Physics for Scientists and Engineers with Modern Physics

Raymond A. Serway, John W. Jewett, Jr.

Chapter 46

Particle Physics and Cosmology - all with Video Answers

Educators

Chapter Questions

Problem 1

A photon produces a proton-antiproton pair according to the reaction $\gamma \rightarrow \mathrm{p}+\overline{\mathrm{p}}$ . What is the minimum possible frequency of the photon? What is its wavelength?

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Problem 2

At some time in your life, you may find yourself in a hospital to have a PET, or positron-emission tomography, scan. In the procedure, a radioactive element that undergoes $\mathrm{e}^{+}$ decay is introduced into your body. The equipment detects the gamma rays that result from pair annihilation when the emitted positron encounters an electron in your body's tissue. Suppose you receive an injection of glucose containing on the order of $10^{10}$ atoms of $14 \mathrm{O},$ with half-life 70.6 s. Assume the oxygen remaining after 5 min is uniformly distributed through 2 $\mathrm{L}$ of blood. What is then the order of magnitude of the oxygen atoms' activity in 1 $\mathrm{cm}^{3}$ of the blood?

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Problem 3

Your two cents' worth. Model a penny as 3.10 $\mathrm{g}$ of copper. Consider an anti-penny minted from 3.10 $\mathrm{g}$ of copper anti-atoms, each with 29 positrons in orbit around a nucleus comprising 29 antiprotons and 34 or 36 antineutrons. (a) Find the energy released if the two coins collide. (b) Find the value of this energy at the unit price of $\$ 0.14 / \mathrm{kWh}$ , a representative retail rate for energy from the electric company.

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Problem 4

Two photons are produced when a proton and antiproton annihilate each other. In the reference frame in which the center of mass of the proton-antiproton system is stationary, what are the minimum frequency and corresponding wavelength of each photon?

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Problem 5

A photon with an energy $E_{\gamma}=2.09$ GeV creates a protonantiproton pair in which the proton has a kinetic energy of 95.0 MeV. What is the kinetic energy of the antiproton? Note: $m_{p} c^{2}=938.3 \mathrm{MeV}$ .

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Problem 6

Occasionally, high-energy muons collide with electrons and produce two neutrinos according to the reaction $\mu^{+}+\mathrm{e}^{-} \rightarrow 2 \nu .$ What kind of neutrinos are they?

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Problem 7

A One mediator of the weak interaction is the $\mathrm{Z}^{0}$ boson, with mass $91 \mathrm{GeV} / c^{2} .$ Use this information to find the order of magnitude of the range of the weak interaction.

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Problem 8

(a) Prove that the exchange of a virtual particle of mass $m$ can be associated with a force with a range given by
$$d=\frac{1240 \mathrm{eV} \cdot \mathrm{nm}}{4 \pi m c^{2}}=\frac{98.7 \mathrm{eV} \cdot \mathrm{nm}}{m c^{2}}$$
(b) State the pattern of dependence of the range on the mass. (c) Of the interactions listed in Table $46.1,$ which are associated with field particles according to this rule and which are not? Explain your answer. (d) What is the range of the force that might be produced by the virtual exchange of a proton?

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Problem 9

A neutral pion at rest decays into two photons according to $\pi^{0} \rightarrow \gamma+\gamma$ Find the energy, momentum, and frequency of each photon.

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Problem 10

When a high-energy proton or pion traveling near the speed of light collides with a nucleus, it travels an average distance of $3 \times 10^{-15} \mathrm{m}$ before interacting. From this information, find the order of magnitude of the time interval required for the strong interaction to occur.

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Problem 11

A free neutron beta decays by creating a proton, an electron, and an antineutrino according to the reaction
$\mathrm{n} \rightarrow \mathrm{p}+\mathrm{e}^{-}+\overline{\nu} .$ What If? Imagine that a free neutron were to decay by creating a proton and electron according to the reaction $\mathrm{n} \rightarrow \mathrm{p}+\mathrm{e}^{-}$ and assume the neutron is initially at rest in the laboratory. (a) Determine the energy released in this reaction. (b) Energy and momentum are conserved in the reaction. Determine the speeds of the proton and electron after the reaction. (c) Is either of these particles moving at a relativistic speed? Explain.

