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

ISODISTORT (isotropy subgroup distortions) is a graphical user interface (GUI) that utilizes the computer algorithms of the Isotropy Software Suite. This program explores a variety of physical distortions (atomic displacements, atomic orderings, magnetic moments and strains) of structural phase transitions induced by irreducible representations (single or superposed, commensurate or incommensurate). The output includes (1) interactive java applets for 3D visualization of structural distortions and the corresponding diffraction patterns, (2) a detailed description of each of the symmetry-modes of the parent that contribute to the distorted structure, (3) both CIF and TOPAS structure files describing the distorted crystal structure in terms of symmetry mode amplitudes (the TOPAS.STR file facilitates direct symmetry-mode refinements), (4) a list of the equivalent domains of the distorted structure, and (5) a list of the order-parameter combinations that are potentially primary.

See the **EXAMPLE** which guides the user through a simple case.

(1) Introduction
to Isotropy Subgroups and Displacive Phase Transitions

(2) B. J. Campbell, H. T. Stokes, D. E. Tanner, and
D. M. Hatch, "ISODISPLACE: a web-based tool for exploring structural
distortions," *J. Appl. Cryst.*
**39**, 607-614 (2006). Download PDF
reprint

(3) H. T. Stokes, B. J. Campbell, and D. M. Hatch, "Order parameters
for phase transitions to structures with one-dimensional incommensurate
modulations," * Acta Cryst. A* **63**, 365-373 (2007).
href="aca63_365reprint.pdf">Download PDF
reprint

(4) B. J. Campbell, J. S. O. Evans, F. Perselli, and H. T. Stokes,
"Rietveld refinement of structural distortion-mode amplitude,"
*IUCr
Computing Commission Newsletter* No. 8, 81-95 (2007).
Download PDF
reprint

**CIF**. Crystallographic Information File containing
information about the structure of a crystal using a standardized
format.

**IR**. An irreducible representation of the space-group symmetry
of the parent structure.

** International Tables**.

**Kovalev**. Tables of irreducible representations found in (1)
O. V. Kovalev, *Irreducible Representations of the Space Groups*
(Gordon and Breach, New York, 1965) or (2) O. V. Kovalev,
*Representations of the Crystallographic Space Groups: Irreducible
Representations, Induced Representations and Corepresentations* (Gordon and
Breach, New York, 1993).

**Miller and Love**. Tables of irreducible representations
found in (1) S. C. Miller and W. F. Love, *Tables of Irreducible
Representations of Space Groups and Co-Representations of Magnetic
Space Groups* (Pruett, Boulder, 1967) or (2) A. P. Cracknell, B. L.
Davies, S. C. Miller and W. F. Love, *Kronecker Product Tables*,
Vol. 1 (Plenum, New York, 1979).

**Stokes and Hatch**. Tables of isotropy subgroups found in
H. T. Stokes and D. M. Hatch, *Isotropy Subgroups of the 230
Crystallographic Space Groups* (World Scientific, Singapore, 1988).
These tables are limited to special *k* points.

When a crystal experiences a symmetry-lowering structural transition,
some of the symmetry elements of the parent space group are lost,
while others persist in the distorted superstructure. Those elements
that persist form the space group symmetry of the superstructure, and
comprise a subgroup of the parent space-group. We refer to such a
subgroup as an *isotropy subgroup* of the parent space group, or as a
*distortion symmetry* of the parent symmetry. The purpose of ISODISTORT
is to make it easy to identify and explore the distortion symmetries
available to a user-provided parent structure. With distortions
comprised of atomic-displacements in mind, this software tool was
originally called ISODISPLACE . Because it now handles several
different types of physical order parameters, it has been renamed as
ISODISTORT.

You don't have to be a group theory expert to use ISODISTORT, though a
working knowledge of crystallographic information (i.e. how to use a
space-group, cell parameters and atomic coordinates to describe a
structure) is needed. The internal design attempts to avoid prompting
the user for information that they are not likely to know, and where
possible, to predetermine the options available to the user at each
step, displaying those options in menus that can be explored
one possibility at a time. After receiving user input, ISODISTORT
performs crystallographic and group-theoretical computations, and
returns the details of the selected distortion in a variety of
user-selected formats. Among other resources, a server-based Java
applet allows you to interactively visualize and manipulate the free
parameters (modes) of your distortion. For a more technical
description of the relevant group-theoretical concepts, see
H. T. Stokes and D.M. Hatch, *Isotropy Subgroups of the 230
Crystallographic Space Groups* (World Scientific, Singapore, 1988).

A symmetry-motivated change of basis: The internal degrees of freedom
available within a distortion-induced superstructure can be viewed as
basis vectors in a generalized distortion space. The traditional
crystallographic structural parameters (cell parameters, the *xyz*
atomic coordinates, site occupancies and magnetic moments) constitute
one oft-used basis of this space. In general, ISODISTORT will generate
new symmetry-motivated basis vectors (i.e. modes) to describe
distortion space which are linear combinations of the familiar
crystallographic basis vectors. In practice, the final product of an
ISODISTORT calculation is a square matrix that transforms the
description of the distorted structure from the traditional basis to
the symmetry-motivated basis; the number of free structural parameters
does not change. The new symmetry-motivated structural parameters are
precisely the natural order parameters of the associated structural
transition, and therefore have special physical and geometric meaning
with respect to the energetics of the transition. For example,
ISODISTORT will often generate modes that include polyhedral
stretches, rotations, shears, buckles, and other familiar displacement
patterns, being closely related to the more complicated frozen phonon
basis which further depends on masses and interatomic forces. In
summary, the principal component of many physical distortions can be
captured by a relatively small number of symmetry-motivated degrees of
freedom for which ISODISTORT is an ideal exploratory tool.

Primary and secondary order parameters: A primary IR is an IR that can single-handedly generate the final distortion symmetry. If two IRs must be superposed in order to achieve a given distortion symmetry, then we refer to them as superposed primary IRs. A primary order parameter is a structural degree of freedom associated with a primary IR. In addition to the primary order parameter(s), ISODISTORT calculates and displays any secondary order parameters that can coexist with the primary order parameter(s). Without going into too much detail, a secondary IR generates secondary order parameters which can coexist within the final superstructure but do not necessarily of themselves generate the final distortion symmetry (i.e. they don't lower the symmetry enough). For example, any internal degrees of freedom within the parent structure will also exist in the superstructure as secondary order parameters.

