research papers
Algorithms for magnetic
search and identification of magnetic from magnetic crystal structure^{a}Department of Materials Science and Engineering, Kyoto University, Sakyo, Kyoto 6068501, Japan, ^{b}Research and Services Division of Materials Data and Integrated System, National Institute for Materials Science, Tsukuba, Ibaraki 3050047, Japan, ^{c}Center for Elements Strategy Initiative for Structural Materials, Kyoto University, Sakyo, Kyoto 6068501, Japan, and ^{d}Nanostructures Research Laboratory, Japan Fine Ceramics Center, Nagoya 4568587, Japan
^{*}Correspondence email: tanaka.isao.5u@kyotou.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 spacegroup 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 spacegroup types and transformations to the BNS setting are identified by a twostep approach combining spacegrouptype 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 spacegroup 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 highthroughput firstprinciples calculations (Horton et al., 2019).
Owing to the development of a computerfriendly description of space groups (Hall, 1981; Shmueli et al., 2010) and algorithms (Opgenorth et al., 1998; GrosseKunstleve, 1999; GrosseKunstleve & Adams, 2002; Eick & Souvignier, 2006), we can automatically perform the crystal symmetry search nowadays. For example, spglib implements the symmetrysearch 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 timereversal operations or magnetic crystal structures. To the best of our knowledge, existing implementations only partly provide MSG functionalities. AFLOWSYM (Hicks et al., 2018) proposed and implemented a robust spacegroup analysis algorithm; however, it does not seem to support MSGs yet. IDENTIFY MAGNETIC GROUP (PerezMato et al., 2015) in the Bilbao Crystallographic Server (Aroyo et al., 2011) and CrysFML2008 (RodríguezCarvajal, 1993; GonzálezPlatas 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 spacegroup 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. ISOMAG (https://iso.byu.edu/iso/magneticspacegroups.php) provides tables of MSGs in both human and computerreadable formats. Magnetic Hall symbols (GonzálezPlatas et al., 2021) and unified (UNI) MSG symbols (Campbell et al., 2022) have been developed to represent MSGs or magnetic spacegroup 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 GonzálezPlatas et al. (2021). The implementation is distributed under the BSD 3clause 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 spacegroup 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 spacegroup 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 timereversal operation and call an indextwo group generated from a timereversal group , where 1 represents an identity operation. Let be a W a matrix part, w a translation part and a timereversal part of the magnetic In particular, is called an antisymmetry operation. A translation of is defined as
of a of threedimensional Euclidean group E(3) and . An element of is called a magnetic where we callwhere 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 timereversal parts in magnetic symmetry operations:A maximal space
(XSG) of 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 indextwo translationengleiche of . [The notation indicates a complement set, = .] Thus, translation subgroups of and are identical.
(Type IV) and is an indextwo klassengleiche of . Thus, point groups of and are identical.
For a typeIII 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 P4_{2}/mnm (No. 136) and Pnnm (No. 58), respectively.
For a typeIV MSG example, Fig. 1(b) shows an AFM bodycentered 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 spacegroup 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 orientationpreserving. We assume that a magnetic
is transformed into by asWe refer to the criteria to choose a representative of each spacegroup type as a setting. The standard ITA setting is one of the conventional descriptions for each spacegroup 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 orientationpreserving transformations is called a magnetic spacegroup type.
2.3. BNS setting
The BNS symbol represents each magnetic spacegroup type (Belov et al., 1957). We refer to a setting of the BNS symbol as a BNS setting: for typesI, II and III MSGs, it uses the same setting as the standard ITA setting of the FSG. For a typeIV 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 timereversal 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 N is the number of sites in the unit cell.
represented by basis vectors, point coordinates, atomic types and magnetic moments within a Formally, our input for the magnetic search is the following four objects: (1) basis vectors of its lattice , (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 , whereWe search for a magnetic 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 ,
that preserves the magnetic . Therefore, thewhere 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 timereversal 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 of the FSG because the latter simply ignores timereversal parts of . 
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 nonzero .
