University Physics with Modern Physics

Hugh D. Young

Chapter 44

Particle Physics and Cosmology - all with Video Answers

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Chapter Questions

Problem 1

A neutral pion at rest decays into two photons. Find the 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.)

Joshua Young

Joshua Young

Numerade Educator

Problem 2

Two equal-energy photons collide head-on and annihilate each other, producing a $\mu^+\mu^-$ pair. The muon mass is given in terms of the electron mass in Section 44.1. (a) Calculate the maximum wavelength of the photons for this to occur. If the photons have this wavelength, describe the motion of the $\mu^+$ and $\mu^-$ immediately after they are produced. (b) If the wavelength of each photon is half the value calculated in part (a), what is the speed of each muon after they have moved apart? Use correct relativistic expressions for momentum and energy.

Joshua Young

Numerade Educator

Problem 3

A positive pion at rest decays into a positive muon and a neutrino. (a) Approximately how much energy is released in the decay? (Assume the neutrino has zero rest mass. Use the muon and pion masses given in terms of the electron mass in Section 44.1.) (b) Why can't a positive muon decay into a positive pion?

Joshua Young

Numerade Educator

Problem 4

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 620 MeV.

Kai Chen

Princeton University

Problem 5

For the nuclear reaction given in Eq. (44.2) assume that the initial kinetic energy and momentum of the reacting particles are negligible. Calculate the speed of the $\alpha$ particle immediately after it leaves the reaction region.

Joshua Young

Numerade Educator

Problem 6

Estimate the range of the force mediated by an $\omega$$^0$ meson that has mass 783 MeV/$c$$^2$.

Joshua Young

Numerade Educator

Problem 7

The starship $Enterprise$, of television and movie fame, is powered by combining matter and antimatter. If the entire 400-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 $1.0 \times 10^{20}$ J ?

Joshua Young

Numerade Educator

Problem 8

An electron with a total energy of 30.0 GeV collides with a stationary positron. (a) What is the available energy? (b) If the electron and positron are accelerated in a collider, what total energy corresponds to the same available energy as in part (a)?

Joshua Young

Numerade Educator

Problem 9

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

Joshua Young

Numerade Educator

Problem 10

The magnetic field in a cyclotron that accelerates protons is 1.70 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 m. What is the maximum speed of the proton? (c) Through what potential difference must the proton be accelerated from rest to give it the speed that you calculated in part (b)?

Joshua Young

Numerade Educator

Problem 11

(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?

Joshua Young

Numerade Educator

Problem 12

Let $\omega_{nr}$ be the nonrelativistic cyclotron angular frequency given by Eq. (44.7), and let $\omega_r$ be the corresponding relativistic value, $\omega_r = (|q|B/m) \sqrt{1- v^2/c^2}$. (a) What is the speed $v$ of a proton for which $\omega_r = 0.90 \omega_{nr}$ so that the two expressions differ by 10%? (b) What is the kinetic energy (in MeV) of a proton with the speed calculated in part (a)? Use the nonrelativistic expression for kinetic energy.

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Erika Lynn

Numerade Educator

Problem 13

(a) What is the speed of a proton that has total energy 1000 GeV? (b) What is the angular frequency $\omega$ of a proton with the speed calculated in part (a) in a magnetic field of 4.00 T? Use both the nonrelativistic Eq. (44.7) and the correct relativistic expression, and compare the results.

Kai Chen

Princeton University

Problem 14

Calculate the minimum beam energy in a proton-proton collider to initiate the $p + p \rightarrow p + p + \eta^0$ reaction. The rest energy of the $\eta^0$ is 547.3 MeV (see Table 44.3).

Kai Chen

Princeton University

Problem 15

In Example 44.3 it was shown that a proton beam with an 800-GeV beam energy gives an available energy of 38.7 GeV for collisions with a stationary proton target. (a) You are asked to design an upgrade of the accelerator that will double the available energy in stationary-target collisions. What beam energy is required? (b) In a colliding-beam experiment, what total energy of each beam is needed to give an available energy of $2(38.7 GeV) = 77.4 GeV$ ?

