Solved: (a) A high-energy beam of alpha particles collides

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.)

CP Two equal-energy photons collide head-on and annihilate each other, producing a 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 and 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.

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 cant a positive muon decay into a positive pion?

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

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 particle immediately after it leaves the reaction region.

Estimate the range of the force mediated by an meson that has mass .

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 * 1020 J?

An electron with a total energy of 20.0 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)?

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.

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 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)?

(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?

(a) What is the speed of a proton that has total energy 1000 GeV? (b) What is the angular frequency 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.

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 2138.7 GeV2 = 77.4 GeV?

Calculate the minimum beam energy in a protonproton collider to initiate the reaction. The rest energy of the is 547.3 MeV (see Table 44.3).

A \(\mathrm{K}^{+}\) 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.

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.

What is the mass (in kg) of the What is the ratio of the mass of the to the mass of the proton?

Table 44.3 shows that a decays into a and a photon. (a) Calculate the energy of the photon emitted in this decay, if the 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 Explain

If a at rest decays into a proton and a what is the total kinetic energy of the decay products?

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

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

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) \(\mathrm{p} \rightarrow \mathrm{n}+\mathrm{e}^{-}+\bar{\nu}_{\mathrm{e}}\); (d) \(p+\bar{p} \rightarrow 2 \gamma\).

In which of the following decays are the three lepton numbers conserved? In each case, explain your reasoning. (a) \(\mathrm{K}^{+} \rightarrow \ \mu^{+}+\nu_{\mu}\) (b) \(\mathrm{n}+\mathrm{K}^+\rightarrow\ \mathrm{p}+\pi^0\) (c) \(\mathrm{K}^{+}+\mathrm{K}^-\rightarrow\ \pi^0+\pi^0\) (d) \(p+\mathrm{K}^-\rightarrow\ \Lambda^0+\pi^0\)

CP (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 = 1orbit is equal to times the coupling constant \(e^{2} / 4 \pi \epsilon_{0} \hbar c\). Text Transcription: e^2 / 4 pi epsilon_0 hbar c

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

Nine of the spin \(-\frac{3}{2}\) baryons are four \(\Delta\) particles, each with mass \(1232 \mathrm{MeV} / c^{2}\), strangeness 0, and charges +2e, +e, 0, and -e; three \(\Sigma^{*}\) particles, each with mass \(1385 \mathrm{MeV} / c^{2}\), strangeness -1, and charges +e, 0, and -e; and two \(\Xi *\) particles, each with mass \(1530 \mathrm{MeV} / c^{2}\), 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 \(-\frac{3}{2}\) baryon, the \(\Omega^{-}\) particle, and place it on your diagram. Also label the particles with their masses. The mass of the \(\Omega^{-}\) is \(1672 \mathrm{MeV} / c^{2}\); 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?

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

Determine the electric charge, baryon number, strangeness quantum number, and charm quantum number for the following quark combinations: (a) uus, (b) (c) and (d) cb

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.

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

The quark content of the neutron 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 content of the is Is the its own antiparticle? Explain your reasoning.

Given that each particle contains only combinations of u, d, s, \(\bar{u}\), \(\bar{d}\), and \(\bar{s}\), 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.

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

Redshift Parameter. The definition of the redshift parameter is given in Example 44.8. (a) Show that Eq. (44.13) may be written as where (b) The observed redshift parameter for a certain galaxy is Find the speed of the galaxy relative to the earth, if the redshift is due to the Doppler shift. (c) Use the Hubble law to find the distance of this galaxy from the earth

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 is expected for light from this galaxy?

(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).

The critical density of the universe is (a) Assuming that the universe is all hydrogen, express the critical density in the number of H atoms per cubic meter. (b) If the density of the universe is equal to the critical density, how many atoms, on the average, would you expect to find in a room of dimensions (c) Compare your answer in part (b) with the number of atoms you would find in the same room under normal conditions on the earth

(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).

Calculate the energy released in each reaction: (a) (b)

Calculate the energy (in MeV) released in the triple-alpha process 3 4 He S 12C

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

Calculate the reaction energy (in MeV) for the nucleosynthesis reaction Is this reaction endoergic or exoergic?

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?

