Problems & Exercises

Calculate the mass in $\text{GeV/}{c}^2$ of a virtual carrier particle that has a range limited to ${\text{10}}^{-\text{30}}$ m by the Heisenberg uncertainty principle. Such a particle might be involved in the unification of the strong and electroweak forces.

The ratio of the strong to the weak force and the ratio of the strong force to the electromagnetic force become 1 under circumstances where they are unified. What are the ratios of the strong force to those two forces under normal circumstances?

Suppose a $W^{-}$ created in a bubble chamber lives for $5\text{.}\text{00}\times {\text{10}}^{-\text{25}}\text{s}.$ What distance does it move in this time if it is traveling at 0.900 c? Since this distance is too short to make a track, the presence of the $W^{-}$ must be inferred from its decay products. Note that the time is longer than the given $W^{-}$ lifetime, which can be due to the statistical nature of decay or time dilation.

The 3.20-km-long SLAC produces a beam of 50.0-GeV electrons. If there are 15,000 accelerating tubes, what average voltage must be across the gaps between them to achieve this energy?

A proton and an antiproton collide head-on, with each having a kinetic energy of 7.00 TeV (such as in the LHC at CERN). How much collision energy is available, taking into account the annihilation of the two masses? (Note that this is not significantly greater than the extremely relativistic kinetic energy.)

The primary decay mode for the negative pion is ${\pi }^{-}\to {\mu }^{-}+{\stackrel{-}{\nu }}_{\mu }$. What is the energy release in MeV in this decay?

The decay mode of the negative muon is ${\mu }^{-}\to e^{-}+{\stackrel{-}{\nu }}_e+{\nu }_{\mu }$.

(a) Find the energy released in MeV.

(b) Verify that charge and lepton family numbers are conserved.

The principal decay mode of the sigma zero is ${\text{Σ}}^0\to {\text{Λ}}^0+\gamma$.

(a) What energy is released?

(b) Considering the quark structure of the two baryons, does it appear that the ${\text{Σ}}^0$ is an excited state of the ${\text{Λ}}^0$?

(c) Verify that strangeness, charge, and baryon number are conserved in the decay.

(d) Considering the preceding and the short lifetime, can the weak force be responsible? State why or why not.

(a) What is the uncertainty in the energy released in the decay of a ${\tau }^{-}$ due to its short lifetime?

(b) Is the uncertainty in this energy greater than or less than the uncertainty in the mass of the tau neutrino? Discuss the source of the uncertainty.

Accelerators such as the Triangle Universities Meson Facility (TRIUMF) in British Columbia produce secondary beams of pions by having an intense primary proton beam strike a target. Such “meson factories” have been used for many years to study the interaction of pions with nuclei and, hence, the strong nuclear force. One reaction that occurs is ${\pi }^{+}+p\to {\text{Δ}}^{\text{++}}\to {\pi }^{+}+p$, where the ${\text{Δ}}^{\text{++}}$ is a very short-lived particle. The graph in Figure 33.26 shows the probability of this reaction as a function of energy. The width of the bump is the uncertainty in energy due to the short lifetime of the ${\text{Δ}}^{\text{++}}$.

(a) Find this lifetime.

(b) Verify from the quark composition of the particles that this reaction annihilates and then re-creates a d quark and a $\stackrel{-}{d}$ antiquark by writing the reaction and decay in terms of quarks.

(c) Draw a Feynman diagram of the production and decay of the ${\text{Δ}}^{\text{++}}$ showing the individual quarks involved.

Figure 33.26 This graph shows the probability of an interaction between a ${\pi }^{+}$ and a proton as a function of energy. The bump is interpreted as a very short lived particle called a ${\text{Δ}}^{\text{++}}$. The approximately 100-MeV width of the bump is due to the short lifetime of the ${\text{Δ}}^{\text{++}}$.

One of the decay modes of the omega minus is ${\Omega }^{-}\to {\text{Ξ}}^0+{\pi }^{-}$.

(a) What is the change in strangeness?

(b) Verify that baryon number and charge are conserved, while lepton numbers are unaffected.

(c) Write the equation in terms of the constituent quarks, indicating that the weak force is responsible.

One decay mode for the eta-zero meson is ${\eta }^0\to \gamma +\mathrm{\gamma .}$

(a) Find the energy released.

