32–2 to 32–6 Particle Interactions, Particle Exchange(II)

Give a reaction between two nucleons, similar to Eq. 32-4, that could produce a \(\pi^{-}\).

Problem 2P (I) Calculate the wavelength of 28-GeV electrons.

If a proton is moving at very high speed, so that its kinetic energy is much greater than its rest energy \(\left(m_{0} c^{2}\right)\), can it then decay via \(p \rightarrow n+\pi^{+}\)? Equation Transcription: Text Transcription: (m_0 c^2) p \rightarrow n+\pi^+

(I) What strength of magnetic field is used in a cyclotron in which protons make \(2.8 \times 10^{7}\) revolutions per second? Equation Transcription: Text Transcription: 2.8 x 10^7

Problem 3Q What would an ‘'antiatom."made up of the antiparticles to the constituents of normal atoms, consist of? What might happen if antimatter, made of such antiatoms, came in contact with our normal world of matter?

Problem 4Q What particle in a decay signals the electromagnetic interaction?

(I) If \(\alpha\) particles are accelerated by the cyclotron of Example 32-2, what must be the frequency of the voltage applied to the dees?

Problem 4P (I) What is the time for one complete revolution for a very high-energy proton in the 1.0-km-radius Fermilab accelerator?

Problem 5Q (a) Does the presence of a neutrino among the decay products of a particle necessarily mean that the decay occurs via the weak interaction? (P) Do all decays via the weak interaction produce a neutrino? Explain.

(II) If the cyclotron of Example 32-2 accelerated \(\alpha\) particles, what maximum energy could they attain? What would their speed be? (b) Repeat for deuterons \({ }_{1}^{2} H\). (c) In each case, what frequency of voltage is required? Equation Transcription: Text Transcription: \alpha _1^2 H

Why is it that a neutron decays via the weak interaction even though the neutron and one of its decay products (proton) are strongly interacting?

Problem 7P Particles and Accelerators (II) Which is better for resolving details of the nucleus: 25-MeV alpha particles or 25-MeV protons? Compare each of their wavelengths with the size of a nucleon in a nucleus.

Problem 7Q Which of the four interactions (strong, electromagnetic, weak, gravitational) does an electron take part in? A neutrino? A proton?

Problem 8P Particles and Accelerators (II) What is the wavelength (= minimum resolvable size) of 7.0-TeV protons at the LHC?

Check that charge and baryon number are conserved in each of the decays in Table .

Which of the particle decays in Table 32-2 occur via the electromagnetic interaction?

Problem 9P Particles and Accelerators (II) The 1.0-km radius Fermilab Tevatron took about 20 seconds to bring the energies of the stored protons from 150 GeV to 1.0 TeV. The acceleration was done once per turn. Estimate the energy given to the protons on each turn. (You can assume that the speed of the protons is essentially c the whole time.)

Which of the particle decays in Table occur by the weak interaction?

Problem 11P Particles and Accelerators (III) Show that the energy of a particle (charge e) in a synchrotron, in the relativistic limit (v ? C), is given by E (in eV) = Brcwhere B is the magnetic field and r is the radius of the orbit (SI units).

Problem 12P (II) The 1.0-km radius Fermilab Tevatron took about 20 seconds to bring the energies of the stored protons from 150 GeV to 1.0 TeV. The acceleration was done once per turn. Estimate the energy given to the protons on each turn. (You can assume that the speed of the protons essentially c the whole time.)

Problem 10P (II) A cyclotron with a radius of 1.0 m is to accelerate deuterons (jH) to an energy of 12 MeV. (a) What is the required magnetic field? (b) What frequency is needed for the voltage between the dees? (c) If the potential difference between the dees averages 22 kV, how many revolutions will the particles make before exiting? (d) How much time does it take for one deuteron to go from start to exit? (e) Estimate how far it travels during this time.

The \(\Delta\) baryon has spin \(\frac{3}{2}\), baryon number 1 , and charge \(Q=+2\), +1, 0, or -1. Why is there no charge state \(Q=-2\)? ________________ Equation Transcription: Text Transcription: Delta frac{3}{2} Q=+2 Q=-2

Problem 13P (III) Show that the energy of a particle (charge e) in a synchrotron, in the relativistic limit (v * C), is given by E (in eV) = Brcwhere B is the magnetic field and r is the radius of the orbit (SI units).

Which of the particle decays in Table occur via the electromagnetic interaction?

Which of the particle decays in Table occur by the weak interaction?

(I) How much energy is released in the decay \(\pi^{+} \rightarrow \mu^{+}+\nu_{\mu}\) ? See Table 32-2

Problem 15Q Quarks have spin ½. How do you account for the fact that baryons have spin ½ or 3/2, and mesons have spin 0 or 1?

(I) About how much energy is released when a \(\Lambda^0\) decays to \(\mathrm n+\pi^0\)? (See Table 32-2.) ________________ Equation Transcription: Text Transcription: Lambda^0 n+pi^0

Problem 16Q Suppose there were a kind of “neutrinolet'’ that was massless, had no color charge or electrical charge, and did not feel the weak force. Could you say that this particle even exists?

Problem 17P (I) How much energy is required to produce a neutron- antineutron pair?

Problem 17Q Is it possible for a particle to be both (a) a lepton and a baryon? (b) a baryon and a hadron? (c) a meson and a quark? (c() a hadron and a lepton? Explain.

Estimate the range of the strong force if the mediating particle were the Kaon instead of a pion.

Using the ideas of quantum chromodynamics, would it be possible to find particles made up of two quarks and no antiquarks? What about two quarks and two antiquarks?

(II) Two protons are heading toward each other with equal speeds. What minimum kinetic energy must each have if a \(\pi^0\) meson is to be created in the process? (See Table 32-2.) ________________ Equation Transcription: Text Transcription: pi^0

Is the reaction \(\mathrm e^- + \mathrm p \rightarrow \mathrm n+ \overline v_e\) possible? Explain. ________________ Equation Transcription: Text Transcription: e^{-}+p{rightarrow}n+overline{v}_e

(II) What minimum kinetic energy must two neutrons each have if they are traveling at the same speed toward each other, collide, and produce a \(\mathrm{K}^{+} \mathrm{K}^{-}\)pair in addition to themselves? (See Table 32-2.)

Occasionally, the A will decay by the following reaction: \(A^0 \rightarrow \mathrm{p^+ + e^-} +\overline v_e\). Which of the four forces in nature is responsible for this decay? How do you know?

(II) What are the wavelengths of the two photons produced when a proton and antiproton at rest annihilate?

(II) For the decay \(\Lambda^0 \rightarrow \mathrm{p} + \pi^-\), calculate the -value (energy released), and the kinetic energy of the and \(\pi^-\), assuming the \(\Lambda^0\) decays from rest. (Use relativistic formulas.) ________________ Equation Transcription: Text Transcription: \Lambda^0 {rightarrow}p+pi^- pi^- Lambda^0

(II) Which of the following reactions and decays are possible? For those forbidden, explain what laws are violated. \((a)\ \pi^{-}+\mathrm{p} \rightarrow \mathrm{n}+\eta^{0}\) \((b)\ \pi^{+}+\mathrm{p} \rightarrow \mathrm{n}+\pi^{0}\) \((c)\ \pi^{+}+\mathrm{p} \rightarrow \mathrm{p}+\mathrm{e}^{+}\) \((d)\ \mathrm{p} \rightarrow \mathrm{e}^{+}+\nu_{\mathrm{e}}\) \((e)\ \mu^{+} \rightarrow \mathrm{e}^{+}+\bar{\nu}_{\mu}\) \((f)\ \mathrm{p} \rightarrow \mathrm{n}+\mathrm{e}^{+}+\nu_{\mathrm{e}}\) ________________ Equation Transcription: Text Transcription: (a) pi^{-}+p{rightarrow}n+eta^0 (b) pi^{+}+p{rightarrow}n+pi^0 (c) pi^{+}+p{rightarrow}p+e^+ (d) p{rightarrow}e^{+}+v_e (e) mu^+{rightarrow}e^{+}+overline{v}_mu (f) p{rightarrow}n+e^{+}+v_e

(II) What would be the wavelengths of the two photons produced when an electron and a positron, each with 420 keV of kinetic energy, annihilate in a head-on collision?

(III) Calculate the maximum kinetic energy of the electron in the decay \(\mu^{-} \rightarrow \mathrm{e}^{-}+\bar{\nu}_{\mathrm{c}}+\nu_{\mu}\). [Hint: in what direction do the two neutrinos move relative to the electron in order to give the latter the maximum kinetic energy? Both energy and momentum are conserved; use relativistic formulas.] ________________ Equation Transcription: Text Transcription: mu^-{rightarrow}e^-+overline{v}_c+v_mu

Problem 36P (I) The B- meson is bu a quark combination, (a) Show that this is consistent for all quantum numbers, (0) What are the quark combinations for B+, BO. BO

(II) Which of the following decays are possible? For those that are forbidden, explain which laws are violated. \((a)\ \Xi^{0} \rightarrow \Sigma^{+}+\pi^{-}\) \((b)\ \Omega^{-} \rightarrow \Sigma^{0}+\pi^{-}+\nu\) \((c)\ \Sigma^{0} \rightarrow \Lambda^{0}+\gamma+\gamma\) ________________ Equation Transcription: Text Transcription: (a) Xi^0{rightarrow}Sigma^{+}+pi^- (b) Omega^-{rightarrow}Sigma^0+pi^-+v (c) Sigma^0{rightarrow}Lambda^0+gamma+gamma

