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

International Tables for Crystallography (2010). Vol. B, ch. 2.5, pp. 333-334   | 1 | 2 |

Section 2.5.3.3.4. Dynamical extinction in HOLZ reflections

M. Tanakaf

2.5.3.3.4. Dynamical extinction in HOLZ reflections

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Space-group determination as described in the previous sections is carried out using the extinction lines appearing in ZOLZ reflections. Vertical glide planes whose translation vectors are perpendicular to the specimen surface do not cause extinction lines in ZOLZ reflections but cause them in HOLZ reflections. (It is noted that the vertical glide planes with glide translations not parallel to the surface are not the symmetry elements of diperiodic plane figures.) Vertical glide planes whose translation vectors are parallel to the surface cause extinction lines in both ZOLZ and HOLZ reflections. Vertical screw axes are expected to form extinction lines in HOLZ reflections whose vectors are parallel to the screw axes. These reflections, however, cannot be observed by ordinary CBED. Thus, the extinction lines appearing in observable HOLZ reflections are used to identify not screw axes but glide planes. Examination of HOLZ extinction lines together with ZOLZ extinction lines is an efficient way to characterize the glide vectors and determine the space group.

The dynamical extinction lines appearing in HOLZ reflections caused by the glide planes whose glide vectors are not only parallel but also not parallel to the specimen surface were tabulated by Nagasawa (1983[link]) for various incident-beam orientations of all the space groups that have glide planes. The tabulated results appear on pages 214–225 of the book by Tanaka et al. (1988[link]). Table 2.5.3.12[link] shows the results. The meanings of the letters used in the table are explained in Fig. 2.5.3.13[link]. We consider a vertical glide plane with a glide vector perpendicular to the surface as is shown in Fig. 2.5.3.13[link](a). Letter A is given for cases in which the Ewald sphere intersects two circled-cross reflections in the first Laue zone as seen in Fig. 2.5.3.13[link](b), where black circles and circled crosses denote allowed reflections and kinematically forbidden but dynamically allowed reflections due to the glide plane, respectively. A* denotes cases in which the Ewald sphere intersects a circled-cross reflection on one side of the incident beam and a black-circled reflection on the other, as seen in Fig. 2.5.3.13[link](c). This case occurs only in space group [P2_1/a\bar{3}]. Ah denotes cases in which the Ewald sphere intersects a circled-cross reflection on one side but does not intersect on the other, owing to the asymmetric arrangement of reflections with respect to the incident beam.

Table 2.5.3.12| top | pdf |
Dynamical extinction lines appearing in HOLZ reflections for crystal space groups that have mirror and glide planes

Point groups m, 2/m (second setting, unique axis b)

Space groupIncident-beam direction
[u0w]
6 Pm    
7 Pc h0lo Ah
c
8 Cm    
9 Cc he0lo Ah
c
10 P2/m    
11 P21/m    
12 C2/m    
13 P2/c h0lo Ah
c
14 P21/c h0lo Ah
c
15 C2/c he0lo Ah
c

