McGill University. Department of Physics. Ph.D. Preliminary Examination. Long Questions

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1 McGill University Department of Physics Ph.D. Preliminary Examination Long Questions Examiners: Date: Wednesday June 8, 2005 Time: 2:00 p.m. to 5:00 p.m. D. Hanna (Chairman) R. Bennewitz G. Holder S. Lovejoy J. Seuntjens J. Viñals Instructions: 4 questions constitute a complete paper. You must answer any 3 questions from Part A and any 1 question from Part B. If you attempt more than 4 questions, indicate which 4 you wish to be considered for marking. All questions are of equal value. No textbooks or notes are allowed. A data sheet is provided at the end. Approved calculators may be used. Casio fx-300ms Casio fx-991 Casio fx-270w Sharp EL-546 Sharp EL-510 Sharp EL-520 V Sharp EL-531 V TI-30X IIS Use separate answer booklets for each problem. Write clearly your name and student number, as well as the question number, on each answer booklet.

2 Long Questions: Part A (Answer 3 of the following 5 questions.) A.1 Consider electromagnetic waves impinging on a flat surface of a conductor. Show that the energy lost as the wave penetrates the conductor (conductivity σ) equals the Joule heating. A.2 A ball of mass m and radius r rolls down a movable wedge of mass M. The angle of the wedge is φ and it is free to move on a smooth horizontal surface. The contact between the ball and the wedge is perfectly rough (there is rolling but no slipping). Find the acceleration of the wedge. A.3 Consider the following approximate model of a liquid: interactions between molecules are neglected except that all molecules experience a constant binding energy η that confines their motion to a volume v 0 per particle. Show that for a classical liquid of N indistinguishable molecules, the partition function is, Z L = 1 [ ] NL v 0 e βη NL Z p, N L! where Z p denotes the ideal gas contribution arising from the integral over momenta, Z p = 1 h 3 d 3 pe βp2 /2m. (a) Calculate the chemical potential of the liquid µ L and of a vapor phase µ G assumed to be an ideal gas. (b) At coexistence between the liquid and solid phases, µ L = µ G. Find the pressure (vapor pressure as it is called) as a function of temperature at coexistence. A.4 A rocket starts out from earth with a constant acceleration of 1 g in its own frame. (ie This is the acceleration of the rocket as measured at any given instant in a non-accelerating frame of reference travelling at the same instantaneous speed as the rocket.) It accelerates for 10 years (of its own time). At this point, it decelerates at 1 g for 10 years, the people on board get out and look around for a negligible amount of time and then repeat the process to come back (10 years accelerating, 10 years decelerating). a) How far away does the rocket get at the turnaround point (in earth units)? b) How long has the rocket been gone in earth units? 2

3 A.5 A particle with mass m moves in a potential given by V = 0 for 0 < x < L V = elsewhere. The normalized eigenfunctions are given by ψ n (x) = 2 nπx sin L L for 0 < x < L ψ n (x) = 0 elsewhere. The particle is in the lowest-energy (ground) state of this infinite potential well. At time t = 0, the wall located at x = L is suddenly pulled back to a position x = 2L. This change occurs so rapidly that instantaneously the wave function does not change. (a) Calculate the probability that a measurement of the energy will yield the ground-state energy of the new well. (Recall that sin(x) sin(y) = 1 (cos(x y) + cos(x + y))) 2 (b) What is the probability that a measurement of the energy will yield the first-excited energy of the new well? (c) Describe the procedure you would use to determine the time development of the system. Is the system in a stationary state? What development do you expect? 3

4 Long Questions: Part B (Answer 1 of the following 6 questions.) B.1 Explain the fundamental physics and the realization of a HeNe laser along the following guidelines. (a) Discuss the basic elements of coherent light production in a gas laser using the concept of stimulated and spontaneous emission. Why is population inversion a condition for laser activity? Why is a laser producing red light easier to build than a laser producing blue light? 2 1 S0 2 3 S1 He-Necollisions 3s 2s 1150 nm 3390 nm 3p nm 2p Excitation by electron collision Fast radiative transitions 1s De-excitation at walls Helium Ground state Neon Figure 1: Level scheme for the HeNe-laser (b) Using the level scheme in the figure, explain how population inversion is obtained in a HeNe gas laser. You should mention why the excited states of the He are metastable and why optical transitions in Ne are indicated for 3s 3p, 3s 2p, and 2s 2p but not for 3s 2s. What role does the transition 2p 1s play for the laser activity? (c) How do you select the optical transition at nm as the only active laser wavelength? (d) The 3s 2p optical emission is broadened by Doppler broadening. What is Doppler broadening and on what parameters does it depend? What does the emission of a Doppler broadened HeNe laser look like, including the effects of a resonator of 1m length (Draw an intensity versus wavelength plot)? Can you think of an additional element in the laser that would suppress the effects of the Doppler broadening? 4

