Question

The figure below shows a plane PQR in a unit cell. The Miller indices of the plane PQR is ...........

• A. (432)
• B. ($\bar{4}$32)
• C. (234)
• D. (2$\bar{3}$4)

Hint:

For Miller indices: write intercepts, take reciprocals, and clear fractions. Negative intercepts are denoted with a bar.

Answer 1

The Correct Option is C

Step 1: Recall Miller indices definition.
For a crystal plane: 1. Find the intercepts of the plane with the crystallographic axes in terms of the lattice constants $a, b, c$. 2. Take reciprocals of intercepts. 3. Clear fractions to smallest integers → gives Miller indices $(hkl)$.

Step 2: Identify intercepts from figure.
From the figure: - Plane passes through $P$, $Q$, $R$. - Along $a$ axis: intercept at full length $\Rightarrow$ $1a$. - Along $b$ axis: intercept at $\tfrac{2}{3}b$. - Along $c$ axis: intercept at $\tfrac{1}{2}c$. So, intercepts = $(1a, \tfrac{2}{3}b, \tfrac{1}{2}c)$.

Step 3: Take reciprocals.
\[ \frac{a}{a} = 1, \frac{b}{2/3b} = \frac{3}{2}, \frac{c}{1/2c} = 2. \] So reciprocals = $(1, \tfrac{3}{2}, 2)$.

Step 4: Clear fractions.
Multiply through by 2: \[ (2, 3, 4). \] Thus, Miller indices = $(234)$.



Step 5: Check orientation sign.
From the diagram, all intercepts are positive along $+a, +b, +c$ directions. No negative sign needed. So final Miller indices = $(234)$. Final Answer:
\[ \boxed{(234)} \]