International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 3.5, pp. 827-838

## Section 3.5.2. Euclidean and affine normalizers of plane groups and space groups

E. Koch,a W. Fischera and U. Müllerb

### 3.5.2. Euclidean and affine normalizers of plane groups and space groups

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#### 3.5.2.1. Euclidean normalizers of plane groups and space groups

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Since each symmetry operation of the Euclidean normalizer maps the space group onto itself, it also maps the set of all symmetry elements of onto itself. Therefore, the Euclidean normalizer of a space group can be interpreted as the group of motions that maps the pattern of symmetry elements of the space group onto itself, i.e. as the symmetry of the symmetry pattern'.

For most space (plane) groups, the Euclidean normalizers are space (plane) groups again. Exceptions are those groups where origins are not fully fixed by symmetry, i.e. all space groups of the geometrical crystal classes 1, m, 2, 2mm, 3, 3m, 4, 4mm, 6 and 6mm, and all plane groups of the geometrical crystal classes 1 and m. The Euclidean normalizer of each such group contains continuous translations (i.e. translations of infinitesimal length) in one, two or three independent lattice directions and, therefore, is not a space (plane) group but a supergroup of a space (plane) group.

If one regards a certain type of space (plane) group, usually the Euclidean normalizers of all corresponding groups belong also to only one type of normalizer. This is true for all cubic, hexagonal, trigonal and tetragonal space groups (hexagonal and square plane groups) and, in addition, for 21 types of orthorhombic space group (4 types of rectangular plane group), e.g. for Pnma.

In contrast to this, the Euclidean normalizer of a space (plane) group belonging to one of the other 38 orthorhombic (3 rectangular) types may interchange two or even three lattice directions if the corresponding basis vectors have equal length (example: Pmmm with a = b). Then, the Euclidean normalizer of this group belongs to the tetragonal (square) or even to the cubic crystal system, whereas another space (plane) group of the same type but with general metric has an orthorhombic (rectangular) Euclidean normalizer.

For each space (plane)-group type belonging to the monoclinic (oblique) or triclinic system, there also exist groups with specialized metric that have Euclidean normalizers of higher symmetry than for the general case (cf. Koch & Müller, 1990). The description of these special cases, however, is by far more complicated than for the orthorhombic system.

The symmetry of the Euclidean normalizer of a monoclinic (oblique) space (plane) group depends only on two metrical parameters. A clear presentation of all cases with specialized metric may be achieved by choosing the cosine of the monoclinic angle and the related axial ratio as parameters. To cover all different metrical situations exactly once, not all pairs of parameter values are allowed for a given type of space (plane) group, but one has to restrict the study to a certain parameter range depending on the type, the setting and the cell choice of the space (plane) group. Parthé & Gelato (1985) have discussed in detail such parameter regions for the first setting of the monoclinic space groups. Figs. 3.5.2.1 to 3.5.2.4 are based on these studies.

 Figure 3.5.2.1 | top | pdf |Parameter range for space groups of types and (plane groups of types p1 and p2). The information in parentheses refers to unique axis c.
 Figure 3.5.2.2 | top | pdf |Parameter range for space groups of types C2, Pc, Cm, Cc, , , and . They refer to the following settings: unique axis b, cell choice 2: P1n1, , ; unique axis b, cell choice 3: I121, I1m1, I1a1, , ; unique axis c, cell choice 2: P11n, , ; unique axis c, cell choice 3: I112, I11m, I11b, , . The information in parentheses refers to unique axis c.
 Figure 3.5.2.3 | top | pdf |Parameter range for space groups of types C2, Pc, Cm, Cc, , , and : unique axis b, cell choice 1: P1c1, , ; unique axis b, cell choice 2: A121, A1m1, A1n1, , ; unique axis c, cell choice 1: P11a, , ; unique axis c, cell choice 2: B112, B11m, B11n, , . The information in parentheses refers to unique axis c.
 Figure 3.5.2.4 | top | pdf |Parameter range for space groups of types C2, Pc, Cm, Cc, , , and : unique axis b, cell choice 1: C121, C1m1, C1c1, , ; unique axis b, cell choice 3: , , , ; unique axis c, cell choice 1: A112, A11m, A11a, , ; unique axis c, cell choice 3: P11b, , , . The information in parentheses refers to unique axis c.

