phys 3313 – section 001 lecture # 20

Monday, Nov. 19, 2012 PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

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PHYS 3313 – Section 001Lecture #20

Monday, Nov. 19, 2012Dr. Jaehoon Yu

• Quantum Theory of Conductivity• Bose-Einstein Statistics• Liquid Helium• Bose-Einstein Condenstation• Laser• Superconductivity


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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

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.

Wednesday, Nov. 14, 2012

PHYS 3313-001, Fall 2012 Dr. Jaehoon Yu

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Table 9-3 p319

Free Electron Number Density



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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 speed uF 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



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Table 9-4 p319

Fermi energies, temperatures and velocities



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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



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Bose-Einstein StatisticsBlackbody 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



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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:



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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/λ and |dE| = (hc/λ2) dλ

• In the SI system, multiplying by c/4 is required.



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Liquid Helium• Has the lowest boiling point of any element (4.2 K at 1 atmosphere

pressure) and has no solid phase at normal pressureThe density of liquid helium as a function of temperature:



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Liquid HeliumThe 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.



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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



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Liquid Helium Fritz London claimed (1938) that liquid helium 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



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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.



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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,



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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.



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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.



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Stimulated Emission and Lasers


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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).

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 samewavelength and phase traveling in the same direction.Because the incoming photon just triggers emission of the second photon.



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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)



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Stimulated Emission and LasersLaser stands 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

helium-neon laser


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• 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:1) Any photon produced by stimulated emission would have to be

“used up” to excite another atom.2) There may be nothing to prevent spontaneous emission from atoms

in the excited state. The beam would not be coherent.



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Stimulated Emission and LasersUse a multilevel atomic system to see those problems.• Three-level system

1) Atoms in the ground state are pumped to a higher state by some external energy.

2) 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.

3) Population inversion: more atoms are in the metastable than in the ground state


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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.


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Stimulated Emission and Lasers• Four-level system

1) Atoms are pumped from the ground state to E4.

2) They decay quickly to the metastable state E3.

3) The stimulated emission takes atoms from E3 to E2.

4) 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.


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Stimulated Emission and Lasers• The red helium-neon laser uses transitions between energy

levels in both helium and neon.


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Stimulated Emission and LasersTunable laser:• The emitted radiation wavelength can be adjusted as wide

as 200 nm.• Semi conductor lasers are replacing dye lasers.Free-electron laser:


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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.


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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.


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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.


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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!


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HolographyTransmission 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.


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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.


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Quantum Entanglement, Teleportation, and Information

Alice, who does not know the property of the photon, is spatially separated from Bob and tries to transfer information about photons.

1) A beam splitter can be used to produce two additional photons that can be used to trigger a detector.

2) Alice can manipulate her quantum system and send that information over a classical information channel to Bob.

3) Bob then arranges his part of the quantum system to detect information.


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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.


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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

Diamagnetism• The magnetization opposes the applied field.


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• There exist unpaired magnetic moments that can be aligned by an external field.

• The paramagnetic susceptibility χ is strongly temperature dependent.

• Consider a collection of N unpaired magnetic moments per unit volume.N+ moments aligned parallelN− moments aligned antiparallel to the applied field.

• By Maxwell-Boltzmann statistics,

• where A is a normalization constant and β ≡ (kT)−1

Paramagnetism


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ParamagnetismSample magnetization curves

• Curie law breaks down at higher values of B, when the magnetization reaches a “saturation point”


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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 temperature TC a ferromagnet changes to a paramagnet.


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Antiferromagnetism and Ferrimagnetism


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• Antiferromagnetic: Adjacent magnet moments have opposing directions.– The net effect is zero magnetization below the

Neel temperature TN.– 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.

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

– Onnes achieved temperatures approaching 1 K with liquid helium.

– In a superconductor the resistivity drops abruptly to zero at critical (or transition) temperature Tc.

– Superconducting behavior tends to be similar within a given column of the periodic table.


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Superconductivity

2) 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.

Resistivity of a superconductor


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Superconductivity

• The Meissner effect works only to the point where the critical field Bc 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.


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Type I and Type II SuperconductorsThere is a lower critical field, Bc1 and an upper critical

field, Bc2.

Type II: Below Bc1 and above Bc2.

Type I: Below and above Bc.

Behave in the same manner


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• 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.

Type I and Type II Superconductors


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SuperconductivityIsotope 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):1) Electrons form Cooper pairs, which propagate

throughout the lattice.2) Propagation is without resistance because the electrons

move in resonance with the lattice vibrations (phonons).


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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.


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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.1) An isotope effect with an exponent very close to 0.5.2) It gives a critical field.


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SuperconductivityQuantum fluxoid: Magnetic flux through a superconducting ring

3) 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.


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The Search for a Higher Tc • Keeping materials at extremely low temperatures is very

expensive and requires cumbersome insulation techniques.History of transition temperature


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The Search for a Higher Tc• The copper oxide superconductors fall into a category of ceramics.• Most ceramic materials are not easy to mold into convenient shapes.• There is a regular variation of Tc with n.

Tc of thallium-copper oxide with n = 3


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The Search for a Higher Tc• Higher values of n correspond to more stacked layers of

copper and oxygen.thallium-based superconductor


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Superconducting Fullerenes• Another class of exotic superconductors is based on the

organic molecule C60.

• Although pure C60 is not superconducting, the addition of certain other elements can make it so.


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Applications of SuperconductivityJosephson junctions:• The superconductor / insulator / superconductor layer

constitutions.• In the absence of any applied magnetic or electric field, a DC

current will flow across the junction (DC Josephson effect).• Junction oscillates with frequency when a voltage is applied

(AC Josephson effect).

• They are used in devices known as SQUIDs. SQUIDs are useful in measuring very small amounts of magnetic flux.


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Applications of SuperconductivityMaglev: Magnetic levitation of trains


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In an electrodynamic (EDS) system, magnets on the guideway repel the car to lift it.

In an electromagnetic (EMS) system, magnets attached to the bottom of the car lie below the guideway and are attracted upward toward the guideway to lift the car.

Generation and Transmission of Electricity• Significant energy savings if the heavy iron cores used

today could be replaced by lighter superconducting magnets.

• Expensive transformers would no longer have to be used to step up voltage for transmission and down again for use.

• Energy loss rate for transformers is

• MRI obtains clear pictures of the body’s soft tissues, allowing them to detect tumors and other disorders of the brain, muscles, organs, and connective tissues.


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phys 3313 – section 001 lecture # 20

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