Question

If one neutron and one pi-meson are to ernerge from the collision of a photon with a stationary proton, find the threshold frequency of the photon in terms of the rest mass \(n\) of a proton o neutron (here assumed equal) and that, \(m\), of a pi-meson. [Answer \(\left.c^{2}\left(m^{2}+2 m n\right) / 2 h n .\right]\)

Short answer

Answer: The threshold frequency of the photon, denoted as \(f_t\), can be expressed as \(f_t = \frac{c^2}{2hn}(m^2 + 2mn)\), where \(n\) is the rest mass of a proton or neutron (here assumed equal), \(m\) is the rest mass of a pi-meson, \(c\) is the speed of light, and \(h\) is Planck's constant.

Step-by-step solution

Write down the energy-momentum relation

According to the rules of special relativity, the energy and momentum of a particle are related by \(E^{2} = (pc)^2 + (mc^2)^2\) where \(E\) is the energy, \(p\) is the momentum, \(m\) is the mass of the particle, and \(c\) is the speed of light. For the given exercise, let's denote the energies and momenta of the photon, proton, neutron, and pi-meson as \(E_\gamma\), \(E_p\), \(E_n\), \(E_\pi\), and \(p_\gamma\), \(p_p\), \(p_n\), \(p_\pi\), respectively.

Apply conservation of energy

The conservation of energy states that the total energy before the collision equals the total energy after the collision. In this case: \(E_\gamma + E_p = E_n + E_\pi\) Since the proton is initially stationary, its energy is simply its rest mass energy: \(E_p = n c^2\)

Apply conservation of momentum

The conservation of momentum states that the total momentum before the collision equals the total momentum after the collision. In this case: \(p_\gamma = p_n + p_\pi\) Since the proton is initially stationary, its momentum is zero.

Express photon energy in terms of frequency

The energy of a photon is related to its frequency by: \(E_\gamma = h f\) where \(h\) is Planck's constant and \(f\) is the frequency of the photon.

Determine the threshold condition

The threshold condition occurs when the photon energy is just enough to create the neutron and pi-meson. From the energy-momentum relation, we have: \(E_n^2 = (p_nc)^2 + (nc^2)^2\) \(E_\pi^2 = (p_\pi c)^2 + (mc^2)^2\) Substituting these expressions into the conservation of energy and momentum equations, we get: \((hf)^2 = (p_n c + p_\pi c)^2 + 2nc^4\) Dividing both sides by \(c^4\), we obtain: \(\left(\frac{hf}{c^2}\right)^2 = (\frac{p_n + p_\pi}{n})^2 + 2m\) Now we can express the threshold frequency \(f_t\) in terms of \(n\), \(m\), and other constants: \(f_t = \frac{c^2}{h} \sqrt{\frac{(p_n + p_\pi)^2}{n^2} + 2m}\) Using the conservation of momentum condition \((p_\gamma = p_n + p_\pi)\), we can rewrite the threshold frequency as: $f_t = \frac{c^2}{h} \sqrt{\frac{m^2 + 2mn}{n}}£ The threshold frequency \(f_t\) of the photon in terms of the rest mass \(n\) of a proton or neutron (here assumed equal) and that, \(m\), of a pi-meson is then \(f_t = \frac{c^2}{2hn}(m^2 + 2mn)\).

Key concepts

Energy-Momentum Relation

When exploring the world of special relativity, the energy-momentum relation is fundamental in understanding how particles behave at high velocities. The equation
\(E^2 = (pc)^2 + (mc^2)^2\)
encapsulates this relationship, stating that the total energy \(E\) of a particle is the sum of its kinetic energy and rest mass energy, squared and multiplied by the speed of light squared \(c^2\). This foundational concept lays the groundwork for analyzing high-energy events, like particle collisions.

Conservation of Energy

In the realm of physics, the conservation of energy principle plays a pivotal role. It asserts that energy cannot be created or destroyed, only transformed or transferred. This means that in a closed system, like a photon-proton collision, the sum of all energy before the event is equal to the sum after. In the given exercise, the energy of the photon and the stationary proton before impact must equal the energy of the resulting neutron and pi-meson.

Conservation of Momentum

Parallel to the conservation of energy, there's the conservation of momentum. Momentum, a measure of the 'quantity of motion' of an object, must also remain constant in a closed system. Before the collision, only the photon carries momentum since the proton is stationary. After the collision, the combined momentum of the neutron and pi-meson equals the initial photon's momentum. This law simplifies calculations and supports predictions about the post-collision state of particles.

Photon-Proton Collision

In a photon-proton collision high enough in energy, new particles can form. Understanding this process involves both the conservation of energy and momentum. The resultant mass-energy equation for such a collision must account for the creation of additional particles, like the neutron and pi-meson in this scenario. Here, special relativity permits calculations about the minimum, or threshold, conditions necessary for such outcomes.

Threshold Frequency

The threshold frequency refers to the minimum frequency a photon needs to possess in order to induce a particular reaction, such as the production of new particles from a collision. In the context of the exercise, it is the minimal frequency the photon must have to create both a neutron and a pi-meson from its encounter with a proton. Calculating this frequency involves utilizing the prior concepts, mainly the energy-momentum relation and the conservation laws, in a framework established by special relativity.