Table of Magnetic Space Groups

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

This table is based on data from:

(1) Daniel B. Litvin, *Magnetic Group Tables* (International Union of
Crystallography, 2013) www.iucr.org/publ/978-0-9553602-2-0.

(2) C. J. Bradley and A. P. Cracknell, *The Mathematical Theory of
Symmetry in Solids* (Clarendon Press, Oxford, 1972).

The 1651 Shubnikov/Heesch magnetic space groups combine the 230 crystallographic space groups with the time reversal operator. We denote space-group operators as (p|t), where p is a point operator and t is a translation. The time reversal operator is expressed as (1|0,0,0)' or 1'. To make our descriptions more intuitive, we occasionally refer to a symmetry operation as "primed" or "black" if it includes "time reversal", or refer to it as "unprimed" or "white" if it does not. Thus, a primed operator like (p|t)' is understood to include time reversal.

Each magnetic space group G is classified as one of four different types depending on how it is constructed from the one of the 230 crystallographic space groups F:

(1) G = F (Federov groups). For this type, no operator contains time reversal. The magnetic space group symbol is the same as that for the corresponding crystallographic space group.

(2) G = F + F1'. The magnetic space group symbol ends in 1' because the pure time reversal (1|000)' is an operator in the group. The magnetic group contains two copies of every spatial operator, one with a time reversal and one without. Because each operator is both black and white, we call this a "grey" group. A grey group can describe a paramagnetic material in zero external magnetic field -- the average magnetic moment of every atom is zero.

(3) G = D + (F - D)1', where D is an equi-translational subgroup of F of index 2. Each operator of a type-3 magnetic space group is either black or white, so that exactly half of the point operators include a time reversal. We thus say that a type-3 magnetic space group has a "colored" point group. The lattice translations, however, do not include time-reversal. In the magnetic space-group symbol, the generators of the magnetic point group that include a time reversal are indicated by the prime symbol.

(4) G = D + (F - D)1', where D is an equi-class subgroup of F of index 2. Each translational symmetry of the magnetic lattice is either black or white, so that exactly half of the lattice translations include time reversal. We thus say that a type-4 magnetic space group has a "colored" lattice. After the translations and point symmetries are combined, each point operator appears both with and without a time-reversal, so that the magnetic point group of a type-4 magnetic space group is actually a grey group. The magnetic space-group symbol adds a subscript to the leading letter (i.e. the lattice indicator) to indicate the magnetic lattice type. In some cases (described below), a type-4 group symbol will also have primes on some of the point-group generators.

Historically, the magnetic space groups have been classified using two different settings:

**BNS**: N. V. Belov, N. N. Neronova and T. S. Smirnova

**OG**: W. Opechowski and R. Guccione

The BNS and OG settings and symbols are identical for type-1, type-2 and type-3 groups, and are derived from subgroup F described above.

For type-4 groups, however, the BNS setting and symbols are derived
from D, while the OG setting and symbols are derived from F. In G,
pure lattice translations such as (1|u) are contained in D, while
lattice translations with time reversal such as (1|v)' are contained
in (F-D)1'. Both u and v are lattice translations in F, but only u is
a lattice translation in D. Therefore, in the BNS setting, operators
containing 1' are accompanied only by fractional translations, whereas
in the OG setting, they are accompanied by integral lattice
translations. For example, consider BNS:67.509 C_{a}mma and
OG:47.11.357 P_{C}mmm', which are the same group. In this
case, F=Pmmm and D=Cmma. BNS bases its symbol C_{a}mma on
D=Cmma (#67), while OG bases its symbol P_{C}mmm' on F=Pmmm
(#47). The BNS setting contains the operator (1|1/2,0,0)', while the
OG setting expresses the same operator as (1|0,1,0)'.

We refer to the BNS and OG descriptions of a given magnetic structure as two different "settings" of the same magnetic symmetry because there always exists an affine transformation matrix that allows one to transform the structural details (lattice vectors, atomic coordinates, magnetic moments, thermal parameters, etc.) back and forth between the two descriptions. This procedure puts the transformation between the BNS and OG settings on par with other common setting changes such as axes permutations, origin shifts and centering transformations.

For type-4 groups in the BNS setting, the magnetic space-group symbol is the crystallographic space-group symbol for D with a subscript added to the first letter, denoting the type of colored lattice. The BNS lattice is defined only by the white points, which are the lattice points t of D. The black points are at t+v, where v is a fractional translation. Below we define the symbols for the colored lattices by giving the fractional v.

