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The magnetic section of the Bilbao Crystallographic Server –

1 st -Tutorial

J. Manuel Perez-Mato and Luis Elcoro

Dept. Física de la Materia Condensada, Facultad de Ciencia y Tecnología, Universidad del Pais Vasco (UPV/EHU), Apdo. 644, 48080 Bilbao This first tutorial includes practical introductory examples for the following programs:

MGENPOS : Section2 IDENTIFY MAGNETIC GROUP : Section 3 MWYCKPOS: Section 4 MPOINT: Section 5 MTENSOR: Section 6

For other programs of the magnetic section see the 2 nd rd and 3 tutorials

1. Introduction

Below you can see a screenshot showing the databases, tools and programs that are available online at the Bilbao Crystallographic Server

that are available online at the Bilbao Crystallographic Server (www.cryst.ehu.es) in its second section entitled: “Magnetic Symmetry and Applications”. In this initial tutorial, we shall first become familiar with the databases and tools in this section (MGENPOS, MWYCKPOS, IDENTIFY MAGNETIC GROUP, MPOINT) which are purely informative about magnetic space groups (Shubnikov groups) and magnetic point groups. This will help us to get deeper into the mathematical structure of magnetic symmetry groups, and their consequences on the physical properties. We shall explore then the features of the program MTENSOR, which allows to know the symmetry adapted form of any kind of crystal tensor property for any magnetic point-group symmetry. Further information on some of these programs and the theory behind can be found in [1] and [2].

2. MGENPOS: General Positions of Magnetic Space Groups

This tool is a database of the 1651 different possible magnetic space groups (MSGs), also called Shubnikov groups. It lists for any chosen MSG the set of symmetry operations defining the group. These symmetry operations are also called the “general positions of the group”, as they represent for a general position with arbitrary magnetic moment: (x,y,z; mx,my,mz), all the positions (and magnetic moments) that are symmetry related (apart from those related by the lattice translations). The list of general positions or symmetry operations of the MSGs are given in the setting, which was first provided in computer readable form by the ISOTROPY webpage [3], and we take as the “standard setting” of the MSGs.

The symmetry operations describing the symmetry group of any commensurate magnetic structure can always be put, by choosing an appropriate unit cell and origin, in the standard form of one MSG (and only one!) of the 1651 MSGs, as listed here (see below the tool IDENTIFY MAGNETIC GROUP).

MGENPOS can list the operations of any MSG in either the BNS or the OG setting (this latter coinciding with the one used in Litvin’s ebook [4]) , but for the moment in this tutorial we will restrict ourselves to the BNS setting. We leave some additional considerations about the OG setting for a more advanced tutorial.

Any desired MSG can be introduced in MGENPOS by means of its BNS numerical label, but the desired MSG can also be decided by first choosing its mathematically underlying space group:

2a) Click on the button “choose it”. We then get a list of the 230 ordinary space groups. Click on one of them, say Pcca (N. 54). You will then obtain a listing of all possible MSGs derived in the BNS approach from the space group Pcca (N. 54), in the following form (disregard the second listing corresponding

Pcca (N. 54), in the following form (disregard the second listing corresponding to the OG description): The BNS numerical labels of all the listed MSGs include the number 54 showing their relation with the ordinary space group Pcca (N. 54). For each group the type of MSG is indicated at the end of the line. In between the corresponding labelling used for the same group in the OG description is also shown.

2b) Click on one MSG of type IV, say 54.352 P I cca, and click next on “standard/default setting”. The list of the symmetry operations of the MSG

“standard/default setting”. The list of the symmetry operations of the MSG appears. It has 16 operations and we show below a partial view: The heading lists a set of lattice generators, showing in this case that the lattice is primitive (The P from the label). A set of generators of the black & white lattice, i.e. the lattice formed by both translations and “antitranlations” (translations plus time reversal) is also listed. Being a MSG of type IV, this list includes one antitranslation, which in this case is a body centered antitranslation (this corresponds to the “I” subindex in the MSG label). The symmetry operations are listed in four different formats in four consecutive columns: i) as a transformed (x, y, z) position in relative coordinates with respect to the lattice unit cell, and the corresponding transformation of the associated magnetic moment (mx , my , mz ) ; ii) as a 3x3 transformation matrix and a column vector indicating the rotation or roto-inversion plus translation of a general point (x, y, z) expressed in unit cell relative coordinates; iii) a geometrical interpretation of the operation using the symbols proposed in the International Tables for Crystallography; and iv) in a Seitz notation extended to magnetic group operations. The operations that include time reversal are distinguished by either a “-1” index, instead of “+1”, or by the addition of a “prime” on the symbol of the operation. They are also listed in red, instead of black. For our purposes it is sufficient to understand and consider the last column where the operations are given Seitz notation.

