In the decay process:
`A overset(- alpha)rarr B overset(-beta)rarr C overset(-beta)rarr D`

Options

A. `A` and `B` are isodiaphers
B. `A` and `C` are isotones
C. `A` and `C` are isotopes
D. `B, C` and `D` are isobars

Answer

To solve the problem regarding the decay process \( A \overset{\alpha}{\rightarrow} B \overset{\beta}{\rightarrow} C \overset{\beta}{\rightarrow} D \), we will analyze the changes in atomic number (Z) and mass number (A) at each step of the decay.

### Step-by-step Solution:

1. **Identify the Decay Process**:
- The decay process involves three steps:
- \( A \) decays to \( B \) by emitting an alpha particle.
- \( B \) decays to \( C \) by emitting a beta particle.
- \( C \) decays to \( D \) by emitting another beta particle.

2. **Decay of A to B (Alpha Decay)**:
- In alpha decay, an alpha particle (which consists of 2 protons and 2 neutrons) is emitted.
- This results in:
- Atomic number of \( B \): \( Z_B = Z_A - 2 \)
- Mass number of \( B \): \( A_B = A_A - 4 \)

3. **Decay of B to C (Beta Decay)**:
- In beta decay, a neutron in the nucleus is converted into a proton, and a beta particle (electron) is emitted.
- This results in:
- Atomic number of \( C \): \( Z_C = Z_B + 1 = (Z_A - 2) + 1 = Z_A - 1 \)
- Mass number of \( C \): \( A_C = A_B = A_A - 4 \)

4. **Decay of C to D (Beta Decay)**:
- Again, in beta decay, another neutron is converted into a proton, and another beta particle is emitted.
- This results in:
- Atomic number of \( D \): \( Z_D = Z_C + 1 = (Z_A - 1) + 1 = Z_A \)
- Mass number of \( D \): \( A_D = A_C = A_A - 4 \)

5. **Summarizing the Relationships**:
- From the above steps, we can summarize:
- \( A \) has atomic number \( Z_A \) and mass number \( A_A \).
- \( B \) has atomic number \( Z_A - 2 \) and mass number \( A_A - 4 \).
- \( C \) has atomic number \( Z_A - 1 \) and mass number \( A_A - 4 \).
- \( D \) has atomic number \( Z_A \) and mass number \( A_A - 4 \).

6. **Conclusion**:
- **Isotopes**: \( A \) and \( D \) have the same atomic number \( Z_A \) but different mass numbers, hence they are isotopes.
- **Isobars**: \( B \), \( C \), and \( D \) have the same mass number \( A_A - 4 \) but different atomic numbers, hence they are isobars.

### Final Answer:
- \( A \) and \( D \) are isotopes.
- \( B \), \( C \), and \( D \) are isobars.