Chapter 43

What is the total energy of a proton whose kinetic energy is \(4.65 \mathrm{GeV} ?\)

(1) What is the total energy of a proton whose kinetic energy is 4.65 \(\mathrm{GeV}\) ?

Calculate the wavelength of 28 -GeV electrons.

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

(II) What magnetic field is required for the \(7.0-\) Te \(V\) protons in the 4.25 -km-radius Large Hadron Collider \((\mathrm{LHC})\) ?

(II) A cyclotron with a radius of 1.0 \(\mathrm{m}\) is to accelerate deuterons \(\left(\frac{2}{1} \mathrm{H}\right)\) to an energy of 12 \(\mathrm{MeV}\) . (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 \(\mathrm{kV}\) , 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 1.0-km radius Fermilab Tevatron takes about 20 seconds to bring the energies of the stored protons from \(150 \mathrm{GeV}\) to \(1.0 \mathrm{TeV}\). The acceleration is 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.)

(II) The \(1.0-\mathrm{km}\) radius Fermilab Tevatron takes about 20 seconds to bring the energies of the stored protons from 150 \(\mathrm{GeV}\) to 1.0 \(\mathrm{TeV}\) . The acceleration is 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.)

Show that the energy of a particle (charge \(e\) ) in a synchrotron, in the relativistic limit \((v \approx c),\) is given by \(E(\) in \(\mathrm{eV})=B r c,\) where \(B\) is the magnetic field and \(r\) is the radius of the orbit (SI units).

(II) Show that the energy of a particle (charge e) in a synchrotron, in the relativistic limit \((v \approx c),\) is given by \(E(\) in eV \()=B r c,\) where \(B\) is the magnetic field and \(r\) is the radius of the orbit (SI units).

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

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

(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) In the rare decay \(\pi^{+} \rightarrow \mathrm{e}^{+}+\nu_{\mathrm{e}},\) what is the kinetic energy of the positron? Assume the \(\pi^{+}\) decays from rest.

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) \(\mu^{+} \rightarrow \mathrm{e}^{+}+\bar{\nu}_{\mu}\) \((f) \mathrm{p} \rightarrow \mathrm{n}+\mathrm{e}^{+}+\nu_{\mathrm{e}}\)

(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) \(\mathrm{p} \rightarrow \mathrm{e}^{+}+\nu_{\mathrm{c}}\) \((e) \mu^{+} \rightarrow \mathrm{e}^{+}+\overline{v}_{\mu}\) (f) \(\mathrm{p} \rightarrow \mathrm{n}+\mathrm{e}^{+}+\nu_{\mathrm{e}}\)

(II) Antiprotons can be produced when a proton with sufficient energy hits a stationary proton. Even if there is enough energy, which of the following reactions will not happen? \(p+p \rightarrow p+\overline{p}\) \(p+p \rightarrow p+p+\overline{p}\) \(p+p \rightarrow p+p+p+\overline{p}\) \(p+p \rightarrow p+e^{+}+c^{+}+\overline{p}\)

Calculate the maximum kinetic energy of the electron when a muon decays from rest via \(\mu^{-} \rightarrow \mathrm{e}^{-}+\bar{\nu}_{\mathrm{e}}+\nu_{\mu}\). [Hint: In what direction do the two neutrinos move relative to the electron in order to give the electron the maximum kinetic energy? Both energy and momentum are conserved; use relativistic formulas.

(III) Calculate the maximum kinetic energy of the electron when a muon decays from rest via \(\mu^{-} \rightarrow \mathrm{e}^{-}+\overline{v}_{\mathrm{e}}+\nu_{\mu}\) [Hint. In what direction do the two neutrinos move relative to the electron in order to give the electron the maximum kinetic energy? Both energy and momentum are conserved; use relativistic formulas.]

The mean life of the \(\Sigma^{0}\) particle is \(7 \times 10^{-20} \mathrm{~s}\). What is the uncertainty in its rest energy? Express your answer in MeV.

(1) The mean life of the \(\Sigma^{0}\) particle is \(7 \times 10^{-20}\) s. What is the uncertainty in its rest energy? Express your answer in MeV.

The measured width of the \(\psi(3686)\) meson is about \(300 \mathrm{keV}\). Estimate its mean life.