Ren Jie Tuieng

Numerade Educator

Problem 16

You work for a start-up company that is planning to use antiproton annihilation to produce radioactive isotopes for medical applications. One way to produce antiprotons is by the reaction p + pS p + p + p + $\overline{p}$ in proton-proton collisions. (a) You first consider a colliding-beam experiment in which the two proton beams have equal kinetic energies. To produce an antiproton via this reaction, what is the required minimum kinetic energy of the protons in each beam? (b) You then consider the collision of a proton beam with a stationary proton target. For this experiment, what is the required minimum kinetic energy of the protons in the beam?

Robert Schnibbe

Robert Schnibbe

Numerade Educator

Problem 17

A $K^+$ meson at rest decays into two $\pi$ mesons. (a) What are the allowed combinations of $\pi^0$, $\pi^+$, and $\pi^-$ as decay products? (b) Find the total kinetic energy of the $\pi$ mesons.

Kai Chen

Princeton University

Problem 18

How much energy is released when a $\mu^-$ muon at rest decays into an electron and two neutrinos? Neglect the small masses of the neutrinos.

Kai Chen

Princeton University

Problem 19

What is the mass (in kg) of the $Z^0$? What is the ratio of the mass of the $Z^0$ to the mass of the proton?

Kai Chen

Princeton University

Problem 20

Table 44.3 shows that a $\Sigma^0$ decays into a $\Lambda^0$ and a photon. (a) Calculate the energy of the photon emitted in this decay, if the $\Lambda^0$ is at rest. (b) What is the magnitude of the momentum of the photon? Is it reasonable to ignore the final momentum and kinetic energy of the $\Lambda^0$? Explain.

Kai Chen

Princeton University

Problem 21

If a $\Sigma^+$ at rest decays into a proton and a $\pi^0$, what is the total kinetic energy of the decay products?

Kai Chen

Princeton University

Problem 22

The discovery of the $\Omega^-$ particle helped confirm Gell-Mann's eightfold way. If an $\Omega^-$ decays into a $\Lambda^0$ and a $K^-$, what is the total kinetic energy of the decay products?

Kai Chen

Princeton University

Problem 23

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

Kai Chen

Princeton University

Problem 24

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

Kai Chen

Princeton University

Problem 25

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

Kai Chen

Princeton University

Problem 26

Determine the electric charge, baryon number, strangeness quantum number, and charm quantum number for the following quark combinations: (a) $uus$, (b) c$\overline{s}$, (c) $\overline{dd}$ $\overline{u}$, and (d) $\overline{c}$b.

Kai Chen

Princeton University

Problem 27

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

Kai Chen

Princeton University

Problem 28

What is the total kinetic energy of the decay products when an upsilon particle at rest decays to $\tau^+ + \tau^-$?

Tara Appleyard

Numerade Educator

Problem 29

Given that each particle contains only combinations of u, d, s, $\overline{u}$, $\overline{d}$, and $\overline{s}$, use the method of Example 44.7 to deduce the quark content of (a) a particle with charge +e, baryon number 0, and strangeness +1; (b) a particle with charge $+e$, baryon number -1, and strangeness +1; (c) a particle with charge 0, baryon number +1, and strangeness -2.

Kai Chen

Princeton University

Problem 30

Section 44.5 states that current experiments show that the mass of the Higgs boson is about 125 GeV/$c$$^2$. What is the ratio of the mass of the Higgs boson to the mass of a proton?

Kai Chen

Princeton University

Problem 31

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 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.

