The frquencies of `K_alpha, K_beta` and `L_alpha` X-rays of a materail are `gamma_1, gamma_2 and gamma_3` respectively. Which of the following relation holds good?

Options

A. `gamma_2 = sqrt(gamma_1+gamma_3)`
B. `gamma_2 = gamma_1+gamma_3`
C. `gamma_2 = (gamma_1+gamma_3)/(2)`
D. `gamma_3 = sqrt(gamma_1 gamma_2)`

Explanation

To solve the problem, we need to establish the relationship between the frequencies of K_alpha (γ1), K_beta (γ2), and L_alpha (γ3) X-rays.

### Step-by-Step Solution:

1. **Understanding the X-ray Transitions**:
- K_alpha (γ1) and K_beta (γ2) are transitions from higher energy levels to the K shell (n=1).
- L_alpha (γ3) is a transition from higher energy levels to the L shell (n=2).

2. **Energy Relationships**:
- The energy of X-rays can be expressed using the formula:
\[ E = h \nu \]
where \(E\) is energy, \(h\) is Planck's constant, and \(\nu\) is frequency.

3. **Energy of X-rays**:
- For K_alpha, the energy can be written as:
\[ E_{K\alpha} = h \gamma_1 \]
- For K_beta, the energy is:
\[ E_{K\beta} = h \gamma_2 \]
- For L_alpha, the energy is:
\[ E_{L\alpha} = h \gamma_3 \]

4. **Applying Energy Conservation**:
- According to the theory of X-ray emissions, the energy of K_beta can be expressed as the sum of the energies of K_alpha and L_alpha:
\[ E_{K\beta} = E_{K\alpha} + E_{L\alpha} \]

5. **Substituting the Energy Expressions**:
- Substituting the expressions for energy, we have:
\[ h \gamma_2 = h \gamma_1 + h \gamma_3 \]

6. **Canceling Planck's Constant**:
- Since \(h\) is a common factor, we can cancel it from both sides:
\[ \gamma_2 = \gamma_1 + \gamma_3 \]

7. **Conclusion**:
- The relation that holds good is:
\[ \gamma_2 = \gamma_1 + \gamma_3 \]
- Thus, the correct answer is option B.