Harold T. Stokes and Branton J. Campbell,
Department of Physics and Astronomy, Brigham
Young University, Provo, Utah, 84602, USA, branton_campbell@byu.edu
Sander van Smaalen, University of Bayreuth, Bayreuth, Germany,
smash@uni-bayreuth.de
ITC-A: International Tables for Crystallography, Vol. A, Edited by T. Hahn (Kluwer, Dordrecht, 2002).
ITC-C: T. Janssen, A. Janner, A. Looijenga-Vos, and P. M. de Wolff, "Incommensurate and commensurate modulated structures," International Tables for Crystallography, Vol. C, pp. 907-955. Edited by E. Prince (Kluwer, Dordrecht, 2004).
JJdW: A. Janner, T. Janssen and P. M. de Wolff, "Bravais Classes for Incommensurate Crystal Phases," Acta Cryst. A39, 658-666 (1983).
I. P. Orlov and G. Chapuis, "List of (3+1) Dimensional Superspace Groups," superspace.epfl.ch/groups/ (2005).
S. van Smaalen, Incommensurate Crystallography, IUCr Monographs on Crystallography #21 (Oxford University Press, Oxford, 2007).
A. Yamamoto, "Superspace groups for 1D, 2D and 3D modulated structures," quasi.nims.go.jp/yamamoto/. A. Yamamoto, Acta Crystallogr. A52, 509-560, (1996)
d=1, 24 Bravais classes, 775 Superspace groups
d=2, 83 Bravais classes, 3338 Superspace groups
d=3, 215 Bravais classes, 12584 Superspace groups
Any lattice in three dimensions will belong to one of the 14 crystallographic Bravais classes. Similarly, Bravais classes are used to identify lattice types in (3+d)D superspace. The identifier of a Bravais class consists of two integers separated by a period. The first integer denotes the dimension d of the modulations involved (i.e. d=1, d=2 or d=3). The second number simply enumerates the Bravais classes for a given dimension. For d=1, we list the Bravais classes in the same order as those listed in ITC-C and JJdW. For d=2, we list the Bravais classes in the same order as those listed in JJdW. For d=3, however, we have corrected several errors in JJdW (four redundant classes and two missing classes) and renumbered the d=3 classes accordingly. Because a renumbering was necessary, we also found it convenient to make other changes to the order of the d=3 classes for aesthetic reasons.
The identifier of a superspace group consists of four integers separated by periods. The first integer refers to the basic space group of the superspace group, and corresponds to one of the 230 three-dimensional space groups in ITC-A. The second and third integers indicate the Bravais class of superspace groups to which the superspace group belongs. The fourth integer enumerates the superspace groups associated with a given basic space group.
ITC-A uses a coordinate triplet to indicate a vector position within the 3D lattice of the crystal. In treating (3+1)D superspace-groups, ITC-C uses the symbol "t" to indicate the phase-shift coordinate. This pattern can be extended to (3+2) and (3+3) dimensions by using "u" and "v" as additional phase-shift coordinates. On the unmixed basis, which strictly separates the 3D and phase-coordinate axes, (x,y,z,t,u,v) indicates a point (x,y,z) in 3D space as well as phase shifts of t, u, and v along three independent modulation directions. Within the superspace construction, however, an external (3+d)D basis vector can have non-zero projections onto both 3D and phase-space coordinates. To distinguish them from unmixed-basis vector components, van Smaalen uses a (xs1,xs2,xs3,xs4,xs5,xs6) notation for vector components defined on the superspace-lattice basis. ITC-C also uses the "s" subscript to indicate superspace-lattice basis, but omits the "s" subscript when indicating their components, so that the result appears as (x1,x2,x3,x4,x5,x6). The JANA software package also uses this notation. In contrast, the tables of Yamamoto employed the (x,y,z,t,u,v) notation to indicate superspace-lattice vector components; this form is especially easy to read but can easily be confused with components on the unmixed basis when comparing against most modern sources. The real-time tools available on the ISO(3+d)D site are flexible in allowing the user to use any of these three notations as either input or output. Static tables and information pages, however, use only the more readable (x,y,z,t,u,v) notation. Confusion can be avoided by understanding that the ISO(3+d)D site only treats vectors and operators on the superspace-lattice basis -- it never employs the unmixed basis.
