Figure 44-14 shows part of the experimental arrangement in

A positively charged pion decays by Eq. 44-7: . What must be the decay scheme of the negatively charged pion? (Hint:The p# is the antiparticle of the p'.)

Certain theories predict that the proton is unstable,with a half-life of about 1032 years.Assuming that this is true, calculate the number of proton decays you would expect to occur in one year in the water of an Olympic-sized swimming pool holding 4.32 " 105 L of water.

An electron and a positron undergo pair annihilation (Eq. 44-5). If they had approximately zero kinetic energy before the annihilation, what is the wavelength of each g produced by the annihilation?

A neutral pion initially at rest decays into two gamma rays: . Calculate the wavelength of the gamma rays. Why must they have the same wavelength?

An electron and a positron are separated by distance r. Find the ratio of the gravitational force to the electric force between them. From the result, what can you conclude concerning the forces acting between particles detected in a bubble chamber? (Should gravitational interactions be considered?)

(a) A stationary particle 1 decays into particles 2 and 3, which move off with equal but oppositely directed momenta. Show that the kinetic energy K2 of particle 2 is given by where E1, E2, and E3 are the rest energies of the particles. (b) A stationary positive pion p' (rest energy 139.6 MeV) can decay to an antimuon m' (rest energy 105.7 MeV) and a neutrino n (rest energy approximately 0). What is the resulting kinetic energy of the antimuon?

The rest energy of many short-lived particles cannot be measured directly but must be inferred from the measured momenta and known rest energies of the decay products. Consider the r meson, which decays by the reaction . Calculate the rest energy of the r0 meson given that the oppositely directed momenta of the created pions each have magnitude 358.3 MeV/c. See Table 44-4 for the rest energies of the pions.

A positive tau (t', rest energy ! 1777 MeV) is moving with 2200 MeV of kinetic energy in a circular path perpendicular to a uniform 1.20 T magnetic field. (a) Calculate the momentum of the tau in kilogram- meters per second. Relativistic effects must be considered. (b) Find the radius of the circular path

Observations of neutrinos emitted by the supernova SN1987a (Fig. 43-12b) place an upper limit of 20 eV on the rest energy of the electron neutrino. If the rest energy of the electron neutrino were, in fact, 20 eV, what would be the speed difference between light and a 1.5 MeV electron neutrino?

A neutral pion has a rest energy of 135 MeV and a mean life of If it is produced with an initial kinetic energy of 80 MeV and decays after one mean lifetime, what is the longest possible track this particle could leave in a bubble chamber? Use relativistic time dilation

Which conservation law is violated in each of these proposed decays? Assume that the initial particle is stationary and the decay products have zero orbital angular momentum. (a) m# : e# ' nm; (b) m# : e' ' ne ' nm; (c) m' : p' ' nm.

The particle and its products decay according to the scheme (a) What are the final stable decay products? From the evidence, (b) is the particle a fermion or a boson and (c) is it a meson or a baryon? (d) What is its baryon number? A2 ' m# : e p # ' n ' n. ' : m' ' n, p# : m r # ' n, 0 : p' ' p#, m' : e A ' ' n ' n, 2 ' : r0 ' p'

Show that if, instead of plotting strangeness S versus charge q for the spin- baryons in Fig. 44-3a and for the spin-zero mesons in Fig. 44-3b, we plot the quantity versus the quantity , we get the hexagonal patterns without using sloping axes. (The quantity Y is called hypercharge, and Tz is related to a quantity called isospin.)

Calculate the disintegration energy of the reactions (a) p' ' p : ' ' K' and (b) K# ' p : ?0 ' p0

Which conservation law is violated in each of these proposed reactions and decays? (Assume that the products have zero orbital angular momentum.) (a) ?0 :p ' K#;(b) 8# : # ' p0 (S ! #3,q ! #1,m ! 1672 MeV/c2 ,and for 8#);(c) K# ' p :?0 ' p'.

Does the proposed reaction p ' p : ?0 ' ' ' e# conserve (a) charge, (b) baryon number, (c) electron lepton number, (d) spin angular momentum, (e) strangeness, and (f) muon lepton number?

Does the proposed decay process @# : p# ' n ' K# ' p conserve (a) charge, (b) baryon number, (c) spin angular momentum, and (d) strangeness?