The list includes 16 symmetry operations divided into 8 which do not include time reversal and 8 that are generated by multiplying these initial 8 operations

time reversal and 8 that are generated by multiplying these initial 8 operations by the body centered antitranslation {1’| ½ ½ ½ }: This is the typical layout of the symmetry operations of a type IV MSG: a set of operations that define an ordinary space group F without time reversal, plus the same set of operations, but multiplied by an antitranslation, such that all operations in this second set include time reversal. The total set G then can be expressed as:

G = F + {1’| t} F

In our case the ordinary space group F is the number 54, Pcca. All type IV MSGs in the initial list have therefore the same set of “black” symmetry operations in the list given by MGENPOS, as all of them have Pcca as the group F in the above decomposition. You can check this fact by looking at the list given by the program for any other MSG of type IV, and see that the set of black operations, i.e. operations without time reversal coincide with those obtained previously for 54.352.

2c) Come back to the list of possible MSGs derived from the space group 54, and click on a group of type III, say 54.343 Pcc’a’ :

The lattice of type III MSGs trivially coincides with the one of the ordinary space group used in their labelling, and there are no antitranslations. Therefore the list heading does not include any information on the lattice generators, etc, as it happens with type IV MSGs. The list of operations does not include any antitranslation and the total set of operations is now reduced to 8. They correspond to the ordinary space group Pcca, but half of them include time reversal. The set of 4 “black” operations without time reversal generate a subgroup of Pcca. It is in fact a subgroup of type P2/c, with the monoclinic axis along a. Hence, being a type III MSG, this group can be described as:

G = F + {R’|t} F

where F is a subgroup of Pcca, and {R|t} (without time reversal) is an operation of the ordinary space group Pcca, not belonging to F.

We can check that F in this example is a group of type P2 1 /c using a tool in the Bilbao Crystallographic Server for ordinary space groups, which is outside its magnetic section, namely the program IDENTIFY GROUP in the first section: “Space Group Symmetry”.

2d) Doing copy/paste, just take the “black” operations in the list deleting the “+1”, to convert them in operations of an ordinary space group:

x, y, z x+1/2, -y, -z+1/2 -x, -y, -z -x+1/2, y, z+1/2

Paste this set of four operations in the input window of “IDENTIFY GROUP”, and submit. The program will then identify the group as P2 1 /c, but with the monoclinic axis along a, as it is shown in the transformation it proposes to pass to the standard setting of this space group. If we follow the same procedure, but including the 8 operations, disregarding the “-1” of the other 4 operations, what space group will be identified by the program, and which transformation to standard?

The MSG of a magnetic phase defines the symmetry constraints of all degrees of freedom of the structure, including the non-magnetic ones, as the atomic positions. But for non magnetic degrees of freedom the presence of time reversal or not in a symmetry operation is irrelevant, and the symmetry constraints can therefore be described by an ordinary space group, formed by all operations of the MSG disregarding the presence or not in them of time reversal. We shall call this ordinary space group, the effective space group of the MSG. In the case of the MSG Pcc’a’, it is clear that this space group is Pcca, both groups having the same lattice. But what is the effective space group for the type IV MSG 54.352 P I cca, which we have considered at the beginning? We shall obtain this effective space group using again “IDENTIFY GROUP”

2e) Come back to the listing of the MSG 54.352 P I cca and copy/paste in the input window of “IDENTIFY GROUP” all the listed symmetry operations, disregarding the “-1” or “+1”: x, y, z x+1/2, -y, -z+1/2 -x, y, -z+1/2 -x+1/2, -y, z -x, -y, -z -x+1/2, y, z+1/2 x, -y, z+1/2 x+1/2, y, -z x+1/2, y+1/2, z+1/2 x, -y+1/2, -z -x+1/2, y+1/2, -z -x, -y+1/2, z+1/2 -x+1/2, -y+1/2, -z+1/2 -x, y+1/2, z x+1/2, -y+1/2, z x, y+1/2, -z+1/2

and submit. In fact, the program only needs as input a minimal set of generators, and therefore this list can be strongly shortened, if you make a appropriate choice of generators (only 4 generators are really needed). The program then identifies the effective space group:

The effective space group of PI c ca is therefore identified as Ibca (N. 73), with its standard origin shifted by ( ¼ ¼ ¼ ) with respect to the one being used in the standard setting of PI c ca. The effective space group is therefore not Pcca as in type III MSGs, but a different space group with a different lattice. If we come back to the listing of operations of PI c ca, we can see on its heading that the alternative OG label for this MSG is Ip b ’ca (73.7.649). This shows a nice general property of the OG labelling and description: the reference space group for the OG description and labelling of a MSG is in all cases the effective space group for the non-magnetic degrees of freedom. For type IV MSGs, this space group is in many cases different from the one considered by the BNS labelling, and also the reference lattice is different.