A (3+d)D Bravais class symbol contains a 3D space-group symbol to identify the three-dimensional symmetry of the external part of the class, followed by d rationally-independent modulation (q) vectors. As in ITC-A, a 3D space group symbol contains a capital letter (e.g. P, A, B, C, F, I, R) to indicate a centering type and a point-group symbol to indicate the Laue group. The 3D external portions of these Bravais class symbols are identical to those used in JJdW and ITC-C, except that we place the centering type before, rather than after, the point-group symbol. In contrast to JJdW, our symbols include a full d-dimensional basis of q vectors rather than including only the generating q vectors. The irrational components of these q vectors are denoted by a=alpha, b=beta and g=gamma, which are then subscripted with integers (e.g. 1,2,3) to denote q vectors that belong to independent sets of q that are not related to each other via any of the rotational operators in the group. Following the conventions of JJdW, we frequently use nonstandard settings (e.g. Ammm rather than Cmmm) for the 3D portion of a class in order to obtain modulations vectors with a desired form. One could also imagine giving precedence to standard 3D space group settings, and letting the q vectors fall where they may, though such an approach would be a departure from the past.
Example: 3.128 Immm(a1,b1,0)(-a1,b1,0)(0,0,g2) is a (3+3)D Bravais class. From "Immm", we learn that the external 3D part of the class is body-centered orthorhombic. The basis of the internal part of the class contains a two-dimensional modulation defined by vectors (a1,b1,0) and (-a1,b1,0), as well as a one-dimensional modulation defined by vector (0,0,g2). Because the parameters a1 and b1 have the same values in both of the first two q vectors, these two q vectors are related to each other via a rotational operator in Immm, while the third q vector is not.
A (3+d)D superspace group symbol can be viewed as a construction that builds upon the (3+d)D Bravais class symbol to which it belongs. To begin with, the 3D space-group symbol of the external portion of the Bravais class is replaced by the 3D space-group symbol of the basic space group (i.e. the external part of the superspace group symmetry). But the basic space group can be added in two stages. First, the external point group is chosen from within the Laue family of the point group of the external Bravais class, at which point, both the internal and external symmetry operators are still free of non-lattice translations (i.e. both the basic space group and the superspace group are symmorphic). Then, a self-consistent set of non-lattice translations can be added to the generators of the external point group to obtain a non-symmorphic basic space group. And finally, a self-consistent set of internal non-lattice translations can be added to the point-group generators to obtain a fully general superspace group.
Historically, superspace group symbols have been designed to reflect the intrinsic (i.e. origin-independent) part of the non-lattice translation associated with each point-group generator. Note that the symbols for the 3D basic space groups, which are taken from ITC-A, already accomplish this via subscripted integers for screw axes and labels like "a" and "n" for glide planes. We chose to adopt and extend the approach of ITC-C, which associates non-zero internal intrinsic translations with lowercase alphabet letters (0, s=1/2, t=1/3, q=1/4, h=1/6) in a one-line symbol. Our symbols are similar to those in ITC-C except that we explicitly include all zeros, even when there are no non-zero translations. In order to appropriately represent all unique intrinsic internal translations, we also use -t=2/3, -q=3/4 and -h=5/6 (van Smaalen, 2007).
For d > 1 groups, the primary difference between our symbols and those of Yamamoto (1996) is that (1) we display each internal intrinsic translation immediately after the modulation vector to which it corresponds, rather then grouping all of the internal translations at the end of the symbol and (2) we use only the simple symbols s,t,q,h for intrinsic translations rather than symbols like m,g,d,a,b,c,n that are meant to denote specific operations in d-dimensional internal space. Note that the internal intrinsic translations are determined from the generators in the supercentered setting.
Example: 44.3.128.51 I2mm(a1,b1,0)000(-a1,b1,0)000(0,0,g2)0s0 is a (3+3)-dimensional superspace group based on 3D basic space group I2mm and Bravais class Immm(a1,b1,0)(-a1,b1,0)(0,0,g2). The generators of the basic space group, as given in the symbol I2mm, are 2=(x,-y,-z), m=(x,-y,z) and m=(x,y,-z). The corresponding superspace-group generators are (x,-y,-z,t,-u,-v), (x,-y,z,t,-u,v+1/2) and (x,y,-z,t,u,-v+1/2). While the basic space group is symmorphic in this case, there are still some non-zero internal intrinsic translations. Because there are no intrinsic translations associated with u or v coordinates, the translations are displayed as "000" for the first two modulation vectors. Note, however, that the second generator has an intrinsic 1/2 on the v coordinate, which causes the translations of the third modulation vector to be displayed as "0s0".
