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

International Tables for Crystallography (2010). Vol. B, ch. 2.5, pp. 318-344   | 1 | 2 |

Section 2.5.3.3. Space-group determination

M. Tanakaf

2.5.3.3. Space-group determination

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2.5.3.3.1. Lattice-type determination

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When the point group of a specimen crystal is determined, the crystal axes may be found from a spot diffraction pattern recorded at a high-symmetry zone axis, using the orientations of the symmetry elements determined in the course of point-group determination. Integral-number indices are assigned to the spots of the diffraction patterns. The systematic absence of reflections indicates the lattice type of the crystal. It should be noted that reflections forbidden by the lattice type are always absent, even if dynamical diffraction takes place. (This is true for all sample thicknesses and accelerating voltages.) By comparing the experimentally obtained absences and the extinction rules known for the lattice types [P, C (A, B), I, F and R], a lattice type may be identified for the crystal concerned.

2.5.3.3.2. Identification of screw axes and glide planes

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There are three space-group symmetry elements of diperiodic plane figures: (1) a horizontal screw axis [2_1'], (2) a vertical glide plane g with a horizontal glide vector and (3) a horizontal glide plane g′. These are related to the point-group symmetry elements 2′, m and m′ of diperiodic plane figures, respectively. (It is noted that these symmetry elements and ten point-group symmetry elements form 80 space groups.)

The ordinary extinction rules for screw axes and glide planes hold only in the approximation of kinematical diffraction. The kinematically forbidden reflections caused by these symmetry elements appear owing to Umweganregung of dynamical diffraction. Extinction of intensity, however, does take place in these reflections at certain crystal orientations with respect to the incident beam (i.e. in certain regions within a CBED disc). This dynamical extinction was first predicted by Cowley & Moodie (1959[link]) and was discussed by Miyake et al. (1960[link]) and Cowley et al. (1961[link]). Goodman & Lehmpfuhl (1964[link]) first observed the dynamical extinction as dark cross lines in kinematically forbidden reflection discs of CBED patterns of CdS. Gjønnes & Moodie (1965[link]) discussed the dynamical extinction in a more general way considering not only ZOLZ reflections but also HOLZ reflections. They completely clarified the dynamical extinction rules by considering the exact cancellation which may occur along certain symmetry-related multiple-scattering paths. Based on the results of Gjønnes & Moodie (1965[link]), Tanaka, Sekii & Nagasawa (1983[link]) tabulated the dynamical extinctions expected at all the possible crystal orientations for all the space groups. These were later tabulated in a better form on pages 162 to 172 of the book by Tanaka & Terauchi (1985[link]).

Fig. 2.5.3.10[link](a) illustrates Umweganregung paths to a kinematically forbidden reflection. The 0k0 (k = odd) reflections are kinematically forbidden because a b-glide plane exists perpendicular to the a axis and/or a 21 screw axis exists in the b direction. Let us consider an Umweganregung path a in the zeroth-order Laue zone to the 010 forbidden reflection and path b which is symmetric to path a with respect to axis k. Owing to the translation of one half of the lattice parameter b caused by the glide plane and/or the 21 screw axis, the following relations hold between the crystal structure factors:[\eqalignno{F(h,k) &= F(\bar{h},k)\quad \hbox{for }k=2n,&\cr F(h,k) &= -F(\bar{h},k)\quad\hbox{for }k=2n+1. &(2.5.3.1)}]That is, the structure factors of reflections hk0 and [\bar{h}k0] have the same phase for even k but have opposite phases for odd k.

[Figure 2.5.3.10]

Figure 2.5.3.10 | top | pdf |

Illustration of the production of dynamical extinction lines in kinematically forbidden reflections due to a b-glide plane and a 21 screw axis. (a) Umweganregung paths a, b and c. (b) Dynamical extinction lines A are formed in forbidden reflections 0k0 (k = odd). Extinction line B perpendicular to the lines A is formed in the exactly excited 010 reflection.