By examining strangeness, determine which of the following decays or reactions proceed via the strong interaction: (a) K0 : p' ' p#; (b) ?0 ' p : ' ' n; (c) ?0 : p ' p#; (d) K# ' p : ?0 ' p0 .

The reaction p' ' p : p ' p ' n proceeds via the strong interaction. By applying the conservation laws, deduce the (a) charge quantum number, (b) baryon number, and (c) strangeness of the antineutron

There are 10 baryons with spin .Their symbols and quantum numbers for charge q and strangeness S are as follows: qS qS )# #10*0 0 #1 )0 00*' '1 #1 )' '1 0 @*# #1 #2 )'' '2 0 @*0 0 #2 *# #1 #1 8# #1 #3 Make a chargestrangeness plot for these baryons, using the sloping coordinate system of Fig. 44-3. Compare your plot with this figure.

Use the conservation laws and Tables 44-3 and 44-4 to identify particle x in each of the following reactions, which proceed by means of the strong interaction: (a) p ' p : p ' ?0 ' x; (b) p ' p : n ' x; (c) p# ' p : @0 ' K0 ' x

A 220 MeV # particle decays: # : p# ' n. Calculate the total kinetic energy of the decay products.

Consider the decay ?0 : p ' p# with the ?0 at rest. (a) Calculate the disintegration energy. What is the kinetic energy of (b) the proton and (c) the pion? (Hint: See Problem 6.)

The spin- baryon (see table in Problem 24) has a rest energy of 1385 MeV (with an intrinsic uncertainty ignored here); the spin- baryon has a rest energy of 1192.5 MeV. If each of these particles has a kinetic energy of 1000 MeV, (a) which is moving faster and (b) by how much?

The quark makeups of the proton and neutron are uud and udd, respectively. What are the quark makeups of (a) the antiproton and (b) the antineutron?

From Tables 44-3 and 44-5, determine the identity of the baryon formed from quarks (a) ddu, (b) uus, and (c) ssd. Check your answers against the baryon octet shown in Fig. 44-3a

7 What is the quark makeup of ?

What quark combination is needed to form (a) ?0 and (b) @0 ?

Which hadron in Tables 44-3 and 44-4 corresponds to the quark bundles (a) ssu and (b) dds?

Using the up, down, and strange quarks only, K0 construct, if possible, a baryon (a) with q ! '1 and strangeness S ! #2 and (b) with q ! '2 and strangeness S ! 0

In the laboratory, one of the lines of sodium is emitted at a wavelength of 590.0 nm. In the light from a particular galaxy, however, this line is seen at a wavelength of 602.0 nm. Calculate the distance to the galaxy, assuming that Hubbles law holds and that the Doppler shift of Eq. 37-36 applies.

Because of the cosmological expansion, a particular emission from a distant galaxy has a wavelength that is 2.00 times the wavelength that emission would have in a laboratory. Assuming that Hubbles law holds and that we can apply Doppler-shift calculations, what was the distance (ly) to that galaxy when the light was emitted?

What is the observed wavelength of the 656.3 nm (first Balmer) line of hydrogen emitted by a galaxy at a distance of 2.40 " 108 ly? Assume that the Doppler shift of Eq. 37-36 and Hubbles law apply

An object is from us and does not have any motion relative to us except for the motion due to the expansion of the universe. If the space between us and it expands according to Hubbles law, with H ! 21.8 mm/s$ly, (a) how much extra distance (meters) will be between us and the object by this time next year and (b) what is the speed of the object away from us?

If Hubbles law can be extrapolated to very large distances, at what distance would the apparent recessional speed become equal to the speed of light?

What would the mass of the Sun have to be if Pluto (the outermost planet most of the time) were to have the same orbital speed that Mercury (the innermost planet) has now? Use data from Appendix C, express your answer in terms of the Suns current mass MS, and assume circular orbits

The wavelength at which a thermal radiator at temperature T radiates electromagnetic waves most intensely is given by Wiens law: lmax ! (2898 mm$K)/T. (a) Show that the energy E of a photon corresponding to that wavelength can be computed from E ! (4.28 " 10 #10 MeV/K)T. (b) At what minimum temperature can this photon create an electronpositron pair (as discussed in Module 21-3)?