Detailed information can be obtained about a superspace group by (1) choosing the group from a list, (2) searching for a specific group number, or (3) by using the SSG equivalence finder tool. In each case, the information displayed is identical. We consider a specific example, superspace group 44.3.128.51, in detail. Below, we give each line from the detailed information about this group, followed by an explanation.
Superspace group: 44.3.128.51 I2mm(a1,b1,0)000(-a1,b1,0)000(0,0,g2)0s0 [Y:3.3821]
The superspace group number and symbol are followed by a cross reference to group #3821 in Yamamoto's d=3 superspace group tables. When multiple entries turn out to be equivalent in Yamamoto's tables, we cross reference all of them. When a superspace group is missing from Yamamoto's tables, we display [Y:none]. Note that there are a number of entries in Yamamoto's d=3 tables that do not obey multiplication tables, and cannot therefore be cross referenced to our tables.
Bravais class: 3.128 Immm(a1,b1,0)(-a1,b1,0)(0,0,g2) [JJdW:3.130]
The Bravais class number and symbol are followed by a cross reference to the JJdW tables of Bravais classes. Because our Bravais-class numberings only differ from those of JJdW for d=3 classes, we only cross reference JJdW for d=3 classes. When multiple entries turn out to be equivalent in the JJdW tables, we cross reference all of them. When a class is missing from the JJdW tables, we display [JJdW:none].
Note that when a superspace group is a member of an enantiomorphic pair, the other member of the pair is displayed. For example, the information displayed for 76.1.19.1 P41(0,0,g)0 contains the following line:
Enantiomorph: 78.1.19.1 P43(0,0,g)0
Transformation to supercentered setting: A1=a1, A2=a2, A3=a3, A4=a4-a5, A5=a4+a5, A6=a6
Let a1,a2,a3,a4,a5,a6 be the basis vectors of the superspace lattice in the setting of the 3D basic space group, which is the setting that puts the external part of each operator into a form that matches the corresponding entry in ITC-A. Let A1,A2,A3,A4,A5,A6 be the basis vectors of the superspace lattice in the supercentered setting (i.e. a setting that eliminates all rational components of the q vectors and simplifies the forms of the q vectors). We explicitly define the transformation between the setting of the basic space group and the supercentered setting.
Note that when the transformation does not change any of the basis vectors of the lattice (for example, A1=a1, A2=a2, A3=a3, A4=a4, A5=a5, A6=a6), the entry looks like
Transformation to supercentered setting: none
BASIC SPACE GROUP SETTING
Setting where the 3D part of the operators are the same as those listed for the basic space group in ITC-A. The internal coordinates are in a primitive setting so that no centering translation involves those coordinates.
Modulation vectors: q1=(a1,b1,0), q2=(-a1,b1,0), q3=(0,0,g2)
The modulation vectors in the basic space-group setting are taken directly from the superspace group symbol.
Centering: (0,0,0,0,0,0), (1/2,1/2,1/2,0,0,0)
Centering translations that arise in the conventional setting of the basic space group are the same as those in ITC-A. Note that they do not include any internal components.
Non-lattice generators: (x,-y,-z,-u,-t,-v); (x,-y,z,-u,-t,v+1/2); (x,y,-z,t,u,-v+1/2)
The superspace group generators correspond precisely to (and are listed in the same order as) the non-lattice generators displayed within the basic space group symbol. Here, I2mm denotes three generators: (2) a 2-fold rotation about the a axis, (m) a reflection through the ac plane, and (m) a reflection through the ab plane. The lattice translations required to completely generate a superspace group are not listed here because they are trivial to obtain.
Non-lattice operators: (x,y,z,t,u,v); (x,-y,-z,-u,-t,-v); (x,-y,z,-u,-t,v+1/2); (x,y,-z,t,u,-v+1/2)
In addition to the generators, we list the full set of operators associated with the general Wyckoff position.
Note that if the supercentered setting is the same as the basic space group setting, the reflection conditions are listed here using lower-case letters h,k,l,m,n,p.
SUPERCENTERED SETTING
Setting where the q-vectors are in a simplified uniform form. This often results in centering translations that have nonzero internal coordinates. If the supercentered setting is the same as the basic space group setting, this section is omitted.
Modulation vectors: Q1=(A1,0,0), Q2=(0,B1,0), Q3=(0,0,G2), where A1=a1, B1=b1, G2=g2
The supercentered setting not only eliminates rational q-vector components, it often simplifies the forms of the q vectors. In this example, multi-component modulation vectors like (a1,b1,0) and (-a1,b1,0) are separated into single-component vectors. When the modulation vectors in the supercentered setting are defined relative to the original modulations vectors in the basic space-group setting, we use uppercase letters for the supercentered setting and lowercase letters for the basic space-group setting.
