Rodney W. Forcade

Algorithm for Generating Derivative Structures

We present an algorithm for generating all derivative superstructures—for arbitrary parent structures and for any number of atom types. This algorithm enumerates superlattices and atomic configurations in a geometry-independent way. The key concept is to use the quotient group associated with each superlattice to determine all unique atomic configurations. The run time of the algorithm scales linearly with the number of unique structures found. We show several applications demonstrating how the algorithm can be used in materials design problems. We predict an altogether new crystal structure in Cd-Pt and Pd-Pt, and several new ground states in Pd-rich and Pt-rich binary systems.

UNCLE: a code for constructing cluster expansions for arbitrary lattices with minimal user-input

We present a new implementation of the cluster expansion formalism. The new code, UNiversal CLuster Expansion (UNCLE), consolidates recent advances in the methodology and leverages one new development in the formalism itself. As a core goal, the package reduces the need for user intervention, automating the method to reduce human error and judgment. The package extends standard cluster expansion formalism to the more complicated cases of ternary compounds, as well as surfaces, including adsorption and inequivalent sites.

On the spectral radius of a (0,1) matrix related to Merten's function

Abstract Define n× n matrices Dn = (dij) and Cn = (cij) by dij = 1 if i∣j, 0 otherwise and Cn = (0, 1, 1,…, 1)T(1, 0, 0,…, 0). Let An = Dn + Cn. We use the directed graph of An −In to obtain the characteristic polynomial of An. Then we show that all but [log2n]+1 of the eigenvalues of An are equal to 1 and that ϱ(An) is asymptotically equal to √n as n → ∞.

Lattice-Simplex Coverings and the 84-Shape

It is known that diameters of abelian Cayley graphs with n generators are related to Manhattan diameters of a particular type of lattice tile called a Cayley tile and that determining the lower bound of those diameters is related to the density of lattice-simplex coverings of Rn. We construct a natural tile associated with each lattice-simplex covering and show that the 84-shape (discovered in [R. Dougherty and V. Faber, The Degree-Diameter Problem for Several Varieties of Cayley Graphs, http://www.c3.lanl.gov/dm/pub/laces.html (1994) and C. M. Fiduccia, J. S. Zito, and E. Mann, Network Interconnection Architectures and Translational Tilings, Tech. report, Center for Computing Science, Bowie, MD, 1994]) represents a local mininum of the density of lattice coverings of R3 by a particular simplex D (the convex hull of the unit basis vectors).

Numerical Algorithm for Pólya Enumeration Theorem

Although the Pólya enumeration theorem has been used extensively for decades, an optimized, purely numerical algorithm for calculating its coefficients is not readily available. We present such an algorithm for finding the number of unique colorings of a finite set under the action of a finite group.

Diameter series of lattice covering simplices

We develop an interesting relationship between finite sets in a lattice and the minimal density of simplex coverings of n-space.