What is (a) the minimum energy of 1 photon that can produce an electron pair?

(b) what is the photon’s wavelength

To determine the minimum energy of a photon that can produce an electron pair, we need to consider the process known as pair production. This phenomenon occurs when a photon interacts with a strong electromagnetic field, typically near a nucleus, and creates a particle-antiparticle pair, such as an electron and a positron. Let's break this down step by step.

Minimum Energy Required for Pair Production

The minimum energy required to create an electron-positron pair can be calculated using Einstein's famous equation, E=mc². In this case, we need to account for the rest mass energy of both the electron and the positron. The rest mass energy of an electron (or positron) is approximately 0.511 MeV (mega-electronvolts).

• Energy of electron: 0.511 MeV
• Energy of positron: 0.511 MeV

Since both particles are created, the total minimum energy required is:

E_total = E_electron + E_positron = 0.511 \text{ MeV} + 0.511 \text{ MeV} = 1.022 \text{ MeV}

Photon Energy and Wavelength

Now that we have established that the minimum energy of a photon required for pair production is 1.022 MeV, we can find the corresponding wavelength using the relationship between energy and wavelength given by the equation:

E = \frac{hc}{\lambda}

Where:

• E is the energy of the photon
• h is Planck's constant (approximately 4.135667696 × 10^-15 eV·s)
• c is the speed of light (approximately 3 × 10^8 m/s)
• λ is the wavelength of the photon

Rearranging this equation to solve for wavelength gives us:

λ = \frac{hc}{E}

Substituting in the values:

• h = 4.135667696 × 10^-15 eV·s
• c = 3 × 10^8 m/s
• E = 1.022 MeV = 1.022 × 10^6 eV

Now, plugging these values into the equation:

λ = \frac{(4.135667696 × 10^{-15} \text{ eV·s})(3 × 10^8 \text{ m/s})}{1.022 × 10^6 \text{ eV}}

Calculating this gives:

λ ≈ 1.22 × 10^{-10} \text{ m} = 0.122 \text{ nm}

Summary

To summarize, the minimum energy of a photon that can produce an electron pair is approximately 1.022 MeV, and the corresponding wavelength of this photon is about 0.122 nm. This wavelength falls within the gamma-ray region of the electromagnetic spectrum, which is consistent with the high energy required for pair production.