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All Solutions of Type: Advanced Topics
 0 Click here to jump to the problem! GR8677 #16 Advanced Topics}Particle Physics The properties of a muon most closely resembles that of an electron. In the usual -decay, an electron is emitted along with either a neutrino or an antineutrino. In a wildcard -decay, a muon is emitted instead of an electron. Click here to jump to the problem!
 2 Click here to jump to the problem! GR8677 #23 Advanced Topics}Solid State Actually, one can figure this one out with only knowledge of lower div baby physics. (A) Electrical conductivities for conductors, semiconductors, and insulators go (in general), like this . Thus, copper should be more conductive than silicon. This is true (but one is trying to find a false statement). (B) The resistivity , thus the conductivity, . As increases, decreases. This is true. (C) Silicon is a semi-conductor, and thus it probably does not follow the same relations as (B). In fact, semiconductors have negative temperature coefficient of resistivities. Thus, , which implies that for temperature increase. (D) Doping a conductor like copper will just make it cheap. Think of cheap wire. (E) Doping a semiconductor, however, can make it more conductive. Click here to jump to the problem!
 3 Click here to jump to the problem! GR8677 #38 Advanced Topics}Logical Circuit (A) This is the OR gate. Triangle with fat-end on input side denotes OR. (B) Triangle with fat-end closer to output side denotes AND. (Pointy tip points to each input.) (C) A 2 bit-adder involves more operations than this... (D) A flip-flop is a sort of sandal that flips and flops. It might also flip the floating point. (E) A fan-out describes the maximum number of outputs a circuit can excrete. (Fan-in would be inputs.) Click here to jump to the problem!
 5 Click here to jump to the problem! GR8677 #50 Advanced Topics}Solid State Physics Recall , where the sign shows that the charge carrier is negative. ( is the electron density, electron charge, speed of light) The Hall effect has to do with a bar of metal placed in a magnetic field. An electric current runs through it perpendicular to the field direction. However, the magnetic field term in the Lorentz force creates a force perpendicular to both the magnetic field and electric field. This pulls" the current astride from its original straight path. Charge starts accumulating on one side of the slab from this force, even as current continues to flow perpendicular to the accumulation direction. In equilibrium, the magnetic force balances the electric force from this charge accumulation. Apply some basic EM theory, and one can derive the Hall coefficient () defined above. Click here to jump to the problem!
 6 Click here to jump to the problem! GR8677 #85 Advanced Topics}Nuclear Physics Pair production only occurs above a certain energy on the order of MeV. Thus, all except choices (B) and (C) remain. The photoelectric effect is dominant for low energies, so its cross-section must be line (1). Choose choice (B). Click here to jump to the problem!
 7 Click here to jump to the problem! GR8677 #10 Advanced Topics}Particle Physics Recall that in gamma-ray production, the excited nucleus jumps to a lower level and emits a photon . In internal conversion, however, an orbital electron is absorbed and ejected along with an X-ray. Click here to jump to the problem!
 8 Click here to jump to the problem! GR8677 #29 Advanced Methods}Dimensional Analysis The current author is fortunate enough to have taken a String Theory course as an undergraduate, and thus know by heart that the Planck length is . However, the problem can also be solved via dimensional analysis (A) has units of , which doesn't have the units of . (B) , which doesn't have the units of . (C) , which doesn't have the units of . (D) , which doesn't have the units of . (E) This is the last one. Take it! Click here to jump to the problem!
 10 Click here to jump to the problem! GR8677 #34 Advanced Topics}Particle Physics Regularly, electrons are emitted in any direction. Thus, there is infinite symmetry. In the case of a magnetic field, electrons are more likely to be emitted in a direction opposite to the spin direction of the decaying atom. Place the atom in an x-y plane, with its spin-direction pointing along the z-axis. If the electron is mostly emitted in the -z axis, then reflection symmetry is violated since it's not (mostly) emitted in the +z axis, i.e., not mirrored across the x-y plane. Choice (D). (This is due to Joe Bradley.) Click here to jump to the problem!
 11 Click here to jump to the problem! GR8677 #53 Advanced Topics}Particle Physics One can ignore baryon numbers and lepton numbers and all that and just deal with spin conservation. For the positronium-electron spin singlet state, one has, initially, . The decay must conserve spin. Thus, one must have the final spin as . Since a photon is its own antiparticle (and antiparticles have the negation of the usual particle's quantum number), the photon has spin and the antiphoton (just another photon) has spin . Thus, two photons are emitted to conserve spin. (Wheee... can one get more ad hoc than the Standard Model?) Click here to jump to the problem!
 13 Click here to jump to the problem! GR8677 #64 Advanced Topics}Nuclear Physics In symmetric fission, the change in kinetic energy is just the change in binding energy. The change in binding energy for a heavy nucleus is the difference in energy between the initial un-fissioned heavy nucleus and the final 2 medium-sized nuclei, . For a heavy nucleus, one has , and thus one arrives at choice (C). (This is due to David Schaich.) Click here to jump to the problem!
