

research papers

Algorithms for magnetic
search and identification of magnetic from magnetic crystal structureaDepartment of Materials Science and Engineering, Kyoto University, Sakyo, Kyoto 606-8501, Japan, bResearch and Services Division of Materials Data and Integrated System, National Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan, cCenter for Elements Strategy Initiative for Structural Materials, Kyoto University, Sakyo, Kyoto 606-8501, Japan, and dNanostructures Research Laboratory, Japan Fine Ceramics Center, Nagoya 456-8587, Japan
*Correspondence e-mail: tanaka.isao.5u@kyoto-u.ac.jp
A crystal symmetry search is crucial for computational crystallography and materials science. Although algorithms and implementations for the crystal symmetry search have been developed, their extension to magnetic space groups (MSGs) remains limited. In this paper, algorithms for determining magnetic symmetry operations of magnetic crystal structures, identifying magnetic space-group types of given MSGs, searching for transformations to a Belov–Neronova–Smirnova (BNS) setting, and symmetrizing the magnetic crystal structures using the MSGs are presented. The determination of magnetic symmetry operations is numerically stable and is implemented with minimal modifications from the existing crystal symmetry search. Magnetic space-group types and transformations to the BNS setting are identified by a two-step approach combining space-group-type identification and the use of affine normalizers. Point coordinates and magnetic moments of the magnetic crystal structures are symmetrized by projection operators for the MSGs. An implementation is distributed with a permissive free software license in spglib v2.0.2: https://github.com/spglib/spglib.
Keywords: magnetic space group; magnetic space-group type; magnetic structure; crystal structure analysis; affine normalizer.
1. Introduction
A crystal symmetry search and the standardization of crystal structures play crucial roles in computational materials science. For example, symmetry operations are required in irreducible representations of electronic states (Gao et al., 2021), band paths (Hinuma et al., 2017
), phonon calculations (Togo & Tanaka, 2015
; Togo et al., 2015
), a random structure search (Fredericks et al., 2021
) and description (Ganose & Jain, 2019
). Moreover, the standardization of crystal structures is indispensable for comparing crystal structures in different settings and analyzing magnetic crystal structures in high-throughput first-principles calculations (Horton et al., 2019
).
Owing to the development of a computer-friendly description of space groups (Hall, 1981; Shmueli et al., 2010
) and algorithms (Opgenorth et al., 1998
; Grosse-Kunstleve, 1999
; Grosse-Kunstleve & Adams, 2002
; Eick & Souvignier, 2006
), we can automatically perform the crystal symmetry search nowadays. For example, spglib implements the symmetry-search algorithm and an iterative method to robustly determine crystal symmetries (Togo & Tanaka, 2018
), which one of the authors has developed and maintained.
On the other hand, algorithms and implementations for magnetic space groups (MSGs) (Litvin, 2016) are limited. MSGs are essential when we consider time-reversal operations or magnetic crystal structures. To the best of our knowledge, existing implementations only partly provide MSG functionalities. AFLOW-SYM (Hicks et al., 2018
) proposed and implemented a robust space-group analysis algorithm; however, it does not seem to support MSGs yet. IDENTIFY MAGNETIC GROUP (Perez-Mato et al., 2015
) in the Bilbao Crystallographic Server (Aroyo et al., 2011
) and CrysFML2008 (Rodríguez-Carvajal, 1993
; González-Platas et al., 2021
) can identify MSGs from magnetic symmetry operations; however, the determination of magnetic symmetry operations from magnetic crystal structures is not supported. FINDSYM (Stokes & Hatch, 2005
; Stokes et al., 2022
) supports the determination of magnetic symmetry operations and the identification of MSGs; however, the source code is not freely available.
Here, we present algorithms for determining magnetic symmetry operations of given magnetic crystal structures, identifying magnetic space-group types of given MSGs, searching for transformations to a Belov–Neronova–Smirnova (BNS) setting, and symmetrizing the magnetic crystal structures on the basis of the determined MSGs. Note that the implementation of these algorithms is virtually unattainable without recent developments in crystallography. Litvin (2014) provided extensive tables for the 1651 MSGs. ISO-MAG (https://iso.byu.edu/iso/magneticspacegroups.php) provides tables of MSGs in both human- and computer-readable formats. Magnetic Hall symbols (González-Platas et al., 2021
) and unified (UNI) MSG symbols (Campbell et al., 2022
) have been developed to represent MSGs or magnetic space-group types unambiguously, which are based on BNS symbols (Belov et al., 1957
; Bradley & Cracknell, 2009
). In this paper, we use the magnetic Hall symbols and the MSG data sets tabulated by González-Platas et al. (2021
). The implementation is distributed under the BSD 3-clause license in spglib v2.0.2.
