Born's three-dimensional lattice-stability problem: classifying when long-wave (elastic) stability implies all-wave (dynamical) stability, the necessity of his sign conditions, and stability at finite temperature

In 'On the stability of crystal lattices. I' (1940), Max Born founded the elastic theory of crystal stability: he expressed the second-order change of lattice free energy as a quadratic form, obtained the elastic constants as lattice sums, proved that for a one-dimensional chain of equal particles with pair potential, macroscopic (long-wave) stability implies all-wave (microscopic) stability in the harmonic approximation, under the sign conditions that the second derivative of the potential is positive at the nearest-neighbour distance and negative at all further multiples of it, and stated explicitly that he had not succeeded in proving the analogous theorem for three-dimensional lattices. Eighty-five years of subsequent literature settles the unrestricted three-dimensional claim negatively: Wallace and Patrick (1965) exhibited a counterexample in the diamond structure, and modern phonon studies find short-wavelength instabilities abundant; the contemporary canon (Mouhat and Coudert, 2014) treats elastic stability and dynamical (phonon) stability as independent necessary conditions. What survives as an open, precisely posed problem is the classification question: for exactly which three-dimensional lattices and interaction classes does positive definiteness of the Born elastic-constant matrix imply positivity of the entire phonon spectrum? The only general positive result remains Power's 1942 theorem for the face-centred cubic lattice with central pair forces truncated at the twelve nearest neighbours; the Lennard-Jones classification of Ono and Ito (2021) is numerical and case-by-case. Two further items of Born's own list remain open: whether his one-dimensional sign conditions are necessary, and the formulation of rigorous stability conditions at finite temperature beyond the harmonic approximation, where anharmonic stabilization (e.g. simple-cubic calcium under pressure) shows that harmonic instability does not decide physical stability.

Background

A Bravais lattice is an infinite set of points generated by integer combinations of three non-coplanar translation vectors; common three-dimensional examples are the simple cubic (sc), face-centred cubic (fcc), body-centred cubic (bcc), and hexagonal close-packed (hcp) lattices. A crystal with a basis, such as the diamond structure, has two or more atoms per primitive cell. A pair potential is an interaction energy that depends only on the distance between each pair of atoms; the Lennard-Jones family $\varphi(r)=r^{-m}-r^{-n}$ with $m>n$ (a repulsive and an attractive inverse-power term) is the standard model class. The harmonic approximation expands the crystal's potential energy to second order in the displacements of atoms from their equilibrium positions. The coefficients of this quadratic form define, for each wave vector in the first Brillouin zone (the set of points in reciprocal space closer to the origin than to any other reciprocal-lattice point), a dynamical matrix whose eigenvalues are the squared frequencies of the lattice's normal vibrational modes (phonons); dynamical (all-wave) stability means every eigenvalue of the dynamical matrix is positive at every wave vector. In lattices with a basis, some branches (optical branches) have no long-wavelength continuum analogue. Expanding the same energy instead for long-wavelength (macroscopic, homogeneous strain) deformations yields the elastic constants; Born's elastic stability criteria require the elastic-constant matrix to be positive definite (for cubic symmetry: the three conditions on elastic constants stated below in the problem statement). Long-wave (elastic) stability is thus the small-wave-vector limit of dynamical stability, and the question of whether the former implies the latter is nontrivial because contributions from different neighbour shells enter the dynamical matrix with distance-dependent signs and trigonometric weights. Born proved in one dimension that macroscopic (long-wave) stability implies all-wave (microscopic) stability, under the sign conditions that the second derivative of the pair potential is positive at the nearest-neighbour distance and negative at every larger multiple of it (conditions he noted are 'satisfied in most practical cases'); his macroscopic-stability premise is the lattice-sum inequality $\sum_{l\ge1} b_l l^2>0$ on the second derivatives $b_l=\varphi''(la)$ of the potential. His paper series (Parts II-IV and VII-XI, with Misra, Fürth, Power, Peng, and Bradburn, together with Fürth's 1941 companion paper) extended the framework to power-law lattices, finite deformation, rhombohedral and hexagonal lattices, and thermodynamics; the 1954 book by Born and Huang canonized the theory. Related later notions: the Born melting criterion (a crystal melts when a shear modulus, i.e. a long-wave modulus, vanishes; the 'rigidity catastrophe'), which distinguishes mechanical from thermodynamic melting; the quasi-harmonic approximation (harmonic phonons with a volume-dependent equilibrium), and anharmonic effects, where phonon-phonon interactions stabilize or destabilize structures that the harmonic approximation misjudges.