A positronium atom consists of an electron and a positron. In the Bohr model the two particles orbit around their common center of mass. In the Bohr model, what is the ionization energy for a positronium atom when it is in its ground state?

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 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.

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.

Calculate the threshold kinetic energy for the reaction if a beam is incident on a stationary proton target. The has a mass of

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

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

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) (b) (c) (d)

Estimate the energy width (energy uncertainty) of the \(\psi\) if its mean lifetime is \(7.6\times10^{-21}\mathrm{\ s}\). 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.

A \(\phi\) meson (see Problem 44.53) at rest decays via \(\phi \rightarrow \mathrm{K}^{+}+\mathrm{K}^{-}\). It has strangeness 0. (a) Find the kinetic energy of the \(\mathrm{K}^{+}\) meson. (Assume that the two decay products share kinetic energy equally, since their masses are equal.) (b) Suggest a reason the decay \(\phi \rightarrow \mathrm{K}^{+}+\mathrm{K}^{-}+\pi^{0}\) has not been observed. (c) Suggest reasons the decays \(\phi \rightarrow \mathrm{K}^{+}+\pi^{-}\) and \(\phi \rightarrow \mathrm{K}^{+}+\mu^{-}\) have not been observed.

BIO One proposed proton decay is which violates both baryon and lepton number conservation, so the proton lifetime is expected to be very long. Suppose the proton half-life were (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 molecule would decay in the manner shown; do you see why? Assume that the decays to two 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 * 1018 y?

A particle at rest decays to a and a (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.)

CALC Consider the spherical balloon model of a twodimensional expanding universe (see Fig. 44.17 in Section 44.6). The shortest distance between two points on the surface, measured along the surface, is the arc length where As the balloon expands, its radius increases, but the angle between the two points remains constant. (a) Explain why, at any given time, is the same for all points on the balloon. (b) Show that is directly proportional to at any instant. (c) From your answer to part (b), what is the expression for the Hubble constant in terms of and (d) The expression for you found in part (c) is constant in space. How would have to depend on time for to be constant in time? (e) Is your answer to part (d) consistent with the observed rate of expansion of the universe?

Suppose all the conditions are the same as in Problem 44.57, except that v = dr/dt is constant for a given \(\theta\), rather than \(H_{0}\) being constant in time. Show that the Hubble constant is \(H_{0}=1 / t\) and, hence, that the current value is 1/T, where T is the age of the universe.

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 todays value of the scale factor, what was the scale factor at that time? (c) We see the galaxy clusters in Figs. 44.15b and 44.19 as they were 300 million years ago and 10.2 billion years ago. Was the expansion of the universe slowing down or speeding up at these times? (Hint: See the caption for Fig. 44.19.)

The meson has rest energy 497.7 MeV. A meson moving in the ?x-direction with kinetic energy 225 MeV decays into a and a , which move off at equal angles above and below the ?x-axis. Calculate the kinetic energy of the and the angle it makes with the ?x-axis. Use relativistic expressions for energy and momentum.

A particle moving in the ?x-direction with kinetic energy 180 MeV decays into a and a neutron. The 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.

Consider a collision in which a stationary particle with mass M is bombarded by a particle with mass m, speed \(v_{0}\), and total energy (including rest energy) \(E_{m}\). (a) Use the Lorentz transformation to write the velocities \(v_{m}\) and \(v_{M}\) of particles m and M in terms of the speed \(v_{\mathrm{cm}}\) 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 \(v_{\mathrm{cm}}\) in terms of m, M, and \(v_{0}\). (c) Combine the results of parts (a) and (b) to obtain Eq. (44.9) for the total energy in the center-of-momentum frame.

Problem 8DQ According to the standard model of the fundamental particles, what are the similarities between quarks and leptons? What are the most important differences?

Problem 9DQ What are the main advantages of colliding-beam accelerators compared with those using stationary targets? What are the main disadvantages?

Problem 1DQ Is it possible that some parts of the universe contain antimatter whose atoms have nuclei made of antiprotons and antineutrons, surrounded by positrons? How could we detect this condition without actually going there? Can we detect these antiatoms by identifying the light they emit as composed of anti-photons? Explain. What problems might arise if we actually did go there?

Problem 1E 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.)

