Effect of finite system size on thermal fluctuations: Implications for melting

Lattice-dynamics calculations and molecular-dynamics simulations are used to study the variation with particle number N in the mean squared displacement 〈(δu${)}^2$ ${\mathrm{〉}}_{\mathit{N}}$ for systems with Lennard-Jones and repulsive Yukawa and ${\mathit{r}}^{\mathrm{-}\mathit{n}}$ interactions. In the low-temperature harmonic regime, the leading correction to the N→∞ limit is found to have the known form 〈(δu${)}^2$ ${\mathrm{〉}}_{\mathit{N}}$/〈(δu${)}^2$ ${\mathrm{〉}}_{\mathrm{\infty }}$=1+κ${\mathit{N}}^{\mathrm{-}1/3}$. However, the sign of κ is not always negative as is indicated by simple arguments. For fcc crystals, κ≃-1 for all potentials. Thus finite-size errors are ∼10% for N=1000. In the bcc phase, errors may be more than 5 times larger and of either sign. We show that positive values of κ result from large anisotropies in the sound velocities. Anharmonic effects at higher temperatures change the value of κ, but not the scaling with N. For both structures κ becomes more negative, but the changes are much more pronounced in the bcc phase where κ may change sign. These results indicate that one must be careful in using 〈(δu${)}^2$ ${\mathrm{〉}}_{\mathit{N}}$ for typical values of N in calculations of the Debye-Waller factor or a Lindemann criterion for melting. The variation with N of the temperature where melting is observed indicates that low-frequency shear modes are important in destabilizing the solid phase.