Point group mm2

Space groupIncident-beam direction
[100][010][001][0vw][u0w]
25 Pmm2                    
26 Pmc21 h0lo A     h0lo A     h0lo Ah
c     c     c
27 Pcc2 h0lo A 0klo A 0klo A 0klo Ah h0lo Ah
c2 c1 c1 c1 c2
    h0lo    
    c2    
28 Pma2 ho0l A     ho0l A     ho0l Ah
a     a     a
29 Pca21 ho0l A 0klo A 0klo A 0klo Ah ho0l Ah
a c c c a
    ho0l    
    a    
30 Pnc2 h0lo
c
A 0kl: k + l = 2n + 1
n
A 0kl: k + l = 2n + 1
n
A 0kl: k + l = 2n + 1
n
Ah h0lo
c
Ah
    h0lo    
    c    
31 Pmn21 h0l: h + l = 2n + 1 A     h0l: h + l = 2n + 1 A     h0l: h + l = 2n + 1 Ah
n     n     n
32 Pba2 ho0l A 0kol A 0kol A 0kol Ah ho0l Ah
a b b b a
    ho0l    
    a    
33 Pna21 ho0l
a
A 0kl: k + l = 2n + 1
n
A 0kl: k + l = 2n + 1
n
A 0kl: k + l = 2n + 1
n
Ah ho0l
a
Ah
    ho0l      
    a      
34 Pnn2 h0l: h + l = 2n + 1 A 0kl: k + l = 2n + 1 A 0kl: k + l = 2n + 1 A 0kl: k + l = 2n + 1 Ah h0l: h + l = 2n + 1 Ah
n2 n1 n1 n1 n2
    h0l: h + l = 2n + 1    
    n2    
35 [\matrix{\hfill Cmm2\cr \hfill ba2}]                    
36 [\matrix{\hfill Cmc2_1\cr\hfill bn2_1}] he0lo A     he0lo A     he0lo Ah
c     c     c
37 [\matrix{\hfill Ccc2\cr\hfill nn2}] he0lo A 0kelo A 0kelo A 0kelo Ah he0lo Ah
c2 c1 c1 c1 c2
    he0lo    
    c2    
38 [\matrix{\hfill Amm2\phantom{_1}\cr\hfill nc2_1}]                    
39 [\matrix{\hfill Abm2\phantom{_1}\cr\hfill cc2_1}]     0kolo A 0kolo A 0kolo Ah    
    b b b    
40 [\matrix{\hfill Ama2\phantom{_1}\cr\hfill nn2_1}] ho0le A     ho0le A     ho0le Ah
a     a     a
41 [\matrix{\hfill Aba2\phantom{_1}\cr\hfill cn2_1}] ho0le A 0kolo A 0kolo A 0kolo Ah ho0le Ah
a b b b a
    ho0le    
    a    
42 Fmm2                    
43 [\matrix{\hfill Fdd2\phantom{_1}\cr\hfill dd2_1}] he0le: he + le = 4n + 2 A 0kele: ke + le = 4n + 2 A 0kele: ke + le = 4n + 2 A 0kele: ke + le = 4n + 2 Ah he0le: he + le = 4n + 2 Ah
d2 d1 d1 d1 d2
    he0le: he + le = 4n + 2    
    d2    
44 [\matrix{\hfill Imm2\phantom{_1}\cr\hfill nn2_1}]                    
45 [\matrix{\hfill Iba2\phantom{_1}\cr\hfill cc2_1}] ho0lo A 0kolo A 0kolo A 0kolo Ah ho0lo Ah
a b b b a
    ho0lo    
    a    
46 [\matrix{\hfill Ima2\phantom{_1}\cr\hfill nc2_1}] ho0lo A     ho0lo A     ho0lo Ah
a     a     a

Point group mmm

Space groupIncident-beam direction
[100][010][001][uv0][0vw][u0w]
47 P2/m2/m2/m                        
48 P2/n2/n2/n h0l: h + l = 2n + 1 A 0kl: k + l = 2n + 1 A 0kl: k + l = 2n + 1 A hk0: h + k = 2n + 1 Ah 0kl: k + l = 2n + 1 Ah h0l: h + l = 2n + 1 Ah
n2 n1 n1 n3 n1 n2
hk0: h + k = 2n + 1 hk0: h + k = 2n + 1 h0l: h + l = 2n + 1      
n3 n3 n2      
49 P2/c2/c2/m h0lo A 0klo A 0klo A     0klo Ah h0lo Ah
c2 c1 c1     c1 c2
    h0lo        
    c2        
50 P2/b2/a2/n ho0l
a
hk0: h + k = 2n + 1
n
A 0kol
b
hk0: h + k = 2n + 1
n
A 0kol
b
ho0l
a
A hk0: h + k = 2n + 1
n
Ah 0kol
b
Ah ho0l
a
Ah
51 P21/m2/m2/a hok0 A hok0 A     hok0 Ah        
a a     a        
52 P2/n21/n2/a h0l: h + l = 2n + 1
n2
A 0kl: k + l = 2n + 1
n1
A 0kl: k + l = 2n + 1
n1
A hok0
a
Ah 0kl: k + l = 2n + 1
n1
Ah h0l: h + l = 2n + 1
n2
Ah
hok0
a
hok0
a
h0l: h + l = 2n + 1
n2
     
