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Space Groups

•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 extend indefinitely in all directions, we must also include translations (the Bravais lattices) in our description of . Space groups: formed when combining a point symmetry with a set of vectors (the Bravais lattices), i.e. self-consistent set of symmetry operations acting on a . ( lattice types and translations have no meaning in symmetry.)

Space group numbers for all the 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

•The combination of point group 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 Class21/2 or (can’t superimpose image on another, e.g., mirror image) to the pattern. Screw Axes (continued)

The 11 possible screw axes: alternate plane projection: oblique projection: plane projection:

A A A’ A’ A’ A

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 ? ? Class21/3 Screw Axes (continued)

Class21/4 Glide Planes

•Glide is the combination of a mirror () and a translation. Recall: •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). •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 glide (d) w/ translation components of a/4+b/4, b/4+c/4 or a/4+c/4. two b-glide operations: two n-glide operations: a, b A A and [010] [110] d: d or in [100] 2-D: A’ (or c) (displacement vector) (b or c-axis is ┴ to g) (a or c-axis is ┴ to g) can’t be ┴ to = net glide direction (a or b-axis is ┴ to g)

(c-axis is ┴ to g) (a-axis is ┴ to g) (b-axis is ┴ to g) *Diamond glides (d-glide) can only occur in F and I-centered lattices, e.g. crystal (C, Si, Ge) structure is Fd3m (see next slide Class21/5 Diamond Glide Planes in Diamond Cubic

d = 1/4b + 1/4c d = 1/4a + 1/4c [101] d = 1/4a + 1/4b [011] [011] [101]

[110]

[110]

Class21/6 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. You should be able to look at any one of the The 13 unique monoclinic 230 3-D Space groups space groups that are and identify its 3-D derived from the 3 Point group and monoclinic point groups 3-D Bravais lattice and the 2 monoclinic Bravais lattices: Class21/7 Space Group Pnma

Class21/8 The 230 3-D Space Groups categorized according to from Rohrer Also good website: http://img.chem.ucl.ac.uk/sgp/large/sgp.htm

Class21/9 Alternative Notation for Crystal Structures

Also listed in DeGraef Structure appendix .pdf

The 230 space groups categorized according to crystal system with examples: Class21/10 http://www.aflowlib.org/CrystalDatabase/space_groups.html Example from International Tables for (a) a. 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 (b) 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 More Examples from International Tables for Crystallography

•No. 122 has (E11) Structure (CuFeS2, AgAlTe2, AlCuSe2, CdGeP2,etc. •No. 60 has no examples of real crystals at all! Class21/12 http://www.aflowlib.org/CrystalDatabase/space_groups.html