(b) What is the uncertainty in the energy due to the short lifetime?

(c) Write the decay in terms of the constituent quarks.

(d) Verify that baryon number, lepton numbers, and charge are conserved.

Is the decay $n\to e^{+}+e^{-}$ possible considering the appropriate conservation laws? State why or why not.

(a) Is the decay ${\text{Λ}}^0\to n+{\pi }^0$ possible considering the appropriate conservation laws? State why or why not.

(b) Write the decay in terms of the quark constituents of the particles.

The only combination of quark colors that produces a white baryon is RGB. Identify all the color combinations that can produce a white meson.

(a) Show that the conjectured decay of the proton, $p\to {\pi }^0+e^{+}$, violates conservation of baryon number and conservation of lepton number.

(b) What is the analogous decay process for the antiproton?

Verify the quantum numbers given for the proton and neutron in Table 33.2 by adding the quantum numbers for their quark constituents as given in Table 33.4.

(a) Find the charge, baryon number, strangeness, charm, and bottomness of the $J/\Psi$ particle from its quark composition.

(b) Do the same for the $\Upsilon$ particle.

There are particles called bottom mesons or B-mesons. One of them is the $B^{-}$ meson, which has a single negative charge; its baryon number is zero, as are its strangeness, charm, and topness. It has a bottomness of $-1$. What is its quark configuration?

(a) Show that all combinations of three quarks produce integral charges. Thus baryons must have integral charge.

(b) Show that all combinations of a quark and an antiquark produce only integral charges. Thus mesons must have integral charge.

Integrated Concepts

Assuming conservation of momentum, what is the energy of each $\gamma$ ray produced in the decay of a neutral at rest pion, in the reaction ${\pi }^0\to \gamma +\gamma$?

Integrated Concepts

(a) Calculate the relativistic quantity $\gamma =\frac{1}{\sqrt{1-v^2/c^2}}$ for 1.00-TeV protons produced at Fermilab. (b) If such a proton created a ${\pi }^{+}$ having the same speed, how long would its life be in the laboratory? (c) How far could it travel in this time?

Integrated Concepts

Plans for an accelerator that produces a secondary beam of K-mesons to scatter from nuclei, for the purpose of studying the strong force, call for them to have a kinetic energy of 500 MeV. (a) What would the relativistic quantity $\gamma =\frac{1}{\sqrt{1-v^2/c^2}}$ be for these particles? (b) How long would their average lifetime be in the laboratory? (c) How far could they travel in this time?

Integrated Concepts

In supernovas, neutrinos are produced in huge amounts. They were detected from the 1987A supernova in the Magellanic Cloud, which is about 120,000 light years away from the Earth (relatively close to our Milky Way galaxy). If neutrinos have a mass, they cannot travel at the speed of light, but if their mass is small, they can get close. (a) Suppose a neutrino with a $7\text{-eV/}{c}^2$ mass has a kinetic energy of 700 keV. Find the relativistic quantity $\gamma =\frac{1}{\sqrt{1-v^2/c^2}}$ for it. (b) If the neutrino leaves the 1987A supernova at the same time as a photon and both travel to Earth, how much sooner does the photon arrive? This is not a large time difference, given that it is impossible to know which neutrino left with which photon and the poor efficiency of the neutrino detectors. Thus, the fact that neutrinos were observed within hours of the brightening of the supernova only places an upper limit on the neutrino’s mass. (Hint: You may need to use a series expansion to find v for the neutrino, since its $\gamma$ is so large.)

Construct Your Own Problem  

Consider an ultrahigh-energy cosmic ray entering the Earth’s atmosphere (some have energies approaching a joule). Construct a problem in which you calculate the energy of the particle based on the number of particles in an observed cosmic ray shower. Among the things to consider are the average mass of the shower particles, the average number per square meter, and the extent (number of square meters covered) of the shower. Express the energy in eV and joules.

Construct Your Own Problem  

Consider a detector needed to observe the proposed, but extremely rare, decay of an electron. Construct a problem in which you calculate the amount of matter needed in the detector to be able to observe the decay, assuming that it has a signature that is clearly identifiable. Among the things to consider are the estimated half life (long for rare events), and the number of decays per unit time that you wish to observe, as well as the number of electrons in the detector substance.