About crystal symmetry, Miller indices

and parentheses

 
 

CLEAVAGE is the preferred planar direction of breakage that many minerals possess. It is due to planes of weakness that exist in some minerals because the bonding strength between the atoms or molecules is not the same in every direction. Because crystals are composed of orderly arrangements of atoms or molecules, we really should expect cleavage to be present in many crystals. The notation that denotes cleavage is derived in much the same manner as Miller indices, but is expressed in braces.
The Miller indices when used as face symbols are enclosed in parentheses, as the (111) face for example.
Form symbols are enclosed in braces, as the {111} form for example.
Zone symbols are enclosed in brackets, [111] for example and denote a zone axis in the crystal.
So in the discussion of cleavage, you must use braces to denote cleavage.

The rules for Miller indices are: 

  • Determine the intercepts of the face along the crystallographic axes,in terms of unit cell dimensions.

  • Take the reciprocals

  • Clear fractions

  • Reduce to lowest terms
    For example, if the x-, y-, and z- intercepts are 2,1, and 3, the Miller indices are calculated as:

  •     Take reciprocals: 1/2, 1/1, 1/3

  •     Clear fractions (multiply by 6): 3, 6, 2

  •     Reduce to lowest terms (already there)
    Thus, the Miller indices are 3,6,2. If a plane is parallel to an axis, its intercept is at infinity and its Miller
    index is zero. A generic Miller index is denoted by (hkl). For hexagonal and trigonal minerals, there are three possible axis directions, spaced 120 degrees apart. Obviously, any two intercepts specify the face. Also, there will be two intercepts of one sign and one of the other. The Miller indices for a hexagonal mineral are often written hikl. Indices h, i and k are related by h + i + k = 0.

    A family of faces all parallel to some particular line is called a zone, and the line is called the zone axis.

    Crystal faces are defined by indicating their intercepts on the crystallographic axes.
    Form: group of crystal faces, all of which have the same relation to the elements of symmetry. {111}
    Zones: A group of faces whose lines of intersection are parallel. [100]