Generating derivative structures from multilattices: Algorithm and application to hcp alloys

We present an algorithm for generating all derivative superstructures of a nonprimitive parent lattice. The algorithm has immediate application in important materials design problems such as modeling hexagonal-close-packed (hcp) alloys. Extending the work of Hart and Forcade [Phys. Rev. B 77, 224115 (2008)] (which applies only to Bravais lattices), this approach applies to arbitrary multilattices. The algorithm enumerates superlattices and atomic configurations using permutation groups rather than direct geometric comparisons. The key concept is to use the quotient group associated with each superlattice to determine all unique atomic configurations. The algorithm is very efficient; the run time scales linearly with the number of unique structures found. We demonstrate the algorithm in the important case of hcp-derived superstructures. In the list of enumerated hexagonal-close-packed derivative superstructures, we predict several as-yet-unobserved structures as likely candidates for new intermetallic prototypes.

Figure 1

A two-dimensional example of a multilattice. In each unit cell there are two atomic sites. Because this set of atomic sites cannot be defined by just two vectors (i.e., the unit cell only), it is not a lattice in the proper sense. Rather we refer to it as a multilattice or, more specifically, as a two-lattice: it is two square lattices superimposed—one placed at the origin and another translated slightly in the direction $\widehat{x}+\widehat{y}$. The origins of each lattice constitute the $D$ set, (0,0), and $\left(\frac{1}{3},\frac{1}{3}\right)$ in this case.

Figure 2

(Color online) (a) A square parent lattice and (b) and (c) two derivative structures. The dotted lines in the superstructures show the superlattices, which are multiples of the parent lattice. That is, the dotted lines indicate the unit cell of the superstructure. The lattice vectors are indicated by arrows. The superstructure lattice vectors are integer combinations of the parent cell lattice vectors. In the superstructures, the atoms (colored circles) lie on the lattice points of the original parent lattice and have a periodicity that matches the superlattice.

Figure 3

Two real examples of three-dimensional superstructures. The $L{1}_0$ structure (top right) is a doubled superstructure derived from the fcc lattice (top left). The $D{0}_{19}$ structure (bottom right) is a quadrupled superstructure derived from an hcp parent multilattice (bottom left).

Figure 4

(Color online) An example demonstrating that the geometric equivalence of multilattice labelings can be expressed as a permutation group. (a) is a multilattice, a square lattice with two points in the $D$ set: (0,0) and $\left(\frac{1}{3},\frac{1}{3}\right)$. (b) is a superlattice derived from (a) using the given transformation matrix $\mathbb{H}$. (c) shows that if any point is labeled, all equivalent points (translates) must have the same label. (The set of such points is a coset.) (d)–(f) show how lattice points are permuted by a symmetry operation of the multilattice, an orthogonal transformation followed by a fractional translation. (d) is the starting configuration. (e) shows how the points are moved by a reflection about the $x+y=0$ line. (f) shows the reflected points of (e) translated so the permuted points are now coincident with the original superlattice. This symmetry operation permutes the elements of the first $D\times G$ table to form the second table (rows reversed, second and third columns exchanged). The middle table depicts the permutation mapping, $\pi$, that maps each point, $p$, of the superlattice to another. The bottom left picture shows one particular but arbitrary labeling of the multilattice; the bottom right shows how this labeling is permuted by the symmetry operation. The second labeling is not identical to the first but it is equivalent.

Figure 5

Two equivalent multilattices. Though the bases are different, the first multilattice can be transformed into the second by a reflection about the line $x-y=0$. (Note that a $90°$ rotation is not a symmetry of the multilattice but the reflection is.)

Figure 6

(Color online) Seven binary hcp-derived superstructures (Ref. 5). These structures are all of the experimentally known hcp-derived structures with eight atoms/cell or less. $B_h$ (WC) has in index of $n=1$; $B19$ (AuCd), ${\text{Ag}}_3\text{Sb}$, and ${\text{LiI}}_3$ all have index $n=2$; the ${\text{Au}}_5\text{Sb}$ structure has $n=3$; and the last two, $D{0}_{19}$ $\left({\text{Ni}}_3\text{Sn}\right)$ and $D{0}_a$ $\left(\beta {\text{Cu}}_3\text{Ti}\right)$, have index $n=4$. In the pictures, the red (thick) lines indicate the primitive unit cells; the thin black lines are the underlying parent hcp cells. For an index of $n\le 4$, our algorithm finds 201 hcp-derived structures, but only these 7 have been observed.

Figure 7

(Color online) The likelihood measure of hcp derivative structures (index $n=1–4$). A few enumerated structures have higher measures than the seven experimentally observed structures.