Serie 06 - Crystallography

Übungsaufgaben zur Kristallographie
HS14
Serie 6
„International Tables for Crystallography“
Betrachten Sie die Kopien aus den „International Tables“ zu den Raumgruppen
P4/mmm und Pmmm.
a) Welche Symmetrieelemente sind im Symmetriegerüst der Raumgruppe P4/mmm zu
erkennen? Ordnen Sie diese jeweils den kristallographisch relevanten Richtungen des
Kristallsystems zu.
b) Welches sind die Atomlagen mit höchster Symmetrie in P4/mmm und wie lautet ihre
Symmetrie?
c) Aus welchen kristallographischen Richtungen betrachtet ist an einem Objekt mit
Symmetrie P4/mmm nur eine orthorhombische Symmetrie erkennbar? Welcher
Ebenengruppe entspricht diese Symmetrie?
Die Raumgruppe Pmmm ist eine Untergruppe von P4/mmm. Der Index der GruppeUntergruppe-Beziehung ist 2.
d) Welche Symmetrieelemente entfallen im Vergleich zu P4/mmm?
e) Zu welchen Atomlagen gehört die Position 0,0,0 in den Raumgruppen P4/mmm und
Pmmm und was unterscheidet diese?
f) Wie verhält es sich mit der Position 0.10, 0.33, 0? Zu welcher Lage gehören diese
Koordinaten? Berechnen Sie die Koordinaten aller entstehenden Atompositionen in den
Raumgruppen P4/mmm und Pmmm. (Geben Sie die Koordinaten innerhalb einer
Elementarzelle an; 0 ≤ x, y, z < 1.)
g) In Raumgruppe Pmmm wird von einem Atom an 0.10, 0.33, 0 nur ein Teil der
Positionen erzeugt, die in P4/mmm entstehen. Welche zusätzliche Atomlage müsste
man in Raumgruppe Pmmm hinzufügen um die übrigen Koordinaten zu erzeugen?
Welche der Positionen befindet sich innerhalb der asymmetrischen Einheit?
Walter Steurer, Thomas Weber, Julia Dshemuchadse, Laboratorium für Kristallographie
http://www.crystal.mat.ethz.ch/education/courses/HS2014/Kristallographie
International Tables for Crystallography (2006). Vol. A, Space group 123, pp. 430–431.
P 4/m m m
D14h
No. 123
P 4/m 2/m 2/m
4/m m m
Tetragonal
Patterson symmetry P 4/m m m
Origin at centre (4/m m m)
Asymmetric unit
0 ≤ x ≤ 12 ;
0 ≤ y ≤ 21 ;
0 ≤ z ≤ 12 ;
x≤y
Symmetry operations
(1)
(5)
(9)
(13)
1
2 0, y, 0
1̄ 0, 0, 0
m x, 0, z
(2)
(6)
(10)
(14)
2
2
m
m
0, 0, z
x, 0, 0
x, y, 0
0, y, z
(3)
(7)
(11)
(15)
4+
2
4̄+
m
0, 0, z
x, x, 0
0, 0, z; 0, 0, 0
x, x̄, z
(4)
(8)
(12)
(16)
4−
2
4̄−
m
0, 0, z
x, x̄, 0
0, 0, z; 0, 0, 0
x, x, z
Maximal non-isomorphic subgroups
I
[2] P 4̄ m 2 (115)
1; 2; 7; 8; 11; 12; 13; 14
IIa
IIb
[2] P 4̄ 2 m (111)
1; 2; 5; 6; 11; 12; 15; 16
[2] P 4 m m (99)
1; 2; 3; 4; 13; 14; 15; 16
[2] P 4 2 2 (89)
1; 2; 3; 4; 5; 6; 7; 8
[2] P 4/m 1 1 (P4/m, 83)
1; 2; 3; 4; 9; 10; 11; 12
[2] P 2/m 1 2/m (C m m m, 65) 1; 2; 7; 8; 9; 10; 15; 16
[2] P 2/m 2/m 1 (P m m m, 47) 1; 2; 5; 6; 9; 10; 13; 14
none
[2] P 42 /m c m (c# = 2c) (132); [2] P 42 /m m c (c# = 2c) (131); [2] P 4/m c c (c# = 2c) (124);
[2] C 4/e m m (a# = 2a, b# = 2b) (P 4/n m m, 129); [2] C 4/m m d (a# = 2a, b# = 2b) (P 4/m b m, 127);
[2] C 4/e m d (a# = 2a, b# = 2b) (P 4/n b m, 125); [2] F 4/m m c (a# = 2a, b# = 2b, c# = 2c) (I 4/m c m, 140);
[2] F 4/m m m (a# = 2a, b# = 2b, c# = 2c) (I 4/m m m, 139)
Maximal isomorphic subgroups of lowest index
IIc [2] P 4/m m m (c# = 2c) (123); [2] C 4/m m m (a# = 2a, b# = 2b) (P 4/m m m, 123)
Minimal non-isomorphic supergroups
I
[3] P m 3̄ m (221)
II
[2] I 4/m m m (139)
Copyright  2006 International Union of Crystallography
430
No. 123
CONTINUED
P 4/m m m
Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5); (9)
Positions
Coordinates
Multiplicity,
Wyckoff letter,
Site symmetry
16
u
1
8
t
8
Reflection conditions
General:
(1)
(5)
(9)
(13)
x, y, z
x̄, y, z̄
x̄, ȳ, z̄
x, ȳ, z
(2)
(6)
(10)
(14)
(3)
(7)
(11)
(15)
x̄, ȳ, z
x, ȳ, z̄
x, y, z̄
x̄, y, z
(4)
(8)
(12)
(16)
ȳ, x, z
y, x, z̄
y, x̄, z̄
ȳ, x̄, z
y, x̄, z
ȳ, x̄, z̄
ȳ, x, z̄
y, x, z
no conditions
Special:
.m.
