Harold T. Stokes and Dorian M. Hatch, Department of Physics and Astronomy, Brigham Young University, Provo, Utah, 84602, USA, branton_campbell@byu.edu

COPL produces a complete order parameter listing for a phase transition, given the parent and subgroup symmetries.

**Background**

In a crystalline phase transition, the space-group symmetry of the
high- and low- symmetry phases often exhibit a group-subgroup
relation. Let *H* be the space-group symmetry of the
high-symmetry phase, and let *L* be the space-group symmetry of
the low-symmetry phase. In the Landau theory of phase transitions,
this transition is described by a set of order parameters (OPs). Each
such OP is an *n*-dimensional vector **v** in the space
defined by an irreducible representation (IR) of *H*. The IR
maps each operator *h* in *H* onto an *n*x*n*
matrix *D(h)*. For a given OP **v** of the transition, the
operators *h* in *H* which satisfy
*D(h)***v**=**v** form a group *L'* such that its
symmetry is greater than or equal to that of *L*. The OPs for
which *L'=L* are primary OPs. They determine the symmetry of the
low-symmetry phase. The other OPs in the set are secondary OPs.
(This situation can become more complex in the case of coupled OPs.)
The aim of COPL is to generate a complete list (set) of OPs, both
primary and secondary, associated with a given group-subgroup pair of
space groups, *L* and *H*. The generation of this list is a
solution to a generalization of the inverse Landau problem [Ascher and
Kobayashi, J. Phys. C **10**, 1349 (1977); Ascher, J. Phys. C **
10**, 1365 (1977)].

Further information about COPL: D. M. Hatch and H. T. Stokes, "The Complete Listing of Order Parameters for a Crystalline Phase Transition: A Solution to the Generalized Landau Problem." Phys. Rev. B 65, 014113-1-4 (2002).

**Input:**

(1) Parent space group: settings from the International Tables of Crystallography are listed.

(2) Subgroup space group: settings from the International Tables of Crystallography are listed.

(3) Lattice vectors of subgroup in terms of lattice vectors of parent.

(4) Origin of subgroup with respect to parent in terms of lattice vectors of parent. This information is sometimes a little difficult to obtain. If you know the position of a particular atom in both the parent and subgroup space groups, then take its position in the parent space group and subtract its position in the subgroup space group (written in terms of the lattice vectors of the parent).

**Output:**

(1) Irrep: irreducible representation of the parent space group involved in the phase transition. We use the notation of Miller and Love.

(2) k params: parameters that define the k vector when it is not at a point of symmetry in the first Brillouin zone. For example, in the simple cubic lattice, the DT (delta) irreps are associated with k vectors on a line of symmetry connecting the GM (gamma) and X points in the first Brillouin zone. For a point halfway between the GM and X points, the "k params" column would show "1/2". For planes of symmetry in the first Brillouin zone, two parameters are given. For a general point, three parameters are given. Again, we use the convection of Miller and Love. To obtain an unambiguous interpretation of entries in this column, you may need to run ISOTROPY. See the first part of Session 3 in the ISOTROPY Tutorial for assistance in using ISOTROPY for obtain information about k vectors. If all of the irreps listed in the output of COPL are associated with k vectors at points of symmetry, then this column will be missing from the output.

(3) Dir: direction of the order parameter. Arbitrary constants a,b,c,... appear in these vectors. The order parameter is a vector in representation space. It is zero in the high-symmetry phase. At the phase transition, the order parameter becomes nonzero and points in the direction given here.

(4) Subgroup and Size: The symmetry of the subgroup (and the size of its unit cell relative to the unit cell of the parent) determined by the order parameter. Primary order parameters determine the exact symmetry of the lower-symmetry phase. Secondary order parameters determine subgroup symmetries which are supergroups of the symmetry of the lower-symmetry phase. It is easy to determine from this column which order parameters are primary and which are secondary.