Problem Statement

Let $L$ be a three-dimensional Bravais lattice (or a crystal structure with a basis) whose atoms interact through a pair potential $\varphi(r)$ with an absolutely convergent lattice sum, at an equilibrium lattice parameter. Define two properties. Long-wave (elastic, Born) stability: the elastic-constant matrix computed from Born's lattice-sum formulas is positive definite (for cubic symmetry: $C_{11}-C_{12}>0$, $C_{11}+2C_{12}>0$, $C_{44}>0$). All-wave (dynamical) stability: the dynamical matrix $D(\mathbf{k})$, whose entries are lattice sums of second derivatives of $\varphi$ weighted by trigonometric factors, has positive eigenvalues for every wave vector $\mathbf{k}$ in the first Brillouin zone, excepting only the three zero eigenvalues at $\mathbf{k}=0$ that translational invariance forces (the acoustic branches). Three open items constitute the problem; they are independent sub-claims. (1) CLASSIFICATION: determine exactly the pairs (lattice, interaction class) for which long-wave stability implies all-wave stability. Here an interaction class is a family of pair potentials fixed by a functional form together with an interaction range; the classification is required to range over a bounded domain: the lattice set is the finite list sc, fcc, bcc, hcp and diamond, and the interaction-class set is the finite list consisting of (i) central pair forces with support on exactly the first two neighbour shells, the two shell second derivatives of the pair potential carrying independent signs, and (ii) the Lennard-Jones $(m,n)$ family with integer exponents in the bounded range $6\le n<m\le 12$. The classification ranges over these ten (lattice, class) cases rather than over arbitrary pair potentials. Known: the implication holds for the face-centred cubic lattice with central pair forces truncated at the twelve nearest neighbours (Power 1942); it fails in the diamond structure (Wallace-Patrick 1965); fcc Lennard-Jones crystals are stable for all exponent ratios while bcc is unstable except for small ones (Ono-Ito 2021, numerically). A rigorous and complete classification, even for the single family of Lennard-Jones $(m,n)$ potentials across all three-dimensional Bravais lattices, does not exist. (2) SIGN CONDITIONS: Born's one-dimensional theorem, that macroscopic (long-wave) stability implies all-wave (microscopic) stability for a linear chain of equal particles with pair potential, is proved under the conditions $b_1=\varphi''(a)>0$ and $b_k=\varphi''(ka)<0$ for $k\ge 2$, where $a$ is the nearest-neighbour distance; Born noted these are 'satisfied in most practical cases'. This item comprises two independently standing sub-claims: (2a) prove or refute the necessity of these one-dimensional conditions within a fixed analytic, single-well pair potential with an absolutely convergent lattice sum (such as a Lennard-Jones $(m,n)$ potential): does there exist a one-dimensional lattice that is both long-wave and all-wave stable with some $\varphi''(ka)>0$ for $k\ge 2$? (2b) determine a three-dimensional analogue of these conditions: a sign or inequality condition-set $C$ on the shell second derivatives $\varphi''$, or on the potential's parameters, that is independent of the number of interacting shells (not a mere restriction on the potential's support or range), does not presuppose all-wave stability, holds across a class of three-dimensional lattices rather than a single one, and is such that long-wave stability together with $C$ implies all-wave stability, and determine whether that condition-set $C$ is necessary, that is, whether every all-wave-stable three-dimensional lattice must satisfy it. (3) FINITE TEMPERATURE: formulate rigorous necessary and sufficient stability conditions at temperature $T>0$ beyond the harmonic (and quasi-harmonic) approximation, in particular for lattices that are unstable in the harmonic approximation but anharmonically stabilized (such as simple-cubic calcium under pressure), where no accepted criterion currently exists. Here stability at $T>0$ means thermodynamic stability: the structure is a (meta)stable minimum of the free energy at temperature $T$.