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Problem 12

Identify the unknown particle on the left side of the reaction $\quad ?+p \rightarrow n+\mu^{+}$

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Problem 13

Each of the following reactions is forbidden. Determine a conservation law that is violated for each reaction.
(a) $\mathrm{p}+\overline{\mathrm{p}} \rightarrow \mu^{+}+\mathrm{e}^{-}$
(b) $\pi^{-}+\mathrm{p} \rightarrow \mathrm{p}+\pi^{+}$
(c) $\mathrm{p}+\mathrm{p} \rightarrow \mathrm{p}+\pi^{+}$
(d) $\mathrm{p}+\mathrm{p} \rightarrow \mathrm{p}+\mathrm{p}+\mathrm{n}$
(e) $\gamma+\mathrm{p} \rightarrow \mathrm{n}+\pi^{0}$

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Problem 14

(a) Show that baryon number and charge are con- served in the following reactions of a pion with a proton:
$$\begin{array}{l}{\text { (1) } \pi^{+}+\mathrm{p} \rightarrow \mathrm{K}^{+}+\Sigma^{+}} \\ {\text { (2) } \pi^{+}+\mathrm{P} \rightarrow \pi^{+}+\Sigma^{+}}\end{array}$$
(b) The first reaction is observed, but the second never occurs. Explain.

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Problem 15

A The following reactions or decays involve one or more neutrinos. In each case, supply the missing neutrino $\left(\nu_{e},\right.$ $\nu_{\mu},$ or $\nu_{\tau} )$ or antineutrino.
$$\begin{array}{ll}{\text { (a) } \pi^{-} \rightarrow \mu^{-}+?} & {\text { (b) } \mathrm{K}^{+} \rightarrow \mu^{+}+?} \\ {\text { (c) } ?+\mathrm{p} \rightarrow \mathrm{n}+\mathrm{e}^{+}} & {\text { (d) } ?+\mathrm{n} \rightarrow \mathrm{p}+\mathrm{e}^{-}} \\ {\text { (e) } ?+\mathrm{n} \rightarrow \mathrm{p}+\mu^{-}} & {\text { (f) } \mu^{-} \rightarrow \mathrm{e}^{-}+?+?}\end{array}$$

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Problem 16

The first of the following two reactions can occur, but the second cannot. Explain.
$$\begin{aligned} \mathbf{K}_{\mathrm{S}}^{0} & \rightarrow \pi^{+}+\pi^{-} & \text { (can occur) } \\ \Lambda^{0} & \rightarrow \pi^{+}+\pi^{-} & \text { (cannot occur) } \end{aligned}$$

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Problem 17

A Determine which of the following reactions can occur. For those that cannot occur, determine the conservation law (or laws) violated.
$$\begin{array}{ll}{\text { (a) } \mathrm{p} \rightarrow \pi^{+}+\pi^{0}} & {\text { (b) } \mathrm{p}+\mathrm{p} \rightarrow \mathrm{p}+\mathrm{p}+\pi^{0}} \\ {\text { (c) } \mathrm{p}+\mathrm{p} \rightarrow \mathrm{p}+\pi^{+}} & {\text { (d) } \pi^{+} \rightarrow \mu^{+}+\nu_{\mu}} \\ {\text { (e) } \mathrm{n} \rightarrow \mathrm{p}+\mathrm{e}^{-}+\overline{\nu}_{e}} & {\text { (f) } \pi^{+} \rightarrow \mu^{+}+\mathrm{n}}\end{array}$$

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Problem 18

A $\mathrm{K}_{\mathrm{s}}^{0}$ particle at rest decays into a $\pi^{+}$ and a $\pi^{-} .$ What is the speed of each of the pions? The mass of the $\mathrm{K}_{\mathrm{S}}^{0}$ is $497.7 \mathrm{MeV} / c^{2},$ and the mass of each $\pi$ is 139.6 $\mathrm{MeV} / c^{2}$ .

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Problem 19

(a) Show that the proton-decay reaction $\mathrm{p} \rightarrow \mathrm{e}^{+}+\gamma \mathrm{can}^{-}$ not occur because it violates conservation of baryon number. (b) What If? Imagine that this reaction does occur, and the proton is initially at rest. Determine the energy and momentum of the positron and photon after the reaction. Suggestion: Recall that energy and momentum must be conserved in the reaction. (c) Determine the speed of the positron after the reaction.

