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Quantum Theory of Conductivity: Bose-Einstein Statistics and Liquid Helium

This lecture discusses the quantum theory of conductivity, focusing on Bose-Einstein statistics and its application to liquid helium. Topics include Bose-Einstein condensation, lasers, and superconductivity.

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Quantum Theory of Conductivity: Bose-Einstein Statistics and Liquid Helium

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  1. PHYS 3313 – Section 001Lecture #20 Monday, Nov. 19, 2012 Dr. Jaehoon Yu • Quantum Theory of Conductivity • Bose-Einstein Statistics • Liquid Helium • Bose-Einstein Condenstation • Laser • Superconductivity PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  2. Announcements • Research team member list have been updated on the web! • Remember the deadline for your research paper is Monday, Nov. 26!! • Reminder: homework #6 • CH6 end of chapter problems: 34, 39, 46, 62 and 65 • Due on Monday, Nov. 12, in class • Reading assignments • CH7.6 and the entire CH8 • Colloquium today • At 4pm, Wednesday, Nov. 7, in SH101 • Dr. Nick White of Goddard Space Center, NASA PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  3. Classical Theory of Electrical Conduction • Paul Drude (1900) showed that the current in a conductor should be linearly proportional to the applied electric field that is consistent with Ohm’s law. • Prediction of the electrical conductivity • Mean free path is • True electrical conductivity: • According to the Drude model, the conductivity should be proportional to T−1/2. • But for most conductors is very nearly proportional to T−1 • The heat capacity of the electron gas is R. • This is not consistent with experimental results. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  4. Free Electron Number Density PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu Table 9-3 p319

  5. Quantum Theory of Electrical Conduction • Arnold Sommerfield used correct distribution n(E) at room temperature and found a value for α of π2 / 4. • With the value TF = 80,000 K for copper, we obtain cV≈ 0.02R, which is consistent with the experimental value! Quantum theory has proved to be a success. • Replace mean speed in the previous page by Fermi speeduF defined from . • Conducting electrons are loosely bound to their atoms • these electrons must be at the high energy level • at room temperature the highest energy level is close to the Fermi energy • We should use PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  6. Fermi energies, temperatures and velocities PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu Table 9-4 p319

