What is the first step in finding derivative structures of a cell?

Given a “parent” cell any lattice , the first step in finding all derivative structures of that cell is to enumerate all derivative superlattices.

What is the first step to generate the acf triplets?

The second step, generating each set of values of b, d, and e for each diagonal set of acf triplets , can be accomplished simply by three nested loops that start at zero and terminate at b c and d ,e f .

How many symmetrically distinct superlattices are there for the H=2 case?

For the case of cubic symmetry, the seven superlattices for the H=2 case mentioned above reduce to only two symmetrically distinct superlattices.

What is the general idea of the enumeration step in the FWZ algorithm?

The general idea of the superlattice enumeration step in the FWZ algorithm is to find the most compact form for a unit cell of given volume and then generate all possible triads of unit vectors subject to this constraint.

What is the effect of translating the lattice by adding +2 to every point moving?

Translating the lattice by adding +2 to every point moving the origin by two units has the same effect on the labeling as if the authors had merely labeled the four elements of the quotient group, and then added +2 to every element of the group, producing the permutation 0→2, 1→3, 2→0, and 3→1, denoted 2,3,0,1 .

What is the number of fcc/bcc derivatives?

For the “stackable” structures, the quotient group is a single cyclic group, Z4.Table VII lists the number of fcc/bcc ternary and quaternary derivative structures.

How many HNF matrices are generated for each case?

generating the off-diagonal terms for each of these three cases, the authors find seven HNF matrices:case 1: 2 0 00 1 0 0 0 1 , case 2: 1 0 00 2 0 0 0 1 1 0 01 2 0 0 0 1, case 3: 1 0 00 1 00 0 2 1 0 00 1 0 0 1 2 1 0 00 1 0 1 0 2 1 0 00 1 0 1 1 2 .For increasing index, n= H =1,2 ,3 ,¯, the number of HNF matrices generates an interesting sequence: 1, 7, 13, 35, 31, 91, ¯.

What is the way to eliminate duplicate structures?

The algorithm is exceptionally efficient due to the use of i perfect, minimal hash tables and ii a group-theoretical approach to eliminating duplicate structures.

Open AccessJournal Article10.1103/PHYSREVB.77.224115

Algorithm for Generating Derivative Structures

Gus L. W. Hart, +1 more

- 26 Jun 2008

- Physical Review B

- Vol. 77, Iss: 22, pp 224115

438

TL;DR: In this paper, the authors present an algorithm for generating all derivative superstructures for arbitrary parent structures and for any number of atom types in a geometrically independent way.

Abstract: 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.

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Most frequently asked questions

1. What is the function of the transformation of the HNF matrix?

The row and column operations required to transform the HNF matrix of a superlattice into its SNF can be represented by two integer transform matrices, L and R, so that LHR=S, where S is the SNF.

2. What is the role of the points of the lattice in a crystal?

The points of the lattice play the role of positions in a crystal, and the colors play the role of atoms placed at those positions.

3. What is the name of the derivative lattices?

In the mathematical literature, and in some of the crystallogra-phy literature, these derivative lattices are referred to as sublattices.

4. What is the simplest algorithm for finding the greatest common divisor of two numbers?

So the mapx PA−1x mod D11D22 D33meaning that each row of the resulting column matrix is reduced modulo the corresponding diagonal element of Dmaps from L into the direct-sum group G, with its kernel being the superlattice S.As for computing the SNF of a matrix, there are special algorithms designed to compute it efficiently when M is very large but the simplest algorithm, effective for small e.g., 3 3 matrices, is basically an extension of Euclid’s algorithm for finding the greatest common divisor of two numbers.

TABLE II. Two representations of the Cayley table for the quotient group G cosets of the subgroup S . On the right, the elements of the group have been denoted 0,1,2,3 for notational convenience.

FIG. 12. Two-dimensional superlattices of index n=8. The first and third share the same SNF, while the second has the trivial SNF, implying a purely cyclic quotient group. In contrast to the first and third superlattices, the lattice points in layers parallel to the superlattice edge dotted line must all have the same label.

