We consider pairwise interaction energies and we investigate their minimizers among lattices with prescribed minimal vectors (length and coordination number), i.e. the one corresponding to the crystal’s bonds. In particular, we show the universal minimality—i.e. the optimality for all completely monotone interaction potentials—of strongly eutactic lattices among these structures. This gives new optimality results for the square, triangular, simple cubic (sc), face-centred-cubic (fcc) and body-centred-cubic (bcc) lattices in dimensions 2 and 3 when points are interacting through completely monotone potentials. We also show the universal maximality of the triangular and fcc lattices among all lattices with prescribed bonds. Furthermore, we apply our results to Lennard-Jones type potentials, showing the minimality of any universal minimizer (resp. maximizer) for small (resp. large) bond lengths, where the ranges of optimality are easily computable. Finally, a numerical investigation is presented where a phase transition of type ‘square–rhombic–triangular’ (resp. ‘sc-rhombic-bcc-rhombic-fcc’) in dimension d = 2 (resp. d = 3) among lattices with more than 4 (resp. 6) bonds is observed.

我们考虑成对相互作用能，并研究它们在具有指定最小矢量（长度和配位数）的晶格中的极小值，即对应于晶体键的那个。特别地，我们展示了这些结构中强共晶晶格的普遍极小性——即所有完全单调相互作用势的最优性。当点通过完全单调势相互作用时，这为 2 维和 3 维的方形、三角形、简单立方 (sc)、面心立方 (fcc) 和体心立方 (bcc) 晶格提供了新的最优性结果。我们还展示了具有指定键的所有晶格中三角形和 fcc 晶格的普遍极大值。此外，我们将我们的结果应用于 Lennard-Jones 型势，显示了任何通用最小化器（resp. maximer）用于小（或大）键长，其中最优范围很容易计算。最后，提出了一个数值研究，其中尺寸上的“方形-菱形-三角形”（分别为“sc-rhombic-bcc-rhombic-fcc”）类型的相变在具有超过 4 个（分别为 6 个）键的晶格中观察到d = 2（分别为d = 3）。