Fig. 3.5.2.1 shows a suitably chosen parameter region for the five space-group types P2, , Pm, and and for the plane-group types p1 and p2. Each such space (plane) group with general metric may be uniquely assigned to an inner point of this region and any metrical specialization corresponds either to one of the three boundary lines or to one of their points of intersection and gives rise to a symmetry enhancement of the respective Euclidean normalizer.

For each of the other eight types of monoclinic space groups, i.e. C2, Pc, Cm, Cc, , , and , and for each setting three possibilities of cell choice are listed in Chapter 2.3 , which can be distinguished by different space-group symbols (example: , , , , , ). For each setting, there exist two ways to choose a suitable range for the metrical parameters such that each group corresponds to exactly one point:

 (i) One arbitrarily restricts oneself to cell choice 1, 2 or 3. Then, the suitable parameter range (displayed in one of the Figs. 3.5.2.2, 3.5.2.3 or 3.5.2.4) is larger than the range shown in Fig. 3.5.2.1 because, in contrast to the space-group types discussed above, some of the possible metrical specializations do not give rise to any symmetry enhancement of the Euclidean normalizers. These special metrical cases refer to the light lines subdividing the parameter regions of Figs. 3.5.2.2 to 3.5.2.4. Again, all inner points of these regions correspond to space groups with Euclidean normalizers without enhanced symmetry, and all points on the heavy-line boundaries refer to space groups, the Euclidean normalizers of which show symmetry enhancement. (ii) For all types of monoclinic space groups, one regards only the small parameter region shown in Fig. 3.5.2.1, but in return takes into consideration all three possibilities for the cell choice. Then, however, not all boundaries of this small parameter region correspond to Euclidean normalizers with enhanced symmetry. (Similar considerations are true for oblique plane groups.)

For triclinic space groups, five metrical parameters are necessary and, therefore, it is impossible to describe the special metrical cases in an analogous way.

In general, between a space group (or plane group) and its Euclidean normalizer , two uniquely defined intermediate groups and exist, such that holds. is that klassengleiche supergroup of that is at the same time a translationengleiche subgroup of . It is well defined according to the theorem of Hermann (1929). The group differs from only if is noncentrosymmetric but is centrosymmetric; then is that centrosymmetric supergroup of of index 2 that is again a subgroup of . It belongs to the Laue class of . If is noncentrosymmetric, an intermediate group cannot exist.

The chirality-preserving Euclidean normalizer of a Sohncke space group is the unique noncentrosymmetric sub­group of which is a supergroup of : If is centrosymmetric, is a subgroup of index 2 of . If is noncentrosymmetric, and are identical.

With the aid of its chirality-preserving Euclidean normalizer it is possible to determine all equivalent sets of coordinates of a chiral crystal structure, excluding the opposite enantiomorph (cf. Section 3.5.3.2).

The groups and are of special interest in connection with direct methods for structure determination: they cause the parity classes of reflections; defines the permissible origin shifts and the parameter ranges for the phase restrictions in the specification of the origin; and gives information on possible phase restrictions for the selection of the enantiomorph. For any space (plane) group , the translation subgroups of , , and even coincide.

The Euclidean normalizers of the plane groups are listed in Table 3.5.2.1, those of triclinic space groups in Table 3.5.2.2. The Euclidean and the chirality-preserving Euclidean normalizers of monoclinic and orthorhombic space groups are in Tables 3.5.2.3 and 3.5.2.4, those of all other space groups in Table 3.5.2.5. Herein all settings and choices of cell and origin as tabulated in Chapters 2.2 and 2.3 are taken into account and, in addition, all metrical specializations giving rise to Euclidean normalizers with enhanced symmetry. Each setting, cell choice, origin or metrical specialization corresponds to one line in the tables. (Exceptions are some orthorhombic space groups with tetragonal metric: if as well as and give rise to a symmetry enhancement of the Euclidean normalizer, only the case is listed in Table 3.5.2.4.)

