Question

For simple cubic crystal edge length is expressed as

• A. $a = 2r$
• B. $a = \frac{r}{2}$
• C. $a = \sqrt{2}r$
• D. $a = \frac{r}{\sqrt{2}}$

Hint:

Visualizing the lattice helps! A simple cubic cell is the most straightforward arrangement because there are no extra interior or face-centered atoms to push the corners apart. The edge is just two radii side-by-side!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks for the mathematical relationship between the cube edge length ($a$) and the atomic radius ($r$) in a standard simple cubic (SC) crystal lattice structure.

Step 2: Key Formula or Approach:
In a simple cubic unit cell, the constituent particles (spheres) are located at the eight corners of the cube. These spheres touch each other along the edges of the unit cell. Therefore, the total length of any single edge equals the combined radii of two adjacent touching spheres.

Step 3: Detailed Explanation:
Let's consider one edge of a simple cubic unit cell:
The edge connects two corner lattice points.
Each corner accommodates one atom, contributing a distance equal to its radius $r$ along the edge path before reaching the point of contact.
Since the two corner atoms are in direct physical contact with each other, the total edge length ($a$) is simply the sum of these two identical atomic radii:
$$a = r + r = 2r$$ This direct relation corresponds perfectly to option (A).

Step 4: Final Answer:
The edge length for a simple cubic crystal is expressed as $a = 2r$, which corresponds to option (A).