Centering: (0,0,0,0,0,0), (1/2,1/2,1/2,0,0,0), (0,0,0,1/2,1/2,0), (1/2,1/2,1/2,1/2,1/2,0)
Supercentering operations generally add additional centering translations that include components in the internal space. We list the entire group of centering translations.
Non-lattice generators: (X,-Y,-Z,T,-U,-V); (X,-Y,Z,T,-U,V+1/2);
(X,Y,-Z,T,U,-V+1/2)
Non-lattice operators: (X,Y,Z,T,U,V); (X,-Y,-Z,T,-U,-V);
(X,-Y,Z,T,-U,V+1/2); (X,Y,-Z,T,U,-V+1/2)
The generators and operators from the basic space-group setting are transformed into the supercentered setting and presented in exactly the same order. We use uppercase coordinates (X,Y,Z,T,U,V) for the supercentered setting and lowercase coordinates (x,y,z,t,u,v) for the basic space-group setting. Note that the internal intrinsic translations denoted in the superspace-group symbol are calculated in the supercentered setting.
Reflection conditions: HKLMNP:M+N=2n; HKLMNP:H+K+L=2n; H0LM0P:P=2n
The list of reflection conditions includes both the centering translations of the Bravais class and any non-lattice translations possessed by non-symmorphic groups. The list of conditions is always minimal, which means that any other reflections conditions that can be defined will not add any extinctions to those already created by the conditions in the list. When a supercentered setting is available, we only express the reflection conditions in the supercentered setting, and use uppercase letters (H,K,L,M,N,P) in the conditions. When no supercentered setting is available, we express the reflection conditions in the basic space-group setting, and use lowercase letters (h,k,l,m,n,p) in the conditions. We use "p" instead of "o" in the reflection conditions because "o" and "O" appear very similar to zero.
Due to their dependence on the intrinsic translational components of the generators, space-group symbols can be ambiguous when multiple groups can have the same intrinsic translations. This ambiguity arises only in centered groups, where one can combine a non-lattice generator with a centering translation to produce an equivalent generator that may have different intrinsic translational components. As a classic example, there are eight ways to combine the (1/2,1/2,1/2) centering translation with the three two-fold rotation generators of space group #23 (I222), resulting in eight different sets of intrinsic translations and eight possible symbols for the group: I222, I2221, I2212, I22121, I2122, I21221, I21212, I212121. Surprisingly, the same procedure applied to space group #24 (I212121) produces the same eight symbols. Thus, the traditional choice of I222 as the symbol for space group #23 and I212121 as the symbol for space group #24 is not grounded purely in the intrinsic translations, but requires other considerations. The only other 3D space groups subject to this problem are I23 and I213. For superspace groups, which have a large number of rather complex centering types, this problem is greatly compounded. While the present system of symbols is still based on intrinsic translations, we have endeavored to define conventions that ensure that distinct groups are given distinct symbols, and that the symbol assigned to each group is as simple as possible. We describe these conventions below.
Generators of basic space group
The primary restriction that we place on the lattice translations to be combined with any given generator are that the resulting intrinsic translation must strictly match the symbol of the basic space group in ITC-A. In cases where ITC-A gives more than one setting for a space group, we use the following setting conventions: (1) monoclinic: cell choice 1, (2) trigonal: hexagonal axes, (3) origin choice 2.
The symbol for the basic space group contains information about the generating symmetry operators. For example, Pmc21 has three generators: (1) m=mirror reflection through (100), (2) c=glide reflection through (010) with an intrinsic translation along [001], and (3) 21=screw rotation along [001] with an intrinsic translation also along [001]. The part of the space group symbol corresponding to a given generator does not uniquely identify a symmetry operator of the space group, but rather identifies a class of operators (e.g. all of the c-glide planes perpendicular to the b axis). Sometimes, the symbol of the point-group generator doesn't uniquely specify the orientation of the operator (e.g. the first "2" in P422 could represent a rotation around either the a or b axis). Because the intrinsic translation of the generator is often influenced by its orientation, we list the point groups with generator ambiguities below, together with the generators that we have chosen for them.
