UNIVERSITY OF TORONTO Faculty of Arts and Science APRIL 2018 EXAMINATIONS. PHY357H1S (Solutions) [grades] Duration 3 hours

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1 UNIVERSITY OF TORONTO Faculty of Arts and Science APRIL 2018 EXAMINATIONS PHY357H1S (Solutions) [grades] Duration 3 hours Examination Aids: Non-Programmable scientific calculator, without text storage One, personally prepared (not mechanically photo-copied/photo-reduced), aid-sheet (materials can appear on both sides of the sheet) PLEASE read carefully the following instructions. There are five questions on this exam. You must answer four of them. Each answer is worth 25 % of the full examination grade. If you answer more than four please indicate which four you want to count, otherwise only the first four attempted will be graded and included in your final score. Partial credit will be given for partially correct answers, so show any intermediate calculations that you do and write down, in a clear fashion, any relevant assumptions you are making along the way. There are three pages of background material, not all of which you ll need to answer the questions, that are found on pages two to four of this paper. The questions start on page five and continue to page seven. Good luck! Page 1 of 9 pages

2 PAGE 5 1. (a) Explain the significance of the following terms used in the classification of subatomic particles: i. lepton; Spin 1/2 matter particles that interact only weakly and (some) electromagnetically. I guess the most succinct way to answer this was they do not undergo strong interactions. Some people listed all the leptons without really explain why they were leptons. They didn t get full credit for doing that. [1] ii. isotope; Nucleii that have the same charge (number of protons) but differing numbers of neutrons. [1] iii. boson; Integer spin particles or nucleii. Obey Bose-Einstein statistics (ie. can have many in the same state lowest energy state of a system. For the fundamental particles these include all the force carriers (photon, gluons, W and Z). But all mesons also have integer spin and so behave as bosons too.[1] iv. isotone; Nucleii that have the same total number of neutrons but differing the number of protons. These families of nucleii are typically linked together by weak beta decay transitions, where one proton converts to a neutron (or vice versa).[1] v. meson; Integer spin composite particles (made up of a quark-anti-quark pair) that interact strongly. Some decay weakly, if one (or both of the quarks) is forced to decay weakly to get to a lighter family of mesons, or in the case of π mesons decaying to the only thing lighter (a pair of leptons).[1] vi. left-handed helicity. Particles whose spin is anti-aligned with their direction of travel. Massless weakly interacting particles only interact if they have left-handed helicity. This is because the W boson only couples to the left-handed component of the particle s wavefunction. Massless particles, travel at the speed of light and will always be in a purely left-handed or righthanded helicity state. All observers will agree on this (no observer can be travelling faster than them).[1] (b) Which of the terms from the list above applies to each of the following particles? i. τ + Lepton, some people said partially left-handed which was OK.[1] ii. Z 0 Boson[1] iii P b and Bi Isotones[1] iv. K + Meson, boson[1] v Ne and 22 10Ne Isotopes[1] vi. ν e Lepton and left-handed helicity.[1] In some cases more than one term from part a) may apply. To get full credit you must list all that apply. 2. There are three generations of charged leptons e, µ, and τ. Their masses and lifetimes are given on page 3 of this exam. (a) Compare the lifetimes of the three charged leptons. At a given momentum, which travels furthest before half the beam decays? Which travels the least distance before half the beam decays? The electron lifetime is infinite. It travels the furthest and never decays. The tau lifetime is shortest so it travels the least distance at a given momentum before it decays. Some people tried to say the tau was also the heaviest so it was harder to get up to that given momentum. But that wasn t part of this, this part of the, question. Naturally the muon is in between. Very few people actually said that, but I didn t deduct points if they didn t.[1.5]