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Problem 20

Determine the type of neutrino or antineutrino involved in each of the following processes:
$$\begin{array}{ll}{\text { (a) } \pi^{+} \rightarrow \pi^{0}+\mathrm{e}^{+}+?} & {\text { (b) } ?+\mathrm{p} \rightarrow \mu^{-}+\mathrm{p}+\pi^{+}} \\ {\text { (c) } \Lambda^{0} \rightarrow \mathrm{p}+\mu^{-}+?} & {\text { (d) } \tau^{+} \rightarrow \mu^{+}+?+?}\end{array}$$

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Problem 21

Determine whether or not strangeness is conserved in the following decays and reactions:
$$\begin{array}{ll}{\text { (a) } \Lambda^{0} \rightarrow \mathrm{p}+\pi^{-}} & {\text { (b) } \pi^{-}+\mathrm{p} \rightarrow \Lambda^{0}+\mathrm{K}^{0}} \\ {\text { (c) } \overline{\mathrm{p}}+\mathrm{p} \rightarrow \overline{\Lambda}^{0}+\Lambda^{0}} & {\text { (d) } \pi^{-}+\mathrm{p} \rightarrow \pi^{-}+\Sigma^{+}} \\ {\text { (e) } \Xi^{-} \rightarrow \Lambda^{0}+\pi^{-}} & {\text { (f) } \Xi^{0} \rightarrow \mathrm{p}+\pi^{-}}\end{array}$$

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Problem 22

The neutral meson $\rho^{0}$ decays by the strong interaction into two pions:
$$\rho^{0} \rightarrow \pi^{+}+\pi^{-} \quad\left(T_{1 / 2} \sim 10^{-23} \mathrm{s}\right)$$
The neutral kaon also decays into two pions:
$$\mathrm{K}_{\mathrm{S}}^{0} \rightarrow \pi^{+}+\pi^{-}\left(T_{1 / 2} \sim 10^{-10} \mathrm{s}\right)$$
How do you explain the difference in half-lives?

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Problem 23

For each of the following forbidden decays, determine which conservation law is violated:
$$\begin{array}{ll}{\text { (a) } \mu^{-} \rightarrow \mathrm{e}^{-}+\gamma} & {\text { (b) } \mathrm{n} \rightarrow \mathrm{p}+\mathrm{e}^{-}+\nu_{e}} \\ {\text { (c) } \Lambda^{0} \rightarrow \mathrm{p}+\pi^{0}} & {\text { (d) } \mathrm{p} \rightarrow \mathrm{e}^{+}+\pi^{0}}\end{array}$
(e) $\Xi^{0} \rightarrow \mathrm{n}+\pi^{0}$$

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Problem 24

Which of the following processes are allowed by the strong interaction, the electromagnetic interaction, the weak interaction, or no interaction at all?
$$\begin{array}{ll}{\text { (a) } \pi^{-}+p \rightarrow 2 \eta} & {\text { (b) } K^{-}+n \rightarrow \Lambda^{0}+\pi^{-}} \\ {\text { (c) } K^{-} \rightarrow \pi^{-}+\pi^{0}} & {\text { (d) } \Omega^{-} \rightarrow \Xi^{-}+\pi^{0}} \\ {\text { (e) } \eta \rightarrow 2 \gamma}\end{array}$$

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Problem 25

Identify the conserved quantities in the following processes:
$$\begin{array}{ll}{\text { (a) } \Xi^{-} \rightarrow \Lambda^{0}+\mu^{-}+\nu_{\mu}} & {\text { (b) } \mathrm{K}_{\mathrm{S}}^{0} \rightarrow 2 \pi^{0}} \\ {\text { (c) } \mathrm{K}^{-}+\mathrm{p} \rightarrow \Sigma^{0}+\mathrm{n}} & {\text { (d) } \Sigma^{0} \rightarrow \Lambda^{0}+\gamma} \\ {\text { (e) } \mathrm{e}^{+}+\mathrm{e}^{-} \rightarrow \mu^{+}+\mu^{-}} & {\text { (f) } \overline{\mathrm{p}}+\mathrm{n} \rightarrow \overline{\Lambda}^{0}+\Sigma^{-}}\end{array}$$