Kai Chen

Princeton University

Problem 32

In an experiment done in a laboratory on the earth, the wavelength of light emitted by a hydrogen atom in the $n = 4$ to $n = 2$ transition is 486.1 nm. In the light emitted by the quasar 3C273 (see Problem 36.60), this spectral line is redshifted to 563.9 nm. Assume the redshift is described by Eq. (44.14) and use the Hubble law to calculate the distance in light-years of this quasar from the earth.

Kai Chen

Princeton University

Problem 33

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?

Kai Chen

Princeton University

Problem 34

$\textbf{Redshift}$. The definition of the redshift $z$ is given in Example 44.8. (a) Show that Eq. (44.13) can be written as 1 + $z$ $=$ ([1 $+$ $\beta$]/[1 $-$ $\beta$])$^1$$^/$$^2$, where $\beta$ $=$ $v$/$c$. (b) The observed redshift for a certain galaxy is $z$ $=$ 0.700. Find the speed of the galaxy relative to the earth; assume the redshift is described by Eq. (44.14). (c) Use the Hubble law to find the distance of this galaxy from the earth.

Kai Chen

Princeton University

Problem 35

Calculate the reaction energy Q (in MeV) for the reaction $e^- + p \rightarrow n + \nu_e$. Is this reaction endoergic or exoergic?

Kai Chen

Princeton University

Problem 36

Calculate the energy (in MeV) released in the triplealpha process 3 $^4He \rightarrow ^{12}C$.

Kai Chen

Princeton University

Problem 37

The 2.728-K blackbody radiation has its peak wavelength at 1.062 mm. What was the peak wavelength at $t = 700,000$ y when the temperature was 3000 K?

Kai Chen

Princeton University

Problem 38

Calculate the reaction energy $Q$ (in MeV) for the nucleosynthesis reaction $^{12}_{6}C + ^{4}_{2}He \rightarrow ^{16}_{8}O$ Is this reaction endoergic or exoergic?

Kai Chen

Princeton University

Problem 39

$\textbf{Radiation Therapy with $\pi^-$ Mesons.}$ Beams of $\pi^-$ mesons are used in radiation therapy for certain cancers. The energy comes from the complete decay of the $\pi^-$ to stable particles. (a) Write out the complete decay of a $\pi^-$ meson to stable particles. What are these particles? (b) How much energy is released from the complete decay of a single $\pi^-$ meson to stable particles? (You can ignore the very small masses of the neutrinos.) (c) How many $\pi^-$ mesons need to decay to give a dose of 50.0 Gy to 10.0 g of tissue? (d) What would be the equivalent dose in part (c) in Sv and in rem? Consult Table 43.3 and use the largest appropriate RBE for the particles involved in this decay.

Kai Chen

Princeton University

Problem 40

A proton and an antiproton collide head-on with equal kinetic energies. Two $\gamma$ rays with wavelengths of 0.720 fm are produced. Calculate the kinetic energy of the incident proton.

Kai Chen

Princeton University

Problem 41

Calculate the threshold kinetic energy for the reaction $p + p \rightarrow p + p + K^+ + K^-$ if a proton beam is incident on a stationary proton target.

Kai Chen

Princeton University

Problem 42

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

Kai Chen

Princeton University

Problem 43

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

Kai Chen

Princeton University

Problem 44

An $\eta^0$ meson at rest decays into three $\pi$ mesons. (a) What are the allowed combinations of $\pi^0$, $\pi^+$, and $\pi^-$ as decay products? (b) Find the total kinetic energy of the $\pi$ mesons.

Kai Chen

Princeton University

Problem 45

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

Kai Chen

Princeton University

Problem 46

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

Kai Chen

Princeton University

Problem 47

One proposed proton decay is $p^+ \rightarrow e^+ + \pi^0$, which violates both baryon and lepton number conservation, so the proton lifetime is expected to be very long. Suppose the proton half-life were $1.0 \times 10^{18} y$. (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 $H_2O$ 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 guideline for industrial workers. Based on your calculation, can the proton lifetime be as short as $1.0 \times 10^{18} y$?

