Modern Physics Departmental Exam Last updated November 213 87 1. Recently, 2 rubidium atoms ( 37 Rb ), which had been compressed to a density of 113 atoms/cm 3, were observed to undergo a Bose-Einstein condensation as a function of temperature. This occurs when the debroglie wavelengths overlap and the system forms a single macroscopic quantum state. Estimate the temperature at which Bose-Einstein condensation occurs for this system. 2. Rutile, TiO 2, is a tetragonal crystal a = 4.4923Å. The position of the Ti atoms in the unit cell are,, ; 1/2, 1/2, 1/2 and the position of the oxygen atoms are.31,.31, ;.81,.19,.5;.69,.69, ;.19,.81,.5. a. What would be the d-spacing of the (31) planes of Rutile? b. Using CuK α x-radiation, λ = 1.54 Å, what would be the diffraction angle from the (31) planes of Rutile? c. Assuming atomic scattering factors of f Ti for the 2 titanium atoms and f O for the 4 oxygen atoms, what is the crystalline scattering factor from the (31) planes of TiO 2? 3. Copper is a monovalent metal with density of 8 g/cm 3 and atomic weight of 64 g/mole. Using the free electron approximation, and considering a cubic system of side L, a. find a formula for the number of states dn within an energy interval d ε. b. Use your expression to evaluate the Fermi energy for copper in ev, at OK. 4. A white dwarf is a star which has used up its nuclear fuel and contracted under its own weight to a state of high density and arbitrarily low temperature. It is held up by the kinetic energy of its degenerate electrons. Assume that all electrons in the star behave like free electrons in a metal, that all the electron states are full up to the Fermi energy, that the star is spherical and made of helium, and that the electrons are nonrelativistic. a. Discuss briefly the physical principles needed to find the equilibrium radius of the star. Write down a set of equations which, when combined, will determine this radius. Don't worry too much about factors of 2. b. Show that this radius is given approximately by 2 h 1/3 R = C M, 5/3 Gmemp where C is a purely numerical constant, G is the gravitational constant, m e and m p are the electron and proton masses, and M is the mass of the star. 5. Since the neutron has no charge, its mass must be found indirectly. From the data given below determine the mass difference m n -m H. It might not be necessary to use all of the information provided. a. 2 H + 2 H 3 H + 1 H + 4.31 MeV b. 1H + n 2 H + γ, E γ = 2.225 MeV c. 3 H 3 He + e +.185 MeV d. 2 H + 2 H 3 He + n + 3.267 MeV 6. A helium-neon laser emits light with wavelength 6328 Å in a transition between two states of neon atoms. What would be the ratio of population of the upper state relative to the lower state of the neon atoms if they were in thermal equilibrium at 3 K.
7. In HCl a number of absorption lines with wave numbers (in cm -1 ) 83.3, 13.73, 124.3, 145.3, 165.51, 185.86 have been observed. a. Carefully justify whether or not these transitions are vibrational or rotational transitions. b. If these transitions represent vibrational transitions, determine the characteristic frequency and effective "spring" constant. If these transitions represent rotational transitions determine the J values, moment of inertia of HCl and separation distance between the nuclei. 8. A slab of lead shielding 1. cm thick reduces the intensity of 15 MeV γ rays by a factor of 2. a. By what factor will 5. cm of lead reduce the beam? b What is the assumption cross-section for 15 MeV γ 's in lead nuclei? Lead has a density of 11.4g/cm 3, and an atomic weight of 27 g/mol. 9. The potential energies of two diatomic molecules of the same reduced mass are shown. From the graph determine which molecule has the larger a. inter-nuclear distance, b. rotational inertia (moment of inertia), c. separation between rotational energy levels, d. binding energy, e. zero-point energy, f. separation between low-lying vibrational states. 1. A K o meson (mass 497.7 MeV/c 2 + ) decays to a π, π meson pair with a mean life of.89 x 1-1 s. a. Which one of the fundamental interactions is responsible for this decay? b. Suppose the K o has a kinetic energy of 276 MeV when it decays, and that the two π mesons emerge at equal angles to the original K o direction. Find the kinetic energy of each π meson and the opening angle between them. The mass of a π meson is 139.6 MeV/c 2. 11. A rocket with a proper length of 2 m moves at a speed of.85c directly away from an observer on earth. An astronaut standing at the center of the rocket fires two electrons (rest mass.511 MeV/c 2 ) at a speed of.95c through a vacuum pipe; one electron is aimed toward the front of the rocket, the other toward the rear. a. In the astronaut's frame, which electron reaches its end of the rocket first? How much time elapses between the two collisions? b. In the earth observer's frame, which electron reaches its end of the rocket first? How much time elapses between the two collisions? c. What is the momentum of each electron in the astronaut's frame? d. What is the momentum of each electron in the earth observer's frame? 12. One of the strongest emission lines observed from distant galaxies comes from hydrogen and has a wavelength of 122 nm (in the ultraviolet region). a. How fast must a galaxy be moving away from us in order for that line to be observed in the visible region at 366 nm? b. What would be the wavelength of the line if that galaxy were moving toward us at the same speed?
