International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 2.5, pp. 323-333   | 1 | 2 |

## Section 2.5.3.3.3. Space-group determination

M. Tanakaf

#### 2.5.3.3.3. Space-group determination

| top | pdf |

We now describe a space-group determination method which uses the dynamical extinction lines caused by the horizontal screw axis and the vertical glide plane g of an infinitely extended parallel-sided specimen. We do not use the extinction due to the glide plane g′ because observation of the extinction requires a laborious experiment. It should be noted that a vertical glide plane with a glide vector not parallel to the specimen surface cannot be a symmetry element of a specimen of finite thickness; however, the component of the glide vector perpendicular to the incident beam acts as a symmetry element g. (Which symmetry elements are observed by CBED is discussed in Section 2.5.3.3.5.) The 21, 41, 43, 61, 63 and 65 screw axes of crystal space groups that are set perpendicular to the incident beam act as a symmetry element because two or three successive operations of 41, 43, 61, 63 and 65 screw axes make them equivalent to a 21 screw axis: (41)2 = (43)2 = (61)3 = (63)3 = (65)3 = 21. The 42, 31, 32, 62 and 64 screw axes that are set perpendicular to the incident beam do not produce dynamical extinction lines because the 42 screw axis acts as a twofold rotation axis due to the relation (42)2 = 2, the 31 and 32 screw axes give no specific symmetry in CBED patterns, and the 62 and 64 screw axes are equivalent to 31 and 32 screw axes due to the relations (62)2 = 32 and (64)2 = 31. Modifications of the dynamical extinction rules were investigated by Tanaka, Sekii & Nagasawa (1983) when more than one crystal symmetry element (that gives rise to dynamical extinction lines) coexists and when the symmetry elements are combined with various lattice types. Using these results, dynamical extinction lines A2, A3, B2 and B3 expected from all the possible crystal settings for all the space groups were tabulated.

Table 2.5.3.9 shows all the dynamical extinction lines appearing in the kinematically forbidden reflections for all the possible crystal settings of all the space groups. The first column gives space groups. In each of the following pairs of columns, the left-hand column of the pair gives the reflection indices and the symmetry elements causing the extinction lines and the right-hand column gives the types of the extinction lines. The (second) suffixes 1, 2 and 3 of a 21 screw axis in each column distinguish the first, the second and the third screw axis of the space group (as in the symbols 211 and 212 of space group P21212). The glide symbols in the [001] column for space group P4/nnc have two suffixes (n21 and n22). The first suffix 2 denotes the second glide plane of the space group. The second suffixes 1 and 2, which appear in the tetragonal and cubic systems, distinguish two equivalent glide planes which lie in the x and y planes. The equivalent planes are distinguished only for the cases of [100], [010] and [001] electron-beam incidences, for convenience. The c-glide planes of space group Pcc2 are distinguished with symbols c1 and c2 (the first suffix only), because the equivalent planes do not exist. The glide symbol in the [001] column for space group P4/mbm has only one suffix 1 or 2. The suffix distinguishes the equivalent glide planes lying in the x and y planes. The first suffix to distinguish the first and the second glide planes is not necessary because the space group has only one glide symbol b. When the index of the incident-beam direction is expressed with a symbol like [h0l] for point groups 2, m and 2/m, the index h or l can take a value of zero. That is, the extinction rules are applicable to the [100] and [001] electron-beam incidences. However, if columns for [100], [010] and [001] incidences are present, as in the case of point group mm2, [hk0], [0kl] and [h0l] incidences are only for nonzero h, k and l. The reflections in which the extinction lines appear are always perpendicular to the corresponding incident-beam directions . The indices of the reflections in which extinction lines appear are odd if no remark is given. For c-glide planes of space groups R3c and and for d-glide planes, the reflections in which extinction lines appear are specified as 6n + 3 and 4n + 2 orders, respectively.