Symbol | Fractional |

P_{S} | (0,0,1/2) triclinic only |

P_{a} | (1/2,0,0) |

P_{b} | (0,1/2,0) |

P_{c} | (0,0,1/2) |

P_{A} | (0,1/2,1/2) |

P_{B} | (1/2,0,1/2) |

P_{C} | (1/2,1/2,0) |

P_{I} | (1/2,1/2,1/2) |

A_{a} | (1/2,0,0) |

A_{b} | (0,1/2,0) |

A_{B} | (1/2,0,1/2) |

B_{b} | (0,1/2,0) |

B_{a} | (1/2,0,0) |

B_{A} | (0,1/2,1/2) |

C_{c} | (0,0,1/2) |

C_{a} | (1/2,0,0) |

C_{A} | (0,1/2,1/2) |

F_{S} | (1/2,1/2,1/2) |

I_{a} | (1/2,0,0) |

I_{b} | (0,1/2,0) |

I_{c} | (0,0,1/2) |

R_{I} | (0,0,1/2) |

For type-4 groups in the OG setting, the magnetic space-group symbol is the crystallographic space-group symbol for F with a subscript added to the first letter, denoting the type of colored lattice. In the OG setting, the lattice of F is decorated by both black and white points, so that the white points form a sublattice of F. Below we define the symbols for each colored lattice by giving the basis vectors of the corresponding primitive sublattice.

Symbol | Sublattice |

P_{2s} | (1,0,0,),(0,1,0),(0,0,2) triclinic only |

P_{2a} | (2,0,0,),(0,1,0),(0,0,1) |

P_{2b} | (1,0,0),(0,2,0),(0,0,1) |

P_{2c} | (1,0,0),(0,1,0),(0,0,2) |

P_{A} | (1,0,0),(0,1,1),(0,-1,1) |

P_{B} | (1,0,1),(0,1,0),(-1,0,1) |

P_{C} | (1,1,0),(-1,1,0),(0,0,1) non-tetragonal only |

P_{P} | (1,1,0),(-1,1,0),(0,0,1) tetragonal only |

P_{F} | (0,1,1),(1,0,1),(1,1,0) non-tetragonal only |

P_{I} | (0,1,1),(1,0,1),(1,1,0) tetragonal only |

A_{2a} | (2,0,0),(0,1/2,1/2),(0,-1/2,1/2) |

A_{P} | (1,0,0),(0,1,0),(0,0,1) |

A_{I} | (1,1/2,1/2),(0,1,0),(0,0,1) |

B_{2b} | (1/2,0,1/2),(0,2,0),(-1/2,0,1/2) |

B_{P} | (1,0,0),(0,1,0),(0,0,1) |

B_{I} | (1,0,0),(1/2,1,1/2),(0,0,1) |

C_{2c} | (1/2,1/2,0),(-1/2,1/2,0),(0,0,2) |

C_{P} | (1,0,0),(0,1,0),(0,0,1) |

C_{I} | (1,0,0),(0,1,0),(1/2,1/2,1) |

F_{A} | (1,0,0),(0,1/2,1/2),(0,-1/2,1/2) |

F_{B} | (1/2,0,1/2),(0,1,0),(-1/2,0,1/2) |

F_{C} | (1/2,1/2,0),(-1/2,1/2,0),(0,0,1) |

I_{P} | (1,0,0),(0,1,0),(0,0,1) |

I_{A} | (1/2,1/2,1/2),(0,1,1),(0,-1,1) |

I_{B} | (1,0,1),(1/2,1/2,1/2),(-1,0,1) |

I_{C} | (1,1,0),(-1,1,0),(1/2,1/2,1/2) |

R_{R} | (1/3,2/3,2/3),(-2/3,-1/3,2/3),(1/3,-1/3,2/3) |

For example, the representative operators in ITC52 for 49 Pccm are
chosen to be (1|0,0,0), (2_{x}|0,0,1/2),
(2_{y}|0,0,1/2), (2_{z}|0,0,0), (-1|0,0,0),
(m_{x}|0,0,1/2), (m_{y}|0,0,1/2),
(m_{z}|0,0,0). The consecutive symbols c,c,m in Pccm refer to
the generators (m_{x}|0,0,1/2), (m_{y}|0,0,1/2),
(m_{z}|0,0,0), respectively. The magnetic space groups based
on F=Pccm and the P_{2a} lattice are listed below. In each
case, the presence of absence of time reversal on each of these
generators determines the OG symbol.