5 B.2 The partition function for the normal modes of lattice vibration (harmonic lattice) is given by, Z(N, T, V ) = k e β hω(k)/2 1 e β hω(k). (a) Obtain the free energy F (N, T, V ). (b) An approximate treatment of a weakly anharmonic lattice can be carried as follows: Assume that the free energy of the anharmonic system is that given in the previous item plus a known function φ(v ) that accounts for the elastic energy of uniform solid compression. Assume further that the normal mode frequencies are not constant, but that they depend on the volume of the system ω = ω(v ) for all k. Calculate the equation of state of this solid by computing the pressure p = ( F/ V ) T. By defining the so-called Gruneisen parameter γ = d ln(ω)/d ln(v ), show that the equation of state is, p = dφ dv + γ E V. B.3 a) Estimate the electron Fermi energy of a relativistic white dwarf, assuming a constant density carbon/oxygen composition star of radius R. b) Estimate the gravitational binding energy. c) What is the maximum mass (approximately, in solar masses) that can be supported by electron degeneracy pressure in a relativistic white dwarf? B.4 The 7/2 + state at 1.72 MeV in 21 Ne has a half-life of 48 fs. This state decays 94% of the time to the 0.33 MeV 5/2 + state, with a mixing ratio of the two possible multipoles of δ = 0.14, and 6% of the time to the 3/2 + ground state. What are the B(M1) and B(E2) for these transitions? (Decays rates for E2 and M1 are: W (E2) = E 5 B(E2) W (M1) = E 3 B(M1) where E is in MeV, B(E2) in e 2 fm 4, and B(M1) in µ 2 N.) 5

6 B.5 (a) Draw the first-order Feynman diagrams for (i) e + e annihilation into muon pairs. (ii) e + e annihilation into quark pairs. Use the diagrams, along with quark counting and coupling constant values, to make a rough estimate of the values of R (the ratio of multi-hadron events to muon-pair events in e + e collisions) at centre-of-mass energies 2.0 GeV and 6.0 GeV. (b) Compare the merits of measuring the τ lifetime at CESR (E CMS 10 GeV ) and LEP (E CMS 90 GeV ). Which collider would you choose to make the most accurate measurement and what technology would you use? B.6 A water calorimeter is used to calibrate a waterproof thimble type, cylindrical ionization chamber in terms of absorbed dose to water. To do so, a thermistor, positioned at the point of measurement in water in the calorimeter, is connected to the bridge circuit depicted in Figure 2 where nominal values of all electrical components are shown. The voltage change measured across the bridge (point A and point B in Fig. 2) after finalization of a calorimeter measurement amounts to 20 µv. The thermistor sensitivity amounts to S = K 1. The heat defect of the calorimeter is -2.2%. Questions: (a) Determine the absorbed dose to water (in Gy) used to calibrate the ionization chamber. (b) The calorimeter experiment is followed by measurements with the ionization chamber for the same irradiation time with the chamber positioned at the same point in water as where the thermistors were located during the calorimeter experiment. The chamber charge, corrected for environmental conditions and for polarity and recombination, amounts to Q = 90 nc. Determine the absorbed dose to water calibration coefficient, N D,w. (c) The calorimeter and chamber experiments described above were carried out at a depth of 10 cm in water in a 6 MV clinical photon beam and the chamber characteristics are as follows: wall material: graphite; wall thickness: g cm 2 cavity inner radius: 3 mm Determine the effective volume of the chamber. Hints: determine the absorbed dose-to-cavity air calibration coefficient first. Assume 6

7 in your calculations that the chamber is waterproof. Assume also that ratios of unrestricted collision mass stopping powers evaluated at appropriate electron energies can be used to calculate the restricted collision mass stopping power ratio. Finally, approximate ratios of average mass-energy absorption coefficients by ratios of mass-energy absorption coefficients, evaluated at an appropriate photon energy. Figure 2: Schematic drawing of the calorimeter electrical setup. Useful information: (a) Specific heat capacity of water at constant pressure (at the calorimeter operating temperature), c p = 4180 J kg 1 K 1. (b) The heat defect of a calorimeter is defined as: h = E a E h E h, (1) where E a is the energy absorbed and E h the energy appearing as a temperature rise. 7

