= Single-phonon neutron scattering model
{title2=$\varepsilon(q)=\varepsilon_*|\sin(qa/2)|$}
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 is
$$
M\ddot u_r=K(u_{r+1}+u_{r-1}-2u_r).
$$
Substitute the <normal mode> $u_r=u_0e^{i(qra-\omega t)}$. Since $e^{iqa}+e^{-iqa}-2=-4\sin^2(qa/2)$,
$$
\omega(q)=2\sqrt{K/M}|\sin(qa/2)|,\qquad
\varepsilon(q)=\hbar\omega(q)=\varepsilon_*|\sin(qa/2)|.
$$
Choose $a=4$ <ångströms> and $\varepsilon_*=10\,\mathrm{meV}$. This is one <acoustic phonon branch>, distinct from the two-atom elastic example. It has $\varepsilon(\pi/(2a))=5\sqrt2\,\mathrm{meV}$ and $\varepsilon(\pi/a)=10\,\mathrm{meV}$.
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 (physics)>, 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, $e^{iQ(ra+u_r)}\simeq e^{iQra}(1+iQu_r)$. 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 $Q^2/\varepsilon(Q)$, apart from common factors. Normalize this weight to one at $Q=\pi/a$:
$$
x=\frac{Qa}{\pi},\qquad
w(x)=\frac{x^2}{\sin(\pi x/2)},\qquad 0<x\leq1.
$$
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>:
$$
J(Q,\Delta E)=\frac{w(x)}{\sqrt{2\pi}\sigma}
\exp\!\left[-\frac{(\Delta E-\varepsilon(Q))^2}{2\sigma^2}\right].
$$
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.
\Image[/neutron-bragg-peaks-phonon-model.png]
{title=Chosen elastic reflection weights and a single-phonon dispersion/intensity model}
{description=Original models, not experimental data. Left: positive reciprocal-lattice orders for the two-atom cell, omitting forward scattering. Middle: the separate monatomic-chain branch. Right: the same branch with normalized longitudinal one-phonon weights and chosen Gaussian broadening. Elastic and inelastic intensities have independent arbitrary normalizations.}
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.
Back to article page
