A metallic crystal cystallizes into a lattice containing a sequence of layers `ABABAB…`. Any packing of spheres leaves out voids in the lattice. What percentage by volume of this lattice is empty spece?

Options

A. `74%`
B. `26%`
C. `50%`
D. None of these

Explanation

To solve the problem of determining the percentage by volume of empty space in a metallic crystal lattice that crystallizes into a sequence of layers `ABABAB...`, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Type of Lattice**:
The given sequence of layers `ABABAB...` indicates that the lattice is a close-packed structure, specifically a cubic close-packed (CCP) lattice.

2. **Determine the Effective Number of Atoms (Z effective)**:
In a cubic close-packed (CCP) lattice, the effective number of atoms per unit cell (Z effective) is 4.

3. **Understand the Relationship Between Radius and Edge Length**:
For a CCP lattice, the relationship between the radius (R) of the spheres and the edge length (a) of the unit cell is given by:
\[ 4R = \sqrt{2}a \]
From this, we can express the radius in terms of the edge length:
\[ R = \frac{\sqrt{2}}{4}a \]

4. **Calculate the Packing Fraction**:
The packing fraction (PF) can be calculated using the formula:
\[ \text{Packing Fraction} = \frac{Z \cdot \frac{4}{3} \pi R^3}{a^3} \]
Substituting \( Z = 4 \) and \( R = \frac{\sqrt{2}}{4}a \):
\[ \text{Packing Fraction} = \frac{4 \cdot \frac{4}{3} \pi \left(\frac{\sqrt{2}}{4}a\right)^3}{a^3} \]
Simplifying this:
\[ = \frac{4 \cdot \frac{4}{3} \pi \cdot \frac{2\sqrt{2}}{64} a^3}{a^3} = \frac{4 \cdot \frac{4}{3} \pi \cdot \frac{2\sqrt{2}}{64}}{1} \]
\[ = \frac{4 \cdot \frac{4}{3} \pi \cdot \frac{2\sqrt{2}}{64}}{1} = \frac{4 \cdot \frac{4 \cdot 2\sqrt{2}}{192}}{1} = \frac{32\sqrt{2}}{192} = \frac{\sqrt{2}}{6} \]

5. **Convert the Packing Fraction to a Percentage**:
The packing fraction is approximately \( 0.74 \) or \( 74\% \).

6. **Calculate the Percentage of Empty Space**:
The percentage of empty space is calculated by subtracting the packing fraction from 1 (or 100%):
\[ \text{Empty Space} = 100\% - \text{Packing Fraction} = 100\% - 74\% = 26\% \]

### Final Answer:
The percentage by volume of the lattice that is empty space is **26%**.

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