53 P2/m2/n21/a h0l: h + l = 2n + 1
n
A hok0
a
A h0l: h + l = 2n + 1
n
A hok0
a
Ah     h0l: h + l = 2n + 1
n
Ah
hok0            
a            
54 P21/c2/c2/a h0lo A 0klo A 0klo A hok0 Ah 0klo Ah h0lo Ah
c2 c1 c1 a c1 c2
hok0 hok0 h0lo      
a a c2      
55 P21/b21/a2/m ho0l A 0kol A 0kol A     0kol Ah ho0l Ah
a b b     b a
    ho0l        
    a        
56 P21/c21/c2/n h0lo
c2
hk0: h + k = 2n + 1
n
A 0klo
c1
hk0: h + k = 2n + 1
n
A 0klo
c1
h0lo
c2
A hk0: h + k = 2n + 1
n
Ah 0klo
c1
Ah h0lo
c2
Ah
57 P2/b21/c21/m h0lo A 0kol A 0kol A     0kol Ah h0lo Ah
c b b     b c
    h0lo        
    c        
58 P21/n21/n2/m h0l: h + l = 2n + 1 A 0kl: k + l = 2n + 1 A 0kl: k + l = 2n + 1 A     0kl: k + l = 2n + 1 Ah h0l: h + l = 2n + 1 Ah
n2 n1 n1     n1 n2
    h0l: h + l = 2n + 1        
    n2        
59 P21/m21/m2/n hk0: h + k = 2n + 1 A hk0: h + k = 2n + 1 A     hk0: h + k = 2n + 1 Ah        
n n     n        
60 P21/b2/c21/n h0lo
c
hk0: h + k = 2n + 1
n
A 0kol
b
hk0: h + k = 2n + 1
n
A 0kol
b
h0lo
c
A hk0: h + k = 2n + 1
n
Ah 0kol
b
Ah h0lo
c
Ah
61 P21/b21/c21/a h0lo A 0kol A 0kol A hok0 Ah 0kol Ah h0lo Ah
c b b a b c
hok0 hok0 h0lo      
a a c      
62 P21/n21/m21/a hok0
a
A 0kl: k + l = 2n + 1
n
A 0kl: k + l = 2n + 1
n
A hok0
a
Ah 0kl: k + l = 2n + 1
n
Ah    
  hok0          
  a          
63 C2/m2/c21/m he0lo A     he0lo A         he0lo Ah
c     c         c
64 C2/m2/c21/a he0lo A hoko0 A he0lo A hoko0 Ah     he0lo Ah
c a c a     c
hoko0            
a            
65 C2/m2/m2/m                        
66 C2/c2/c2/m he0lo A 0kelo A 0kelo A     0kelo Ah he0lo Ah
c2 c1 c1     c1 c2
    he0lo        
    c2        
67 C2/m2/m2/a hoko0 A hoko0 A     hoko0 Ah        
a a     a        
68 C2/c2/c2/a he0lo A 0kelo A 0kelo A hoko0 Ah 0kelo Ah he0lo Ah
c2 c1 c1 a c1 c2
hoko0 hoko0 he0lo      
a a c2      
69 F2/m2/m2/m                        
70 F2/d2/d2/d he0le: he + le = 4n + 2 A heke0: he + ke = 4n + 2 A 0kele: ke + le = 4n + 2 A heke0: he + ke = 4n + 2 Ah 0kele: ke + le = 4n + 2 Ah he0le: he + le = 4n + 2 Ah
d2 d3 d1 d3 d1 d2
heke0: he + ke =4n + 2 0kele: ke + le = 4n + 2 he0le: he + le = 4n + 2      
d3 d1 d2      
71 I2/m2/m2/m                        
72 I2/b2/a2/m ho0lo A 0kolo A 0kolo A     0kolo Ah ho0lo Ah
a b b     b a
    ho0lo        
    a        
73 I21/b21/c21/a ho0lo A hoko0 A 0kolo A hoko0 Ah 0kolo Ah ho0lo Ah
c a b a b c
hoko0 0kolo ho0lo      
a b c      
74 I21/m21/m21/a hoko0 A hoko0 A     hoko0 Ah        
a a     a        

Point group 4/m

Space groupIncident-beam direction
[100], [110][uv0]
83 P4/m        
84 P42/m        
85 P4/n hk0: h + k = 2n + 1 A hk0: h + k = 2n + 1 Ah
n n
86 P42/n hk0: h + k = 2n + 1 A hk0: h + k = 2n + 1 Ah
n n
87 I4/m        
88 I41/a hoko0 A hoko0 Ah
a   a

Point group 4mm. The symbol a in the column [u0w] is equivalent to the symbol b in the space groups of the first column.

