## NOMENCLATURE, SYMBOLS AND CLASSIFICATION OF THE SUBPERIODIC GROUPS

This is a report presented to the Commission on Crystallographic Nomenclature of the International Union of Crystallography. An announcement of this report is publication #62: “Nomenclature, Symbols, and Classification of the Subperiodic Groups,” by V. Kopsky and D.B. Litvin, Acta Crystallographica A49 594 (1993).

This material is based on work supported by the National Science Foundation under grant DMR-9100418.

Download IUCr Report.pdf (293 KB)

## TENSOR DISTINCTION OF NON-FERROELASTIC MAGNETOELECTRIC DOMAIN PAIR

Of the 380 classes of magnetic completely transposable twin laws, 141 classes are non-ferroelastic completely transposable magnetoelectric twin laws. For each of these classes of domain pairs we give the tensor distinction for physical property tensors of the type V, aeV, [V2], aeV2,V[V2], aeV[V2], V3 and ae[V3] , where V denotes a polar vector tensor and “a” and “e” rank zero tensors that change sing, respectively, under time inversion and spatial inversion. The file is in PDF format. See publication #64: “Non-ferroelastic Magnetoelectric Twin Laws” by D.B. Litvin, V. Janovec, and S.Y. Litvin, Ferroelectrics (1994) 162, 275-280.

This material is based on work supported by the National Science Foundation under grants DMR-9100418, DMR-9305825.

Download TDNFMEDP.zip (249 KB)

## MAGNETIC COMPLETELY TRANSPOSABLE TWIN LAWS AND TENSOR DISTINCTION

PDF files of publication #66: “Magnetic Completely Transposable Twin Laws and Tensor Distinction” by D.B. Litvin, S.Y. Litvin, and V. Janovec, Acta Cryst. (1995) A51, 524-529. along with the unpublished tables of the 380 classes of magnetic completely transposable twin laws.

This material is based on work supported by the National Science Foundation under grants DMR-9100418, DMR-9305825, and INT-8922251.

Download mcttws.zip (146 KB)

## NON-MAGNETIC TWIN LAWS

For a phase transition between a high symmetry phase of non-magnetic point group symmetry G and low symmetry phase of non-magnetic point group F, this program calculates:

The coset and double coset decomposition of the point group G with respect to the point group subgroup F.

Indexes the domain states S(i) which arise in such a transition.

Calculates the point group symmetry F(i) of each domain state.

Calculates the permutations of the domain states S(i) under the action of elements g of the point group G.

Calculates all domain pair {S(i),S(j)} in each class of domain pairs.

Calculates a representative domain pair {S(1),S(j)} from each class of domain pairs.

Calculates for each representative domain pair its symmetry group J(1,j) and its twinning group K(1,j).

Calculates the domain pair {S(1), gS(1)} and its symmetry and twinning groups.

Calculates the symmetry and twinning groups of an arbitrary domain pair {S(i),S(j)}.

See publication #68: “Non-Magnetic Twin Laws” by J. Schlessman and D.B. Litvin, Acta Cryst. (1995) A51, 947-949.

This material is based on work supported by the National Science Foundation under grant DMR-9305825.

Download twinlaws.zip (63KB)

## BRIEF PRACTICAL GUIDE TO THE SCANNING TABLES

If a crystal is transected by a plane then the subgroup of all elements of the space group of the crystal which leaves the plane invariant is a layer group. The layer groups corresponding to all planes of a crystallographic orientation for all space groups are tabulated in the scanning tables to be published in the forthcoming volume of the International Tables for Crystallography: Volume E: Subperiodic Groups by V. Kopsky and D.B. Litvin. This is my personal guide to interpreting the information in the scanning tables.

This material is based on work supported by the National Science Foundation under grant DMR-9305825 and DMR-9510335.