Problem 2DQ Given the Heisenberg uncertainty principle, is it possible to create particle–antiparticle pairs that exist for extremely short periods of time before annihilating? Does this mean that empty space is really empty?

Problem 2E CP Two equal-energy photons collide head-on and annihilate each other, producing a ?+?-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 ?+ and ?- 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.

Problem 3DQ When they were first discovered during the 1930s and 1940s, there was confusion as to the identities of pions and muons. What are the similarities and most significant differences?

Problem 3E 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?

Problem 4DQ The gravitational force between two electrons is weaker than the electric force by the order of 10-40. Yet the gravitational interactions of matter were observed and analyzed long before electrical interactions were understood. Why?

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

Problem 5DQ When a ?0 decays to two photons, what happens to the quarks of which it was made?

CP 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. Equation transcription: Text transcription: alpha

Problem 6DQ Why can’t an electron decay to two photons? To two neutrinos?

Problem 6E Estimate the range of the force mediated by an ?0 meson that has mass 783 MeV/c2.

According to the standard model of the fundamental particles, what are the similarities between baryons and leptons? What are the most important differences?

Problem 7E 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 × 1020 J?

What are the main advantages of colliding-beam accelerators compared with those using stationary targets? What are the main disadvantages?

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.

Problem 10DQ Does the universe have a center? Explain.

Problem 10E 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 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)?

Problem 11E (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?

Does it make sense to ask, “If the universe is expanding, what is it expanding into?”

Problem 12DQ Assume that the universe has an edge. Placing yourself at that edge in a thought experiment, explain why this assumption violates the cosmological principle.

(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.

Explain why the cosmological principle requires that \(H_0\) must have the same value everywhere in space, but does not require that it be constant in time.

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 \(2(138.7 \mathrm{GeV} 2)=77.4 \mathrm{GeV} ?\) Equation transcription: Text transcription: 2(138.7{GeV} 2)=77.4{GeV} ?

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

Problem 15E A K+ meson at rest decays into two ? mesons. (a) What are the allowed combinations of ?0, ?+, and ?- as decay products? (b) Find the total kinetic energy of the ? mesons.

Problem 16E How much energy is released when a ?- muon at rest decays into an electron and two neutrinos? Neglect the small masses of the neutrinos.

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?

Table 44.3 shows that a ?0 decays into a ?0 and a photon. (a) Calculate the energy of the photon emitted in this decay, if the ?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 ?0 ? Explain.

If a \(\Sigma^{+}\) at rest decays into a proton and a \(\pi^{0}\), what is the total kinetic energy of the decay products? Equation transcription: Text transcription: Sigma^{+} pi^{0}

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

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

Which of the following reactions obey the conservation of baryon number? (a) \(p+p \rightarrow p+e^{+}\) (b) \(p+n \rightarrow 2 e^{+}+e^{-}\); (c) \(p \rightarrow n+e^{-}+v_{e}^{-}\) (d) \(p+\bar{p} \rightarrow 2 y\) Equation transcription: Text transcription: p+p rightarrow p+e^{+} p+n rightarrow 2 e^{+}+e^{-} p rightarrow n+e^{-}+v{e}^{-} p+bar{p} rightarrow 2 y

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

Problem 24E (a) Show that the coupling constant for the electro-magnetic interaction, e2/4??0hc, 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 e2/4??0hc.

Show that the nuclear force coupling constant f2/hc is dimensionless.

Nine of the spin- \(\frac{3}{2}\) baryons are four \(\Delta\) particles, each with mass \(1232 \mathrm{MeV} / \mathrm{c}^{2}\), strangeness 0, and charges \(+2 e,+e, 0\), and \(-e\) three \(\Sigma^{0}\) particles, each with mass \(1385 \mathrm{MeV} / \mathrm{c}^{2}\), strangeness , and charges , and \(-e\) and two particles, each with mass \(1530 \mathrm{MeV} / \mathrm{c}^{2}\), strangeness , and charges 0 and \(-e\) (a) Place these particles on a plot of versus Deduce the and 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 \(1672 \mathrm{MeV} / c^{2}\) is this value consistent with your diagram? (b) Deduce the three-quark combinations (of , and ) that make up each of these ten particles. Redraw the plot of versus from part (a) with each particle labeled by its quark content. What regularities do you see? Equation transcription: Text transcription: frac{3}{2} Delta 1232{MeV} /{c}^{2} +2 e,+e, 0 -e Sigma^{0} 1385{MeV} /{c}^{2} 1530{MeV} /{c}^{2} Omega^{-} 1672{MeV} / c^{2}

