A nucelus with atomic number `Z` and neutron number `N` undergoes two decay processes. The result is a nucleus with atomic number `Z-3` and neutron `N-1`. Which decay processes took place?

Options

A. Two `beta^(bar)` decays
B. Two `beta^(+)` decays
C. An `alpha` decay and a `beta^(bar)` decays
D. `An `alpha` decay and a `beta^(+)` decays

Explanation

To solve the problem, we need to analyze the decay processes that lead to a nucleus with an atomic number of \( Z-3 \) and a neutron number of \( N-1 \) from an initial nucleus with atomic number \( Z \) and neutron number \( N \).

### Step-by-Step Solution:

1. **Understanding the Initial and Final States**:
- Initial nucleus: Atomic number \( Z \), Neutron number \( N \).
- Final nucleus after two decay processes: Atomic number \( Z-3 \), Neutron number \( N-1 \).

2. **Identifying the Changes in Atomic and Neutron Numbers**:
- The atomic number decreases from \( Z \) to \( Z-3 \), which means a total decrease of 3 protons.
- The neutron number decreases from \( N \) to \( N-1 \), which means a total decrease of 1 neutron.

3. **Considering Alpha Decay**:
- In alpha decay, a nucleus emits an alpha particle, which consists of 2 protons and 2 neutrons.
- Therefore, after one alpha decay:
- New atomic number = \( Z - 2 \)
- New neutron number = \( N - 2 \)

4. **Considering Beta Minus Decay**:
- In beta minus decay, a neutron is converted into a proton, emitting an electron (beta particle).
- This process increases the atomic number by 1 and decreases the neutron number by 1.
- Therefore, if we perform a beta minus decay after an alpha decay:
- New atomic number = \( (Z - 2) + 1 = Z - 1 \)
- New neutron number = \( (N - 2) + 1 = N - 1 \)

5. **Combining the Two Decay Processes**:
- After the alpha decay, we have:
- Atomic number = \( Z - 2 \)
- Neutron number = \( N - 2 \)
- After the beta minus decay, we have:
- Atomic number = \( Z - 1 \)
- Neutron number = \( N - 1 \)

6. **Final Result**:
- We need to achieve an atomic number of \( Z - 3 \) and a neutron number of \( N - 1 \).
- Therefore, the first decay must be an alpha decay (which reduces the atomic number by 2), followed by a beta minus decay (which reduces the atomic number by 1).

### Conclusion:
The decay processes that took place are:
- **First decay**: Alpha decay (reducing atomic number by 2).
- **Second decay**: Beta minus decay (reducing atomic number by 1).