Chapter 44

A neutral pion at rest decays into two photons. Find the approximate energy, frequency, and wavelength of each photon. In which part of the electromagnetic spectrum does each photon lie? (Use the pion mass given in terms of the electron mass in Section 44.1.)

A proton and an antiproton annihilate, producing two photons. Find the energy, frequency, and wavelength of each photon emitted in the center-of-momentum reference frame (a) if the initial kinetic energies of the proton and antiproton are negligible and (b) if each particle has an initial kinetic energy of 830 \(\mathrm{MeV}\) .

"Maximum Power, Scotty" The starship Enterprise, of television and movie fame, is powered by the controlled combination of matter and antimatter. If the entire \(400-\) kg antimater fuel supply of the Enterprise combines with matter, how much energy is released? How does this compare to the U.S. yearly energy use, which is roughly \(1.0 \times 10^{20} \mathrm{J} ?\)

The magnetic field in a cyclotron that accelerates protons is 1.30 T. (a) How many times per second should the potential across the dees reverse? (This is twice the frequency of the circulating protons.) (b) The maximum radius of the cyclotron is 0.250 \(\mathrm{m}\) . What is the maximum speed of the proton? (c) Through what potential difference would the proton have to be accelerated from rest to give it the same speed as calculated in part (b)?

In which of the following decays are the three lepton numbers conserved? In each case, explain your reasoning. (a) \(\mu^{-} \rightarrow \mathrm{e}^{-}+\nu_{\mathrm{e}}+\overline{\nu}_{\mu} ;(\mathrm{b}) \tau^{-} \rightarrow \mathrm{e}^{-}+\overline{\nu}_{\mathrm{e}}+\nu_{\tau} ;(\mathrm{c}) \pi^{+} \rightarrow \mathrm{e}^{+}+\overline{\gamma}\) \((\mathrm{d}) \mathrm{n} \rightarrow \mathrm{p}+\mathrm{e}^{-}+\overline{\nu}_{\mathrm{e}}\)

In which of the following reactions or decays is strangeness conserved? In each case, explain your reasoning. (a) \(\mathbf{K}^{+} \rightarrow \boldsymbol{\mu}^{+}+\boldsymbol{\nu}_{\boldsymbol{\mu}}\) (b) \(\mathbf{n}+\mathbf{K}^{+} \rightarrow \mathbf{p}+\boldsymbol{\pi}^{0} ;(\mathbf{c}) \mathbf{K}^{+}+\mathbf{K}^{-} \rightarrow \boldsymbol{\pi}^{0}+\boldsymbol{\pi}^{0} ;(\mathrm{d}) \mathbf{p}+\mathbf{K}^{-} \rightarrow\) \(\Lambda^{0}+\pi^{0}\)

(a) Show that the coupling constant for the electromagnetic interaction, \(e^{2} / 4 \pi \epsilon_{0} \hbar c,\) is dimensionless and has the numerical value 1\(/ 137.0\) . (b) Show that in the Bohr model (see Section 38.5 ) the orbital speed of an electron in the \(n=1\) orbit is equal to \(c\) times the coupling constant \(e^{2} / 4 \pi \epsilon_{0} \hbar c .\)

Show that the nuclear force coupling constant \(f^{2} / \hbar c\) is dimensionless.

Determine the electric charge, baryon number, strangeness, and charm quantum numbers for the following quark combinations: (a) \(u d s ;(b) c \overline{u}\) (c) \(d d d ;\) (d) \(d c .\) Explain your reasoning.

The quark content of the neurron is udd. (a) What is the quark content of the antineutron? Explain your reasoning. (b) Is the neutron its own antiparticle? Why or why not? (c) The quark \(k\) content of the \(\psi\) is \(c\) . Is the \(\psi\) its own antiparticle? Explain your reasoning.

The weak force may change quark flavor in an interaction. Explain how \(\boldsymbol{\beta}^{+}\) decay changes quark flavor. If a proton undergoes \(\boldsymbol{\beta}^{+}\) decay, determine the decay reaction.

