A `pi^(o)` meson at rest decays into two photons of equal energy. If the wavelength (in `m`) of the photons is `1.8xx10^(-n)` then find `n//2` ( The mass of the `pi^(o) = 135 Me V//c^(2)`)

Explanation

To solve the problem, we need to find the wavelength of the photons produced from the decay of a π^0 meson at rest. The mass of the meson is given as 135 MeV/c². We will use the principles of energy-mass equivalence and the properties of photons.

### Step-by-step Solution:

1. **Understand the decay process**:
The π^0 meson decays into two photons. Since the meson is at rest, the total energy before the decay is equal to the rest energy of the meson.

2. **Use the energy-mass equivalence**:
According to Einstein's equation, the energy (E) of the meson can be expressed as:
\[ E = mc^2 \]
where \( m \) is the mass of the meson and \( c \) is the speed of light.

3. **Calculate the energy of the meson**:
Given that the mass of the π^0 meson is 135 MeV/c², we can write:
\[ E = 135 \, \text{MeV} \]

4. **Energy of the photons**:
The total energy of the two photons produced in the decay is equal to the energy of the meson:
\[ E_{\text{total}} = E_{\text{photon1}} + E_{\text{photon2}} = 2E_{\text{photon}} \]
Therefore, we have:
\[ 2E_{\text{photon}} = 135 \, \text{MeV} \]
This gives:
\[ E_{\text{photon}} = \frac{135}{2} \, \text{MeV} = 67.5 \, \text{MeV} \]

5. **Relate energy to wavelength**:
The energy of a photon is also related to its wavelength (\( \lambda \)) by the equation:
\[ E = \frac{hc}{\lambda} \]
where \( h \) is Planck's constant (\( 6.63 \times 10^{-34} \, \text{Js} \)) and \( c \) is the speed of light (\( 3 \times 10^8 \, \text{m/s} \)).

6. **Rearranging for wavelength**:
We can rearrange the equation to find \( \lambda \):
\[ \lambda = \frac{hc}{E} \]

7. **Convert energy from MeV to Joules**:
To use the above formula, we need to convert the energy from MeV to Joules. We know:
\[ 1 \, \text{MeV} = 1.6 \times 10^{-13} \, \text{J} \]
Therefore:
\[ E_{\text{photon}} = 67.5 \, \text{MeV} \times 1.6 \times 10^{-13} \, \text{J/MeV} = 1.08 \times 10^{-12} \, \text{J} \]

8. **Calculate the wavelength**:
Now substituting the values into the wavelength formula:
\[ \lambda = \frac{(6.63 \times 10^{-34} \, \text{Js})(3 \times 10^8 \, \text{m/s})}{1.08 \times 10^{-12} \, \text{J}} \]
\[ \lambda = \frac{1.989 \times 10^{-25}}{1.08 \times 10^{-12}} \approx 1.84 \times 10^{-14} \, \text{m} \]

9. **Express the wavelength in the required form**:
We can express the wavelength as:
\[ \lambda = 1.8 \times 10^{-14} \, \text{m} \]
This matches the form \( 1.8 \times 10^{-n} \).

10. **Find the value of n**:
From \( 1.8 \times 10^{-14} \), we can see that \( n = 14 \).

11. **Calculate \( n/2 \)**:
Finally, we need to find \( n/2 \):
\[ \frac{n}{2} = \frac{14}{2} = 7 \]

### Final Answer:
The value of \( n/2 \) is \( 7 \).