 Table 2.5.3.9| top | pdf | Dynamical extinction lines appearing in ZOLZ reflections for all crystal space groups except Nos. 1 and 2
 Point groups 2, m, 2/m (second setting, unique axis b)
Space groupIncident-beam direction
[h0l]
3 P2
4 P21 0k0 A2 B2
21   B3
5 C2
6 Pm
7 Pc h0lo A2 B2
c A3
8 Cm
9 Cc he0lo A2 B2
c A3
10 P2/m
11 P21/m 0k0 A2 B2
21   B3
12 C2/m
13 P2/c h0lo A2 B2
c A3
14 P21/c 0k0 A2 B2
21   B3
h0lo A2 B2
c A3
15 C2/c he0lo A2 B2
c A3
 Point group 222
Space groupIncident-beam direction
[100][010][001][hk0][0kl][h0l]
16 P222
17 P2221 00l A2 B2 00l A2 B2       00l A2 B2
21   B3 21   B3       21   B3
18 P21212 0k0 A2 B2 h00 A2 B2 h00 A2 B2       h00 A2 B2 0k0 A2 B2
212   B3 211   B3 211   B3       211   B3 212   B3
0k0
212
19 P212121 0k0 A2 B2 h00 A2 B2 h00 A2 B2 00l A2 B2 h00 A2 B2 0k0 A2 B2
212   B3 211   B3 211   B3 213   B3 211   B3 212   B3
00l     00l     0k0
213     213     212
20 C2221 00l A2 B2 00l A2 B2       00l A2 B2
21   B3 21   B3       21   B3
21 C222
22 F222
23 I222
24 I212121
 Point group mm2
Space groupIncident-beam direction
[100][010][001][hk0][0kl][h0l]
25 Pmm2
26 Pmc21 00l A2 B2 00l           00l A2 B2       h0lo A2 B2
c, 21 A3 B3 21   B3       21   B3       c A3
27 Pcc2 00l     00l                 0klo A2 B2 h0lo A2 B2
c2 A3   c1 A3               c1 A3   c2 A3
28 Pma2             h00 A2 B2             ho0l A2 B2
a A3               a A3
29 Pca21 00l     00l A2 B2 h00 A2 B2 00l A2 B2 0klo A2 B2 ho0l A2 B2
21   B3 c, 21 A3 B3 a A3   21   B3 c A3   a A3
30 Pnc2 00l
c

A3
00l
n

A3
0k0
n
A2
A3
B2       0kl: k + l = 2n + 1
n
A2
A3
B2 h0lo
c
A2
A3
B2
31 Pmn21 00l
n, 21
A2
A3
B2
B3
00l
21

B3
h00
n
A2
A3
B2 00l
21
A2 B2
B3
h0l: h + l = 2n + 1
n
A2
A3
B2
32 Pba2             h00 A2 B2       0kol A2 B2 ho0l A2 B2
a A3         b A3   a A3
0k0
b
33 Pna21 00l
21

B3
00l
n, 21
A2
A3
B2
B3
h00
a
0k0
n
A2
A3
B2 00l
21
A2 B2
B3
0kl: k + l = 2n + 1
n
A2
A3
B2 ho0l
a
A2
A3
B2
34 Pnn2 00l
n2

A3
00l
n1

A3
h00
n2
0k0
n1
A2
A3
B2       0kl: k + l = 2n + 1
n1
A2
A3
B2 h0l: h + l = 2n + 1
n2
A2
A3
B2
35
36 00l A2 B2 00l           00l A2 B2       he0lo A2 B2
c, 21 A3 B3 21   B3       21   B3       c A3
37 00l     00l                 0kelo A2 B2 he0lo A2 B2
c2 A3   c1 A3               c1 A3   c2 A3
38
39                         0kolo A2 B2
b A3
40             h00 A2 B2             ho0le A2 B2
a A3               a A3
41             h00 A2 B2       0kolo A2 B2 ho0le A2 B2
a A3         b A3   a A3
42 Fmm2
43 00l: l = 4n + 2
d2

A3
00l: l = 4n + 2
d1

A3
h00: h = 4n + 2
d2
0k0: k = 4n + 2
d1
A2
A3
B2       0kele: ke + le = 4n + 2
d1
A2
A3
B2 he0le: he + le = 4n + 2
d2
A2
A3
B2
44
45                         0kolo A2 B2 ho0lo A2 B2
b A3   a A3
46                               ho0lo A2 B2
a A3
 Point group mmm
Space groupIncident-beam direction
[100][010][001][hk0][0kl][h0l]
47 P2/m2/m2/m
48 P2/n2/n2/n 00l
n2
0k0
n3

A3
00l
n1
h00
n3

A3
0k0
n1
h00
n2

A3
hk0: h + k = 2n + 1
n3
A2
A3
B2 0kl: k + l = 2n + 1
n1
A2
A3
B2 h0l: h + l = 2n + 1
n2
A2
A3
B2
49 P2/c2/c2/m 00l     00l                 0klo A2 B2 h0lo A2 B2
c2 A3   c1 A3               c1 A3   c2 A3
50 P2/b2/a2/n 0k0
n