1
2
1
2
x, , z
x̄, , z̄
x̄, , z
x, , z̄
1
2
1
2
, x̄, z
, x̄, z̄
no extra conditions
s
.m.
x, 0, z
x̄, 0, z̄
x̄, 0, z
x, 0, z̄
0, x, z
0, x, z̄
0, x̄, z
0, x̄, z̄
no extra conditions
8
r
..m
x, x, z
x̄, x, z̄
x̄, x̄, z
x, x̄, z̄
x̄, x, z
x, x, z̄
x, x̄, z
x̄, x̄, z̄
no extra conditions
8
q
m..
x, y, 21
x̄, y, 21
x̄, ȳ, 12
x, ȳ, 12
ȳ, x, 12
y, x, 12
y, x̄, 21
ȳ, x̄, 21
no extra conditions
8
p
m..
x, y, 0
x̄, y, 0
x̄, ȳ, 0
x, ȳ, 0
ȳ, x, 0
y, x, 0
y, x̄, 0
ȳ, x̄, 0
no extra conditions
4
o
m 2 m.
x, 12 , 21
x̄, 21 , 12
1
2
, x, 12
1
2
, x̄, 21
no extra conditions
4
n
m 2 m.
x, 12 , 0
x̄, 21 , 0
1
2
, x, 0
1
2
, x̄, 0
no extra conditions
4
m
m 2 m.
x, 0, 21
x̄, 0, 12
0, x, 21
0, x̄, 12
no extra conditions
4
l
m 2 m.
x, 0, 0
x̄, 0, 0
0, x, 0
0, x̄, 0
no extra conditions
4
k
m . 2m
x, x, 12
x̄, x̄, 12
x̄, x, 12
x, x̄, 21
no extra conditions
4
j
m . 2m
x, x, 0
x̄, x̄, 0
x̄, x, 0
x, x̄, 0
no extra conditions
4
i
2 m m.
0, 12 , z
1
2
, 0, z
0, 21 , z̄
1
2
2
h
4mm
1
2
, 21 , z
1
2
, 21 , z̄
no extra conditions
2
g
4mm
0, 0, z
0, 0, z̄
no extra conditions
2
f
m m m.
0, 12 , 0
1
2
, 0, 0
hkl : h + k = 2n
2
e
m m m.
0, 12 , 21
1
2
, 0, 21
hkl : h + k = 2n
1
d
4/m m m
1
2
, 21 , 12
no extra conditions
1
c
4/m m m
1
2
, 21 , 0
no extra conditions
1
b
4/m m m
0, 0, 12
no extra conditions
1
a
4/m m m
0, 0, 0
no extra conditions
1
2
1
2
, x, z
, x, z̄
1
2
1
2
, 0, z̄
hkl : h + k = 2n
Symmetry of special projections
Along [001] p 4 m m
a# = a
b# = b
Origin at 0, 0, z
Along [110] p 2 m m
b# = c
a# = 21 (−a + b)
Origin at x, x, 0
Along [100] p 2 m m
b# = c
a# = b
Origin at x, 0, 0
(Continued on preceding page)
431
International Tables for Crystallography (2006). Vol. A, Space group 47, pp. 262–263.