Download guide to scanning.pdf(104 KB)

## VRML GENERAL POSITION / SYMMETRY DIAGRAMS OF THE 80 LAYER GROUPS

This program generates VRML General Position/Symmetry Diagrams of the 80 Layer Groups. See publication #74: “VRML General Position / Symmetry Diagrams of the 80 Layer Groups,” by D.K. Tshudy and D.B. Litvin (Journal of Applied Crystallography, 31 973 (1998). Needed to view the diagrams are a web browser with a VRML viewer plug-in (e.g. Netscape with Cosmo). An Adobe Acrobat readable Symbol file is also included.

This material is based on work supported by the National Science Foundation under grants DMR-9510335 and DMR-9722799.

Download LG1-52.zip (1227 KB) Download LG53-80.zip (1392 KB) Download Symbols.pdf (43 KB)

## TENSOR DISTINCTION OF DOMAINS IN FERROIC CRYSTALS

This supplementary material contains (1) a listing of the 247 classes of non-magnetic ferroic phase transitions from G to F. For one transition in each class we give G, F, gij, and the twinning group Kij (2) the distinct double coset classes of non-magnetic domain pairs (3) the 137 tensor classes of non magnetic domain pairs and (4) for 22 tensor types and each of the 139 tensor classes listed is if the tensor type can or can not distinguish between the domains of domain pairs belonging to the tensor class. See publication #77: Tensor Distinction of Domains in Ferroic Crystals,” by D.B. Litvin, Acta Crystallographica A55 884-890 (1999).

This material is based on work supported by the National Science Foundation under grant DMR-9722799.

Download Supplemental Material.pdf (319 KB)

## MAGNETIC SUBPERIODIC GROUPS

This file contains a short introduction to magnetic subperiodic groups. Symbols for the types of these groups are constructed in analogy to the Opechowski-Guccione symbols for magnetic space groups. Tables are given that list one group from each type. Each group is specified not only by its symbol but also by explicitly listing the coset representatives of the coset decomposition of the group with respect to its translational subgroup. See publication #78: “Magnetic Subperiodic Groups,” by D.B. Litvin, Acta Crystallographica (1999) A55 963-964. [Version 2 of the Tables has corrections.]

This material is based on work supported by the National Science Foundation under grant DMR-9722799.

This material has now been revised and corrected and incorporated into MAGNETIC GROUP TABLES below.

## SUBPERIODIC GROUPS ISOMORPHIC TO FACTOR GROUPS OF REDUCIBLE SPACE GROUPS

This pdf file contains the listings of the subperiodic groups isomorphic to factor groups of reducible space groups. See publication #79 “Subperiodic Groups Isomorphic to Factor Groups of Reducible Space Groups,” D.B. Litvin and V. Kopsky, Acta Crystallographica (2000) A56 370-374.

This material is based on work supported by the National Science Foundation under grant DMR-9722799.

Download 79 Supplementary Material.pdf (687 KB)

## PROPERTIES OF THE MAGNETIC POINT GROUPS

This program gives the notation, elements, subgroups, centralizers, normalizers, normal subgroups, and coset and double coset decompositions of the magnetic point groups. See publication #82 “Coset and Double Coset Decomposition of Magnetic Point Groups,” by J. Schlessman and D.B. Litvin, Acta Crystallographica (2001) A57 114-115.

This material is based on work supported by the National Science Foundation under grant DMR-9722799.

Download the program.zip (390 KB)

## MAGNETIC SPACE GROUP TYPES

The interpretation of Opechowski-Guccione symbols for magnetic space group types is based on coordinate triplets given in the now superceded International Tables for X-Ray Crystallography (1952). Changes to coordinate triplets in the International Tables for Crystallography (1983) lead to misinterpretations of these symbols. We provide here a list of Opechowski-Guccione symbols for the 1651 magnetic space group types with their original definitions given independent of the International Tables. See publication #86 “Magnetic Space Group Types,” Acta Crystallographica (2001) A57 729-730.

This material is based on work supported by the National Science Foundation under grant DMR-0074550.