| 422 | (-y,x,z) (x,-y,-z) (-y,-x,-z) |
| 4mm | (-y,x,z) (-x,y,z) (y,x,z) |
| -42m | (y,-x,-z) (x,-y,-z) (y,x,z) |
| -4m2 | (y,-x,-z) (-x,y,z) (-y,-x,-z) |
| 4/mmm | (-y,x,z) (x,y,-z) (-x,y,z) (y,x,z) |
| 312 | (-y,x-y,z) (x,y,z) (-y,-x,-z) |
| 321 | (-y,x-y,z) (-x,-x+y,-z) (x,y,z) |
| 3m1 | (-y,x-y,z) (x,x-y,z) (x,y,z) |
| 31m | (-y,x-y,z) (x,y,z) (y,x,z) |
| 622 | (x-y,x,z) (-x,-x+y,-z) (-y,-x,-z) |
| 6mm | (x-y,x,z) (x,x-y,z) (y,x,z) |
| -6m2 | (-x+y,-x,-z) (x,x-y,z) (-y,-x,-z) |
| -62m | (-x+y,-x,-z) (-x,-x+y,-z) (y,x,z) |
| 6/mmm | (x-y,x,z) (x,y,-z) (x,x-y,z) (y,x,z) |
| 23 | (-x,-y,z) (z,x,y) |
| m-3 | (x,y,-z) (-z,-x,-y) |
| 432 | (-y,x,z) (z,x,y) (-y,-x,-z) |
| -43m | (y,-x,-z) (z,x,y) (y,x,z) |
| m-3m | (x,y,-z) (-z,-x,-y) (y,x,z) |
The translational components in each generating operator are generally chosen so that they are positive and less than 1. However, we often find it necessary to use equivalent translation components (related by a lattice translation) outside this range in order to ensure that the generators operate in a way that exactly matches the space group symbol. The ITC-A does not generally concern itself with this level of detail. For example, space group 100 P4bm has both mirror and glide planes parallel to the [110] directions. In the symbol, the third generator is a mirror reflection, though the symmetry operator listed in ITC-A is (y+1/2,x+1/2,z), which has an intrinsic translation of (1/2,1/2,0) and is therefore actually a glide reflection. Because the operator (y+1/2,x-1/2,z) has an intrinsic transition of (0,0,0), and is therefore a true mirror reflection, we use it instead. A similar situation arises in space groups with centered lattices. For example, the third generator in 67 Cmma is a glide reflection (x+1/2,y,-z). In ITC-A, this operator is listed as (x,y+1/2,-z) which would be the third generator for Cmmb (an alternate symbol for Cmma). In order to strictly represent the symbol Cmma, we use (x+1/2,y,-z) as the third generator rather than (x,y+1/2,-z).
Transformation to supercentered setting
We specify the transformation of the superspace group from the basic space group setting to the supercentered setting for each Bravais class. We use this same transformation for every superspace group that belongs to the same Bravais class.
Choice of internal intrinsic translations
We use Eq. (9.8.3.5) in ITC-C to calculate the origin-invariant part of the translation associated with each operator in the superspace group. We call this the intrinsic translation. If we denote an operator by {R|v}, then, using a simplified notation, the intrinsic translation vi for this operator is given by
vi = (1/n) * sum(m=1,n) (Rm)*v,
where n is the order of R, i.e., Rn=1.
The intrinsic translations are not unique since adding a lattice translation to v can change vi. The symbol for the superspace group is also not unique, since the nonzero internal components of the intrinsic translations of the generators, denoted by the symbols (0,s,t,q,h), become part of the symbol itself.
For each group, we generate every possible symbol for the superspace group and then choose the one with the "nicest" appearance. We generate candidate symbols by adding a variety of different lattice translations (both centering translations and conventional lattice translations) to the translational part v of each generator. In order to strictly respect the basic space group symbol, we only explore lattice translations which preserve the external parts (mod 1) of the intrinsic translations vi in the basic space group setting. While the set of lattice translations is infinite, the set that needs to be tested is limited by the fact that adding the cyclic order n of R to any component of v does not affect vi. We choose the "nicest"-looking symbol by applying the following criteria to the internal translational components, in order of priority:
(1) minimum number of negative components
(2) maximum number of zero components
(3) minimum value of the maximum denominator among the components
(4) smallest denominators occur first
(5) smallest numerators occur first
Sometimes, a number of superspace groups will all share the same candidate symbols, and therefore, the same “nicest” symbol. We say that these superspace groups are “degenerate” with respect to their symbols. We find that we can almost always lift this degeneracy by assigning the nicest symbol to the first group in the degenerate set, and then simply requiring that the external translational components of corresponding generators be exactly identical (not just equivalent mod 1) for each of the groups in the degenerate set.