Space groupIncident-beam direction
[100][001][110][u0w][uuw]
99 P4mm                    
100 P4bm ho0l A 0kol A     ho0l Ah    
a2 b1     a    
  ho0l          
  a2          
101 P42cm h0lo A 0klo A     h0lo Ah    
c2 c1     c    
  h0lo          
  c2          
102 P42nm h0l: h + l = 2n + 1 A 0kl: k + l = 2n + 1 A     h0l: h + l = 2n + 1 Ah    
n2 n1     n    
  h0l: h + l = 2n + 1          
  n2          
103 P4cc h0lo A 0klo A hhlo A h0lo Ah hhlo Ah
c12 c11 c2 c1 c2
  h0lo      
  c12      
  [hhl_{\rm o}, \bar h h l_{\rm o}]      
  c2      
104 P4nc h0l: h + l = 2n + 1
n2
A 0kl: k + l = 2n + 1
n1
A hhlo
c
A h0l: h + l = 2n + 1
n
Ah hhlo
c
Ah
  h0l: h + l = 2n + 1      
  n2      
  [hhl_{\rm o}, \bar h h l_{\rm o}]      
  c      
105 P42mc     [hhl_{\rm o }, \bar h h l_{\rm o}] A hhlo A     hhlo Ah
    c c     c
106 P42bc ho0l A 0kol A hhlo A ho0l Ah hhlo Ah
a2 b1 c a c
  ho0l      
  a2      
  [hhl_{\rm o }, \bar h h l_{\rm o}]      
  c      
107 I4mm                    
108 I4cm ho0lo A 0kolo A     ho0lo Ah    
c2 c1     c    
  ho0lo          
  c2          
109 I41md     hhle, [\bar h h l_{\rm e}]: 2h + le = 4n + 2 A hhle: 2h + le = 4n + 2 A     hhle: 2h + le = 4n + 2 Ah
    d d       d
110 I41cd ho0lo
c2
A 0kolo
c1
ho0lo
c2
hhle, [\bar h h l_{\rm e}]: 2h + le = 4n + 2
d
A hhle: 2h + le = 4n + 2
d
A ho0lo
c
Ah hhle: 2h + le = 4n + 2
d
Ah

Point group [\bar 4 2 m]. The symbol a in the column [u0w] is equivalent to the symbol b in the space groups of the first column.

Space groupIncident-beam direction
[100][001][110][u0w][uuw]
111 [P\bar 4 2 m]                    
112 [P\bar 4 2 c]     hhlo, [\bar h h l_{\rm o}] A hhlo A     hhlo Ah
    c c     c
113 [P\bar 4 2_1 m]                    
114 [P\bar 4 2_1 c]     hhlo, [\bar h h l_{\rm o}] A hhlo A     hhlo Ah
    c c     c
115 [P \bar 4 m 2]                    
116 [P\bar 4 c 2] h0lo A 0klo A     h0lo Ah    
c2 c1     c    
  h0lo          
  c2          
117 [P\bar 4 b 2] ho0l A 0kol A     ho0l Ah    
a2 b1     a    
  ho0l          
  a2          
118 [P\bar 4 n 2] h0l: h + l = 2n + 1 A 0kl: k + l = 2n + 1 A     h0l: h + l = 2n + 1 Ah    
n2 n1     n    
  h0l: h + l = 2n + 1          
  n2          
119 [I\bar 4 m 2]                    
120 [I\bar 4 c 2] ho0lo A 0kolo A     ho0lo Ah    
c2 c1     c    
  ho0lo          
  c2          
121 [I\bar 4 2 m]                    
122 [I\bar 4 2 d]     hhle, [\bar h h l_{\rm e}]: 2h + le = 4n + 2 A hhle: 2h + le = 4n + 2 A     hhle: 2h + le = 4n + 2 Ah
    d d     d

Point group 4/mmm. The symbol a in the column [u0w] is equivalent to the symbol b in the space groups of the first column.