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Problem 26

Fill in the missing particle. Assume (a) occurs via the strong interaction and (b) and (c) involve the weak interaction.
(a) $\mathrm{K}^{+}+\mathrm{p} \rightarrow ?+\mathrm{P} \quad$ (b) $\Omega^{-} \rightarrow ?+\pi^{-}$
(c) $\mathrm{K}^{+} \rightarrow ?+\mu^{+}+\nu_{\mu}$

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Problem 27

If a $\mathrm{K}_{\mathrm{s}}^{0}$ meson at rest decays in $0.900 \times 10^{-10} \mathrm{s},$ how far
does a $\mathrm{K}_{\mathrm{S}}^{0}$ meson travel if it is moving at 0.960 $\mathrm{c} ?$

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Problem 28

The particle decay $\Sigma^{+} \rightarrow \pi^{+}+n$ is observed in a bubble chamber. Figure $P 46.28$ represents the curved tracks of the particles $\Sigma^{+}$ and $\pi^{+},$ and the invisible track of the neutron, in the presence of a uniform magnetic field of 1.15 T directed out of the page. The measured radii of curvature are 1.99 $\mathrm{m}$ for the $\Sigma^{+}$ particle and 0.580 $\mathrm{m}$ for the $\pi^{+}$ particle. ( a) Find the momenta of the $\Sigma^{+}$ and the $\pi^{+}$ particles in units of $\mathrm{MeV} / \mathrm{c}$ . (b) The angle between the momenta of the $\Sigma^{+}$ and the $\pi^{+}$ particles at the moment of decay is $64.5^{\circ} .$ Find the momentum of the neutron. (c) Calculate the total energy of the $\pi^{+}$ particle and of the neutron from their known masses $\left(m_{\pi}=\right.$ $139.6 \mathrm{MeV} / c^{2}, m_{n}=939.6 \mathrm{MeV} / c^{2} )$ and the relativistic energy-momentum relation. What is the total energy of the $\Sigma^{+}$ particle? (d) Calculate the mass and speed of the $\Sigma^{+}$ particle.

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Problem 29

A particle of mass $m_{1}$ is fired at a stationary particle of mass $m_{2},$ and a reaction takes place in which new particles are created out of the incident kinetic energy. Taken together, the product particles have total mass $m_{3} .$ The minimum kinetic energy the bombarding particle must have so as to induce the reaction is called the threshold energy. At this energy, the kinetic energy of the products is a minimum, so the fraction of the incident kinetic energy that is available to create new particles is a maximum. This condition is met when all the product particles have the same velocity and the particles have no kinetic energy of motion relative to one another. (a) By using conservation of relativistic energy and momentum, and the relativistic energy-momentum relation, show that the threshold energy is
$$K_{\min }=\frac{\left[m_{3}^{2}-\left(m_{1}+m_{2}\right)^{2}\right] c^{2}}{2 m_{2}}$$
Calculate the threshold energy for each of the following reactions: (b) $\mathrm{p}+\mathrm{p} \rightarrow \mathrm{p}+\mathrm{p}+\mathrm{p}+\overline{\mathrm{p}}$ (One of the initial
protons is at rest. Antiprotons are produced.) (c) $\pi^{-}+$ $\mathrm{p} \rightarrow \mathrm{K}^{0}+\Lambda^{0}$ (The proton is at rest. Strange particles areproduced.) (d) $\mathrm{p}+\mathrm{p} \rightarrow \mathrm{p}+\mathrm{p}+\pi^{0}$ (One of the initial protons is at rest. Pions are produced.) (e) $\mathrm{p}+\overline{\mathrm{p}} \rightarrow \mathrm{Z}^{0}$ (One of the initial particles is at rest. $\mathrm{Z}^{0}$ particles of mass 91.2 $\mathrm{GeV} / c^{2}$ are produced.)

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Problem 30

(a) Find the number of electrons and the number of each species of quarks in 1 $\mathrm{L}$ of water. (b) Make an order-of magnitude estimate of the number of each kind of fundamental matter particle in your body. State your assumptions and the quantities you take as data.

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Problem 31

The quark composition of the proton is uud and that of the neutron is udd. Show that in each case the charge, baryon number, and strangeness of the particle equal, respectively, the sums of these numbers for the quark constituents.