Example 1: 48.2.51.12 Pnnn(1/2,b,g)q0q(1/2,-b,g)qq0. The generators in the supercentered setting are (-X,Y+1/2,Z+1/2,T+1/4,U+1/4), (X+1/4,-Y,Z+1/2,-T,U+1/4), and (X+1/4,Y+1/2,-Z,T+1/4,-U). There are four centering translations for this Bravais class: (0,0,0,0,0), (1/2,0,0,0,1/2), (0,0,0,1/2,1/2), (1/2,0,0,1/2,0). We find the following possible internal intrinsic translations for each generator:
| Original generator | Lattice translation | New generator | IIT | IIT Symbols |
| (-X,Y+1/2,Z+1/2,T+1/4,U+1/4) | (0,0,0,0,0) | (-X,Y+1/2,Z+1/2,T+1/4,U+1/4) | 1/4,1/4 | q,q |
| (1/2,0,0,0,-1/2) | (-X+1/2,Y+1/2,Z+1/2,T+1/4,U-1/4) | 1/4,-1/4 | q,-q | |
| (0,0,0,-1/2,-1/2) | (-X,Y+1/2,Z+1/2,T-1/4,U-1/4) | -1/4,-1/4 | -q,-q | |
| (1/2,0,0,-1/2,0) | (-X+1/2,Y+1/2,Z+1/2,T-1/4,U+1/4) | -1/4,1/4 | -q,q | |
| (X+1/4,-Y,Z+1/2,-T,U+1/4) | (0,0,0,0,0) | (X+1/4,-Y,Z+1/2,-T,U+1/4) | 0,1/4 | 0,q |
| (-1/2,0,0,0,-1/2) | (X-1/4,-Y,Z+1/2,-T,U-1/4) | 0,-1/4 | 0,-q | |
| (0,0,0,1/2,-1/2) | (X+1/4,-Y,Z+1/2,-T+1/2,U-1/4) | 0,-1/4 | 0,-q | |
| (-1/2,0,0,1/2,0) | (X-1/4,-Y,Z+1/2,-T+1/2,U+1/4) | 0,1/4 | 0,q | |
| (X+1/4,Y+1/2,-Z,T+1/4,-U) | (0,0,0,0,0) | (X+1/4,Y+1/2,-Z,T+1/4,-U) | 1/4,0 | q,0 |
| (-1/2,0,0,0,1/2) | (X-1/4,Y+1/2,-Z,T+1/4,-U+1/2) | 1/4,0 | q,0 | |
| (0,0,0,-1/2,1/2) | (X+1/4,Y+1/2,-Z,T-1/4,-U+1/2) | -1/4,0 | -q,0 | |
| (-1/2,0,0,-1/2,0) | (X-1/4,Y+1/2,-Z,T-1/4,-U) | -1/4,0 | -q,0 |
Based on four symbol sets for the first generator and two unique symbol sets for the second and third generators, we have sixteen possible symbols for this superspace group. Of these, the nicest looking one is the one without any minus signs in the symbols.
Example 2: 100.2.68.12 P4bm(a,a,0)00s(-a,a,0)000. In this case, though there are no centering translations (thus no distinction between the basic space-group setting and the supercentered setting), we can still obtain different symbols by using integer lattice translations with the generators.
| Original generator | Lattice translation | New generator | Intrinsic translation | IIT Symbols |
| (-y,x,z,-u,t) | (0,0,0,0,0) | (-y,x,z,-u,t) | (0,0,0,0,0) | 0,0 |
| (-x+1/2,y+1/2,z,u+1/2,t+1/2) | (0,0,0,0,0) | (-x+1/2,y+1/2,z,u+1/2,t+1/2) | (0,1/2,0,1/2,1/2) | s,s |
| (0,0,0,0,-1) | (-x+1/2,y+1/2,z,u+1/2,t-1/2) | (0,1/2,0,0,0) | 0,0 | |
| (y+1/2,x-1/2,z,t+1/2,-u+1/2) | (0,0,0,0,0) | (y+1/2,x-1/2,z,t+1/2,-u+1/2) | (0,0,0,1/2,0) | s,0 |
In this case, there are two possible symbols. We choose the symbol with the greatest number of zeros.