This paper is organized as follows. In Section 2, we recall the mathematical structures of MSGs and present definitions and terminology for describing MSGs. In Section 3
, we provide an algorithm for determining magnetic symmetry operations of a given magnetic on the basis of equivalence relationships between sites in the magnetic In Section 4
, we provide an algorithm to identify a magnetic space-group type of the determined MSG and to search for a transformation from the determined MSG to one in the BNS setting. In Section 5
, we provide an algorithm to symmetrize point coordinates and magnetic moments of the magnetic from the determined MSG.
2. Definitions
Before we discuss algorithms for MSGs and magnetic crystal structures, we describe definitions and terminology for MSGs. In Section 2.1, we define MSGs and derived space groups, which are essential in identifying a magnetic space-group type and searching for a transformation between MSGs. In Section 2.2
, we define equivalence relationships between MSGs. In Section 2.3
, we mention BNS symbols and their settings, which specify representatives of MSGs, and we use them to standardize given MSGs. Finally, in Section 2.4
, we give examples of actions of magnetic symmetry operations for magnetic moments.
2.1. MSG and its construct type
We consider a time-reversal operation and call an index-two group generated from
a time-reversal group
, where 1 represents an identity operation. Let
be a of a of three-dimensional Euclidean group E(3) and
. An element
of
is called a magnetic where we call W a matrix part, w a translation part and
a time-reversal part of the magnetic In particular,
is called an antisymmetry operation. A translation of
is defined as
where E represents the identity matrix. The is called a MSG when
is generated from three independent translations. We write a magnetic of
as
We consider two derived space groups from . A family (FSG) of
is a obtained by ignoring time-reversal parts in magnetic symmetry operations:
A maximal space is a obtained by removing antisymmetry operations:
The MSGs are classified into the following four construct types (Bradley & Cracknell, 2009; Campbell et al., 2022
):
(Type I) : the MSG
does not have antisymmetry operations.
(Type II) : the MSG
has antisymmetry operations and corresponding ordinary symmetry operations.
(Type III) and
is an index-two translationengleiche of
. [The notation
indicates a complement set,
=
.] Thus, translation subgroups of
and
are identical.
(Type IV) and
is an index-two klassengleiche of
. Thus, point groups of
and
are identical.
For a type-III MSG example, Fig. 1(a) shows an antiferromagnetic (AFM) rutile structure whose MSG is
(BNS No. 136.498) in the BNS symbol. The FSG and XSG of
are P42/mnm (No. 136) and Pnnm (No. 58), respectively.
![]() | Figure 1 Examples of antiferromagnetic (AFM) crystal structures with (a) type-III and (b) type-IV MSGs. The red arrows represent collinear spins with the same magnitudes. |
For a type-IV MSG example, Fig. 1(b) shows an AFM body-centered cubic (b.c.c.) structure whose MSG is
(BNS No. 221.97) in the BNS symbol. The FSG and XSG of
are
(No. 229) and
(No. 221), respectively.
2.2. Magnetic space-group type
We consider a transformation between two coordinate systems specified with basis vectors
with origin
and basis vectors
with origin
. A transformation
with
is called orientation-preserving. We assume that a magnetic
is transformed into
by
as
We refer to the criteria to choose a representative of each space-group type as a setting. The standard ITA setting is one of the conventional descriptions for each space-group type used in the International Tables for Crystallography, Vol. A (Aroyo, 2016): unique axis b setting, cell choice 1 for monoclinic groups, hexagonal axes for rhombohedral groups and origin choice 2 for centrosymmetric groups. Similarly to space groups, each equivalent class of MSGs up to orientation-preserving transformations is called a magnetic space-group type.
2.3. BNS setting
The BNS symbol represents each magnetic space-group type (Belov et al., 1957). We refer to a setting of the BNS symbol as a BNS setting: for types-I, -II and -III MSGs, it uses the same setting as the standard ITA setting of the FSG. For a type-IV MSG, it uses that of the XSG. In Section 4
, we consider standardizing a given magnetic by applying a transformation to an MSG in the BNS setting.
