eigenstates of the problem . They
overlap and thus can transform into each other. Since wave functions decay exponentially into
a potential wall (Gamow factor) the overlap or tunnelling matrix element is very sensitive to
the strength of the potential . The eigenvalue matrix obtained from the Hamiltonian is
1123> 1231>
~123> t
~231> t A
312 > t t
The characteristic polynom yields a unique eigenvalue A = 21, related to the totally symmetric
A groundstate and a doubly degenerate eigenvalue A = -t related to the right and left handed
E states, respectively . The matrix is formally identical to a jump matrix . The meaning of the
eigenvalues is very different, however.
A tetrahedron like methane requires 12 pocket states . The 9 eigenvalues are partially degenerate
depending on the environmental symmetry . The mathematics becomes more complicated
.
To obtain the scattering function including intensities of transitions the influence of proton
spins via the Pauli principle bas to be taken into account . The complete theory with inclusion
of spin wavefunctions is found in ref.[8] . The resulting scattering function is normalized te, the
number of protons in the rotor
si ,(Q, ~) = (1 + 29o(Qd»s(~) + (3 - 3 j o(Qd»s( ~,j )
2 2
+ (3 - 3.7o(Qd))(s( ~,i + wc) + s( üj - üi c))
(17 .44)
The first term represents purely elastic scattering . Its intensity is called elastic incoherent structure
factor (EISF) . The second term is due to transitions between different but degenerate E-
states . Finally there are inelastic A -- E transitions between tunnelling substates . The latter
terms are ô-functions only at low temperature . By coupling to phonons they broaden and shift
[10] until they merge into the single classical quasielastic Lorentzian. The width of tunnelling
lines can be interpreted as a lifetime broadening due to transitions into the first excited librational
level Eol of the same symmetry. Eol acts as activation energy and can be obtained
from an Arrhenius plot. The fig .17 .8 shows this transition for acetamide CH3C0_NH2 , the
17-17