What will be the ratio of the total volume of bcc to simple cubic structure is?

Options

A. `3 sqrt(3) :8`
B. `1 : 24 sqrt(3)`
C. `24 sqrt(3) : 1`
D. `8:3 sqrt(3)`

Explanation

To find the ratio of the total volume of body-centered cubic (BCC) to simple cubic (SCC) structures, we can follow these steps:

### Step 1: Calculate the volume of the BCC structure
In a BCC structure, the relationship between the edge length (a) and the radius (r) of the atoms is given by:
\[ a = \frac{4r}{\sqrt{3}} \]
The volume (V) of the BCC unit cell can be calculated using the formula for the volume of a cube:
\[ V_{BCC} = a^3 \]
Substituting the expression for a:
\[ V_{BCC} = \left(\frac{4r}{\sqrt{3}}\right)^3 = \frac{64r^3}{3\sqrt{3}} \]

### Step 2: Calculate the volume of the SCC structure
In a simple cubic structure, the relationship between the edge length (a) and the radius (r) of the atoms is:
\[ a = 2r \]
The volume of the SCC unit cell is:
\[ V_{SCC} = a^3 = (2r)^3 = 8r^3 \]

### Step 3: Calculate the ratio of the volumes
Now, we can find the ratio of the volume of BCC to the volume of SCC:
\[ \text{Ratio} = \frac{V_{BCC}}{V_{SCC}} = \frac{\frac{64r^3}{3\sqrt{3}}}{8r^3} \]
Simplifying this expression:
\[ \text{Ratio} = \frac{64r^3}{3\sqrt{3} \cdot 8r^3} = \frac{64}{24\sqrt{3}} = \frac{8}{3\sqrt{3}} \]

### Final Answer
Thus, the ratio of the total volume of BCC to simple cubic structure is:
\[ \frac{8}{3\sqrt{3}} \]

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