A3
h00
n

A3
0k0
b
h00
a

A3
hk0: h + k = 2n + 1
n
A2
A3
B2 0kol
b
A2
A3
B2 ho0l
a
A2
A3
B2
51 P21/m2/m2/a       h00 A2 B2 h00     hok0 A2 B2 h00 A2 B2
21, a A3 B3 21   B3 a A3   21   B3
52 P2/n21/n2/a 00l
n2

A3
00l
n1
h00
a

A3
0k0
n1, 21
A2
A3
B2
B3
hok0
a
A2
A3
B2 0kl: k + l = 2n + 1
n1
A2
A3
B2 h0l: h + l = 2n + 1
n2
A2
A3
B2
0k0           h00                 0k0 A2 B2
21   B3       n2 A3               21   B3
53 P2/m2/n21/a 00l
n, 21
A2
A3
B2
B3
h00
a

A3
h00
n

A3
hok0
a
A2
A3
B2       h0l: h + l = 2n + 1
n
A2
A3
B2
00l           00l A2 B2
21   B3       21   B3
54 P21/c2/c2/a 00l     00l     h00     hok0 A2 B2 0klo A2 B2 h0lo A2 B2
c2 A3   c1 A3   21   B3 a A3   c1 A3   c2 A3
h00 A2 B2             h00 A2 B2
a, 21 A3 B3             21   B3
55 P21/b21/a2/m 0k0     h00     0k0 A2 B2       0kol A2 B2 ho0l A2 B2
212   B3 211   B3 b, 212 A3 B3       b A3   a A3
h00           h00 A2 B2 0k0 A2 B2
a, 211           211   B3 212   B3
56 P21/c21/c2/n 00l
c2

A3
00l
c1

A3
0k0
212
h00
211

B3
hk0: h + k = 2n + 1
n
A2
A3
B2 0klo
c1
A2
A3
B2 h0lo
c2
A2
A3
B2
0k0 A2 B2 h00 A2 B2             h00 A2 B2 0k0 A2 B2
212, n A3 B3 211, n A3 B3             211   B3 212   B3
57 P2/b21/c21/m 00l A2 B2 00l     0k0 A2 B2 00l A2 B2 0kol A2 B2 h0lo A2 B2
c, 212 A3 B3 212   B3 b, 211 A3 B3 212   B3 b A3   c A3
0k0                             0k0 A2 B2
211   B3                         211   B3
58 P21/n21/n2/m 00l
n2

A3
00l
n1

A3
0k0
n1, 212
h00
n2, 211
A2
A3
B2
B3
0kl: k + l = 2n + 1
n1
A2
A3
B2 h0l: h + l = 2n + 1
n2
A2
A3
B2
0k0     h00                 h00 A2 B2 0k0 A2 B2
212   B3 211   B3             211   B3 212   B3
59 P21/m21/m2/n 0k0
n, 212
A2
A3
B2
B3
h00
n, 211
A2
A3
B2
B3
0k0
212
h00
211

B3
hk0: h + k = 2n + 1
n
A2
A3
B2 h00
211
A2 B2
B3
0k0
212
A2 B2
A3
60 P21/b2/c21/n 00l
c, 212
A2
A3
B2
B3
h00
n, 211
A2
A3
B2
B3
0k0
b

A3
hk0: h + k = 2n + 1
n
A2
A3
B2 0kol
b
A2
A3
B2 h0lo
c
A2
A3
B2
0k0     00l     h00     00l A2 B2 h00 A2 B2
n A3   212   B3 211   B3 212   B3 211   B3
61 P21/b21/c21/a 00l A2 B2 00l     0k0 A2 B2 hok0 A2 B2 0kol A2 B2 h0lo A2 B2
c, 213 A3 B3 213   B3 b, 212 A3 B3 a A3   b A3   c A3
0k0     h00 A2 B2 h00     00l A2 B2 h00 A2 B2 0k0 A2 B2
212   B3 a, 211 A3 B3 211   B3 213   B3 211   B3 212   B3
62 P21/n21/m21/a 00l
213
0k0
212