Use Wiens law (see Problem 37) to answer the following questions: (a) The cosmic background radiation peaks in intensity at a wavelength of 1.1 mm. To what temperature does this correspond? (b) About 379 000 y after the big bang, the universe became transparent to electromagnetic radiation. Its temperature then was 1.5 " 104 ly 2970 K.What was the wavelength at which the background radiation was then most intense?

Will the universe continue to expand forever? To attack this question, assume that the theory of dark energy is in error and that the recessional speed v of a galaxy a distance r from us is determined only by the gravitational interaction of the matter that lies inside a sphere of radius r centered on us. If the total mass inside this sphere is M, the escape speed ve from the sphere is (Eq. 13-28). (a) Show that to prevent unlimited expansion, the average density r inside the sphere must be at least equal to (b) Evaluate this critical density numerically; express your answer in terms of hydrogen atoms per cubic meter. Measurements of the actual density are difficult and are complicated by the presence of dark matter

Because the apparent recessional speeds of galaxies and quasars at great distances are close to the speed of light, the relativistic Doppler shift formula (Eq. 37-31) must be used. The shift is reported as fractional red shift z ! )l/l0. (a) Show that, in terms of z, the recessional speed parameter b ! v/c is given by (b) A quasar detected in 1987 has z ! 4.43. Calculate its speed parameter. (c) Find the distance to the quasar, assuming that Hubbles law is valid to these distances.

An electron jumps from n ! 3 to n ! 2 in a hydrogen atom in a distant galaxy, emitting light. If we detect that light at a wavelength of 3.00 mm, by what multiplication factor has the wavelength, and thus the universe, expanded since the light was emitted?

Due to the presence everywhere of the cosmic background radiation, the minimum possible temperature of a gas in interstellar or intergalactic space is not 0 K but 2.7 K. This implies that a significant fraction of the molecules in space that can be in a lowlevel excited state may, in fact, be so. Subsequent de-excitation would lead to the emission of radiation that could be detected. Consider a (hypothetical) molecule with just one possible excited state. (a) What would the excitation energy have to be for 25% of the molecules to be in the excited state? (Hint: See Eq. 40-29.) (b) What would be the wavelength of the photon emitted in a transition back to the ground state?

Suppose that the radius of the Sun were increased to 5.90 1012 m (the average radius of the orbit of Pluto), that the density of this expanded Sun were uniform, and that the planets revolved within this tenuous object. (a) Calculate Earths orbital speed in this new configuration. (b) What is the ratio of the orbital speed calculated in (a) to Earths present orbital speed of 29.8 km/s? Assume that the radius of Earths orbit remains unchanged. (c) What would be Earths new period of revolution? (The Suns mass remains unchanged.)

Suppose that the matter (stars, gas, dust) of a particular galaxy, of total mass M, is distributed uniformly throughout a sphere of radius R. A star of mass m is revolving about the center of the galaxy in a circular orbit of radius r / R. (a) Show that the orbital speed v of the star is given by v ! r 2GM/R3 and therefore that the stars period T of revolution is independent of r. Ignore any resistive forces. (b) Next suppose that the galaxys mass is concentrated near the galactic center, within a sphere of radius less than r. What expression then gives the stars orbital period?

There is no known meson with charge quantum number T ! 2p 2R3 /GM, Figure 44-13 Problem 48. 7 8 9 6 2 4 5 1 3 q ! '1 and strangeness S ! #1 or with q ! #1 and S ! '1. Explain why in terms of the quark model.

Figure 44-12 is a hypothetical plot of the recessional speeds v of galaxies against their distance r from us; the best-fit straight line through the data points is shown. From this plot determine the age of the universe, assuming that Hubbles law holds and that Hubbles constant has always had the same value. Figure 44-12 Problem 46. 0.40c 5.3 Speed v Distance r (109 ly)

How much energy would be released if Earth were annihilated by collision with an anti-Earth?