Kai Chen

Princeton University

Problem 48

A $\phi$ meson (see Problem 44.45) at rest decays via $\phi \rightarrow K^+ + K^-$. It has strangeness 0. (a) Find the kinetic energy of the $K^+$ meson. (Assume that the two decay products share kinetic energy equally, since their masses are equal.) (b) Suggest a reason the decay $\phi \rightarrow K^+ + K^- + \pi^0$ has not been observed. (c) Suggest reasons the decays $\phi \rightarrow K^+ + \pi^-$ and $\phi \rightarrow K^+ + \mu^-$ have not been observed.

Kai Chen

Princeton University

Problem 49

$\textbf{Cosmic Jerk}$. The densities of ordinary matter and dark matter have decreased as the universe has expanded, since the same amount of mass occupies an ever-increasing volume. Yet observations suggest that the density of dark energy has remained constant over the entire history of the universe. (a) Explain why the expansion of the universe actually slowed down in its early history but is speeding up today. "Jerk" is the term for a change in acceleration, so the change in cosmic expansion from slowing down to speeding up is called $cosmic$ $jerk$. (b) Calculations show that the change in acceleration took place when the combined density of matter of all kinds was equal to twice the density of dark energy. Compared to today's value of the scale factor, what was the scale factor at that time? (c) We see the galaxies in Figs. 44.16b and 44.20 as they were 300 million years ago and 13.1 billion years ago. Was the expansion of the universe slowing down or speeding up at these times? (Hint: See the caption for Fig. 44.20.)

Jacob Shpiece

Numerade Educator

Problem 50

A $\Xi^-$ particle at rest decays to a $\Lambda^0$ and a $\pi^-$. (a) Find the total kinetic energy of the decay products. (b) What fraction of the energy is carried off by each particle? (Use relativistic expressions for momentum and energy.)

Kai Chen

Princeton University

Problem 51

A $\Sigma^-$ particle moving in the $+x-$ direction with kinetic energy 180 MeV decays into a $\pi^-$ and a neutron. The $\pi^-$ moves in the $+y-$ direction. What is the kinetic energy of the neutron, and what is the direction of its velocity? Use relativistic expressions for energy and momentum.

Kai Chen

Princeton University

Problem 52

The $K^0$ meson has rest energy 497.7 MeV. A $K^0$ 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 $\pi^+$ andthe angle it makes with the $+x-$ axis. Use relativistic expressions for energy and momentum.

Keshav Singh

Numerade Educator

Problem 53

While tuning up a medical cyclotron for use in isotope production, you obtain the data given in the table. $B$ is the uniform magnetic field in the cyclotron, and $K_{max}$ is the maximum kinetic energy of the particle being accelerated, which is a proton. The radius $R$ of the proton path at maximum kinetic energy has the same value for each magnetic-field value. (a) Compare the kinetic energy values in the table to the rest energy $mc^2$ of a proton. Is it necessary to use relativistic expressions in your analysis?Explain. (b) Graph your data as $K_{max}$ versus $B^2$. Use the slope of the best-fit straight line to your data to find $R$. (c) What is the maximum kinetic energy for a 0.25-T magnetic field? (d) What is the angular frequency $\omega$ of the proton when $B = 0.40 T$?

Keshav Singh

Numerade Educator

Problem 54

The decay products from the decay of shortlived unstable particles can provide evidence that these particles have been produced in a collision experiment. As an initial step in designing an experiment to detect short-lived hadrons, you make a literature study of their decays. Table 44.3 gives experimental data for the mass and typical decay modes of the particles $\Sigma^-$, $\Xi^0$, $\Delta^ + +$, and $\Omega^-$. (a) Which of these four particles has the largest mass? The smallest? (b) By the decay modes shown in the table, for which of these particles do the decay products have the greatest total kinetic energy? The least?