Example 3: 47.2.36.60 Pmmm(1/2,b1,1/2)000(1/2,0,g2)000 is the first member of a set of groups with the same nicest symbol ("degenerate set"). For brevity's sake, let us just consider the first generator: (-x,y,z,-x+t,-x+u) in the basic space group setting and (-X,Y,Z,T,U) in the supercentered setting with each of three centering translations: (1/2,0,0,1/2,1/2), (0,0,1/2,1/2,0), and (1/2,0,1/2,0,1/2).
| Lattice translation (supercentered setting) | New generator (supercentered setting) |
New generator (basic space group setting) | Intrinsic translation (supercentered setting) | IIT Symbols |
| (0,0,0,0,0,0) | (-X,Y,Z,T,U) | (-x,y,z,-x+t,-x+u) | (0,0,0,0,0) | 0,0 |
| (1/2,0,0,1/2,1/2) | (-X+1/2,Y,Z,T+1/2,U+1/2) | (-x+1,-y,z,-x+t+1,-x+u+1) | (0,0,0,1/2,1/2) | s,s |
| (0,0,1/2,1/2,0) | (-X,Y,Z+1/2,T+1/2,U) | (-x,y,z+1,-x+t+1,-x+u) | (0,0,1/2,1/2,0) | s,0 |
| (1/2,0,1/2,0,1/2) | (-X+1/2,Y,Z+1/2,T,U+1/2) | (-x+1,y,z+1,-x+t+1,-x+u+1) | (0,0,1/2,0,1/2) | 0,s |
Note that although the last two centering translations change the external part of the intrinsic translational components in the supercentered setting, they do not affect the external part (mod 1) of the intrinsic translational components in the basic space group setting. So we are allowed to consider them because they still agree with the basic space group symbol. Of course, we have chosen the symbol with the greatest number of zeros. Once we choose the symbol for 47.2.36.60, the symbols for each of the other three groups in the degenerate set are also determined since (0,0,0,0,0,0) is the only lattice translation that won't alter the external translational components of the generators. The four superspace groups in this degenerate set are:
47.2.36.60 Pmmm(1/2,b1,1/2)000(1/2,0,g2)000
47.2.36.61 Pmmm(1/2,b1,1/2)000(1/2,0,g2)0s0
47.2.36.62 Pmmm(1/2,b1,1/2)000(1/2,0,g2)s00
47.2.36.63 Pmmm(1/2,b1,1/2)000(1/2,0,g2)ss0
Example 4: 16.3.137.108 P222(0,b,g)000(a,0,g)000(a,b,0)000 and 16.3.137.109 P222(0,b,g)000(a,0,g)000(a,b,0)00s. There is one centering translation (0,0,0,1/2,1/2,1/2) associated with this Bravais class. We find the following possible internal intrinsic translations for each generator:
| Original generator 16.3.137.108 | Lattice translation | New generator | Intrinsic translation | IIT Symbols |
| (X,-Y,-Z,T,-U,-V) | (0,0,0,0,0,0) | (X,-Y,-Z,T,-U,-V) | (0,0,0,0,0,0) | 0,0,0 |
| (0,0,0,1/2,1/2,1/2) | (X,-Y,-Z,T+1/2,-U+1/2,-V+1/2) | (0,0,0,1/2,0,0) | s,0,0 | |
| (-X,Y,-Z,-T,U,-V) | (0,0,0,0,0,0) | (-X,Y,-Z,-T,U,-V) | (0,0,0,0,0,0) | 0,0,0 |
| (0,0,0,1/2,1/2,1/2) | (-X,Y,-Z,-T+1/2,U+1/2,-V+1/2) | (0,0,0,0,1/2,0) | 0,s,0 | |
| (-X,-Y,Z,-T,-U,V) | (0,0,0,0,0,0) | (-X,-Y,Z,-T,-U,V) | (0,0,0,0,0,0) | 0,0,0 |
| (0,0,0,1/2,1/2,1/2) | (-X,-Y,Z,-T+1/2,-U+1/2,V+1/2) | (0,0,0,0,0,1/2) | 0,0,s |
| Original generator 16.3.137.109 | Lattice translation | New generator | Intrinsic translation | IIT Symbols |
| (X,-Y,-Z,T,-U+1/2,-V) | (0,0,0,0,0,0) | (X,-Y,-Z,T,-U+1/2,-V) | (0,0,0,0,0,0) | 0,0,0 |
| (0,0,0,1/2,-1/2,1/2) | (X,-Y,-Z,T+1/2,-U,-V+1/2) | (0,0,0,1/2,0,0) | s,0,0 | |
