An Analytical Self-consistent Solution for the Free Energy of a 1-D Chain of Atoms Including Anharmonic Effects

Abstract

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.

Appendix

In this appendix, we derive the relations in (31). First, we rewrite Φ ₂ in (8) and δ u in (16) using vector notation as

(A.1)

(A.2)

where

(A.3)

(A.4)

(A.5)

and

$$ \mathbf{Q}_k= \begin{bmatrix} e^{i k X_0}\\ e^{i k X_1}\\ \vdots\\ e^{i k X_{N-1}} \end{bmatrix} . $$

(A.6)

Note that

(A.7)

(A.8)

Then, by substituting (A.2) into (A.1) we have

$$ \tilde{\varPhi}_2=\frac{1}{2} \frac{1}{N m} \boldsymbol{q}^T \mathbf{Q}^T \overset{o}{\mathbf{H}}\mathbf{Q} \boldsymbol{q}. $$

(A.9)

Also, from (23) $\tilde{\varPhi}_{2}$ is given by

(A.10)

where

(A.11)

Then, from (A.9) and (A.10) we have

(A.12)

and we find that $m \omega_{k}^{2}$ and $\mathbf{Q}_{k} / \sqrt{N}$ are an eigenvalue and an eigenvector of the Hessian.

For a k (>0), which is neither zero nor a boundary value, $\omega _{k}^{2}$ ($=\omega_{-k}^{2}$) is degenerate and has two independent eigenvectors of $\mathbf{Q}_{k}/ \sqrt{N}$ and $\mathbf{Q}_{-k}/ \sqrt {N}$. Since these eigenvectors consist of complex numbers, we reconstruct the real eigenvectors by the linear combination such that

(A.13)

(A.14)

We can show that $\mathbf{R}_{k}^{+}\cdot\boldsymbol {R}_{k}^{+}=1$,$\mathbf{R}_{k}^{-}\cdot\mathbf{R}_{k}^{-}=1$, and $\mathbf{R}_{k}^{+}\cdot\mathbf{R}_{k}^{-}=0$, i.e. $\boldsymbol {R}_{k}^{+}$ and $\mathbf{R}_{k}^{-}$ are the orthonormal eigenvectors of the Hessian.

Next, we transform δ u using

(A.15)

where

(A.16)

(A.17)

(A.18)

Note that the parentheses are used for the last terms in (A.16) and (A.17) to indicate that these terms appear only when N is even and the +/− terms are defined only for positive k’s.

Then, from (A.2) and (A.15) we have

(A.19)

and by matching the terms with the same eigenvalue we have

(A.20)

(A.21)

(A.22)

Finally, we find the following relations

(A.23)

(A.24)

which appear in (31).