3 PAGE 6 (b) What momentum beam of τ leptons would be needed if they were to have the same average decay length (in the lab frame) as a beam of p = 1 GeV/c muons? The decay length for particle is given by l = γβcτ. But γβ = p/m so the muon and tau decay lengths will be the same when m p τ = p τ τ µ µ m µτ τ = GeV/c for a 1 GeV/c muon. Yes this is an unrealistically high momentum tau beam. I just wanted some math in this problem...[3.5] A linear accelerator is a device that accelerates particles up to high energy, by passing them through an electric field. There are plans to make a very high energy e + e linear collider, where one arm accelerates electrons and collides them, head-on, with positrons accelerated by the other arm. If the e + and e collide with equal and opposite momenta, they will annihilate to produce some final state with a non-zero total relativistic energy, but zero total linear momentum. Say we want to produce a total relativistic energy in the final state of 1000 GeV. (c) What are the energy and momenta of the two electron beams? In computing this you neglect the electron mass if you do, explain why the assumption is reasonable. The electron mass is only 0.5 MeV. So to get energies 10 6 times larger we can surely neglect it (it will be a quadratic correction (ie ) to the final answer. So we can treat these electrons as ultra relativistic which means E = p. To get 1000 GeV from two counter rotating beams we need 2p = E cm or p = 500 GeV/c. There were many ways to get this. Most people did. A few got lost along the way and lost 0.5 or 1 points. [1.5] (d) Assume the accelerator has an electric field gradient of 35 MV/m in the direction of motion of the particles. How long would each arm of this linear collider have to be to reach this final state energy? The electrons have a charge of 1 in units where 1V of acceleration gives 1 ev of energy. So the length required was just 500 GeV/35 MV/m = 14.3 km. Some people put e = C in to try to fix the units when they didn t need to and got some other answer (and lost 0.5 points for their trouble). But most survived this. If you got some creative energy/momentum requirement for the beam from part c) but then translated that to a length correctly here you got full marks, here.[1.5] (e) Electrons are stable particles, so they won t decay while being accelerated. If we injected a beam of muons in to this collider that already had a momentum of 10 GeV/c, would they be accelerated to the same collision energy? Explain qualitatively what would limit the performance of such a muon collider. Feel free to use numerical arguments in your answer but, to get full credit, you must explain the physics that might limit the energy or luminosity of such a machine. I wasn t looking for calculations of the precise energy of the muon beam. Yes, the muons are heavier than the electrons, but they have the same charge, so if they pass through a 35 MV electric potential drop they gain 35 MeV of energy. Is suppose they gain a little less in momentum, but by the time you have 500GV of drop (in the 14km long accelerator arm) the difference in momenta between electrons and muons is tiny. You can almost treat the muons as massless in this regime too. What I was looking for was some discussion of the decay length of the muons. Most people eventually got around to saying that the number of muons reaching the collision point would be limited by their lifetime. The 10 GeV/c injection momentum was supposed to get you thinking about the proper decay length of muons already at that momentum (it s actually 6km at that point). But the fact remains, some fraction of the muons will decay during their 14km journey in the lab frame. So the collision intensity (or luminosity) will be limited in a machine like this.[2] Page 6 of 9 pages

4 PAGE 7 3. CERN s Large Electron Positron collider (LEP) had a diameter of 8 km and produced beams of energy 45 GeV. Each beam consisted of 12 bunches, and each bunch contained particles (electrons or positrons, depending on the beam). The bunches had a cross-sectional area of 0.02 mm 2. (a) What was the luminosity of this machine in units of cm 2 s 1? The 8km radius LEP/LHC tunnel takes particles 90 µs to get around (at the speed of light). So the bunch crossing frequency is f = (n b = 12)/(90µs) = /s. And L = n b fn 2 e /(4πσ 2 ) = cm 2 s 1. There was some confusion about whether the cross-sectional area of the beams given in the question was σ 2 (ie. some kind of rectangular beam overlap with sides σ x and σ y, or 4πσ 2 (ie. some kind of beam falling off gently with a size sigma x by σ y. We accepted both interpretations to get full credit here. [2] (b) If the cross-section to produce a Z-boson at a centre-of-mass energy of 90 GeV is 1 nb, how many Z bosons were produced per second, when the LEP collider was running? The event rate is given by R = σl so the number of Z bosons created per second is 1 nb = 0.06 per second. Of course if you got the other luminosity, you got a different rate here but still got full credit.[2] There were four experiments located around the LEP ring that were designed to study the decay of the resulting Z (and W bosons) that LEP produced. The LEP experiments used similar detector strategies to ATLAS, as discussed in class. What would the experimental signatures have been for the following decays: For the following three parts there were several different answers. You needed to have at least two of them to get full credit. (c) Z b b; Two jets tagged with a displaced vertex evidence for a long lived b quark decay; invariant mass of the two jets near 90 GeV; charged particle tracks, Energy deposited in EM and Had sections of the calorimeter; possible µ from the B meson decays. [2] (d) W eν; Energy deposits in the front (EM part of the calorimeter) and not in the back (Had part of the calorimeter); momentum imbalance (usually measured by imbalanced energy flow) in the calorimeter system, indicative of an energetic neutrino that didn t interact in the detector; at least one track pointing at the EM energy deposit (the charged electron track); invariant mass reconstructed from electron and missing momentum consistent with 80 GeV W boson. [2] (e) W µν. Track segments in the outer ( muon ) tracking layers; track segment in inner tracker pointing at muon tracks, energy/momentum imbalance characteristic of un-measurable neutrino, invariant mass consistent with W boson.[2] To get full credit you must list at least two detector signatures for each of these decays. 4. This problem contains two independent questions based on our study of nuclear physics in the course. To get full credit you must answer both sets of questions. I) Assume that, in the shell model, the nucleon energy levels are ordered as shown in the figure on p4 of this exam. Write down the shell model configuration of the following nucleii, and state what the shell model would predict for their spin and parity: (a) 7 3 Li; 4 neutrons: 2 in 1S state, 2 in 1P 3/2 state; 3 protons: 2 in 1S state, 1 in P 3/2 state. This leads to J = 3/2 (from the single proton in that state), l = 1 gives J P = 3/2 [1] + [1] (b) Nb. 52 neutrons fill all states leaving 2 in the D 5/2 state. 41 protons fill all states leaving one in the G 9/2 state. This leads to J = 9/2 from the single proton and l = 4 giving J P = 9/2 + [2] + [1] II) A space probe, designed to land on another planet, is to be powered by P u. This isotope decays via α emission to the stable isotope U with a release of 5.5 MeV of kinetic energy and a lifetime of 127 years.