B3
00l
n, 213
h00
a, 211
A2
A3
B2
B3
0k0
n, 212
A2
A3
B2
B3
hok0
a
A2
A3
B2 0kl: k + l = 2n + 1
n
A2
A3
B2 0k0
212
A2 B2
B3
h00     00l A2 B2 h00 A2 B2
211   B3 213   B3 211   B3
63 C2/m2/c21/m 00l A2 B2 00l           00l A2 B2       he0lo A2 B2
c, 21 A3 B3 21   B3       21   B3       c A3
64 C2/m2/c21/a 00l A2 B2 00l           hoko0 A2 B2       he0lo A2 B2
c, 21 A3 B3 21   B3       a A3         c A3
00l A2 B2
21   B3
65 C2/m2/m2/m
66 C2/c2/c2/m 00l     00l                 0kelo A2 B2 he0lo A2 B2
c2 A3   c1 A3               c1 A3   c2 A3
67 C2/m2/m2/a                   hoko0 A2 B2
a A3
68 C2/c2/c2/a 00l     00l           hoko0 A2 B2 0kelo A2 B2 he0lo A2 B2
c2 A3   c1 A3         a A3   c1 A3   c2 A3
69 F2/m2/m2/m
70 F2/d2/d2/d 00l: l =
4n + 2
d2
0k0: k =
4n + 2
d3

A3
h00: h =
4n + 2
d3
00l: l =
4n + 2
d1

A3
0k0: k =
4n + 2
d1
h00: h =
4n + 2
d2

A3
heke0: he + ke = 4n + 2
d3
A2
A3
B2 0kele: ke + le = 4n + 2
d1
A2
A3
B2 he0le: he + le = 4n + 2
d2
A2
A3
B2
71 I2/m2/m2/m
72 I2/b2/a2/m                         0kolo A2 B2 ho0lo A2 B2
b A3   a A3
73 I21/b21/c21/a                   hoko0 A2 B2 0kolo A2 B2 ho0lo A2 B2
a A3   b A3   c A3
74 I21/m21/m21/a                   hoko0 A2 B2
a A3
 Point groups
Space groupIncident-beam direction
[hk0]
75 P4
76 P41 00l A2 B2
41   B3
77 P42
78 P43 00l A2 B2
43   B3
79 I4
80 I41
81
82
83 P4/m
84 P42/m
85 P4/n hk0: h + k = 2n + 1 A2 B2
n A3
86 P42/n hk0: h + k = 2n + 1 A2 B2
n A3
87 I4/m
88 I41/a hoko0 A2 B2
a A3
 Point group 422
Space groupIncident-beam direction
[hk0][0kl]
89 P422
90 P4212       h00 A2 B2
21   B3
91 P4122 00l A2 B2
41   B3
92 P41212 00l A2 B2 h00 A2 B2
41   B3 21   B3
93 P4222
94 P42212       h00 A2 B2
21   B3
95 P4322 00l A2 B2
43   B3
96 P43212 00l A2 B2 h00 A2 B2
43   B3 21   B3
97 I422
98 I4122
 Point group 4mm. The symbol a in the column [h0l] is equivalent to the symbol b in the space groups of the first column.
Space groupIncident-beam direction
[100][001][110][h0l][hhl]
99 P4mm
100 P4bm     h00 A2 B2     ho0l A2 B2
a2 A3         a A3
0k0
b1
101 P42cm 00l                 h0lo A2 B2
c2 A3               c A3
102 P42nm 00l     h00 A2 B2       h0l: h + l = 2n + 1 A2 B2
n2 A3   n2 A3         n A3
0k0
n1
103 P4cc 00l           00l     h0lo A2 B2 hhlo A2 B2
c12 A3         c2 A3   c1 A3   c2 A3
104 P4nc 00l     h00 A2 B2 00l     h0l: h + l = 2n + 1 A2 B2 hhlo A2 B2
n2 A3   n2 A3   c A3   n A3   c A3
0k0
n1
105 P42mc             00l           hhlo A2 B2
c A3         c A3
106 P42bc       h00 A2 B2 00l     ho0l A2 B2 hhlo A2 B2
a2 A3   c A3   a A3   c A3
0k0
b1
107 I4mm
108 I4cm                   ho0lo A2 B2
c A3
109 I41md       hh0, A2 B2 00l: l = 4n + 2           hhle: 2h + le = 4n + 2 A2 B2
d A3   d A3         d A3
110 I41cd       hh0, A2 B2 00l: l = 4n + 2     ho0lo A2 B2 hhle: 2h + le = 4n + 2 A2 B2
d A3   d A3   c A3   d A3
 Point group . The symbol a in the column [h0l] is equivalent to the symbol b in the space groups of the first column.
Space groupIncident-beam direction
[100][001][110][h0l][hhl]
111
112             00l         hhlo A2 B2
c A3         c A3
113 0k0 A2 B2 h00 A2 B2       0k0 A2 B2
212   B3 211   B3       21   B3
0k0
212
114 0k0 A2 B2 h00 A2 B2 00l     0k0 A2 B2 hhlo A2 B2
212   B3 211   B3 c A3   21   B3 c A3
0k0
212
115
116 00l                 h0lo A2 B2
c2 A3               c A3
117       h00 A2 B2       ho0l A2 B2
a2 A3         a A3
0k0
b1
118 00l     h00 A2 B2       h0l: h + l = 2n + 1 A2 B2
n2 A3   n2 A3         n A3
0k0
n1
119
120                   ho0lo A2 B2
c A3
121
122       hh0, A2 B2 00l: l = 4n + 2           hhle: 2h + le = 4n + 2 A2 B2
d A3   d A3         d A3
 Point group 4/mmm. The symbol a in the column [h0l] is equivalent to the symbol b in the space groups of the first column.
Space groupIncident-beam direction
[100][001][110][h0l][hhl][hk0]
123 P4/mmm
P4/m2/m2/m
124 P4/mcc 00l           00l     h0lo A2 B2 hhlo A2 B2
P4/m2/c2/c c12 A3         c2 A3   c1 A3   c2 A3
125 P4/nbm
P4/n2/b2/m
0k0
n

