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 10Molecules, 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 SolidsCondensed 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 amorphousand 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 A and to the temperature gradient dT/dx.

K is the thermal conductivity The ratio of thermal conductivity K and electrical conductivity :

: Classical Lorentz number

:Quantum-mechanicallycorrect 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) Net magnetic moment per unit volume

Magnetic susceptibility ( : Positive for paramagnetsNegative for diamagnets

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 TC ,ferromagnet changes to paramagnet.

A net magnetic moment exists without an applied magnetic field

Antiferromagnetism and FerrimagnetismAntiferromagnetic: 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.

10.5: SuperconductivitySuperconductivity is characterized by

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

10.5: SuperconductivitySuperconductivity 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 Tc.

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 1933The 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 Bc

Superconductivity is lost until B is reduced to below Bc.

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

Only work below Bc.

Type II Superconducting in alloys: YBa2Cu3O7 and many others There are two critical fields: Bc1 and Bc2. Below Bc1 and above Bc2, type II behaves as type I. Between Bc1 and Bc2 (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 Bc2 can sometimes be very high. The bad news is that Bc1 is seldom more than hundredth tesla.

Type I Type II

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

Only work below Bc.

Superconductor

Type II SuperconductorsType II There are two critical fields: Bc1 and Bc2.

Below Bc1 : Superconductor state with no B flux. Between Bc1 and Bc2 : Superconductor state with some B flux

Superconductor

Ordinary conductor

Ceramic Alloys

BCS theory and Cooper pairsBCS theory John Bardeen, Leon Cooper, and Robert Schrieffer in 1957.(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, Tc – 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 Tc 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 Tc

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

Entr

opy

Temperature

adiabatic demagnetization

isothermalmagnetization

(First step) isothermal magnetizationA 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 demagnetizationThe helium gas in contact with the salt ispumped away No further heat transfer takesplace 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 SuperconductivityJosephson 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 (superconducting quantum interference devices)for measuring very small amounts of magnetic flux.

(for example, Brain signals)

SQUIDs (Superconducting QUantum Interference Devices)

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)