We have traditionally referred to any distortion symmetry of the
parent space group as an isotropy subgroup of the parent, even if
multiple primary IRs must be superposed to obtain it. Some, however,
reserve the term *isotropy subgroup* for distortion symmetries obtained
with a single primary IR.

There are numerous distinct classes of physical order parameters that can be associated with isotropy subgroups. ISODISTORT specifically employs isotropy subgroups that yield any of four types of order parameters: (e.g. displacements, occupancies, strains, magnetic moments). Any order parameter type will have specific tensor properties. Lattice strains, for example, are described as macroscopic polar rank-2 tensors. Atomic displacement modes are described using microscopic polar vectors. Magnetic moments are described with microscopic axial vectors. And atomic occupancies are described with microscopic scalar parameters. Atomic occupancy modes, which are relevant to compositional order-disorder transitions, cause a single Wyckoff site in the parent cell to split into multiple sites in the supercell.

Magnetic order parameters are a new addition to ISODISTORT. Note that in the current version of the software, the parent structure must be non-magnetic. Internally, the program replaces a nonmagnetic parent space group with the corresponding a magnetic gray group before generating magnetic distortions. Though a gray group does not support magnetic moments itself, many of its distortion symmetries can.

**Structure of parent phase**. You will usually
begin an ISODISTORT session by providing information about the
crystalline structure of the parent (undistorted) phase. This includes
the space-group symmetry, the cell parameters, and the positions of
the atoms. ISODISTORT requires that you upload this information from a
CIF file. If you do not have a CIF file for the parent structure, you
can create one (or modify an existing one) using a related tool called
ISOCIF. If you just want to do some exploring, ISODISTORT also allows
you to start with a predefined cubic perovskite structure. You can
also begin an ISODISTORT session by uploading a previously-saved
ISODISTORT distortion file. Once again, if you just want to do some
exploring, you can start with a predefined distorted cubic perovskite
structure.

**CIF file**. Structural information must
be input in CIF format. There are two options for uploading a CIF
file. (1) You may upload the CIF file directly from a local drive, or
(2) you may copy and paste the contents of a CIF file into a text
field. If you don't have your structure in CIF format, first use
ISOCIF to prepare a CIF file, and then proceed to ISODISTORT. If you
have a CIF file that cannot be interpreted by ISODISTORT (this can
happen if, for example, the space-group setting used in the CIF file
is not one of the settings found in *International Tables*),
ISOCIF can also be used to modify your CIF file to render it suitable
for use in ISODISTORT. ISOCIF and ISODISTORT now read and write CIF
files containing magnetic structures. In the absence of an
international standard, we have employed a handful of magnetic CIF
tags that extend the core CIF dictionary in a logical way. To learn
more about magnetic space groups, and their symbols, settings and
operators, go to (link not yet available).

**Space-group
preferences**. The *International Tables* gives more than one
setting for some space groups. When you upload a parent structure, you
can choose "default" space-group setting preferences that will affect
any distorted structures generated using that parent. The setting of
the space-group symmetry of the parent structure, however, is
determined by the parent CIF file rather than by your "default"
choices. If you want a different setting for your parent structure,
modify the associated CIF file using ISOCIF.

Monoclinic space groups have settings for six different orientations of the
axes. Choose axes *a(b)c*, *c(-b)a*, *ab(c)*, *ba(-c)*,
*(a)bc*, or *(-a)cb*. Unique axes are in parentheses. See
Table 4.3.1 in *International Tables* for more details.

Most monoclinic space groups also have settings for different cell choices. Choose cell choice 1, 2, or 3.

Orthorhombic space groups have 6 different choices for the
orientation of axes. Choose axes *abc*, *ba-c*, *cab*,
*-cba*, *bca*, or *a-cb*. See Table 4.3.1 in
*International Tables* for more details.

Trigonal space groups (for example, #146, R3) have settings using hexagonal axes and rhombohedral axes. Choose one of these.

Some orthorhombic, tetragonal, and cubic space groups (for example, #227 Fd-3m) have two choices for the position of the origin, one of which (origin choice 2) in located at a point of inversion. Choose origin choice 1 or 2.

For (3+1)-dimensional superspace groups, choose either the standard
setting listed in Vol. C of *International Tables* or the setting
of the basic space group as given by the above choices in Vol. A
of *International Tables*.

Magnetic space-groups respect the same space-group settings as non-magnetic space groups.

**IR matrices**. An IR maps each
space-group operator onto a matrix. In 1988, a set of matrices was
published in Stokes and Hatch for IRs at special *k* points.
Subsequently, matrices for IRs at non-special *k* points were
made available through the isotropy software suite. In 2007, a set of
matrices for operators extended to (3+1)-dimensional superspace was
made available at the ISO(3+1)D website. We call all of these choices of
matrices the "1988 version".

In 2011, a new set of IR matrices was introduced.
The matrices were chosen to have a specific "block"
form so that the contributions from different modulation
*k* vectors would appear separated from each other in the order
parameters. For example, consider a *k* point (a,0,0) in a cubic
space group. The space group operators generate three modulation
vectors from this
*k* point: (a,0,0), (0,a,0), (0,0,a). Suppose that one of the
IRs at this *k* point is six-dimensional. Then two of the
dimensions would be associated with each modulation. An order
parameter (a,b,0,0,0,0) would generate a distortion with a modulation
vector (a,0,0), an order parameter (0,0,a,b,0,0) would generate a
distortion with a modulation vector (0,a,0), an order parameter
(a,b,c,d,0,0) would generate a distortion with two superposed
modulation vectors, (a,0,0) and (0,a,0), etc. You would be able to
see by inspection from the order parameter direction which modulation
vectors were involved in the distortion. To accomplish this,
IR matrices must be chosen to have a certain form. We call this
new choice of matrices the "2011 version".

In addition to putting these matrices into "block" form, we also chose the matrices so that those representing pure translations would have a specific form, and we chose the matrices for types 2 and 3 IRs so that a simple transformation would bring them into complex block-diagonal form. As a result, about 70% of the IRs are different from those in the 1988 version, and about 15% of the IRs are different from those in the 2007 version of ISO(3+1)D.

