Single-phonon neutron scattering model ($\epsilon (q)={\epsilon }_{\ast }|\sin (qa/2)|$)

Consider an original ideal one-dimensional monatomic harmonic oscillator chain with neighbor separation $a$, mass $M$ and nearest-neighbor restoring coefficient $K>0$. Its displacement equation isSubstitute the normal mode $u_r=u_0{e}^{i(qra-\omega t)}$. Since $e^{iqa}+e^{-iqa}-2=-4{\sin }^2(qa/2)$,Choose $a=4$ ångströms and ${\epsilon }_{\ast }=10\mathrm{meV}$. This is one acoustic phonon branch, distinct from the two-atom elastic example. It has $\epsilon (\pi /(2a))=5\sqrt{2}\mathrm{meV}$ and $\epsilon (\pi /a)=10\mathrm{meV}$.

For coherent creation of one phonon in a crystal, the excitation's wave vector $q$ and the measured transfer obey $Q=G+q$, with $G$ a reciprocal lattice vector. To connect the dispersion relation to an illustrative intensity, take zero temperature, longitudinal displacements and the first interval $Q=q\in (0,\pi /a]$, choosing $G=0$. Expand the scattering phase to first order, $e^{iQ(ra+u_r)}\simeq e^{iQra}(1+iQ{u}_r)$. The constant term supplies elastic scattering; the linear term creates one phonon. Its displacement matrix element is proportional to $1/\sqrt{\omega }$, so the squared amplitude has weight $Q^2/\epsilon (Q)$, apart from common factors. Normalize this weight to one at $Q=\pi /a$:The limit $w(0)=0$ follows from $\sin (\pi x/2)\sim \pi x/2$. Set $\sigma =0.4\mathrm{meV}$ and replace the ideal sharp line by a normalized Gaussian function:This chosen broadening illustrates readout width; it is not a measured resolution or a full spectrometer resolution function. Integrating over all $\Delta E$ gives $w(x)$. The small negative-energy tail is a broadening artifact, not a zero-temperature phonon-annihilation signal.

These panels separate three kinds of information: allowed reciprocal lattice positions, cancellation within a crystal basis structure factor, and an excitation's energy as a function of wave vector. The chain has no magnetic degrees of freedom, optical branches, damping or multi-phonon processes. It illustrates how one excitation could be read; it proves nothing about the completeness of a real material's excitation spectrum.