Introduction

The problem asks to identify the pair of particles (X, Y) that correctly completes the given nuclear transmutation reaction: $_4^9Be + X \to _4^8Be + Y$ To solve this, we must apply the laws of conservation of mass number (A) and atomic number (Z) in nuclear reactions.

1. Conservation of Mass Number (A): The sum of the mass numbers of the reactants must equal the sum of the mass numbers of the products.
2. Conservation of Atomic Number (Z): The sum of the atomic numbers (charges) of the reactants must equal the sum of the atomic numbers of the products.

Step-by-Step Solution

Let's represent the incoming particle X as $_Z^A X$ and the outgoing particle Y as $_z^a Y$. The nuclear reaction can be written as: $_4^9Be + _Z^A X \to _4^8Be + _z^a Y$

Applying Conservation Laws:

1. Conservation of Mass Number (A): $9 + A = 8 + a$ $a - A = 9 - 8$ $a - A = 1$

2. Conservation of Atomic Number (Z): $4 + Z = 4 + z$ $z - Z = 4 - 4$ $z - Z = 0 \implies z = Z$

Now, we will evaluate each option to see if it satisfies these two conditions: a - A = 1 and z = Z.

Particle Notations:

• Gamma photon ($\\\gamma$): $_0^0\gamma$ (A=0, Z=0)
• Neutron (n): $_0^1n$ (A=1, Z=0)
• Proton (p): $_1^1p$ (A=1, Z=1)
• Deuteron (D): $_1^2D$ (A=2, Z=1)

Evaluation of Options

A: $(\gamma ,n)$

• X is a gamma photon ($_0^0\gamma$), so A=0 and Z=0.
• Y is a neutron ($_0^1n$), so a=1 and z=0.

Check conditions:

1. a - A = 1 - 0 = 1. This condition is satisfied.
2. $z = Z \implies 0 = 0$. This condition is satisfied. Since both conditions are met, option A is a valid solution. The reaction is: $_4^9Be + _0^0\gamma \to _4^8Be + _0^1n$. This is a photodisintegration reaction.

B: (p, D)

• X is a proton ($_1^1p$), so A=1 and Z=1.
• Y is a deuteron ($_1^2D$), so a=2 and z=1.

Check conditions:

1. a - A = 2 - 1 = 1. This condition is satisfied.
2. $z = Z \implies 1 = 1$. This condition is satisfied. Since both conditions are met, option B is also a valid solution. The reaction is: $_4^9Be + _1^1p \to _4^8Be + _1^2D$.

C: (n, D)

• X is a neutron ($_0^1n$), so A=1 and Z=0.
• Y is a deuteron ($_1^2D$), so a=2 and z=1.

Check conditions:

1. a - A = 2 - 1 = 1. This condition is satisfied.
2. $z = Z \implies 1 = 0$. This is false. Therefore, option C is incorrect. For verification, the reaction $_4^9Be + _0^1n \to _4^8Be + _1^2D$ shows the atomic number is not conserved (4+0=4 on the left, 4+1=5 on the right).

D: $(\gamma ,p)$

• X is a gamma photon ($_0^0\gamma$), so A=0 and Z=0.
• Y is a proton ($_1^1p$), so a=1 and z=1.

Check conditions:

1. a - A = 1 - 0 = 1. This condition is satisfied.
2. $z = Z \implies 1 = 0$. This is false. Therefore, option D is incorrect. For verification, the reaction $_4^9Be + _0^0\gamma \to _4^8Be + _1^1p$ shows the atomic number is not conserved (4+0=4 on the left, 4+1=5 on the right).

Conclusion

Based on the analysis, options A and B both satisfy the principles of conservation of mass number and atomic number for the given nuclear transmutation. Thus, both are correct answers.