Question

. Show that a photon cannot spontaneously disintegrate into an electron- positron pair. [Hint: four-momentum conservation.] But in the presence of a stationary nucleus (acting as a kind of catalyst) it can. If the rest mass of the nucleus is \(N\), and that of the electron (and positron) is \(m\), what is the threshold frequency of the photon? Verify that for large \(N\) the efficiency is \(\sim 100\) per cent (cf. the preceding problem), so that the nucleus then comes close to being a pure catalyst.

Short answer

A: The threshold frequency of the photon (ν) can be found using the equation: $$ \nu = \frac{2mc^2 + (M_N^{'}-M_N)c^2}{h} $$ Q: Verify the efficiency of the process for a large nucleus mass N. A: For large N, the efficiency of the process can be defined as: $$ \eta = \frac{2mc^2}{2mc^2 + (M_N^{'}-M_N)c^2} \approx 1 $$ This means that for large N, the efficiency is close to 100% and the nucleus comes close to being a pure catalyst.

Step-by-step solution

Spontaneous disintegration without a nucleus

We need to check if energy and momentum conservation laws are upheld for spontaneous disintegration of a photon into an electron-positron pair. The energy conservation equation is: $$ E_{\gamma} = E_{e^-} + E_{e^+} $$ The momentum conservation equation is: $$ \vec{p}_{\gamma} = \vec{p}_{e^-} + \vec{p}_{e^+} $$ where \(E_{\gamma}\) and \(\vec{p}_{\gamma}\) are the energy and momentum of the photon, and \(E_{e^-}\), \(E_{e^+}\), \(\vec{p}_{e^-}\) and \(\vec{p}_{e^+}\) are those of the electron and positron. For the photon, we have \(E_{\gamma}=p_{\gamma}c\) and for each of electron and positron \(E_{\text{particle}}^2 = (m_{\text{particle}}c^2)^2+(\vec{p}_{\text{particle}}c)^2\) Since the photon has no mass, its four-momentum conservation equation is not upheld in this case, and the photon cannot spontaneously disintegrate into an electron-positron pair.

Disintegration with a stationary nucleus

In the presence of a stationary nucleus, we have a photon, a nucleus, an electron, and a positron. We can write the energy and momentum conservation equations as: $$ E_{\gamma} + M_Nc^2 = E_{e^-} + E_{e^+} + M_N^{'}c^2 $$ and $$ \vec{p}_{\gamma} = \vec{p}_{e^-} + \vec{p}_{e^+} + \vec{p}_N^{'} $$ Where \(M_N\) and \(M_N^{'}\) are the rest mass of the nucleus before and after the interaction, respectively. To satisfy momentum conservation, the final sum of the nucleus and electron-positron pair momenta should be equal to the initial photon momentum. As a result, this process can occur.

Threshold frequency of the photon

To find the threshold frequency of the photon, we will analyze the energy conservation equation, where the nucleus's kinetic energy (and thus momentum change) is minimized: $$ E_{\gamma} = E_{e^-} + E_{e^+} + (M_N^{'}-M_N)c^2 $$ In the threshold condition, we have \(E_{\text{particle}}=m_{\text{particle}}c^2\) for electron and positron $$ E_{\gamma} = 2mc^2 + (M_N^{'}-M_N)c^2 $$ The photon's energy is \(E_{\gamma} = h\nu\), with \(\nu\) being the frequency and \(h\) being the Planck constant. The threshold frequency is: $$ \nu = \frac{2mc^2 + (M_N^{'}-M_N)c^2}{h} $$

Efficiency verification

The efficiency of the process can be defined as: $$ \eta = \frac{2mc^2}{E_{\gamma}} $$ In the limit of large N, we have \(M_N^{'} \approx M_N\) $$ \eta = \frac{2mc^2}{2mc^2 + (M_N^{'}-M_N)c^2} \approx 1 $$ Therefore, for large N, the efficiency is close to 100%, and the nucleus comes close to being a pure catalyst.

Key concepts

Four-Momentum Conservation

In particle physics, four-momentum conservation is a crucial principle that extends the classical conservation laws of energy and momentum into a single framework. Four-momentum combines the energy and the three components of momentum into a four-vector, \( P^{\mu} = (E/c, \mathbf{p}) \), where \( E \) is energy, \( c \) is the speed of light, and \( \mathbf{p} \) is the momentum vector.
This allows us to analyze particle interactions, such as the disintegration of photons. When a photon encounters a stationary nucleus, the combined system of photon and nucleus must conserve four-momentum both before and after the interaction. Without the nucleus, a free photon lacks mass, and the resulting particles, an electron and a positron, cannot fulfill the four-momentum conservation condition due to mass difference, making spontaneous disintegration impossible.

Electron-Positron Pair Production

Electron-positron pair production is a fascinating quantum process where energy, typically from a photon, gets converted into mass, resulting in the creation of an electron-positron pair.
It's vital to remember that energy and momentum conservation must be satisfied for this process to occur. A lone photon cannot spontaneously transform into a pair in a vacuum because it can’t conserve both energy and momentum. However, when a photon interacts with an external object like a nucleus, the nucleus can absorb part of the momentum, allowing the transformation.
This interaction is similar to a catalyst in a chemical reaction. The presence of the nucleus doesn't supply energy, but it provides a medium for momentum conservation, essential for the pair creation.

Energy Conservation

Energy conservation is a universal principle stating that energy cannot be created or destroyed, only transformed from one form to another. In the context of particle interactions, like photon disintegration, this means the energy before and after any interaction must remain constant.
For pair production near a nucleus, the energy equation initially includes the photon's energy and the rest energy of the nucleus. Conservation ensures that this energy equals the combined rest energy and kinetic energies of the resultant particles (electron, positron) and any energy accounting for changes in the nucleus.
This helps us determine the photon's threshold frequency (minimum energy needed) to produce an electron-positron pair in the presence of a nucleus. Mathematically, the photon's energy, \( E_{\gamma} = hu \), must meet or exceed the total rest mass energy resulting in the equation used to find the threshold frequency.

Momentum Conservation

Momentum conservation is another key principle in physics, stipulating that the total momentum of an isolated system remains constant over time unless acted upon by external forces.
In terms of photon disintegration with a stationary nucleus, the momentum of the initial system (photon and stationary nucleus) must equal the momentum of the final system (electron, positron, and possibly a recoiling nucleus).
In the free-space disintegration of a photon, there's no way to balance the equations of momentum conservation, because a photon alone cannot transform its momentum into two particles without introducing an imbalance. That is why a stationary nucleus is necessary; it can absorb the "extra" momentum that enables the electron and positron to travel with their respective momenta, ensuring the conservation law is satisfied. This creates a dynamic bridge, facilitating energy conversion into the mass of the electron and positron pair.