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Problem 32

What If? Imagine that binding energies could be ignored. Find the masses of the u and d quarks from the masses of the proton and neutron.

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Problem 33

The quark compositions of the $\mathrm{K}^{0}$ and $\Lambda^{0}$ particles are $\overline{\mathrm{s}} \mathrm{d}$ and uds, respectively. Show that the charge, baryon number, and strangeness of these particles equal, respectively, the sums of these numbers for the quark constituents.

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Problem 34

The reaction $\pi^{-}+\mathrm{p} \rightarrow \mathrm{K}^{0}+\Lambda^{0}$ occurs with high probability, whereas the reaction $\pi^{-}+\mathrm{p} \rightarrow \mathrm{K}^{0}+\mathrm{n}$ never occurs. Analyze these reactions at the quark level. Show that the first reaction conserves the total number of each
type of quark and the second reaction does not.

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Problem 35

Analyze each of the following reactions in terms of constituent quarks:
(a) $\pi^{-}+p \rightarrow K^{0}+\Lambda^{0}$
(b) $\pi^{+}+p \rightarrow K^{+}+\Sigma^{+}$
(c) $K^{-}+p \rightarrow K^{+}+K^{0}+\Omega^{-}$
(d) $p+p \rightarrow K^{0}+p+\pi^{+}+?$
In the last reaction, identify the mystery particle.

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Problem 36

A $\Sigma^{0}$ particle traveling through matter strikes a proton; then a $\Sigma^{+}$ and a gamma ray, as well as a third particle, emerge. Use the quark model of each to determine the identity of the third particle.

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Problem 37

Identify the particles corresponding to the quark combinations (a) suu, (b) $\overline{\mathrm{u}} \mathrm{d},(\mathrm{c}) \overline{\mathrm{s} \mathrm{d}},$ and $(\mathrm{d})$ ssd.

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Problem 38

What is the electrical charge of the baryons with the quark compositions (a) $\overline{u} \overline{u} \overline{d}$ and $(b) \overline{u} \overline{d} \overline{d}$ ? What are these baryons called?

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Problem 39

Imagine that all distances expand at a rate described by the Hubble constant of $17.0 \times 10^{-3} \mathrm{m} / \mathrm{s} \cdot$ ly. (a) At what rate would the $1.85-\mathrm{m}$ height of a basketball player be increasing? (b) At what rate would the distance between the Earth and the Moon be increasing? In fact, gravitation and other forces prevent the Hubble's-law expansion from taking place except in systems larger than clusters of galaxies.

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Problem 40

Review problem. Refer to Section $39.4 .$ Prove that the Doppler shift in wavelength of electromagnetic waves is described by
$$\lambda^{\prime}=\lambda \sqrt{\frac{1+v / c}{1-v / c}}$$
where $\lambda^{\prime}$ is the wavelength measured by an observer moving at speed $v$ away from a source radiating waves of wavelength $\lambda .$

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Problem 41

A distant quasar is moving away from the Earth at such high speed that the blue 434 -nm $\mathrm{H}_{\gamma}$ line of hydrogen is observed at 510 $\mathrm{nm}$ , in the green portion of the spectrum (Fig. P46.41). (a) How fast is the quasar receding? You may use the result of Problem $40 .$ (b) Edwin Hubble discovered that all objects outside the local group of galaxies are moving away from us, with speeds proportional to their distances. Hubble's law is expressed as $v=H R,$ where Hubble's constant has the approximate value $H=$ $17 \times 10^{-3} \mathrm{m} / \mathrm{s}$ !y. Determine the distance from the Earth to this quasar.
Figure $P 46.41$ (a) Image of the quasar 3 $\mathrm{C} 273$ . (b) Spectrum of the quasar above a comparison spectrum emitted by stationary hydrogen and helium atoms. Both parts of the figure are printed as black-and-white photographic negatives to reveal detail.

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Problem 42

The various spectral lines observed in the light from a distant quasar have longer wavelengths $\lambda_{n}^{\prime}$ than the wavelengths $\lambda_{n}$ measured in light from a stationary source.
Here $n$ is an index taking different values for different spectral lines. The fractional change in wavelength toward the red is the same for all spectral lines. That is, the red-shift parameter $Z$ defined by
$$Z=\frac{\lambda_{n}^{\prime}-\lambda_{n}}{\lambda_{n}}$$
is common to all spectral lines for one object. In terms of $Z,$ determine (a) the speed of recession of the quasar and (b) the distance from the Earth to this quasar. Use the result of Problem 40 and Hubble's law.