Since an Umweganregung path to the kinematically forbidden reflection 0k0 contains an odd number of reflections with odd k, the following relations hold:[\eqalignno{&F(h_1,k_1)F(h_2,k_2)\ldots F(h_n,k_n) \quad\hbox{for path }a&\cr &\quad = -F(\bar{h}_1,k_1)F(\bar{h}_2,k_2)\ldots F(\bar{h}_n,k_n)\quad\hbox{for path }b,&\cr&&(2.5.3.2)}]where[\textstyle\sum\limits_{i=1}^n h_i=0,\quad \textstyle\sum\limits_{i=1}^n k_i = k \,\,(k={\rm odd})]and the functions including the excitation errors are omitted because only the cases in which the functions are the same for all the paths are considered. The excitation errors for paths a and b become the same when the projection of the Laue point along the zone axis concerned, L, lies on axis k. Since the two waves passing through paths a and b have the same amplitude but opposite signs, these waves are superposed on the 0k0 discs (k = odd) and cancel out, resulting in dark lines A in the forbidden discs, as shown in Fig. 2.5.3.10[link](b). The line A runs parallel to axis k passing through the projection point of the zone axis.

In path c, the reflections are arranged in the reverse order to those in path b. When the 010 reflection is exactly excited, two paths a and c are symmetric with respect to the bisector m′–m′ of the 010 vector having the same excitation errors. The following equation holds:[\eqalignno{&F(h_1,k_1)F(h_2,k_2)\ldots F(h_n,k_n) \quad\hbox{for path }a&\cr &\quad = -F(\bar{h}_n,k_n)F(\bar{h}_{n-1},k_{n-1})\ldots F(\bar{h}_1,k_1)\quad\hbox{for path }c.&\cr&&(2.5.3.3)}]Since the waves passing through these paths have the same amplitude but opposite signs, these waves are superposed on the 010 discs and cancel out, resulting in dark line B in this disc, as shown in Fig. 2.5.3.10[link](b). Line B appears perpendicular to line A at the exact Bragg positions. When Umweganregung paths are present only in the zeroth-order Laue zone, the glide plane and screw axis produce the same dynamical extinction lines A and B. We call these lines A2 and B2 lines, subscript 2 indicating that the Umweganregung paths lie in the zeroth-order Laue zone.

The dynamical extinction effect is analogous to interference phenomena in the Michelson interferometer. That is, the incident beam is split into two beams by Bragg reflections in a crystal. These beams take different paths, in which they suffer a relative phase shift of π and are finally superposed on a kinematically forbidden reflection to cancel out.

When the paths include higher-order Laue zones, the glide plane produces only extinction lines A but the screw axis causes only extinction lines B. These facts are attributed to the different relations between structure factors for a 21 screw axis and a glide plane,[\eqalignno{F(hkl)&=(-1)^kF(\bar{h}k\bar{l})\,\,\,\hbox{for a }2_1\hbox{ screw axis in the [010] direction,}&\cr &&(2.5.3.4)\cr F(hkl)&=(-1)^kF(\bar{h}kl)\,\,\,\hbox{for a }b\hbox{ glide in the (100) plane.}&\cr&&(2.5.3.5)}%fd2o5o3o5]In the case of the glide plane, extinction lines A are still formed because two waves passing through paths a and b have opposite signs to each other according to equation (2.5.3.5)[link], but extinction lines B are not produced because equation (2.5.3.4)[link] holds only for the 21 screw axis. In the case of the 21 screw axis, only the waves passing through paths a and c have opposite signs according to equation (2.5.3.4)[link], forming extinction lines B only. We call these lines A3 and B3 dynamical extinction lines, suffix 3 indicating the Umweganregung paths being via higher-order Laue zones.

It was predicted by Gjønnes & Moodie (1965[link]) that a horizontal glide plane g′ gives a dark spot at the crossing point between extinction lines A and B (Fig. 2.5.3.10[link]b) due to the cancellation between the waves passing through paths b and c. Tanaka, Terauchi & Sekii (1987[link]) observed this dynamical extinction, though it appeared in a slightly different manner to that predicted by Gjønnes & Moodie (1965[link]). Table 2.5.3.8[link] summarizes the appearance of the dynamical extinction lines for the glide planes g and g′ and the 21 screw axis. The three space-group symmetry elements can be identified from the observed extinctions because these three symmetry elements produce different kinds of dynamical extinctions.

