Neutron Scattering

decreased barrier increases the probability for quantum mechanical tunnelling of the proton
and thus accelerates the diffusion process .
Diffusion in the presence of traps
Hydrogen traps in a metal lattice can be created by chemical impurities . In a lattice with traps
the proton dynamics splits into a local motion around the trap and a diffusion between traps . A
corresponding two state model [3] yields a scattering function consisting of 2 Lorentzians . At
small Q diffusion, at large Q jumps around the trap dominate scattering . Since jumps around a
trap are qualitatively very similar to rotations of a molecule around ils center of mass the saure
theory can be applied to get classical jump rates and local librations .
17.2 .5 Librations
Usually the 3-dimensional potential of a proton at equilibrium site is expanded harmonically
and completed by anharmonic terras consistent with symmetry requirements . It determines ils
eigenenergies . Vice versa the librations allow to deduce the potential . This information refines
the potential beyond the pure knowledge of the barrier height obtained from QNS .
17.2 .6 Translational tunnelling
At low temperatures the proton localises in a pocket of the potential . If the barrier between such
pockets is weak, the proton wavefunctions of neighbouring pockets overlap and the degenerate
librational states split into tunnelling substates . This translational tunnelling is formally almost
equivalent to the rotational tunnelling to be described below .
NbOo .oo,Ho .ooi (Fig . 17 .5) represents an especially clear case . The oxygen defect distorts
the lattice locally and makes exactly two hydrogen sites - almost - equivalent . Almost : the
presence of the particle itself in one minimum introduces an asymmetry . Thus one has to
calculate the scattering function of an atom in an asymmetric double minimum [3] .
Wave
functions ~P are set up from basis functions I01 > and 110 > which describe the two possible
proton sites . The two configurations can transform into each other by tunnelling due to a finite
tunnel matrix element t. The corresponding Schrödinger equation H~P = EW in matrix form
leads to the eigenvalues problem (symmetric case assumed for simplicity!)
t
\ t
The characteristic polynom yields eigenvalues 1~ 1,2 = ±t . They are connected with the totally
symmetric eigenvectorr e( l ) = 1~2 (I 1, 0 > + 10,1 >) and the antisymmetric eigenvector
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