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Problem 43

Using Hubble's law, find the wavelength of the 590 -nm sodium line emitted from galaxies (a) $2.00 \times 10^{6}$ ly away from the Earth, (b) $2.00 \times 10^{8}$ ly away, and $(\mathrm{c}) 2.00 \times 10^{9} \mathrm{ly}$ away. You may use the result of Problem $40 .$

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Problem 44

The visible section of the Universe is a sphere centered on the bridge of your nose, with radius 13.7 billion light-years. (a) Explain why the visible Universe is getting larger, with its radius increasing by one light-year in every year. (b) Find the rate at which the volume of the visible section of the Universe is increasing.

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Problem 45

Assume dark matter exists throughout space with a uniform density of $6.00 \times 10^{-28} \mathrm{kg} / \mathrm{m}^{3} .$ (a) Find the amount of such dark matter inside a sphere centered on the Sun, having the Earth's orbit as its equator. (b) Explain whether the gravitational field of this dark matter would have a measurable effect on the Earth's revolution.

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Problem 46

Review problem. The cosmic background radiation is blackbody radiation from a source at a temperature of 2.73 $\mathrm{K}$ (a) Use Wien's law to determine the wavelength at which this radiation has its maximum intensity. (b) In what part of the electromagnetic spectrum is the peak of the distribution?

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Problem 47

Review problem. Use Stefan's law to find the intensity of the cosmic background radiation emitted by the fireball of the Big Bang at a temperature of 2.73 $\mathrm{K}$ .

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Problem 48

It is mostly your roommate's fault. Nosy astronomers have discovered enough junk and clutter in your dorm room to constitute the missing mass required to close the Universe. After observing your floor, closet, bed, and computer files, they extrapolate to slobs in other galaxies and calculate the average density of the observable Universe as 1.20$\rho_{\mathrm{c}}$ . How many times larger will the Universe become before it begins to collapse? That is, by what factor will the distance between remote galaxies increase in the future?

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Problem 49

The early Universe was dense with gamma-ray photons of energy $\sim k_{\mathrm{B}} T$ and at such a high temperature that protons and antiprotons were created by the process $\gamma \rightarrow \mathrm{p}+\overline{\mathrm{p}}$ as rapidly as they annihilated each other. As the Universe cooled in adiabatic expansion, its temperature fell below a certain value and proton pair production became rare. At that time, slightly more protons than antiprotons existed, and essentially all the protons in the Universe today date from that time. (a) Estimate the order of magnitude of the temperature of the Universe when protons condensed out. (b) Estimate the order of magnitude of the temperature of the Universe when electrons condensed out.

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Problem 50

If the average density of the Universe is small compared with the critical density, the expansion of the Universe described by Hubble's law proceeds with speeds that are nearly constant over time. (a) Prove that in this case the age of the Universe is given by the inverse of Hubble's constant. (b) Calculate 1$/ H$ and express it in years.

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Problem 51

Assume the average density of the Universe is equal to the critical density. (a) Prove that the age of the Universe is given by $2 / 3 H .$ (b) Calculate 2$/ 3 H$ and express it in years.

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Problem 52

Hubble's law can be stated in vector form as $\overrightarrow{\mathbf{v}}=H \overrightarrow{\mathbf{R}} .$ Outside the local group of galaxies, all objects are moving away from us with velocities proportional to their positions relative to us. In this form, it sounds as if our location in the Universe is specially privileged. Prove that Hubble's law is equally true for an observer elsewhere in the Universe. Proceed as follows. Assume we are at the origin of coordinates, one galaxy cluster is at location $\overline{\mathbf{R}}_{1}$ and has velocity $\overrightarrow{\mathbf{v}}_{1}=H \overrightarrow{\mathbf{R}}_{1}$ relative to $u s,$ and another galaxy cluster has position vector $\overrightarrow{\mathbf{R}}_{2}$ and velocity $\overrightarrow{\mathbf{v}}_{2}=H \overrightarrow{\mathbf{R}}_{2} .$ Suppose the speeds are nonrelativistic. Consider the frame of reference of an observer in the first of these galaxy clusters. Show that our velocity relative to her, together with the position vector of our galaxy cluster from hers, satisfies Hubble's law. Show that the position and velocity of cluster 2 relative to cluster 1 satisfy Hubble's law.

