For the radioactive nuclei that undergo either `alpha or beta` decay, which one of the following cannot occur?

Options

A. Isobar of original nucleus is produce
B. Isotop of the original nucleus is produced
C. Nuclei with higher atomic number than that of the original nucleus is produced
D. Nuclei with lower atomic number than that of the original nucleus is produced

Answer

To solve the question regarding which decay cannot occur for radioactive nuclei undergoing either alpha or beta decay, we need to analyze the outcomes of each type of decay.

### Step-by-Step Solution:

1. **Understand Alpha Decay:**
- In alpha decay, a nucleus with atomic number \( Z \) and mass number \( A \) emits an alpha particle (which is a helium nucleus, \( ^4_2He \)).
- The resulting nucleus will have:
- Atomic number: \( Z - 2 \)
- Mass number: \( A - 4 \)
- This means that the nucleus produced has a lower atomic number and a lower mass number than the original nucleus.

2. **Understand Beta Decay:**
- In beta decay, there are two types: beta minus (β-) and beta plus (β+).
- **Beta Minus Decay (β-):**
- A nucleus with atomic number \( Z \) and mass number \( A \) converts into a nucleus with:
- Atomic number: \( Z + 1 \)
- Mass number: \( A \)
- This results in a nucleus with a higher atomic number than the original nucleus.
- **Beta Plus Decay (β+):**
- A nucleus with atomic number \( Z \) and mass number \( A \) converts into a nucleus with:
- Atomic number: \( Z - 1 \)
- Mass number: \( A \)
- This results in a nucleus with a lower atomic number than the original nucleus.

3. **Analyze the Options:**
- **Option A:** Isobar of the original nucleus is produced.
- This can occur because isobars have the same mass number but different atomic numbers. Both alpha and beta decay can produce isobars.
- **Option B:** Isotope of the original nucleus is produced.
- This cannot occur because isotopes have the same atomic number but different mass numbers. Neither alpha nor beta decay produces a nucleus that retains the same atomic number.
- **Option C:** Nuclei with lighter atomic number than that of the original nucleus is produced.
- This can occur with both alpha decay (produces a nucleus with \( Z - 2 \)) and beta plus decay (produces a nucleus with \( Z - 1 \)).
- **Option D:** Nuclei with lower atomic number than that of the original nucleus is produced.
- This can also occur with both alpha decay and beta plus decay.

4. **Conclusion:**
- The only option that cannot occur is **Option B: Isotope of the original nucleus is produced**.

### Final Answer:
The correct answer is **Option B**: Isotope of the original nucleus is produced.