We can describe the slope of crystal faces with reference to the intercepts they make on the crystallographic axes. The methods that are commonly used for expressing the intercepts of crystal faces are : (a) Weiss Parameter System, (b) Index System of Miller. These two are the most common parameters used to determine the slopes of crystal faces.
(1). Wiess’s Parameter System:
As per Weiss Parameter System, the parameters of the crystal face system are written along with the notation of crystallographic axes. For example, a face that cuts the a-axis at unit distance, b-axis at a distance of 2 units and lies parallel to the c-axis, its Weiss symbol is written as follows.
(2). Miller’s Index System:
As per Miller’s index system, the notations of crystal faces are known as ”Miller indices”. The Miller Indices of a face consist of a series of the whole numbers that have been derived from the parameters by taking their reciprocals and then by clearing of fractions. The numbers of the Miller indices are always written in the axial order a,b and c. For example, let us consider the crystal face PQR shown in the following figure. Its Miller indices are determined as follows.
For the face PQR, which cuts the positive ends of the crystallographic axes, the parameters are 3a, 3b, and 2c respectively. Since the intercepts are always written the axial order a, b, and c, the letters themselves are omitted and the parameters are written as: 3, 3, 2.
The reciprocal of the parameters are taken which leads to 1/3, 1/3, and 1/2 respectively.
The fractions are cleared by multiplying all by 6 and the Miller symbol for the face QPR is obtained, which is ”223”. The Miller indices of the octahedron are shown in the following figure.
For faces that intersect negative ends of crystallographic axes, a bar is placed over the corresponding index figure. For example, index 223 suggests that this face intercepts the c-axis on the negative end. Faces, which are parallel to an axis cut that axis at infinity. For example, if the intercepts of a face 1, ∞, ∞, it miller symbol will be 100.
A Miller symbol, which is not bracketed symbol represents represent all the faces of a form. The general symbol for a face that cuts all the axes at different lengths is ”hkl” and for the hexagonal system, it is ”hkil”.
It may be noted that for given faces of the crystal, the Miller indices could always be expressed by simple whole numbers or zero. This is known as the ”law of rational indices”.