10.3 Structural Properties of 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

CHAPTER 10

Molecules, Lasers and Solids

The secret of magnetism, now explain that to me! There is no greater secret, except love and hate.

- Johann Wolfgang von Goethe

10.3: Structural Properties of Solids

Condensed matter physics:

 Studies of solids and liquids.

 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.

Crystal lattices found in solids.

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 a particular crystal lattice?

 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’s use the sodium chloride crystal as an example.

The spatial symmetry (cubic) results because there is no preferred direction for bonding.

NaCl crystal

 The net potential energy 

 At the equilibrium position 

 Typically the repulsive is very short range

 The ratio ρ / r₀ is much less than 1.

NaCl crystal

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

 the x³ term is responsible for the anharmonicity of the oscillation (that is, the deviation from the standard harmonic oscillator)

At T = 0, the ion is nearly “frozen solid” at r = r₀ For T > 0, oscillating between r₁ and r₂

Thermal Expansion: a quantitative model

 Mean displacement from the Maxwell-Boltzmann distribution: e^-V

(Nominator) By a Taylor expansion for x³ term

 Only the even (x⁴) term survives integration from −∞ to ∞

(Denominator)



where β = (kT)⁻¹ and

 thermal expansion is nearly linear with temperature

Thermal Conductivity

 A measure of how well they transmit thermal energy.



The flow of heat per unit time along the rod is proportional to

 K is the thermal conductivity

 The ratio of thermal conductivity K and electrical conductivity :

: Classical Lorentz number :Quantum-mechanically

correct Lorenz number

Magnetic Properties



Solids are characterized by their intrinsic magnetic moments and their responses to applied magnetic fields.

Ferromagnet

 a net magnetic moment without an applied magnetic field

Paramagnet

 A net magnetic moment only in the presence of an applied field.

Diamagnet

 a (usually weak) tendency to have an induced magnetic moment opposite to the applied field.

Magnetization (M)



Magnetic susceptibility (



:

Diamagnetism



The magnetization opposes the applied field.

 Normal material characteristic responding to the applied field by Faraday’s law.

 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 produce a torque:

(  ^{< 0 )}

 To a direction out of the page For a magnetic field that increases from 0 to B, directed out of the page,

 The change in magnetic moment is opposite to the applied field,

 Characteristic of diamagnetism.

Paramagnetism

 There exist unpaired magnetic moments (for rare-earth elements and for many transition metals) that can be aligned by an external field.

 The paramagnetic susceptibility  is strongly temperature dependent.

(  > 0 )

(Curie constant) Curie’s Law:

N : unpaired magnetic moments per unit volume

 Nearly linear over a wide range of magnetic fields.

 Curie law breaks down at higher B

Ferromagnetism

 Only five (Fe, Ni, Co, Gd, and Dy) are ferromagnetic.

 A number of compounds (such as Nd₂Fe₁₄B) are ferromagnetic, including some that do not contain any of 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, above Curie temperature T_C , ferromagnet changes to paramagnet.

 A net magnetic moment exists without an applied magnetic field

Antiferromagnetism and Ferrimagnetism

Antiferromagnetic:

 Adjacent magnet moments have opposing directions.

 The net effect is zero magnetization below the Neel temperature T_N.

 Above T_N, 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.

10.5: Superconductivity

Superconductivity is characterized by

(1) Absence of electrical resistance (Zero resistivity)

(2) Expulsion of magnetic flux (Meissner effect)

10.5: Superconductivity

Superconductivity is characterized by

(1) Absence of electrical resistance (Zero resistivity) (2) Expulsion of magnetic flux (Meissner effect)

(1) Zero resistivity

 In 1911, Heike Kamerlingh Onnes achieved

temperatures approaching 1 K with liquid helium.

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

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

 The resistivity is not merely very low; it really is zero.

Superconductivity

(2) Meissner Effect:

Discovered by W. Meissner and R. Ochsenfeld in 1933

The complete expulsion of magnetic flux from within a superconductor.

 It is necessary for the superconductor to generate screening currents.

 One can view the superconductor as a perfect diamagnet, with  = −1.

T > Tc T < Tc

Meissner Effect

 The Meissner effect works below the critical field B_c

 Superconductivity is lost until B is reduced to below B_c.

 The critical field varies with temperature.

 Just below Tc the critical field is low;

that is, it takes very little magnetic field to eliminate the superconductivity.

 Current-carrying wires generate magnetic fields, both inside and outside the wire. Therefore,

 To use a superconducting wire to carry current without resistance, there will be a limit (critical current) to the current that can be used.