(1) The measured width of the \(\psi(3686)\) meson is about 300 \(\mathrm{keV}\) . Estimate its mean life.

(I) The B \(^{-}\) meson is a b\overline{u} \text { quark combination. } ( a ) \text { Show that } this is consistent for all quantum numbers. (b) What are the quark combinations for \(\mathrm{B}^{+}, \mathrm{B}^{0}, \mathrm{B}^{0} ?\)

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

What quark combinations produce \((a)\) a \(\Xi^{0}\) baryon and (b) a \(\Xi^{-}\) baryon?

What are the quark combinations that can form \((a)\) a neutron, \((b)\) an antineutron, (c) a \(\Lambda^{0},(d)\) a \(\bar{\Sigma}^{0} ?\)

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

What is the quark combination needed to produce a \(\mathrm{D}^{0}\) meson \((Q=B=S=0, c=+1) ?\)

(II) What is the quark combination needed to produce a D' meson \((Q=B=S=0, c=+1) ?\)

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

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

Draw a Feynman diagram for the reaction \(\mathrm{n}+\nu_{\mu} \rightarrow \mathrm{p}+\mu^{-}\)

(1I) Draw a Feynman diagram for the reaction \(n+\nu_{\mu} \rightarrow p+\mu^{-}\)

Assume there are \(5.0 \times 10^{13}\) protons at \(1.0 \mathrm{TeV}\) stored in the 1.0 -km-radius ring of the Tevatron. (a) How much current (amperes) is carried by this beam? (b) How fast would a \(1500-\mathrm{kg}\) car have to move to carry the same kinetic energy as this beam?

(a) How much energy is released when an electron and a positron annihilate each other? (b) How much energy is released when a proton and an antiproton annihilate each other? (All particles have \(K \approx 0 .)\)

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

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) \(\mathrm{K}^{-}+\mathrm{p} \rightarrow \Lambda^{0}+\pi^{0}\) (c) \(\mathrm{K}^{+}+\mathrm{n} \rightarrow \Sigma^{+}+\pi^{0}+\gamma\) (d) \(\mathrm{K}^{+} \rightarrow \pi^{0}+\pi^{0}+\pi^{+}\) (e) \(\pi^{+} \rightarrow \mathrm{e}^{+}+\nu_{\mathrm{e}}\)

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

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

Symmetry breaking occurs in the electroweak theory at about \(10^{-18} \mathrm{~m}\). Show that this corresponds to an energy that is on the order of the mass of the \(\mathrm{W}^{\pm}\).

How many fundamental fermions are there in a water molecule?

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 \(0.60 \mathrm{MeV}\). Use conservation of energy and momentum to determine the \(\pi^{0}\) mass.

(a) Show that the so-called unification distance of \(10^{-31} \mathrm{~m}\) in grand unified theory is equivalent to an energy of about \(10^{16} \mathrm{GeV}\). Use the uncertainty principle, and also de Broglie's wavelength formula, and explain how they apply. (b) Calculate the temperature corresponding to \(10^{16} \mathrm{GeV}\)

For the reaction \(p+p \rightarrow 3 p+\bar{p},\) where one of the initial protons is at rest, use relativistic formulas to show that the threshold energy is \(6 m_{\mathrm{p}} c^{2},\) 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.

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 6\(m_{\mathrm{p}} c^{2}\) , equal to three times the magnitude of the \(Q\) -value of the reaction, where \(m_{p}\) is the proton mass. [Hint : Assume all final particles have the same velocity.]

Use the quark model to describe the reaction $$ \overline{\mathrm{p}}+\mathrm{n} \rightarrow \pi^{-}+\pi^{0} $$

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

What fraction of the speed of light \(c\) is the speed of a \(7.0-\mathrm{TeV}\) proton?

A particle at rest, with a rest energy of \(m c^{2},\) decays into two fragments with rest energies of \(m_{1} c^{2}\) and \(m_{2} c^{2}\) . Show that the kinetic energy of fragment 1 is \(K_{1}=\frac{1}{2 m c^{2}}\left[\left(m c^{2}-m_{1} c^{2}\right)^{2}-\left(m_{2} c^{2}\right)^{2}\right]\)