Modern Physics

Kenneth S. Krane

Chapter 14

Elementary Particles - all with Video Answers

Educators

Chapter Questions

Problem 1

Identify the interaction responsible for the following decays (approximate half-lives are given in parentheses):

Suzanne W.

Suzanne W.

Numerade Educator

Problem 2

What is the range of the $\mathrm{W}^{-}$ particle that is responsible for the weak interaction of a proton and a neutron?

Suzanne W.

Numerade Educator

Problem 3

Give one possible decay mode of the following mesons:
$(a) x^{-}$
$(b) \rho^{-}$
$(c) D^{-}$
$(d) \mathrm{K}^{0}$

Suzanne W.

Numerade Educator

Problem 4

Give one possible decay mode of the following antibaryons:
$(a) \bar{n}$
$(b) \bar{\Lambda}^{0}$
$(c) \bar{\Omega}$
$(d) \bar{\Sigma}^{0}$

Suzanne W.

Numerade Educator

Problem 5

Suggest a possible decay mode for the $\mathrm{K}^{0}$ meson that involves the emission of
$(a) v_{c}$
(b) $\bar{v}_{e}$
$(c) v_{\mu}$
$(d) \nabla_{\mu}$
Is it possible to have a decay mode of the $\mathrm{K}^{0}$ that involves the emission of $v_{x}$ or $\nabla_{x} ?$

Suzanne W.

Numerade Educator

Problem 6

Name the conservation law that would be violated in each
of the following decays:
$(a) \pi^{4} \rightarrow \mathrm{e}^{+}+\gamma$
$(b) \Lambda^{0} \rightarrow \mathrm{p}+\mathrm{K}^{-}$
$(c) \Omega^{-} \rightarrow \Sigma^{-}+\pi^{0}$
$(d) \Lambda^{0} \rightarrow \pi^{-}+\pi^{+}$
$(e) \Lambda^{0} \rightarrow \mathrm{n}+\gamma$
$\text { (f) } \Omega^{-} \rightarrow \Xi^{0}+K^{-}$
$(g) \Xi^{0} \rightarrow \Sigma^{0}+\pi^{0}$
$(h) \mu^{-} \rightarrow \mathrm{e}^{-}+\gamma$

Suzanne W.

Numerade Educator

Problem 7

Each of the following reactions violates one (or more) of the conservation laws. Name the conservation law violated in each case:
$(a) v_{e}+p \rightarrow n+e^{+}$
$(b) \mathrm{p}+\mathrm{p} \rightarrow \mathrm{p}+\mathrm{n}+\mathrm{K}^{+}$
$(c) \mathrm{p}+\mathrm{p} \rightarrow \mathrm{p}+\mathrm{p}+\Lambda^{0}+\mathrm{K}^{0}$
$(d) \pi^{-}+\mathrm{n} \rightarrow \mathrm{K}^{-}+\Lambda^{0}$
(e) $\mathrm{K}^{-}+\mathrm{p} \rightarrow \mathrm{n}+\Lambda^{\circ}$

Suzanne W.

Numerade Educator

Problem 8

Supply the missing particle in each of the following decays:
(a) $\mathrm{K}^{-} \rightarrow \pi^{0}+\mathrm{e}^{-}+$
$(b) \mathrm{K}^{0} \rightarrow \pi^{0}+\pi^{0}+$
$(c) \eta \rightarrow \pi^{+}+\pi^{-}+$

Ivan Kochetkov

Numerade Educator

Problem 9

Each of the reactions below is missing a single particle. Supply the missing particle in each case.
$(a) p+p \rightarrow p+\Lambda^{0}+$
$(b) \mathrm{p}+\overline{\mathrm{p}} \rightarrow \mathrm{n}+$
$(c) \pi^{-}+p \rightarrow \Xi^{0}+K^{0}+$
$(d) \mathrm{K}^{-}+\mathrm{n} \rightarrow \Lambda^{0}+$
$(e) \bar{v}_{n}+p \rightarrow n+$
$(f) \mathrm{K}^{-}+\mathrm{p} \rightarrow \mathrm{K}^{+}+$

Suzanne W.

Numerade Educator

Problem 10

Carry out the calculations of $m c^{2}$ for the three decays of Figure 14.6.

Suzanne W.

Numerade Educator

Problem 11

\begin{aligned}
&\text { Determine the energy uncertainty or width of (a) } \eta\\
&\text { (b) } \eta^{\prime} ;(c) \Sigma^{0} ;(d) \Delta^{*}
\end{aligned}

Suzanne W.

Numerade Educator

Problem 11

Determine the energy uncertainty or width of (a) $\eta$
(b) $\eta^{\prime} ;(c) \Sigma^{0} ;(d) \Delta^{*}$

Suzanne W.