Determine the electric charge, baryon number, strangeness quantum number, and charm quantum number for the following quark combinations: (a) \(\text { uds; }\) (b) \(\mathrm{CU}^{-}\) (c) \(\text { ddd; }\) and (d) \(d c^{-}\). Explain your reasoning. Equation transcription: Text transcription: text { uds; } {CU}^{-} text { ddd; } d c^{-}

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

The weak force may change quark flavor in an interaction. Explain how ?+ decay changes quark flavor. If a proton undergoes ?+ decay, determine the decay reaction.

Problem 30E What is the total kinetic energy of the decay products when an upsilon particle at rest decays to ?+ + ?-?

The quark content of the neutron 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 content of the \(\psi\) is \(c \overline c\) Is the \(\psi\) its own antiparticle? Explain your reasoning.

Given that each particle contains only combinations of \(u, d, s, \bar{u}, \bar{d}\), and \(\bar{s}\), use the method of Example to deduce the quark content of (a) a particle with charge , baryon number 0 , and strangeness (b) a particle with charge , baryon number , and strangeness a particle with charge 0 , baryon number , and strangeness Equation transcription: Text transcription: u, d, s, bar{u}, bar{d} bar{s}

Problem 33E 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.

Redshift Parameter. The definition of the redshift parameter z is given in Example 44.8. (a) Show that Eq. (44.13) may be written as \(1+z=([1+\beta] /[1-\beta])^{1 / 2}\), where \(\beta=v / c\). (b) The observed redshift parameter for a certain galaxy is z = 0.500. Find the speed of the galaxy relative to the earth, if the redshift is due to the Doppler shift. (c) Use the Hubble law to find the distance of this galaxy from the earth.

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 \(\lambda_{0} / \lambda_{\mathrm{S}}\) is expected for light from this galaxy? Text Transcription: Lambda_0 / lambda_S

(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).

Problem 37E The critical density of the universe is 9.5 × 10–27 kg/m3. (a) Assuming that the universe is all hydrogen, express the critical density in the number of H atoms per cubic meter. (b) If the density of the universe is equal to the critical density, how many atoms, on the average, would you expect to find in a room of dimensions 4 m × 7 m × 3 m? (c) Compare your answer in part (b) with the number of atoms you would find in the same room under normal conditions on the earth.

(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).

Problem 39E Calculate the energy released in each reaction: (a) p + 2H ? 3He; (b) n + 3He ? 4He.

Calculate the energy (in MeV) released in the triple-alpha process \(3\ ^4\mathrm{He}\ \rightarrow\ ^{12}\mathrm{C}\). Text Transcription: 3 ^4He rightarrow ^12C

Calculate the reaction energy Q (in MeV) for the \(\mathrm{e}^{-}+\mathrm{p} \rightarrow \mathrm{n}+\nu_{\mathrm{e}}\) reaction Is this reaction endoergic or exoergic?

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

Problem 43P CP 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?

A positronium atom consists of an electron and a positron. In the Bohr model the two particles orbit around their common center of mass. In the Bohr model, what is the ionization energy for a positronium atom when it is in its ground state?

Problem 45P 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 Ea in a collision? (b) In a fixed-target experiment in which a beam of protons is incident on a stationary pr ton 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)?

Problem 46P A proton and an antiproton collide head-on with equal kinetic energies. Two ? rays with wavelengths of 0.780 fm are produced. Calculate the kinetic energy of the incident proton.

CP BIO Radiation Therapy with ?? Mesons. Beams of ?? mesons are used in radiation therapy for certain cancers. The energy comes from the complete decay of the ?? to stable particles. (a) Write out the complete decay of a ?? meson to stable particles. What are these particles? (b) How much energy is released from the complete decay of a single ?? meson to stable particles? (You can ignore the very small masses of the neutrinos.) (c) How many ?? mesons need to decay to give a dose of 50.0Gy to 10.0g 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.

