Part 5 Space groups - Startseite 5 Space groups 5.1 Glide planes 5.2 Screw axes ... 5.5 Space group...
Transcript of Part 5 Space groups - Startseite 5 Space groups 5.1 Glide planes 5.2 Screw axes ... 5.5 Space group...
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Part 5 Space groups
5.1 Glide planes
5.2 Screw axes
5.3 The 230 space groups
5.4 Properties of space groups
5.5 Space group and crystal structure
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Glide planes and screw axes
32 point groups are symmetry groups of all crystals, so long as only morphology.Space groups give the symmetry only of crystal lattices, but also crystal structures.
For example, space groups of centered lattices contains compounds symmetryoperations which arised through: 1) reflection and translation
2) rotation and translation
Glide reflection:reflection through a plane (---)followed by a translation (b/2)
symmetry element: glide plane
1/4,y,z
b/2
Screw rotation:a 180° rotation about an axis ( )followed by a translation (c/2)
symmetry element: screw axis
B-glide plane:move 0,0,0 to 1/2,1/2,0
0,0,0
1/2,1/2,1/2
2-fold screw axis
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Glide planesGlide reflection implies:
1) a reflection and2) a translation by the vector g parallel to the glide plane g is called glide component.
mirror glide plane
g is one-half of a latticetranslation (τ: a,b,c) parallel tothe glide plane,
τrr
21=g
v.s.
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Glide planes - orthorhombic system
Glide planes can only occur in an orientation that is possible for a mirror.
In orthorhombic system, glide planes only occur parallel to (100), (010), (001).For (100) glide plane, the glide components are:
br
21 cr
21 cb rr
+21 cb rr
+41
Glide planes are designated by symbols indicating the relationship of theirglide components to the lattice vectors a, b and c.
a-glide, b-glide, c-glide:
n-glide:
d-glide: for centered lattices
av21 b
r
21
cr21
2121 ττ rv +
2141 ττ rr +
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Glide planes - orthorhombic projected on x, y, 0
a-glide at x,1/4,z b-glide at x,y,0
c-glide at x,1/2,zX,1/4,z
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Glide planes - orthorhombic projected on x, y, 0
n-glide at x,y,1/4 n-glide at 0,y,z
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Screws axes
Screw rotation implies:1) a rotation of an angle ; and (X=1, 2, 3, 4, 6)2) a translation by a vector s parallel to the axis, s is called screw component.
X°= 360ε
Screw rotation is direction sensitive:right-handed axial system
6-fold screw axis (ε=60°)
Lattice translation
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Screws axes - screw component
°=⋅ 360εX
τσ rr =⋅ sX τσ rr
Xs =
τrr <sX<σ
σ = 0, 1, 2, ...X-1
= 0, sr τrX1
τrX2
τrX
X 1−
Screw axes are designated: Xσ=X0, X1, X2, ... XX-1screw axes can only occur parallel to the rotation axes.
or σ is integral
,...
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Screw axes - 4-fold axis
Perspective views
Projections on x,y,0
40 41 42 43 ε=360/X=90°
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Possible rotation and screw axes - enantiomorphous
61 65 62 64
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The 230 space groups
Point groups of highest symmetry in each crystal system
Remain one point subgroups - 32 crystallographic point groups
Screw axes can replace rotation axes:2 213 31, 324 41, 42, 436 61, 62, 63, 64, 65
Glide planes can replace mirror planes:m a, b, c, n, d
230 space groups
Space groups of highest symmetry in each crystal system, 14 Bravais lattices
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Example - Space group of point group 2/mMonoclinic space groups of highest symmetry: P2/m and C2/mMonoclinic subgroups of point group 2/m are 2 and m.The point symmetry element 2 and m can be replaced by 21 and a glide plane.
Replace 2 by 21
P2/m C2/m
P21/m P2/c C2/c P21/c
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Example - Space group of point group 2 and m
Space groups of point group m
Space groups of point group 2
M is replaced by c
2 is replaced by 21
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The 230 space groups
The entire 230 space groups can be derived from point groups by:
The rotation axis are replaced by corresponding screw axes
The mirror planes are replaced by glide planes
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Properties of space groupsThe number of equivalent points in the unit cell is called its multiplicity.Multiplicity of a special position is an integral factor of the multiplicity of 1 position.
A general position is a set of equivalent points with point symmetry 1.
A special position is a set of equivalent points with point symmetry higher than 1.
General position Special position Special position
Pmm2
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Properties of space groups
Pna21-orthorhombic
Glide planes and screw axes do not alter the multiplicity of a point.
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Properties of space groups - P2/m
General positionspecial positions on m, 2, and 2/m
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Asymmetric unit of a space group
The asymmetric unit of a space group is the smallest part of the unit cell fromwhich the whole cell may be filled exactly by the operation of all thesymmetry operations. Its volume is given by:
Vunit cell Vasym, unit=
multiplicity of the general position
An asymmetric unit contains all the information necessary for the completedescription of a crystal structure.
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Hexagonal space group P61
Operation of a 61-screwaxis at 0,0,z on a point ina general site x,y,z.
Equivalent points of a in a single unit cell21 at 1/2,1/2,z, 31 at 2/3,1/3,z and 1/3,2/3,z
Space group P61
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Cubic system - P4/m 3 2/mSpace group P4/m 3 2/m projected on x,y,0
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Cubic system - P4/m 3 2/m
3-fold rotation axis
Projection ofequivalent pointson x,y,0
Operation of mirrorplane at x,x,z
mirror
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Cubic system - P4/m 3 2/m
Operation of 4-fold axis at 0,0,z and mirror plane at x,y,0 in c completethe full set of 48 equivalent points of the general positions.
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Number of faces - multiplicity
1. For space groups with a P-lattice, the multiplicity of the general position isequal to the number of faces in the general form for the point group.
2. For space groups with C-, A- and I-lattices, the multiplicity of the generalposition is twice as great as the number of faces
3. For those with F-lattice, four times.
4. If the point group includes an inversion center, all the correspondingspace groups will be centrosymmetric.
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Space group and crystal structureA: Crystal structure = lattice + basisB: Crystal structure can be described by its space group and the occupationof general or special positions by atoms.
Rutile TiO2