13. In his original experiment, Mössbauer used the decay of an excited state of 191 Ir, which has a mean lifetime of 1.4 x 1-1 sec., to the ground state. a. Estimate the shift downwards in photon energy due to the recoil of the nucleus if the photon is emitted by a free nucleus and the excitation energy is 129 kev. (b) Compare this shift to the expected line width, and discuss whether you would expect to see resonant absorption of these photons by a gas of 191 Ir. (c) What velocity range of the source is required to scan over the width of the absorption line using the Doppler shift? (d) What is the Mossbauer effect? 238 26 14. The natural decay chain 92 U 82 Pb includes several alpha and negative beta decays. a. How many alpha decays and how many negative beta decays are there in this chain? b. The decay chain actually has several different branches, corresponding to different orders of the alpha and beta decays. Explain why the answer to (a) does not depend on which branch is taken. (3) 15. A pure single crystal of the metal sodium is placed in a static magnetic field of 1T as shown. The metal is illuminated by microwave energy of frequency ω. Excess microwave energy is absorbed at the cyclotron resonance frequency ω c. a. For an ideal free electron gas determine a value for ω c. b. Determine the characteristic size of the orbital motion. c. Explain what is occurring at the resonance frequency. d. Calcium has a similar atomic density as sodium but it has a valence of 2 while sodium has a valence of 1. Estimate the characteristic size for orbital motion for calcium. (6) 16. For a molecule such as CO, which has a permanent electric dipole moment, radiative transitions obeying the selection rule l = ± 1 between two rotation energy levels of the same vibrational energy state are allowed. (that is, the selection rule ν = ± 1does not hold.) a. Find the moment of inertia of CO for which r =. 113nm, and calculate the characteristic rotation energy E or, in electron volts? b. Make an energy-level diagram for the rotational levels l= to l=5 for some vibrational level. c. Indicate on your diagram transitions that obey l = 1and calculate the energy of the photons emitted. d. Find the wavelength of the photon emitted for each transition in (c). In what region of the electromagnetic spectrum are these photons? (87) 17. From the value 294.8 cm -1 for the reciprocal wavelength equivalent to the fundamental vibration of a molecule Cl 2, each of whose atoms has an atomic weight 35, a. Determine the corresponding reciprocal wavelength for Cl 2 in which one atom has atomic weight 35 and the other 37. b. Determine the separation of spectral lines, in reciprocal wavelengths, due to this isotope effect. (68)
18. The resistivity ρ of a metal due to scattering of the quasi-free electrons as per the Drude model is given by 2 ( mev av ) / ( ne ) ρ= λ. a. Derive the expression above. What are the quantities n andλ, and explain qualitatively is ρ inversely proportional to them? b. The average electron velocity v av is very different in the classical and quantum-mechanical treatments. Explain the origin of the difference (why the quantum-mechanical velocity is much larger at room temperature, and is nearly independent of temperature). c. How is λ related to the cross-section σ for scattering of conduction electrons in the metal? d. Naively, one might take σ to be the geometrical area of the lattice ions. A more correct treatment gives a much smaller cross section which depends on temperature. Explain how this arises. 19. A jet plane circles the Earth at the equator with a speed of Mach 4. By how much time is the pilot s watch slow when it lands? 2. When sodium metal is illuminated with light of wavelength 4.2 x 1 2 nm, the stopping potential is found to be.65 V; when the wavelength is changed to 3.1 x 1 2 nm, the stopping potential is 1.69V. Using only these data and the values of the speed of light and the electronic charge, find the work function of sodium and a value of Planck's constant. 