References

1. Max Born, 'On the stability of crystal lattices. I', Mathematical Proceedings of the Cambridge Philosophical Society 36, 160-172 (1940). DOI: 10.1017/s0305004100017138, URL: https://doi.org/10.1017/s0305004100017138
2. Max Born, 'Thermodynamics of Crystals and Melting', The Journal of Chemical Physics 7, 591 (1939). DOI: 10.1063/1.1750497, URL: https://doi.org/10.1063/1.1750497
3. Rama Dhar Misra, 'On the stability of crystal lattices. II', Mathematical Proceedings of the Cambridge Philosophical Society 36, 173 (1940). DOI: 10.1017/s030500410001714x, URL: https://doi.org/10.1017/s030500410001714x
4. M. Born and R. Fürth, 'The stability of crystal lattices. III', Mathematical Proceedings of the Cambridge Philosophical Society 36, 454-465 (1940). DOI: 10.1017/s0305004100017503, URL: https://doi.org/10.1017/s0305004100017503
5. Max Born and Rama Dhar Misra, 'On the stability of crystal lattices. IV', Mathematical Proceedings of the Cambridge Philosophical Society 36, 466 (1940). DOI: 10.1017/s0305004100017515, URL: https://doi.org/10.1017/s0305004100017515
6. R. Fürth, 'The Stability of Crystal Lattices', Mathematical Proceedings of the Cambridge Philosophical Society 37, 34-54 (1941). DOI: 10.1017/s0305004100021514, URL: https://doi.org/10.1017/s0305004100021514
7. S. C. Power, 'On the stability of crystal lattices VII. Long-wave and short-wave stability for the face-centred cubic lattice', Mathematical Proceedings of the Cambridge Philosophical Society 38, 61-66 (1942). DOI: 10.1017/s0305004100022222, URL: https://doi.org/10.1017/s0305004100022222
8. H. W. Peng and S. C. Power, 'On the stability of crystal lattices VIII. Stability of rhombohedral Bravais lattices', Mathematical Proceedings of the Cambridge Philosophical Society 38, 67 (1942). DOI: 10.1017/s0305004100022234, URL: https://doi.org/10.1017/s0305004100022234
9. M. Born, 'On the stability of crystal lattices IX. Covariant theory of lattice deformations and the stability of some hexagonal lattices', Mathematical Proceedings of the Cambridge Philosophical Society 38, 82-99 (1942). DOI: 10.1017/s0305004100022246, URL: https://doi.org/10.1017/s0305004100022246
10. Max Born, 'The thermodynamics of crystal lattices', Mathematical Proceedings of the Cambridge Philosophical Society 39, 100-103 (1943). DOI: 10.1017/S0305004100017746, URL: https://doi.org/10.1017/S0305004100017746
11. Max Born and Mary Bradburn, 'The thermodynamics of crystal lattices II', Mathematical Proceedings of the Cambridge Philosophical Society 39, 104-113 (1943). DOI: 10.1017/S0305004100017758, URL: https://doi.org/10.1017/S0305004100017758
12. M. Born and K. Huang, 'Dynamical Theory of Crystal Lattices', Oxford University Press (1954; 1996 printing). ISBN: 9780192670083, DOI: 10.1093/oso/9780192670083.001.0001, URL: https://doi.org/10.1093/oso/9780192670083.001.0001
13. D. C. Wallace and J. L. Patrick, 'Stability of Crystal Lattices', Physical Review 137, A152-A160 (1965). DOI: 10.1103/PhysRev.137.A152, URL: https://doi.org/10.1103/PhysRev.137.A152
14. F. W. de Wette and B. R. A. Nijboer, 'On the stability of close-packed crystals with Lennard-Jones interaction', Physics Letters 18, 19-20 (1965). DOI: 10.1016/0031-9163(65)90012-0, URL: https://doi.org/10.1016/0031-9163(65)90012-0
15. F. Milstein, 'Mechanical Stability of Crystal Lattices with Two-Body Interactions', Physical Review B 2, 512 (1970). DOI: 10.1103/PhysRevB.2.512, URL: https://doi.org/10.1103/PhysRevB.2.512
