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Functional Materials SaarlandUniversity

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


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


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.


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 +


Glide planes - orthorhombic system - glide designation


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


Glide planes - orthorhombic projected on x, y, 0

n-glide at x,y,1/4 n-glide at 0,y,z


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


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

,...


Screw axes - 4-fold axis

Perspective views

Projections on x,y,0

40 41 42 43 ε=360/X=90°


Possible rotation and screw axes - enantiomorphous2 21


Possible rotation and screw axes - enantiomorphous

61 65 62 64


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


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


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


Point and space subgroups of monoclinic system


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


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


Properties of space groups

Pna21-orthorhombic

Glide planes and screw axes do not alter the multiplicity of a point.


Properties of space groups - P2/m

General positionspecial positions on m, 2, and 2/m


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.


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


Cubic system - P4/m 3 2/mSpace group P4/m 3 2/m projected on x,y,0


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


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.


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.


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


Relationship between point group and space group

part 5 space groups - startseite 5 space groups 5.1 glide planes 5.2 screw axes ... 5.5 space group...

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