For pair production i.e. for the production of electron and positron, the incident photon must have a minimum frequency of the order of

Options

A. `10^(18) s^(-1)`
B. `10^(21) s^(-1)`
C. `10^(25) s^(-1)`
D. `10^(30) s^(-1)`

Answer

To solve the problem of determining the minimum frequency of a photon required for pair production (the creation of an electron and a positron), we will follow these steps:

### Step 1: Understand the energy requirement for pair production
For pair production to occur, the energy of the incident photon must be at least equal to the combined rest mass energy of the electron and positron. The rest mass energy \(E\) can be expressed using the formula:
\[ E = m c^2 \]
where \(m\) is the mass and \(c\) is the speed of light.

### Step 2: Calculate the rest mass energy of an electron and positron
The rest mass of an electron (and positron) is approximately \(0.511 \, \text{MeV}\). Therefore, the total rest mass energy for both particles is:
\[ E_{\text{total}} = 2 \times 0.511 \, \text{MeV} = 1.022 \, \text{MeV} \]

### Step 3: Convert the energy from MeV to Joules
To convert MeV to Joules, we use the conversion factor \(1 \, \text{MeV} = 1.6 \times 10^{-13} \, \text{J}\):
\[ E_{\text{total}} = 1.022 \, \text{MeV} \times 1.6 \times 10^{-13} \, \text{J/MeV} = 1.6352 \times 10^{-13} \, \text{J} \]

### Step 4: Relate energy to frequency using Planck's equation
The energy of a photon can also be expressed in terms of its frequency \(f\) using Planck's equation:
\[ E = h f \]
where \(h\) is Planck's constant (\(h \approx 6.626 \times 10^{-34} \, \text{J s}\)).

### Step 5: Solve for the minimum frequency
Rearranging the equation gives:
\[ f = \frac{E}{h} \]
Substituting the values we calculated:
\[ f = \frac{1.6352 \times 10^{-13} \, \text{J}}{6.626 \times 10^{-34} \, \text{J s}} \approx 2.47 \times 10^{20} \, \text{Hz} \]

### Step 6: Conclusion
The minimum frequency of the incident photon required for pair production is approximately \(2.47 \times 10^{20} \, \text{Hz}\).