**This material has now been revised and corrected and incorporated into MAGNETIC GROUP TABLES below.**

## SYMMETRY RELATIONS OF MAGNETIC DOMAIN PAIRS ( TWIN LAWS )

Symmetry relationships between two simultaneously observed domain states (domain pair) are used to determine physical properties which can distinguish between the observed domains. A computer program generates these symmetry relationships, in terms of magnetic point groups, i.e. we determine and tabulate all possible magnetic symmetry groups and magnetic twinning groups of domain pairs. See publication #87 ” Symmetry Relations of Magnetic Domain Pairs,” by J. Schlessman and D.B. Litvin, Acta Crystallographica, A57 731 – 732 (2001).

This material is based on work supported by the National Science Foundation under grants DMR-9722799 and No. DMR-0074550.

Download this program.zip (203 KB)

## DOMAIN AVERAGE ENGINEERING

In a domain average engineered sample of a multidomain ferroic, the sample is divided into a very large number of domains, representing m domain states where m is less than the theoretically allowed maximum number n of domain states. The response to external fields is described by tensorial properties averaged over all the involved domain states. We have developed a program to classify subsets of m < n domain states which can arise in a ferroic phase transition. We calculate properties of these subsets of domain states, including the symmetry of the subset, the subset domain polarizations and magnetizations, and, if they exist, the poling directions which gives rise to the subsets of domain states. See Publication #90 ” Domain Average Engineering in Ferroics,” by B. Shaparenko, J. Schlessman, and D.B. Litvin, Ferroelectrics, 269 9 – 14 (2002).

This material is based on work supported by the National Science Foundation under grant No. DMR-0074550.

Download this program.zip (390 KB)

## MAGNETIC SUBPERIODIC GROUPS: VRML GENERAL POSITION DIAGRAMS

VRML general position diagrams for the 31 magnetic frieze groups, 394 magnetic rod groups, and 528 magnetic layer groups are given. These diagrams are viewable in a web brouser with a VRML plug-in, e.g. Cortona VRML Client or a VRML viewer, e.g. VRMLview . (The NIST website lists many plug-ins, applets, and programs to view VRML files on Windows, Linux, and Mac operating systems.)

This material is based on work supported by the National Science Foundation under grant No. DMR-0074550.

Download Frieze and Rod.zip (10.4 MB)

Download Frieze and Rod.zip (11.2 MB)

Download Documentation (260 KB)

## SPACE-GROUP SCANNING TABLES

Due to page limitations, in Volume E: “Subperiodic Groups” of the International Tables for Crystallography not all scanning tables were explicitly given. Instead, auxiliary tables were given providing information from which to construct the additional tables. We have constructed these additional tables and present them here (See publication #95.)

This material is based on work supported by the National Science Foundation under grant No. DMR-0074550.

Download the additional scanning tables (11.1MB)

## MAGNETIC SUBPERIODIC GROUPS (Electronic Book)

Tables of properties of the magnetic subperiodic groups: The content and format is similar to that on non-magnetic subperiodic groups and space groups given in the International Tables for Crystallography. Additional content for each group type includes a diagram of general positions with corresponding general magnetic moments, Seitz notation used as a second notation for symmetry operations, and General and Special Positions listed with the corresponding magnetic moments allowed by symmetry.Also available at: The International Union of Crystallography electronic archives as Supplementary Material to the paper “Tables of Properties of Magnetic Subperiodic Groups” Acta Crystallographica A61 382-385 (2005) at http://journals.iucr.org/ ,

This material is based on work supported by the National Science Foundation under grants No. DMR-9722799 and No. DMR-0074550.

**This material has now been revised and corrected and incorporated into MAGNETIC GROUP TABLES below.**

## MAGNETIC SPACE GROUPS: VRML GENERAL POSITION DIAGRAMS

VRML general position diagrams for the 1502 non-cubic magnetic space groups are given. These diagrams are viewable in a web brouser with a VRML plug-in, e.g. Cortona VRML Client or a VRML viewer, e.g. VRMLview . (The NIST website lists many plug-ins, applets, and programs to view VRML files on Windows, Linux, and Mac operating systems.) Examples of individual magnetic space group VRML General Position Diagrams are given below. The complete set of diagrams, on CD, is available on request, or can be downloaded from the Non-Cubic Magnetic Space Group General Position Diagram download page.

