For characteristic `X` - ray of some material

Options

A. `E(K_(gamma))ltE(K_(beta))ltE(K_(alpha))`
B. `E(K_(alpha))ltE(L_(alpha))ltE(M_(alpha))`
C. `lambda(K_(gamma))ltlambda(K_(beta))ltlambda(K_(alpha))`
D. `lambda(M_(alpha))ltlambda(L_(alpha))lt lambda (K_(alpha))`

Explanation

To solve the question regarding the characteristic X-rays of a material, we need to analyze the given options based on our understanding of X-ray energies and wavelengths.

### Step-by-Step Solution:

1. **Understanding Characteristic X-rays**:
Characteristic X-rays are emitted by atoms when electrons transition between energy levels. The most common transitions involve the K and L shells (energy levels).

2. **Energy Levels**:
- The K shell is the closest shell to the nucleus and has the highest energy.
- The L shell is the next shell, and the M shell follows.
- The transitions that produce X-rays are labeled as Kα, Kβ, Kγ, etc., where Kα corresponds to an electron transition from the L shell to the K shell, Kβ corresponds to a transition from the M shell to the K shell, and so forth.

3. **Energy Relationships**:
- The energy of the emitted X-ray is higher for transitions involving outer shells to inner shells.
- Therefore, we have the following energy relationships:
- \( E_{K\gamma} > E_{K\beta} > E_{K\alpha} \)

4. **Wavelength and Energy Relationship**:
- The energy (E) of a photon is inversely proportional to its wavelength (λ), expressed by the equation:
\[ E = \frac{hc}{\lambda} \]
where \( h \) is Planck's constant and \( c \) is the speed of light.
- This means that as the energy increases, the wavelength decreases.

5. **Wavelength Relationships**:
- From the energy relationships, we can deduce:
- Since \( E_{K\gamma} > E_{K\beta} > E_{K\alpha} \), it follows that:
- \( \lambda_{K\gamma} < \lambda_{K\beta} < \lambda_{K\alpha} \)

6. **Evaluating the Options**:
- **Option A**: \( E_{\gamma} < E_{K\beta} < E_{K\alpha} \) - Incorrect.
- **Option B**: \( E_{K\alpha} < E_{L\alpha} < E_{M\alpha} \) - This is not a direct comparison of K and L shells.
- **Option C**: \( \lambda_{K\gamma} < \lambda_{K\beta} < \lambda_{K\alpha} \) - Correct based on our analysis.
- **Option D**: \( \lambda_{M\alpha} < \lambda_{L\alpha} < \lambda_{K\alpha} \) - This is not directly relevant to the K series.

7. **Conclusion**:
- The correct answer is **Option C**: The wavelength of \( K\gamma \) is less than the wavelength of \( K\beta \), which is less than the wavelength of \( K\alpha \).