Mixed-basis cluster expansion for thermodynamics of bcc alloys

Abstract

To predict the ground-state structures and finite-temperature properties of an alloy, the total energies of many different atomic configurations ϵ͕ i ; i =1, . . . ,N͖, with N sites i occupied by atom A ͑ i =−1͒, or B ͑ i = +1͒, must be calculated accurately and rapidly. Direct local-density approximation (LDA) calculations provide the required accuracy, but are not practical because they are limited to small cells and only a few of the 2 N possible configurations. The "mixed-basis cluster expansion" (MBCE) method allows to parametrize LDA configurational energetics E LDA ͓ i ; i =1, . . . ,N͔ by an analytic functional E MBCE ͓ i ; i =1, . . . ,N͔. We extend the method to bcc alloys, describing how to select N ordered structures (for which LDA total energies are calculated explicitly) and N F pair and multibody interactions, which are fit to the N energies to obtain a deterministic MBCE mapping of LDA. We apply the method to bcc Mo-Ta. This system reveals an unexpectedly rich ground-state line, pitting Mo-rich (100) superlattices against Ta-rich complex structures. Predicted finite-T properties such as order-disorder temperatures, solid-solution short-range order and the random alloy enthalpy of mixing are consistent with experiment. PACS number(s): 61.66.Dk, 71.15.Nc I. INTRODUCTION A. The cluster expansion method: Definition and scopeAlloy thermodynamics, including properties at T = 0, require the knowledge of the excess energy ͑1͒of the solid A 1−x B x relative to the total energy of its constituents A and B. For a given underlying lattice, the degrees of freedom form a configurational vector , whose components i = ±1 record whether a lattice site i is occupied by element A ͑ i =−1͒ or B ͑ i = +1͒. Since ⌬H direct ͑͒ is difficult to calculate quantum mechanically for an exhaustive set of structures , it is often described by way of a cluster expansion (CE) Hamiltonian 1-6 ͑2͕͒J͖ are the interaction parameters for each pair or many-body combination of lattice sites i , j , k, etc. The cluster expansion 7-10 attempts to describe the energies of all different configurations with the accuracy of present-day densityfunctional methods. It is based on the fact that Eq.(2) allows to map an arbitrarily complex Hamiltonian with electronic degrees of freedom exactly onto a simple sum over crystallographic degrees of freedom. 8 For practical purposes, Eq.(2) is typically recast in terms of symmetry-inequivalent figures. 11 Also, without a loss of generality one may subtract 9 a configuration-dependent reference energy E ref from ⌬H. We may expand ⌬H LDA = ⌬H LDA − E ref as ⌬H CE , so thatThe interaction parameters ͕J f ͖ now signify all possible inequivalent pairs and many-body (MB) figures f. They are configuration independent, 8 with D f as each figure's symmetry degeneracy per site-any configuration dependence is contained in the lattice-averaged correlation functions ⌸ f ͑͒. Although Eqs.(2) and (3) contain, in principle, many interactions, the energetics of bonding is usually determined by relatively short length sca...

First-principles calculation of phase equilibrium of V-Nb, V-Ta, and Nb-Ta alloys

“…8,9 Recent first-principles study of phase equilibrium of similar refractory bcc binary alloys, such as Nb-Mo, Nb-W, Ta-W, and Ta-Mo, has however predicted several intermediate ground states. 10 For V-Nb, V-Ta, and Nb-Ta systems, while there exist a few CALPHAD studies, first-principles studies of phase equilibrium do not exist in the literature.…”

Bayesian approach to cluster expansions

“…Forms of leave-many-out cross validation may also be considered. 37 Use a regularization function with few parameters. The use of regularization functions with a large number of degrees of freedom, such as cluster selection or the independent regularization function, can lead to overfitting of the crossvalidation score.…”

“…To limit the degrees of freedom of the mixed-basis cluster expansion, constraints may be applied to the number of multisite cluster functions with nonzero ECI allowed in the expansion. 37 In contrast, the inverse-density regularization function applies no such limit but reduces the number of degrees of freedom through parameterization while simultaneously encouraging smaller ECI for clusters with a larger number of sites.…”

“…It has been suggested that forms of leave-many-out cross validation, in which multiple elements are left out of the training set at once, may be used in place of leave-one-out cross validation. 37 There is evidence that for cluster selection in systems with only a few significant ECI, leave-many-out cross validation can significantly reduce prediction error compared to leave-one out cross validation but the difference between the two approaches rapidly diminishes as the number of significant ECI increases. 38 In addition, leave-manyout cross validation can be an unreliable indicator of prediction error for cluster selection.…”

A generalized Ising model for studying alloy evolution under irradiation and its use in kinetic Monte Carlo simulations

“…This problem can quickly become intractable, given the combinatorial nature of arranging N distinguishable atomic sites, and a number of approaches have been proposed to reduce the dimensional complexity of the problem [34][35][36]. A common simplification is to assume that the Hamiltonian H of the system can be calculated as the sum of all possible pair interactions, defined by their bond energies: …”