 These effects severely limited the applications of superconductors

Type I and Type II Superconductors

Type I  Superconducting in pure metals: Hg, Al, and many others

 Only work below B_c.

Type II  Superconducting in alloys: YBa₂Cu₃O₇ and many others

 There are two critical fields: B_c1 and B_c2.

 Below B_c1 and above B_c2, type II behaves as type I.

 Between B_c1 and B_c2 (known as the vortex state), there is a partial penetration of magnetic flux,

but, the zero resistivity property is generally not lost.

 The good news is that B_c2 can sometimes be very high.

 The bad news is that B_c1 is seldom more than hundredth tesla.

Type I Type II

Type I Superconductors

Type I  Superconducting in pure metals: Hg, Al, and many others

 Only work below B_c.

Superconductor

Type II Superconductors

Type II  There are two critical fields: B_c1 and B_c2.

 Below B_c1 : Superconductor state with no B flux.

 Between B_c1 and B_c2 : Superconductor state with some B flux

Superconductor

Ordinary conductor

Ceramic Alloys

BCS theory and Cooper pairs

BCS theory



(Nobel Prize in Physics, 1972)

 Two principal features of the BCS theory:

(1) Electrons form pair (Cooper pairs), which propagate throughout the lattice.

(2) Such propagation is without resistance because the electrons move in resonance with the lattice vibrations (phonons).

 Hence, BCS theory is known as the electron-phonon interaction

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.



With a net spin of zero, the two fermions combine to form a boson.



Then the collection of these bosons is analogous to a Bose- Einstein condensation  Superconducting state.

BCS theory: Electron-Phonon interaction

BCS theory: Electron-Phonon interaction

How can the zero resistivity property be explained?

 Even at low temperatures there is some ionic motion.

 That is why one would expect some resistance.

 But if we neglect for a moment the second electron in the pair,

 we can understand how a single electron can travel without scattering.

 The Coulomb attraction between the electron and ions causes a deformation of the lattice, which propagates along with the electron.

 This propagating wave is associated with phonon transmission.

 The electron-phonon resonance allows the electron to move without resistance.

 The complete BCS theory contains sophisticated mathematics and is based solidly on the foundations of the quantum theory.

 The BCS theory predicts several other observed phenomena.

1) An isotope effect with an exponent very close to 0.5.

2) A critical field varies with temperature as

BCS theory: Electron-Phonon interaction

(M - the atomic mass, T_c – critical T)

3) The metals with higher resistivity at room T be better superconductors.

4) The magnetic flux through a superconducting ring is quantized.

 It is the basis for the Josephson junction (10.6)

The Search for a Higher T _c



Keeping materials at extremely low temperatures is very expensive and requires cumbersome insulation techniques.

 In 1987, a group at the University of Houston led by Paul Chu,

Tc of about 93 K for YBa₂Cu₃O₇ (Type II)  Tc > 77 K (Liquid Nitrogen)

The Search for a Higher T _c

138 K

Special Topic: Low-Temperature Methods

 Very cold (cryogenic) liquids are used as a low-temperature bath.

Double-dewar apparatus (Low-T pioneer James Dewar, 1892)

 Pumping the vapor above the helium bath

 Remaining liquid cool by adiabatic expansion.

 T limit of about 1 K.

In 1926, Giauque and Debye (working independently) developed the idea of adiabatic demagnetization

Entropy

Temperature adiabatic demagnetization

isothermal magnetization

(First step) isothermal magnetization A paramagnetic salt is put into a vessel containing helium gas  magnetization occurs at a fixed T (isothermal)  Salt’s entropy decreases (more ordered)

(Second step) adiabatic demagnetization The helium gas in contact with the salt is

pumped away  No further heat transfer takes place  The demagnetization of the salt takes place without heat transfer (adiabatically)

   T decreases!

Lowest temperatures on record, 500 pK (5 x 10^-10 K) reported by a MIT research group in 2003

10.6: Applications of Superconductivity

Josephson junctions (1962, Brian Josephson)



Superconductor / Insulator / Superconductor junction



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.



used in SQUIDs (

for measuring very small amounts of magnetic flux.

(for example, Brain signals)

SQUIDs (

Maglev: Magnetic levitation of trains



Electrodynamic (EDS) system,

 Magnets on the guideway repel the car to lift it.



Electromagnetic (EMS) system,

 Magnets on the train are attracted upward toward the guide way 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.

 MRI (magnetic resonance imaging)