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Chapter 7 Space Group
Reading Assignment:1. W. B-Ott, Crystallography-chapter 9
Symmetry Aspects of M. C. Escher’s Periodic Drawings
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Contents
Screw AxisScrew Axis
17 Plane Group17 Plane Group
Space GroupSpace Group
3
1
2
4
Glide PlaneGlide Plane
5 ExamplesExamples
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Space Group
- 32 point groups- symmetry groups of many molecules
and of all crystals so long as morphology
is considered
- space group- symmetry of crystal lattices and
crystal structures
14 Bravais lattice
centered lattices- new symmetry operations
reflection + translation
rotation + translation
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Space Lattice
- 14 Bravais lattice
The 14 Bravais lattice represent the 14 and only way in whichit is possible to fill space by a three-dimensional periodic arrayof points.
W. B-Ott, Crystallography
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정수진, 결정학개론
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New Symmetry Operations
i) orthorhombic C-lattice
reflection at
+ translation
glide reflectionglide plane (b-glide)
14 , ,y z
2b
1 12 20,0,0 , ,0→
ii) orthorhombic I-lattice
rotation about at
+translation
screw rotationscrew axis (21-screw)
1 14 4, , z
2c
1 1 12 2 20,0,0 , ,→
W. B-Ott, Crystallography
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Compound Symmetry Operation
W. B-Ott, Crystallography
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Glide Plane
i) reflectionii) translation by the vector parallel to the plane of
glide reflection where is called glide component
gg
is one half of a lattice translation parallel to the glide plane
g12g t=
정수진, 결정학개론
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Mirror Plane vs. Glide Plane
- glide plane can occur in an orientation that is possiblefor a mirror plane
W. B-Ott, Crystallography
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Glide Plane- orthorhombic 2 2 2
m m mP(100), (010), (001) possible
glide plane parallel to (100)1 1 1 12 2 2 4glide component , , , b c b c b c+ ±
b-glide c-glide n-glide d-glide
W. B-Ott, Crystallography
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12 along the line parallel to the projection plane
12 normal to the projection plane
12 in the direction of the arrow
Glide Plane
W. B-Ott, Crystallography
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12 in the direction of the arrow
12
1 2
along the line parallel to the projection plane
combined with normal to the projection plane
Glide Plane
W. B-Ott, Crystallography
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Screw Axis
- rotation
- translation by a vector parallel to the axiswhere is called the screw component
ss
φ
2 ( =1,2,3,4,6)XXπφ =
s
s
t
p=0,1,2...,X-1ps tX
=
0 1 1, ,.....p XX X X X −=W. B-Ott, Crystallography
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Screw Axis
W. B-Ott, Crystallography
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Screw Axis
W. B-Ott, Crystallography
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Screw Axis
W. B-Ott, Crystallography
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17 Plane Groups- 5 plane lattices + 10 plane point groups+glide plane- criterion: lattice itself must possess at least
the symmetry of the motif
A. Putnis, Introduction to Mineral Sciences
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17 Plane Groups
A. Putnis, Introduction to Mineral Sciences
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17 Plane Groups
A. Putnis, Introduction to Mineral Sciences
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17 Plane Groups
A. Putnis, Introduction to Mineral Sciences
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17 Plane Groups
A. Putnis, Introduction to Mineral Sciences
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17 Plane Groups
A. Putnis, Introduction to Mineral Sciences
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17 Plane Groups
A. Putnis, Introduction to Mineral Sciences
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17 Plane Groups
A. Putnis, Introduction to Mineral Sciences
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17 Plane Groups
A. Putnis, Introduction to Mineral Sciences
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17 Plane Groups
A. Putnis, Introduction to Mineral Sciences
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17 Plane Groups
A. Putnis, Introduction to Mineral Sciences
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17 Plane Groups
A. Putnis, Introduction to Mineral Sciences
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International Tables for X-ray Crystallography
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International Tables for X-ray Crystallography
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International Tables for X-ray Crystallography
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123
456
7
International Tables for X-ray Crystallography
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1
2
3
short international (Hermann-Mauguin) symbolfor the plane group
short international (Hermann-Mauguin) symbolfor the point group
crystal systemsequential number of plane groupfull international (Hermann-Mauguin) symbol
for the plane grouppatterson symmetrydiagram for the symmetry elements and thegeneral position
45
67
International Tables for X-ray Crystallography
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,, , ,,
International Tables for X-ray Crystallography
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International Tables for X-ray Crystallography
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Flow Diagram for Identifying Plane Groups
A. Putnis, Introduction to Mineral Sciences
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Example I
3 1mP
Symmetry Aspects of M. C. Escher’s Periodic Drawings
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Example II
2ggPhttp://www.clarku.edu/~djoyce/wallpaper/
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Pg
Symmetry Aspects of M. C. Escher’s Periodic Drawings
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Space Groups
-Bravais lattice + point group 230 space groups+ screw axis+ glide plane
- Bravais lattice + point group= 73- Bravais lattice + screw axis = 41- Bravais lattice + glide plane = 116
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Space Groups-monoclinic
1 34 4
1 1 1 3 3 11 4 4 2 4 4 2
2 2, (a-glide at x, , , x, ,
2 screw axis at , ,0, , , , , ,0, , , )
P C z zm m
y y y y
- monoclinic system- highest symmetry
c
a 12
0,1
14
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Space Groups-monoclinic
1
2 subgroup 2,
2 2, c-glide m
mm
−
→ →
-13 monoclinic space groups glide plane parallel to (101)
W. B-Ott, Crystallography
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Space Groups-monoclinic
W. B-Ott, Crystallography
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2mP
International Tables for X-ray Crystallography
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12cP
International Tables for X-ray Crystallography
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2mC
a
c
+
+−
− ,,
+
+−
− ,,
International Tables for X-ray Crystallography
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International Tables for X-ray Crystallography
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Space Groups-Pmm2
short space group symbolSchoenflies symbolpoint groupcrystal systemnumber of space groupfull space group symbol
projection of symmetryelements
projection of generalposition
International Tables for X-ray Crystallography
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International Tables for X-ray Crystallography
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Space Groups-Pmm2
- origin
(i) all centrosymmetric space groups are describedwith an inversion centers as origin.
