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

**INVARIANTS** produces a list of invariant polynomials of the
components of order parameters associated with one or more
irreducible representations of a space group.

In a crystalline phase transition, the space-group symmetry of the
high- and low-symmetry phases often exhibit a group-subgroup
relationship. In the Landau theory of phase transitions, the
distortion which accompanies the transition is described by a primary
order parameter (OP). Such an OP is an *n*-dimensional vector
in the space defined by an irreducible representation (irrep)
of the space group *H* of the high-symmetry phase. In addition
to the primary OP, there are also secondary OPs belonging to other
irreps of *H* and corresponding to addtional distortions that accompany the
transition.

The free energy of the crystal (often called the Landau potential) in
both the high- and low-symmetry phases is written as a linear
combination of sets of invariant polynomials in the components of the
order parameters. Each polynomial is invariant under the action of
any operator in *H*. Some polynomials contain components of
both the primary and secondary OPs and describe the coupling between
those OPs. The purpose of this program is to generate the invariant
polynomials for a given set of OPs.

A more complete discussion of the role of invariants, including example polynomial forms, can be found in:

(1) H. T. Stokes and D. M. Hatch,
*Isotropy Subgroups of the 230 Crystallographic Space Groups,*
(World Scientific, Singapore, 1988).

(2) A. Saxena, G. R. Barsch, and D. M. Hatch, *Phase Transitions*
**46**, 89 (1994).

(3) H. T. Stokes and D. M. Hatch, *Phase Transitions*
**34**, 53 (1991).

**Step 1**

Choose a space group from a list. We use the notation of
*International Tables of Crystallography*.

**Step 2**

Choose a point in the first Brillouin zone from a list.

Points in the first Brillouin zone and irreducible representations are
given using the notation of Miller and Love (S. C. Miller, and
W. F. Love, *Tables of Irreducible Representations of Space Groups
and Co-Representations of Magnetic Space Groups,* Pruett, Boulder,
1967; see also A. P. Cracknell, B. L. Davies, S. C. Miller, and
W. F. Love, *Kronecker Product Tables,*, Vol. 1, Plenum, New
York, 1979) followed by the notation of Kovalev (O. V. Kovalev,
*Representations of the Crystallographic Space Groups: Irreducible
Representations, Induced Representations and Corepresentations,*
Gordon-Breach, Amsterdam, 1993). The Kovalev notation is given only
for points of symmetry (no parameters in the coordinates of the
point). We have not yet worked out the correspondance between the
Miller-Love and Kovalev notations for the other points in the first
Brillouin zone.

The coordinates for the points in the first Brillouin zone are given
in terms of "conventional" vectors of the reciprocal lattice, as
defined by Miller and Love. Such vectors are derived from the
conventional unit cell in real space, as given in the *International
Tables for Crystallography*. For example, the conventional unit
cell for cubic space groups is a cube of side *a*, and therefore
the conventional vectors of the reciprocal lattice are mutually
orthogonal with lengths 2pi/*a*.

Some points require one or more additional parameters to specify it.
For example, the delta point in the simple cubic lattice is (0,*a*,0)
which means anywhere along the line from the gamma point (0,0,0) to
the X point (0,1/2,0). If you want to specify the point half-way between,
then enter 1/4 for the variable *a*. All points must be specified
using rational numbers, i.e., 1/4 instead of 0.25.

**Step 3**

Choose an irreducible representation (irrep) from a list. See Step 2 above for an explanation about the notation used.

**Step 4**

Choose a direction of the order parameter from a list. The variables
*a,b,c,*... have arbitrary values. (These variables should not
be confused with the variables *a,b,c* in Step 2 used for
specifying a point in the first Brillouin zone.)

Each direction of the order parameter listed determines a structure of different symmetry. The space-group symmetry (called the isotropy subgroup) determined by the order parameter is listed following the direction, along with the basis vectors of its lattice (in terms of the basis vectors of the parent space group) and the origin of the standard setting of the subgroup (relative to the origin of the parent). The invariant polynomials will be evaluated for the direction selected. If you want complete invariant polynomials as a function of every component of the order parameter, then choose the general direction of the order parameter (the last item on the list).

The labeling (P1, P2, C1, etc.) for the direction of order parameters
is from Stokes and Hatch (H. T. Stokes and D. M. Hatch, *Isotropy
Subgroups of the 230 Crystallographic Space Groups,* World
Scientific, Singapore, 1988).

The directions of order parameters for the case of points of symmetry in the first Brillouin zone are obtained from a data base. For other points in the first Brillouin zone, these directions are obtained by calculating the isotropy subgroups associated with the irreducible represenation chosen. In some cases this calculation may require a few minutes. You just have to be patient and wait for the results, even if the browser claims to be "stalled". In some rare cases, this calculation could require hours. However, once the calculation has been completed, the results are stored in a file which the program accesses the next time you want results for the same irreducible representation.

**Step 5**

Choose one of the equivalent directions of the order parameter.