Space groupIncident-beam direction
[100][001][110][u0w][uuw][uv0]
123 P4/mmm                        
P4/m2/m2/m                        
124 P4/mcc h0lo A 0klo A hhlo A h0lo Ah hhlo Ah    
P4/m2/c2/c c12 c11 c2 c1 c2    
    h0lo          
    c12          
    [hhl_{\rm o}, \bar h h l_{\rm o}]          
    c2          
125 P4/nbm
P4/n2/b2/m
hk0: h + k = 2n + 1
n
ho0l
a2
A 0kol
b1
ho0l
a2
A hk0: h + k = 2n + 1
n
A ho0l
a
Ah     hk0: h + k = 2n + 1
n
Ah
126 P4/nnc
P4/n2/n2/c
hk0: h + k = 2n + 1
n1
A 0kl: k + l = 2n + 1
n21
A hk0: h + k = 2n + 1
n1
A h0l: h + l = 2n + 1
n2
Ah hhlo
c
Ah hk0: h + k = 2n + 1
n1
Ah
  h0l: h + l = 2n + 1
n22
h0l: h + l = 2n + 1
n22
hhlo
c
     
    [hhl_{\rm o}, \bar h h l_{\rm o}]        
    c        
127 P4/mbm ho0l A 0kol A     ho0l Ah        
P4/m21/b2/m a2 b1     a        
    ho0l              
    a2              
128 P4/mnc
P4/m21/n2/c
h0l: h + l = 2n + 1
n2
A 0kl: k + l = 2n + 1
n1
A hhlo
c
A h0l: h + l = 2n + 1
n
Ah hhlo
c
Ah    
    h0l: h + l = 2n + 1          
    n2          
    [hhl_{\rm o}, \bar h h l_{\rm o}]          
    c          
129 P4/nmm
P4/n21/m2/m
hk0: h + k = 2n + 1
n
A     hk0: h + k = 2n + 1
n
A         hk0: h + k = 2n + 1
n
Ah
130 P4/ncc
P4/n21/c2/c
hk0: h + k = 2n + 1
n
h0lo
c12
A 0klo
c11
h0lo
c12
[hhl_{\rm o}, \bar h h l_{\rm o}]
c2
A hk0: h + k = 2n + 1
n
hhlo
c2
A h0lo
c1
Ah hhlo
c2
Ah hk0: h + k = 2n + 1
n
Ah
131 P42/mmc     [hhl_{\rm o}, \bar h h l_{\rm o}] A hhlo A     hhlo Ah    
P42/m2/m2/c     c c     c    
132 P42/mcm h0lo A 0klo A     h0lo Ah        
P42/m2/c2/m c2 c1     c        
    h0lo              
    c2              
133 P42/nbc
P42/n2/b2/c
hk0: h + k = 2n + 1
n
ho0l
a2
A 0kol
b1
ho0l
a2
[hhl_{\rm o}, \bar h h l_{\rm o}]
c
A hk0: h + k = 2n + 1
n
hhlo
c
A ho0l
a
Ah hhlo
c
Ah hk0: h + k = 2n + 1
n
Ah
134 P42/nnm
P42/n2/n2/m
hk0: h + k = 2n + 1
n1
A 0kl: k + l = 2n + 1
n21
A hk0: h + k = 2n + 1
n1
A h0l: h + l = 2n + 1
n2
Ah     hk0: h + k = 2n + 1
n1
Ah
  h0l: h + l = 2n + 1 h0l: h + l = 2n + 1          
  n22 n22          
135 P42/mbc ho0l A 0kol A hhlo A ho0l Ah hhlo Ah    
P42/m21/b2/c a2 b1 c a c    
    ho0l          
    a2          
    [hhl_{\rm o}, \bar h h l_{\rm o}]          
    c          
136 P42/mnm
P42/m21/n2/m
h0l: h + l = 2n + 1
n2
A 0kl: k + l = 2n + 1
n1
A     h0l: h + l = 2n + 1
n
Ah        
    h0l: h + l = 2n + 1              
    n2              
137 P42/nmc
P42/n21/m2/c
hk0: h + k = 2n + 1
n
A [hhl_{\rm o}, \bar h h l_{\rm o}]
c
A hhlo
c
hk0: h + k = 2n + 1
n
A     hhlo
c
Ah hk0: h + k = 2n + 1
n
Ah
138 P42/ncm
P42/n21/c2/m
hk0: h + k = 2n + 1
n
A 0klo
c1
h0lo
c2
A hk0: h + k = 2n + 1
n
A h0lo
c
Ah     hk0: h + k = 2n + 1
n
Ah
  h0lo            
  c2            
139 I4/mmm                        
I4/m2/m2/m                        
140 I4/mcm ho0lo A 0kolo A     ho0lo Ah        
I4/m2/c2/m c2 c1     c        
    ho0lo              
    c2              
141 I41/amd
I41/a2/m2/d
hoko0
a
A [hhl_{\rm e}, \bar h h l_{\rm e}]: 2h + le = 4n + 2
d
A hoko0
a
hhle: 2h + le = 4n + 2
d
A     hhle: 2h + le = 4n + 2
d
Ah hoko0
a
Ah
142 I41/acd
I41/a2/c2/d
hoko0
a
ho0lo
c2
A 0kolo
c1
ho0lo
c2
[hhl_{\rm e}, \bar h h l_{\rm e}]: 2h + le = 4n + 2
d
A hoko0
a
hhle: 2h + le = 4n + 2
d
A ho0lo
c
Ah hhle: 2h + le = 4n + 2
d
Ah hoko0
a
Ah