The "Search" page contains (1) Information about the parent structure, including nearest-neighbor distances between different types of atoms, (2) a "View parent" tool, (3) a check box for each type of physical distortion that you may wish to explore, and (4) four different search methods for finding distorted structures.

**View parent**. If you click on "View
parent," you will view a three-dimensional rendition of the unit
cell of the parent structure. If no image appears, you may need to
install a new version of Java on your computer. Each type of atom is
represented by a different color. There are two input
parameters. Reasonable default values for these parameters are already
entered and may be used without any adjustment. After viewing the
graphical rendition, you may return to this page and adjust the values
of these parameters if you wish. (1) Radius of the atoms
(Angstroms). This is the radius used in the graphical rendition and
should, for visual clarity, be somewhat smaller than the actual atomic
radii. (2) Maximum bond length (Angstroms). A line will be drawn
between any two atoms with a center-to-center distance less than this
value. This parameter determines which bonds are displayed.
(3) Applet width (pixels). This
parameter allows you to adjust the size of the applet so that it fits
on your computer screen.

**Types of distortions to be
considered**. By default, the program considers only atomic
displacements and strains, though site-occupancies and magnetic
moments are also available. Use the check boxes available to make your
selection if different from the default. Note that you MUST click on
OK to implement any changes to these check boxes. If you check
displacements, occupancies and strains, all appropriate isotropy
subgroups will be generated that contain any one of these order
parameter types (logical OR rather than AND). If you call for
magnetic distortions, however, no isotropy subgroup that fails to
produce a magnetic distortion will be generated.

Reciprocal-space points that contain no variable
parameters are called *special k points*. The isotropy subgroups
associated with a single IR of any special k point have been
pre-calculated and placed in a readily-accessible database so that
users can simultaneously search over all special *k*-points, filtered by
user-specified constraints such as (1) crystal system, (2) space-group
symmetry, (3) conventional or primitive direct sublattice, and/or (4)
whether the space-group symmetry of the distorted phase is a maximal
subgroup of the parent space group. If your distortion is fairly
simple, there is a good chance that this method will be sufficient to
obtain it. Your search will not be affected by any drop-down list
that is set to "no choice". If you use more than one filter
criterion, all selections will apply simultaneously (logical AND
rather than OR). Note that each lattice selection is specified by
three basis vectors and also includes lattices which are rotated by
any point operators of the parent space group. The search should
produce a drop-down list of the corresponding isotropy subgroups and
their associated *k* points, IRs, and order parameter directions as
described below.

Choose a
*k* point in the first Brillouin zone. This choice affects the
possible superlattices which can result from the phase transition.
Each line in the drop-down menu contains (1) the label of the *k*
point using the notation of Miller and Love, (2) the label of the
*k* point using the notation of Kovalev (only included for
special *k* points), and (3) the coordinates of the point in
terms of the basis vectors of the reciprocal lattice of the
conventional lattice defined in *International Tables*. Some
points contain one or more of the parameters
*a*, *b*, or *g* (for example, *a*,0,0). You must
enter the values of the parameters needed for fully specifying the
position of the point. If no parameters are needed (for example, the
*k* point 0,0,0), you do not need to enter any
values. You *must* enter all parameters as rational numbers (for
example, 1/2 instead of 0.5).

**Incommensurate modulations**. Incommensurate *k*-points
are points with one or more irrational components. If you want to
explore an incommensurate modulation at a given *k*-point, enter
"1" for the number of modulations to include (the current version only
supports one modulation), and enter the irrational parameters as
decimal numbers (for example, 0.5 instead of 1/2). An incommensurate
distortion arising due to *d* incommensurate modulations will
possess the symmetry of a (3+*d*)-dimensional superspace group.
The ISO(3+1)D tables at
stokes.byu.edu/incommensurate.html list all of the isotropy subgroups
arising from a single modulation at an incommensurate
*k*-point.

**Superposed IRs**. If you want to
superimpose distortions from more than one primary order parameter,
you need to couple two or more IRs. Enter the number of superposed IRs
and click on OK. This will take you to a page where you will choose
a *k* point for each of the superposed IRs. After making your
initial selections, you will see two additional pages to select and IR
and an order-parameter direction.

**IR**.
Choose an irreducible representation
(IR). The list in the drop-down menu contains IRs associated with the
*k* point you selected. The choice of IR affects the symmetry of
the atomic displacements you will obtain. The list contains only IRs
which allow atomic displacements or atomic ordering at one or more of
the unique atomic positions you selected. Each line in the drop-down
menu contains the label of the IR using the notation of (1) Miller and
Love and (2) Kovalev (only included for IRs associated with special
*k* points). Type-2 and type-3 IRs are complex. We want real IRs
since atomic displacements induced by the IR must be real. In these
cases, we obtain the *physical IR* from the direct sum of the IR
and its complex conjugate. These are indicated in the notation by a
pair of IR symbols (for example, P1P1, where P1 is a type-2 IR which
is equivalent to its own complex conjugate, and A2A3, where A2 and A3
are type-3 IRs which are complex conjugates of each other). Note that
physical IRs are reducible with respect to complex numbers but
irreducible with respect to real numbers. When dealing with magnetic
distortions, IRs that produce magnetic moments have an "m" prepended
to their labels.

**Order parameter direction (OPD)**.
Choose the direction of the primary order parameter. An order
parameter direction (OPD) is a vector in representation space and has
the same dimension as the IR. The isotropy subgroup is actually
defined as the subgroup of parent space-group operators which leave
the direction of the order parameter invariant. For the IR that you
selected, a radio-button list appears that contains all possible
OPDs. Each entry in the menu contains (1) the OPD symbol (notation of
Stokes and Hatch), (2) the OPD vector components in representation
space, (3) the space group type of the resulting isotropy subgroup,
(4) the basis vectors of the resulting sublattice (i.e. supercell) in
parent-lattice units, (5) the origin of the resulting supercell in
parent lattice units, (6) the size *s* of the primitive unit cell
of the isotropy subgroup relative to the parent space group, and (7)
the index *i* of the isotropy subgroup relative to the parent
space group. For incommensurate structures, the (3+1)-dimensional
superspace group symmetry is given, and the basis vectors of the
lattice as well as the origin of the superspace group is given in
(3+1)-dimensional space with four components.

