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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

What is the de Broglie wavelength of an electron with kinetic energy $7.0 \mathrm{TeV} ?$
When a proton and an antiproton annihilate, the annihilation products are usually pions. (a) Suppose three pions are produced. What combination(s) of \(\pi^{+}, \pi^{-},\) and \(\pi^{0}\) are possible? (b) Suppose five pions are produced. What combination(s) of \(\pi^{+}, \pi^{-},\) and \(\pi^{0}\) are possible? (c) What is the maximum number of pions that could be produced if the kinetic energies of the proton and antiproton are negligibly small? The mass of a charged pion is \(0.140 \mathrm{GeV} / c^{2}\) and the mass of a neutral pion is \(0.135 \mathrm{GeV} / c^{2}.\)
What is the quark content of an antiproton? [Hint: Replace each of the three quarks that compose a proton with its corresponding antiquark.]
A muon decay is described by $\mu^{-} \rightarrow \mathrm{e}^{-}+v_{\mu}+\bar{v}_{c} .$ What is the maximum kinetic energy of the electron, if the muon was at rest? Assume that the electron is extremely relativistic and ignore the small masses of the neutrinos.
Show that the charge of the neutron and the charge of the proton can be derived from their constituent quark content.
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