A particle game. Figure 44-13 is a sketch of the tracks made by particles in a fictional cloud chamber experiment (with a uniform magnetic field directed perpendicular to the page), and Table 44-6 gives fictional quantum numbers associated with the particles making the tracks. Particle A entered the chamber at the lower left, leaving track 1 and decaying into three particles. Then the particle creating track 6 decayed into three other particles, and the particle creating track 4 decayed into two other particles, one of which was electrically unchargedthe path of that uncharged particle is represented by the dashed straight line because,being electrically neutral,it would not actually leave a track in a cloud chamber.The particle that created track 8 is known to have a seriousness quantum number of zero. By conserving the fictional quantum numbers at each decay point and by noting the directions of curvature of the tracks, identify which particle goes with track (a) 1, (b) 2, (c) 3, (d) 4, (e) 5, (f) 6, (g) 7, (h) 8, and (i) 9. One of the listed particles is not formed; the others appear only once each

Figure 44-14 shows part of the experimental arrangement in which antiprotons were discovered in the 1950s. A beam of 6.2 GeV protons emerged from a particle accelerator and collided with nuclei in a copper target. According to theoretical predictions at the time, collisions between protons in the beam and the protons and neutrons in those nuclei should produce antiprotons via the reactions and However, even if these reactions did occur, they would be rare compared to the reactions p ' p : p ' p ' -' ' -# andThus, most of the particles produced by the collisions between the6.2 GeV protons and the copper target were pions.To prove that antiprotons exist and were produced by somelimited number of the collisions, particles leaving the target weresent into a series of magnetic fields and detectors as shown inFig. 44-14. The first magnetic field (M1) curved the path of anycharged particle passing through it; moreover, the field was arranged so that the only particles that emerged from it to reach the second magnetic field (Q1) had to be negatively charged (either a p or a p#) and have a momentum of 1.19 GeV/c. Field Q1 was a special type of magnetic field (a quadrapole field) that focused the particles reaching it into a beam, allowing them to pass through a hole in thick shielding to a scintillation counter S1. The passage of a charged particle through the counter triggered a signal, with each signal indicating the passage of either a 1.19 GeV/c p# or (presumably) a 1.19 GeV/c p . After being refocused by magnetic field Q2, the particles were directed by magnetic field M2 through a second scintillation counter S2 and then through two Cerenkov counters C1 and C2. These latter detectors can be manufactured so that they send a signal only when the particle passing through them is moving with a speed that falls within a certain range. In the experiment, a particle with a speed greater than 0.79c would trigger C1 and a particle with a speed between 0.75c and 0.78c would trigger C2. There were then two ways to distinguish the predicted rare antiprotons from the abundant negative pions. Both ways involved the fact that the speed of a 1.19 GeV/c p differs from that of a 1.19 GeV/c p#: (1) According to calculations, a p would trigger one of the Cerenkov counters and a p# would trigger the other. (2) The time interval )t between signals from S1 and S2, which were separated by 12 m, would have one value for a p and another value for a p#. Thus, if the correct Cerenkov counter were triggered and the time interval )t had the correct value, the experiment would prove the existence of antiprotons. What is the speed of (a) an antiproton with a momentum of 1.19 GeV/c and (b) a negative pion with that same momentum? (The speed of an antiproton through the Cerenkov detectors would actually be slightly less than calculated here because the antiproton would lose a little energy within the detectors.) Which Cerenkov detector was triggered by (c) an antiproton and (d) a negative pion? What time interval )t indicated the passage of (e) an antiproton and (f) a negative pion? [Problem adapted from O. Chamberlain, E. Segr, C.Wiegand, and T.Ypsilantis,Observation of Antiprotons,Physical Review, Vol. 100, pp. 947950 (1955).]

Verify that the hypothetical proton decay scheme in Eq. 44-14 does not violate the conservation law of (a) charge, (b) energy, and (c) linear momentum. (d) How about angular momentum?