**Real-time calculations**. ISODISTORT
uses precomputed data tables containing the isotropy subgroups for
Method 1 (single IRs at special *k* points) and for single IRs
associated with incommensurate *k* points. For any other case (IRs at
non-special k points or superposed IRs), the isotropy subgroups must be
generated on demand and saved to a temporary file on the server. If
the isotropy subgroups of the IR(s) that you selected have not been
recently generated, the list of OPDs will contain only the general
direction of the order parameter. You can generate the file containing
the other directions and their isotropy subgroups by clicking on
"Generate isotropy subgroups" at the bottom of the page. The
generation of isotropy subgroups may take anywhere from a few seconds
to many hours. Be prepared to wait while they are being generated.
Factors that increase the time required include a high-symmetry
parent, a low-symmetry distortion, or the coupling of multiple
IRs. Couple more than three IRs of a cubic parent with caution.
Calculations on the server are automatically killed if they have not
run to completion within one hour, and all temporary files on the
server are automatically deleted once a week. Contact us if you need
help with a special case that warrants an exception to these policies.

This algorithm searches for any distortion symmetries consistent with a user-selected point-group or space-group type and supercell. After you click on OK, the next page will contain a list containing all possible distortion symmetries consistent with your selections, each of which will possess a unique combination of space-group type, supercell basis and supercell origin. You must either select a point-group or a space-group type. If you attempt to select both, the space-group selection will supercede the point-group selection. In the case of magnetic distortions, only magnetic space-groups whose BNS symbols reduce to the selected non-magnetic point group or space group will be generated.

To employ this method, you must provide a set of representative basis vectors with which to define a supercell. All crystallographically-appropriate orientations and origin shifts of this supercell relative to the parent cell will be computed and tested automatically. Note that the basis must be specified as a transformation matrix containing only integers and/or rational fractions (e.g. 1/2 instead of 0.5).

The basis vectors that you enter are used to identify a primitive sublattice of the direct parent lattice without regards to its final symmetry. If you enter conventional direct-lattice vectors (which is often more convenient than specifying primitive lattice vectors), you must also indicate what type of centering you have assumed so that a primitive sublattice can be unambiguously identified. In selecting a centering type, switch off your brain's autopilot function and carefully consider that you are NOT choosing the centering of the supercell, but instead deciding which parent lattice points are going to define the sublattice! This is a very convenient, but potentially confusing feature. Because the basis vectors are only used to identify a lattice, and do not reflect the true symmetry of the resulting supercell, any set of basis vectors that corresponds to the desired lattice will be sufficient. You can even use non-standard centering types (e.g. base-centered tetragonal) if you find them convenient. If the program finds any distortion symmetries that match the lattice and space-group type you selected, the results will be automatically transformed into a standard space-group setting. If you don't indicate a centering type, the program assumes a default centering that matches the space-group type that you selected, or simply primitive if you selected a point group instead. Instead of choosing a direct sublattice, note that it is sometimes more conveient to specify a primitive reciprocal superlattice, from which the corresponding primitive direct sublattice can easily be determined.

Basis Example 1: Start with a primitive cubic parent cell and consider the formation of a C-centered supercell with twice the conventional volume, but the same primitive volume. Thus, no new reciprocal-lattice peaks will arise from this transformation. The simplest approach would be to enter (1,0,0),(0,1,0),(0,0,1) as the direct-lattice basis with no (i.e. primitive) centering. You could also specify the primitive reciprocal-lattice basis as (1,0,0),(0,1,0),(0,0,1). If you prefer to specify a centered basis, you could enter (1,1,0),(-1,1,0),(0,0,1) as the direct-lattice basis with C-type centering, or (1,0,0),(0,1,1),(0,-1,1) with A-type centering, or (1,0,1),(0,1,0),(-1,0,1) with B-type centering. Any of these options would be equally effective.

Basis Example 2: Start with a body-centered tetragonal parent cell and consider the formation of a primitive tetragonal supercell with the same cell parameters, so that the primitive volume doubles while the conventional volume stays the same. This will produce intensity at the h+k+l = 2n+1 reciprocal-lattice positions that were systematically absent for the parent structure. Enter (1,0,0),(0,1,0),(0,0,1) as the direct-lattice basis with no centering, or else use it as the reciprocal-lattice basis. No centering option for the direct lattice is appropriate to this situation.

Basis Example 3: Start with a body-centered tetragonal parent cell and consider the formation of a primitive triclinic supercell that coincides with the primitive parent cell, so that the conventional volume gets cut in half while the primitive volume stays the same. The primitive direct sublattice will be described as (-1/2,1/2,1/2),(1/2,-1/2,1/2),(1/2,1/2,-1/2); and the primitive reciprocal superlattice will be described as (0,1,1),(1,0,1),(1,1,0). But the same direct sublattice is more conveniently specified by entering (1,0,0),(0,1,0),(0,0,1) as the basis with conventional I-type centering.

With this method, you will upload a distorted structure from a CIF file and automatically decompose it into symmetry modes of your parent structure. This method is especially useful for analyzing a structure that has already been refined against experimental data. If numerical uncertainties are included with any of the variable parameters in the CIF, they will also be calculated for the corresponding mode amplitudes, which can be useful in determining which mode and strain amplitudes are significant. There are two options: (1) You may upload the CIF, or (2) you may copy and paste the contents of the CIF into a text field. In either case, the CIF is preprocessed and displayed in a text field where you may edit it if desired. The setting of the space-group symmetry of the distorted structure is determined by the CIF, and not by the "default" choices you made earlier. Note that if you selected to include magnetic order-parameters, the distorted structure to be decomposed must be magnetic.

**Basis**. After uploading a distorted
structure, you will be asked to enter the sublattice basis (i.e. the
real-space lattice vectors of the conventional cell of the distorted
structure in terms of the lattice vectors of the conventional cell of
the parent structure). There are two options: (1) Choose the
transformation in the drop-down menu containing all of the unique
possibilities that are not prohibited by symmetry and which generate
cell parameters similar (~10% tolerance) to those in your daughter
CIF. If the strains are large, the correct transformation may not
appear in the list. (2) Enter the transformation matrix in the fields
provided. If non-integer values need to be entered, enter them as
rational numbers (for example, 1/2 instead of 0.5). Be careful: the
basis vectors that you use must reflect the exact (but undistorted) shape and
orientation of your supercell relative to the parent cell. Also, the
types of atoms in the CIF must match the types of atoms in the parent
phase.

