Calculate the minimum energy needed to separate a neutron form a nucleus with Z protons and N neutrons in terms of the masses `M_(Z,N),M_(Z,N-1)` and the mass of the neutron.

Explanation

To calculate the minimum energy needed to separate a neutron from a nucleus with Z protons and N neutrons, we can follow these steps:

### Step 1: Understand the Masses Involved
We have:
- \( M_{Z,N} \): Mass of the nucleus with Z protons and N neutrons.
- \( M_{Z,N-1} \): Mass of the nucleus with Z protons and (N-1) neutrons (after removing one neutron).
- \( m_n \): Mass of the neutron.

### Step 2: Write the Reaction
The process of separating a neutron from the nucleus can be represented as:
\[ M_{Z,N} \rightarrow M_{Z,N-1} + m_n \]
This indicates that the initial nucleus with Z protons and N neutrons splits into a nucleus with Z protons and (N-1) neutrons plus a free neutron.

### Step 3: Calculate the Change in Mass
The change in mass (\( \Delta m \)) during this reaction can be expressed as:
\[ \Delta m = M_{Z,N-1} + m_n - M_{Z,N} \]

### Step 4: Calculate the Energy Required
According to Einstein's mass-energy equivalence principle, the energy (\( \Delta E \)) required to separate the neutron is given by:
\[ \Delta E = \Delta m \cdot c^2 \]
Substituting the expression for \( \Delta m \):
\[ \Delta E = \left( M_{Z,N-1} + m_n - M_{Z,N} \right) c^2 \]

### Final Expression
Thus, the minimum energy needed to separate a neutron from the nucleus is:
\[ \Delta E = \left( M_{Z,N-1} + m_n - M_{Z,N} \right) c^2 \]