Cosmological red shift. The expansion of the universe is often represented with a drawing like Fig. 44- 15a. In that figure, we are located at the symbol labeled MW (for the Milky Way galaxy), at the origin of an r axis that extends radially away from us in any direction. Other, very distant galaxies are also represented. Superimposed on their symbols are their velocity vectors as inferred from the red shift of the light reaching us from the galaxies. In accord with Hubbles law, the speed of each galaxy is proportional to its distance from us. Such drawings can be misleading because they imply (1) that the red shifts are due to the motions of galaxies relative to us, as they rush away from us through static (stationary) space, and (2) that we are at the center of all this motion. Actually, the expansion of the universe and the increased separation of the galaxies are due not to an outward rush of the galaxies into pre-existing space but to an expansion of space itself throughout the universe. Space is dynamic, not static. Figures 44-15b, c, and d show a different way of representing the universe and its expansion. Each part of the figure gives part of a one-dimensional section of the universe (along an r axis); the other two spatial dimensions of the universe are not shown. Each of the three parts of the figure shows the Milky Way and six other galaxies (represented by dots); the parts are positioned along a time axis, with time increasing upward. In part b, at the earliest time of the three parts, the Milky Way and the six other galaxies are represented as being relatively close to one another. As time progresses upward in the figures, space expands, causing the galaxies to move apart. Note that the figure parts are drawn relative to the Milky Way, and from that observation point all the other galaxies move away because of the expansion. However, there is nothing special about the Milky Waythe galaxies also move away from any other observation point we might have chosen. Figures 44-16a and b focus on just the Milky Way galaxy and one of the other galaxies, galaxy A, at two particular times during the expansion. In part a, galaxy A is a distance r from the Milky Way and is emitting a light wave of wavelength l. In part b, after a time interval )t, that light wave is being detected at Earth. Let us represent the universes expansion rate per unit length of space with 9, which we assume to be constant during time interval )t. Then during )t, every unit length of space (say, every meter) expands by an amount 9 )t; hence, a distance r expands by r9 )t. The light wave of Figs. 44.16a and b travels at speed c from galaxy A to Earth. (a) Show that The detected wavelength of the light is greater than the emitted wavelength l because space expanded during time interval )t. This increase in wavelength is called the cosmological red shift; it is not a Doppler effect. (b) Show that the change in wavelength is given by (c) Expand the right side of this equation using the binomial expansion (given in Appendix E). (d) If you retain only the first term )l l ! ra c # ra . )l (!l4 # l) l4 )t ! r c # ra . 0 rA rB rC r MW A B C (a) (d) (c) (b) t3 t2 t1 r r r MW MW MW A B C A B C A B C t Figure 44-15 Problem 51. (b) (a) MW MW A A t MW MW MW A A A B B B (c) Figure 44-16 Problem 51. of the expansion, what is the resulting equation for )l/l? If, instead, we assume that Fig. 44-15a applies and that )l is due to a Doppler effect, then from Eq. 37-36 we have where v is the radial velocity of galaxy A relative to Earth. (e) Using Hubbles law, compare this Doppler-effect result with the cosmological-expansion result of (d) and find a value for a. From this analysis you can see that the two results, derived with very different models about the red shift of the light we detect from distant galaxies, are compatible. Suppose that the light we detect from galaxy A has a red shift of and that the expansion rate of the universe has been constant at the current value given in the chapter. (f) Using the result of (b), find the distance between the galaxy and Earth when the light was emitted. Next, determine how long ago the light was emitted by the galaxy (g) by using the result of (a) and (h) by assuming that the red shift is a Doppler effect. (Hint: For (h), the time is just the distance at the time of emission divided by the speed of light, because if the red shift is just a Doppler effect, the distance )l/l ! 0.050 )l l ! v c , does not change during the lights travel to us. Here the two models about the red shift of the light differ in their results.) (i) At the time of detection, what is the distance between Earth and galaxy A? (We make the assumption that galaxy A still exists; if it ceased to exist, humans would not know about its death until the last light emitted by the galaxy reached Earth.) Now suppose that the light we detect from galaxy B (Fig. 44-16c) has a red shift of . (j) Using the result of (b), find the distance between galaxy B and Earth when the light was emitted. (k) Using the result of (a), find how long ago the light was emitted by galaxy B. (1) When the light that we detect from galaxy A was emitted, what was the distance between galaxy A and galaxy B?

Calculate the difference in mass, in kilograms, between the muon and pion of Sample Problem 44.01

What is the quark formation that makes up (a) the xi-minus particle and (b) the anti-xi-minus particle? The particles have no charm, bottom, or top.

An electron and a positron, each with a kinetic energy of 2.500 MeV, annihilate, creating two photons that travel away in opposite directions. What is the frequency of each photon?