Point groups [3m, \bar 3m]

Space groupIncident-beam direction
[0001][[11\bar2 0]][[1\bar 1 00]][[11\bar 2 w]][[1\bar 1 0 w]]
156 P3m1                    
157 P31m                    
158 P3c1 [h\bar h 0 l_{\rm o}, 0 h \bar h l_{\rm o}, \bar h 0 h l_{\rm o}] A     [h\bar h 0 l_{\rm o}] A     [h\bar h 0 l_{\rm o}] Ah
c     c     c
159 P31c [hh\overline{2h} l_{\rm o}, h \overline{2 h}h l_{\rm o}, \overline{2 h}h h l_{\rm o}] A [hh\overline{2h} l_{\rm o}] A     [hh\overline{2h} l_{\rm o}] Ah    
c c     c    
160 R3m                    
161 R3c [h\bar h 0 l_{\rm o}, 0 h \bar h l_{\rm o}, \bar h 0 h l_{\rm o}]: h + lo = 3n Ah     [h\bar h 0 l_{\rm o}]: h + lo = 3n Ah     [h\bar h 0 l_{\rm o}]: h + lo = 3n Ah
c     c     c
162 [P\bar 3 1 m]                    
163 [P\bar 3 1 c] [hh\overline{2h} l_{\rm o}, h \overline{2 h}h l_{\rm o}, \overline{2 h}h h l_{\rm o}] A [hh\overline{2h} l_{\rm o}] A     [hh\overline{2h} l_{\rm o}] Ah    
c c     c    
164 [P\bar 3 m 1]                    
165 [P\bar 3 c 1] [h\bar h 0 l_{\rm o}, 0 h \bar h l_{\rm o}, \bar h 0 h l_{\rm o}] A     [h\bar h 0 l_{\rm o}] A     [h\bar h 0 l_{\rm o}] Ah
c     c     c
166 [R\bar 3 m]                    
167 [R\bar 3 c] [h\bar h 0 l_{\rm o}, 0 h \bar h l_{\rm o}, \bar h 0 h l_{\rm o}]: h + lo = 3n Ah     [h\bar h 0 l_{\rm o}]: h + lo = 3n Ah     [h\bar h 0 l_{\rm o}]: h + lo = 3n Ah
  c     c     c