Symbol | Generators |

49.8.371 P_{2a}ccm | (m_{x}|0,0,1/2),(m_{y}|0,0,1/2),(m_{z}|0,0,0) |

49.10.373 P_{2a}ccm' | (m_{x}|0,0,1/2),(m_{y}|0,0,1/2),(m_{z}|0,0,0)' |

49.11.374 P_{2a}c'c'm | (m_{x}|0,0,1/2)',(m_{y}|0,0,1/2)',(m_{z}|0,0,0) |

49.12.375 P_{2a}c'c'm' | (m_{x}|0,0,1/2)',(m_{y}|0,0,1/2)',(m_{z}|0,0,0)' |

As another example, the representative operators for 24
I2_{1}2_{1}2_{1} are listed as (1|0,0,0),
(2_{x}|1/2,1/2,0), (2_{y}|0,1/2,1/2,),
(2_{z}|1/2,0,1/2) in ITC52. The consecutive symbols
2_{1},2_{1},2_{1} in
I2_{1}2_{1}2_{1} refer to the generators
(2_{x}|1/2,1/2,0), (2_{y}|0,1/2,1/2,),
(2_{z}|1/2,0,1/2), respectively. The magnetic space groups
based on F=I2_{1}2_{1}2_{1} and the
I_{P} lattice are as follows:

Symbol | Generators |

24.4.153 I_{P}2_{1}2_{1}2_{1} | (2_{x}|1/2,1/2,0),(2_{y}|0,1/2,1/2),(2_{z}|1/2,0,1/2) |

24.5.154 I_{P}2_{1}'2_{1}'2_{1} | (2_{x}|1/2,1/2,0)',(2_{y}|0,1/2,1/2)',(2_{z}|1/2,0,1/2) |

Note that we could have just as well ignored ITC52 and chosen the representative
operators for 24 I2_{1}2_{1}2_{1} to be (1|0,0,0),
(2_{x}|0,0,1/2), (2_{y}|1/2,0,0),
(2_{z}|0,1/2,0). In that case the symbols would have been
24.4.153 I_{P}2_{1}'2_{1}'2_{1}' and
24.5.154 I_{P}2_{1}2_{1}2_{1}'. It is
not obvious by inspection which operators are to be primed.
Only by relying on the operators explicitly given in ITC52 can we see how the
primes were placed by OG.

Changes to the form of the representative operators listed in subsequent versions of the International Tables of Crystallography (ITC) have led to some confusion (Litvin).

The strict use of ITC52 operators is not always sufficient
for determining the OG symbol. For example, the representative
operators for 93 P4_{2}22 are chosen to be (1|0,0,0),
(4_{z}|0,0,1/2), (4_{z}^{-1}|0,0,1/2),
(2_{x}|0,0,0), (2_{y}|0,0,0), (2_{z}|0,0,0),
(2_{xy}|0,0,1/2), (2_{-xy}|0,0,1/2) in ITC52. For
this space group symbol, the consecutive point-operator symbols
{4_{2},2,2} refer to sets of related operators:
{(4_{z}|0,0,1/2), (4_{z}^{-1}|0,0,1/2)},
{(2_{x}|0,0,0), (2_{y}|0,0,0)}, and
{(2_{xy}|0,0,1/2), (2_{-xy}|0,0,1/2)}, rather than to
individual operators. Usually, when one operator in such a set
contains time reversal, all of the operators in that set do. But this
is not always the case. For 93.6.781 P_{2c}4_{2}22',
(2_{y}|0,0,0)' has the time reversal while
(2_{x}|0,0,0) does not. This leaves us with an arbitrary
decision to make: should we prime the corresponding 2 in the symbol or
not?
In 2011, Litvin recommended that this ambiguity be resolved by using
the representative operator associated with a given generator symbol
from the table, Representatives for the sets of lattice
symmetry directions in the various crystal families, Chapter 12,
"Space-Group
Symbols and Their Use," by H. Burzlaff and H. Zimmerman, in ITC, Vol. A.
(Note that the table number is different in various editions of ITC.)
For the cases of interest here,
that table gives the following representative operators:

Point Group | Operators |

422 | 4_{z}, 2_{x}, 2_{-xy} |

321 | 3_{z}, 2_{x}, 1 |

312 | 3_{z}, 1, 2_{3} |

622 | 6_{z}, 2_{x}, 2_{3} |

m-3 | -2_{z}, -3_{xyz} |

The following table lists all of the magnetic space groups for which sets of related operators contain the time reversal and some do not. Though OG did not use the above-convention in treating these groups historically, we have followed Litvin by applying the new convention here in all but two cases (153.4.1270 and 154.4.1274 -- see discussion below), which changed the OG symbols for seven of the magnetic space groups (only the new symbols are shown here).