8 (c) The thermistor sensitivity is defined as: S = 1 dr R dt, (2) where R represents the electrical resistance and T the temperature of the thermistor bead. (d) If needed, use the following equation to evaluate the replacement correction factor for the ionization chamber in all derivations: P repl = r cyl (3) where r cyl represents the inner radius of the chamber in mm. (e) If needed, use the following equation to evaluate the contribution to the cavity ionization in the chamber from photon interactions in the wall, when the chamber is positioned in water: α(t wall ) = 1 exp( t wall ) (4) where t wall represents the thickness of the ionization chamber wall (in units of g/cm 2 ). (f) See Table 1 and Table 2 for numerical values of mass attenuation coefficients, mass-energy absorption coefficients and mass collision stopping powers. (g) The average energy required to create an ion pair in (dry) air per unit of charge is (W/e) air = J/C 8

9 Table 1: Mass attenuation coefficients and mass-energy absorption coefficients for graphite, air and water as a function of photon energy. Photon graphite air water energy µ/ρ µ en /ρ µ/ρ µ en /ρ µ/ρ µ en /ρ (MeV) (cm 2 /g) (cm 2 /g) (cm 2 /g) 1.000E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-02 9

10 Table 2: Unrestricted collision mass stopping powers for graphite, air and water as a function of electron kinetic energy. Electron energy S coll /ρ (MeV cm 2 /g) (MeV) graphite air water 1.000E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00 Information which may be useful 1. Constants: e = Coulombs hc = 197 MeV-Fermi h = Joule-second = MeV-second c = meter/second m p = MeV/c 2 = kg m n = MeV/c 2 = kg m e = MeV/c 2 = kg u = MeV/c 2 10

11 k B = MeV/K = J/K N A = atoms/mole ɛ o = Coulombs/(Newton-meter 2 ) µ o = 4π 10 7 Newtons/Ampere 2 g = 9.8 m/s 2 Stefan-Boltzmann constant: σ = J/sec/m 2 /K 4 G N = m 3 kg 1 s 2 L = W M = kg R = m AU = m 2. Maxwell s equations in linear isotropic media: ɛ E = ρ free E = µ H t µ H = 0 H = ɛ E t + J free 3. Cylindrical Coordinates: (r, φ, z) ψ = ψ r ˆr + 1 ψ r φ ˆφ + ψ z ẑ, 2 ψ = 1 ( r ψ ) ψ r r r r 2 φ + 2 ψ 2 z 2 A = 1 r r (ra r) + 1 A φ r φ + A z z ( A 1 A z = r φ A ) ( φ Ar ˆr + z z A ) z r ˆφ + ( 1 r r (ra φ) 1 r ) A r φ ẑ 4. Time-dependent Perturbation Theory: If H = H 0 + V, then S = U(, ) = 1 ī h where V (τ) = e ih0τ/ h V e ih0τ/ h. dτ V (τ) + O(V 2 ) 5. Hydrogen Wavefunctions: r, θ, φ n; l; m are given by: r 1; 0; 0 = 2 a e r/a 0 r 2; 1; ±1 = ( r (2a 0 ) 3 2 a 0 8π ) e r/(2a 0) sin θ e ±iφ 11

12 6. Gibbs Free Energy Relation: 7. Lorentz Transforms etc x = γ(x vt) y = y z = z t = γ(t vx/c 2 ) l = l 0 /γ t = γτ u x = u y = u z = ux v 1 vu x/c 2 u y γ(1 vu x/c 2 ) u z γ(1 vu x/c 2 ) E = γm 0 c 2 E 2 = p 2 c 2 + m 2 0c 4 S = ( ) G T pn 12

McGill University. Department of Physics. Ph.D. Preliminary Examination. Long Questions

McGill University. Department of Physics. Ph.D. Preliminary Examination. Long Questions McGill University Department of Physics Ph.D. Preliminary Examination Long Questions Examiners: Date: Tuesday May 27, 2003 Time: 2:00 p.m. to 5:00 p.m. C. Burgess (Chairman) S. Das Gupta V. Kaspi J. Strom