Introduction and point groups Stereographic projections Low symmetry systems Space groups
Space groups
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1 Space groups Ex: 222 1 C 2 C 2 ' C 2 " 1 1 C 2 C 2 ' C 2 " C 2 C 2 1 C 2 " C 2 ' C 2 ' C 2 ' C 2 " 1 C 2 C 2 " C 2 " C 2 ' C 2 1
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Space groups. Ex: 222. Space groups. Ex: 222. represents all elements in the lattice translation group. Space groups. Ex: 222. represents all elements in the lattice translation group. C 2 C 2 ' = C 2 " 1 t 2 t = 3 t. Space groups. Ex: 222. represents all elements in - PowerPoint PPT Presentation
Transcript of Space groups
1
Space groupsEx: 222
1 C2 C2' C2"
1 1 C2 C2' C2"C2 C2 1 C2" C2'
C2' C2' C2" 1 C2
C2" C2" C2' C2 1
2
Space groupsEx: 222
1t 2t 3t
1t 2t 3t1t 1t 3t 2t2t 2t 3t 1t3t 3t 2t 1t
represents all elements inthe lattice translation group
1 C2 C2' C2"
1 1 C2 C2' C2"C2 C2 1 C2" C2'
C2' C2' C2" 1 C2
C2" C2" C2' C2 1
3
Space groupsEx: 222
represents all elements inthe lattice translation group
C2 C2' = C2"
1t 2t = 3t
1t 2t 3t
1t 2t 3t1t 1t 3t 2t2t 2t 3t 1t3t 3t 2t 1t
1 C2 C2' C2"
1 1 C2 C2' C2"C2 C2 1 C2" C2'
C2' C2' C2" 1 C2
C2" C2" C2' C2 1
4
Space groupsEx: 222
represents all elements inthe lattice translation group
C2 C2' = C2"
1t 2t = 3t
C2,1 C2',2
= C2", 3
1t 2t 3t
1t 2t 3t1t 1t 3t 2t2t 2t 3t 1t3t 3t 2t 1t
1 C2 C2' C2"
1 1 C2 C2' C2"C2 C2 1 C2" C2'
C2' C2' C2" 1 C2
C2" C2" C2' C2 1
5
Space groupsEx: 222
C2 C2' = C2"
1t 2t = 3t
C2,1 C2',2
= C2", 3
x 1t 0 a/2 b/2 c/2 –– a/2+b/2
y 2t 0 a/2 b/2 c/2 c/2 a/2+b/2
z 3t 0 –– –– –– c/2 ––P222 P212'2 P2'212 P2'2'2 P22'21 P21212
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Space groupsEx: 422
1 C4 C42 C4
3 C2 C2' C2" C2"'
1 1 C4 C42 C4
3 C2 C2' C2" C2"'C4 C4 C4
2 C43 1 C2"' C2 C2' C2"
C42 C4
2 C43 1 C4 C2" C2"' C2 C2'
C43 C4
3 1 C4 C42 C2' C2" C2"' C2
C2 C2 C2' C2" C2"' 1 C4 C42 C4
3
C2' C2' C2" C2"' C2 C43 1 C4 C4
2
C2" C2" C2"' C2 C2' C42 C4
3 1 C4
C2"' C2"' C2 C2' C2" C4 C42 C4
3 1
C2t1
C2't2
C2"t3
C2"'t4
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Space groupsEx: 422
tz tz2 tz
3 t1 t2 t3 t4
tz tz2 tz
3 t1 t2 t3 t4
tz tz tz2 tz
3 t4 t1 t2 t3
tz2 tz
2 tz3 tz t3 t4 t1 t2
tz3 tz
3 tz tz2 t2 t3 t4 t1
t1 t1 t2 t3 t4 tz tz2 tz
3
t2 t2 t3 t4 t1 tz3 tz tz
2
t3 t3 t4 t1 t2 tz2 tz
3 tz
t4 t4 t1 t2 t3 tz tz2 tz
3
C2t1
C2't2
C2"t3
C2"'t4
8
Space groupsEx: 422
tz tz2 tz
3 t1 t2 t3 t4
tz tz2 tz
3 t1 t2 t3 t4
tz tz tz2 tz
3 t4 t1 t2 t3
tz2 tz
2 tz3 tz t3 t4 t1 t2
tz3 tz
3 tz tz2 t2 t3 t4 t1
t1 t1 t2 t3 t4 tz tz2 tz
3
t2 t2 t3 t4 t1 tz3 tz tz
2
t3 t3 t4 t1 t2 tz2 tz
3 tz
t4 t4 t1 t2 t3 tz tz2 tz
3
C2t1
C2't2
C2"t3
C2"'t4
[100]
[110]
[110]
[010]
tz = c/4, 2c/4, 3c/4, 0 41 42 43
t1 = a/2, b/2, c/2, 0
but t2 must be, e.g., a/2+b/2tz t1 = t2
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Space groupsEx: 422