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

Space groupIncident-beam direction
[0001][[11\bar2 0]][[1\bar 1 00]][[11\bar 2 w]][[1\bar10w]]
183 P6mm                    
184 P6cc [h\bar h 0 l_{\rm o}, 0 h \bar h l_{\rm o}, \bar h 0 h l_{\rm o}] A [hh\overline{2h} l_{\rm o}] A [h\bar h 0 l_{\rm o}] A [hh\overline{2h} l_{\rm o}] Ah [h\bar h 0 l_{\rm o}] Ah
c1 c2 c1 c2 c1
[hh\overline{2h} l_{\rm o}, h \overline{2 h}h l_{\rm o}, \overline{2 h}h h l_{\rm o}]        
c2        
185 P63cm [h\bar h 0 l_{\rm o}, 0 h \bar h l_{\rm o}, \bar h 0 h l_{\rm o}] A     [h\bar h 0 l_{\rm o}] A     [h\bar h 0 l_{\rm o}] Ah
c     c     c
186 P63mc [hh\overline{2h} l_{\rm o}, h \overline{2 h}h l_{\rm o}, \overline{2 h}h h l_{\rm o}] A [hh\overline{2h} l_{\rm o}] A     [hh\overline{2h} l_{\rm o}] Ah    
c c     c    
187 [P\bar 6 m 2]                    
188 [P\bar 6 c 2] [h\bar h 0 l_{\rm o}, 0 h \bar h l_{\rm o}, \bar h 0 h l_{\rm o}] A     [h\bar h 0 l_{\rm o}] A     [h\bar h 0 l_{\rm o}] Ah
c     c     c
189 [P\bar 6 2 m]                    
190 [P\bar 6 2 c] [hh\overline{2h} l_{\rm o}, h \overline{2 h}h l_{\rm o}, \overline{2 h}h h l_{\rm o}] A [hh\overline{2h} l_{\rm o}] A     [hh\overline{2h} l_{\rm o}] Ah    
c c     c    
191 P6/mmm                    
192 P6/mcc [h\bar h 0 l_{\rm o}, 0 h \bar h l_{\rm o}, \bar h 0 h l_{\rm o}] A [hh\overline{2h} l_{\rm o}] A [h\bar h 0 l_{\rm o}] A [hh\overline{2h} l_{\rm o}] Ah [h\bar h 0 l_{\rm o}] Ah
c1 c2 c1 c2 c1
[hh\overline{2h} l_{\rm o}, h \overline{2 h}h l_{\rm o}, \overline{2 h}h h l_{\rm o}]        
c2        
193 P63/mcm [h\bar h 0 l_{\rm o}, 0 h \bar h l_{\rm o}, \bar h 0 h l_{\rm o}] A     [h\bar h 0 l_{\rm o}] A     [h\bar h 0 l_{\rm o}] Ah
c     c     c
194 P63/mmc [hh\overline{2h} l_{\rm o}, h \overline{2 h}h l_{\rm o}, \overline{2 h}h h l_{\rm o}] A [hh\overline{2h} l_{\rm o}] A     [hh\overline{2h} l_{\rm o}] Ah    
c c     c    

Point group m3

Space groupIncident-beam direction
[100][110][uv0]
200 Pm3            
[P2/m\bar 3]            
201 Pn3 h0l: h + l = 2n + 1 A hk0: h + k = 2n + 1 A hk0: h + k = 2n + 1 Ah
[P2/n\bar 3] n2 n3 n
  hk0: h + k = 2n + 1    
  n3    
202 Fm3            
[F2/m\bar 3]            
203 Fd3 he0le: he + le = 4n + 2 A heke0: he + ke = 4n + 2 A heke0: he + ke = 4n + 2 Ah
[F2/d\bar 3] d2 d3 d
  heke0: he + ke = 4n + 2    
  d3    
204 Im3            
[I2/m\bar 3]            
205 Pa3 h0lo A hok0 A* hok0 Ah
[P2_1/a\bar 3] c2 a3 a
  hok0    
  a3    
206 Ia3 ho0lo A hoko0 A hoko0 Ah
[I2_1/a\bar 3] c2 a3 a
  hoko0    
  a3    

Point group [\bar 4 3 m]. The symbol a in the column [100] is equivalent to the symbol c in the space groups of the first column.

Space groupIncident-beam direction
[100][110][uuw]
215 [P\bar 4 3 m]            
216 [F\bar 4 3 m]            
217 [I \bar 4 3 m]            
218 [P\bar 4 3 n] [h_{\rm o}kk, h_{\rm o}\bar kk] A hhlo A hhlo Ah
n n n
219 [F\bar 4 3 c] [h_{\rm o}k_{\rm o}k_{\rm o}, h_{\rm o}\bar k_{\rm o}k_{\rm o}] A hoholo A hoholo Ah
a c c
220 [I\bar 4 3 d] [h_{\rm e}kk, h_{\rm e}\bar k k]: 2k + he = 4n + 2 A hhle: 2h + le = 4n + 2 A hhle: 2h + le = 4n + 2 Ah
d d d

Point group m3m. The symbol a in the column [100] is equivalent to the symbol c in the space groups of the first column.