Pmmm
D12h
No. 47
P 2/m 2/m 2/m
mmm
Orthorhombic
Patterson symmetry P m m m
Origin at centre (m m m)
Asymmetric unit
0 ≤ x ≤ 12 ;
0 ≤ y ≤ 21 ;
0≤z≤
1
2
Symmetry operations
(1) 1
(5) 1̄ 0, 0, 0
(2) 2 0, 0, z
(6) m x, y, 0
(3) 2 0, y, 0
(7) m x, 0, z
(4) 2 x, 0, 0
(8) m 0, y, z
Maximal non-isomorphic subgroups (continued)
IIa none
IIb [2] P m m a (a" = 2a) (51); [2] P m a m (a" = 2a) (P m m a, 51); [2] P m a a (a" = 2a) (P c c m, 49); [2] P b m m (b" = 2b) (P m m a, 51);
[2] P m m b (b" = 2b) (Pm m a, 51); [2] P b m b (b" = 2b) (P c c m, 49); [2] Pc m m (c" = 2c) (P m m a, 51);
[2] P m c m (c" = 2c) (Pm m a, 51); [2] P c c m (c" = 2c) (49); [2] A e m m (b" = 2b, c" = 2c) (C m m e, 67);
[2] A m m m (b" = 2b, c" = 2c) (C m m m, 65); [2] B m e m (a" = 2a, c" = 2c) (C m m e, 67); [2] B m m m (a" = 2a, c" = 2c) (C m m m, 65);
[2] C m m e (a" = 2a, b" = 2b) (67); [2] C m m m (a" = 2a, b" = 2b) (65); [2] F m m m (a" = 2a, b" = 2b, c" = 2c) (69)
Maximal isomorphic subgroups of lowest index
IIc [2] P m m m (a" = 2a or b" = 2b or c" = 2c) (47)
Minimal non-isomorphic supergroups
I
[2] P 4/m m m (123); [2] P 42 /m m c (131); [3] P m 3̄ (200)
II
[2] A m m m (C m m m, 65); [2] B m m m (C m m m, 65); [2] C m m m (65); [2] I m m m (71)
Copyright  2006 International Union of Crystallography
262
No. 47
CONTINUED
Pmmm
Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (3); (5)
Positions
Coordinates
Multiplicity,
Wyckoff letter,
Site symmetry
8
α
1
Reflection conditions
General:
(1) x, y, z
(5) x̄, ȳ, z̄
(2) x̄, ȳ, z
(6) x, y, z̄
(3) x̄, y, z̄
(7) x, ȳ, z
(4) x, ȳ, z̄
(8) x̄, y, z
no conditions
Special: no extra conditions
4
z
..m
x, y, 12
x̄, ȳ, 12
x̄, y, 21
x, ȳ, 12
4
y
..m
x, y, 0
x̄, ȳ, 0
x̄, y, 0
x, ȳ, 0
4
x
.m.
x, 21 , z
x̄, 12 , z
x̄, 12 , z̄
x, 21 , z̄
4
w
.m.
x, 0, z
x̄, 0, z
x̄, 0, z̄
x, 0, z̄
4
v
m..
1
2
4
u
m..
0, y, z
2
t
mm2
1
2
, 21 , z
1
2
, 12 , z̄
2
s
mm2
1
2
, 0, z
1
2
, 0, z̄
2
r
mm2
0, 12 , z
0, 12 , z̄
2
q
mm2
0, 0, z
0, 0, z̄
2
p
m2m
1
2
, y, 12
1
2
, ȳ, 21
1
h
mmm
1
2
2
o
m2m
1
2
, y, 0
1
2
, ȳ, 0
1
g
mmm
0, 12 , 21
2
n
m2m
0, y, 12
0, ȳ, 21
1
f
mmm
1
2
2
m
m2m
0, y, 0
0, ȳ, 0
1
e
mmm
0, 21 , 0
2
l
2mm
x, 21 , 12
x̄, 21 , 21
1
d
mmm
1
2
2
k
2mm
x, 12 , 0
x̄, 12 , 0
1
c
mmm
0, 0, 12
2
j
2mm
x, 0, 21
x̄, 0, 21
1
b
mmm
1
2
2
i
2mm
x, 0, 0
x̄, 0, 0
1
a
mmm
0, 0, 0
, y, z
1
2
, ȳ, z
1
2
0, ȳ, z
, y, z̄
0, y, z̄
1
2
, ȳ, z̄
0, ȳ, z̄
, 12 , 21
, 12 , 0
, 0, 21
, 0, 0
Symmetry of special projections
Along [010] p 2 m m
a" = c
b" = a
Origin at 0, y, 0
Along [100] p 2 m m
a" = b
b" = c
Origin at x, 0, 0
Along [001] p 2 m m
a" = a
b" = b
Origin at 0, 0, z
Maximal non-isomorphic subgroups
I
[2] P m m 2 (25)
1; 2; 7; 8
[2] P m 2 m (P m m 2, 25)
[2] P 2 m m (P m m 2, 25)
[2] P 2 2 2 (16)
[2] P 1 1 2/m (P2/m, 10)
[2] P 1 2/m 1 (P2/m, 10)
[2] P 2/m 1 1 (P2/m, 10)
1;
1;
1;
1;
1;
1;
3;
4;
2;
2;
3;
4;
6;
6;
3;
5;
5;
5;
8
7
4
6
7
8
(Continued on preceding page)
263