A wave-mechanical model of incoherent neutron scattering II. Role of the momentum transfer
We recently introduced a wave-mechanical model for quasi-elastic neutron scattering (QENS) in proteins. We call the model ELM for "Energy Landscape Model". We postulate that the spectrum of the scattered neutrons consists of lines of natural width shifted from the center by fluctuations. ELM is based on two facts: Neutrons are wave packets; proteins have low-lying substates that form the free-energy landscape (FEL). Experiments suggest that the wave packets are a few hundred micrometers long. The interaction between the neutron and a proton in the protein takes place during the transit of the wave packet. The wave packet exerts the force $F(t) = dQ(t)/dt$ on the protein moiety, a part of the protein surrounding the struck proton. $Q(t)$ is the wave vector (momentum) transferred by the neutron wave packet to the proton during the transit. The ensuing energy is stored in the energy landscape and returned to the neutron as the wave packet exits. Kinetic energy thus is changed into potential energy and back. The interaction energy is proportional to $Q$, not to $Q^2$. To develop and check the ELM, we use published work on dehydrated proteins after reversing improper normalizations. In such proteins only vibrations are active and the effects caused by the neutron momentum can be studied undisturbed by external fluctuations. ELM has predictive power. For example it quantitatively predicts the observed inelastic incoherent fraction $S(Q, T)$ over a broad range of temperature and momentum $Q$ with one coefficient if $S(0, T)$ is known.