5 PAGE 8 (d) How much power would 1 kg of this isotope produce if the energy released could be converted to useful work with 100% efficiency? Quite a lot actually. This is an exercise in unit conversion. 238 g of P u would have 1 mole of atoms, so 1 g has 1/238 moles. If it all decays and could be converted to work with 100% efficiency, then you d get P = 1000/ MeV J/eV 127yrs s = 555W or MeV/s I accepted either answer. Some of you tried to use the approximation 5.5 MeV 5 MeV in solving this. That lost you half a point... Others did more of the calculation in part c) Together they were worth 8 points, I tried to make sure if you made a mistake here and carried it through to part c) you didn t lose points in both places. [2] (e) If it takes 40 years for the space probe to reach it s destination, how much of the P u initially launched will reach the destination? (Note: you can assume the space probe travels at non-relativistic speeds as it makes it s way to the other planet). During the journey exp( t/τ) = exp( 40/127) = So 73% of the plutonium launched will survive, the rest decays away during the journey and won t be available to the lander upon arrival. [1] (f) If the probe requires 200 W of power to perform its landing tasks when it gets there, and the energy released can only be converted to useful work with 5% efficiency, how much P u should be on the probe at launch? The power that can be generated when the probe lands is given by P = P 40yrs 1kg N kg ɛ, where ɛ is the stated 5% efficiency for using the power during landing tasks. So I get: N kg = 200W (1/ɛ)/(P 40yrs 1kg = 4000W/(555W 0.73)) = 9.85kg. [2] 5. Examine the following processes and state for each one whether it represents a possible reaction, or if it is impossible according to the Standard Model. If the interaction is possible state, which force mediates it (ie. strong, electromagnetic or weak) and draw the Feynman diagram for the process. If the process is a weak decay involving quarks, state whether it is Cabbibo favoured, or Cabbibo suppressed. If the reaction is impossible, cite a conservation law that prevents it from occurring. π 0 γ + γ Ξ Λ 0 + π n p + e + ν e Λ 0 Σ + + π D 0 K + + π π + + n π + + π 0 B D 0 + π ++ p + π + p + p J/Ψ + π + + π τ e + ν e + ν τ Λ b Λ + c + µ + ν µ Σ 0 p + π Starting at the top left and working down each column: The standard π 0 EM decay, allowed; This would be an allowed strong interaction, but the Λ 0 is lighter than the Σ +, forbidden; This is one of the dominant decays of the B meson, but it is a weak decay, allowed; this would be the standard τ lepton decay but then there would have to be ν τ in the final state, so tau lepton number is not conserved, forbidden; This is a strong decay of the Ξ, the strange quark re-appears in the final state, allowed ; This is the standard weak decay of a D 0 meson, allowed; This is a strong decay of the doubly charged resonance, allowed; This is an allowed weak decay of the Λ b, the muon number is zero in the final state, allowed; This is the standard weak decay of the neutron; allowed, This would be a standard strong scattering except that there is no baryon in the final state, forbidden; This is the strong production of J/P si mesons, several people asked me about the J/Psi in the test since it wasn t on the list of particles given at the front. Even if you didn t remember that it was a c c-bar resonance you could have made

6 PAGE 9 some guess and arranged the quark lines to match up, allowed; this is a weak decay of the Σ 0 where one strange quark converts to a u/d, allowed. Some of you hoped I would forget that I asked you to distinguish between Cabbibo favoured and Cabbibo suppressed weak decays. The general rule of thumb is that if the W boson couples to quarks in different families then that is suppressed (like us or even bc couplings). The only favoured one was the neutron decay where the W couples fo ud and then materialises as an electron and neutrino. If either end of the W crosses generations then it is suppressed (even more than just the fact that it is a weak decay. But if you got a little confused here you didn t lose points on all questions just 1 point overall on the question if you neglected to answer this or got several of them wrong. Getting one wrong didn t lose you any points (!) There were 12 reactions here, for 10 points, so if you got one or two wrong (or even 3 half-wrong ) you didn t lose any points. This was my way of putting some bonus points back in to the system after the, tougher than I had planned, midterm [10] Partial credit will be given for listing all the quantities that are conserved even in cases where you might miss the one quantity that prevents a reaction from proceeding. For those involving hadrons, make sure you show all the quarks that are involved and which hadrons they belong to in the initial/final states. End of examination Total pages: 9