a second description is given if a space groupcontains points of high site symmetry that do notcoincide with a center of symmetry
(ii) for non-centrosymmetric space groups, the originis at a point of highest site symmetry.if no site symmetry is higher than 1, the origin isplaced on a screw axis or a glide plane, or atthe intersection of several such symmetry elements
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International Tables for X-ray Crystallography
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International Tables for X-ray Crystallography
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International Tables for X-ray Crystallography
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Space Groups-Pmm2
- Pmm2- for a point (general point)symmetry element generates
- The number of equivalent points in the unit cell iscalled its multiplicity.
- A general position is a set of equivalent points withpoint symmetry (site symmetry) 1.
, ,x y z, , ; , , ; , ,x y z x y z x y z
, , ; , , ; , , ; , , are equivalent multi ( plicity f )o 4x y z x y z x y z x y z
W. B-Ott, Crystallography
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Space Groups-Pmm2
- move a point on to mirror plane at
multiplicity of 2as long as the point remains on the mirror plane,its multiplicity is unchanged- degree of freedom 2
- A special position is a set of equivalent points withpoint symmetry (site symmetry) higher than 1.
, ,x y z12, , and 1- , , coalesce to , ,x y z x y z y z
12 , ,y z
12,1 , and 1- ,1 , coalesce to ,1 ,x y z x y z y z− − −
W. B-Ott, Crystallography
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Space Groups-Pmm2
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Space Groups-multiplicity
- screw axis and glide plane do not alter the multiplicityof a point
-Pna21: orthorhombic n-glide normal to a-axisa-glide normal to b-axis
21 screw axis along c-axis
no special position
W. B-Ott, Crystallography
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Space Groups-asymmetric unit
-The asymmetric unit of a space group is the smallestpart of the unit cell from which the whole cell may befilled by the operation of all the symmetry operations.It volume is given by:
ex) Pmm2- multiplicity of 4, vol. of asymm=1/4 unit cell
-An asymmetric unit contains all the information necessaryfor the complete descript of a crystal structure.
unit cellasymm.unit multiplicity of general position
VV =
1 12 20 , 0 , 0 1x y z≤ ≤ ≤ ≤ ≤ ≤
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Space Groups-P61
31
21
W. B-Ott, Crystallography
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International Tables for X-ray Crystallography
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International Tables for X-ray Crystallography
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orthographic representation
International Tables for X-ray Crystallography
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International Tables for X-ray Crystallography
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International Tables for X-ray Crystallography
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3-fold rotation
projection on x,y,0
mirror plane at x,x,z
4-fold rotation at 0,0,z
48-fold general position
W. B-Ott, Crystallography
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Rutile, TiO2
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Rutile, TiO2
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International Tables for X-ray Crystallography
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International Tables for X-ray Crystallography
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Perovskite, CaTiO3
- Ca-cornerO- face centeredTi- body centered
- high temperature- cubicPm3m (No.221)
- Ti- 6OO- 4Ca+2TiCa- 12O
1 1 12 2 2
1 1 1 1 1 10 0 02 2 2 2 2 2
: 1 , 3 , 0,0,0
: 1 , 3 , , ,
: 3 , 4 / , , , , , , ,
Ca a m m
Ti b m m
O c mmm
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International Tables for X-ray Crystallography
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International Tables for X-ray Crystallography
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Perovskite, BaTiO3
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Diamond, C 4 21 3 (No.227)d mF
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International Tables for X-ray Crystallography
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International Tables for X-ray Crystallography
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Zinc Blende, ZnS
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Zinc Blende, ZnS
-diamond derivative structure-Zn and S replace the C atoms-Zn cubic close packingS ½ tetrahedral site
-Zn and S cubic close packing displaced by-Space group
1 1 1, ,4 4 4
1 1 1, ,4 4 4
43 (No.216)
Zn: 4a, 43 , 0,0,0
Zn: 4c, 43 ,
F m
m
m
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Zinc Blende, ZnS
International Tables for X-ray Crystallography
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Zinc Blende, ZnS