Table 2.5.3.8| top | pdf |
Dynamical extinction rules for an infinitely extended parallel-sided specimen

Symmetry elements of plane-parallel specimenOrientation to specimen surfaceDynamical extinction lines
Two-dimensional (zeroth Laue zone) interactionThree-dimensional (HOLZ) interaction
Glide planes perpendicular: g A2 and B2 A3
parallel: g intersection of A3 and B3
Twofold screw axis perpendicular: 21
parallel: [2_1^\prime] A2 and B2 B3

In principle, a horizontal screw axis and a vertical glide plane can be distinguished by observations of the extinction lines A3 and B3. It is, however, not easy to observe the extinction lines A3 and B3 because broad extinction lines A2 and B2 appear at the same time. The presence of the extinction lines A3 and B3 can be revealed by inspecting the symmetries of fine defect HOLZ lines appearing in the forbidden reflections instead of by direct observation of the lines A3 and B3 (Tanaka, Sekii & Nagasawa, 1983[link]). That is, if HOLZ lines form lines A3 and B3, HOLZ lines are symmetric with respect to the extinction lines A2 and B2. If HOLZ lines do not form lines A3 and B3, HOLZ lines are asymmetric with respect to the extinction lines A2 and B2. When the HOLZ lines are symmetric about the extinction lines A2, the specimen crystal has a glide plane. When the HOLZ lines are symmetric with respect to lines B2, a 21 screw axis exists. It should be noted that a relatively thick specimen area has to be selected to observe HOLZ lines in ZOLZ reflection discs.

Fig. 2.5.3.11[link] shows CBED patterns taken from (a) thin and (b) thick areas of FeS2, whose space group is [P2_1/a\bar{3}], at the 001 Bragg setting with the [100] electron-beam incidence. In the case of the thin specimen (Fig. 2.5.3.11[link]a), only the broad dynamical extinction lines formed by the interaction of ZOLZ reflections are seen in the odd-order discs. On the other hand, fine HOLZ lines are clearly seen in the thick specimen (Fig. 2.5.3.11[link]b). The HOLZ lines are symmetric with respect to both A2 and B2 extinction lines. This fact proves the presence of the extinction lines A3 and B3, or both the c glide in the (010) plane and the 21 screw axis in the c direction, this fact being confirmed by consulting Table 2.5.3.9[link]. Fig. 2.5.3.12[link] shows a [110] zone-axis CBED pattern of FeS2. A2 extinction lines are seen in both the 001 and [\bar{1}10] discs. Fine HOLZ lines are symmetric with respect to the A2 extinction lines in the [\bar{1}10] disc but asymmetric about the A2 extinction line in the 001 disc, indicating formation of the A3 extinction line only in the [\bar{1}10] disc. This proves the existence of a 21 screw axis in the [001] direction and an a glide in the (001) plane. The appearance of HOLZ lines is easily changed by a change of a few hundred volts in the accelerating voltage. Steeds & Evans (1980[link]) demonstrated for spinel changes in the appearance of HOLZ lines in the ZOLZ discs at accelerating voltages around 100 kV.

[Figure 2.5.3.11]

Figure 2.5.3.11 | top | pdf |

CBED patterns obtained from (a) thin and (b) thick areas of (001) FeS2. (a) Dynamical extinction lines A2 and B2 are seen. (b) Extinction lines A3 and B3 as well as A2 and B2 are formed because HOLZ lines are symmetric about lines A2 and B2.

[Figure 2.5.3.12]

Figure 2.5.3.12 | top | pdf |

CBED pattern of FeS2 taken with the [110] electron-beam incidence. In the 001 and [00\bar 1] discs, HOLZ lines are asymmetric with respect to extinction lines A2, indicating the existence of a 21 screw axis parallel to the c axis. In the [\bar 1 10] and [1\bar10] discs, HOLZ lines are symmetric with respect to extinction lines A2, indicating existence of a glide plane perpendicular to the c axis.

Another practical method for distinguishing between glide planes and 21 screw axes is that reported by Steeds et al. (1978[link]). The method is based on the fact that the extinction lines are observable even when a crystal is rotated with a glide plane kept parallel and with a 21 screw axis perpendicular to the incident beam. With reference to Fig. 2.5.3.10[link](a), extinction lines A3 produced by a glide plane remain even when the crystal is rotated with respect to axis h but the lines are destroyed by a rotation of the crystal about axis k. Extinction lines B3 originating from a 21 screw axis are not destroyed by a crystal rotation about axis k but the lines are destroyed by a rotation with respect to axis h.

