10.1Molecular Bonding and Spectra

1
CHAPTER 10Molecules and Solids

• 10.1 Molecular Bonding and Spectra
• 10.2 Stimulated Emission and Lasers
• 10.3 Structural Properties of Solids
• 10.4 Thermal and Magnetic Properties of Solids
• 10.5 Superconductivity
• 10.6 Applications of Superconductivity

The secret of magnetism, now explain that to me!
There is no greater secret, except love and
hate. - Johann Wolfgang von Goethe
2
10.1 Molecular Bonding and Spectra

• The Coulomb force is the only one to bind atoms.
• The combination of attractive and repulsive
  forces creates a stable molecular structure.
• Force is related to potential energy F -dV /
  dr, where r is the distance separation.
• it is useful to look at molecular binding using
  potential energy V.
• Negative slope (dV / dr lt 0) with repulsive
  force.
• Positive slope (dV / dr gt 0) with attractive
  force.

3
Molecular Bonding and Spectra

• An approximation of the force felt by one atom in
  the vicinity of another atom is
• where A and B are positive constants.
• Because of the complicated shielding effects of
  the various electron shells, n and m are not
  equal to 1.

• Eq. 10.1 provides a stable equilibrium for total
  energy E lt 0. The shape of the curve depends on
  the parameters A, B, n, and m. Also n gt m.

4
Molecular Bonding and Spectra

• Vibrations are excited thermally, so the exact
  level of E depends on temperature.
• A pair of atoms is joined.
• One would have to supply energy to raise the
  total energy of the system to zero in order to
  separate the molecule into two neutral atoms.
• The corresponding value of r of a minimum value
  is an equilibrium separation. The amount of
  energy to separate the two atoms completely is
  the binding energy which is roughly equal to the
  depth of the potential well.

5
Molecular Bonds

• Ionic bonds
• The simplest bonding mechanisms.
• Ex Sodium (1s22s22p63s1) readily gives up its 3s
  electron to become Na, while chlorine
  (1s22s22p63s23p5) readily gains an electron to
  become Cl-. That forms the NaCl molecule.
• Covalent bonds
• The atoms are not as easily ionized.
• Ex Diatomic molecules formed by the combination
  of two identical atoms tend to be covalent.
• Larger molecules are formed with covalent bonds.

6
Molecular Bonds

• Van der Waals bond
• Weak bond found mostly in liquids and solids at
  low temperature.
• Ex in graphite, the van der Waals bond holds
  together adjacent sheets of carbon atoms. As a
  result, one layer of atoms slides over the next
  layer with little friction. The graphite in a
  pencil slides easily over paper.
• Hydrogen bond
• Holds many organic molecules together.
• Metallic bond
• Free valence electrons may be shared by a number
  of atoms.

7
Rotational States

• Molecular spectroscopy
• We can learn about molecules by studying how
  molecules absorb, emit, and scatter
  electromagnetic radiation.
• From the equipartition theorem, the N2 molecule
  may be thought of as two N atoms held together
  with a massless, rigid rod (rigid rotator model).
• In a purely rotational system, the kinetic energy
  is expressed in terms of the angular momentum L
  and rotational inertia I.

8
Rotational States

• L is quantized.
• The energy levels are
• Erot varies only as a function of the quantum
  number l.

9
Vibrational States

• There is the possibility that a vibrational
  energy mode will be excited.
• No thermal excitation of this mode in a diatomic
  gas at ordinary temperature.
• It is possible to stimulate vibrations in
  molecules using electromagnetic radiation.
• Assume that the two atoms are point masses
  connected by a massless spring with simple
  harmonic motion.

10
Vibrational States

• The energy levels are those of a
  quantum-mechanical oscillator.
• The frequency of a two-particle oscillator is
• Where the reduced mass is µ m1m2 / (m1 m2)
  and the spring constant is ?.
• If it is a purely ionic bond, we can compute ? by
  assuming that the force holding the masses
  together is Coulomb.
• and

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Vibration and Rotation Combined

• It is possible to excite the rotational and
  vibrational modes simultaneously.
• Total energy of simple vibration-rotation system
• Vibrational energies are spaced at regular
  intervals.
• emission features due to vibrational
  transitions appear at regular intervals.
• Transition from l 1 to l
• Photon will have an energy

12
Vibration and Rotation Combined

• An emission-spectrum spacing that varies with l.
• the higher the starting energy level, the
  greater the photon energy.
• Vibrational energies are greater than rotational
  energies. This energy difference results in the
  band spectrum.