Group | Operators |

93.6.781 P_{2c}4_{2}22' | {(2_{x}|0,0,0),(2_{y}|0,0,0)'} & {(2_{xy}|0,0,1/2),(2_{-xy}|0,0,1/2)'} |

93.8.783 P_{I}4_{2}22' | {(2_{x}|0,0,0),(2_{y}|0,0,0)' & {(2_{xy}|0,0,1/2),(2_{-xy}|0,0,1/2)'} |

93.9.784 P_{2c}4_{2}'22 | {(2_{x}|0,0,0),(2_{y}|0,0,0)'} & {(2_{xy}|0,0,1/2)',(2_{-xy}|0,0,1/2)} |

94.6.791 P_{2c}4_{2}2_{1}2 | {(2_{x}|1/2,1/2,1/2),(2_{y}|1/2,1/2,1/2)'} & {(2_{xy}|0,0,0)',(2_{-xy}|0,0,0)} |

94.7.792 P_{2c}4_{2}'2_{1}'2 | {(2_{x}|1/2,1/2,1/2)',(2_{y}|1/2,1/2,1/2)} & {(2_{xy}|0,0,0)',(2_{-xy}|0,0,0)} |

98.6.819 I_{P}4_{1}22 | {(2_{x}|0,1/2,1/4),(2_{y}|0,1/2,1/4)'} & {(2_{xy}|0,0,0)',(2_{-xy}|0,0,0)} |

98.7.820 I_{P}4_{1}'22' | {(2_{x}|0,1/2,1/4),(2_{y}|0,1/2,1/4)'} & {(2_{xy}|0,0,0),(2_{-xy}|0,0,0)'} |

98.8.821 I_{P}4_{1}2'2' | {(2_{x}|0,1/2,1/4)',(2_{y}|0,1/2,1/4)} & {(2_{xy}|0,0,0),(2_{-xy}|0,0,0)'} |

98.9.822 I_{P}4_{1}'2'2 | {(2_{x}|0,1/2,1/4)',(2_{y}|0,1/2,1/4)} & {(2_{xy}|0,0,0)',(2_{-xy}|0,0,0)} |

151.4.1262 P_{2c}3_{2}12 | {(2_{1}|0,0,0),(2_{2}|0,0,1/3)',(2_{3}|0,0,2/3)} |

152.4.1266 P_{2c}3_{2}21 | {(2_{x}|0,0,2/3),(2_{xy}|0,0,0),(2_{y}|0,0,1/3)'} |

153.4.1270 P_{2c}3_{1}12 | {(2_{1}|0,0,0),(2_{2}|0,0,2/3),(2_{3}|0,0,1/3)'} |

154.4.1274 P_{2c}3_{1}21 | {(2_{x}|0,0,1/3)',(2_{xy}|0,0,0),(2_{y}|0,0,2/3)} |

180.6.1401 P_{2c}6_{2}22' | {(2_{x}|0,0,0),(2_{xy}|0,0,2/3),(2_{y}|0,0,1/3)'} |

& {(2_{1}|0,0,1/3),(2_{2}|0,0,0)',(2_{3}|0,0,2/3)'} | |

180.7.1402 P_{2c}6_{2}'22 | {(2_{x}|0,0,0),(2_{xy}|0,0,2/3),(2_{y}|0,0,1/3)'} |

& {(2_{1}|0,0,1/3)',(2_{2}|0,0,0),(2_{3}|0,0,2/3)} | |

181.6.1408 P_{2c}6_{4}22' | {(2_{x}|0,0,0),(2_{xy}|0,0,1/3)',(2_{y}|0,0,2/3)} |

& {(2_{1}|0,0,2/3),(2_{2}|0,0,0),(2_{3}|0,0,1/3)'} | |

181.7.1409 P_{2c}6_{4}'2'2' | {(2_{x}|0,0,0)',(2_{xy}|0,0,1/3),(2_{y}|0,0,2/3)'} |

& {(2_{1}|0,0,2/3),(2_{2}|0,0,0),(2_{3}|0,0,1/3)'} | |

206.4.1541 I_{P}a-3' | {(-3_{xyz}|0,0,0)',(-3_{xyz}^{-1}|0,0,0)',(-3_{-xyz}|1/2,0,0),(-3_{-xyz}^{-1}|0,1/2,0), |

(-3_{x-yz}|0,1/2,0),(-3_{x-yz}^{-1}|0,0,1/2),(-3_{xy-z}|0,0,1/2),(-3_{xy-z}^{-1}|1/2,0,0)} |