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Problem 53

Classical general relativity views the structure of space- time as deterministic and well defined down to arbitrarily small distances. On the other hand, quantum general relativity forbids distances smaller than the Planck length given by $L=\left(\hbar G / c^{3}\right)^{1 / 2}$ (a) Calculate the value of the Planck length. The quantum limitation suggests that after the Big Bang, when all the presently observable section of the Universe was contained within a point-like singularity, nothing could be observed until that singularity grew larger than the Planck length. Because the size of the singularity grew at the speed of light, we can infer that no observations were possible during the time interval required for light to travel the Planck length. (b) Calculate this time interval, known as the Planck time $T,$ and state how it compares with the ultrahot epoch mentioned in the text. (c) Does this reasoning suggest we may never know what happened between the time $t=0$ and the time $t=T$ ? Explain.

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Problem 54

Review problem. Supernova Shelton 1987 $\mathrm{A}$ , located approximately 170000 ly from the Earth, is estimated to have emitted a burst of neutrinos carrying energy $\sim 10^{46} \mathrm{J}$ (Fig. P46.54). Suppose the average neutrino energy was 6 $\mathrm{MeV}$ and your mother's body presented cross-sectional area 5000 $\mathrm{cm}^{2}$ . To an order of magnitude, how many of these neutrinos passed through her?

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Problem 55

The most recent naked-eye supernova was Supernova Shelton 1987 $\mathrm{A}$ (Fig. P46.54). It was 170000 ly away in the next galaxy to ours, the Large Magellanic Cloud. Approximately 3 $\mathrm{h}$ before its optical brightening was noticed, two neutrino detection experiments simultaneously registered the first neutrinos from an identified source other than the Sun. The Irvine-Michigan-Brookhaven experiment in a salt mine in Ohio registered 8 neutrinos over a $6-\mathrm{s}$ period, and the Kamiokande II experiment in a zinc mine in Japan counted 11 neutrinos in 13 s. (Because the supernova is far south in the sky, these neutrinos entered the detectors from below. They passed through the Earth before they were by chance absorbed by nuclei in the detectors.) The neutrino energies were between approximately 8 MeV and 40 MeV. If neutrinos have no mass, neutrinos of all energies should travel together at the speed of light, and the data are consistent with this possibility. The arrival times could vary simply because neutrinos were created at different moments as the core of the star collapsed into a neutron star. If neutrinos have nonzero mass, lower-energy neutrinos should move comparatively slowly. The data are consistent with a 10 -MeV
neutrino requiring at most approximately 10 s more than a photon would require to travel from the supernova to us. Find the upper limit that this observation sets on the mass of a neutrino. Other evidence sets an even tighter limit.

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Problem 56

Name at least one conservation law that prevents each of the following reactions: (a) $\pi^{-}+\mathrm{p} \rightarrow \Sigma^{+}+\pi^{0}$ (b) $\mu^{-} \rightarrow$ $\pi^{-}+\nu_{e}(\mathrm{c}) \mathrm{p} \rightarrow \pi^{+}+\pi^{+}+\pi^{-}$

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Problem 57

A The energy flux carried by neutrinos from the Sun is estimated to be on the order of 0.4 $\mathrm{W} / \mathrm{m}^{2}$ at the Earth's surface. Estimate the fractional mass of the Sun over $10^{9}$ yr due to the emission of neutrinos. The mass of the Sun is $2 \times 10^{30} \mathrm{kg}$ . The Earth-Sun distance is $1.5 \times 10^{11} \mathrm{m}$ .

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Problem 58

Two protons approach each other head-on, each with 70.4 MeV of kinetic energy, and engage in a reaction in which a proton and positive pion emerge at rest. What third particle, obviously uncharged and therefore difficult to detect, must have been created?