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Vibration and Rotation Combined

• The positions and intensities of the observed
  bands are ruled by quantum mechanics. Note two
  features in particular
• 1) The relative intensities of the bands are due
  to different transition probabilities.
• - The probabilities of transitions from an
  initial state to final state are not necessarily
  the same.
• 2) Some transitions are forbidden by the
  selection rule that requires ?l 1.
• Absorption spectra
• Within ?l 1 rotational state changes,
  molecules can absorb photons and make transitions
  to a higher vibrational state when
  electromagnetic radiation is incident upon a
  collection of a particular kind of molecule.

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Vibration and Rotation Combined

• ?E increases linearly with l as in Eq. (10.8).

15
Vibration and Rotation Combined

• In the absorption spectrum of HCl, the spacing
  between the peaks can be used to compute the
  rotational inertia I. The missing peak in the
  center corresponds to the forbidden ?l 0
  transition.
• The central frequency

16
Vibration and Rotation Combined

• Fourier transform infrared (FTIR) spectroscopy
• Data reduction methods for the sole purpose of
  studying molecular spectra.
• A spectrum can be decomposed into an infinite
  series of sine and cosine functions.
• Random and instrumental noise can be reduced in
  order to produce a clean spectrum.
• Raman scattering
• If a photon of energy greater than ?E is absorbed
  by a molecule, a scattered photon of lower energy
  may be released.
• The angular momentum selection rule becomes ?l
  2.

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Vibration and Rotation Combined

• A transition from l to l 2.
• Let hf be the Raman-scattered energy of an
  incoming photon and hf is the energy of the
  scattered photon. The frequency of the scattered
  photon can be found in terms of the relevant
  rotational variables
• Raman spectroscopy is used to study the
  vibrational properties of liquids and solids.

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10.2 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 ?E, then an upper
  bound estimate of the lifetime is ?t h / (2 ?E).

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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.

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Stimulated Emission and Lasers

• Einsteins analysis
• Consider transitions between two molecular states
  with energies E1 and E2 (where E1 lt 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 Lasers

• Once the system has reached equilibrium with the
  incoming radiation, the total number of downward
  and upward transitions must be equal.
• In the thermal equilibrium each of Ni are
  proportional to their Boltzmann factor .
• In the classical time limit T ? 8. Then
  and u(f) becomes very large.
• the probability of stimulated emission is
  approximately equal to the probability of
  absorption.

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Stimulated Emission and Lasers

• Solve for u(f),
• or, use Eq. (10.12),
• This closely resembles the Planck radiation law,
  but Planck law is expressed in terms of
  frequency.
• Eqs.(10.13) and (10.14) are required
• The probability of spontaneous emission (A) is
  proportional to the probability of stimulated
  emission (B) in equilibrium.

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Stimulated Emission and Lasers

• Laser
• An acronym 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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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.

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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 ?l 1 selection
  rule.E2 is said to be metastable.
• 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.
• It is undesirable.

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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.

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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 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

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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 arent
  dependent upon atomic energy levels and can be
  tuned to wavelengths well into the UV part of the
  spectrum.

31
Scientific Applications of Lasers

• 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.

32
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!

34
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.

35
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 spacially
• separated 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.
• Ex. The polarization status, about the unknown
  quantum state at his detector.

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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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10.3 Structural Properties of Solids

• Condensed matter physics
• The study of the electronic properties of solids.
• Crystal structure
• The atoms are arranged in extremely regular,
  periodic patterns.
• Max von Laue proved the existence of crystal
  structures in solids in 1912, using x-ray
  diffraction.
• The set of points in space occupied by atomic
  centers is called a lattice.

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Structural Properties of Solids

• Most solids are in a polycrystalline form.
• They are made up of many smaller crystals.
• Solids lacking any significant lattice structure
  are called amorphous and are referred to as
  glasses.
• Why do solids form as they do?
• When the material changes from the liquid to the
  solid state, the atoms can each find a place that
  creates the minimum energy configuration.

Let us use the sodium chloride crystal. The
spatial symmetry results because there is no
preferred direction for bonding. The fact that
different atoms have different symmetries
suggests why crystal lattices take so many
different forms.
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Structural Properties of Solids

• Each ion must experience a net attractive
  potential energy.
• where r is the nearest-neighbor distance.
• a is the Madelung constant and it depends on the
  type of crystal lattice.
• In the NaCl crystal, each ion has 6 nearest
  neighbors.
• There is a repulsive potential due to the Pauli
  exclusion principle.
• The value e-r /? diminishes rapidly for r gt ?.
• ? is roughly regarded as the range of the
  repulsive force.