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 \(497.7 \mathrm{MeV} / \mathrm{c}^{2}\). Equation Transcription: Text Transcription: pi^- K^0 497.7 MeV/c^2

Calculate the threshold kinetic energy for the reaction \(p+p \rightarrow+p+K^{+}+K^{-}\) if a proton beam is incident on a stationary proton target. Equation transcription: Text transcription: p+p rightarrow+p+K^{+}+K^{-}

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.

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) \(p+\bar{p} \rightarrow n+?\); (d) \(v_{\mu}^{-}+p \rightarrow n+?\) Equation transcription: Text transcription: p+p rightarrow p+Lambda^{0}+? K^{-}+n rightarrow Lambda^{0}+? p+bar{p} rightarrow n+? v{\mu}^{-}+p rightarrow n+?

Problem 52P Estimate the energy width (energy uncertainty) of the ? if its mean lifetime is 7.6 × 10-21 s. What fraction is this of its rest energy?

Problem 53P The ? meson has mass 1019.4 MeV/c2 and a measured energy width of 4.4 MeV/c2. Using the uncertainty principle, estimate the lifetime of the ? meson.

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

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 guideline for industrial workers. Based on your calculation, can the proton lifetime be as short as \(1.0\times10^{18\ }\mathrm{y}\)?

CP \(A E^{-}\) 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.) Equation transcription: Text transcription: A E^{-} Lambda^{0} pi^{-}

CALC Consider the spherical balloon model of a two-dimensional expanding universe (see Fig. in Section ). The shortest distance between two points on the surface, measured along the surface, is the arc length , where As the balloon expands, its radius increases, but the angle between the two points remains constant. (a) Explain why, at any given time, \((d R / d f) / R\) is the same for all points on the balloon. (b) Show that \(v=d r / d t\) is directly proportional to at any instant. (c) From your answer to part (b), what is the expression for the Hubble constant \(H_{0}\) in terms of and \(d R / d t ?\) (d) The expression for \(H_{0}\) you found in part. (c) is constant in space. How would have to depend on time for \(H_{0}\) to be constant in time? (e) Is your answer to part. (d) consistent with the observed rate of expansion of the universe? Equation transcription: Text transcription: (d R / d f) / R v=d r / d t H_{0} d R / d t ?

CALC Suppose all the conditions are the same as in Problem , except that \(v=d r / d t\) is constant for a given , rather than \(H_{0}\) being constant in time. Show that the Hubble constant is \(H_{0}=1 / t\) and, hence, that the current value is \(1 / T\), where is the age of the universe. Equation transcription: Text transcription: v=d r / d t H{0} H{0}=1 / t 1 / T

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 galaxy clusters in Figs. and as they were 300 million years ago and billion years ago. Was the expansion of the universe slowing down or speeding up at these times? (Hint: See the caption for Fig. 44.19.)

Problem 60P CP The K0 meson has rest energy 497.7 MeV. A K0 meson moving in the +x-direction with kinetic energy 225 MeV decays into a ?+ and a ?-, which move off at equal angles above and below the +x-axis. Calculate the kinetic energy of the ?+ and the angle it makes with the +x-axis. Use relativistic expressions for energy and momentum.

CP A \(\Sigma^{-}\) particle moving in the -direction with kinetic energy \(180 \mathrm{MeV}\) decays into a \(\pi^{-}\) and a neutron. The \(\pi^{-}\) moves in the -direction. What is the kinetic energy of the neutron, and what is the direction of its velocity? Use relativistic expressions for energy and momentum. Equation transcription: Text transcription: Sigma^{-} 180{MeV} pi^{-}

CP Consider a collision in which a stationary particle with mass M is bombarded by a particle with mass m, speed \(v_{0}\), and total energy (including rest energy) \(v_{m}\) (a) Use the Lorentz transformation to write the velocities \(v_{m}\) and \(v_{M}\) of particles m and M in terms of the speed \(v_{c m}\) 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 (v_{c m}\) in terms of m, M, and \(v_{0}\). (c) Combine the results of parts (a) and (b) to obtain Eq. (44.9) for the total energy in the center-of-momentum frame.