A metal crystallizes in a body-centered cubic structure with a unit cell edge length of 4.31 Å. What is the radius of the atoms in Å?

• A. 1.87 Å
• B. 2.30 Å
• C. 2.66 Å
• D. 1.33 Å

Answer: (A)

Verified step by step guidance

1. Understand the structure: A body-centered cubic (BCC) unit cell has atoms at each corner and one atom at the center of the cube.
2. Visualize the geometry: In a BCC structure, the diagonal of the cube passes through the center atom and connects opposite corner atoms. This diagonal is equal to four times the radius of the atom.
3. Use the geometry of the cube: The diagonal of the cube can be calculated using the Pythagorean theorem in three dimensions. The formula for the diagonal length (d) in terms of the edge length (a) is: $\sqrt{3}$ times the edge length.
4. Set up the equation: Since the diagonal is equal to four times the radius (r), we have: $4r$ = $\sqrt{3}a$, where $a$ is the edge length.
5. Solve for the radius: Rearrange the equation to solve for the radius: $r$ = $\frac{\sqrt{3}a}{4}$. Substitute the given edge length to find the radius.