Numerade Educator

Problem 12

A $\Sigma^{-}$ baryon is produced in a certain reaction with a kinetic energy of 3642 MeV. If the particle decays after one mean lifetime, what is the longest possible track this particle could leave in a detector?

Suzanne W.

Numerade Educator

Problem 13

Repeat the calculation of Example 14.5 for the case in which the $\pi$ meson has zero kinetic energy, and show that the electron energy in this case is less than the maximum value.

Suzanne W.

Numerade Educator

Problem 13

Repeat the calculation of Example 14.5 for the case in which the $\pi$ meson has zero kinetic energy, and show that the electron energy in this case is less than the maximum value.

Suzanne W.

Numerade Educator

Problem 14

Find the $Q$ values of the following decays:
$(a) \pi^{0} \rightarrow \gamma+\gamma$
$(b) \Sigma^{+} \rightarrow p+\pi^{0}$
$(c) \mathrm{D}^{+} \rightarrow \mathrm{K}^{-}+\pi^{+}+\pi^{+}$

Suzanne W.

Numerade Educator

Problem 15

Find the $Q$ values of the following decays:
$(a) \pi^{-} \rightarrow \mu^{-}+\bar{v}_{\mu}$
(b) $\mathrm{K}^{0} \rightarrow \pi^{+}+\pi^{-}$
(c) $\Sigma^{0} \rightarrow \Lambda^{0}+\gamma$

Suzanne W.

Numerade Educator

Problem 16

Find the kinetic energies of each of the two product particles in the following decays (assume the decaying particle is at rest):
(a) $\mathrm{K}^{0} \rightarrow \pi^{+}+\pi^{-}$
(b) $\Sigma^{-} \rightarrow n+\pi^{-}$

Suzanne W.

Numerade Educator

Problem 17

Find the kinctic energies of each of the two product particles in the following decays (assume the decaying particle is at rest):
$(a) \Omega^{-} \rightarrow \Lambda^{0}+K^{-}$
(b) $\pi^{+} \rightarrow \mu^{+}+v_{\mu}$

Suzanne W.

Numerade Educator

Problem 18

A $\Sigma^{-}$ with a kinetic energy of 0.250 GeV decays into $\pi^{-}+\mathrm{n.}$ The $\pi^{-}$ moves at $90^{\circ}$ to the original direction of travel of the $\Sigma$. . Find the kinetic energies of $\pi^{-}$ and $n$ and the direction of travel of n.

Suzanne W.

Numerade Educator

Problem 19

A K "with a kinetic energy of 276 MeV decays in flight into $\pi^{+}$ and $\pi^{-}$, which move off at equal angles with the original direction of the $\mathrm{K}^{0}$. Find the energies and directions of motion of the $\pi^{+}$ and $\pi^{-}$

Suzanne W.

Numerade Educator

Problem 20

(a) What interaction is responsible for the decay $\Omega^{-} \rightarrow$ $\Lambda^{0}+\mathrm{K}^{-2}(b)$ Find the kinetic energies of the product particles for $\Omega$ decays at rest. (Hint: Check the total kinctic energy available for the two product particles to see if it is a good approximation to use nonrelativistic kinematics.)

Declan Nell

Numerade Educator

Problem 21

(a) In the decay $\Sigma^{0} \rightarrow \Lambda^{0}+\gamma$ with the $\Sigma^{0}$ initially at rest in the laboratory, what are the kinetic energy of the $\Lambda^{\circ}$ and the energy of the $\gamma$ ray? $(b)$ If you were to measure the energy of this $\gamma$ ray, what would you expect the width of the peak to be? Would the peak be considered sharp or broad?

Ren Jie Tuieng

Numerade Educator

Problem 22

Show Equation 14.6 reduces to Equation 13.14 in the nonrelativistic limit.

Suzanne W.

Numerade Educator

Problem 22

Show Equation 14.6 reduces to Equation 13.14 in the nonrelativistic limit.

Suzanne W.

Numerade Educator

Problem 23

Determine the $Q$ values of the following reactions:
(a) $\mathrm{K}^{-}+\mathrm{p} \rightarrow \Lambda^{0}+\pi^{\circ}$
$(b) x^{+}+p \rightarrow \Sigma^{+}+K^{+}$
$(c) \mathrm{p}+\mathrm{p} \rightarrow \mathrm{p}+\pi^{+}+\Lambda^{0}+\mathrm{K}^{0}$

Suzanne W.

