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Chapter 30: Problem 15

A neutral pion (mass \(0.135 \mathrm{GeV} / \mathrm{c}^{2}\) ) decays via the electromagnetic interaction into two photons: $\pi^{0} \rightarrow \gamma+\gamma$ What is the energy of each photon, assuming the pion was at rest?

Short Answer

Expert verified
Answer: The energy of each photon emitted in the decay process is \(0.0675 \, \mathrm{GeV}\).

Step by step solution

01

Note Down The Given Information

We are given: 1. Mass of neutral pion: \(0.135 \,\mathrm{GeV}/\mathrm{c}^2\) 2. Decay process: \(\pi^0 \rightarrow \gamma + \gamma\)
02

Recap Conservation of Energy and Momentum

In this decay process, we need to make sure that both energy and momentum are conserved. In other words, the total energy before the decay should be equal to the total energy after the decay, and the total momentum before the decay should be equal to the total momentum after the decay.
03

Apply Conservation of Energy

The energy of the neutral pion is equal to its mass-energy, which means that the sum of the energies of the two photons must be equal to this mass-energy. Since the pion is initially at rest, we know that the energy of each photon must be half of the total energy. So, the energy of each photon is: \(E_\gamma = \frac{1}{2} E_{\pi^0}\), where \(E_{\pi^0}\) is the energy of the neutral pion.
04

Convert Mass-Energy to Energy

We can convert the mass-energy of the pion to energy using the famous equation of Einstein, \(E = mc^2\). Here, c is the speed of light, which will be squared. For the given mass of the pion, it follows \(E_{\pi^0} = (0.135 \,\mathrm{GeV}/\mathrm{c}^2)(c^2) = 0.135 \,\mathrm{GeV}\)
05

Calculate the Energy of Each Photon

Now we can find the energy of each photon using the expression from Step 3: \(E_\gamma = \frac{1}{2} E_{\pi^0} = \frac{1}{2} (0.135 \,\mathrm{GeV}) = 0.0675 \,\mathrm{GeV}\) So, the energy of each photon emitted in the decay process is \(0.0675 \, \mathrm{GeV}\).

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Most popular questions from this chapter

Two factors that can determine the distance over which a force can act are the mass of the exchange particle that carries the force and the Heisenberg uncertainty principle \([\mathrm{Eq} .(28-3)] .\) Assume that the uncertainty in the energy of an exchange particle is given by its rest energy and that the particle travels at nearly the speed of light. What is the range of the weak force carried by the \(\mathbf{Z}\) particle that has a mass of $92 \mathrm{GeV} / \mathrm{c}^{2} ?$ Compare this with the range of the weak force given in Table 30.3
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