(II) What particles do the following quark combinations produce: \((a)\ \text {uud}\), \((b)\ \mathrm {\overline u \overline u \overline s}\), \((c)\ \overline {\mathrm u}\mathrm s\), \((d)\ \mathrm d\overline {\mathrm u}\), \((d)\ \overline {\mathrm c}\mathrm s\)? ________________ Equation Transcription: Text Transcription: (a) uud (b) overline{u}overline{u}overline{s} (c) overline{u}s (d) {d}overline{u} (e) overline{c}s

Problem 40P 32–7 to 32–11 Resonances, Standard Model, Quarks, QCD, GUT (II) What is the quark combination needed to produce a D0 meson (Q = B = S = 0, c = +1)?

(II) The \(\mathrm D_S^+\) meson has S = c = +1, B = 0. What quark combination would produce it?

Draw a possible Feynman diagram using quarks (as in Fig. 32-13c) for the reaction \(\pi^{-}+\mathrm{p} \rightarrow \pi^0+\mathrm{n}\).

(II) Draw a Feynman diagram for the reaction \(\mathrm n+\nu_\mu \rightarrow \mathrm p+\mu^-\). ________________ Equation Transcription: Text Transcription: n+v_mu{rightarrow}p+mu^-

Protons are injected into the 1.0-km-radius Fermilab Tevatron with an energy of 150 GeV. If they are accelerated by 2.5 MV each revolution, how far do they travel and approximately how long does it take for them to reach 1.0 TeV?

(a) How much energy is released when an electron and a positron annihilate each other? How much energy is released when a proton and an antiproton annihilate each other? (AIl particles \(\mathrm {KE} \approx 0\).) ________________ Equation Transcription: Text Transcription: KE{approx}0

Problem 51GP Symmetry breaking occurs in the electroweak theory at about 10-18 Show that this corresponds to an energy that is on the order of the mass of the W±.

One decay mode for a \(\pi^{+}\) is \(\pi^{+} \rightarrow \mu^{+}+v_\mu\). What would be the equivalent decay for a \(\pi^{-?}\) ? Check conservation rules.

Calculate the Q-value for each of the reactions Eq. 32-4, for producing a pion.

Calculate the -value for the reaction \(\pi^{-}+\mathrm p \rightarrow \Lambda^0+\mathrm K^0\), when negative pions strike stationary protons. Estimate the minimum pion kinetic energy needed to produce this reaction. [Hint: assume \(\Lambda^0\) and \(\mathrm K^0\) move off with the same velocity. ________________ Equation Transcription: Text Transcription: pi^{-}+p{rightarrow}Lambda^0+K^0 Lambda^0 K^0

How many fundamental fermions are there in a water molecule?

A proton and an antiproton annihilate each other at rest and produce two pions, \(\pi^-\) and \(\pi^+\) What is the Kinetic energy of each pion?

Problem 57GP (a) Show that the so-called unification distance of 10-31 in grand unified theory is equivalent to an energy of about 1016 GeV. Use the uncertainty principle, and also de Broglie’s wavelength formula, and explain how they apply, (b) Calculate the temperature corresponding to 1016 GeV.

For the reaction \(\mathrm p +\mathrm p \rightarrow 3\mathrm p+ \overline {\mathrm p}\), where one of the initial protons is at rest, use relativistic formulas to show that the threshold energy is \(6m_{\mathrm{p}}c^2\), equal to three times the magnitude of the Q-value of the reaction, where \(m_{\mathrm{p}}\) is the proton mass. [Hint: assume all final particles have the same velocity.]

Identify the missing particle in the following reactions. \((a)\ \mathrm {p+p} \rightarrow \mathrm {p+n}+\pi^{+}+?\) \((b)\ \mathrm {p+?} \rightarrow \mathrm {n}+\mu^{+}\) ________________ Equation Transcription: Text Transcription: (a) p+p{rightarrow}p+n+pi^{+}+? (b) p+?{rightarrow}n+mu^{+}

Use the quark model to describe the reaction \(\overline{\mathrm p}+\mathrm n \rightarrow \pi^{-}+\pi^{0}\). ________________ Equation Transcription: Text Transcription: overline{p}+n{rightarrow}pi^{-}+pi^{0}

(I) What is the total energy of a proton whose kinetic energy is 6.35 GeV?

Problem 11Q By what interaction, and why, does ?± sdecay to ?0? What about ?0 decaying to ?0?