A3
h00
a2
0k0
b1

A3
ho0l
a
A2
A3
B2       hk0: h + k = 2n + 1
n
A2
A3
B2
126 P4/nnc
P4/n2/n2/c
0k0
n1
00l
n22

A3
h00
n22
0k0
n21

A3
00l
c

A3
h0l: h + l = 2n + 1
n2
A2
A3
B2 hhlo
c
A2
A3
B2 hk0: h + k = 2n + 1
n1
A2
A3
B2
127 P4/mbm 0k0     h00 A2 B2       ho0l A2 B2
P4/m21/b2/m 212   B3 a2, 211 A3 B3       a A3
0k0           0k0 A2 B2
b1, 212           21   B3
128 P4/mnc
P4/m21/n2/c
00l
n2

A3
h00
n2, 211
0k0
n1, 212
A2
A3
B2
B3
00l
c

A3
h0l: h + l = 2n + 1
n
A2
A3
B2 hhlo
c
A2
A3
B2
0k0                 0k0 A2 B2
212   B3             21   B3
129 P4/nmm
P4/n21/m2/m
0k0
n, 212
A2
A3
B2
B3
h00
211
0k0
212

B3
0k0
21
A2 B2
B3
hk0: h + k = 2n + 1
n
A2
A3
B2
130 P4/ncc
P4/n21/c2/c
0k0
n, 212
A2
A3
B2
B3
h00
211
0k0
212

B3
00l
c2

A3
h0lo
c1
A2
A3
B2 hhlo
c2
A2
A3
B2 hk0: h + k = 2n + 1
n
A2
A3
B2
00l                 0k0 A2 B2
c12 A3               21   B3
131 P42/mmc             00l           hhlo A2 B2
P42/m2/m2/c             c A3         c A3
132 P42/mcm 00l                 h0lo A2 B2
P42/m2/c2/m c2 A3               c A3
133 P42/nbc
P42/n2/b2/c
0k0
n

A3
h00
a2
0k0
b1

A3
00l
c

A3
ho0l
a
A2
A3
B2 hhlo
c
A2
A3
B2 hk0: h + k = 2n + 1
n
A2
A3
B2
134 P42/nnm
P42/n2/n2/m
0k0
n1
00l
n22

A3
h00
n22
0k0
n21

A3
h0l: h + l = 2n + 1
n2
A2
A3
B2       hk0: h + k = 2n + 1
n1
A2
A3
B2
135 P42/mbc 0k0     h00 A2 B2 00l     ho0l A2 B2 hhlo A2 B2
P42/m21/b2/c 212   B3 a2, 211 A3 B3 c A3   a A3   c A3
0k0           0k0 A2 B2
b1, 212           21   B3
136 P42/mnm
P42/m21/n2/m
00l
n2

A3
h00
n2, 211
0k0
n1, 212
A2
A3
B2
B3
h0l: h + l = 2n + 1
n
A2
A3
B2
0k0                 0k0 A2 B2
212   B3             21   B3
137 P42/nmc
P42/n21/m2/c
0k0
n, 212
A2
A3
B2
B3
h00
211
0k0
212