2.4. Action of magnetic symmetry operations
In general, we can arbitrarily choose how magnetic symmetry operations act on objects as long as they satisfy the definition of actions. For a m, a acts on m as an axial vector, and the time-reversal operation
reverses the sign of m. When we choose the Cartesian coordinates for m, the matrix part of
is expressed as
in Cartesian coordinates with basis vectors
. Therefore, the magnetic symmetry operations act on m as
3. Magnetic search
We provide a procedure to search for magnetic symmetry operations from a given magnetic , (2) an array of point coordinates of sites in its
, (3) an array of atomic types of sites in its
, and (4) an array of magnetic moments of sites in its
, where N is the number of sites in the unit cell.
We search for a magnetic that preserves the magnetic
. Therefore, the g should map point coordinates
into
up to translations, where
is a permutation of N sites induced by g. Also,
should equate a
with a mapped one
. Such a magnetic forms an MSG of the magnetic as a of
,
where is a symmetric group of degree N. [We recall that the condition of
in equation (8
) can be read as there exists a permutation
such that point coordinates, atomic types and magnetic moments are preserved for every site i.]
Because the domain of the g in equation (8) is not restricted, we cannot search thoroughly for g at this point. To narrow down the candidates for g, we consider a
obtained by ignoring the magnetic moments of
. A of
is written as a of
that preserves
:
As shown in Fig. 2,
may not be a of
in general because the former ignores magnetic moments. Because time-reversal operations do not change point coordinates and atomic types, we can restrict the domain of symmetry operations g to
,
The symmetry operations for can be obtained from existing crystal symmetry search algorithms such as those of Stokes & Hatch (2005
), Togo & Tanaka (2018
) and Hicks et al. (2018
).
![]() | Figure 2 Group–subgroup relationship of MSGs and related space groups. The nodes represent space groups or MSGs. Each edge indicates that a lower group is a subgroup of an upper group in the diagram. Although it is not exploited in this study, the XSG |
Based on the formulation of the MSG in equation (10), we can search for magnetic symmetry operations using the following procedure:
(i) We compute by the existing crystal symmetry search algorithms.
(ii) If all magnetic moments are zero, both g1 and belong to
for all
and we skip the remaining steps (in this case, the MSG is type II). Otherwise, we choose a site
with a non-zero
.
(iii) For each , we search for the time-reversal part as follows:
(a) We compute a permutation and solve
for . We denote the solution of equation (11
) as
if it exists. If the solution does not exist, we skip the g.
(b) We check if the condition holds for other sites. If the condition holds for all sites,
belongs to
.
Note that the comparison of point coordinates and magnetic moments should be performed within tolerances in practice (Grosse-Kunstleve et al., 2004). We use an absolute tolerance parameter ε for point coordinates (Togo & Tanaka, 2018
) and another absolute tolerance
for magnetic moments. Then, the comparisons in this section are replaced with the following inequalities:
Here, takes a remainder with modulo one in the range
.
4. Identification of magnetic space-group type and transformation to BNS setting
For the detected MSG in the previous section, we provide an algorithm to identify its magnetic space-group type and search for a transformation from
to a magnetic space-group representative
in the BNS setting. The algorithms presented in this section are applied to a list of magnetic symmetry operations in the matrix form either obtained through the magnetic search in Section 3
or provided from outside the software package as predetermined operations.
For all the 1651 magnetic space-group types, a magnetic space-group representative in the BNS setting has already been tabulated (González-Platas et al., 2021
; Campbell et al., 2022
). Thus, we search for
with the same magnetic space-group type as
and an orientation-preserving transformation
while satisfying
In Section 4.1, we identify a construct type of
to choose a candidate
, which is one of the magnetic space-group representatives in the BNS setting. In Section 4.2
, we try to obtain
from affine normalizers of
or
.
4.1. Identification of construct type of MSG
The construct type of can be determined from orders of the magnetic and point groups of FSG and XSG. We write a of
as
When ,
is type I or II. Then, when
,
is type I. When
,
is type II.
When ,
is type III or IV. For a type-III or type-IV MSG, we consider a decomposition of
by
:
If the can be taken as an anti-translation,
is a klassengleiche of
and
is type IV. If not,
is a translationengleiche of
and
is type III.
4.2. Transformation of MSG to BNS setting
For each magnetic space-group representative with the same construct type as
, we consider searching for
from two consecutive transformations
and
with
as described below. If such a transformation is found,
belongs to the same magnetic space-group type as
.
