Atomic displacements in quantum crystals

Displacements of atoms and molecules away from lattice sites in helium and parahydrogen solids at low temperature have been studied by means of quantum Monte Carlo simulations. In the bcc phases of ${}^3\mathrm{He}$ and ${}^4\mathrm{He}$, atomic displacements are largely quantum-mechanical in character, even at melting. The computed Lindemann ratio at melting is found to be in good agreement with experimental results for ${}^4\mathrm{He}$. Unlike the case of helium, in solid parahydrogen there exists near melting a significant thermal contribution to molecular vibrations, accounting for roughly half of the total effect. Although the Lindemann ratio at melting is in quantitative agreement with experiment, computed molecular mean square fluctuations feature a clear temperature dependence, in disagreement with recent experimental observations.

Figure 1

Snapshot of a many-particle configuration (world lines) for bcc ${}^4\mathrm{He}$, at temperature $T=1.6\mathrm{K}$, density $\rho =0.02854{\AA }^{-3}$. View is along one of the three main (equivalent) crystallographic directions. The total number of particles is 1024. Only the traces of the 128 in two adjacent planes are visible. Distances are in Å.

Figure 2

Pair correlation function $g\left(r\right)$ for bcc ${}^4\mathrm{He}$ at $T=1.6\mathrm{K}$ and density $\rho =0.02854{\AA }^{-3}$, computed for a system of $N=1024$ atoms.

Figure 3

Mean square atomic displacements (in ${\AA }^2$) away from lattice sites in bcc ${}^4\mathrm{He}$ at the melting temperature $T=1.6\mathrm{K}$ and density $\rho =0.02854{\AA }^{-3}$. Shown here are the results for the three system sizes simulated (see text). Curve through the points is a fit to the data based on the expression $\alpha +\beta N^{-2/3}$. The box at the left upper corner represents the experimental estimate from Ref. [5].

Figure 4

Snapshot of a many-particle configuration (world lines) for bcc ${}^3\mathrm{He}$, at temperature $T=0.65\mathrm{K}$, density $\rho =0.02458{\AA }^{-3}$. View is along one of the three main (equivalent) crystallographic directions. The total number of particles is 1024. Only the traces of the 128 in two adjacent planes are visible. Distances are in Å.

Figure 5

Pair correlation function $g\left(r\right)$ for bcc ${}^3\mathrm{He}$ at $T=0.65\mathrm{K}$ and density $\rho =0.02458{\AA }^{-3}$, computed for a system of $N=1024$ atoms.

Figure 6

Mean square atomic displacements (in ${\AA }^2$) away from lattice sites in bcc ${}^3\mathrm{He}$ at the melting temperature $T=0.65\mathrm{K}$ and density $\rho =0.02458{\AA }^{-3}$. Shown here are the results for the three system sizes simulated (see text). Curve through the points is a fit to the data based on the expression $\alpha +\beta N^{-2/3}$.

Figure 7

Mean square atomic displacements (in ${\AA }^2$) away from lattice sites in hcp $p-{\mathrm{H}}_2$ at different temperatures, at density $\rho =0.0261{\AA }^{-3}$. Shown here are results for two simulated systems, comprising $N=512$ (squares) and $N=4096$ (circles) molecules. When not explicitly shown, statistical errors are smaller of the symbol size. Inset shows the size extrapolation of the results obtained at temperature $T=13.8\mathrm{K}$. Curve through data points is a fit obtained using the expression $\alpha +\beta N^{-1/3}$. Star represents the value extrapolated to the thermodynamic limit.