**Origin**. The transformation from parent lattice to daughter
sublattice also has an origin shift (possibly zero). The program
automatically attempts to compute all possible origin shifts consistent
with the form of the operators of the isotropy subgroup. However, if
you feel that you can speed things up by specifying the correct origin
shift, and happen to know what it is, you may do so. For isotropy
subgroups whose origins have a sliding degree of freedom, a value is
selected that minimizes the average distance between the atoms in the
distorted and undistorted structures.

**Atom-matching method**. In many cases, the
most time-consuming task is not the decomposition itself, but the
process of matching up the atoms in the distorted and undistorted
supercells. Remember that the parent and distorted structures come
from different files that may have originated in very different ways,
and may not be compatible at all. We can only perform a decomposition
if there is a one-to-one mapping of atoms from the undistorted
supercell into the distorted supercell that preserves element types,
Wyckoff sites, and approximate positions. There are two algorithms
available for this process. (1) For a given origin candidate, the
"nearest-site method" simply attempts to map each atom in the
undistorted structure onto the nearest atom in the distorted
structure. If the nearest match for any atom is of the wrong element
type or Wyckoff site, the method fails, but does so quickly. The
"robust" method, on the other hand, tries to match an atom in the
undistorted structure to every atom in the distorted structure
separated by less than a user-selected threshold distance. Suppose
that there are 4 candidate origins and 20 atoms in the undistorted
structure, each of which has 2 neighbors in the distorted structure
that lie within the threshold distance and have the correct types.
This yields 4*2^{20} = 4194304 possible mappings to be
evaluated one at a time. Only the mapping that minimizes the average
distance between the atoms in the undistorted and distorted structures
will be used for the decomposition. If the distortion involves large
atomic displacements, a large threshold distance is necessary, which
can result in a rather long computation time. Extremely long
atom-matching calculations often indicate that you need to try
different options. Decompositions that take longer than 30 minutes
are killed automatically by the server.

The "distortion" page will appear in a new window. It contains input boxes for the amplitudes of all of the displacive, occupancy, and strain modes. All amplitudes are set to zero by default unless they were predetermined via mode decomposition.

The modes are grouped according to IR on this page, though the order
in which the IRs are presented depends on the "search" method used.
Any IR selected by the user via Methods 1 or 2 will be classified as
"primary," so that their modes are listed first, followed by the modes
of all "secondary" IRs that contribute to the distortion. One line of
descriptive information is provided for each IR and contains (1) the
space-group symmetry (short Hermann-Mauguin symbol) of the parent
phase, (2) the components of the k vector, (3) the IR, (4) the
components of the order parameter direction vector, (5) the
space-group symmetry (space-group number and the short Hermann-Mauguin
symbol) of the distortion that would result if this order parameter
acted alone, together with the (6) the basis vectors and (7) origin of
the resulting supercell, (8) the size s of the primitive unit cell of
the supercell relative to the primitive parent cell, and (9) the index
*i* of the subgroup relative to the parent space group.

Displacive modes. Each displacive mode corresponds to a set of
displacements experienced by some or all the atoms in the
superstructure associated with one of the symmetry-unique atoms of the
parent structure. The total number of displacive modes available to
the distorted structure is equal to the total number of its unique
displacive structural variables (i.e. variable atomic *xyz*
coordinates). Thus, the transformation from the *xyz*-atomic-coordinate
basis to the symmetry-motivated distortion-mode basis conserves the
total number of structural degrees of freedom. Each mode has a label
that contains (1) the identity of the parent atom and its Wyckoff
position, (2) the IR of the point-group symmetry of the local Wyckoff
position, and (3) one of the free variables from the order parameter
direction (*a*,*b*,*c*,...).

Occupancy modes. Compositional ordering (i.e. occupancy) modes cause a parent Wyckoff site to split into two or more daughter Wyckoff sites in the distorted phase and alter their occupancies relative to the parent phase. The mode label of an occupancy mode can be easily distinguished from that of a displacive mode because we replace the Wyckoff-site-symmetry with the word "order." The total number of occupancy modes will equal the total number of unique atoms in the supercell.

Strain modes. These modes are distinguished from the
other modes by the word, "strain," contained in the mode label
next to the input box. Each strain mode is some linear combination
of the six strain components, *e _{xx}*,

Magnetic modes. Each magnetic mode corresponds to a set of magnetic
moments acquired by some or all the atoms in the superstructure
associated with one of the symmetry-unique atoms of the parent
structure. The total number of magnetic modes available to the
distorted structure is equal to its total number of unique magnetic
structural variables (i.e. variable
atomic *m _{x}*,

In ferroelectric phase transitions, at least one of the order parameters will be labeled "ferroelectric." We obtain a proper ferroelectric if one of the ferroelectric order parameters is primary, otherwise we obtain an improper ferroelectric. Ferroelectric modes are also infrared active.

In ferroelastic phase transitions, at least one of the order parameters allows strain (except for the identity IR, GM1 or GM1+). This causes the crystal system of the distorted phase to be different from that of the parent. We obtain a proper ferroelastic if one of the primary order parameters allows strain, otherwise we obtain an improper ferroelastic. Order parameters that allow both atomic displacements and strain are Raman active.

Near the top of the page are a variety of choices: "View distortion", "View diffraction", "CIF file", "Distortion file", "Domains", "Primary order parameters", "Modes details", and "TOPAS.STR". Your selection here will determine the type of output.

For the "View distortion" and "View diffraction" options, there are additional input parameters which appear near the bottom of the page. Reasonable default values for these parameters are already entered and may be used without any adjustment. After viewing the graphical rendition, you may return to this page and adjust the values of these parameters if you wish. Parameters for the "View distortion" option: (1) Atomic radius (Angstroms). This is the radius used in the graphical rendition and should, for visual clarity, be somewhat smaller than the actual atomic radii. (2) Maximum bond length (Angstroms). A line will be drawn between any two atoms with a center-to-center distance less than this value. (3) The length of magnetic moment vectors (Angstroms/magneton). (4) Applet width (pixels). This parameter allows you to adjust the size of the applet so that it fits on your computer screen. (5) Maximum slide-bar amplitudes for each type of physical distortion. A large value allows large amplitudes but also makes the distortions very sensitive to movements of the slide bars. These maximum values also apply to the "View diffraction" option.