We rewrite equation (14) to an equivalent one in terms of derived space groups because we would like to use an existing transformation search algorithm to obtain a transformation between space groups with the same space-group type proposed by Grosse-Kunstleve (1999
). As shown in Appendix A
, the condition of equation (14
) is equivalent to satisfying the following two conditions:
Note that a transformation satisfying equation (18a) does not necessarily satisfy equation (18b
) in general, and vice versa.
The present algorithm, based on the new conditions, is outlined as follows. First, we obtain a temporal transformation to match FSGs or XSGs of
and
by the existing transformation search algorithm. Then, we search for a correction transformation
to match FSGs and XSGs simultaneously.
We divide the transformation search into cases by the construct type of in more detail.
4.2.1. When
is type I or II
When is type I or II,
with the same construct type as
uses the standard ITA setting of
. Thus, we need to obtain an orientation-preserving transformation
such that
. The temporal transformation
can be obtained by the existing transformation search algorithm. We write an MSG transformed by
as
By construction, and
are identical as sets,
.
In this case, the XSGs are also identical to one another, =
=
=
. Thus, we do not need to search for a correction transformation because
also satisfies
=
.
4.2.2. When
is type III
When is type III,
with type III uses the standard ITA setting of
. Thus, we need to obtain an orientation-preserving transformation
such that
=
. Then, the FSG of the transformed MSG in equation (19
),
, is the space-group representative in the standard ITA setting.
A correction transformation should satisfy the following conditions to simultaneously satisfy equations (18a
) and (18b
),
The condition of equation (20a) indicates that
belongs to an affine of
(Koch et al., 2016
). The situation is shown in Fig. 3
(a), where we write the affine of a
as
and the three-dimensional affine group as . If a correction transformation
satisfies equation (20b
), the combined transformation in equation (17
) transforms
to
.
![]() | Figure 3 Group–subgroup relationship of conjugated MSGs and affine normalizers. |
Finally, we describe how to prepare transformations in the affine in practice. Because
is a of
, an operation in
does not give another conjugated of
. Also, although the affine may have continuous translations, the continuous translations do not give another conjugated of
. Thus, it is sufficient to consider representatives of
other than continuous translations. We divide the affine computation into cases according to whether the number of representatives other than continuous translations is finite or not.
When is triclinic or monoclinic, the number of representatives other than continuous translations is infinite and we cannot check transformations thoroughly. However, there are no such conjugate space groups with
because
does not have a pair of proper conjugate subgroups in its affine Therefore, we do not need to compute the affine in this case.
When belongs to other crystal systems, the number of representatives other than continuous translations is finite. To simplify the present algorithm and implementation, instead of using a list of affine normalizers as given by Koch et al. (2016
), we enumerate matrix parts and origin shifts of orientation-preserving transformations in the representatives other than continuous translations as follows. For matrix parts, we enumerate integer matrices
whose elements are −1, 0 or 1, and their determinants are equal to one. For origin shifts, we enumerate vectors
by restricting their vector components to one of
. These will be sufficient because they cover all orientation-preserving representatives of
up to translations (Koch et al., 2016
). Since
can be tabulated for each space-group representative in the standard ITA setting, we can precompute them before performing the transformation search in practice.
4.2.3. When
is type IV
When is type IV,
with type IV uses the standard ITA setting of
. Thus, we need to obtain an orientation-preserving transformation
such that
=
. Then, the XSG of the transformed MSG in equation (19
),
, is the space-group representative in the standard ITA setting.
Similarly to type-III MSGs, we need to search for an orientation-preserving transformation
such that
The situation is shown in Fig. 3(b). [The FSG
is a of
: because
is a of
, every operation in
stabilizes
and belongs to
. Similarly,
is a of
.]
When is neither triclinic nor monoclinic, the brute-force tabulation in Section 4.2.2
also works for
. For triclinic and monoclinic cases, a
is not finite, and we cannot prove the completeness in the same manner. Thus, we show that the enumerated
covers all conjugated type-IV MSGs by explicitly listing
and the conjugated MSGs in Appendix B
.