This material is based on work supported by the National Science Foundation under grant No. DMR-0074550.

All Examples can be downloaded from Magnetic_Space_Files.zip:

## MAXIMAL SUBGROUPS OF INDEX < 4 OF 1,2, AND 3 DIMENSIONAL MAGNETIC SPACE GROUPS AND 2 AND 3 DIMENSIONAL MAGNETIC SUBPERIODIC GROUPS

Supplementary material to publication #100 “Magnetic group maximal subgroups of index < 4.”

**This material has now been incorporated into MAGNETIC GROUP TABLES below.**

## SUPPLEMENTARY MATERIAL TO : FERROIC CLASSIFICATIONS EXTENDED TO FERROTOROIDIC CRYSTALS

Supplementary material to publication #101 ” Ferroic Classifications extended to ferrotoroidic crystals.”

Download Supplementary Material (0.8 MB)

## MAGNETIC SPACE GROUPS (Electronic Book)

Tables of properties of the one-, two-, and three-dimensional magnetic space groups: The content and format is similar to that on non-magnetic space groups given in the International Tables for Crystallography. Additional content for each group type includes a diagram of general positions with corresponding general magnetic moments, Seitz notation used as a second notation for symmetry operations, and General and Special Positions listed with the corresponding magnetic moments allowed by symmetry.Also available at: The International Union of Crystallography electronic archives as Supplementary Material to the paper #103 “Tables of Properties of Magnetic Space Groups” Acta Crystallographica A64 419-424 (2008) at http://journals.iucr.org/ ,

This material is based in part on work supported by the National Science Foundation under grant No. DMR-0074550.

**This material has now been revised and corrected and incorporated into MAGNETIC GROUP TABLES below.**

## SUPPLEMENTARY TABLES TO : TENSOR DISTINCTION OF DOMAINS IN FERROIC CRYSTALS

Supplementary tables to publication #106 ” Tensor Distinction of Domains in Ferroic Crystals.”

Download Supplementary Tables (1.2 MB)

## MAGNETIC GROUP TABLES 1-, 2- AND 3-DIMENSIONAL MAGNETIC SUBPERIODIC GROUPS AND MAGNETIC SPACE GROUPS

Survey of magnetic groups.

Properties of magnetic groups

Comparison of OG and BNS magnetic group type symbols

Maximal subgroups of index < 4

Download The Introduction (7.2 MB)

Download The Magnetic Group Tables [**file too large**] (93.2 MB)

## SUPPLEMENTARY TABLES TO : TENSOR COMPONENT DISTINCTION OF MAGNETIC NON-FERROELASTIC DOMAIN PAIRS

Supplementary tables to publication #108 “Tensor component distinction of magnetic non-ferroelastic domain pairs.”

Download Supplementary Tables (2.4 MB)

## SUPPLEMENTARY TABLES TO : SEITZ NOTATION FOR SYMMETRY OPERATIONS OF SPACE GROUPS

Supplementary tables to publication #111 “Seitz notation for symmetry operations of space groups.”

Download Supplementary Tables (0.4 MB)

## SUPPLEMENTARY TABLES TO : SPONTANEOUS TENSOR PROPERTIES FOR MULTIFERROIC PHASES

Supplementary tables to publication #113: Spontaneous tensor property tables for the 1601 Aizu species.

Download Supplementary Tables (3.3 MB)

## SUPPLEMENTARY TABLES TO : ON THE SYMMETRY ANALYSIS OF SURFACE INDUCED EFFECTS

Supplementary tables to publication #114: On the symmetry analysis of surface induced effects

Download Supplementary Tables (1.0 MB)