Numerade Educator

Problem 24

Determine the $Q$ values of the following reactions:
$(a) \gamma+n \rightarrow \pi^{-}+p$
$(b) \mathrm{K}^{-}+\mathrm{p} \rightarrow \Omega^{-}+\mathrm{K}^{+}+\mathrm{K}^{0}$
$(c) p+p \rightarrow p+\Sigma^{+}+K^{0}$

Suzanne W.

Numerade Educator

Problem 25

Find the threshold kinetic energy for the following reactions. In each case the first particle is in motion and the second is at rest.
$(a) p+p \rightarrow n+\Sigma^{4}+K^{0}+\pi^{+}$
$(b) \pi^{-}+p \rightarrow \Sigma^{0}+K^{0}$

Suzanne W.

Numerade Educator

Problem 26

Find the threshold kinetic energy for the following reactions. In each case the first particle is in motion and the second is at rest.
$(a) p+n \rightarrow p+\Sigma^{-}+K^{+}$
$(b) \pi^{+}+p \rightarrow p+p+\bar{n}$

Suzanne W.

Numerade Educator

Problem 27

In the reaction $p+p \rightarrow n+\Lambda^{0}+K^{+}+\pi^{+},$ find the threshold energy if $(a)$ a beam of protons strikes a fixed target of protons, and $(b)$ two beams of protons collide head-on with equal momenta.

Kai Chen

Princeton University

Problem 28

Analyze the following reactions in terms of the quark content of the particles and reduce them to fundamental processes involving the quarks:
$(a) \mathrm{K}^{-}+\mathrm{p} \rightarrow \Omega^{-}+\mathrm{K}^{+}+\mathrm{K}^{0}$
$(b) \pi^{+}+p \rightarrow \Sigma^{+}+K^{+}$
$(c) \gamma+n \rightarrow \pi^{-}+p$

Suzanne W.

Numerade Educator

Problem 29

Analyze the following reactions in terms of the quark content of the particles and reduce them to fundamental processes involving the quarks:
$(a) \mathrm{K}^{-}+\mathrm{p} \rightarrow \Lambda^{0}+\pi^{0}$
$(b) \mathrm{p}+\mathrm{p} \rightarrow \mathrm{p}+\pi^{+}+\Lambda^{0}+\mathrm{K}^{0}$
$(c) \gamma+p \rightarrow D^{+}+\bar{D}^{0}+n$

Suzanne W.

Numerade Educator

Problem 30

Analyze the following decays in terms of the quark content of the particles and reduce them to fundamental processes involving the quarks:
$(a) \Omega^{-} \rightarrow \Lambda^{0}+K^{-}$
$(b) \mathbf{n} \rightarrow \mathbf{p}+\mathbf{e}^{-}+\bar{v}_{e}$
(c) $\pi^{0} \rightarrow y+\gamma$
$(d) \mathrm{D}^{+} \rightarrow
\mathrm{K}^{-}+\pi^{+}+\pi^{+}$

Suzanne W.

Numerade Educator

Problem 31

Analyze the following decays in terms of the quark content of the particles and reduce them to fundamental processes involving the quarks:
(a) $\mathrm{K}^{0} \rightarrow \pi^{+}+\pi^{-}$
$(b) \Delta^{4+4} \rightarrow p+\pi^{+}$
(c) $\Sigma^{-} \rightarrow n+\pi^{-}$
$(d) \mathrm{D}^{0} \rightarrow \mathrm{K}^{+}+\pi^{-}$

Suzanne W.

Numerade Educator

Problem 32

Based on Figure 14.15 , give the quark content of the six D mesons.

Suzanne W.

Numerade Educator

Problem 33

(a) Arrange the 10 spin $-\frac{3}{2}$ baryons listed in Table 14.6 into a strangeness vs. electric charge diagram similar to Figures 14.11 and 14.12
(b) Arrange the 10 three-quark spin- $\frac{3}{2}$ combinations from Table 14.9 into a similar diagram.

Dominador Tan

Numerade Educator

Problem 34

(a) Suppose we add a plane above the spin- $\frac{1}{2}$ baryon diagrams of Figures 14.12 or 14.14 (similar to what is done in Figure 14.15 ) that would show the spin- $\frac{1}{2}$ baryons with a single charmed quark. How many particles would be in that plane and what would be their quark contents? (Hint:
Review Example 14.10 about the number of particles associated with a combination of three different quarks.)
(b) How many particles would appear in the next higher plane with doubly charmed baryons and what are their quark contents?
(c) Recently the first doubly charmed baryon $\Xi_{\mathrm{cc}}^{++}$ with a charge of $+2 e$ was discovered at the LHC. What is its quark content?

Ren Jie Tuieng

Numerade Educator

Problem 35

Table 14.5 lists the most likely decay mode of the $\mathrm{K}^{+}$ meson; Example 14.5 gives another possible decay. List four other possible decays that are allowed by the conservation laws.