4.2.4. Examples of conjugated MSGs
We present examples of conjugated MSGs for type III and type IV. For a type-III MSG example, consider for
in the BNS setting (BNS No. 17.10) as follows:
There is another MSG with the same magnetic space-group type as
and identical FSG to
:
Although and
belong to the same space-group type (No. 3), these XSGs are different. The following transformation maps
to
while satisfying equation (20b
):
For a type-IV MSG example, consider for Ccc in the BNS setting (BNS No. 9.40) as follows:
There is another MSG with the same magnetic space-group type as
and identical XSG to
:
Although and
belong to the same space-group type (No. 8), these FSGs are different. The following transformation maps
to
while satisfying equation (22
):
5. Symmetrization of magnetic crystal structure
We symmetrize the magnetic by magnetic symmetry operations of the determined MSG
. For convenience, we consider its decomposition with a finite index as follows. Let
be a translation group formed by basis vectors
, which may not be vectors. We write a decomposition of
by
as
We write the set of
representatives asA centering operation , where
, may belong to
.
A procedure to symmetrize the array of point coordinates by
is essentially the same as those used by Grosse-Kunstleve & Adams (2002
) and Togo & Tanaka (2018
). For the κth magnetic
, we denote that its inverse maps the
th point coordinates to the ith point coordinates. Then,
should be close to
up to lattice translations in
. With this observation, each of the point coordinates
can be symmetrized to
by a projection operator:
The modulo is required because the original and mapped point coordinates in the
may be displaced by lattice translations.A procedure to symmetrize the array of magnetic moments is similar to that to symmetrize the array of point coordinates. Each
can be symmetrized to
by the following projection operator:
6. Conclusion
We have presented the algorithms for determining magnetic symmetry operations for a given magnetic spglib under the BSD 3-clause license. The present algorithms and their implementations are expected to contribute to computational crystallography and materials science, including high-throughput first-principles calculations and predictions.
identifying a magnetic space-group type for a given MSG, searching for a transformation to the BNS setting, and symmetrizing the magnetic on the basis of the determined MSG. Matrix and translation parts of magnetic symmetry operations are determined from the by ignoring magnetic moments. A transformation between the determined MSG and a BNS-setting MSG is obtained by considering affine normalizers: that of the FSG for type-III MSGs and that of the XSG for type-IV MSGs. In particular, we provide exhaustive tables of conjugated MSGs with triclinic or monoclinic type-IV MSGs in the BNS setting and corresponding transformations. Projection operators of the determined MSG symmetrize point coordinates and magnetic moments of the magnetic These algorithms are designed comprehensively and implemented inAPPENDIX A
The condition that two MSGs are identical
For two MSGs and
, we write the FSG and XSG of
as
and
, respectively. When
and
have the same construct types, they are identical if and only if their FSGs and XSGs are also identical, that is,
and
. Although it is trivial, we give proof of this fact for completeness.
When and
are identical, their FSGs and XSGs are also identical by definition. We check the converse for each construct type. For type-I MSGs,
=
=
=
. For type-II MSGs,
=
=
=
. For type-III or type-IV MSGs,
=
=
=
. Thus, if FSGs and XSGs are identical, the two MSGs are also identical.
APPENDIX B
Correction transformations for triclinic or monoclinic type-IV MSGs
We give all anti-translations in conjugated MSGs and corresponding transformations for a type-IV MSG
in the BNS setting, where
is triclinic or monoclinic. Because an anti-translation
in a type-IV MSG is index-two up to translations, it is sufficient to consider the following seven anti-translations:
When is a triclinic or monoclinic P-centering (Nos. 1, 2, 3, 4, 6, 7, 10, 11, 13 and 14), Tables 1, 2 and 4 show transformations for
, which are obtained by the brute force described in Section 4.2.2
, and anti-translations in the transformed MSGs. Because each table contains the seven anti-translations, we confirm that these transformations are sufficient to search for conjugated type-IV MSGs.
For other cases, is a monoclinic C-centering (Nos. 5, 8, 9, 12 and 15). Tables 3, 5 and 6
show transformations for
and anti-translations in the transformed MSGs. Note that the anti-translation
should not be contained in the conjugated MSGs because
is not type II and
is C-centering. Then, each table contains the six anti-translations other than
. Thus, we also confirm that these transformations are sufficient.
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Acknowledgements
We thank Juan Rodríguez-Carvajal for providing magnetic space-group data sets tabulated by González-Platas et al. (2021).
Funding information
The following funding is acknowledged: Japan Society for the Promotion of Science (grant No. 21J10712 to Kohei Shinohara).
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