B3
00l
c

A3
0k0
21
A2 B2
B3
hhlo
c
A2
A3
B2 hk0: h + k = 2n + 1
n
A2
A3
B2
138 P42/ncm
P42/n21/c2/m
0k0
n, 212
A2
A3
B2
B3
h00
211
0k0
212

B3
h0lo
c
A2
A3
B2       hk0: h + k = 2n + 1
n
A2
A3
B2
00l                 0k0 A2 B2
c2 A3               21   B3
139 I4/mmm
I4/m2/m2/m
140 I4/mcm                   ho0lo A2 B2
I4/m2/c2/m                   c A3
141 I41/amd
I41/a2/m2/d
hh0,
d

A3
00l: l = 4n + 2
d

A3
hhle: 2h + le = 4n + 2
d
A2
A3
B2 hok0
a
A2
A3
B2

a
142 I41/acd
I41/a2/c2/d
hh0,
d

A3
00l: l = 4n + 2
d

A3
ho0lo
c
A2
A3
B2 hhle: 2h + le = 4n + 2
d
A2
A3
B2 hok0
a
A2
A3
B2

a
 Point groups
Space groupIncident-beam direction
Nos. 143–155: no GM line
156 P3m1
157 P31m
158 P3c1       00l A2 B2
c A3
159 P31c 00l A2 B2
c A3
160 R3m
161 R3c       00l: l = 6n + 3 A2 B2
c A3
162
163 00l A2 B2
c A3
164
165       00l A2 B2
c A3
166
167       00l: l = 6n + 3 A2 B2
c A3
 Point groups
Space groupIncident-beam direction
168 P6
169 P61 00l A2 B2 00l A2 B2
61   B3 61   B3
170 P65 00l A2 B2 00l A2 B2
65   B3 65   B3
171 P62
172 P64
173 P63 00l A2 B2 00l A2 B2
63   B3 63   B3
174
175 P6/m
176 P63/m 00l A2 B2 00l A2 B2
63   B3 63   B3
177 P622
178 P6122 00l A2 B2 00l A2 B2
61   B3 61   B3
179 P6522 00l A2 B2 00l A2 B2
65   B3 65   B3
180 P6222
181 P6422
182 P6322 00l A2 B2 00l A2 B2
63   B3 63   B3
183 P6mm
184 P6cc 00l     00l
c2 A3   c1 A3
185 P63cm 00l     00l A2 B2
63   B3 63, c A3 B3
186 P63mc 00l A2 B2 00l
63, c A3 B3 63   B3
187
188       00l A2 B2
c A3
189
190 00l A2 B2
c A3
191 P6/mmm
192 P6/mcc 00l     00l
c2 A3   c1 A3
193 P63/mcm 00l     00l A2 B2
63   B3 63, c A3 B3
194 P63/mmc 00l A2 B2 00l
63, c A3 B3 63   B3
 Point groups 23, m3
Space groupIncident-beam direction
[100] (cyclic)[110] (cyclic)[hk0] (cyclic)
195 P23
196 F23
197 I23
198 P213 00l A2 B2 00l A2 B2 00l A2 B2
213   B3 213   B3 21   B3
0k0
212
199 I213
200 Pm3

201 Pn3 00l           A2 B2
n2 A3         n A3
0k0
n3
202 Fm3

203 Fd3 00l: l = 4n + 2           : h + k = 4n + 2 A2 B2
d2 A3         d A3
0k0: k = 4n + 2
d3
204 Im3

205 Pa3 00l A2 B2 00l A2 B2 00l A2 B2
c2, 213 A3 B3 213   B3 21   B3
0k0     A2 B2 A2 B2
212   B3 a3 A3   a A3
206 Ia3       A2 B2 A2 B2
a3 A3   a A3
 Point group 432
Space groupIncident-beam direction
[hk0] (cyclic)
207 P432
208 P4232
209 F432
210 F4132
211 I432
212 P4332 00l A2 B2
43   B3
213 P4132 00l A2 B2
41   B3
214 I4132
 Point group
Space groupIncident-beam direction
[100] (cyclic)[110] (cyclic)[hhl] (cyclic)
215
216
217
218       00l   hhlo A2 B2
n A3   n A3
219             hoholo A2 B2
c A3
220 0kk, A2 B2 00l: l = 4n + 2     hhle: 2h + le = 4n + 2 A2 B2
d A3   d A3   d A3
 Point group m3m
Space groupIncident-beam direction
[100] (cyclic)[110] (cyclic)[hk0] (cyclic)[hhl] (cyclic)
221 Pm3m