**View distortion**. This option opens a
new window that runs a Java applet containing an interactive
three-dimensional rendition of the unit cell of the isotropy subgroup.
Slide bars allow you to vary the amplitude of each symmetry mode
available to your distortion. A
master slide bar at the top allows you to simultaneously multiply
every slide bar by a factor between 0 and 1, which has the effect of
varying the amplitude of the overall distortion. Similarly, at the
bottom of the slide-bar panel, there are single-IR master slide
bars that have a similar effect on the modes belonging to a
specific IR.

Left-click dragging the mouse across the image changes the orientation of the rendered structure, while a right-click or a middle-click drag translates (i.e. pans) the rendered structure within the view window. The x axis points to the right, the y axis points up, and the z axis points out of the screen. When reorienting the structure, and up or down movement of the mouse rotates the rendition about the x axis, while a side-to-side movement rotates the rendition about the y axis.

If you try to view a distorted structure with an exceptionally large number of atoms, you browser may return an error due to a lack of Java memory. If you feel that you have this problem, try increasing the run-time memory allocated to Java. On a Windows system, adding "-Xmx256m" as a Java applet run-time setting in your Java control panel will increase the allocation to 256 megabytes. The same approach works on a Mac, though the Java preferences have moved around a bit in recent releases of OSX.

Each symmetry mode includes a slide bar, an abbreviated mode label, and a mode-amplitude indicator. Each unique atom in the supercell also has an associated checkbox that allows it to be rendered in a different color (to single it out for closer inspection). This checkbox includes an atom label, a total displacement indicator (in Angstroms), a total occupancy indicator, and a total moment (magneton) indicator.

Below the slide bars, there are several options that affect the rendition:

Atoms: If checked (default), atoms are displayed.

Bonds: If checked (default), bonds between atoms are displayed.

Cells: If checked (default), unit cell boundaries are displayed.

Spin: If checked, dragging the mouse across the image causes the rendition to continuously spin with a rate that depends on mouse drag velocity.

Axes: If checked, the arrows appear to indicate the parent cell and supercell lattice directions (black = a, white = b, grey = c).

Animate: If checked, all modes will be animated via the continuous variation of the master slide bar position.

Color: Appears when more than one parent atom has the same element type, and allows you to make atoms of the same element type to have the same color. Otherwise (default) each unique atom of the parent structure is represented by a different color. In each case, atom colors are assigned so as to uniformly spread the color spectrum as widely as possible.

Yrot: If checked, dragging the mouse right and left across the image results in a pure y-axis rotation.

Zrot: If checked, dragging the mouse clockwise or counterclockwise about the center of the image produces a pure z-axis rotation.

Zoom: If checked, dragging the mouse up and down across the image zooms the rendition in or out.

Note that the Normal, Xrot, Yrot, Zrot and Zoom buttons form a set of related options, only one of which can be selected.

If you want to view the rendition from a particular crystallographic direction, check one of the following: SupHKL for a direction perpendicular to the (hkl) plane of the supercell, SupUVW for a direction [uvw] in the supercell, ParHKL for a direction perpendicular to the (hkl) plane of the parent unit cell, ParUVW for a direction [uvw] in the parent unit cell. Then enter the view coordinates and click on "Apply View" to orient the rendition in the direction you specified.

Save Image: allows you to save an image of the current distortion to a file. This feature involves security issues. In general, you don't want a web-based Java applet to have access to your local computer, though this is necessary if you want to save the image in the applet window. By popular request, we provide this feature for those who trust our RSA security certificate.

Press "r" to reset applet to its initial state.

Press "z" to zero all of the mode amplitudes.

Press "i" to reset the mode amplitudes to their initial values without resetting other parameters.

Press "s" to toggle the amplitudes of the single-IR master sliders (all to zero or all to 1).

Press "n" to reverse the panning direction so as to be opposite the mouse-drag motion.

Press "c" to recenter the rendered structure within the view window after panning it to an off-center position.

Some users will discover that they can (1) use their browser to save the applet window to an HTML file on a local computer, (2) use a text editor to eliminate "/iso" from the applet path, (3) download the applet itself ( isodistort.jar) to the same directory, and thereby review or present a distortion offline. This can be useful provided that you understand that we occasionally modify the content and format of the data sent by ISODISTORT to the applet. If you one day find that an older version of the applet does not work with a newer html file or vice versa, you will need to update your offline html and jar files. Note that you cannot save a distortion from within the applet, but must return to the "distortion" page to save it.

**View diffraction**. This option produces
a new window with a rendition of Bragg peak intensities due to x-ray
or neutron diffraction. The righthand side contains the same slide
bars and other information panels that are available in the "View
distortion" window. But here, the slide bars allow you to vary the
amplitudes of the modes while viewing their effect on the parent and
superlattice peaks in the diffraction pattern. Each parent or
superlattice peak is indicated by a colored marker (a small open
circle in the single-crystal pattern, or a vertical stick in the
powder pattern). Parent peaks come in two colors: red (systematically
absent) and green (normal). Superlattice peaks also come in two
colors: orange (systematically absent) and blue (normal). When you move
your mouse over one of these markers, the parent and superlattice
indices of the corresponding peak will appear.
The "Crystal" and "Powder" buttons move the display back and forth
between the single-crystal pattern and the powder-diffraction pattern.
In powder mode, peaks are represented as fixed-width gaussians with
linearly-scaled intensities. In single-crystal mode, each peak
intensity is represented on a log scale by the size of a filled white
circle. The central peak at (000) has maximum intensity
*I*_{max} and is represented by a filled yellow circle of
maximum radius *r*_{max}. Any peak whose intensity
*I* becomes equal (or nearly equal) to that of the central peak
changes from white to yellow. Any peak with intensity *I* less
than 10^{-4}*I*_{max} has zero radius. All other
peaks are represented by filled white circles with radius
*r*=*r*_{max}[1+log_{10}(*I*/*I*_{max})/4].

