Analyzing Statement A: Electron Positron Mass Equivalence

The rest mass energy of an electron is approximately $0.511$ MeV. Its antiparticle, the positron, has the same rest mass energy.

The total rest mass energy for an electron-positron pair is the sum of their individual rest mass energies:

Total Energy $= m_e c^2 + m_p c^2 \approx 0.511 \text{ MeV} + 0.511 \text{ MeV} = 1.022 \text{ MeV}$

This value is commonly approximated as $1.02$ MeV. Therefore, statement A is correct.

Analyzing Statement B: Nucleus Role in Pair Annihilation

Pair annihilation is the process where a particle and its antiparticle (e.g., an electron and a positron) collide and convert their mass into energy, typically producing photons ($e^+ + e^- \rightarrow \gamma + \gamma$).

While a nucleus can facilitate momentum conservation in specific scenarios (like single-photon annihilation), it is not fundamentally required for the annihilation process itself. Statement B is correct.

Analyzing Statement C: Maximum Photon Wavelength for Pair Production

Pair production requires a minimum photon energy ($E_{min}$) equal to the combined rest mass energy of the electron-positron pair, which is approximately $1.022$ MeV.

The energy ($E$) and wavelength ($\lambda$) of a photon are related by $E = \frac{hc}{\lambda}$, where $h$ is Planck's constant and $c$ is the speed of light. The maximum wavelength ($\lambda_{max}$) corresponds to the minimum energy ($E_{min}$):

$\lambda_{max} = \frac{hc}{E_{min}}$

Using $h \approx 6.626 \times 10^{-34}$ J·s, $c \approx 3 \times 10^8$ m/s, and $E_{min} \approx 1.022 \text{ MeV} \approx 1.636 \times 10^{-13}$ J:

$\lambda_{max} \approx \frac{(6.626 \times 10^{-34} \text{ J·s}) \times (3 \times 10^8 \text{ m/s})}{1.636 \times 10^{-13} \text{ J}} \approx 1.215 \times 10^{-12} \text{ m}$

This value is approximately $1.2$ pm (picometers). Thus, statement C is correct.

Analyzing Statement D: Pair Production in Empty Space

Pair production requires the conservation of both energy and momentum. A single photon interacting in empty space cannot create an electron-positron pair because momentum conservation cannot be satisfied without interaction with another particle or field (like a nucleus) to absorb momentum. Statement D is incorrect.

Analyzing Statement E: Necessity of Nucleus for Pair Production

As explained for statement D, a nucleus or similar interacting particle is necessary for pair production to conserve momentum. Therefore, statement E is incorrect.

Conclusion on Correct Statements

Statements A, B, and C are correct based on the analysis of fundamental physics principles governing pair production and annihilation.

• Statement A: Correct.
• Statement B: Correct.
• Statement C: Correct.
• Statement D: Incorrect.
• Statement E: Incorrect.

The combination of correct statements is A, B, and C.