The self-consistent method of lattice dynamics (SCLD) is used to obtain an analytical solution for the free energy of a periodic, one-dimensional, mono-atomic chain accounting for fourth-order anharmonic effects. For nearest-neighbor interactions, a closed-form analytical solution is obtained. In the case where more distant interactions are considered, a system of coupled nonlinear algebraic equations is obtained (as in the standard SCLD method) however with the number of equations dramatically reduced. The analytical SCLD solutions are compared with a numerical evaluation of the exact solution for simple cases and with molecular dynamics simulation results for a large system. The advantages of SCLD over methods based on the harmonic approximation are discussed as well as some limitations of the approach.
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Acknowledgements We thank Ryan Elliott for helpful discussions and for carefully reviewing this manuscript. This work was supported in part by the U.S. Department of Energy under Award Number DE-SC0002085.
- Anharmonic effects
- Free energy
- Lattice dynamics
- One-dimensional chain
- Self-consistent method
- Statistical mechanics