Derivation of the 32 Point Groups Crystal Structure and Crystallography of Materials Chapter 8: Derivation of the 32 Point Groups (1st Method)

Repetition: Derivation of the 32 Crystal Classes: (basis for the lattice structure) produced by a combination of the geometrical motions of a rotation, a translation, and any operation producing enantimorphous figure. → improper rotation Now consider the particular kind of repetitions than can be produced if translations are omitted. → proper and improper rotations are used. rotations about axes should intersect. →

Derivation of the 32 Crystal Classes: What is the nature of repetitions which can be produced by rotations about intersecting axes?

Derivation of the 32 Crystal Classes: • A set of equivalent points is confined to the surface of a sphere The same is true for planes (geometrical) → repeated planes constitutes a polyhedra. ∴ The possible sets of intersecting rotational axis are the possible symmetries which polyhedra can have. → All repetitions not involving translations can be produced by rotations, proper and improper, all symmetries not involving translations can be described by rotations, proper and improper.

Plane lattices n-fold symmetry 1, 2, 3, 4, and 6 Principles for Discovering the Permissible Crystal Symmetries: Plane lattices n-fold symmetry 1, 2, 3, 4, and 6 n-fold symmetry 1, 2, 3, 4, and 6 (since all lattices are inherently centrosymmetrical)

The 32 Crystal Classes: • Crystallographically permissible symmetries involving sets of axes all of which intersect in a common point → 32 point groups → 32 crystal classes (solid polyhedral body) if a • Plane lattice net has n-fold : symmetry about that axis can be either n, n, or both n, n simultaneously.

6 permissible combinations : The 32 Crystal Classes: Therefore, all the axial symmetries can be derived by imposing these three alternatives on a) on each of the 5 possible individual symmetry axes such as 1, 2, 3, 4, and 6 b) on each of the 6 permissible combinations of intersecting sets of axes, but within the limitations that these combinations be consistent. 6 permissible combinations : 222, 322, 422, 622, 332, 432

Coincident Rotation and Rotoinversion Axes: • Lattice can be consistent with an axis n and an axis n simultaneously. → n-fold pure rotation axis coincident with an n-fold rotoinversion axes. → written as Reduction of n=1, = n=2, = n=3, = n=4, = n=6, =

Derivation of the Crystal Classes: The monoaxial classes: single n-fold axis can be either n, n, or

Permissible Combinations of Proper and Improper Rotations: Consider a connected set of rotation axes ABC. When two of these are proper rotations? A requires 1R → 2R B requires 2R → 3R C requires 1R → 3R Thus PP = P (P : proper rotations) When two of these (A and B) be improper rotations. A requires 1R → 2L B requires 2L → 3R Thus II = P (I : improper rotations) Thus, PPP IIP Permissible Sets. IPI III is impossible. PII

The Polyhedral Symmetry Combinations:

Derivation of the Crystal Classes: a) Proper combinations b) Improper combinations 1) In the event that two of the three are symmetrically equivalent, as in 332, obviously these must always be the same. → if the first 3 is I, so is the second. 2) n22 and n22 are not distinct

3 322 Derivation of the Crystal Classes: e.g. Why not 322 ? Should be 2. not 2

• The two numbers within each pair of parentheses are The Polyaxial Classes Conforming to both Proper and Improper Combinations: When n1, n2, n3 is combined with n1, n2, n3 • The two numbers within each pair of parentheses are coincident axes. → the lattice is simultaneously consistent with all these axis. → impossible if we choose ( )(n2)(n3) Unless there is n1

The Polyaxial Classes Conforming to both Proper and Improper Combinations: Thus the only permissible complete combinations must have the form

Derivation of the 32 Point Groups: