A virtual particle having an approximate mass of 10×1014 GeV/c2 may be associated with the unification of the strong and electroweak forces. For what length of time could this virtual particle exist (in temporary violation of the conservation of mass-energy as allowed by the Heisenberg uncertainty principle)?
Calculate the mass in GeV/c2 of a virtual carrier particle that has a range limited to 10−30 m by the Heisenberg uncertainty principle. Such a particle might be involved in the unification of the strong and electroweak forces.
Another component of the strong nuclear force is transmitted by the exchange of virtual K−mesons. Taking K− mesons to have an average mass of 495 MeV/c2, what is the approximate range of this component of the strong force?
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?
At full energy, protons in the 2.00-km-diameter Fermilab synchrotron travel at nearly the speed of light, since their energy is about 1000 times their rest mass energy.
(a) How long does it take for a proton to complete one trip around?
(b) How many times per second will it pass through the target area?
Suppose a W− created in a bubble chamber lives for 5.00×10−25 s. What distance does it move in this time if it is travelling at 0.900c? 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 due to time dilation.
What length track does a π+ traveling at 0.100c leave in a bubble chamber if it is created there and lives for 2.60×10−8 s ? (Those moving faster or living longer may escape the detector before decaying.)
Because of energy loss due to synchrotron radiation in the LHC at CERN, only 5.00 MeV is added to the energy of each proton during each revolution around the main ring. How many revolutions are needed to produce 7.00-TeV (7000 GeV) protons, if they are injected with an initial energy of 8.00 GeV?
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.)
When an electron and positron collide at the SLAC facility, they each have 50.0 GeV kinetic energies. What is the total collision energy available, taking into account the annihilation energy? Note that the annihilation energy is insignificant, because the electrons are highly relativistic.
The mass of a theoretical particle that may be associated with the unification of the electroweak and strong forces is 10×1014 GeV/c2 .
(a) How many proton masses is this?
(b) How many electron masses is this? (This indicates how extremely relativistic the accelerator would have to be in order to make the particle, and how large the relativistic quantity γ would have to be.)
The decay mode of the positive tau is τ+→μ++νμ+νˉτ.
(a) What energy is released?
(b) Verify that charge and lepton family numbers are conserved.
(c) The τ+ is the antiparticle of the τ−.Verify that all the decay products of the τ+ are the antiparticles of those in the
decay of the τ− given in the text.
The principal decay mode of the sigma zero is
(a) What energy is released?
(b) Considering the quark structure of the two baryons, does it appear that the Σ0 is an excited state of the Λ0 ?
(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 π0 due to its short lifetime?
(b) What fraction of the decay energy is this, noting that the decay mode is π0→γ+γ (so that all the π0 mass is destroyed)?
(a) What is the uncertainty in the energy released in the decay of a τ− 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.
(a) Verify from its quark composition that the Δ+ particle could be an excited state of the proton.
(b) There is a spread of about 100 MeV in the decay energy of the Δ+, interpreted as uncertainty due to its short lifetime. What is its approximate lifetime?
(c) Does its decay proceed via the strong or weak force?
Note: at 2:50 I misspoke saying 10 to the minus "six", when it should have been 10 to the minus "sixteen".
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
π++p→Δ++→π++p, where the Δ++ is a 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 Δ++.
(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 d
antiquark by writing the reaction and decay in terms of quarks.
(c) Draw a Feynman diagram of the production and decay of the Δ++ showing the individual quarks involved.
The reaction π+p→Δ++ (described in the preceding problem) takes place via the strong force. (a) What is the baryon number of the Δ++ particle?
(b) Draw a Feynman diagram of the reaction showing the individual quarks involved.
One of the decay modes of the omega minus is
(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 η0→γ+γ.
(a) Find the energy released.
(b) What is the uncertainty in the energy due to the short
(c) Write the decay in terms of the constituent quarks.
(d) Verify that baryon number, lepton numbers, and charge are conserved.
One decay mode for the eta-zero meson is η0→π0+π0
(a) Write the decay in terms of the quark constituents.
(b) How much energy is released?
(c) What is the ultimate release of energy, given the decay mode for the pi zero is π0→γ+γ?
(a) How much energy would be released if the proton did decay via the conjectured reaction p→π0+e+ ?
(b) Given that the π0 decays to two γs and that the e+ will find an electron to annihilate, what total energy is ultimately produced in proton decay?
(c) Why is this energy greater than the proton's total mass (converted to energy)?
There are particles called D-mesons. One of them is the D+ meson, which has a single positive charge and a baryon number of zero, also the value of its strangeness, topness, and bottomness. It has a charm of +1. What is its quark configuration?
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.
The intensity of cosmic ray radiation decreases rapidly with increasing energy, but there are occasionally extremely energetic cosmic rays that create a shower of radiation from all the particles they create by striking a nucleus in the atmosphere as seen in the figure given below. Suppose a cosmic ray particle having an energy of 1010 GeV converts its energy into particles with masses averaging 200 MeV/c2. (a) How many particles are created? (b) If the particles rain down on a 1.00 km2 area, how many particles are there per square meter?
(a) Calculate the relativistic quantity γ=1−v2/c21 for
1.00-TeV protons produced at Fermilab. (b) If such a proton created a π+ having the same speed, how long would its life be in the laboratory? (c) How far could it travel in this time?
The primary decay mode for the negative pion is π−→μ−+νˉμ. (a) What is the energy release in MeV in this decay?
(b) Using conservation of momentum, how much energy does each of the decay products receive, given the π− is at rest when it decays? You may assume the muon
antineutrino is massless and has momentum p=cE, just like a photon.
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 γ=1−v2/c21 be for these particles? (b) How long would their average lifetime be in the laboratory? (c) How far could they travel in this time?
Suppose you are designing a proton decay experiment and you can detect 50 percent of the proton decays in a tank of water. (a) How many kilograms of water would you need to
see one decay per month, assuming a lifetime of 1031 y ? (b) How many cubic meters of water is this? (c) If the actual lifetime is 1033 y, how long would you have to wait on an average to see a single proton decay?
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 eV/c2 mass has a kinetic energy of 700 keV. Find the relativistic quantity γ=1−v2/c21 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 γ is so large.)
Consider four forces: the gravitational force between the Earth and the Sun; the electrostatic force between the Earth and the Sun; the gravitational force between the proton and electron in a hydrogen atom, and the electrostatic force between the proton and electron in a hydrogen atom. What is the proper ordering of the magnitude of these forces, from greatest to least?
What is the strength of the magnetic field created by the orbiting Moon, at the center of the orbit, in the system in the previous problem? (Treat the charge going around in orbit as a current loop.) How does this compare with the strength of the Earth's intrinsic magnetic field?
The atomic number of an atom is the number of protons in that atom's nucleus. Make a prediction as to what happens to electromagnetic repulsion as the atomic number gets larger. Then, make a further prediction about what this implies about the number of neutrons in heavy nuclei.
How fast would two 7.0-kg bowling balls each have to be going in a collision to have enough spare energy to create a 0.10-kg tennis ball? (Ignore relativistic effects.) Can you explain why we don't see this in daily situations?
Use the information in Table 33.2 to answer the following questions.
Taking only energy and mass into consideration, what is the minimum amount of kinetic energy a K− must have when colliding with a stationary proton to produce an Ω?
Notice in Table 33.2 that the neutron has a half-life of 882 seconds. This is only for a free neutron, not bound with other neutrons and protons in a nucleus. Given the other particles in the table, and using both their charge and masses, what do you think the most likely decay products for a neutron are? Justify your answer.