Chapter 44

A proton and an antiproton annihilate, producing two photons. Find the energy, frequency, and wavelength of each photon (a) if the \(p\) and \(\overline{p}\) are initially at rest and \((b)\) if the \(p\) and \(\overline{p}\) collide head-on, each with an initial kinetic energy of 830 MeV.

Estimate the range of the force mediated by an \(\omega^{0}\) meson that has mass 783 MeVle?

The starship Enterprise, of television and movie fame, is powered by combining matter and antimatter. If the entire \(400-\mathrm{kg}\)\ antimatter 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 \(\times 10^{20} \mathrm{J} ?\)

Deuterons in a cyclotron travel in a circle with radius Deuterons in a cyclotron travel in a circle with radius 32.0 \(\mathrm{cm}\) just before emerging from the dees. The frequency of the applied alternating voltage is 9.00 \(\mathrm{MHz}\) . Find (a) the magnetic field and (b) the kinetic energy and speed of the deuterons upon emergence.

(a) A high-energy beam of alpha particles collides with a stationary helium gas target. What must the total energy of a beam particle be if the available energy in the collision is 16.0 GeV? (b) If the alpha particles instead interact in a colliding-beam experiment, what must the energy of each beam be to produce the same available energy?

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

Which of the following reactions obey the conservation of baryon number? (a) \(\mathrm{p}+\mathrm{p} \rightarrow \mathrm{p}+\mathrm{e}^{+} ;\) (b) \(\mathrm{p}+\mathrm{n} \rightarrow 2 \mathrm{e}^{+}+\mathrm{e}^{-}\) ; (c) \(\mathrm{p} \rightarrow \mathrm{n}+\mathrm{e}^{-}+\overline{\nu}_{\mathrm{c}} ;(\mathrm{d}) \mathrm{p}+\overline{\mathrm{p}} \rightarrow 2 \gamma\)

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

(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 the orbital speed of an electron in the \(n=1\) orbit is equal to \(c\) times the coupling constant \(e^{2} / 4 \pi \epsilon_{0} \hbar c .\)

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

The quark content of the neutron is \(u d d .(a)\) What is the quark content of the antineutron? Explain your reasoning. (b) Is the neutron its own antiparticle? Why or why not? (c) The quarkcontent of the \(\psi\) is \(c \overline{c} .\) Is the \(\psi\) its own antiparticle? Explain your reasoning.

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 change in the expansion rate of the universe due to gravitational attraction or dark energy).

(a) Show that the expression for the Planck length, \(\sqrt{\hbar G / c^{3}},\) has dimensions of length. (b) Evaluate the numerical value of \(\sqrt{\hbar G / c^{3}},\) and verify the value given in Eq. \((44.21) .\)

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

In the LHC, each proton will be accelerated to a kinetic energy of 7.0 \(\mathrm{TeV}\) (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 particles 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. Two \(\gamma\) rays with wavelengths of 0.780 fm are produced. Calculate the kinetic energy of the incident proton.

Calculate the threshold kinetic energy for the reaction \(\pi^{-}+\mathrm{p} \rightarrow \Sigma^{0}+\mathrm{K}^{0}\) if a \(\pi^{-}\) beam is incident on a stationary proton target. The \(\mathrm{K}^{0}\) has a mass of 497.7 \(\mathrm{MeV} / c^{2} .\)

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) \(\mathrm{K}^{-}+\mathrm{n} \rightarrow \Lambda^{0}+? ;(\mathrm{c}) \mathrm{p}+\overline{\mathrm{p}} \rightarrow \mathrm{n}+? ;\) (d) \(\overline{\nu}_{\mu}+\mathrm{p} \rightarrow \mathrm{n}+?\)

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

The \(\phi\) meson has mass 1019.4 \(\mathrm{MeV} / c^{2}\) and a measured energy width of 4.4 \(\mathrm{MeV} / c^{2} .\) Using the uncertainty principle, estimate the lifetime of the \(\phi\) meson.

One proposed proton decay is \(\mathrm{p}^{+} \rightarrow \mathrm{e}^{+}+\pi^{0}\) 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 an 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 guidelinefor industrial workers. Based on your calculation, can the proton lifetime be as short as \(1.0 \times 10^{18} \mathrm{y} ?\)

The \(\mathrm{K}^{0}\) meson has rest energy 497.7 MeV. A \(\mathrm{K}^{0}\) meson moving in the \(+x\) -direction with kinetic energy 225 MeV decays into a \(\pi^{+}\) and a \(\pi^{-}\) , which move off at equal angles above and below the \(+x\) -axis. Calculate the kinetic energy of the \(\pi^{+}\) and the angle it makes with the \(+x\) -axis. Use relativistic expressions for energy and momentum.