 15 Click here to jump to the problem! GR8677 #78 Advanced Topics}Solid State Physics A n-type semiconductor is a material with negative-charge carriers, such as electrons. A p-type semiconductor is a material with positive-charge carriers, such as holes (positrons). In band theory, n-type semiconductor impurities are (electron) donors, while p-type semiconductor impurities are (electron) acceptors. The setup is as follows: Impurities add in more levels to the energy bands. Without impurities, one has just a valance band and a conduction band with an energy gap in between. The impurities supply an extra energy level in between the conduction and valance bands. In an n-type semiconductor, the material becomes conducting when there are electrons in the conduction band; the impurity helps the material become conducting by supplying it with electrons. Essentially, one starts with a lattice of pure semiconductor atoms, say Silicon. Silicon has four valance electrons and forms a decent crystal lattice. Pluck out a few silicon atoms and replace them with some impurities, like Arsenic, which five valance electrons. The extra electron from each impurity atom is free to roam around. In fact, these extra electrons act as donors to the conduction band. This is choice (E). Click here to jump to the problem!
 16 Click here to jump to the problem! GR8677 #95 Advanced Topics}Solid State Physics The specific heat of a superconductor jumps at the critical temperature (c.f. with its resistivity jump). Ordinarily, the specific heat of a metal is . When it is superconducting, the first term, the electronic-contribution, is replaced by . The revised plot of the specific heat has a jump from an exponentially increasing specific heat to a much lower value somewhere in the range for positive T. Reference: Ibach and Luth p 270ff Click here to jump to the problem!
 17 Click here to jump to the problem! GR8677 #19 Advanced Topics}Astrophysics It takes 4 H's to create a Helium nucleus in the sun's primary thermonuclear reactions. One either remembers this or can derive it from conservation of mass. The atomic mass of Hydrogen is 1 (since it has just 1 proton), while the atomic mass of Helium is 2 (2 protons, 2 neutrons). Click here to jump to the problem!
 18 Click here to jump to the problem! GR8677 #52 Advanced Topics}Solid State Physics Recall the definition of a primitive cell to be the unit cell divided by the number of lattice points in a Bravais lattice. A Bravais lattice is just a lattice that looks isotropic from any point---everywhere the same no matter the point-perspective. Simple cubic has 1 lattice point to generate its Bravais lattice. Body-centered cubic has 2 lattice points to generate its Bravais lattice. (One can keep on tesselating the a lattice point on one corner and the lattice point on the body center to generate the whole BCC lattice.) Face-centered cubic has 4 lattice points to generate its Bravais lattice. (One can keep on tesselating the lattice point on each face surrounding a corner lattice point.) So, anyway, from the above, one finds choice (C) for BCC's unit cell. Click here to jump to the problem!
 19 Click here to jump to the problem! GR8677 #53 Advanced Topics}Solid State Physics From Ibach and Luch, one finds that the resistivity of a semiconductor varies as . (Also, from elementary electrodynamics, one recalls the resistivity equation . Semiconductors have a negative coefficient of resistivity , and thus the resistivity should decrease with increasing T. The only graph that shows this behavior of decreasing resistivity with T is (B).) Click here to jump to the problem!
 20 Click here to jump to the problem! GR8677 #66 Advanced Topics}Radioactivity The problem supplies the details for -emission and -emission to be: (Why? Well, one starts with the equation , where one integrates both sides. Plugging in the condition for half-life, one has , where T is the half-life decay time.) The total decay is given by . Thus, . Solve for T to get choice (D). Click here to jump to the problem!
 21 Click here to jump to the problem! GR8677 #67 Advanced Topic}Binding Energy The binding energy is basically the energy that keeps a nucleus together; and makes its mass slightly different than its constituent particles. (If its mass were exactly the same as its constituent particles, its binding energy would be 0, and it would be unstable.) The initial binding energy is , which is the number of nucleons times the binding energy per nucleon. The final binding energy is , where the daughter nuclei have half the number of nucleons from the given fact that the original nucleus splits into 2 equal fragments. The difference in binding energy is equal to the kinetic energy. . Solving, one finds that , as in choice (E). Click here to jump to the problem!
 22 Click here to jump to the problem! GR8677 #68 Advanced Topics}Decay Since Lithium has one less electron than Be, one might think this decay is just -decay. However, -decay always occurs with a neutrino or anti-neutrino, and since none of the choices show this. Beta-decay emits either an electron or positron with an antineutrino or neutrino: (The bit on antineutrinos and neutrinos has to do with conservation of Electron Lepton number . Namely, electrons and neutrinos have , while positrons and antineutrinos have .) (Also, -decay emits a Helium atom with 4 neutrons and 2 protons, so the numbers won't work out here.) The remaining choice is (E). One can check that it's right by noting that it is the only choice that conserves the electron-lepton number. Click here to jump to the problem!
 23 Click here to jump to the problem! GR8677 #76 Advanced Topics}Solid State Physics Elimination time: (A) Electrons have less degrees of freedom than free atoms, since electrons are bound to potential wells. (B) Why not? (C) The electrons do indeed form a degenerate Fermi gas, since the ratio of the fraction of electrons in the ground-state is given by , where K for most metals. (D) The electrons in metal travel at a drift velocity of about . Not quite fast enough to be relativistic. (E) Electron interaction with phonons has nothing to do with their mean kinetic energy. Click here to jump to the problem!