3. IDENTIFY MAGNETIC GROUP: Identification of a MSG from a set of

generators in an arbitrary setting

Magnetic structures are usually described with the same origin and axes orientation as those used for its parent paramagnetic phase, which is taken as reference. In many cases, this means that the symmetry of the magnetic structure must be described by a MSG in a non-standard setting. In general, the transformation to a description with the MSG in its standard form requires a change of origin and a different choice of unit cell.

We shall see here a simple example of this quite general situation, using IDENTIFY MAGNETIC GROUP to identify the actual MSG describing the symmetry of a specific magnetic structure.

3a) Open MAGNDATA and search structures containing Sr and Ir using the corresponding element search option in the menu. From the list of structures appearing choose the entry #1.77 corresponding to Sr 2 IrO 4 .

This structure has a paramagnetic phase with space group I4 1 /acd (N. 142), and its magnetic phase has a propagation vector k = (1,1,1). This means that the magnetic ordering breaks the bcc lattice centering of the paramagnetic structure but keeps the periodicity given by the tetragonal unit cell. The

structure but keeps the periodicity given by the tetragonal unit cell. The description of the magnetic structure is then done using the same unit cell. 3b) Among the options appearing at the head of the output for the entry #1.77, click on “download mcif file”, and download or just open in your computer the file in magCIF format. 3c) Localize within the .mcif file the text, which defines the magnetic space group of the structure in the following form: loop_ _space_group_symop_magn_operation.id _space_group_symop_magn_operation.xyz 1 x,y,z,+1 2 -x+1/2,-y,z+1/2,+1 3 -x+1/2,y,-z,+1 4 x,-y,-z+1/2,+1 5 -x,-y,-z,+1 6 x+1/2,y,-z+1/2,+1 7 x+1/2,-y,z,+1 8 -x,y,z+1/2,+1 loop_ _space_group_symop_magn_centering.id _space_group_symop_magn_centering.xyz 1 x,y,z,+1 2 x+1/2,y+1/2,z+1/2,-1 3d) Copy/paste these operations (or a minimal set that generates all of them) in the input window of IDENTIFY MAGNETIC GROUP and submit (in the default option: BNS setting). (the numerical labels of the operations have to be deleted to be an acceptable input format) Check that the program then identifies the MSG as PI c ca (54.352), i.e. the one we have discussed in the previous section. The output also shows a change of unit cell and origin to transform the symmetry operations to its standard setting. As this transformation is not the identity, it means that the MSG with the basis used is

in a non-standard form, i.e. the operations do not coincide with those listed for this MSG in MGENPOS.

Note that the basis transformation to standard in the ouput of the program is given as a matrix with the transformed cell vectors as columns, such that the

given as a matrix with the transformed cell vectors as columns, such that the basis transformation to standard proposed by the program is (-c,b,a; 0,0,0): Notice that identification of the label of the MSG and its transformation to standard, unambiguously defines the symmetry of the phase, and the list of symmetry operations becomes redundant as these latter could be reconstructed from the standard listing of the MSG and the application of the inverse of the identified transformation. The transformation to the standard setting of a MSG is in general not unique. 3e) Check for instance, using the corresponding option of the program, that a change of basis with an origin shift, such as (c,b,-a; 0 ,0,0 ) or (c,b,- a; ½ ,½,½ ) would also be appropriate to transform the MSG to its standard setting.

4. MWYCKPOS: Wyckoff positions of magnetic space groups

4a) Introduce the numerical label 54.352 correponding to the space group

4a) Introduce the numerical label 54.352 correponding to the space group PI c ca, and submit for the standard setting. This program then provides a listing of all the possible Wyckoff positions for this MSG: Similarly as in the International Tables for Crystallography, the list includes por each possible special position the set of symmetry related ones, forming the so- called orbit of the Wyckoff position. Apart from the relative coordinates of the position, the information includes the components of a magnetic moment at this position, showing their symmetry constraints and relations within the orbit. Each Wyckoff orbit, defined in this form, can be interpreted as a set of spin basis modes compatible with the MSG. The Wyckoff orbit corresponding to a general arbitrary position coincides with the list of symmetry operations or general positions, listed in MGENPOS. The output also indicates the point group symmetry corresponding to each special position. Note, for instance the Wyckoff position 8b ( ¼, ¾, ¼ ) lies on a

point with point group symmetry -1’, and this forbids the presence of a non-zero magnetic moment.

