NoteTaking_B3: Crystal & Crystal Structures

CH2 TWO-DIMENSIONAL PATTERNS 

A point group describes the symmetry of an isolated shape

A plane group describes the symmetry of a two-dimensional repeating pattern. 

1. Rotation axes

monad, diad, triad,  tetrad, pentad,  hexad,  center of symmetry (inversion center)

Only rotation axes that can occur in a lattice are 1, 2, 3, 4, 6. 

The five-fold rotation axis and all axes with $n$ higher than 6, cannot occur in a plain lattice.

A unit cell can not show overall five-fold rotation symmetry does not mean that a unit cell cannot obtain a pentagonal arrangement of atoms. 

2. Five plane lattices

Oblique (mp)

Rectangular primitive (op)

Rectangular centered (oc)

Square (tp)

Hexagonal primitive (hp)

3. Ten plane crystallographic point symmetry groups

rotation axes + mirror planes

1, 2, 3, 4, 6, $m$, 2$mm$, 3$m$, 4$mm$, 6$mm$

See figures here.

4. Symmetry of patterns: the 17 plane groups (Wallpaper group)

1) 17 plane groups = 5 plane lattices + 10 point symmetry groups

2) glide line = translation + reflection

     The glide displacement is restricted to one half of the lattice repeat. 

3) Name of plane groups

Taken from Wikipedia:

For wallpaper groups the full notation begins with either p or c, for a primitive cell or a face-centred cell.  This is followed by a digit, n, indicating the highest order of rotational symmetry: 1-fold (none), 2-fold, 3-fold, 4-fold, or 6-fold. The next two symbols indicate symmetries relative to one translation axis of the pattern, referred to as the "main" one; if there is a mirror perpendicular to a translation axis we choose that axis as the main one (or if there are two, one of them). The symbols are either m, g, or 1, for mirror, glide reflection, or none. The axis of the mirror or glide reflection is perpendicular to the main axis for the first letter, and either parallel or tilted 180°/n (when n > 2) for the second letter. Many groups include other symmetries implied by the given ones. The short notation drops digits or an m that can be deduced, so long as that leaves no confusion with another group.

The first letter gives the lattice type. The number in the second place gives the rotation axis

Mirrors and glide lines are placed last. 

5. General and special positions

general position: position does not fall on a symmetry element, high multiplicity, low site symmetry

special position: position falls on a symmetry element, low multiplicity, high site symmetry