Keshav Singh

Numerade Educator

Problem 55

You have entered a graduate program in particle physics and are learning about the use of symmetry. You begin by repeating the analysis that led to the prediction of the $\Omega$$^-$ particle. Nine of the spin $-\frac{3}{2}$ baryons are four $\Delta$ particles, each with mass 1232 $MeV/c^2$, strangeness 0, and charges $+2e$, $+e$, 0, and $-e$; three $\Sigma^*$ particles, each with mass 1385 $MeV/c^2$, strangeness -1, and charges $+e$, 0, and $-e$; and two $\Xi^*$ particles, each with mass 1530 $MeV/c^2$, strangeness -2, and charges 0 and $-e$. (a) Place these particles on a plot of $S$ versus $Q$. Deduce the $Q$ and $S$ values of the tenth spin $-\frac{3}{2}$ baryon, the $\Omega^-$ particle, and place it on your diagram. Also label the particles with their masses. The mass of the $\Omega^-$ is 1672 $MeV/c^2$; is this value consistent with your diagram? (b) Deduce the three-quark combinations (of $u$, $d$, and $s$) that make up each of these ten particles. Redraw the plot of $S$ versus $Q$ from part (a) with each particle labeled by its quark content. What regularities do you see?

Ajay Singhal

Numerade Educator

Problem 56

Consider a collision in which a stationary particle with mass $M$ is bombarded by a particle with mass $m$, speed $\upsilon_0$ , and total energy (including rest energy) $E_m$. (a) Use the Lorentz transformation to write the velocities $\upsilon_m$ and $\upsilon_M$ of particles $m$ and $M$ in terms of the speed $\upsilon_{cm}$ of the center of momentum. (b) Use the fact that the total momentum in the center-of-momentum frame is zero to obtain an expression for $\upsilon_{cm}$ in terms of $m$, $M$, and $\upsilon_0$ . (c) Combine the results of parts (a) and (b) to obtain Eq. (44.9) for the total energy in the center-of-momentum frame. to near-zero velocity; when it encounters an electron, they may annihilate each other and emit two photons in opposite directions. The patient is enclosed in a circular array of detectors, with the tissue to be imaged centered in the array. If two photons of the proper energy strike two detectors simultaneously (within 10 ns), we can conclude that the photons were produced by positron-electron annihilation somewhere along a line connecting the detectors. By observing many such simultaneous events, we can create a map of the distribution of positron-emitting atoms in the tissue. However, photons can be absorbed or scattered as they pass through tissue. The number of photons remaining after they travel a distance $x$ through tissue is given by $N = N_{0}e ^{-\mu x} $, where $N_0$ is the initial number of photons and $\mu$ is the attenuation coefficient, which is approximately 0.1 $cm^{-1}$ for photons of this energy. The index of refraction of biological tissue for x rays is 1.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 57

What is the energy of each photon produced by positron-electron annihilation? (a) $\frac{1}{2} m_e\upsilon^2$, where $\upsilon$ is the speed of the emitted positron; (b) $m_e\upsilon^2$; (c) $\frac{1}{2} m_ec^2$; (d) $m_ec^2$.

Eduard Sanchez

Numerade Educator

Problem 58

Suppose that positron-electron annihilations occur on the line 3 cm from the center of the line connecting two detectors. Will the resultant photons be counted as having arrived at these detectors simultaneously? (a) No, because the time difference between their arrivals is 100 ms; (b) no, because the time difference is 200 ms; (c) yes, because the time difference is 0.1 ns; (d) yes, because the time difference is 0.2 ns.

Eduard Sanchez

Numerade Educator

Problem 59

If the annihilation photons come from a part of the body that is separated from the detector by 20 cm of tissue, what percentage of the photons that originally travelled toward the detector remains after they have passed through the tissue? (a) 1.4%; (b) 8.6%; (c) 14%; (d) 86%.

Bettina Hanlon

Numerade Educator