2.5.3.3.3. Space-group determination

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We now describe a space-group determination method which uses the dynamical extinction lines caused by the horizontal screw axis [2_1'] 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[link].) 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 [2_1'] 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[link]) 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[link] 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 [(0k'l'\perp [0kl],\, h'k'0 \perp [hk0],\ldots)]. 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 [R\bar{3}c] 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 [\matrix{\hfill Cmm2\cr \hfill ba2}]                                  
36 [\matrix{\hfill Cmc2_1\cr\hfill bn2_1}] 00l A2 B2 00l           00l A2 B2       he0lo A2 B2
c, 21 A3 B3 21   B3       21   B3       c A3  
37 [\matrix{\hfill Ccc2\cr\hfill nn2}] 00l     00l                 0kelo A2 B2 he0lo A2 B2
c2 A3   c1 A3               c1 A3   c2 A3  
38 [\matrix{\hfill Amm2\phantom{_1}\cr\hfill nc2_1}]                                    
39 [\matrix{\hfill Abm2\phantom{_1}\cr\hfill cc2_1}]                         0kolo A2 B2      
                        b A3        
40 [\matrix{\hfill Ama2\phantom{_1}\cr\hfill nn2_1}]             h00 A2 B2             ho0le A2 B2
            a A3               a A3  
41 [\matrix{\hfill Aba2\phantom{_1}\cr\hfill cn2_1}]             h00 A2 B2       0kolo A2 B2 ho0le A2 B2
            a A3         b A3   a A3  
42 Fmm2                                    
43 [\matrix{\hfill Fdd2\phantom{_1}\cr\hfill dd2_1}] 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 [\matrix{\hfill Imm2\phantom{_1}\cr\hfill nn2_1}]                                    
45 [\matrix{\hfill Iba2\phantom{_1}\cr\hfill cc2_1}]                         0kolo A2 B2 ho0lo A2 B2
                        b A3   a A3  
46 [\matrix{\hfill Ima2\phantom{_1}\cr\hfill nc2_1}]                               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 [4, \bar 4, 4/m]

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 [P\bar 4]      
82 [I\bar 4]      
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, [\bar h h 0] A2 B2 00l: l = 4n + 2           hhle: 2h + le = 4n + 2 A2 B2
      d A3   d A3         d A3  
110 I41cd       hh0, [\bar h h 0] 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 [\bar 4 2 m]. 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 [P\bar 4 2 m]                              
112 [P\bar 4 2 c]             00l         hhlo A2 B2
            c A3         c A3  
113 [P\bar 4 2_1 m] 0k0 A2 B2 h00 A2 B2       0k0 A2 B2      
212   B3 211   B3       21   B3      
      0k0                      
      212                      
114 [P\bar 4 2_1 c] 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 [P \bar 4 m 2]                              
116 [P\bar 4 c 2] 00l                 h0lo A2 B2      
c2 A3               c A3        
117 [P\bar 4 b 2]       h00 A2 B2       ho0l A2 B2      
      a2 A3         a A3        
      0k0                      
      b1                      
118 [P\bar 4 n 2] 00l     h00 A2 B2       h0l: h + l = 2n + 1 A2 B2      
n2 A3   n2 A3         n A3        
      0k0                      
      n1                      
119 [I\bar 4 m 2]                              
120 [I\bar 4 c 2]                   ho0lo A2 B2      
                  c A3        
121 [I\bar 4 2 m]                              
122 [I\bar 4 2 d]       hh0, [\bar h h 0] 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, [\bar h h 0]
d

A3
  00l: l = 4n + 2
d

A3
        hhle: 2h + le = 4n + 2
d
A2
A3
B2 hok0
a
A2
A3
B2
              [\bar h h 0]                      
              a                      
142 I41/acd
I41/a2/c2/d
      hh0, [\bar h h 0]
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
              [\bar h h 0]                      
              a                      

Point groups [3, \bar 3, 32, 3m, \bar 3m]

