Most of the crystals are three-dimensional polyhedral forms having a number of faces

Most of the crystals are three-dimensional polyhedral forms having a number of faces. The positions of the faces with respect to crystallographic axes are expressed in a fashion almost similar to that of solid geometry.

The character and nature of the faces are expressed by intercepts, parameters and indices.

Intercepts:

Intercepts are the distances between the origin (point of intersection of the crystallographic axes) and the points of intersection of the face and axes. OX, OY and OZ axes are denoted by conventional crystallographic symbols a, b and c respectively. ABC and PQR are two crystal faces. The intercepts of the face ABC on a, b and c axes are OA, OB and OC respectively. Similarly the intercepts of the face PQR on a, b and c axes are OP, OQ and OR respectively.

Parameters:

Parameters are the ratios of the intercepts i.e. the intercepts of the face in consideration are compared with those of a unit form. If ABC is a face of the unit form then OA, OB and OC are unit distances in and c axes respectively.

Parameters of the face PQR are: OP/OA, OQ/OB and OR/OC on a, b and c axes respectively. This is expressed as la, VI b and 2c [Since OP=OA, OQ=’/2 OB and 0R=20C]

This type of representation of parameters is known as Weiss parameter. The symbols la, V2 b and 2c indicate that the face under consideration (PQR) intersects the a-axis at unit distance, b-axis at half of the unit distance and c-axis at twice of the unit distance.

Indices:

The indices are obtained from the Weiss parameters by taking the reciprocals and clearing the fractions. The indices are written in axial order i.e. a, b and c and hence the alphabets a, b and c associated with the parameters are omitted. The indices of the face PQR are 1/1, IM, 1/2 => 1, 2, »/₂ 241.

This system of representation of indices is known as Miller indices. For a face that intersects the a-axis at unit distance, b-axis at twice of the unit distance and remains parallel with the c-axis, the Weiss parameters are la, 2b, °°c (it is assumed that the face intersects the c-axis at infinite distance).

The miller indices are 1/1, 1/2, 1/°° or 210 (read as two, one, naught but not as two one zero or two hundred ten). A face may cut all the three axes at unequal lengths. In such cases the symbols may be 123, 321, 234 or any other numerical depending on the lengths of the intercepts.

If the lengths of intercepts cannot be determined precisely the general symbol “hive may be used. The intercepts and indices are inversely related i.e. the highest intercept corresponds to the lowest index and vice versa. The Miller indices of a face are also known as the symbol of the corresponding face.

Symbol of the crystal:

A simple form consists of a set of fixed number of faces which possess similar characteristics i.e. shape, size and crystallographic orientation. For example, a cube consists of six square faces each of which intersects one crystallographic axis and remains parallel with other two axes.

The Weiss parameters of the face towards the observer that intersects the axis and remains parallel with other two axes are la ⁰⁰b ⁰⁰c and the corresponding Miller indices are 100, which is the symbol of the corresponding face. The symbols of faces to the right and top are 010 and 001 respectively.

The symbols of remaining three faces, which intersect the axes on the negative sides, are 100, 010 and 001. Out of these six symbols one is to be used to represent the form i.e. the cube. There is disagreement among various authors regarding representation of face and form symbols.

Dana and Ford as well as Read indicate the face symbol without bracket and form symbol within ‘()’ bracket. Hurl but and Klein on the other hand, prefer to keep face symbol within ‘( )’ bracket and form symbol within ‘{}’ bracket. Generally the symbol of one of the faces crystal is taken to represent the form symbol.

There is also lack unanimity regarding selection of the form symbol. Dana and Ford as Read prefer to follow the convention h>k>l i.e. the first digit of symbol is the greatest and last digit of the symbol is the least.

Thus, cube is represented by the symbol (100). Hurl but and Klein and o follow the convention h<k<l i.e. the first digit of the symbol is the 1 and last digit of the symbol is the greatest. According to them the say {001} represents the cube. In both the cases the form symbol does indicate the number of faces of the crystal.

Thus, two or more for which bear similar relationship with the crystallographic axes, even to they belong to different classes are represented by the same symbol, avoid these confusions, in this book, the general symbol of the form intersects all the three crystallographic axes is represented by the sym ‘(hill) where h<k<l and ‘n’ is the number of faces.

Thus, the cube represented by the symbol (001)₆, which indicates that it is a form have six faces all of which intersect one crystallographic axis at unit dies and remain parallel with other two axes. Hence, it is a unique symbol t differs from the form symbols like (001)₂ and (001), which ha dissimilar number of faces but similar crystallographic relationship.