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Structural Properties of Solids

• The net potential energy is
• At the equilibrium position (r r0), F -dV /
  dr 0.
• therefore,
• and
• The ratio ? / r0 is much less than 1 and must be
  less than 1.

42
10.4 Thermal and Magnetic Properties of Solids

• Thermal expansion
• Tendency of a solid to expand as its temperature
  increases.
• Let x r - r0 to consider small oscillations of
  an ion about x 0. The potential energy close to
  x 0 is
• where the x3 term is responsible for the
  anharmonicity of the oscillation.

43
Thermal Expansion

• The mean displacement using the Maxwell-Boltzmann
  distribution function
• where ß (kT)-1 and use a Taylor expansion for
  x3 term.
• Only the even (x4) term survived from -8 to 8.
• We are interested only in the first-order
  dependence on T,

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Thermal Expansion

• Combining Eq. (10.24) and (10.25),
• Thermal expansion is nearly linear with
  temperature in the classical limit. Eq. (10.26)
  vanishes as T ? 0.

45
Thermal Conductivity

• Thermal conductivity
• A measure of how well they transmit thermal
  energy. Defining thermal conductivity is in terms
  of the flow of heat along a solid rod of uniform
  cross-sectional area A.
• The flow of heat per unit time along the rod is
  proportional to A and to the temperature gradient
  dT / dx.
• The thermal conductivity K is the proportionality
  constant.

46
Thermal Conductivity

• In classical theory the thermal conductivity of
  an ideal free electron gas is
• Classically , so
  .
• Compare the thermal and electrical
  conductivities
• From classical thermodynamics the mean speed is
• Therefore
• The constant ratio is

47
Thermal Conductivity

• Eq. (10.32) is called the Wiedemann-Franz law,
  and the constant L is the Lorenz number.
• Experiments show that K / st has numerical value
  about 2.5 times higher than predicted by Eq.
  (10.32).
• We should replace Fermi speed uF
• quantum-mechanical result
• Rewrite Eq. (10.28)
• where R NAk and EF ½ muF2.
  

48
Thermal Conductivity

• Now,
• ?Agrees with experimental results

------ Quantum Lorenz number
49
Magnetic Properties

• Solids are characterized by their intrinsic
  magnetic moments and their responses to applied
  magnetic fields.
• Ferromagnets
• Paramagnets
• Diamagnets
• Magnetization M
• The net magnetic moment per unit volume.
• Magnetic susceptibility ?
• Positive for paramagnets
• Negative for diamagnets

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Diamagnetism

• The magnetization opposes the applied field.

• Consider an electron orbiting counterclockwise in
  a circular orbit and a magnetic field is applied
  gradually out of the page.
• From Faradays law, the changing magnetic flux
  results in an induced electric field that is
  tangent to the electrons orbit.
• The induced electric field strength is
• Setting torque equal to the rate of change in
  angular momentum

51
Diamagnetism

• For a magnetic field from 0 to B, directed out of
  the page, the angular moment changes by an amount
• This results in a magnetic moment changed by
• which has a magnitude
• The change in magnetic moment is opposite to the
  applied field.

52
Paramagnetism

• There exist unpaired magnetic moments that can be
  aligned by an external field.
• The paramagnetic susceptibility ? is 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 ß
  (kT)-1.

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Paramagnetism

• Net magnetic moment is
• Eliminate A by considering the mean magnetic
  moment per atom
• It is only valid for T gtgt 0.
• In the classical limit
• It simply stated as ? C / T, where C µ0Nµ2 / k

--------- Curie law
----Curie constant
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Paramagnetism

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

55
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.

56
Antiferromagnetism and Ferrimagnetism

• 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.

57
10.5 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.

58
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 ?
  -1.

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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.

60
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.
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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.
• Lenzs 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.

62
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).

63
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.

64
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.

65
Superconductivity

• Quantum fluxoid
• Magnetic flux through a superconducting ring.

1. 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.

66
The Search for a Higher Tc

• Keeping materials at extremely low temperatures
  is very expensive and requires cumbersome
  insulation techniques.
• History of transition temperature

67
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

68
The Search for a Higher Tc

• Higher values of n correspond to more stacked
  layers of copper and oxygen.
• thallium-based superconductor

69
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.

70
10.6 Applications of Superconductivity

• Josephson 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.

71
Applications of Superconductivity

• Maglev
• Magnetic levitation of trains.

• 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.

72
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 bodys soft
  tissues, allowing them to detect tumors and other
  disorders of the brain, muscles, organs, and
  connective tissues.