In single-crystal mode, you can determine the 2D slice of
reciprocal space to be viewed by specifying the point at the center of
the plot, the direction of the horizontal axis, and another direction
that should be contained in the upper half of the plot. All three
vectors should be specified in reciprocal lattice (i.e. *hkl*)
units. Note that the horizontal and upper directions must be defined
relative to the center of the plot. The "Parent" and "Super" buttons
determine whether these vectors should are interpreted relative to the
parent lattice or the superlattice. Finally, the "Q Range" field
allows you to determine how large of a slice to view. The tick marks
appear at integer multiples of the user-provided horizontal and upper
*hkl* direction vectors, and are intended to help you identify
specific peaks within a pattern.
In powder-mode, there are three choices for the horizontal-axis
parameter: 2*theta, d-spacing, and q = 2*pi/d.
Note that the "Wave" field allows you to set the wavelength (Å) that
defines the 2*theta scale. For each scale choice, the "Min"
and "Max" fields allow you to choose the display range, while the
"Res" field determines the peak width (FWHM). The "Zoom" field allows
you to reduce the vertical scale of the powder pattern in order to
zoom in on weak superlattice peaks.

The "Xray" and "Neut" buttons allow you to toggle between x-ray and neutron diffraction patterns. Note that the scattering strength of each element has been set to the atomic number for x-rays and to the complex coherent scattering length of the natural isotopic composition for neutrons. No angle-dependent form factors have been implemented, though a modest isotropic thermal parameter has been applied -- this tool is only intended for detecting qualitative intensity patterns that arise due to specific modes.

Press "r" to reset applet to its initial state.

Press "z" to zero all of the mode amplitudes.

Press "i" to reset the mode amplitudes to their initial values without resetting other parameters.

Press "s" to toggle the amplitudes of the single-IR master sliders (all to zero or all to 1).

**CIF file**. This option creates a
distortion-mode CIF that allows other software to interpret the
distorted structure in terms of symmetry-motivated distortion-mode
amplitudes. One can also upload this CIF as a new parent phase, so
that further distortions can be considered. For distortions at
commensurate k points, the CIF file output now includes the details of
the linear transformation that relates free atomic coordinates to
displacive mode amplitudes, free atomic occupancies to occupancy-mode
amplitudes, free magnetic-moment vector components to magnetic-mode
amplitudes, and parent-cell strains to strain-mode amplitudes.
The CIF tags used to describe the distorted superstructure are
standard, whereas the tags used to describe the symmetry modes have
been created especially for ISODISTORT. At some future time, we
anticipate that a CIF standard will be established for symmetry-mode
analyses. When that happens, we will likely adopt the new standard.

**Distortion file**. This option creates a
file containing all of the information on the distortion page. If you
save this file, you can load it again from the ISODISTORT home page,
allowing you to immediately return to a previous result. Because old
distortion files tend **not** to be compatible with new releases of
ISODISTORT, they are only useful for short-term information storage.
CIF output, on the other hand, is more permanent.

**Domains**. This option creates a
list of all domains of the distorted structure with respect to the
parent structure. Domains are specified by
equivalent directions of the primary order parameter. They are
generated by operators which are contained in the parent space group
but not in the isotropy subgroup. The number of possible domains is
equal to the index of isotropy subgroup relative to the parent space
group. The generating operators may (1) rotate the lattice of the
subgroup (lattice orientation), (2) rotate the contents of the unit
cell relative to the lattice (internal orientation), and/or (3) move
the origin of the subgroup relative to the parent (origin shift).
Each line in the output contains (1) the domain number, (2) the
lattice orientation number, (3) the internal orientation number, (4)
the origin shift number, (5) the components of the order parameter,
(6) the domain generator, (7) the space-group symmetry of the
subgroup, (8) the basis vectors of the lattice of the subgroup, and
(9) the origin of the subgroup relative to the parent. The first
domain contains the original primary order parameter.

**Primary order parameters**.
This option creates a list of all possible sets of primary
order parameters involving any combination of atomic displacements,
strains, or atomic orderings (to save space, only the IR symbols are
shown). While you typically choose your primary order-parameters before
arriving at the distortion page, you can think of the entries in this
list as alternative sets that would have brought you to the same
distortion symmetry. The true physical primary order parameters, of
course, are those that actually drive the energetics of the transition
-- a topic that ISODISTORT does not address. The list of
potentially-primary order parameters also indicates for each entry
whether or not the phase transition is allowed to be continuous
according to rules of Landau theory and the rules of renormalization
group theory. This list is not implemented
for incommensurate distortions.

**Modes Details**. This option allows
you to see more detailed information about each of the modes available
to the distortion. First, the superstructure is tabulated in terms
of the traditional atomic-*xyz*-coordinate basis, including the
displacement of each unique atom from its undistorted position in the
parent structure. The modes are grouped according to order-parameter
type (e.g. displacements or occupancies), with mode definitions and
mode amplitudes appearing in separate tables. For a given order
parameter type, the modes are presented in the same order in which
they appeared on the "distortion" page, where they were grouped by IR.

For microscopic order parameters (i.e. displacements, occupancies, and magnetic moments), tables of mode amplitudes display the standard supercell-normalized amplitude (As) for each mode, which is also used elsewhere in ISODISTORT, as well as a parent-cell-normalized amplitude (Ap), that only appears on this page. The magnitude of As is the square root of the sum of the squares of the mode-induced changes within the primitive supercell (i.e. the root-summed-squared displacement, the root-summed-squared occupancy change or the root-summed-squared magnetic moment). While this definition of As is very simple, it does scale with the square-root of the primitive supercell volume, which makes it inconvenient for comparing the amplitudes of multiple distortions of the same parent that result in different supercell sizes. For this reason, we define Ap = As*sqrt(Vp/Vs) to be normalized to the parent cell, where Vp and Vs are the respective primitive parent and supercell volumes. The sign of As (or Ap) merely indicates the direction of the resulting distortion relative to the corresponding mode vector (described below). In addition to amplitudes for individual modes, we display a total amplitude for each IR (i.e. the root-summed-squared amplitude over all modes of the same IR) and an overall amplitude for the entire distortion (i.e. the root-summed-squared amplitude over all the IRs). This paragraph represents a significant change from earlier versions of this software (i.e. earlier than April 2009), which used dmax as the definition for displacive mode amplitude. We encourage the use of these new definitions (i.e. As and Ap) in the literature for the sake of standardization. Note that the maximum change experienced by any atom affected by a mode is also displayed next to its mode amplitude (dmax in Angstroms for displacements, mmax in magnetons for magnetic modes and omax for occupancy modes).

