Lattice ground states for embedded-atom models in 2D and 3D

The Embedded-Atom Model (EAM) provides a phenomenological description of atomic arrangements in metallic systems. It consists of a configurational energy depending on atomic positions and featuring the interplay of two-body atomic interactions and nonlocal effects due to the corresponding electronic clouds. The purpose of this paper is to mathematically investigate the minimization of the EAM energy among lattices in two and three dimensions. We present a suite of analytical and numerical results under different reference choices for the underlying interaction potentials. In particular, Gaussian, inverse-power, and Lennard-Jones-type interactions are addressed.

[1]  R. A. Johnson,et al.  Analytic embedded atom method model for bcc metals , 1989 .

[2]  M. Baskes.  Many-Body Effects in fcc Metals: A Lennard-Jones Embedded-Atom Potential , 1999 .

[3]  M. Finnis,et al.  A simple empirical N-body potential for transition metals , 1984 .

[4]  Harmonic Analysis on Symmetric Spaces and Applications II (Audrey Terras) , 1987 .

[5]  M. Baskes,et al.  Embedded-atom method: Derivation and application to impurities, surfaces, and other defects in metals , 1984 .

[6]  S. Foiles.  Embedded-Atom and Related Methods for Modeling Metallic Systems , 1996 .

[7]  Richard Alan Lesar,et al.  Introduction to Computational Materials Science: Fundamentals to Applications , 2013 .

[8]  Laurent Bétermin,et al.  Effect of Periodic Arrays of Defects on Lattice Energy Minimizers , 2020, Annales Henri Poincaré.

[9]  L. Pártay,et al.  Pressure-Temperature Phase Diagram of Lithium, Predicted by Embedded Atom Model Potentials. , 2020, The journal of physical chemistry. B.

[10]  Hugh L. Montgomery,et al.  Minimal theta functions , 1988, Glasgow Mathematical Journal.

[11]  M. Baskes,et al.  Application of the embedded-atom method to covalent materials: A semiempirical potential for silicon. , 1987, Physical review letters.

[12]  Local variational study of 2d lattice energies and application to Lennard–Jones type interactions , 2016, 1611.07820.

[13]  Mathieu Lewin,et al.  The Crystallization Conjecture: A Review , 2015, 1504.01153.

[14]  Peter Sarnak,et al.  Minima of Epstein’s Zeta function and heights of flat tori , 2006 .

[15]  S. G. Srinivasan,et al.  On the Lennard–Jones EAM potential , 2004, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[16]  S. Russo,et al.  On fitting a gold embedded atom method potential using the force matching method. , 2005, The Journal of chemical physics.

[17]  M. Baskes,et al.  Semiempirical, Quantum Mechanical Calculation of Hydrogen Embrittlement in Metals , 1983 .

[18]  A. Sutton,et al.  Long-range Finnis–Sinclair potentials , 1990 .

[19]  C. Bris,et al.  Periodicity of the infinite-volume ground state of a one-dimensional quantum model , 2002 .

[20]  Joshua R. Smith,et al.  Universal features of the equation of state of metals , 1984 .

[21]  Tilmann Leisegang,et al.  Harmonic Principles of Elemental Crystals - From Atomic Interaction to Fundamental Symmetry , 2018, Symmetry.

[22]  J. van Wezel,et al.  The simple-cubic structure of elemental Polonium and its relation to combined charge and orbital order in other elemental chalcogens , 2017, SciPost Physics.

[23]  C. Poole,et al.  Encyclopedic Dictionary of Condensed Matter Physics , 2004 .

[24]  Henry Cohn,et al.  Universally optimal distribution of points on spheres , 2006, math/0607446.

[25]  A. Terras.  Harmonic Analysis on Symmetric Spaces and Applications I , 1985 .

[26]  Joshua R. Smith,et al.  Origins of the universal binding-energy relation. , 1988, Physical review. B, Condensed matter.

[27]  Peng Zhang,et al.  Minimization of energy per particle among Bravais lattices in R^2 : Lennard-Jones and Thomas-Fermi cases , 2014, 1402.2751.

[28]  Henry Cohn,et al.  Universal optimality of the $E_8$ and Leech lattices and interpolation formulas , 2019, Annals of Mathematics.

[29]  Bin Jiang,et al.  Embedded Atom Neural Network Potentials: Efficient and Accurate Machine Learning with a Physically Inspired Representation. , 2019, The journal of physical chemistry letters.

[30]  Laurent Bétermin.  Minimizing lattice structures for Morse potential energy in two and three dimensions , 2019, Journal of Mathematical Physics.

[31]  Mircea Petrache,et al.  Optimal and non-optimal lattices for non-completely monotone interaction potentials , 2018, Analysis and Mathematical Physics.

[32]  Cai,et al.  Simple analytical embedded-atom-potential model including a long-range force for fcc metals and their alloys. , 1996, Physical review. B, Condensed matter.

[33]  Local optimality of cubic lattices for interaction energies , 2016 .

[34]  E. G. COX,et al.  Structural Inorganic Chemistry , 1946, Nature.

[35]  Adarsh Balasubramanian,et al.  Fast, accurate, and transferable many-body interatomic potentials by symbolic regression , 2019, npj Computational Materials.

[36]  Johnson.  Alloy models with the embedded-atom method. , 1989, Physical review. B, Condensed matter.

[37]  Murray S. Daw,et al.  The embedded-atom method: a review of theory and applications , 1993 .

[38]  Laurent Bétermin,et al.  Two-Dimensional Theta Functions and Crystallization among Bravais Lattices , 2016, SIAM J. Math. Anal..