Finally, there are some cases (see list below) where the OG symbol is based on the symbol for D instead of F. Two of these groups (153.4.1270 and 154.4.1274) were also mentioned in the previous paragraph. We propose that at some future time, the symbols for these six groups be brought into compliance with the conventions above: (1) base the OG symbol on F, and (2) use primes when the representative generator contains time reversal.

Group | F | D |

144.3.1236 P_{2c}3_{2} | P3_{1} | P3_{2} |

145.3.1239 P_{2c}3_{1} | P3_{2} | P3_{1} |

151.4.1262 P_{2c}3_{2}12 | P3_{1}12 | P3_{2}12 |

152.4.1266 P_{2c}3_{2}21 | P3_{1}21 | P3_{2}21 |

153.4.1270 P_{2c}3_{1}12 | P3_{2}12 | P3_{1}12 |

154.4.1274 P_{2c}3_{1}21 | P3_{2}21 | P3_{1}21 |

We propose that at some future time, the symbols for these six groups be brought into compliance with convention: base the symbol on F and use primes in cases where the representative generator contains time reversal:

Proposed Symbols |

144.3.1236 P_{2c}3_{1} |

145.3.1239 P_{2c}3_{2} |

151.4.1262 P_{2c}3_{1}12 |

152.4.1266 P_{2c}3_{1}21 |

153.4.1270 P_{2c}3_{2}12' |

154.4.1274 P_{2c}3_{2}2'1 |

There are two tables. In Table 1, each of the 1651 magnetic space groups are listed in order of their BNS numbers. In Table 2, each of the 1651 magnetic space groups are listed in order of their OG numbers. For each group, we include the following information.

(a) Number and symbol, given in both the BNS and OG settings. The symbols are identical except for type-4 groups.

(b) OG-BNS transformation for each type-4 group.. We give the origin and axes of the subgroup D with respect to F. These transformations are from Litvin, though we do make some adjustments to his transformations for space groups which have two origin choices: Litvin uses origin choice 1, while we use origin choice 2. We give the origin and axes of the subgroup D with respect to F. The corresponding 4x4 affine transformation matrix is easily constructed. For example, if the transformation is given as (1/4,0,0;b,c,a), the affine transformation matrix T would be

0 | 0 | 1 | 1/4 |

1 | 0 | 0 | 0 |

0 | 1 | 0 | 0 |

0 | 0 | 0 | 1 |

If g_{BNS} and g_{OG} are operators in the BNS and OG
settings, respectively, then

g_{OG} = T g_{BNS} T^{-1}.

(c) Operators. The operators are listed using symbols from Litvin. The operators follow the conventions in ITC using monoclinic unique axis b, monoclinic cell choice 1, hexagonal axes for trigonal groups, and origin choice 2 for groups with more than one origin choice. For type-4 magnetic space groups, the same operators are expressed in both the BNS and OG settings. This requires the number of OG operators to be double the number associated with subgroup F so as to fill an orbit that spans the entire magnetic repeating unit.

(d) Wyckoff sites. The x,y,z positional coordinates and the mx,my,mz components of magnetic moment are listed for each symmetry-equivalent site of each Wyckoff orbit. The classification of Wyckoff sites depends only on the action of group operators on the x,y,z coordinates. Time reversal only affects the allowed magnetic moments at each site and does not affect the x,y,z coordinates. If we remove time reversal from every operator in G, we obtain subgroup F. Therefore the Wyckoff sites for G are the same as those for F.

For type-4 magnetic space groups, the Wyckoff positions are given for both the BNS and OG settings. The positions in the two settings are related by the OG-BNS transformation described in (b) above. One must be cautious when interpreting the Wyckoff positions of type-4 groups in the OG setting, since not all integer translations are true lattice translations that take a position to an equivalent position.

