The packing efficiency of the face centered cubic (fcc), body centered cubic (bcc), and simple / primitive cubic ) (pc lattices follows the order

Options

A. `fcc gt bcc gt pc`
B. `bcc gt fcc gt pc`
C. `pc gt hcc gt fcc`
D. `bcc gt pc gt fcc`

Explanation

To determine the packing efficiency of face-centered cubic (FCC), body-centered cubic (BCC), and simple/primitive cubic (PC) lattices and to establish their order, we can follow these steps:

### Step-by-Step Solution:

1. **Understand Packing Efficiency**:
Packing efficiency is defined as the fraction of volume in a crystal structure that is occupied by the constituent particles (atoms, ions, or molecules). It is calculated using the formula:
\[ \text{Packing Efficiency} = \left( \frac{\text{Volume of atoms in the unit cell}}{\text{Volume of the unit cell}} \right) \times 100\% \]

2. **Calculate Packing Efficiency for FCC**:
- In an FCC lattice, there are 4 atoms per unit cell.
- The volume occupied by the atoms can be calculated as:
\[ \text{Volume of atoms} = 4 \times \frac{4}{3} \pi r^3 = \frac{16}{3} \pi r^3 \]
- The edge length \( a \) of the FCC unit cell is related to the radius \( r \) of the atoms by \( a = 2\sqrt{2}r \).
- The volume of the unit cell is:
\[ \text{Volume of unit cell} = a^3 = (2\sqrt{2}r)^3 = 16\sqrt{2}r^3 \]
- Therefore, the packing efficiency for FCC is:
\[ \text{Packing Efficiency}_{FCC} = \left( \frac{\frac{16}{3} \pi r^3}{16\sqrt{2}r^3} \right) \times 100\% = 74\% \]

3. **Calculate Packing Efficiency for BCC**:
- In a BCC lattice, there are 2 atoms per unit cell.
- The volume occupied by the atoms is:
\[ \text{Volume of atoms} = 2 \times \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3 \]
- The edge length \( a \) of the BCC unit cell is related to the radius \( r \) by \( a = \frac{4r}{\sqrt{3}} \).
- The volume of the unit cell is:
\[ \text{Volume of unit cell} = a^3 = \left(\frac{4r}{\sqrt{3}}\right)^3 = \frac{64r^3}{3\sqrt{3}} \]
- Therefore, the packing efficiency for BCC is:
\[ \text{Packing Efficiency}_{BCC} = \left( \frac{\frac{8}{3} \pi r^3}{\frac{64r^3}{3\sqrt{3}}} \right) \times 100\% = 68\% \]

4. **Calculate Packing Efficiency for Simple/Primitive Cubic**:
- In a simple cubic lattice, there is 1 atom per unit cell.
- The volume occupied by the atom is:
\[ \text{Volume of atom} = 1 \times \frac{4}{3} \pi r^3 = \frac{4}{3} \pi r^3 \]
- The edge length \( a \) is equal to \( 2r \).
- The volume of the unit cell is:
\[ \text{Volume of unit cell} = a^3 = (2r)^3 = 8r^3 \]
- Therefore, the packing efficiency for simple cubic is:
\[ \text{Packing Efficiency}_{PC} = \left( \frac{\frac{4}{3} \pi r^3}{8r^3} \right) \times 100\% = 52.4\% \]

5. **Order of Packing Efficiency**:
- From the calculations, we find:
- FCC: 74%
- BCC: 68%
- Simple Cubic: 52.4%
- Thus, the order of packing efficiency is:
\[ \text{FCC} > \text{BCC} > \text{Simple Cubic} \]

### Final Answer:
The packing efficiency of the face-centered cubic (FCC), body-centered cubic (BCC), and simple/primitive cubic (PC) lattices follows the order:
\[ \text{FCC} > \text{BCC} > \text{Simple Cubic} \]