Space groupIncident-beam direction
[100][110][uv0] [uuw]
221 Pm3m                
[P4/m\bar 3 2/m]                
222 Pn3n h0l: h + l = 2n + 1 A hk0: h + k = 2n + 1 A hk0: h + k = 2n + 1 Ah hhlo Ah
[P4/n\bar 3 2/n] n12 n13 n1 n2
  hk0: h + k = 2n + 1 hhlo    
  n13 n2    
  [h_{\rm o}kk, h_{\rm o}\bar kk]      
  n2      
223 Pm3n [h_{\rm o}kk, h_{\rm o}\bar kk] A hhlo A     hhlo Ah
[P4_2/m\bar 32/n] n n     n
224 Pn3m h0l: h + l = 2n + 1 A hk0: h + k = 2n + 1 A hk0: h + k = 2n + 1 Ah    
[P4_2/n\bar 3 2/m] n2 n3 n    
  hk0: h + k = 2n + 1        
  n3        
225 Fm3m                
[F4/m\bar 32/m]                
226 Fm3c [h_{\rm o}k_{\rm o}k_{\rm o}, h_{\rm o}\bar k_{\rm o}k_{\rm o}] A hoholo A     hoholo Ah
[F4/m\bar 3 2/c] a c     c
227 Fd3m he0le: he + le = 4n + 2 A heke0: he + ke = 4n + 2 A heke0: he + ke = 4n + 2 Ah    
[F4_1/d\bar 32/m] d2 d3 d    
  heke0: he + ke = 4n + 2        
  d3        
228 Fd3c he0le: he + le = 4n + 2 A hoholo A heke0: he + ke = 4n + 2 Ah hoholo Ah
[F4_1/d\bar 3 2/c] d2 c d c
  heke0: he + ke = 4n + 2 heke0: he + ke = 4n + 2    
  d3 d3    
  [h_{\rm o}k_{\rm o}k_{\rm o}, h_{\rm o}\bar k_{\rm o}k_{\rm o}]      
  a      
229 Im3m                
[I4/m\bar 32/m]                
230 Ia3d hoko0 A hhle: 2h + le = 4n + 2 A hoko0 Ah hhle: 2h + le = 4n + 2 Ah
[I4_1/a\bar 3 2 /d] a3 d a d
  ho0lo hoko0    
  c2 a3    
  [h_{\rm e}kk, h_{\rm e}\bar kk]: 2k + he = 4n + 2      
  d      
[Figure 2.5.3.13]

Figure 2.5.3.13 | top | pdf |

Illustration of dynamical extinction lines appearing in HOLZ reflections due to glide planes. Black circles and circled crosses show kinematically allowed and kinematically forbidden reflections, respectively. (a) a glide in the (001) plane. (b) [100] incidence: dynamical extinction lines are formed in HOLZ reflections on both sides of the incident beam (type A). (c) [110] incidence: an extinction line is formed at a HOLZ reflection on one side of the incident beam because on the other side the Ewald sphere intersects an allowed HOLZ reflection (type A*). (d) An incidence between [100] and [110]: an extinction line is formed at a HOLZ reflection on one side of the incident beam because on the other side the Ewald sphere does not intersect a HOLZ reflection (type Ah).

The first column of Table 2.5.3.12[link] list the space groups and the following columns show the type of the extinction lines for possible incident-beam directions. In each pair of columns, the left-hand column gives the reflection indices of the extinction line and the symmetry elements causing the extinction and the right-hand column gives the type of extinction. The first suffix 1, 2 or 3 of a glide symbol distinguishes the first, the second or the third glide plane of a 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 suffix o of a reflection index implies that the index is odd-order. Figs. 2.5.3.14[link](a) and (b) were taken for FeS2, space group [P2_1/a\bar{3}], with incident-beam directions of [100] and [110]. Inserts show enlarged HOLZ patterns for ease of viewing. Extinction lines of type A are seen in the hok0 HOLZ reflections in Fig. 2.5.3.14[link](a) due to the b-glide plane (equivalent to the a-glide plane in the space-group symbol) parallel to the (001) plane. An extinction line A* is seen in an hok0 HOLZ reflection in Fig. 2.5.3.14[link](b) due to the same glide plane as that of Fig. 2.5.3.14[link](a). It should be noted that extinction lines in HOLZ reflections are better observed in thinner specimen areas than those suitable for the observation of the extinction lines in ZOLZ reflections, because the profiles of HOLZ reflections are concentrated into small areas of CBED discs in thicker specimens.

[Figure 2.5.3.14]

Figure 2.5.3.14 | top | pdf |

HOLZ CBED pattern of FeS2. (a) [100] incidence: type A dynamical extinction lines are seen clearly in the enlarged insets. (b) [110] incidence: a type A* dynamical extinction line is seen clearly in the enlarged insets.

In summary, the use of not only ZOLZ, but also HOLZ extinction lines is recommended for space-group determination.

References

Nagasawa, T. (1983). Master of Physics Thesis, Tohoku University, Japan.
Tanaka, M., Terauchi, M. & Kaneyama, T. (1988). Convergent-Beam Electron Diffraction II. Tokyo: JEOL Ltd.








































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