222 Pn3n 00l   00l   hk0: h + k = 2n + 1 A2 B2 hhlo A2 B2
n12 A3   n2 A3   n1 A3   n2 A3
0k0
n13
223 Pm3n       00l           hhlo A2 B2
n A3         n A3
224 Pn3m 00l           hk0: h + k = 2n + 1 A2 B2
n2 A3         n A3
0k0
n3
225 Fm3m

226 Fm3c                   hoholo A2 B2
c A3
227 Fd3m 00l: l = 4n + 2           heke0: he + ke = 4n + 2 A2 B2
d2 A3         d A3
0k0: k = 4n + 2
d3
228 Fd3c 00l: l = 4n + 2           heke0: he + ke = 4n + 2 A2 B2 hoholo A2 B2
d2 A3         d A3   c A3
0k0: k = 4n + 2
d3
229 Im3m

230 Ia3d 0kk,     00l: l = 4n + 2     hoko0 A2 B2 hhle: 2h + le = 4n + 2 A2 B2
d A3   d A3   a A3   d A3

a3

The number of indistinguishable space groups was first counted by Tanaka, Sekii & Nagasawa (1983) but later corrections were made by Eades & Spence (1987). It was found that 177 space groups out of 230 can be identified using the extinction lines (Tanaka et al., 2002). Another reference for space-group determination is due to Eades (1988). The indistinguishable space-group sets using the extinction lines are listed in Table 2.5.3.10. Most of the sets are caused by the fact that CBED cannot identify 42, 31 (32) and 62 (64) screw axes. However, these sets can be rather easily distinguished in the ordinary way, that is, by observing how the intensities of the reflections which may be kinematically forbidden change when the crystal orientation is varied. If the axis concerned is a screw axis, kinematically forbidden reflections show a sudden decrease in intensity when an orientation change causes the loss of Umweganregung paths. If the axis is a rotation axis, the intensities of the reflections do not change conspicuously for such an orientation change. Using this test, each space group in the 23 sets can be identified except the pairs in parentheses and pairs (16) and (17) in Table 2.5.3.10 (see Eades, 1988).

 Table 2.5.3.10| top | pdf | Space-group sets indistinguishable by dynamical extinction lines
 (1) P3, (P31, P32) (2) P312, (P3112, P3212) (3) P321, (P3121, P3221) (4) P6, (P62, P64) (5) P622, (P6222, P6422) (6) P63, (P61, P65) (7) P6322, (P6122, P6522) (8) P4, P42 (9) (P41, P43) (10) P4/m, P42/m (11) P4/n, P42/n (12) P422, P4222 (13) P4212, P42212 (14) I4, I41 (15) I422, I4122 (16) I23, I213 (17) I222, I212121 (18) P432, P4232 (19) (P4132, P4332) (20) I432, I4132 (21) F432, F4132 (22) (P4122, P4322) (23) (P41212, P43212)

Tsuda et al. (2000) showed theoretically that the coherent CBED method can distinguish between space groups (I23 and I213) and between (I222 and I212121), which are indistinguishable pairs (16) and (17), respectively, in Table 2.5.3.10. The coherent CBED pattern is obtained in such a way that the convergence angle of the incident beam is set to a larger value than usual to make adjacent CBED discs overlap (Dowell & Goodman, 1973). When the focus point is displaced from the specimen, or a certain area is illuminated, sinusoidal interference fringes of the lattice spacing corresponding to the adjacent discs are formed in the overlapping regions if the probe size of the incident beam is smaller than the lattice spacing. (If the focus point of the incident beam is on the specimen, each overlapping region of the CBED discs shows uniform intensity.) Formation of the interference fringes was explained in detail first by Spence & Cowley (1978). Vine et al. (1992) showed distortion-free interference fringes from 6H-SiC and succeeded in observing the fringes with a shift of half a period due to a glide plane. Tsuda et al.'s method distinguishes the difference in the relative arrangements of twofold rotation axes and 21 screw axes along the [111] direction between the two space groups by examining the symmetry of intensity pairs appearing in the overlapping discs of a coherent [111] ZOLZ pattern. Saitoh, Tsuda et al. (2001) extended the method to distinguish the other ten indistinguishable space-group pairs. The method can distinguish between a space group which is composed of a principal rotation axis and a twofold rotation axis like P321 and a space group which is composed of a principal screw axis and a twofold rotation axis like P3121 (or P3221) by investigating the difference in the relative arrangements of the twofold rotation axis with respect to the principal axis. Table 2.5.3.11 shows the 12 space-group pairs which are distinguishable by applying the coherent CBED method.

