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 〈({δ}u)²{〉}N$ for systems with Lennard-Jones and repulsive Yukawa and ${r}^{{{-}}n}interactions. In the low-temperature harmonic regime, the leading correction to the N→∞ limit is found to have the known form 〈(δu{)}²〉N/〈({δ}u)²{〉}_{{{∞}}}=1+κ{N}^{{{-}}1/3}$. 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 〈({δ}u)²{〉}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.