5. MPOINT: Magnetic Point Group Tables

Macroscopic crystal tensor properties of magnetic structures only depend on the magnetic point group of the system. This is just formed by the set of rotational and roto-inversion operations, with time reversal added or not, which are present in its MSG.

5a) Click on MPOINT to see all the possible magnetic point groups: There are 122 possible magnetic point groups. 5b) Choose the group mmm1’ (8.2.25). You can see then that the group has 16 operations, and they are the 8 operations of the point group mmm plus another 8 operations obtained by “multiplying” this operation by time reversal. This can be expressed as: mmm1’ = mmm + 1’ mmm Check that these point group operations are indeed those that are present in the listing of MGENPOS for the MSG PI c ca (considered in section 2), if the translations of the operations are disregarded. Thus, the magnetic point group associated with PI c ca is the grey point group mmm1’, a group that includes time reversal, i.e. the same type of group as those of non-magnetically ordered systems: All type IV MSGs have grey point-group symmetry and correspondingly the symmetry restrictions on their tensor properties do not differ from those of a paramagnetic structure with the same magnetic point group. 5c) Go back and choose the group m’m’m (8.4.27). Check that this is the point group corresponding the MSG Pcc’a’, that we considered in section 2. Determine the transformation of basis required to make the point-group operations of the MSG fully equivalent to those of the point group m’m’m (it can be done by hand or you can use IDENTIFY GROUP by introducing the operations of Pcc’a’ with the translations deleted. In this example, the point group does not include the time reversal operation, and therefore, crystal tensor properties forbidden in the paramagnetic phase will be present.

6. MTENSOR: Symmetry adapted form of crystal tensors in magnetic

phases

MTENSOR is a program that provides the symmetry adapted form of any type of crystal tensor. The desired property can be chosen among a list which specifies more than 100 distinct crystal tensors divided in four classes: equilibrium tensors, optical tensors, non-linear optical susceptibility tensors and transport tensors. In addition, using abstract generalized (so-called Jahn) symbols, which define the transformation properties of the tensor one can also get the symmetry-adapted form of any kind of tensor property for any magnetic point group, as long as its rank is not greater than 8.

6a) Open MTENSOR and choose the space group Pcc’a’. Although the symmetry constraints on a tensor only depend of the system magnetic point group, the program allows to introduce its full MSG, because then it considers the point group with the orientation associated with the standard setting of the space group, which in general may not be the standard setting for the point group, as we have seen in the previous version.

6b) Choose among the equilibrium tensors: “Magnetization vector Mi”, and

6b) Choose among the equilibrium tensors: “Magnetization vector Mi”, and check that this space group allows ferromagnetism along a. 6c) Come back and choose now the piezomagnetic tensor (direct effect): Check that the ouput indicates that a shear xy can induce a magnetization along b, and a shear xz can induce a magnetization along c. Piezomagnetic effects are certainly forbidden in the paramagnetic phase. (The “a” in its Jahn symbol means that the tensor is odd for time reversal, and therefore it must be zero for any group that includes time reversal as symmetry operation, as it happens in all non-magnetically ordered structures) 6d) Verify with some trials that the symmetry-adapted form of any tensor is the same for the MSGs Pmmm1’ and for PI c ca. Why?

REFERENCES [1] J.M. Perez-Mato, S.V. Gallego, E.S. Tasci, L. Elcoro, G. de la Flor, and M.I. Aroyo, “Symmetry-based computational tools for Magnetic Crystallography” Annu. Rev. Mater. Res. (2015), 45:13.1-13.32 [2] S. V. Gallego, J. Etxebarria, L. Elcoro, E.S. Tasci and J.M. Perez-Mato, “Automatic calculation of symmetry-adapted tensors in magnetic and nonmagnetic materials: A new tool of the Bilbao Crystallographic Server”, Acta Cryst. A (2019) 75, 438-447. [3] H.T. Stokes and B. J. Campbell, ISO-MAG: Table of Magnetic Space Groups http://stokes.byu.edu/iso/magneticspacegroups.php [4] D. B. Litvin, Magnetic Group Tables, https://www.iucr.org/publications/iucr/magnetic-group-tables