For example, the transformation between BNS:72.547 I_{b}bam
and OG:67.10.586 C_{I}mma is given by (0,0,0;b,2c,a). In
I_{b}bam, equivalent Wyckoff positions are obtained by adding
any integers to the x, y, or z coordinates or by adding the centering
translation (1/2,1/2,1/2). In C_{I}mma, equivalent Wyckoff
positions are obtained by adding any integer to the x or y
coordinates, any even integer to the z coordinate, and by adding the
"centering" translation (1/2,1/2,1). In contrast, adding an odd
integer to the z component does not take us to an equivalent Wyckoff
position in the OG setting because that operation in C_{I}mma
includes time reversal. In the table, we list a generating set (not
necessarily minimal) of lattice translations in both the BNS and OG
settings. For this example, we list (0,0,0)+ (1/2,1/2,1/2)+ in the
BNS setting, and (0,0,0)+ (0,1,0)+ (0,0,2)+ (1,0,0)+ (1/2,1/2,1)+ in
the OG setting.

We note that changing the origin of a space group does not change its symbol. Also, for trigonal groups, the symbol is the same whether rhombohedral or hexagonal axes are used.

**Monoclinic settings**: Six different permutations of the axes
are possible: a(b)c, c(-b)a, ab(c), ba(-c), (a)bc, (-a)cb. ITC also
gives three different cell choices. For each of these settings, the
primes in the magnetic space group symbol follow the generators to
which they were originally attached. The symbols C2/m', A2/m', B2/m',
I2/m' all refer to different settings of BNS 12.61 = OG
12.4.69. Observe that the prime stays with the m generator rather than
jumping to the 2 generator. As another example, P2/c', P2/a', P2/b',
P2/n' all refer to different settings of BNS 13.68 = OG 13.4.80. In
each case, the prime always occurs on the symbol for the glide plane,
even though it has different labels in different settings.

Now consider magnetic space group BNS 5.16 C_{c}2 = OG
5.4.22 C_{2c}2, which requires considerably more care due to
its colored lattice. F and D are different subgroups of G
here, but both have the same space-group type (#5 C2), which makes the
BNS and OG symbols look more similar that they might otherwise be.
This should make the details of this example easier to appreciate. In
the figures below, open circles ("white points") represent
translations without time reversal, and the filled circles ("black
points") represent translations with time reversal.

BNS setting. The figure below illustrates the colored
lattice and the BNS cell of this space group for each of the three
monoclinic cell choices and special-axis b. For cell choice 1, the
white lattice is C-centered, and the black point at (0,0,1/2) yields BNS
magnetic lattice Cc, so that the magnetic space-group symbol is
C_{c}2. For cell choice 2, the white lattice is B-centered and the black
point is at (1/2,0,1/2), so that the symbol is A_{B}2. For cell
choice 3, the white lattice is body centered and the black point is at
(1/2,0,0), so that the symbol is I_{a}2.

OG setting: The figure below illustrates the colored lattice and
the OG cell of this space group for each of the three monoclinic cell
choices and special-axis b.. Observe that the OG "unit cell" has half
the volume of the BNS cell and has both black and white corners. For
cell choice 1, the basis vectors of the white sublattice relative to
those of the combined black-white lattice are (1/2,1/2,0),
(-1/2,1/2,0), (0,0,2), yielding magnetic lattice C_{2c} and
magnetic space-group symbol C_{2c}2. For cell choice 2, the
basis vectors are (2,0,0), (0,1/2,1/2), (0,-1/2,1/2), and the symbol
for the magnetic space group is A_{2a}2. For cell choice 3,
the basis vectors are (1,0,1), (1/2,1/2,1/2), (-1,0,1) and the symbol
for the magnetic space group is I_{B}2. Note that the
I_{B} colored lattice occurs only for centered
monoclinic settings and cell-choice-3, and is therefore not listed by
Litvin.

**Orthorhombic settings**: Six different permutations of the
axes are possible: abc, ba-c, cab, -cba, bca, a-cb. Primes in the
magnetic space group symbol follow the generators they are attached
to. For example, Pnma', Pmnb', Pb'nm, Pc'mn, Pmc'n, Pna'm all refer
to different settings of BNS 62.445 = OG 62.5.506. For type-4 groups,
the letters in the subscript denoting the magnetic lattice must be
permuted as well. For example, P_{2c}m'mn,
P_{2c}mm'n,P_{2a}nm'm,P_{2a}nmm',P_{2b}mnm',
P_{2b}m'nm all refer to different settings of OG 59.9.486.

