Local optimality of cubic lattices for interaction energies

Abstract

We study the local optimality of simple cubic, body-centred-cubic and face-centred-cubic lattices among Bravais lattices of fixed density for some finite energy per point. Following the work of Ennola (Math Proc Camb 60:855–875, 1964), we prove that these lattices are critical points of all the energies, we write the second derivatives in a simple way and we investigate the local optimality of these lattices for the theta function and the Lennard–Jones-type energies. In particular, we prove the local minimality of the FCC lattice (resp. BCC lattice) for large enough (resp. small enough) values of its scaling parameter and we also prove the fact that the simple cubic lattice is a saddle point of the energy. Furthermore, we prove the local minimality of the FCC and the BCC lattices at high density (with an optimal explicit bound) and its local maximality at low density in the Lennard–Jones-type potential case. We then show the local minimality of FCC and BCC lattices among all the Bravais lattices (without a density constraint). The largest possible open interval of density’s values where the simple cubic lattice is a local minimizer is also computed.

Appendix: General formulas for the second derivatives of \(E_f\)

We give the formulas for the second derivatives of $E_f$, for any f and at any Bravais lattice $L=(u,v,x,y)$ with volume V.

Lemma 7.1

(Second derivatives of the energy) For any $f\in \mathcal {F}$, any fixed volume $V>0$ and any $L=(u,v,x,y)$, we have

$$\begin{aligned} \partial ^2_{uu}E_f[L]&=C^2\sum _{m,n,p}\left( -\frac{(m+xn+yp)^2}{u^2}-\frac{v^2(n+zp)^2}{u^2}+\frac{u}{v^2}p^2 \right) ^2 f''(Q_L)\\&\quad +\,2C\sum _{m,n,p}\left( \frac{(m+xn+yp)^2}{u^3}+\frac{v^2(n+zp)^2}{u^3}+\frac{1}{2v^2}p^2 \right) f'(Q_L),\\ \partial ^2_{vv}E_f[L]&=C^2\sum _{m,n,p}\left( \frac{2v(n+zp)^2}{u}-\frac{u^2}{v^3}p^2 \right) ^2 f''(Q_L)\\&\quad +\,C\sum _{m,n,p}\left( \frac{2(n+zp)^2}{u}+\frac{3u^2}{v^4}p^2 \right) f'(Q_L),\\ \partial _{xx}^2 E_f[L]&=\frac{4C^2}{u^2}\sum _{m,n,p}n^2(m+xn+yp)^2f''(Q_L)+\frac{2C}{u}\sum _{m,n,p}n^2 f'(Q_L),\\ \partial _{yy}^2 E_f[L]&=\frac{4C^2}{u^2}\sum _{m,n,p}p^2(m+xn+yp)^2f''(Q_L)+\frac{2C}{u}\sum _{m,n,p}p^2 f'(Q_L),\\ \partial _{zz}^2 E_f[L]&=\frac{4C^2v^4}{u^2}\sum _{m,n,p}p^2(n+zp)^2f''(Q_L)+\frac{2Cv^2}{u}\sum _{m,n,p}p^2 f'(Q_L),\\ \partial _{uv}^2 E_f[L]&=C^2\sum _{m,n,p}\left( -\frac{(m+xn+yp)^2}{u^2}-\frac{v^2(n+zp)^2}{u^2}+\frac{u}{v^2}p^2 \right) \\&\quad \times \,\left( \frac{2v(n+zp)^2}{u}-\frac{u^2}{v^3}p^2 \right) f''(Q_L)\\&\quad -\,2C\sum _{m,n,p}\left( \frac{v(n+zp)^2}{u^2}+\frac{u}{v^3}p^2 \right) f'(Q_L),\\ \partial _{ux}^2 E_f[L]&=\frac{2C^2}{u}\sum _{m,n,p}n(m+xn+yp)\\&\quad \times \,\left( -\frac{(m+xn+yp)^2}{u^2}-\frac{v^2(n+zp)^2}{u^2}+\frac{u}{v^2}p^2 \right) f''(Q_L)\\&\quad -\,\frac{2C}{u^2}\sum _{m,n,p}n(m+xn+yp)f'(Q_L),\\ \partial _{uy}^2 E_f[L]&=\frac{2C^2}{u}\sum _{m,n,p}p(m+xn+yp)\\&\quad \times \,\left( -\frac{(m+xn+yp)^2}{u^2}-\frac{v^2(n+zp)^2}{u^2}+\frac{u}{v^2}p^2 \right) f''(Q_L)\\&\quad -\frac{2C}{u^2}\sum _{m,n,p}p(m+xn+yp)f'(Q_L),\\ \end{aligned}$$

$$\begin{aligned} \partial _{uz}^2 E_f[L]&=\frac{2C^2v^2}{u}\sum _{m,n,p}p(n+zp)\\&\quad \times \,\left( -\frac{(m+xn+yp)^2}{u^2}-\frac{v^2(n+zp)^2}{u^2}+\frac{u}{v^2}p^2 \right) f''(Q_L)\\&\quad -\,\frac{2Cv^2}{u^2}\sum _{m,n,p}p(n+zp)f'(Q_L),\\ \partial _{vx}^2 E_f[L]&=\frac{2C^2}{u}\sum _{m,n,p}n(m+xn+yp)\left( \frac{2v(n+zp)^2}{u}-\frac{u^2}{v^3}p^2 \right) f''(Q_L),\\ \partial _{vy}^2 E_f[L]&=\frac{2C^2}{u}\sum _{m,n,p}p(m+xn+yp)\left( \frac{2v(n+zp)^2}{u}-\frac{u^2}{v^3}p^2 \right) f''(Q_L),\\ \partial _{vz}^2 E_f[L]&=\frac{2C^2v^2}{u}\sum _{m,n,p}p(n+zp)\left( \frac{2v(n+zp)^2}{u}-\frac{u^2}{v^3}p^2 \right) f''(Q_L)\\&\quad +\,\frac{4Cv}{u}\sum _{m,n,p}p(n+zp)f'(Q_L),\\ \partial _{xy}^2 E_f[L]&=\frac{4C^2}{u^2}\sum _{m,n,p}np(m+xn+yp)^2f''(Q_L)+\frac{2C}{u}\sum _{m,n,p}npf'(Q_L),\\ \partial _{xz}^2 E_f[L]&=\frac{4C^2v^2}{u^2}\sum _{m,n,p}np(n+zp)(m+xn+yp)f''(Q_L),\\ \partial _{yz}^2 E_f[L]&=\frac{4C^2v^2}{u^2}\sum _{m,n,p}p^2(n+zp)(m+xn+yp)f''(Q_L). \end{aligned}$$