•The 32 crystallographic point groups, whose operation have at least one point unchanged, are

sufficient for the description of finite, macroscopic objects.

•However since ideal crystals extend indefinitely in all directions, we must also include translations

(the Bravais lattices) in our description of symmetry.

Space groups: formed when combining a point symmetry group with a set of lattice translation

vectors (the Bravais lattices), i.e. self-consistent set of symmetry operations acting on a Bravais

lattice. (Space group lattice types and translations have no meaning in point group symmetry.)

Space Groups

Space group numbers for all the crystal

structures we have discussed this

semester, and then some, are listed in

DeGraef and Rohrer books and pdf.

document on structures and AFLOW

website, e.g. ZnS (zincblende) belongs

to SG # 216: F43m)

Class21/1

Screw Axes

Class21/2

•The combination of point group symmetries and translations also leads to two additional

operators known as glide and screw.

•The screw operation is a combination of a rotation and a translation parallel to the

rotation axis.

•As for simple rotations, only diad, triad, tetrad and hexad axes,

that are consistent with Bravais lattice translation vectors can

be used for a screw operator.

•In addition, the translation on each rotation must be

a rational fraction of the entire translation.

•There is no combination of rotations

or translations that can transform the pattern

produced by 31 to the pattern of 32 , and 41 to the

pattern of 43, etc.

•Thus, the screw operation results in handedness

or chirality (can’t superimpose image on another,

e.g., mirror image) to the pattern.

Screw Axes (continued)

The 11 possible screw axes:

oblique projection: plane projection:

Class21/3

alternate plane projection:

When going from a space group to the parent point

group, all the screw subscripts are eliminated and thus

are converted back into the n-fold rotation, e.g. 65 6

A

AA A’

A’

A’

? ?

Screw Axes (continued)

Class21/4

Glide Planes

•Glide is the combination of a mirror (reflection) and a translation.

•Glide must be compatible with the translations of the Bravais lattice, thus the

translation components of glide operators must be rational fractions of lattice vectors.

•In practice, the translation components of a glide operation are always ½ or ¼ of

the magnitude of translation vectors.

•If the translation is parallel to a lattice vector, it is called axial glide (glide planes with

translations a/2, b/2 or c/2 are designated with symbols a, b or c, respectively).

Recall:

two b-glide operations:

Glide plane

can’t be ┴ to

glide direction

or in

2-D:

A

A’

A

= net

*Diamond glides (d-glide) can only occur in F and

I-centered lattices, e.g. diamond cubic crystal (C,

Si, Ge) structure is Fd3m (see next slide

•Another type of glide is diagonal glide (n) and has translation components of a/2+b/2, b/2+c/2 or

a/2+c/2. Last type is diamond glide (d) w/ translation components of a/4+b/4, b/4+c/4 or a/4+c/4. two n-glide operations: a, b

and

d: d

[100]

[110][010]

Class21/5

(or c)

(displacement

vector)(b or c-axis is ┴ to g)

(a or c-axis is ┴ to g)

(a or b-axis is ┴ to g)

(c-axis is ┴ to g) (a-axis is ┴ to g)(b-axis is ┴ to g)

[110]

Diamond Glide Planes in Diamond Cubic

d = 1/4a + 1/4b

Class21/6

d = 1/4b + 1/4c

[011]

[110]

[011] [101]

d = 1/4a + 1/4c[101]

Conversion of Space Group (SG) to

Point Group (PG) Symbolism•Eliminate translation from symbol.

•Example: Space group #62: Pnma (Mg,Fe)2SiO4 belongs to point group mmm:

•P=primitive lattice type does not apply to PG symmetry.

•n(net glide plane perpendicular to x or a-axis)=m because the reflection of a net

glide plane has no meaning in PG symmetry.

•m(mirror plane perpendicular to y or b-axis)=m

•a(axial glide plane perpendicular to z or c-axis)=m because the reflection of an

axial glide plane has no meaning in PG symmetry.

•Example: Space group #167: R3c (Al2O3) belongs to point group 3m:

•R=rhombohedral lattice type does not apply to PG symmetry.

•3(3-fold roto-inversion axis)=3(3-fold roto-inversion axis).

•c(axial glide plane parallel to 3)=m because the reflection of an axial glide plane

has no meaning in PG symmetry.

Class21/7

The 13 unique monoclinic

space groups that are

derived from the 3

monoclinic point groups

and the 2 monoclinic

Bravais lattices:

You should be able to

look at any one of the

230 3-D Space groups

and identify its 3-D

Point group and

3-D Bravais lattice

Space Group Pnma

Class21/8

The 230 3-D Space Groups categorized

according to crystal system

Class21/9

from Rohrer

http://img.chem.ucl.ac.uk/sgp/large/sgp.htmAlso good website:

Alternative Notation for Crystal Structures

Class21/10

Also listed in DeGraef

Structure appendix .pdf

The 230 space groups categorized

according to crystal system with examples:http://www.aflowlib.org/CrystalDatabase/space_groups.html

Example from International Tables

for Crystallographya. Identify all the symmetry elements in (a) and describe which

operation they include.

1. Diads-indicate a two-fold rotation about the axis

2. Screw tetrads (42)-indicate a rotational axis of a tetrad plus a

translation of T=½ where T is the lattice translation fraction parallel to

the axis.

3. Axial glide plane( )-indicates that the translation glide vector is

½ lattice spacing along line parallel to the projection plane

4. Axial glide plane( )-indicates that the translation glide vector is

½ lattice spacing along line normal to the projection plane.

5. Diagonal glide plane( )-indicates a translation of ½ of a face

diagonal.

b. In separate plots, apply each symmetry element to a general point

(equipoint) and show which of the points in (b) are generated:

Class21/11

(a)

(b)

Class21/12

More Examples from International

Tables for Crystallography

http://www.aflowlib.org/CrystalDatabase/space_groups.html

•No. 122 has Chalcopyrite (E11) Structure

(CuFeS2, AgAlTe2, AlCuSe2, CdGeP2,etc.

•No. 60 has no examples of real crystals

at all!