## Symmetry

Motif:

- Basic unit that makes pattern in 3-D
- Repetition of faces for internal & external structure of a crystal (xl), using operations to do it.

Symmetry:

- Systematic arrangement of motifs with respect to a point, line, or plane.
Ex: Wall paper or a cloth pattern

Symmetry operation:

- Geometrical movement which brings a given motif into coincidence with the same motif elsewhere.

Symmetry elements:

- Points, lines, or planes to which parts of a xl are symmetrically arranged

1) Symmetry plane (referred to as "m"):

A mirror plane which passes through the center of an object in such a way that half the object is the mirror image of the other half (like a line through the center of your body, top to bottom). Identical points are equal and opposite along a line perpendicular to m.

2) Rotation Axis:

Perpendicular to a plane.

A line through the center of an object about which motifs are repeated as a result of rotation.
(1,2,3,4,6) - these are called the proper rotation axes.
These are part of what's called the Hermann-Mauguin symbols (H-M).

1 = 1-fold or 360 degree rotation

2 = 2-fold or 180 degree rotation
3 = 3-fold or 120 degree rotation
4 = 4-fold or 90 degree rotation
6 = 6-fold or 60 degree rotation

3) Center of symmetry ("c"):

Symmetry with respect to a point so that identical items will be found at equal distances in opposite directions, from the central point. The key thing here is called INVERSION.

Rotary inversion (rotoinversion) combines rotation with inversion.

Symmetry symbols used on stereograms:

Planes: m

Lines:

Proper axes: 1,2,3,4,6

Associated symbols: point, solid football, solid triangle, solid square, and solid circle.

Improper axes: bar 1, bar 2, bar 3, bar 4, bar 6. Rotoinversion axes.

Combination of these symmetry elements produces the 32 Point Groups (Crystal Classes):

proper axes: 222, 422, 622, 23, 32, 432
proper axes + center: 4/m, 2/m, 6/m
combination proper axes + center: 2/m2/m2/m, 4/m2/m2/m, 6/m2/m2/m, mmm. 4/mmm, 6mm, 2/m bar3, bar 3 2/m, 4/m bar 3 2/m
proper + improper axes: 2mm, 4mm, bar 4 2m, 6mm, bar 6 2m, 3m, bar 4 3m

Certain of the classes have common symmetries, thus producing: 6 Crystal Classes

Crystal classes are characterized by symmetry. They are described by 3 crystallographic axes which correspond to symmetry axes.

Crystallographic axes (a,b, & c) are just like x, y, & z axes in trigonometry.

The 6 crystal classes are:

Isometric, Tetragonal, Orthorhombic, Hexagonal, Monoclinic, & Triclinic

1) Isometric (cube):

3 axes of equal length at 90 degrees to one another.
a=b=c
has 4 3-fold axes of rotation or rotoinversion (3 or bar 3).

2) Tetragonal:

3 axes, 2 of which are equal in length, and all at 90 degrees.
a=b c
Has a single 4-fold axis (4 or bar 4).

3) Orthorhombic:

3 axes at 90 degreees and all unequal
a b c, cHas 3 mirror planes and/or 3 2-fold axes.

4) Hexagonal:

4 axes, 3 of which are equal at 120 degrees, plus c perpendicular to the plane. 4 crystallographic axes.
a=b=b2 c
Has single 6-fold axes (6 or bar 6) or 3 or bar 3.

5) Monoclinic:

3 unequal axes.
a b c, a & c form a plane perpendicular to b, but a and c are not at 90 degrees.
Has single 2-fold axes (2 or bar 2) and/or single mirror plane (m).

6) Triclinic:

3 unequal axes, no 90 degree angles.
a b c, cHas a single 1-fold axes (1 or bar 1).