The ratio of packing fraction in fcc, bcc, and cubic structure is, respectively,

Options

A. `1: 0.92: 0.70`
B. `0.70: 0.92: 1`
C. `1: 0.70: 0.92`
D. `0.92: 0.70: 1`

Explanation

To find the ratio of packing fractions in FCC (Face-Centered Cubic), BCC (Body-Centered Cubic), and simple cubic structures, we will follow these steps:

### Step 1: Understand the Packing Fraction Formula
The packing fraction (PF) is defined as the fraction of volume in a crystal structure that is occupied by the constituent particles (atoms, ions, or molecules). The formula for packing fraction is given by:

\[ \text{Packing Fraction} = \frac{Z \cdot \frac{4}{3} \pi r^3}{a^3} \]

Where:
- \(Z\) = number of atoms per unit cell
- \(r\) = radius of the atom
- \(a\) = edge length of the unit cell

### Step 2: Calculate Packing Fraction for FCC
In FCC:
- \(Z = 4\) (4 atoms per unit cell)
- The relationship between the radius \(r\) and the edge length \(a\) is given by \(a = 2\sqrt{2}r\).

Substituting this into the packing fraction formula:

\[ \text{Packing Fraction}_{FCC} = \frac{4 \cdot \frac{4}{3} \pi r^3}{(2\sqrt{2}r)^3} \]

Calculating the denominator:

\[ (2\sqrt{2}r)^3 = 8 \cdot 2\sqrt{2}^3 \cdot r^3 = 16\sqrt{2}r^3 \]

So,

\[ \text{Packing Fraction}_{FCC} = \frac{4 \cdot \frac{4}{3} \pi r^3}{16\sqrt{2}r^3} = \frac{16\pi}{48\sqrt{2}} \approx 0.74 \]

### Step 3: Calculate Packing Fraction for BCC
In BCC:
- \(Z = 2\) (2 atoms per unit cell)
- The relationship between \(r\) and \(a\) is \(a = \frac{4r}{\sqrt{3}}\).

Substituting into the packing fraction formula:

\[ \text{Packing Fraction}_{BCC} = \frac{2 \cdot \frac{4}{3} \pi r^3}{\left(\frac{4r}{\sqrt{3}}\right)^3} \]

Calculating the denominator:

\[ \left(\frac{4r}{\sqrt{3}}\right)^3 = \frac{64r^3}{3\sqrt{3}} \]

So,

\[ \text{Packing Fraction}_{BCC} = \frac{2 \cdot \frac{4}{3} \pi r^3}{\frac{64r^3}{3\sqrt{3}}} = \frac{8\pi\sqrt{3}}{64} \approx 0.68 \]

### Step 4: Calculate Packing Fraction for Simple Cubic
In simple cubic:
- \(Z = 1\) (1 atom per unit cell)
- The relationship between \(r\) and \(a\) is \(a = 2r\).

Substituting into the packing fraction formula:

\[ \text{Packing Fraction}_{Cubic} = \frac{1 \cdot \frac{4}{3} \pi r^3}{(2r)^3} \]

Calculating the denominator:

\[ (2r)^3 = 8r^3 \]

So,

\[ \text{Packing Fraction}_{Cubic} = \frac{\frac{4}{3} \pi r^3}{8r^3} = \frac{\pi}{6} \approx 0.52 \]

### Step 5: Calculate the Ratio of Packing Fractions
Now we have:
- Packing Fraction in FCC = 0.74
- Packing Fraction in BCC = 0.68
- Packing Fraction in Cubic = 0.52

To find the ratio, we can express it as:

\[ \text{Ratio} = \text{Packing Fraction}_{FCC} : \text{Packing Fraction}_{BCC} : \text{Packing Fraction}_{Cubic} = 0.74 : 0.68 : 0.52 \]

To simplify, we divide each term by 0.52:

\[ \text{Ratio} = \frac{0.74}{0.52} : \frac{0.68}{0.52} : \frac{0.52}{0.52} \approx 1.42 : 1.31 : 1 \]

This can be approximated to:

\[ 1 : 0.92 : 0.70 \]

### Final Answer
Thus, the ratio of packing fractions in FCC, BCC, and simple cubic structures is:

\[ \text{Answer: } 1 : 0.92 : 0.70 \]