The spectrum of the sodium atom is detected in the light from a distant galaxy. (a) If the 590.0 -nm line is redshifted to \(658.5 \mathrm{nm},\) at what speed is the galaxy receding from the earth? (b) Use the Hubble law to calculate the distance of the galaxy from the earth.

(a) According to the Hubble law, what is the distance \(r\) from us for galaxies that are receding from us with a speed \(c ?\) (b) Explain why the distance calculated in part (a) is the size of our observable universe (ignoring any slowing of the expansion of the universe due to gravitational attraction).

A galaxy in the constellation Pisces is 5210 Mly from the earth. (a) Use the Hubble law to calculate the speed at which this galaxy is receding from earth. (b) What redshifted ratio \(\lambda_{0} / \lambda_{s}\) is expected for light from this galaxy?

The 2.728 -K blackbody radiation has its peak wavelength at 1.062 \(\mathrm{mm}\) . What was the peak wavelength at \(t=700,000\) y when the temperature was 3000 \(\mathrm{K} ?\)

In the LHC, each proton will be accelerated to a kinetic energy of 7.0 TeV. (a) In the colliding beams, what is the available energy \(E_{\mathrm{a}}\) in a collision? (b) In a fixed-target experiment in which a beam of protons is incident on a stationary proton target, what must the total energy (in TeV) of the paricles in the beam be to produce the same available energy as in part (a)?

A proton and an antiproton collide head-on with equal kinetic energies. In the center-of-momentum frame, two \(\gamma\) rays with wavelengths of 0.780 \(\mathrm{fm}\) are produced. Calculate the kinetic energy of the incident proton.

Pair Annihilation. Consider the case where an electron \(\mathrm{e}^{-}\) and a positron \(\mathrm{e}^{+}\) annihilate each other and produce photons. Assume that these two particles collide head-on with eqnal, but slow, speeds. (a) Show that it is not possible for only one photon to be produced. (Hint: Consider the conservation law that must be true in any collision. (b) Show that if only two photons are produced, they must travel in opposite directions and have equal energy. (c) Calculate the wavelength of each of the photons in part (b). In what part of the electromagnetic spectrum do they lie?

Each of the following reactions is missing a single particle. Calculate the baryon number, charge, strangeness, and the three lepton numbers (where appropriate) of the missing particle, and from this identify the particle. (a) \(p+p \rightarrow p+\Lambda^{0}+?\) (b) \(K^{-1}+n \rightarrow \Lambda^{0}+7 ;(c) p+\overline{p} \rightarrow n+7 ;(d) \overline{\nu}_{\mu}+p \rightarrow n+?\)

Estimate the energy width (energy uncertainty) of the \(\psi\) if its mean lifetime is \(7.6 \times 10^{-21} \mathrm{s}\) . What fraction is this of its rest energy?

Proton Decay. Proton decay is a feature of some grand unification theories. One possible decay could be \(\mathbf{p}^{+} \rightarrow \mathbf{e}^{+}+\pi^{0}\) , which violates both baryon and lepton number conservation, so the proton lifetime is expected to be very long. Suppose the proton half-life were \(1.0 \times 10^{18} \mathrm{y}\) . (a) Calculate the energy deposited per kilogram of body tissue (in rad) due to the decay of the protons in your body in one year. Model your body as consisting entirely of water. Only the two protons in the hydrogen atoms in each \(\mathrm{H}_{2} \mathrm{O}\) molecule would decay in the manner shown; do you see why? Assume that the \(\pi^{0}\) decays to two \(\gamma\) rays, that the positron annihilates with an electron, and that all the energy produced in the primary decay and these secondary decays remains in your body. (b) Calculate the equivalent dose (in rem) assuming a RBE of 1.0 for all the radiation products, and compare with the 0.1 rem due to the natural background and the 5.0 -rem guideline for industrial workers. Based on your calculation, can the proton lifetime be as short as \(1.0 \times 10^{18} \mathrm{y} ?\)