Space groupIncident-beam direction
[[11\bar2 0]][[1\bar 1 00]]
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 [P\bar 3 1 m]            
163 [P\bar 3 1 c] 00l A2 B2      
c A3        
164 [P\bar 3 m 1]            
165 [P\bar 3 c 1]       00l A2 B2
      c A3  
166 [R\bar 3 m]            
167 [R\bar 3 c]       00l: l = 6n + 3 A2 B2
        c A3  

Point groups [6, \bar 6, 6/m, 622, 6mm, \bar 6 m2, 6/mmm]

Space groupIncident-beam direction
[[11\bar2 0]][[1\bar 1 00]]
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 [P\bar 6]            
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 [P\bar 6 m 2]            
188 [P\bar 6 c 2]       00l A2 B2
      c A3  
189 [P\bar 6 2 m]            
190 [P\bar 6 2 c] 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                  
[P2/m\bar 3]                  
201 Pn3 00l           [\bar k h 0] A2 B2
[P2/n\bar 3] n2 A3         n A3  
  0k0                
  n3                
202 Fm3                  
[F2/m\bar 3]                  
203 Fd3 00l: l = 4n + 2           [\bar k h 0]: h + k = 4n + 2 A2 B2
[F2/d\bar 3] d2 A3         d A3  
  0k0: k = 4n + 2                
  d3                
204 Im3                  
[I2/m\bar 3]                  
205 Pa3 00l A2 B2 00l A2 B2 00l A2 B2
[P2_1/a\bar 3] c2, 213 A3 B3 213   B3 21   B3
  0k0     [\bar h h 0] A2 B2 [\bar k h 0] A2 B2
  212   B3 a3 A3   a A3  
206 Ia3       [\bar h h 0] A2 B2 [\bar k h 0] A2 B2
[I2_1/a\bar 3]       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 [\bar 4 3 m]

Space groupIncident-beam direction
[100] (cyclic)[110] (cyclic)[hhl] (cyclic)
215 [P\bar 4 3 m]                  
216 [F\bar 4 3 m]                  
217 [I \bar 4 3 m]                  
218 [P\bar 4 3 n]       00l   hhlo A2 B2
      n A3   n A3  
219 [F\bar 4 3 c]             hoholo A2 B2
            c A3  
220 [I\bar 4 3 d] 0kk, [0\bar k k] 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                        
[P4/m\bar 3 2/m]                        
222 Pn3n 00l   00l   hk0: h + k = 2n + 1 A2 B2 hhlo A2 B2
[P4/n\bar 3 2/n] n12 A3   n2 A3   n1 A3   n2 A3  
  0k0                      
  n13                      
223 Pm3n       00l           hhlo A2 B2
[P4_2/m\bar 32/n]       n A3         n A3  
224 Pn3m 00l           hk0: h + k = 2n + 1 A2 B2      
[P4_2/n\bar 3 2/m] n2 A3         n A3        
  0k0                      
  n3                      
225 Fm3m                        
[F4/m\bar 32/m]                        
226 Fm3c                   hoholo A2 B2
[F4/m\bar 3 2/c]                   c A3  
227 Fd3m 00l: l = 4n + 2           heke0: he + ke = 4n + 2 A2 B2      
[F4_1/d\bar 32/m] 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
[F4_1/d\bar 3 2/c] d2 A3         d A3   c A3  
  0k0: k = 4n + 2                      
  d3                      
229 Im3m                        
[I4/m\bar 32/m]                        
230 Ia3d 0kk, [0\bar k k]     00l: l = 4n + 2     hoko0 A2 B2 hhle: 2h + le = 4n + 2 A2 B2
[I4_1/a\bar 3 2/ d] d A3   d A3   a A3   d A3  
        [\bar h h 0]                
        a3                

The number of indistinguishable space groups was first counted by Tanaka, Sekii & Nagasawa (1983[link]) but later corrections were made by Eades & Spence (1987[link]). It was found that 177 space groups out of 230 can be identified using the extinction lines (Tanaka et al., 2002[link]). Another reference for space-group determination is due to Eades (1988[link]). The indistinguishable space-group sets using the extinction lines are listed in Table 2.5.3.10[link]. 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[link] (see Eades, 1988[link]).