Question

The volume $V$ of a cubic unit cell is related to the lattice parameter $a$. Which of the following expressions is dimensionally and physically correct?

• A. $V = a + a^2$
• B. $V = 2a$
• C. $V = a^2$
• D. $V = a^3$

Hint:

Dimensional analysis is a powerful tool to quickly eliminate incorrect choices.
Volume must always scale with the cube of a linear dimension ($a^3$).

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question asks for the correct geometric relationship for the volume ($V$) of a cubic unit cell in terms of its edge length (lattice parameter $a$).

Step 2: Key Formula or Approach:
A cube has three orthogonal axes of equal length:
\[ a = b = c \]
The volume of any parallelepiped is given by:
\[ V = \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \]
For a cube where all angles are $90^\circ$:
\[ V = a \times a \times a = a^3 \]

Step 3: Detailed Explanation:

• Volume has the physical dimension of length cubed ($[\text{L}]^3$).

• The lattice parameter $a$ represents the side length of the unit cell and has the dimension of length ($[\text{L}]$).

• Checking the dimensions of the given options:
- Option A: $V = a + a^2$ is dimensionally inconsistent (adding length and area).
- Option B: $V = 2a$ has the dimension of length.
- Option C: $V = a^2$ has the dimension of area.
- Option D: $V = a^3$ has the correct dimension of volume ($[\text{L}]^3$).

Step 4: Final Answer:
The physically and dimensionally correct expression is $V = a^3$.