FIG. 1. An example parent lattice left and a superstructure right . The parent lattice is fcc and the superlattice is defined by the doubled unit cell outlined in gray. The two interior points of the superlattice are occupied by one black atom and one gray atom. Together the superlattice and atoms constitute a derivative structure. The superstructure of this example is that of CuAu.

FIG. 6. A superlattice whose SNF is nontrivial noncyclic . Because the quotient group of the superlattice is noncyclic, translations of the lattice are not cyclic permutations. Instead the sites are permuted in groups of two or four with each translation rather than permuting as a single group of 8 . In the figure, three different pictures of the same superlattice are shown. In each case a different origin is chosen. In the second case, the translation permutes two groups of four sites and in the third, four pairs are permuted.

TABLE VI. Simple cubic superstructures for n 4.

FIG. 11. The first 21 binary structures derived from the simple-cubic lattice. All have 4 atoms/cell or less. Structures marked with a crystallographic direction hkl can be characterized as a stacking of pure A and B atomic layers black and white spheres . For n=4 all of the stacked structures occur in pairs, A3B1 and A2B2. The last three structures, labeled 19, 20, and 21, cannot be characterized as pure stackings. These structures have basis vectors whose corresponding quotient group is G=Z2 Z2 rather than Z4 .

Citations

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References

•Book Chapter•10.1007/978-3-540-73086-6_12

The On-Line Encyclopedia of Integer Sequences

Neil J. A. Sloane

- 27 Jun 2007

TL;DR: The On-Line Encyclopedia of Integer Sequences (or OEIS) is a database of some 130000 number sequences which serves as a dictionary, to tell the user what is known about a particular sequence and is widely used.

...read moreread less

Monograph•10.1017/CBO9780511609589

Enumerative Combinatorics: Index

Richard P. Stanley, +1 more

- 01 Jan 1999

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•Journal Article•10.1016/S0364-5916(02)80006-2

The Alloy Theoretic Automated Toolkit: A User Guide

A. van de Walle, +2 more

- 01 Dec 2002

- Calphad-computer Coupling of Phase Diagr...

TL;DR: A concise user guide is presented outlining the steps required to obtain thermodynamic information from ab initio calculations of alloy thermodynamic properties from first-principles.

...read moreread less

1.1K

•Journal Article•10.1361/105497102770331596

Automating first-principles phase diagram calculations

A. van de Walle, +1 more

- 01 Aug 2002

- Journal of Phase Equilibria

TL;DR: In this article, the authors present a statistical basis for the selection of both the interaction parameters to include in the cluster expansion and the structures to use to determine them, which is based on the concepts of cross-validation and variance minimization.

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Automating First-Principles Phase Diagram Calculations

A. van de Walle, +1 more

- 28 Jan 2002

- arXiv: Statistical Mechanics

TL;DR: The proposed algorithm formalizes the most difficult step of phase diagram calculations, namely the determination of the “cluster expanison,” which is a compact representation of the configurational dependence of the alloy’s energy.

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Algorithm for Generating Derivative Structures

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AI Agents for this Paper

Most frequently asked questions

1. What is the function of the transformation of the HNF matrix?

2. What is the role of the points of the lattice in a crystal?

3. What is the name of the derivative lattices?

4. What is the simplest algorithm for finding the greatest common divisor of two numbers?

5. What is the first step in finding derivative structures of a cell?

6. What is the first step to generate the acf triplets?

7. How many symmetrically distinct superlattices are there for the H=2 case?

8. What is the general idea of the enumeration step in the FWZ algorithm?

9. What is the effect of translating the lattice by adding +2 to every point moving?

10. What is the number of fcc/bcc derivatives?

11. How many HNF matrices are generated for each case?

12. What is the way to eliminate duplicate structures?

Figures

Citations

The high-throughput highway to computational materials design

Two-dimensional materials from high-throughput computational exfoliation of experimentally known compounds

AFLOW: An automatic framework for high-throughput materials discovery

Efficient stochastic generation of special quasirandom structures

The structural and chemical origin of the oxygen redox activity in layered and cation-disordered Li-excess cathode materials

References

The On-Line Encyclopedia of Integer Sequences

Enumerative Combinatorics: Index

The Alloy Theoretic Automated Toolkit: A User Guide

Automating first-principles phase diagram calculations

Automating First-Principles Phase Diagram Calculations

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