Directions in Crystal

DIRECTIONS IN CRYSTAL

It is always useful to have a convention or standardized procedure to specify the directions in a crystal. The procedure of finding the directions inside the crystal is explained below.

1. Consider any lattice point that lies on the line as origin. 

2. Write down the position vector of the next nearest point on the line in terms of the fundamental translation vector of the unit cell of the crystal, say, 

3. Now the components of position vector along the three directions of a, b, c are r1, r2, r3 respectively. Then the crystal direction is denoted by [r1 r2 r3].

Let us apply this procedure to find the directions of OP, OQ and OR of an orthorhombic unit cell (a ≠ b ≠ c; α = β = γ = 90°) in Figure 1.26 taking 'O' as origin.

(a) Direction of OP

Position vector of OP =

⸫ r1 = 1; r2 = 1; r3 = 1

⸫ Direction of OP is represented by [111]

(b) Direction of OQ

Position vector of OQ =

⸫ r1 = 1; r2 = 0; r3 = 0

⸫ Direction of OP is represented by [100]

(c) Direction of OR

Position vector of OR =

⸫ r1 = 1; r2 = 0; r3 = 1

⸫ Direction of OP is represented by [101]

It should be understood that the directions [222], [333], [444], [n r1 n r2 n r3] will all coincide with [1 1 1]. In such cases the lowest combination of integers i.e., [1 1 1] is used to specify the direction. If any of the integers is negative, for example - 3, it should be written as 3 which is read 3 bar.

Given three integers of a direction, a family of directions with different possible combinations of them, both positive and negative, is represented with brackets of the type < >. For example, a family of directions such as [132], [312], [123], etc., is represented by <132>.