 Table 3.5.2.1| top | pdf | Euclidean normalizers of the plane groups
 For the restrictions of the cell metric of the two oblique plane groups see text and Fig. 3.5.2.3.
Plane group Euclidean normalizer Additional generators of Index of in
No.Hermann–Mauguin symbolCell metricSymbolBasis vectorsTranslationsTwofold rotationFurther generators
1 p1 General r, 0; 0, s
r, 0; 0, s
, r, 0; 0, s
, r, 0; 0, s y, x
r, 0; 0, s
r, 0; 0, s
2 p2 General p2
p2mm
, c2mm
, c2mm
p4mm
p6mm
3 p1m1
4 p1g1
5 c1m1
6 p2mm p2mm
p4mm   y, x
7 p2mg   p2mm
8 p2gg p2mm
p4mm   y, x
9 c2mm p2mm
p4mm   y, x
10 p4   p4mm   y, x
11 p4mm   p4mm
12 p4gm   p4mm
13 p3   p6mm , y, x
14 p3m1   p6mm ,
15 p31m   p6mm
16 p6   p6mm     y, x
17 p6mm   p6mm
 Table 3.5.2.2| top | pdf | Euclidean normalizers of the triclinic space groups
 Basis vectors of the Euclidean normalizers ( refer to the possibly centred conventional unit cell for the respective Bravais lattice): ; .
Bravais typeEuclidean normalizer of
P1 (1) (2)
aP
mP
mA
oP Pmmm
oC Cmmm
oF Fmmm
oI Immm
tP
tI
hP
hR
cP
cF
cI
 Table 3.5.2.3| top | pdf | Euclidean and chirality-preserving Euclidean normalizers of the monoclinic space groups
 For the restrictions of the cell metric see text and Figs. 3.5.2.1 to 3.5.2.4. The symbols in parentheses following a space-group symbol refer to the location of the origin (origin choice' in Chapter 2.3 ).
Space group Euclidean normalizer and chirality-preserving normalizer Additional generators of and Index of in or
No.Hermann–Mauguin symbolCell metricSymbolBasis vectorsTranslationsInversion through a centre atFurther generators
3 General

3 General

4 General

4 General

5 General

5 General

5 General

5 General

5 General

5 General

6 General

6 General

7 General

7 General

7 General

7 General

7 General

7 General

8 General

8 General

8 General

8 General

8 General

8 General

9 General

9 General

9 General

9 General

9 General

9 General

10 General

10 General

11 General

11 General

12 General

12 General

12 General

12 General

12 General

12 General

13 General

13 General

13 General

13 General

13 General

13 General

14 General

14 General

14 General

14 General

14 General

14 General

15 General

15 General

15 General

15 General

15 General

15 General

 Table 3.5.2.4| top | pdf | Euclidean and chirality-preserving Euclidean normalizers of the orthorhombic space groups
 The symbols in parentheses following a space-group symbol refer to the location of the origin (origin choice' in Chapter 2.3 ).
Space group Euclidean normalizer and chirality-preserving normalizer Additional generators of and Index of in or
No.Hermann–Mauguin symbolCell metricSymbolBasis vectorsTranslationsInversion through a centre atFurther generators
16

17

18

19

20

21

22

23

24

25

26
27

28
29
30
31
32

33
34

35

36
37

38
39
40
41
42

43

44

45

46
47

48 (both
origins)

49

50 (both
origins)
51
52
53
54
55

56

57
58

59 (both
origins)
60
61 or or

62
63
64
65

66

67

68

68

69

70

70
( at )

71

72

73

74

 Table 3.5.2.5| top | pdf | Euclidean and chirality-preserving Euclidean normalizers of the tetragonal, trigonal, hexagonal and cubic space groups
 The symbols in parentheses following a space-group symbol refer to the location of the origin (origin choice' in Chapter 2.3 ).
Space group Euclidean normalizer and chirality-preserving normalizer Additional generators of and Index of in or
No.Hermann–Mauguin symbolSymbolBasis vectorsTranslationsInversion through a centre atFurther generators
75

76
77

78
79

80

81
82
83
84
85
85
86
86
87
88
88
89

90

91 (222 at )