Each direction of the order parameter has associated with it a
number of other equivalent directions. Equivalent directions are
obtained by applying space-group operators to an order parameter. (If
the order parameter represents a phase transition, then these
equivalent order parameters belong to different domains that arise
from the same phase transition.) The invariant polynomials for
equivalent order parameters are identical, and it makes no difference
which one you select (except for the case of coupled order parameters,
which are discussed in the next step). We list only equivalent
directions which are distinct, i.e., are not obtained by just
redefining the variables *a,b,c,*... The numbers in parentheses
refer to the numbering of domains by ISOTROPY.

**Step 6**

At this step, you may choose to add an order parameter associated with another irrep. These are called coupled order parameters. It will result in some invariant polynomials which contain components of order parameters from more than one irrep. (In phase transitions, these coupling polynomials are responsible for the activation of secondary order parameters.) You will be returned to Step 2 so that you can choose the irrep and direction of the additional order parameter.

At this step, you may also choose to finish and generate the invariant polynomials for the order parameter(s) chosen thus far. Here, you may also choose the degree(s) of polynomials you would like to generate.

**Invariant Results**

The invariant polynomials are calculated for each request and are not stored in files for future use. The calculation may require a few minutes for some cases with many degrees of freedom. Some cases may even require hours. For this reason, we have restricted the list of order parameter directions to those with no more than eight degrees of freedom. If you really need invariant polynomials with more than eight degrees of freedom, you should use the ISOTROPY program itself.

Space Group: 221 Pm-3m Oh-1

Point in first Brillouin zone: GM, k12 (0,0,0)

Irreducible Representation: GM4-, k12t10

Order Parameter: S1(1) (a,b,c)

Display invariant polynomials with degree 1 through degree 4

**Invariant Polynomials**

**Degree 2**

a^{2} + b^{2} + c^{2}

**Degree 4**

a^{4} + 2 a^{2} b^{2} + 2 a^{2} c^{2} + b^{4} + 2 b^{2} c^{2} + c^{4}

a^{4} + b^{4} + c^{4}

Space Group: 148 R-3 C3i-2

Point in first Brillouin zone: LD (0,0,3a)

Parameters: a = 1/3

Irreducible Representation: LD2LD3

Order Parameter: 4D1(1) (a,b,c,d)

Display invariant polynomials with degree 1 through degree 4

**Invariant Polynomials**

**Degree 2**

a^{2} + b^{2} + c^{2} + d^{2}

**Degree 3**

a^{3} - 3 a c^{2} + b^{3} - 3 b d^{2}

3 a^{2} c + 3 b^{2} d - c^{3} - d^{3}

**Degree 4**

a^{4} + 2 a^{2} b^{2} + 2 a^{2} c^{2} + 2 a^{2} d^{2} + b^{4} + 2 b^{2} c^{2} + 2 b^{2} d^{2} + c^{4} + 2 c^{2} d^{2} + d^{4}

a^{4} + 2 a^{2} c^{2} + b^{4} + 2 b^{2} d^{2} + c^{4} + d^{4}

Space Group: 228 Fd-3c Oh-8

Point in first Brillouin zone: W, k8 (1/2,1,0)

Irreducible Representation: W1W2, k8t1t2

Order Parameter: C2(1) (a,0,a,0,0,0,0,0,0,0,0,0,b,0,b,0,0,0,0,0,0,0,0,0)

Display invariant polynomials with degree 1 through degree 4

**Invariant Polynomials**

**Degree 2**

a^{2} + b^{2}

**Degree 4**

a^{4} + 2 a^{2} b^{2} + b^{4}

a^{4} + b^{4}

a^{3} b - a b^{3}

Space Group: 221 Pm-3m Oh-1

Irrep 1:
Point in first Brillouin zone: M, k11 (1/2,1/2,0)

Irreducible Representation: M3+, k11t3

Order Parameter: S1(1) (a,b,c)

Irrep 2:
Point in first Brillouin zone: R, k13 (1/2,1/2,1/2)

Irreducible Representation: R4+, k13t9

Order Parameter: S1(1) (d,e,f)

Display invariant polynomials with degree 1 through degree 4

**Invariant Polynomials**

**Degree 2**

a^{2} + b^{2} + c^{2}

d^{2} + e^{2} + f^{2}

**Degree 4**

a^{4} + 2 a^{2} b^{2} + 2 a^{2} c^{2} + b^{4} + 2 b^{2} c^{2} + c^{4}

a^{2} d^{2} + a^{2} e^{2} + a^{2} f^{2} + b^{2} d^{2} + b^{2} e^{2} + b^{2} f^{2} + c^{2} d^{2} + c^{2} e^{2} + c^{2} f^{2}

d^{4} + 2 d^{2} e^{2} + 2 d^{2} f^{2} + e^{4} + 2 e^{2} f^{2} + f^{4}

a^{4} + b^{4} + c^{4}

a^{2} d^{2} + b^{2} e^{2} + c^{2} f^{2}

d^{4} + e^{4} + f^{4}