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Problem 59

A rocket engine for space travel using photon drive and matter-antimatter annihilation has been suggested. Suppose the fuel for a short-duration burn consists of $N$ protons and $N$ antiprotons, each with mass $m$ . (a) Assume all the fuel is annihilated to produce photons. When the photons are ejected from the rocket, what momentum can be imparted to it? (b) What If? If half of the protons and antiprotons annihilate each other and the energy released is used to eject the remaining particles, what momentum could be given to the rocket? (c) Which scheme results in the greater change in speed for the rocket?

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Problem 60

A gamma-ray photon strikes a stationary electron. Determine the minimum gamma-ray energy to make the following reaction occur:
$$\gamma+\mathrm{e}^{-} \rightarrow \mathrm{e}^{-}+\mathrm{e}^{-}+\mathrm{e}^{+}$$

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Problem 61

Determine the kinetic energies of the proton and pion resulting from the decay of a $\Lambda^{0}$ at rest:
$$\Lambda^{0} \rightarrow \mathrm{p}+\pi^{-}$$

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Problem 62

Two protons approach each other with velocities of equal magnitude in opposite directions. What is the minimum kinetic energy of each proton if they are to produce a $\pi^{+}$ meson at rest in the following reaction?
$$\mathrm{P}+\mathrm{P} \rightarrow \mathrm{p}+\mathrm{n}+\pi^{+}$$

Ren Jie Tuieng

Numerade Educator

Problem 63

A $\Sigma^{0}$ particle at rest decays according to $\Sigma^{0} \rightarrow \Lambda^{0}+\gamma$
Find the gamma-ray energy.

Ren Jie Tuieng

Numerade Educator

Problem 64

An unstable particle, initially at rest, decays into a proton (rest energy 938.3 MeV) and a negative pion (rest energy 139.6 MeV). A uniform magnetic field of 0.250 $\mathrm{T}$ exists perpendicular to the velocities of the created particles. The radius of curvature of each track is found to be $1.33 \mathrm{m} .$ What is the mass of the original unstable particle?

Ren Jie Tuieng

Numerade Educator

Problem 65

A $\pi$ -meson at rest decays according to $\pi^{-} \rightarrow \mu^{-}+\overline{\nu}_{\mu}$ .
What is the energy carried off by the neutrino? Assume the neutrino has no mass and moves off with the speed of light. Take $m_{\pi} c^{2}=139.6 \mathrm{MeV}$ and $m_{\mu} c^{2}=105.7 \mathrm{MeV}$ .

Robert Zaballa

Numerade Educator

Problem 66

Review problem. Use the Boltzmann distribution function $e^{-E / l_{\mathrm{B}} T}$ to calculate the temperature at which 1.00$\%$ of a population of photons has energy greater than 1.00 eV. The energy required to excite an atom is on the order of 1 eV. Therefore, as the temperature of the Universe fell below the value you calculate, neutral atoms could form from plasma and the Universe became transparent. The cosmic background radiation represents our vastly red-shifted view of the opaque fireball of the Big Bang as it was at this time and temperature. The fireball surrounds us; we are embers.

Robert Zaballa

Numerade Educator

Problem 67

What processes are described by the Feynman diagrams in Figure $\mathrm{P} 46.67 ?$ What is the exchanged particle in each process?

Robert Zaballa

Numerade Educator

Problem 68

Identify the mediators for the two interactions described in the Feynman diagrams shown in Figure $\mathrm{P} 46.68$ .

Robert Zaballa

Numerade Educator

Problem 69

The cosmic rays of highest energy are mostly protons, accelerated by unknown sources. Their spectrum shows a cutoff at an energy on the order of $10^{20}$ eV. Above that energy, a proton interacts with a photon of cosmic microwave background radiation to produce mesons, for example according to $\mathrm{p}+\gamma \rightarrow \mathrm{P}+\pi^{0} .$ Demonstrate this fact by taking the following steps. (a) Find the minimum photon energy required to produce this reaction in the reference frame where the total momentum of the photon-proton system is zero. The reaction was observed experimentally in the 1950 s with photons of a few hundred MeV. (b) Use Wien's displacement law to find the wavelength of a photon at the peak of the blackbody spectrum of the primordial microwave background radiation, with a temperature of 2.73 $\mathrm{K}$ . (c) Find the energy of this photon. (d) Consider the reaction in part (a) in a moving reference frame so that the photon is the same as that in part (c). Calculate the energy of the proton in this frame, which represents the Earth reference frame.

Ren Jie Tuieng

Numerade Educator