Solid ${}^4\mathrm{He}$ and the diffusion Monte Carlo method: A study of their properties

Properties of helium atoms in the solid phase are investigated using the multiweight diffusion Monte Carlo method. Two different importance function transformations are used in two series of independent calculations. The kinetic energy is estimated for both the solid and liquid phases of ${}^4\mathrm{He}$. We estimate the melting and freezing densities, among other properties of interest. Our estimates are compared with experimental values. We discuss why walkers biased by two distinctly different guiding functions do not lead to noticeable changes in the reported results. Criticisms concerning the bias introduced into our estimates by population control and system size effects are considered.

Figure 1

Linear fit to the total energy per atom as a function of the inverse of the number of particles at the density $\rho {\sigma }^3=0.549$ (left hand scale, red circles and dashed line). Exact values are shown for the tail correction of the two-body potential for the corresponding number of atoms (right hand scale, black squares). The black line is a linear fit to these values. Deviations from a smooth behavior of this quantity are due to a discontinuous change in the smallest size of a simulation cell built to accommodate an hcp crystalline structure.

Figure 2

Kinetic (right hand scale, black squares) and potential energy (left hand scale, red circles) per atom as a function of the inverse of the number of particles at the density $\rho {\sigma }^3=0.549$. The solid line is a linear fit to the kinetic energy results and the dashed line is a linear fit to those of the potential energy.

Figure 3

Total energy per atom as a function of the density in the solid phase. The curves stand for fits to the estimated values of the energies using Eq. (16); the dashed green line stands for those obtained with the SNJ guide function; those for NJ are shown by a blue line. The curves are hard to distinguish at the figure resolution. Experimental data [32, 33, 34] are displayed by red circles.

Figure 4

Kinetic energy per atom as a function of the density in the solid phase. The curves stand for fits to the estimated values. Results for the SNJ guide function are displayed by a dashed green line; those for NJ are shown with a blue solid line. Experimental data [33, 34, 39, 40] are displayed by red circles.

Figure 5

Equation of state as a function of the density in the liquid phase. The curve stands for a fit to the estimated values of the energies using Eq. (16). Our numerical results are displayed by solid black circles. Errors are smaller than the symbol size. Experimental data for the liquid phase [42, 43] are shown by squares. The triangles stand for values determined at $T=0.05$ K [44].

Figure 6

Kinetic energy per atom as a function of the density in the liquid phase. The curve stands for a fit to the estimated values displayed by solid black circles. Errors are smaller than the symbol size. The squares represent experimental results [44].

Figure 7

The Gibbs free energies per particle as a function of pressure. Results in the liquid phase are shown by a black line. In the solid phase the dashed green line stands for values obtained with the SNJ guide function and the blue one stands for those determined with NJ. The left vertical dotted line identifies the transition pressure (the discontinuity) obtained with the NJ guiding function and the right one identifies that with SNJ.

Figure 8

Density as a function of the pressure. The solid black line stands for results in the liquid phase. In the solid phase the green dashed line and the blue line show results for SNJ and NJ guiding functions, respectively. The dotted vertical lines indicate the transition pressures; the left one is for the NJ guiding function and the right one is for SNJ.