Dynamic structure factor of a monatomic cubic crystal

TL;DR

The paper derives an exact analytical expression for the dynamic structure factor $S({\bf q},\omega)$ of a monatomic cubic crystal with vacancies by solving the dissipative hydrodynamic equations and thermodynamic closures. It identifies eight slow modes, including a novel vacancy-diffusion mode, and provides explicit expressions for heat and vacancy diffusivities, sound speeds, and damping coefficients, revealing how vacancies modify the spectrum through a new central peak and shifts in attenuation. The authors validate the theory by comparing with molecular-dynamics simulations of a hard-sphere fcc crystal containing a vacancy, finding excellent agreement across short and long times and demonstrating how vacancy diffusion can be inferred from the intermediate scattering function without tracking vacancies directly. The work extends previous results for perfect crystals to realistic crystals with defects and discusses implications for more complex crystals and interaction potentials, offering a rigorous framework for connecting microscopic dynamics to DSF measurements in solids.

Abstract

The spectral function of density fluctuations, also known as the dynamic structure factor, of a monatomic cubic crystal with vacancies is derived from the macroscopic equations describing transport in crystalline solids. The resonances of the spectral function are identified as a Brillouin doublet of sound propagation, a central Rayleigh peak of heat diffusion, as for perfect crystals, and another central sharp peak associated with vacancy diffusion. Analytical expressions for the heat and vacancy diffusivities, speeds of sound, and sound damping coefficients are obtained. The theoretical results are compared to molecular dynamics simulations of a face-centered cubic crystal of hard spheres.

Figures (4)

• Figure 1: Representation of the dynamic structure factor $S({\bf q},\omega)$ for a cubic monatomic crystal with vacancies, at a wave number value of $q_*=\Vert{\bf q}_*\Vert=0.4$. The solid black line represents the analytical expression, Eq. \ref{['eq:DSF_Full']}, and the dashed black line shows the small $q$ approximation, Eq. \ref{['eq:DSF_approx']}. The colored solid lines represent the peaks identified in Eq. \ref{['eq:DSF_approx']}: vacancy diffusion (blue), heat diffusion (Rayleigh -- green), and sound propagation and damping (Brillouin -- orange). Below, their underlying poles are depicted in the plane of complex frequencies. The parameters are for a hard-sphere crystal of $499$ particles on $500$ lattice sites at a density $n_{0*}=1.05$ (in dimensionless units of the simulation), as given in Ref. MG25, and the ratio $S_{\rm nv}({\bf q})/S({\bf q})$ is taken as $-30/N_0$. For illustrative purposes, the central sharp peak is truncated vertically.
• Figure 2: Representations of the intermediate scattering function, $F({\bf q},t)$, for a cubic monatomic crystal with vacancies at short times [panel (a), left] and long times [panel (b), right], for a wave vector value of $q_*=\Vert{\bf q}_*\Vert=0.4$. The solid black line shows the inverse Fourier transform from frequency $\omega$ to time $t$ of the analytical expression, Eq. \ref{['eq:DSF_Full']}. The dashed black line represents the small $q$ approximation, Eq. \ref{['eq:ISF']}. The colored solid lines are the damping functions identified in Eq. \ref{['eq:ISF']}: vacancy diffusion (blue), heat diffusion (green), and sound propagation and damping (orange). The same parameters as in Fig. \ref{['Fig:DSF_Theo']} are used.
• Figure 3: Representations of the dynamic structure factor, $S({\bf q},\omega)$, [panel (a), left] and the intermediate scattering function, $F({\bf q},t)$, [panel (b), right] for a cubic monatomic crystal with one vacancy (solid black lines) and without vacancies (perfect crystal -- dotted red lines) for a given value of the wave vector $q_*=\Vert{\bf q}_*\Vert=0.4$. The same parameters as in Fig. \ref{['Fig:DSF_Theo']} are used. For illustrative purposes, the central sharp peak is truncated vertically in panel (a).
• Figure 4: Normalized intermediate scattering function $F({\bf q},t)$ at density $n_{0*}=1.05$, with ${\bf q}_*$ in the $[100]$ direction and $q_*=0.8$, for a hard-sphere crystal with $N_0=500$ on short, panel (a), and long, panel (b), time scales. The dashed red line corresponds to a crystal with one vacancy and 499 particles. The dotted blue line corresponds to a perfect crystal with 500 particles. The solid black line is the inverse Fourier transform from frequency to time of the analytical expression \ref{['eq:DSF_Full']}, with parameters given in Ref. MG25 for the thermodynamic, elastic, and transport coefficients, and in Appendix \ref{['app:static-fns']} for the static correlation functions.