Question

The edge lengths of the unit cells in terms of the radius of spheres constituting fcc,bcc and simple cubic unit cell are respectively - (1) 2r, \(\frac{4 \mathrm{r}}{\sqrt{3}}, 2 \sqrt{2 \mathrm{r}}\) (2) \(\frac{4 \mathrm{r}}{\sqrt{3}}, 2 \sqrt{2 r}, 2 \mathrm{r}\) (3) \(2 r, 2 \sqrt{2 r}, \frac{4 r}{\sqrt{3}}\) (4) \(2 \sqrt{2 r}, \frac{4 r}{\sqrt{3}}, 2 r\)

Short answer

(2) \( \frac{4r}{\sqrt{3}}, 2\sqrt{2r}, 2r \)

Step-by-step solution

Identify the Unit Cell Types

Recognize the three types of unit cells mentioned: face-centered cubic (fcc), body-centered cubic (bcc), and simple cubic.

Determine Edge Length of Simple Cubic Unit Cell

For a simple cubic unit cell, the edge length is directly equal to twice the radius of the sphere: \[ a = 2r \]

Determine Edge Length of Body-Centered Cubic (bcc) Unit Cell

For a body-centered cubic unit cell, the spheres touch along the body diagonal. The length of the body diagonal is equal to 4 times the radius. Given that the body diagonal in terms of edge length is \[ a\sqrt{3} = 4r \] Thus, the edge length of the bcc unit cell is \[ a = \frac{4r}{\sqrt{3}} \]

Determine Edge Length of Face-Centered Cubic (fcc) Unit Cell

For a face-centered cubic unit cell, the spheres touch along the face diagonal. The length of the face diagonal is 4 times the radius. Given that the face diagonal in terms of edge length is \[ a\sqrt{2} = 4r \] Thus, the edge length of the fcc unit cell is \[ a = \frac{4r}{\sqrt{2}} = 2\sqrt{2r} \].

Match Calculation with Given Options

Compare the derived edge lengths of the unit cells: - Simple cubic: \( 2r \) - bcc: \( \frac{4r}{\sqrt{3}} \) - fcc: \( 2\sqrt{2r} \) with the given options. The correct matching option is (2).

Key concepts

Face-Centered Cubic (FCC)

In a face-centered cubic (FCC) unit cell, atoms are located at each of the corners and the center of each face of the cube. Because of this, each atom at the face center is shared among two unit cells.
FCC unit cells are quite efficient in packing atoms. In fact, they have a packing efficiency of about 74%. This means that 74% of the space within the unit cell is occupied by atoms, while the rest is empty space.
The edge length of an FCC unit cell can be derived by considering the face diagonal. The face diagonal is made up of two radii from one face center atom and one edge length. Therefore, the face diagonal is equal to 4 radii:
\[ a\sqrt{2} = 4r \]
Solving for the edge length, we get:
\[ a = \frac{4r}{\sqrt{2}} = 2\sqrt{2r} \]
Hence, the edge length of the FCC unit cell is \mathbf{2\sqrt{2r}}.

Body-Centered Cubic (BCC)

In a body-centered cubic (BCC) unit cell, atoms are located at each corner of the cube as well as a single atom in the very center of the cube. This configuration allows for a different calculation of the edge length.
The spheres in a BCC unit cell touch along the body diagonal, which spans from one corner of the cube to the opposite one through the center. The length of the body diagonal consists of four radii, hence:
\[ a\sqrt{3} = 4r \]
Solving for the edge length, we get:
\[ a = \frac{4r}{\sqrt{3}} \]
Therefore, the edge length of the BCC unit cell is \mathbf{\frac{4r}{\sqrt{3}}}.

Simple Cubic

The simple cubic unit cell is the most straightforward and least dense of the three cubic structures. In this configuration, atoms are located only at the corners of the cube. Each atom at a corner is shared among eight adjacent unit cells, meaning each corner atom only contributes 1/8th of its volume to a single unit cell.
The edge length of a simple cubic unit cell is directly related to the radius of the atoms. This is because the atoms touch along the edge. Therefore, the edge length of a simple cubic unit cell is simply:
\[ a = 2r \]
So, the edge length for the simple cubic structure is \mathbf{2r}.