  7. Quantum Theory of Electrical Conduction Drude thought that the mean free path could be no more than several tenths of a nanometer, but it was longer than his estimation. Einstein calculated the value of ℓ to be on the order of 40 nm in copper at room temperature. The conductivity is Sequence of proportions PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  8. Bose-Einstein Statistics Blackbody Radiation • Intensity of the emitted radiation is • Use the Bose-Einstein distribution because photons are bosons with spin 1. • For a free particle in terms of momentum: • The energy of a photon is pc, so PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  9. Bose-Einstein Statistics The number of allowed energy states within “radius”r is Where 1/8 comes from the restriction to positive values of ni and 2 comes from the fact that there are two possible photon polarizations. Energy is proportional to r, The density of states g(E) is The Bose-Einstein factor: PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  10. Bose-Einstein Statistics • Convert from a number distribution to an energy density distribution u(E). • Multiply by a factor E/L3 • For all photons in the range E to E + dE • Using E = hc/λand |dE| = (hc/λ2) dλ • In the SI system, multiplying by c/4 is required. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  11. Liquid Helium • Has the lowest boiling point of any element (4.2 K at 1 atmosphere pressure) and has no solid phase at normal pressure The density of liquid helium as a function of temperature: PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  12. Liquid Helium The specific heat of liquid helium as a function of temperature: The temperature at about 2.17 K is referred to as the critical temperature (Tc), transition temperature, or lambda point. As the temperature is reduced from 4.2 K toward the lambda point, the liquid boils vigorously. At 2.17 K the boiling suddenly stops. What happens at 2.17 K is a transition from the normal phase to the superfluid phase. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  13. Liquid Helium The rate of flow increases dramatically as the temperature is reduced because the superfluid has a low viscosity. Creeping film – formed when the viscosity is very low PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  14. Liquid Helium • Fritz London claimed (1938) that liquidhelium below the lambda point is part superfluid and part normal. • As the temperature approaches absolute zero, the superfluid approaches 100% superfluid. • The fraction of helium atoms in the superfluid state: • Superfluid liquid helium is referred to as a Bose-Einstein condensation. • not subject to the Pauli exclusion principle • all particles are in the same quantum state PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  15. Liquid Helium Such a condensation process is not possible with fermions because fermions must “stack up” into their energy states, no more than two per energy state. 4He isotope is a fermion and superfluid mechanism is radically different than the Bose-Einstein condensation. Use the fermions’ density of states function and substituting for the constant EF yields Bosons do not obey the Pauli principle, therefore the density of states should be less by a factor of 2. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  16. Liquid Helium • m is the mass of a helium atom. • The number distribution n(E) is now • In a collection of N helium atoms the normalization condition is • Substituting u = E / kT, PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  17. Liquid Helium Use minimum value of BBE = 1; this result corresponds to the maximum value of N. Rearrange this, The result is T≥ 3.06 K. The value 3.06 K is an estimate of Tc. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  18. Bose-Einstein Condensation in Gases • By the strong Coulomb interactions among gas particles it was difficult to obtain the low temperatures and high densities needed to produce the condensate. Finally success was achieved in 1995. • First, they used laser cooling to cool their gas of 87Rb atoms to about 1 mK. Then they used a magnetic trap to cool the gas to about 20 nK. In their magnetic trap they drove away atoms with higher speeds and further from the center. What remained was an extremely cold, dense cloud at about 170 nK. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  19. Stimulated Emission and Lasers • Spontaneous emission: A molecule in an excited state will decay to a lower energy state and emit a photon, without any stimulus from the outside. • The best we can do is calculate the probability that a spontaneous transition will occur. • If a spectral line has a width ΔE, then a lower-bound estimate of the lifetime is Δt = ħ / (2 ΔE). PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  20. Stimulated Emission and Lasers Stimulated emission: A photon incident upon a molecule in an excited state causes the unstable system to decay to a lower state. The photon emitted tends to have the same phase and direction as the stimulated radiation. If the incoming photon has the same energy as the emitted photon: The result is two photons of the same wavelength and phase traveling in the same direction. Because the incoming photon just triggers emission of the second photon. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  21. Stimulated Emission and Lasers Einstein’s analysis: Consider transitions between two molecular states with energies E1 and E2 (where E1 < E2). Eph is an energy of either emission or absorption. f is a frequency where Eph = hf = E2−E1. If stimulated emission occurs: The number of molecules in the higher state (N2) The energy density of the incoming radiation (u(f)) the rate at which stimulated transitions from E2 to E1 is B21N2u(f) (where B21 is a proportional constant) The probability that a molecule at E1 will absorb a photon is B12N1u(f) The rate of spontaneous emission will occur is AN2 (where A is a constant) PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  22. Stimulated Emission and Lasers helium-neon laser Laserstands for “light amplification by the stimulated emission of radiation” Masers: Microwaves are used instead of visible light. The first working laser by Theodore H. Maiman in 1960 PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  23. Stimulated Emission and Lasers The body of the laser is a closed tube, filled with about a 9/1 ratio of helium and neon. Photons bouncing back and forth between two mirrors are used to stimulate the transitions in neon. Photons produced by stimulated emission will be coherent, and the photons that escape through the silvered mirror will be a coherent beam. How are atoms put into the excited state? We cannot rely on the photons in the tube; if we did: Any photon produced by stimulated emission would have to be “used up” to excite another atom. There may be nothing to prevent spontaneous emission from atoms in the excited state.  The beam would not be coherent. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  24. Stimulated Emission and Lasers Use a multilevel atomic system to see those problems. Three-level system Atoms in the ground state are pumped to a higher state by some external energy. The atom decays quickly to E2.The transition from E2 to E1 is forbidden by a Δℓ = ±1 selection rule.E2 is said to be metastable. Population inversion: more atoms are in the metastable than in the ground state PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  25. Stimulated Emission and Lasers After an atom has been returned to the ground state from E2, we want the external power supply to return it immediately to E3, but it may take some time for this to happen. A photon with energy E2−E1 can be absorbed. result would be a much weaker beam This is undesirable because the absorbed photon is unavailable for stimulating another transition. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  26. Stimulated Emission and Lasers Four-level system Atoms are pumped from the ground state to E4. They decay quickly to the metastable state E3. The stimulated emission takes atoms from E3 to E2. The spontaneous transition from E2 to E1 is not forbidden, so E2 will not exist long enough for a photon to be kicked from E2 to E3.  Lasing process can proceed efficiently. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  27. Stimulated Emission and Lasers The red helium-neon laser uses transitions between energy levels in both helium and neon. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  28. Stimulated Emission and Lasers Tunable laser: The emitted radiation wavelength can be adjusted as wide as 200 nm. Semi conductor lasers are replacing dye lasers. Free-electron laser: PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  29. Stimulated Emission and Lasers This laser relies on charged particles. A series of magnets called wigglers is used to accelerate a beam of electrons. Free electrons are not tied to atoms; they aren’t dependent upon atomic energy levels and can be tuned to wavelengths well into the UV part of the spectrum. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  30. Scientific Applications of Lasers • An extremely coherent and nondivergent beam is used in making precise determination of large and small distances. The speed of light in a vacuum is defined. c = 299,792,458 m/s. • Pulsed lasers are used in thin-film deposition to study the electronic properties of different materials. • The use of lasers in fusion research • Inertial confinement: A pellet of deuterium and tritium would be induced into fusion by an intense burst of laser light coming simultaneously from many directions. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  31. Holography • Consider laser light emitted by a reference source R. • The light through a combination of mirrors and lenses can be made to strike both a photographic plate and an object O. • The laser light is coherent; the image on the film will be an interference pattern. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  32. Holography After exposure this interference pattern is a hologram, and when the hologram is illuminated from the other side, a real image of O is formed. If the lenses and mirrors are properly situated, light from virtually every part of the object will strike every part of the film.  Each portion of the film contains enough information to reproduce the whole object! PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  33. Holography Transmission hologram: The reference beam is on the same side of the film as the object and the illuminating beam is on the opposite side. Reflection hologram: Reverse the positions of the reference and illuminating beam.The result will be a white light hologram in which the different colors contained in white light provide the colors seen in the image. Interferometry: Two holograms of the same object produced at different times can be used to detect motion or growth that could not otherwise be seen. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  34. Quantum Entanglement, Teleportation, and Information • Schrödinger used the term “quantum entanglement” to describe a strange correlation between two quantum systems. He considered entanglement for quantum states acting across large distances, which Einstein referred to as “spooky action at a distance.” Quantum teleportation: No information can be transmitted through only quantum entanglement, but transmitting information using entangled systems in conjunction with classical information is possible. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  35. Quantum Entanglement, Teleportation, and Information Alice, who does not know the property of the photon, is spatiallyseparated from Bob and tries to transfer information about photons. A beam splitter can be used to produce two additional photons that can be used to trigger a detector. Alice can manipulate her quantum system and send that information over a classical information channel to Bob. Bob then arranges his part of the quantum system to detect information. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  36. Other Laser Applications • Used in surgery to make precise incisions • Ex: eye operations • We see in everyday life such as the scanning devices used by supermarkets and other retailers • Ex. Bar code of packaged product CD and DVD players • Laser light is directed toward disk tracks that contain encoded information. The reflected light is then sampled and turned into electronic signals that produce a digital output. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  37. Diamagnetism • Consider an electron orbiting counterclockwise in a circular orbit and a magnetic field is applied gradually out of the page. • From Faraday’s law, the changing magnetic flux results in an induced electric field that is tangent to the electron’s orbit. • The induced electric field strength is • Setting torque equal to the rate of change in angular momentum • The magnetization opposes the applied field. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  38. Paramagnetism • There exist unpaired magnetic moments that can be aligned by an external field. • The paramagnetic susceptibility χis strongly temperature dependent. • Consider a collection of N unpaired magnetic moments per unit volume. N+ moments aligned parallel N−moments aligned antiparallel to the applied field. • By Maxwell-Boltzmann statistics, • where A is a normalization constant and β ≡ (kT)−1 PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  39. Paramagnetism Sample magnetization curves Curie law breaks down at higher values of B, when the magnetization reaches a “saturation point” PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  40. Ferromagnetism • Fe, Ni, Co, Gd, and Dy and a number of compounds are ferromagnetic, including some that do not contain any of these ferromagnetic elements. • It is necessary to have not only unpaired spins, but also sufficient interaction between the magnetic moments. • Sufficient thermal agitation can completely disrupt the magnetic order, to the extent that above the Curie temperatureTC a ferromagnet changes to a paramagnet. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  41. Antiferromagnetism and Ferrimagnetism • Antiferromagnetic: Adjacent magnet moments have opposing directions. • The net effect is zero magnetization below the Neel temperatureTN. • Above TN, antiferromagnetic → paramagnetic • Ferrimagnetic: A similar antiparallel alignment occurs, except that there are two different kinds of positive ions present. • The antiparallel moments leave a small net magnetization. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  42. Superconductivity • Superconductivity is characterized by the absence of electrical resistance and the expulsion of magnetic flux from the superconductor. It is characterized by two macroscopic features: • zero resistivity • Onnesachieved temperatures approaching 1 K with liquid helium. • In a superconductor the resistivity drops abruptly to zero at critical (or transition) temperatureTc. • Superconducting behavior tends to be similar within a given column of the periodic table. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  43. Superconductivity Resistivity of a superconductor Meissner effect: The complete expulsion of magnetic flux from within a superconductor. It is necessary for the superconductor to generate screening currents to expel the magnetic flux one tries to impose upon it. One can view the superconductor as a perfect magnet, with χ = −1. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  44. Superconductivity The Meissner effect works only to the point where the critical fieldBc is exceeded, and the superconductivity is lost until the magnetic field is reduced to below Bc. The critical field varies with temperature. To use a superconducting wire to carry current without resistance, there will be a limit (critical current) to the current that can be used. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  45. Type I and Type II Superconductors There is a lower critical field, Bc1 and an upper critical field, Bc2. Type II: Below Bc1 and above Bc2. Behave in the same manner Type I: Below and above Bc. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  46. Type I and Type II Superconductors Between Bc1 and Bc2 (vortex state), there is a partial penetration of magnetic flux although the zero resistivity is not lost. Lenz’s law: A phenomenon from classical physics A changing magnetic flux generates a current in a conductor in such way that the current produced will oppose the change in the original magnetic flux. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  47. Superconductivity Isotope effect: M is the mass of the particular superconducting isotope. Tc is a bit higher for lighter isotopes. It indicates that the lattice ions are important in the superconducting state. BCS theory (electron-phonon interaction): Electrons form Cooper pairs, which propagate throughout the lattice. Propagation is without resistance because the electrons move in resonance with the lattice vibrations (phonons). PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  48. Superconductivity How is it possible for two electrons to form a coherent pair? Consider the crude model. Each of the two electrons experiences a net attraction toward the nearest positive ion. Relatively stable electron pairs can be formed. The two fermions combine to form a boson. Then the collection of these bosons condense to form the superconducting state. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  49. Superconductivity • Neglect for a moment the second electron in the pair. The propagation wave that is created by the Coulomb attraction between the electron and ions is associated with phonon transmission, and the electron-phonon resonance allows the electron to move without resistance. • The complete BCS theory predicts other observed phenomena. • An isotope effect with an exponent very close to 0.5. • It gives a critical field. PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

  50. Superconductivity An energy gap Eg between the ground state and first excited state. This means that Eg is the energy needed to break a Cooper pair apart Eg(0) ≈ 3.54kTc at T = 0. Quantum fluxoid: Magnetic flux through a superconducting ring PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

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