C2t1
C2't2
C2"t3
C2"'t4
[100]
[110]
[110]
[010]
tz = c/4, 2c/4, 3c/4, 0 41 42 43
t1 = a/2, b/2, c/2, 0
but t2 must be, e.g., a/2+b/2tz t1 = t2
P422 - tz = ti = 0
10
Space groupsEx: 422
C2t1
C2't2
C2"t3
C2"'t4
[100]
[110]
[110]
[010]
tz = c/4, 2c/4, 3c/4, 0 41 42 43
t1 = a/2, b/2, c/2, 0
but t2 must be, e.g., a/2+b/2tz t1 = t2
P422 - tz = ti = 0
11
Space groupsEx: 422
C2t1
C2't2
C2"t3
C2"'t4
[100]
[110]
[110]
[010]
tz = c/4, 2c/4, 3c/4, 0 41 42 43
t1 = a/2, b/2, c/2, 0
but t2 must be, e.g., a/2+b/2tz t1 = t2
P4212 - tz = 0, t1 = t2 = a/2 + b/2
12
Space groupsEx: 422
C2t1
C2't2
C2"t3
C2"'t4
[100]
[110]
[110]
[010]
tz = c/4, 2c/4, 3c/4, 0 41 42 43
t1 = a/2, b/2, c/2, 0
but t2 must be, e.g., a/2+b/2tz t1 = t2
P4212 - tz = 0, t1 = t2 = a/2 + b/2
13
Space groupsEx: 422
C2t1
C2't2
C2"t3
C2"'t4
[100]
[110]
[110]
[010]
tz = c/4, 2c/4, 3c/4, 0 41 42 43
t1 = a/2, b/2, c/2, 0
but t2 must be, e.g., a/2+b/2
tz t3 = t4 = c/4tz t4 = t1 = c/2tz t1 = t2 = 3c/4
P4122 - tz = c/4, t3 = 0, t4 = c/4
14
Space groupsEx: 422
C2t1
C2't2
C2"t3
C2"'t4
[100]
[110]
[110]
[010]
tz = c/4, 2c/4, 3c/4, 0 41 42 43
t1 = a/2, b/2, c/2, 0
but t2 must be, e.g., a/2+b/2
tz t3 = t4 = c/4tz t4 = t1 = c/2tz t1 = t2 = 3c/4
P4122 - tz = c/4, t3 = 0, t4 = c/4
1/8
1/8
1/81/81/8
1/8
1/8
1/8 1/8 1/8 3/8
3/8
3/8
3/83/8
3/8
3/8
3/8
3/83/8
1/41/41/4
1/41/41/4
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Space groupsPmm2 & Cmm2
1 xm ym z2
1 1 xm ym z2xm xm 1 z2 ymym ym z2 1 xmz2 z2 ym xm 1
1t 2t 3t
1t 2t 3t1t 1t 32 2t2t 2t 3t 1tzt zt 2t 1t
Xm ym = z2
1t 2t = 3t
sm1 sm2
= Aπ, 3
16
Space groups
1t 2t = 3t
consists of translations for C lattice (includes a/2 + b/2)
1 = 0, b/2, c/2
2 = 0, a/2, c/2
3 = 0, c/21t w/ 3t = 0
1t = 2t = 3t = 0 Cmm2
1can only be 0, b/2, c/2, b/2 + c/2
17
Space groups
1t 2t = 3t 1 = 0, b/2, c/2
2 = 0, a/2, c/2
3 = 0, c/21t w/ 3t = 0
1t = a/2 3t = 0 2t = a/2 Cmm2
1can only be 0, b/2, c/2, b/2 + c/2
18
Space groups
1t 2t = 3t 1 = 0, b/2, c/2
2 = 0, a/2, c/2
3 = 0, c/21t w/ 3t = 0
1t = a/2 + b/2 3t = 0 2t = a/2 + b/2 Cmm2
1can only be 0, b/2, c/2, b/2 + c/2
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Space groups
1t 2t = 3t 1 = 0, b/2, c/2
2 = 0, a/2, c/2
3 = 0, c/21t w/ 3t = 0
1t = c/2 3t = 0 2t = c/2 Ccc2
1can only be 0, b/2, c/2, b/2 + c/2
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Space groups
1t 2t = 3t 1 = 0, b/2, c/2
2 = 0, a/2, c/2
3 = 0, c/21t w/ 3t = 0
1t = a/2 + c/2 3t = 0 2t = a/2 + c/2 Ccc2
1can only be 0, b/2, c/2, b/2 + c/2
21
Space groups
1t 2t = 3t 1 = 0, b/2, c/2
2 = 0, a/2, c/2
3 = 0, c/21t w/ 3t = c/2
1t = 0 3t = c/2 2t = c/2 Cmc21
1can only be 0, b/2, c/2, b/2 + c/2
22
Space groups
1t 2t = 3t 1 = 0, b/2, c/2
2 = 0, a/2, c/2
3 = 0, c/21t w/ 3t = c/2
1t = 0 3t = c/2 2t = c/2 Cmc21
1can only be 0, b/2, c/2, b/2 + c/2
All other translation combinations equiv-alent to Cmm2, Ccc2, or Cmc21