 Table 2.5.3.11| top | pdf | Space-group sets distinguishable by coherent CBED
 The space-group pairs in parentheses can not be distinguished by coherent CBED but can be distinguished by a handedness test. An asterisk (*) indicates the incidence at which the distinction is carried out by many-beam interference (Saitoh, Tsuda et al., 2001).
Space-group setIncidence
(2) P312, (P3112, P3212)
(3) P321, (P3121, P3221)
(5) P622, (P6222, P6422)
(7) P6322, (P6122, P6522)
(12) P422, P4222 [321], [211], [112]*
(13) P4212, P42212 [211]
(15) I422, I4122 [111]
(16) I23, I213 [111]
(17) I222, I212121 [111]
(18) P432, P4232 [321], [211]*
(20) I432, I4132 [111]
(21) F432, F4132 [432]

The pairs in parentheses form left- and right-handed space groups. Handedness or chirality may occur in space groups that do not possess mirror and/or inversion symmetry. The handedness of space groups is identified in such a way that the senses of two crystal axes are determined with the aid of kinematical structure-factor calculations and the sense of the third axis is determined with the aid of dynamical calculations. This method was used for quartz by Goodman & Secomb (1977) and Goodman & Johnson (1977) and for MnSi by Tanaka et al. (1985). We also mention that Taftø & Spence (1982) developed a simple but clever method without computation for determining the absolute polarity of the sphalerite structure utilizing multiple-scattering effects on weak beams, which are almost independent of thickness. Because of the importance of structure in the field of semiconductor science, this method is conveniently used nowadays to determine polarity.

It is worth mentioning that space groups that are indistinguishable by CBED (Table 2.5.3.10) do not appear frequently in real inorganic materials. The crystal data collected by Nowacki (1967) on 5572 different inorganic materials shows that the number of materials belonging to space groups among sets (2), (3), (5), (7) and (11) in Table 2.5.3.10 is more than 15 but the number belonging to space groups among the other sets is less than ten. This implies that the probability of finding indistinguishable space groups is very low.

### References

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Eades, J. A. (1988). Microbeam Analysis, edited by D. Newbuly, p. 75. San Francisco Press.
Eades, J. A. & Spence, J. C. H. (1987). Private communication.
Goodman, P. & Johnson, A. W. S. (1977). Identification of enantiomorphically related space groups by electron diffraction – a second method. Acta Cryst. A33, 997–1001.
Goodman, P. & Secomb, T. W. (1977). Identification of enantiomorphously related space groups by electron diffraction. Acta Cryst. A33, 126–133.
Nowacki, W. (1967). Crystal Data. ACA Monograph No. 6. Washington: American Crystallographic Association.
Saitoh, K., Tsuda, K., Terauchi, M. & Tanaka, M. (2001). Distinction between space groups having principal rotation and screw axes, which are combined with twofold rotation axes, using the coherent convergent-beam electron diffraction method. Acta Cryst. A57, 219–230.
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Taftø, T. & Spence, J. C. H. (1982). A simple method for the determination of structure-factor phase relationships and crystal polarity using electron diffraction. J. Appl. Cryst. 15, 60–64.
Tanaka, M., Sekii, H. & Nagasawa, T. (1983). Space group determination by dynamic extinction in convergent beam electron diffraction. Acta Cryst. A39, 825–837.
Tanaka, M., Takayoshi, H., Ishida, M. & Endoh, Y. (1985). Crystal chirality and helicity of the helical spin-density wave in MnSi. 1. Convergent-beam electron diffraction. J. Phys. Soc. Jpn, 54, 2970–2974.
Tanaka, M., Terauchi, M., Tsuda, K. & Saitoh, K. (2002). Convergent-Beam Electron Diffraction IV, p. 13. Tokyo: JEOL.
Tsuda, K., Saitoh, K., Terauchi, M., Tanaka, M. & Goodman, P. (2000). Distinction of space groups (I23 and I213) and (I222 and I212121) using coherent convergent-beam electron diffraction. Acta Cryst. A56, 359–369.
Vine, W. J., Vincent, R., Spellward, P. & Steeds, J. W. (1992). Observation of phase contrast in convergent-beam electron diffraction. Ultramicroscopy, 41, 423–428.