Question

Calculate the miller indices of crystal planes which cut through the crystal axes
$(a,b,c),(2a,b,c),\& (2a, - 3b, - 3c)$.

Answer 1

Miller indices are a set of three numbers indicating the orientation of a set of parallel planes of an atom in crystal. If each atom in a crystal is represented by a point. These points are connected by the lines, the resulting lattice is further divided into a number of blocks which are identical also known as unit cells.

Complete step by step solution:
Miller indices is defined as a plane of crystal that is inversely proportional to the intercepts of the given plane on various axes.
It can be calculated by the following steps below.
Write the fractional intercepts of axes
Reciprocals of the co efficient axes
Multiply by the number to get a whole number
For $(a,b,c)$,

| x- axis| y-axis| z-axis
---|---|---|---
Intercepts| a| b| c
Weiss indices| aa=$1$| bb=$1$| cc=$1$
Miller indices| 1$$$$1=$1$ | 1$$$$1=$1$| 1$$$$1=$1$

Therefore miller indices is $= [111]$
For $(2a,b,c)$

| x- axis| y-axis| z-axis
---|---|---|---
Intercepts| 2a| b| c
Wiess indices| $2$aa=| bb=$1$| cc=$1$
Miller indices| $21 \times 2$=$1$| $11 \times 2$=$2$| $21 \times 2$=$2$

Miller indices = $[122]$
For $(2a, - 3b, - 3c)$

| x- axis| y-axis| z-axis
---|---|---|---
Intercepts| 2a| b| c
Weiss indices| $b2a = 2b$| $- 3b = 3c$ $= [322]$ | $- 3c = - 3$
Reverse| $21$| $3 - 1$| $3 - 1$
Miller indices| $21 \times 6 = 3$| $3 - 1 \times 6 = - 2$| $3 - 1 \times 6 = - 2$ $$$$ $(2a, - 3b, - 3c)$

Therefore miller indices is $= [322]$

Note: The negative sign in the miller indices is indicated by placing a bar on the given integer. They are enclosed within the parentheses. The intersecting edges of unit cells define a set of crystallographic axes, whereas the miller indices can be determined by intersection of the plan with these axes.