| (-X,Y,-Z,-T,U,-V+1/2) | (0,0,0,0,0,0) | (-X,Y,-Z,-T,U,-V+1/2) | (0,0,0,0,0,0) | 0,0,0 |
| (0,0,0,1/2,1/2,-1/2) | (-X,Y,-Z,-T+1/2,U+1/2,-V) | (0,0,0,0,1/2,0) | 0,s,0 | |
| (-X,-Y,Z,-T+1/2,-U,V) | (0,0,0,0,0,0) | (-X,-Y,Z,-T+1/2,-U,V) | (0,0,0,0,0,0) | 0,0,0 |
| (0,0,0,-1/2,1/2,1/2) | (-X,-Y,Z,-T,-U+1/2,V+1/2) | (0,0,0,0,0,1/2) | 0,0,s |
Observe that we obtain the same set of candidate symbols for each group. Thus, our rules of respecting the basic space-group symbol and of maintaining the same external translational components across an entire degenerate set of groups have not entirely prevented symbol collisions. But the collisions are very rare and are easily resolved by manually assigning appropriate generators. The symmorphic group 16.3.137.108 clearly has a greater claim on the symbol with all-zero translations. For 16.3.137.109, we simply choose the next-nicest symbol which includes one "s". All total, there are seven pairs of groups (listed below) that still have identical sets of candidate symbols. In each case, one of the groups is symmorphic and receives the symbol with no translational components, while the other group gets the next-nicest symbol available. It is interesting that the three-dimensional internal portion of each of these (3+3)D superspace groups is identical to one of I222, I212121, I23 or I213.
16.3.137.109 P222(0,b,g)000(a,0,g)000(a,b,0)00s
21.3.142.106 C222(0,b,g)000(a,0,g)000(a,b,0)00s
22.3.144.38 F222(0,b,g)000(a,0,g)000(a,b,0)00s
23.3.141.16 I222(0,b,g)000(a,0,g)000(a,b,0)00s
195.3.210.8 P23(0,b,b)00(b,0,b)00(b,b,0)s0
196.3.212.6 F23(0,b,b)00(b,0,b)00(b,b,0)s0
197.3.211.4 I23(0,b,b)00(b,0,b)00(b,b,0)s0
Symbols in ITC-C
Symbols for the (3+1)-D superspace groups have been listed in ITC-C. There are only eleven cases where the symbols that follow from our conventions disagree with the ITC-C symbols. In eight of these cases, our method found nicer symbols:
| Group | ITC-C Symbol | Symbol Using Our Method |
| 35.1.14.5 | Cmm2(1,0,g)s0s | Cmm2(1,0,g)s00 |
| 36.1.14.4 | Cmc2_1(1,0,g)s0s | Cmc2_1(1,0,g)s00 |
| 37.1.14.4 | Ccc2(1,0,g)s0s | Ccc2(1,0,g)s00 |
| 42.1.18.5 | Fmm2(1,0,g)s0s | Fmm2(1,0,g)s00 |
| 99.1.20.6 | P4mm(1/2,1/2,g)0ss | P4mm(1/2,1/2,g)00s |
| 101.1.20.4 | P4_2cm(1/2,1/2,g)0ss | P4_2cm(1/2,1/2,g)00s |
| 123.1.20.6 | P4/mmm(1/2,1/2,g)00ss | P4/mmm(1/2,1/2,g)000s |
| 132.1.20.4 | P4_2/mcm(1/2,1/2,g)00ss | P4_2/mcm(1/2,1/2,g)000s |
In three of these cases, our method could not obtain the ITC-C symbol without using external intrinsic translations that were inconsistent with the basic space group symbol.
| Group | ITC-C Symbol | Symbol Using Our Method |
| 104.1.20.3 | P4nc(1/2,1/2,g)qq0 | P4nc(1/2,1/2,g)qqs |
| 106.1.20.3 | P42bc(1/2,1/2,g)qq0 | P42bc(1/2,1/2,g)qqs |
| 126.1.20.3 | P4/nnc(1/2,1/2,g)q0q0 | P4/nnc(1/2,1/2,g)q0qs |
We do not advocate changing the symbols established in ITC-C. Therefore, we propose to continue using the ITC-C symbols, and not the new symbols generated by our method.
Comparison with Yamamoto's tables of superspace groups
Comparision with Orlov and Chapuis's tables of superspace groups
Comparison with JJdW's tables of Bravais classes