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 and Solids Molecules and Solids
The secret of magnetism, now explain that to me! There is no greater secret, except love and hate.
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 < 0) with repulsive force.
Positive slope (dV / dr > 0) with attractive force.
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 < 0.
The shape of the curve depends on the parameters A, B, n, and m.
Also n > m.
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.
Molecular Bonds
Ionic bonds:
The simplest bonding mechanisms.
Ex: Sodium (1s
22s
22p
63s
1) readily gives up its 3s electron to become Na
+, while chlorine (1s
22s
22p
63s
23p
5) 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.
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.
Rotational States
Molecular spectroscopy:
We can learn about molecules by studying how molecules absorb, emit, and scatter electromagnetic radiation.
From the equipartition theorem, the N
2molecule 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.
Rotational States
L is quantized.
The energy levels are
E
rotvaries only as a function of the
quantum number l.
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.
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 = m
1m
2/ (m
1+ m
2) and the spring constant is κ.
If it is a purely ionic bond, we can compute κ by assuming that the force holding the masses together is Coulomb.
and
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
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.
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 Δℓ = ±1.
Absorption spectra:
Within Δℓ = ±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.
Vibration and Rotation Combined
ΔE increases linearly with l as in Eq. (10.8).
Vibration and Rotation Combined
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 Δℓ = 0 transition.
The central frequency
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 ΔE is absorbed by a molecule, a scattered photon of lower energy may be released.
The angular momentum selection rule becomes Δℓ = ±2.
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.
Vibration and Rotation Combined
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 an upper bound estimate of the lifetime is Δt = ħ / (2 ΔE).
10.2: 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.
Stimulated Emission and Lasers
Einstein’s analysis:
Consider transitions between two molecular states with energies E
1and E
2(where E
1< E
2).
E
phis an energy of either emission or absorption.
f is a frequency where E
ph= hf = E
2− E
1. If stimulated emission occurs:
The number of molecules in the higher state (N
2).
The energy density of the incoming radiation (u(f)).
the rate at which stimulated transitions from E
2to E
1is B
21N
2u(f) (where B
21is a proportional constant).
The probability that a molecule at E
1will absorb a photon is B
12N
1u(f).
The rate of spontaneous emission will occur is AN
2(where A is a constant).
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 N
iare proportional to their Boltzmann factor .
In the classical time limit T → ∞. Then and u(f) becomes very large.
the probability of stimulated emission is approximately equal to the probability of absorption.
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.
Stimulated Emission and Lasers
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
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.
Stimulated Emission and Lasers
Stimulated Emission and Lasers
Use 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 E
2.
The transition from E
2to E
1is forbidden by a Δℓ = ±1 selection rule.
E
2is said to be metastable.
3)
Population inversion: more atoms are in the metastable than in the
ground state.
Stimulated Emission and Lasers
After an atom has been returned to the ground state from E
2, we want the external power supply to return it immediately to E
3, but it may take some time for this to happen.
A photon with energy E
2− E
1can be absorbed.
result would be a much weaker beam.
It is undesirable.
Stimulated Emission and Lasers
Four-level system
1)
Atoms are pumped from the ground state to E
4.
2)
They decay quickly to the metastable state E
3.
3)
The stimulated emission takes atoms from E
3to E
2.
4)
The spontaneous transition from E
2to E
1is not forbidden, so E
2will
not exist long enough for a photon to be kicked from E
2to E
3.
Stimulated Emission and Lasers
The red helium-neon laser uses transitions between energy
levels in both helium and neon.
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:
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.
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.
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.
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!
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.
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.
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.
1)
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.
2)
Bob then arranges his part of the quantum system to detect information.
Ex. The polarization status, about the unknown quantum state
at his detector.
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.
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.
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
Each ion must experience a net attractive potential energy.
where r is the nearest-neighbor distance.
α 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 /ρdiminishes rapidly for r > ρ.
ρ is roughly regarded as the range of the repulsive force.
Structural Properties of Solids
Structural Properties of Solids
The net potential energy is
At the equilibrium position (r = r
0), F = −dV / dr = 0.
therefore,
and
The ratio ρ / r is much less than 1 and must be less than 1.
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.
Thermal Expansion
The mean displacement using the Maxwell-Boltzmann distribution function:
where β = (kT)
−1and use a Taylor expansion for x
3term.
Only the even (x
4) term survived from −∞ to ∞.
We are interested only in the first-order dependence on T,
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.
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.
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
Thermal Conductivity
Thermal Conductivity
Eq. (10.32) is called the Wiedemann-Franz law, and the constant L is the Lorenz number.
Experiments show that K / σt has numerical value about 2.5 times higher than predicted by Eq. (10.32).
We should replace Fermi speed u
Fquantum-mechanical result
Rewrite Eq. (10.28)
where R = N
Ak and E
F= ½ mu
F2.
Thermal Conductivity
Now,
Agrees with experimental results
--- Quantum Lorenz number
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 χ:
Positive for paramagnets Negative for diamagnets
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 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
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.
Paramagnetism
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 parallel
N
−moments aligned antiparallel to the applied field.
By Maxwell-Boltzmann statistics,
where A is a normalization constant and β ≡ (kT)
−1.
Paramagnetism
Net magnetic moment is
Eliminate A by considering the mean magnetic moment per atom :
It is only valid for T >> 0.
In the classical limit
--- Curie law
Paramagnetism
Sample magnetization curves
Curie law breaks down at higher values of B, when the
magnetization reaches a “saturation point”
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 T
Ca
ferromagnet changes to a paramagnet.
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 the absence of electrical resistance and the expulsion of magnetic flux from the
superconductor.
It is characterized by two macroscopic features:
1)
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 T
c.
- Superconducting behavior tends to be similar within a given
column of the periodic table.
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
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.
Type I and Type II Superconductors
There is a lower critical field B
c1and an upper critical field B
c2.
Type II: Below B
c1and above B
c2.
Type I: Below and above B
c.
Behave in the same manner
Between B
c1and B
c2(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
Superconductivity
Isotope effect:
M is the mass of the particular superconducting isotope. T
cis 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).
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.
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.
Superconductivity
Quantum fluxoid:
Magnetic flux through a superconducting ring.
3)
An energy gap E
gbetween the
ground state and first excited state.
This means that E
gis the energy
needed to break a Cooper pair
apart E
g(0) ≈ 3.54kT
cat T = 0.
The Search for a Higher T
c
Keeping materials at extremely low temperatures is very expensive and requires cumbersome insulation techniques.
History of transition temperature
The Search for a Higher T
c
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 T
cwith n.
T
cof thallium-copper oxide with n = 3
The Search for a Higher T
c
Higher values of n correspond to more stacked layers of copper and oxygen.
thallium-based superconductor
Superconducting Fullerenes
Another class of exotic superconductors is based on the organic molecule C
60.
Although pure C
60is not superconducting, the addition of certain
other elements can make it so.
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.
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.
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