Suzanne W.

Numerade Educator

Problem 36

It is desired to form a beam of $\Lambda^{\circ}$ particles to use for the study of reactions with protons. The $\Lambda^{0}$ are produced by reactions at one target and must be transported to another target $2.0 \mathrm{m}$ away so that at least half of the original $\Lambda^{\circ}$ remain in the beam. Find the speed and the kinetic energy of the $\Lambda^{\circ}$ for this to occur.

Suzanne W.

Numerade Educator

Problem 37

Find a decay mode, other than that listed in Table 14.6 for $(a) \Omega^{-} ;(b) \Lambda^{0} ;(c) \Sigma^{+},$ that satisfies the applicable conservation laws.

Suzanne W.

Numerade Educator

Problem 38

Consider the reaction $p+p \rightarrow p+p+\pi^{0}$ discussed in Example $14.8,$ but viewed instead from a frame of reference in which the two protons collide head-on with equal velocities. $(a)$ At threshold in this frame of reference, the product particles are formed at rest. Find the proton velocities in this case. $(b)$ Use the Lorentz velocity transformation to switch to the laboratory frame of reference in which one of the protons is at rest, and find the velocity of the other proton. (c) Find the kinetic energy of the incident proton in the laboratory frame and compare with the value found in Example 14.8 .

Suzanne W.

Numerade Educator

Problem 39

The $D_{s}^{+}$ meson (rest energy $=1969$ MeV, $S=+1, C=+1$ see Figure 14.15 ) has a lifetime of $0.5 \times 10^{-12} \mathrm{s}$. $(a)$ Which interaction is responsible for the decay? (b) Among the possible decay modes are $\phi+\pi^{+}, \mu^{+}+v_{\mu},$ and $\mathrm{K}^{+}+\mathrm{K}^{0}$
How do the $S$ and $C$ quantum numbers change in these three decays? The $\phi$ meson has a spin of $1,$ a rest energy of $1020 \mathrm{MeV},$ and a quark content of ss.) $(c)$ Analyze the three decay modes according to the quark content of the initial and final particles. (d) Why is the decay into $\mathrm{K}^{+}+$ $\pi^{+}+\pi^{-}$ allowed, while the decay into $\mathrm{K}^{-}+\pi^{+}+\pi^{+}$ is forbidden?

Suzanne W.

Numerade Educator

Problem 40

In the decay $\mathrm{K}^{+} \rightarrow \pi^{+}+\pi^{+}+\pi^{-}$ with the initial $\mathrm{K}$ meson at rest, what is the maximum kinetic energy of the pi mesons?

Suzanne W.

Numerade Educator

Problem 41

A beam of $x^{-}$ mesons with a speed of $0.9980 c$ is incident on a target of protons at rest. The reaction produces 2 particles, one of which is a K $^{\circ}$ meson that is observed to travel with momentum 1561 MeV/c in a direction that makes an angle of $20.6^{\circ}$ with the direction of the incident pions. (a) Find the momentum and the direction of the second product particle. (b) Find the energy of that particle.
(c) Find the rest energy of the second particle and deduce its identity.

Suzanne W.

Numerade Educator

Problem 42

(a) The Large Hadron Collider accelerates protons to an energy of 7 TeV $\left(7 \times 10^{12} \mathrm{eV}\right)$. What is the speed of these protons? Express the result as the difference between the proton speed and the speed of light.
(b) Suppose this beam were directed against a fixed target of protons to obtain the reaction $p+p \rightarrow p+p+X$ where $\mathrm{X}$ represents one or more new particles produced in the reaction. What is the maximum amount of energy available to produce the new particles? (c) At the LHC, two beams of 7 TeV protons collide head on, so the energy available for particles $\mathrm{X}$ is 14 TeV. What energy would be needed for a proton beam colliding with a fixed target to have $14 \mathrm{TeV}$ available to produce new particles?

Dominador Tan

Numerade Educator

Problem 43

(a) In the reaction $\pi^{-}+p \rightarrow n+\pi^{\circ}$ with $\pi$ mesons of momentum 1140 MeV/c incident on protons at rest, the neutron is observed to travel in the direction of the incident pions with very small momentum. (Why would you expect the neutron momentum to be small?) The $\pi^{0}$ decays quickly into two photons that make equal angles. with the pion direction of travel. What are those angles?
(b) The same reaction can produce a different product particle y in place of the $\pi^{0} .$ Particle y also decays into two photons, which make equal angles of $28.6^{\circ}$ with the direction of travel of the original particle. Find the mass of particle y and make a guess as to its identity.

Suzanne W.

Numerade Educator