(III) What magnetic field intensity is needed at the 1.0-km-radius Fermilab synchrotron for 1.0-TeV protons?

Why do neutrons decay when they are free but not when they are inside the nucleus?

Problem 21P 32–2 to 32–6 Particle Interactions, Particle Exchange (II) What would be the wavelengths of the two photons produced when an electron and a positron, each with 420 keV of kinetic energy, annihilate in a head-on collision?

(II) The \(\Lambda\) cannot decay by the following reactions. What conservation law is violated in each of the reactions? \((a)\ \Lambda^0 \rightarrow \mathrm n+\pi^-\) \((b)\ \Lambda^0 \rightarrow \mathrm {p+K}^-\) \((c)\ \Lambda^0 \rightarrow \pi^{+} + \pi^{-}\)

(II) (a) Show, by conserving momentum and energy, that it is impossible for an isolated electron to radiate only a single photon. (b) With this result in mind, how can you defend the photon exchange diagram in Fig. 32-7?

(II) In the rare decay \(\pi{+} \rightarrow e^{+}+\nu_e\), what is the kinetic energy of the positron? Assume the \(\pi^+\) decays from rest. ________________ Equation Transcription: Text Transcription: pi{+}{rightarrow}e^{+}+v_e pi^+

(II) Calculate the kinetic energy of each of the two products in the decay \(\Xi^{-}\rightarrow \Lambda^{0}+\pi^{-}\). Assume the \(\Xi^{-}\) decays from rest. ________________ Equation Transcription: Text Transcription: Xi^{-}{rightarrow}Lambda^{0}+pi^{-} Xi^{-}

Could a \(\pi^+\) meson be produced if a 100-MeV proton struck a proton at rest? What minimum kinetic energy must the incoming proton have?

(I) Use Fig. 32-11 to estimate the energy width and then the lifetime of the \(\Delta\) resonance using the uncertainty principle.

(I) The measured width of the \(\mathrm J/\psi\) meson is 88 keV. Estimate its lifetime. ________________ Equation Transcription: Text Transcription: J/psi

(I) The measured width of the \(\psi\) (3685) meson is 277 keV. Estimate its lifetime. ________________ Equation Transcription: Text Transcription: psi

(I) What is the energy width (or uncertainty) of \((a)\ \eta^0\), and \((b)\ \Sigma^0\)? See Table 32-2.

(II) What are the quark combinations that can form a neutron, an antineutron, a \(\Lambda^0\), a \(\overline {\Sigma}^0\) ? ________________ Equation Transcription: Text Transcription: Lambda^0 overline{Sigma}^0

Problem 44GP If 2 X 1014 protons moving at v ? c, with KE = 4.0 TeV, are stored in the 4.3-km-radius ring of the LHC, (a) how much current (amperes) is carried by this beam? (b) How fast would a 1500-kg car have to move to carry the same kinetic energy as this beam?

Problem 45GP Protons are injected into the 4.3-km-radius Large Hadron Collider with an energy of 450 GeV. If they are accelerated by 8.0 MV each revolution, how far do they travel and approximately how much time does it take for them to reach 4.0 TeV?

Problem 48GP One decay mode for a ?+ is ?+ µ+ +v µ. What would be the equivalent decay for a ?-? Check conservation laws.

Which of the following reactions are possible, and by what interaction could they occur? For those forbidden, explain why. (a) \(\pi^{-}+\mathrm{p} \rightarrow \mathrm{K}^0+\mathrm{p}+\pi^0\) (b) \(\mathrm{K}^{-}+\mathrm{p} \rightarrow \mathrm{\Lambda}^0+\pi^0\) (c) \(\mathrm{K}^{+}+\mathrm{n} \rightarrow \Sigma^{+}+\pi^0+\gamma\) (d) \(\mathrm{K}^{+} \rightarrow \pi^0+\pi^0+\pi^{+}\) (e) \(\pi^{+} \rightarrow \mathrm{e}^{+}+\nu_{\mathrm{e}}\)

The mass of a \(\pi^0\) can be measured by observing the reaction \(\pi^-+\mathrm p \rightarrow \pi^0+\mathrm n\) at very low incident \(\pi^-\) kinetic energy (assume it is zero). The neutron is observed to be emitted with a kinetic energy of 0.60 MeV. Use conservation of energy and momentum to determine the \(\pi^0\) mass. ________________ Equation Transcription: Text Transcription: pi^0 pi^-+p{rightarrow}pi^0+n pi^- pi^0

The lifetimes listed in Table are in terms of proper time, measured in a reference frame where the particle is at rest. If a tau lepton is created with a kinetic energy of , how long would its track be as measured in the lab, on average, ignoring any collisions?

What fraction of the speed of light is the speed of a 7.0-TeV proton?