BNS: 80.32 I_c4_1 OG: 77.6.677 P_I4_2 OG-BNS transformation: (0,1/2,0;a-b,a+b,2c) Operators (BNS): (1|0,0,0) (4z|0,1/2,1/4) (4z-1|1/2,0,3/4) (2z|1/2,1/2,1/2) (1|0,0,1/2)' (4z|0,1/2,3/4)' (4z-1|1/2,0,1/4)' (2z|1/2,1/2,0)' Wyckoff positions (BNS): (0,0,0)+ (1/2,1/2,1/2)+ 16d (x,y,z;mx,my,mz) (-y,x+1/2,z+1/4;-my,mx,mz) (y+1/2,-x,z+3/4;my,-mx,mz) (-x+1/2,-y+1/2,z+1/2;-mx,-my,mz) (x,y,z+1/2;-mx,-my,-mz) (-y,x+1/2,z+3/4;my,-mx,-mz) (y+1/2,-x,z+1/4;-my,mx,-mz) (-x+1/2,-y+1/2,z;mx,my,-mz) 8c (0,0,z;0,0,mz) (0,1/2,z+1/4;0,0,mz) (0,0,z+1/2;0,0,-mz) (0,1/2,z+3/4;0,0,-mz) 8b (1/4,1/4,z;mx,my,0) (3/4,3/4,z+1/4;-my,mx,0) (3/4,3/4,z+3/4;my,-mx,0) (1/4,1/4,z+1/2;-mx,-my,0) 8a (1/4,3/4,z;mx,my,0) (1/4,3/4,z+1/4;-my,mx,0) (1/4,3/4,z+3/4;my,-mx,0) (1/4,3/4,z+1/2;-mx,-my,0) Operators (OG): (1|0,0,0) (4z|0,0,1/2) (4z-1|0,0,3/2) (2z|0,0,1) (1|0,0,1)' (4z|0,0,3/2)' (4z-1|0,0,1/2)' (2z|0,0,0)' Wyckoff positions (OG): (0,0,0)+ (1,-1,0)+ (1,1,0)+ (0,0,2)+ (1,0,1)+ 16d (x,y,z;mx,my,mz) (-y,x,z+1/2;-my,mx,mz) (y,-x,z+3/2;my,-mx,mz) (-x,-y,z+1;-mx,-my,mz) (x,y,z+1;-mx,-my,-mz) (-y,x,z+3/2;my,-mx,-mz) (y,-x,z+1/2;-my,mx,-mz) (-x,-y,z;mx,my,-mz) 8c (0,1/2,z;0,0,mz) (-1/2,0,z+1/2;0,0,mz) (0,1/2,z+1;0,0,-mz) (-1/2,0,z+3/2;0,0,-mz) 8b (1/2,1/2,z;mx,my,0) (-1/2,1/2,z+1/2;-my,mx,0) (1/2,-1/2,z+3/2;my,-mx,0) (-1/2,-1/2,z+1;-mx,-my,0) 8a (0,0,z;mx,my,0) (0,0,z+1/2;-my,mx,0) (0,0,z+3/2;my,-mx,0) (0,0,z+1;-mx,-my,0)

From the OG-BNS transformation, (0,1/2,0;a-b,a+b,2c),
we obtain the affine transformation matrix: T =
((1,1,0,0),(-1,1,0,1/2),(0,0,2,0),(0,0,0,1)).

Because g_{OG} = T g_{BNS} T^{-1}, we also know
that g_{BNS} = T^{-1} g_{OG} T .

The point portion of the transformation is matrix T_{P} =
((1,1,0),(-1,1,0),(0,0,2)).

The purely rotational component of T_{P} is T_{R} =
((0.707,0.707,0),(-0.707,0.707,0),(0,0,1)).

Let the OG description of a two-atom structure be as follows:

OG: 77.6.677 P_{I}4_{2}

Cell parameters: a = 2.5 Å, c = 3.0 Å

A: 16d (x,y,z) = (0.1, 0.2, 0.3); (m_{x},m_{y},m_{z}) = (1.414, 1.414, 0)μ_{B};
(u_{11},u_{22},u_{33},u_{12},u_{13},u_{23}) = (0.11, 0.22, 0.33, 0.12, 0.13, 0.23)

B: 8c (0,1/2,z) = (0, 1/2, 0.4); (0,0,m_{z}) = (0, 0, 1.5)μ_{B};
(u_{11},u_{11},u_{33},u_{12},0,0) = (0.11, 0.22, 0.33, 0.12, 0, 0)

The lattice of the OG cell can be described by a matrix of three
column vectors (a | b | c),

B_{OG} = ((2.5,0,0),(0,2.5,0),(0,0,3.0)), which transforms as

B_{BNS} = B_{OG} T_{3D} = ((2.5,2.5,0),(-2.5,2.5,0),(0,0,6.0)).

