Mark the correct options.

Options

A. An atom with a vacancy has smaller energy than a neutral atom.
B. K X-ray is emitted when a hole makes a jump from the K shell to some other shell.
C. The wavelength of K X-ray is smaller than the wavelength of L X-ray of the same material.
D. The wavelength of `K_alpha` X-ray is smaller than the wavelength of `K_beta` X-ray of the same material.

Answer

To solve the question, we need to analyze each option provided and determine whether they are correct or incorrect based on our understanding of X-rays and atomic structure.

### Step-by-Step Solution:

1. **Evaluate Option A**:
- **Statement**: An atom with a vacancy is smaller than the neutral atom.
- **Analysis**: A vacancy in an atom (i.e., an electron missing from an orbital) leads to a higher energy state compared to a neutral atom. Thus, the statement is incorrect because a vacancy does not make the atom smaller; rather, it increases the energy state.
- **Conclusion**: Option A is **incorrect**.

2. **Evaluate Option B**:
- **Statement**: K X-ray is emitted when a hole makes a jump from the K shell to some other shell.
- **Analysis**: K X-rays are emitted when an electron from a higher energy level (such as L or M shell) falls into the K shell, creating a vacancy (or hole) in the higher shell. Therefore, the statement is correct as it describes the process accurately.
- **Conclusion**: Option B is **correct**.

3. **Evaluate Option C**:
- **Statement**: The wavelength of K X-ray is smaller than the wavelength of L X-ray of the same material.
- **Analysis**: The energy of K X-rays is higher than that of L X-rays because K X-rays result from transitions to the K shell from higher energy levels (L or M), while L X-rays result from transitions to the L shell from higher energy levels (M or N). Since energy is inversely proportional to wavelength, K X-rays will indeed have a smaller wavelength than L X-rays.
- **Conclusion**: Option C is **correct**.

4. **Evaluate Option D**:
- **Statement**: The wavelength of KL alpha X-ray is smaller than the wavelength of K beta X-ray of the same material.
- **Analysis**: The K alpha X-ray results from the transition of electrons from the L shell to the K shell, while K beta X-ray results from transitions from the M shell to the K shell. The energy difference for K beta is greater than that for K alpha, which means K beta has a shorter wavelength than K alpha. Therefore, the statement is incorrect.
- **Conclusion**: Option D is **incorrect**.

### Final Summary of Options:
- Option A: Incorrect
- Option B: Correct
- Option C: Correct
- Option D: Incorrect

### Correct Options:
The correct options are **B and C**.

---