Nonlinear effects in quasielastic neutron scattering: Exact line-shape calculation for a dimer

An exact solution of a discrete nonlinear Schroumldinger equation, obtained recently for the site occupation probabilities in a two-site system such as a molecular dimer, has shown that the probabilities evolve in the form of Jacobian elliptic functions and exhibit a self-trapping transition. On the basis of that solution, we examine the effect of nonlinearities on the quasielastic scattering function in a dimer. The calculation is appropriate to the scattering of probe particles such as neutrons off moving quasiparticles which interact with lattice vibrations strongly enough to produce nonlinear effects while moving in the lattice. A well-known example is provided by hydrogen atoms diffusing among sites around impurities, e.g., oxygen, in metals such as niobium. Our calculation results in explicit expressions for the scattering spectrum. They exhibit the phenomenon of motional narrowing even in the absence of true damping. Comparison of the results for the undamped nonlinear dimer and the damped linear dimer uncover striking similarities as well as differences.