Question

The Miller indices of the shown lattice plane in a simple cubic Bravais lattice are : 

 

• A. (101)
• B. (110)
• C. (001)
• D. (111)

Hint:

If a crystallographic plane is parallel to an axis, its intercept is infinity, and the corresponding Miller index is exactly $0$. Always remember: Intercept $\rightarrow$ Reciprocal $\rightarrow$ Clear Fractions.

Answer 1

The Correct Option is A

Step 1: Concept:
The question requires us to determine the Miller indices of a specific crystallographic plane illustrated inside a simple cubic unit cell. The image defines three crystallographic axes: $a_1$, $a_2$, and $a_3$.

Step 2: Key Formula or Approach:
Miller indices $(hkl)$ are determined through a standardized three-step process:
1. Identify the fractional intercepts that the plane makes with the crystallographic axes ($a_1, a_2, a_3$).
2. Take the reciprocal of each intercept.
3. Clear any fractions to express the reciprocals as the smallest possible set of integers.

Step 3: Step-by-step Explanation:

• Based on standard representations of cubic lattices, let's identify the intercepts from the provided diagram. The axes are given as $a_1$ (typically the x-axis pointing towards the viewer/left), $a_2$ (the y-axis pointing right), and $a_3$ (the z-axis pointing up).

• Observing the shaded plane in the cubic unit cell:
- The plane cuts the $a_1$ axis exactly at the unit cell boundary, so the intercept is $1$.
- The plane cuts the $a_3$ axis exactly at the top of the unit cell, so the intercept is $1$.
- The plane extends completely parallel to the $a_2$ axis without ever intersecting it. Therefore, its intercept on the $a_2$ axis is considered to be infinity ($\infty$).

• The fractional intercepts are thus: $(1, \infty, 1)$.

• Now, we take the reciprocals of these intercepts:
\[ h = \frac{1}{1} = 1 \]
\[ k = \frac{1}{\infty} = 0 \]
\[ l = \frac{1}{1} = 1 \]

• The resulting indices are already integers, so no clearing of fractions is needed.

• The Miller indices are enclosed in parentheses without commas: $(101)$.

Step 4: Final Answer:
The Miller indices for the plane are $(101)$, which corresponds to option (A).