This corresponds to a tetragonal lattice with a = 3.54 Å, c = 6.0 Å.

The first atom position transforms as x_{BNS} = T^{-1} x_{OG} = T^{-1} (0.1,
0.2, 0.3, [1]) = (0.2, -0.1, 0.15, [1])

The corresponding moment transforms as m_{BNS} =
T_{R}^{-1} m_{OG} =
T_{R}^{-1} (1.414, 1.414, 0) = (0, 2.0, 0)

The corresponding thermal ellipsoid transforms as u_{BNS} = T_{R}^{-1} u_{OG} T_{R}

= T_{R}^{-1} ((0.11,0.12,0.13),
(0.12,0.22,0.23), (0.13,0.23,0.33)) T_{R}

= ((0.045, -0.055, -0.071), (-0.055, 0.285, 0.255), (-0.071,
0.255, 0.33)

The second atom position transforms as x_{BNS} =
T^{-1} x_{OG} = T^{-1} (0, 1/2, 0.4, [1]) =
(0, 0, 0.2, [1])

The corresponding moment transforms as m_{BNS} =
T_{R}^{-1} m_{OG} =
T_{R}^{-1} (0, 0, 1.5) = (0, 0, 1.5)

The corresponding thermal ellipsoid transforms as u_{BNS}
= T_{R}^{-1} u_{OG} T_{R}

= T_{R}^{-1} ((0.11, 0.12, 0),(0.12, 0.22, 0),(0,
0, 0.33)) T_{R}

= ((0.045,-0.055,0), (-0.055, 0.285, 0), (0, 0, 0.33))

Thus, the BNS description of this two-atom structure is as follows:

BNS: 80.32 I_{c}4_{1}

Cell parameters: a = 3.54 Å, c = 6.0 Å

A: 16d (x,y,z) = (0.2, -0.1, 0.15); (m_{x},m_{y},m_{z}) = (0, 2.0, 0)μ_{B};
(u_{11},u_{22},u_{33},u_{12},u_{13},u_{23}) = (0.045, 0.285, 0.33, -0.055, -0.071, 0.255)

B: 8c (0,0,z) = (0, 1/2, 0.4); (0,0,m_{z}) = (0, 0, 1.5)μ_{B};
(u_{11},u_{11},u_{33},u_{12},0,0) = (0.045, 0.285, 0.33, -0.055, 0, 0)

The direction of the transform can just as easily be reversed to recover the OG description.

Because the moments are defined as projections along the crystal
axes, they need to be transformed into cartesian coordinates in order
to determine their magnitudes. For this purpose, we define a matrix L
= ((a,0,0),(0,b,0),(0,0,c)), which unlike B, is always a diagonal
matrix containing the three cell edge lengths. In this simple
orthorhombic example, B and L are identical. The magnitudes of the
moments, which are invariant with respect to setting, are computed as

|M| = |B_{OG} L_{OG}^{-1} m_{OG}|
= |(1.414, 1.414, 0)|
= |B_{BNS} L_{BNS}^{-1} m_{BNS}|
= |(0, 2.0, 0)| = 2.0 μ_{B} for the 1st atom, and

|M| = |B_{OG} L_{OG}^{-1} m_{OG}|
= |(0, 0, 1.5)|
= |B_{BNS} L_{BNS}^{-1} m_{BNS}|
= |(0, 0, 1.5)| = 1.5 μ_{B} for the 2nd atom.

In July 2013, we discovered an error in the BNS symbols for eight of the
magnetic space groups:

227.130 Fd'-3m should be Fd'-3'm,

227.132 Fd'-3m' should be Fd'-3'm',

228.136 Fd'-3c should be Fd'-3'c,

228.138 Fd'-3c' should be Fd'-3'c',

229.142 Im'-3m should be Im'-3'm,

229.144 Im'-3m' should be Im'-3'm',

230.147 Ia'-3d should be Ia'-3'd,

230.149 Ia'-3d' should be Ia'-3'd'.

These errors were introduced when we copied the
symbols from Bradley and Cracknell and then changed 3 to -3 to be
consistent with current symbols in *International Tables of
Crystallography*. For example, 227.130 is listed correctly as Fd'3m
in Bradley and Cracknell.
We changed the symbol to Fd'-3m. However, in this space group, 3_{xyz}
is not followed by a time reversal, but -3_{xyz}' is, so we
should have put a prime on the -3 in the symbol. These symbols have now
been corrected in our tables.