To find the maximum energy of the anti-neutrino produced during beta decay, we can follow these steps:
### Step 1: Understand the Process
In negative beta decay, a neutron decays into a proton, an electron, and an anti-neutrino. The reaction can be represented as:
\[ n \rightarrow p + e^- + \bar{\nu} \]
where \( n \) is the neutron, \( p \) is the proton, \( e^- \) is the electron, and \( \bar{\nu} \) is the anti-neutrino.
### Step 2: Apply the Conservation of Energy
According to the conservation of energy, the energy released in the decay process is equal to the difference in mass (converted to energy using Einstein's equation \( E = mc^2 \)):
\[ E = (m_n - m_p - m_e) c^2 \]
where:
- \( m_n \) = mass of the neutron
- \( m_p \) = mass of the proton
- \( m_e \) = mass of the electron
- \( c \) = speed of light
### Step 3: Insert the Mass Values
Using the given masses in atomic mass units (u):
- Mass of neutron \( m_n = 1.00866 \, u \)
- Mass of proton \( m_p = 1.00727 \, u \)
- Mass of electron \( m_e = 0.00055 \, u \)
We can substitute these values into the equation:
\[ E = (1.00866 \, u - 1.00727 \, u - 0.00055 \, u) c^2 \]
### Step 4: Calculate the Mass Difference
Calculating the mass difference:
\[ m_n - m_p - m_e = 1.00866 \, u - 1.00727 \, u - 0.00055 \, u = 0.00084 \, u \]
### Step 5: Convert Mass Difference to Energy
Now, we convert the mass difference into energy:
\[ E = 0.00084 \, u \cdot c^2 \]
Using the conversion factor \( 1 \, u \approx 931.5 \, \text{MeV} \):
\[ E = 0.00084 \, u \cdot 931.5 \, \text{MeV/u} \]
\[ E \approx 0.782 \, \text{MeV} \]
### Step 6: Conclusion
Thus, the maximum energy of the anti-neutrino produced in the decay process is approximately:
\[ E \approx 0.78 \times 10^6 \, \text{eV} \]
### Final Answer
The maximum energy of the anti-neutrino is less than \( 0.8 \times 10^6 \, \text{eV} \).
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