Strain modes are macroscopic, and are therefore treated differently. Each parent lattice strain is defined as a linear combination of strain-mode amplitudes. In this way, a given set of strain-mode amplitudes are used to determine the parent lattice strains, which are in turn used to calculate the cell parameters of the distorted parent cell. Using the basis transform that defines the daughter sublattice relative to the parent lattice, the distorted supercell parameters are also determined. To see how this is accomplished in detail, export your distortion in TOPAS.STR format and view the resulting strain-mode equations.

In the table of mode definitions, each mode has a label and a normalization factor, followed by a multi-atom mode vector with one line for each unique atom in the supercell affected by the mode. Displacive mode vectors indicate the displacement direction and magnitude of each affected atom in unitless superlattice coordinates. Occupancy-mode vectors indicate the relative occupancy-change direction and magnitude of each affected atom. The mode vectors are not normalized in the form presented, but are instead defined such that the largest non-zero component of a given mode has a value of 1.0. We provide a normalization factor (normfactor) in case you want to manually construct normalized mode vectors. If the entire mode vector is multiplied by its normalization factor (normfactor), then the sum of squares of the resulting changes within the supercell (all the atoms, not just the unique ones) will equal 1.0 (Angstrom units for displacive modes, Bohr magneton units for magnetic modes, unitless for occupancy modes).

**Mode-amplitude calculation example:**

Let our parent cell be a cubic ABO_{3} perovskite with a
3.86 Å cell parameter, and consider a distortion that produces a
tetragonal supercell, i.e. sqrt(2)×sqrt(2)×1. Let B' be the matrix of
column vectors that define the unstrained supercell in cartesian coordinates,
which in this case will be diagonal with 5.46, 5.46 and 3.86 Å along the diagonal.
And let B be the matrix of column vectors that define the strained supercell
in cartesian coordinates, though no lattice strains are present in this example.
In general, the normfactor must be computed using B', whereas the actual
atomic displacements must be computed using B.

Suppose that a displacive mode splits the parent oxygen atom into two descendents
with respective multiplicities of 4 and 2, and relative displacements of
(1,-1,0) and (0,0,1) within the supercell. We define the
mode vector to be {(1,-1,0),(0,0,1)}and compute
the normalization factor to be
normfactor = 1/sqrt[4*||B'.(1,-1,0)||^{2}
+2*||B'.(0,0,1)||^{2}] =
1/sqrt[4*(7.72 Å)^{2}+2*(3.86 Å)^{2}] = 0.06106
Å^{-1}. If the displacive mode amplitude is As = 0.16 Å,
then the two
descendent oxygens will experience displacements of d1 =
As*normfactor*||B.(1,-1,0)|| = (0.16 Å)(0.06106 Å^{-1})(5.46 Å)
= 0.0533 Å and d2
= As*normfactor*||B.(0,0,1)|| = (0.16 Å)(0.06106 Å^{-1})(3.86 Å)
= 0.0377 Å,
respectively. And dmax will be defined as the larger of the two
(i.e. 0.0533 Å) displacements.

Now suppose instead that the same parent oxygen, with an occupancy
of 0.8, is split by an occupancy mode that yields a final occupancy of
1.0 (a change of +0.2) for the first descendent oxygen and 0.7 (a change
of -0.1) for the second one. We would define the occupancy mode
vector as {1.0, -0.5}, which has normfactor =
1/sqrt[4(1.0)^{2} + 2(-0.5)^{2}] = 0.4851, where the
occupancy mode amplitude is As = sqrt(4*(0.2)^{2} +
2*(-0.1)^{2}) = 0.42. The occupancy changes are then
recovered as As*normfactor*modevector = (0.42)(0.4851){1.0, -0.5} =
{0.2, -0.1}.

Lattice coordinates are a convenient
international standard for describing atomic positions as unitless
parameters. For magnetic moments, however, lattice coordinates
are unfamiliar and yield weird units (μ_{B}/Å).
Yet, we choose to employ lattice coordinates when describing
magnetic symmetry modes, so that both displacive and magnetic
mode amplitudes are always defined independent of any lattice strains present.
This also allows the magnetic normalization factor to be defined just
as for atomic displacements.

Continuing the example above, suppose that the two B-site atoms
of the supercell are on the same Wyckoff site and have magnetic
moments along the (0,0,1) direction. We then have normfactor =
1/sqrt[2*||B'.(0,0,1)||^{2}] = 1/sqrt[2*(3.86 Å)^{2}]
= 0.1832 Å^{-1}. Let the magnetic mode amplitude be As =
2.2 μ_{B}.
In lattice coordinates, the magnetic-moment vector will then be μ =
(0,0,μ_{z})
= As*normfactor*(0,0,1) = (2.2 μ_{B})(0.1832
Å^{-1})(0,0,1) = (0, 0, 0.4030)μ_{B}/Å.

The most familiar crystallographic description of magnetic moments is
the crystal-axis coordinate system, which consists of projections of
the total moment vector along each of the possibly non-orthogonal
crystal-lattice basis vectors. In the crystal-axis coordinate
system, the magnetic moment vector is described as m = L.μ where L
= ((a,0,0),(0,b,0),(0,0,c)). In this example, we find that m =
(0,0,(3.86 Å)0.4030 μ_{B}/Å) = (0, 0, 1.56)μ_{B}.
Of course, the cartesian coordinate system is also important, but
somewhat inconvenient in a non-orthogonal lattice. In cartesian
coordinates, the magnetic-moment vector is M = B.μ = B.L^{-1}.m.
In this orthogonal-lattice example,
B.L^{-1}
is just the identity matrix, so that M = m. Regardless of the
coordinate system, the total
magnetic moment will be ||M|| =
||B.L^{-1}.m||
= ||B.μ|| = As*normfactor*||B.(0,0,1)|| = (2.2
&mu);_{B}(0.1832
Å^{-1})(3.86
Å) = 1.56
μ_{B}.

When lattice strains coexist with displacive or magnetic modes, the mode amplitudes, normalization factors, and displacement/moment vectors expressed in lattice coordinates are independent of the strains. But in general, the displacement/moment vectors expressed in crystal-axis coordinates or cartesian coordinates will vary as a function of the lattice strains at fixed mode amplitude.

**TOPAS.STR**. This option generates an .STR
file containing everything TOPAS needs to know for the direct
refinement of distortion-mode amplitudes. John Evans at the
University of Durham has posted a simple TOPAS tutorial for performing
distortion-mode refinements
(http://www.dur.ac.uk/)