The coefficient of linear expansion of crystal in one direction is `alpha_(1)` and that in every direction perpendicular to it is `alpha_(2)`. The coefficient of cubical expansion is

Options

A. `alpha_(1) + alpha_(2)`
B. `2alpha_(1) + alpha_(2)`
C. `alpha_(1) + 2alpha_(2)`
D. None of these

Explanation

To find the coefficient of cubical expansion (γ) of a crystal with a linear expansion coefficient of α₁ in one direction and α₂ in every direction perpendicular to it, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Linear Expansion**:
The linear expansion of a crystal in one direction can be expressed as:
\[ L = L_0 (1 + \alpha_1 \Delta T) \]
where \(L_0\) is the original length, \(L\) is the new length after temperature change \(\Delta T\), and \(\alpha_1\) is the coefficient of linear expansion in that direction.

2. **Linear Expansion in Perpendicular Directions**:
In the directions perpendicular to the first, the linear expansion can be expressed as:
\[ L = L_0 (1 + \alpha_2 \Delta T) \]
for both perpendicular directions.

3. **Volume of the Cube**:
The volume \(V\) of a cube is given by:
\[ V = L^3 \]
If we consider a cube where one side expands by \(\alpha_1\) and the other two sides expand by \(\alpha_2\), the new volume can be expressed as:
\[ V = L_1 \cdot L_2 \cdot L_3 \]
where \(L_1\) is the side with expansion \(\alpha_1\) and \(L_2\) and \(L_3\) are the sides with expansion \(\alpha_2\).

4. **Calculating the New Volume**:
The new volume after expansion can be calculated as:
\[ V = (L_0 (1 + \alpha_1 \Delta T))(L_0 (1 + \alpha_2 \Delta T))(L_0 (1 + \alpha_2 \Delta T)) \]
Simplifying this gives:
\[ V = L_0^3 (1 + \alpha_1 \Delta T)(1 + \alpha_2 \Delta T)^2 \]

5. **Expanding the Expression**:
Using the binomial expansion for small \(\Delta T\), we can approximate:
\[ (1 + \alpha_2 \Delta T)^2 \approx 1 + 2\alpha_2 \Delta T \]
Thus, we have:
\[ V \approx L_0^3 (1 + \alpha_1 \Delta T)(1 + 2\alpha_2 \Delta T) \]

6. **Combining the Terms**:
Expanding this product results in:
\[ V \approx L_0^3 (1 + \alpha_1 \Delta T + 2\alpha_2 \Delta T + \text{higher order terms}) \]
Neglecting the higher order terms, we get:
\[ V \approx L_0^3 (1 + (\alpha_1 + 2\alpha_2) \Delta T) \]

7. **Identifying the Coefficient of Cubical Expansion**:
From the expression for volume expansion, we can identify the coefficient of cubical expansion \(γ\) as:
\[ \gamma = \alpha_1 + 2\alpha_2 \]

### Final Result:
Thus, the coefficient of cubical expansion of the crystal is:
\[ \gamma = \alpha_1 + 2\alpha_2 \]