21. Two identical photons are produced when a proton and an antiproton, both at rest, annihilate each other. What are the frequencies and corresponding wavelengths of the photons? It is possible to produce a single photon in proton-antiproton annihilation? Why or why not? 22. A dust particle at rest relative to the sun is in equilibrium between radiation pressure and gravitational attraction. What is the size of the dust particle? (For simplicity you can assume that the dust particle is spherical.) 23. What is the density of a neutron star? Evaluate the size (radius) of a neutron star as massive as the sun. Given: the radius of a 197 79 Au nucleus is 7 x 1-14 m, mass of sun = 2 x 1 3 kg. 24. A typical particle production process is p+ p p+ p +π. Mass of proton = 938 MeV/c 2 ; Mass of π = 135 MeV/c 2 a. In a standard accelerator one of the initial protons would be at rest. In this case what is the minimum laboratory energy for the moving proton so that the reaction listed above occurs. b. In colliding beam accelerators, such as the one proposed for the Superconducting Supercollider, the two initial particles are moving toward each other with equal speeds. Why do colliding beam accelerators have an energetic advantage over the type of accelerator described in part a of this question? c. What energy would be required of each proton in a colliding beams accelerator to produce a pion? 25. In a muonic hydrogen atom, the electron is replaced by a muon, which is 27 times heavier. a. What is the wavelength of the n=2 to n=1 transition in muonic hydrogen? In what region of the electromagnetic spectrum does this wavelength belong? b. How would the fine structure and Zeeman effects differ from those in ordinary atomic hydrogen?
26. Consider a Si semiconductor, in which the dielectric constant κ= 12 and electron effective mass m* =.2m e. a. Calculate the binding energy of the extra electron of an impurity arsenic. b. Suppose the arsenic doping concentration is 1 17 cm -3, what fraction of these donor atoms would supply an electron to the conduction band at room temperature? 27. The potential energy between two atoms in a molecule can often be described rather well by the Lennard- Jones potential, which can be written U ( r) = U a r 12 2 where U and a are constants. a. Find the interatomic separation r in terms of a for which the potential energy is minimum. b. Find corresponding value of U min. c. Use figure to obtain numerical value for r and U for the H 2 molecule. Express your answer in nanometers and electron volts. a r 6 d. Make a plot of the potential energy U (r) versus the internuclear separation r for the H 2 molecules. Plot each term separately, together with the total U (r). 28. The measured conductivity of copper at room temperature is 5.88x1 7 1 Ω m 1, its Fermi energy is 7.3 ev, density is 8.96 gm/cm 3 and molar mass is 63.5 gm/mole. a. Calculate the density of free electrons in copper. b. Calculate the average time between collisions of the conduction electrons. c. Calculate the mean free path of electrons in copper. d. Calculate the smallest possible debroglie wavelength of electrons in copper at T = K. How does this value compare with the atomic separation in copper?
29. The proper mean lifetime of π mesons (pions) 2.6x1-8 s. If a beam of such particles has speed of.9c? a. What would their mean life be as measured in the laboratory? b. How far would they travel (on the average) before they decay? c. What would your answer be to part (b) if you neglect time dilation? d. What is the interval in space-time between creation of a typical pion and its decay? 3. What would happen to the magnetic field of the Earth if the planet suddenly freezes? (What will be eventual value of the magnetic field and how long will it take to get there?) Permeability of free space is 4 1-7 NA -2, iron conductivity is 13 1/mOhm cm.