16. R. Hill, 'On the elasticity and stability of perfect crystals at finite strain', Mathematical Proceedings of the Cambridge Philosophical Society 77, 225 (1975). DOI: 10.1017/S0305004100049549, URL: https://doi.org/10.1017/S0305004100049549
17. Jinghan Wang, Sidney Yip, S. R. Phillpot, and Dieter Wolf, 'Crystal instabilities at finite strain', Physical Review Letters 71, 4182 (1993). DOI: 10.1103/PhysRevLett.71.4182, URL: https://doi.org/10.1103/PhysRevLett.71.4182
18. Jinghan Wang, Ju Li, Sidney Yip, Simon Phillpot, and Dieter Wolf, 'Mechanical instabilities of homogeneous crystals', Physical Review B 52, 12627 (1995). DOI: 10.1103/PhysRevB.52.12627, URL: https://doi.org/10.1103/PhysRevB.52.12627
19. V. Sorkin, E. Polturak, and Joan Adler, 'Molecular dynamics study of melting of the bcc metal vanadium. I. Mechanical melting', Physical Review B 68, 174102 (2003). DOI: 10.1103/PhysRevB.68.174102, arXiv: cond-mat/0304215, URL: https://doi.org/10.1103/PhysRevB.68.174102
20. Ion Errea, Bruno Rousseau, and Aitor Bergara, 'Anharmonic Stabilization of the High-Pressure Simple Cubic Phase of Calcium', Physical Review Letters 106, 165501 (2011). DOI: 10.1103/PhysRevLett.106.165501, arXiv: 1103.0108, URL: https://doi.org/10.1103/PhysRevLett.106.165501
21. Göran Grimvall, Blanka Magyari-Köpe, Vidvuds Ozoliņš, and Kristin A. Persson, 'Lattice instabilities in metallic elements', Reviews of Modern Physics 84, 945 (2012). DOI: 10.1103/RevModPhys.84.945, URL: https://doi.org/10.1103/RevModPhys.84.945
22. Ekaterina Podolskaya, Artem Panchenko, and Anton Krivtsov, 'Stability and Structural Transitions in Crystal Lattices', in 'Surface Effects in Solid Mechanics' (IUTAM Symposium, Advanced Structured Materials vol. 21), Springer, pp. 123-133 (2013). DOI: 10.1007/978-3-642-35783-1_10, URL: https://doi.org/10.1007/978-3-642-35783-1_10
23. Félix Mouhat and François-Xavier Coudert, 'Necessary and sufficient elastic stability conditions in various crystal systems', Physical Review B 90, 224104 (2014). DOI: 10.1103/PhysRevB.90.224104, arXiv: 1410.0065, URL: https://doi.org/10.1103/PhysRevB.90.224104
24. Hossein Salahshoor, Raj Kumar Pal, and Julian J. Rimoli, 'Non-Schmid Effects In Perfect Single Crystal Metals At Zero Temperature', arXiv:1704.08376 (2017). URL: https://arxiv.org/abs/1704.08376
25. Nikolai A. Zarkevich, Hao Chen, Valery I. Levitas, and Duane D. Johnson, 'Lattice Instability during Solid-Solid Structural Transformations under a General Applied Stress Tensor: Example of Si I → Si II with Metallization', Physical Review Letters 121, 165701 (2018). DOI: 10.1103/PhysRevLett.121.165701, arXiv: 1806.00055 (preprint title: 'Deformation, lattice instability, and metallization during solid-solid structural transformations under general applied stress tensor: example of Si I -> Si II'), URL: https://doi.org/10.1103/PhysRevLett.121.165701
26. Shota Ono and Tasuku Ito, 'Theory of dynamical stability for two- and three-dimensional Lennard-Jones crystals', Physical Review B 103, 075406 (2021). DOI: 10.1103/PhysRevB.103.075406, arXiv: 2011.01465, URL: https://doi.org/10.1103/PhysRevB.103.075406
27. Peter Schwerdtfeger, Antony Burrows, and Odile R. Smits, 'The Lennard-Jones Potential Revisited: Analytical Expressions for Vibrational Effects in Cubic and Hexagonal Close-Packed Lattices', The Journal of Physical Chemistry A 125, 3037-3057 (2021). DOI: 10.1021/acs.jpca.1c00012, arXiv: 2012.05413, URL: https://doi.org/10.1021/acs.jpca.1c00012
28. Kevin M. Knowles, 'Necessary and Sufficient Elastic Stability Conditions for Single Crystals', Journal of Elasticity 157, article 19 (2025). DOI: 10.1007/s10659-025-10112-0, URL: https://doi.org/10.1007/s10659-025-10112-0