(iii) For each
, we search for the timereversal 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 (GrosseKunstleve 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 spacegroup type and transformation to BNS setting
For the detected MSG in the previous section, we provide an algorithm to identify its magnetic spacegroup type and search for a transformation from to a magnetic spacegroup 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 or provided from outside the software package as predetermined operations.
search in Section 3For all the 1651 magnetic spacegroup types, a magnetic spacegroup representative in the BNS setting has already been tabulated (GonzálezPlatas et al., 2021; Campbell et al., 2022). Thus, we search for with the same magnetic spacegroup type as and an orientationpreserving transformation while satisfying
In Section 4.1, we identify a construct type of to choose a candidate , which is one of the magnetic spacegroup 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 asWhen , is type I or II. Then, when , is type I. When , is type II.
When , is type III or IV. For a typeIII or typeIV MSG, we consider a
decomposition of by :If the klassengleiche of and is type IV. If not, is a translationengleiche of and is type III.
representative can be taken as an antitranslation, is a4.2. Transformation of MSG to BNS setting
For each magnetic spacegroup 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 spacegroup 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 spacegroup type proposed by GrosseKunstleve (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 orientationpreserving 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 orientationpreserving transformation such that = . Then, the FSG of the transformed MSG in equation (19), , is the spacegroup 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 threedimensional affine group as . If a correction transformation satisfies equation (20b), the combined transformation in equation (17) transforms to .
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 et al. (2016), we enumerate matrix parts and origin shifts of orientationpreserving 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 orientationpreserving representatives of up to translations (Koch et al., 2016). Since can be tabulated for each spacegroup representative in the standard ITA setting, we can precompute them before performing the transformation search in practice.
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 Koch4.2.3. When is type IV
When is type IV, with type IV uses the standard ITA setting of . Thus, we need to obtain an orientationpreserving transformation such that = . Then, the XSG of the transformed MSG in equation (19), , is the spacegroup representative in the standard ITA setting.
Similarly to typeIII MSGs, we need to search for an orientationpreserving 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 bruteforce 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 typeIV 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 typeIII MSG example, consider
representatives of for in the BNS setting (BNS No. 17.10) as follows:There is another MSG with the same magnetic spacegroup type as and identical FSG to :
Although and belong to the same spacegroup type (No. 3), these XSGs are different. The following transformation maps to while satisfying equation (20b):
For a typeIV MSG example, consider C_{c}c in the BNS setting (BNS No. 9.40) as follows:
representatives of forThere is another MSG with the same magnetic spacegroup type as and identical XSG to :
Although and belong to the same spacegroup 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 asWe 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 GrosseKunstleve & 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 3clause license. The present algorithms and their implementations are expected to contribute to computational crystallography and materials science, including highthroughput firstprinciples calculations and predictions.
identifying a magnetic spacegroup 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 BNSsetting MSG is obtained by considering affine normalizers: that of the FSG for typeIII MSGs and that of the XSG for typeIV MSGs. In particular, we provide exhaustive tables of conjugated MSGs with triclinic or monoclinic typeIV 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 typeI MSGs, = = = . For typeII MSGs, = = = . For typeIII or typeIV MSGs, = = = . Thus, if FSGs and XSGs are identical, the two MSGs are also identical.
APPENDIX B
Correction transformations for triclinic or monoclinic typeIV MSGs
We give all antitranslations in conjugated MSGs and corresponding transformations for a typeIV MSG in the BNS setting, where is triclinic or monoclinic. Because an antitranslation in a typeIV MSG is indextwo up to translations, it is sufficient to consider the following seven antitranslations:
When is a triclinic or monoclinic Pcentering (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 antitranslations in the transformed MSGs. Because each table contains the seven antitranslations, we confirm that these transformations are sufficient to search for conjugated typeIV MSGs.
For other cases, is a monoclinic Ccentering (Nos. 5, 8, 9, 12 and 15). Tables 3, 5 and 6 show transformations for and antitranslations in the transformed MSGs. Note that the antitranslation should not be contained in the conjugated MSGs because is not type II and is Ccentering. Then, each table contains the six antitranslations other than . Thus, we also confirm that these transformations are sufficient.






Acknowledgements
We thank Juan RodríguezCarvajal for providing magnetic spacegroup data sets tabulated by GonzálezPlatas 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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