Transcript CHAPTER 10: Molecules and Solids

CHAPTER 10
Molecules and Solids
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10.1
10.2
10.3
10.4
10.5
10.6
Molecular Bonding and Spectra
Stimulated Emission and Lasers
Structural Properties of Solids
Thermal and Magnetic Properties of Solids
Superconductivity
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
1
10.1: Molecular Bonding and Spectra
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The Coulomb force is the only one to bind atoms.
The combination of attractive and repulsive forces creates a
stable molecular structure.
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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.
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Negative slope (dV / dr < 0) with repulsive force.
Positive slope (dV / dr > 0) with attractive force.
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2
Molecular Bonding and Spectra
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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.
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Molecular Bonding and Spectra
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Vibrations are excited thermally, so the exact
level of E depends on temperature.
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A pair of atoms is joined.
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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.
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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.
4
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.
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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.
6
Rotational States
Molecular spectroscopy:
 We can learn about molecules by studying how molecules
absorb, emit, and scatter electromagnetic radiation.
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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).
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In a purely rotational system, the kinetic energy is expressed in
terms of the angular momentum L and rotational inertia I.
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Rotational States
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L is quantized.
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The energy levels are
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Erot varies only as a function of the
quantum number l.
8
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.
9
Vibrational States
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The energy levels are those of a quantum-mechanical oscillator.
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The frequency of a two-particle oscillator is
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Where the reduced mass is μ = m1m2 / (m1 + m2) 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
10
Vibration and Rotation Combined
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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
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Vibration and Rotation Combined
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An emission-spectrum spacing that varies with l.
the higher the starting energy level, the greater the photon
energy.
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Vibrational energies are greater than rotational energies. This
energy difference results in the band spectrum.
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Vibration and Rotation Combined
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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:
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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.
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Vibration and Rotation Combined
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ΔE increases linearly with l as
in Eq. (10.8).
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Vibration and Rotation Combined
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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
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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.
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Vibration and Rotation Combined
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A transition from l to l + 2.
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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:
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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.
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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).
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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.
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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
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 Lasers
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Once the system has reached equilibrium with the incoming radiation,
the total number of downward and upward transitions must be equal.
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In the thermal equilibrium each of Ni are proportional to their
Boltzmann factor
.
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In the classical time limit T → ∞. Then
becomes very large.
and u(f)
the probability of stimulated emission is approximately equal
to the probability of absorption.
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Stimulated Emission and Lasers
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Solve for u(f),
or, use Eq. (10.12),
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This closely resembles the Planck radiation law, but Planck law is
expressed in terms of frequency.
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Eqs.(10.13) and (10.14) are required:
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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.
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The first working laser by Theodore H. Maiman in 1960.
helium-neon laser
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Stimulated Emission and Lasers
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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 Lasers
Use a multilevel atomic system to see those problems.
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Three-level system
1)
2)
3)
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 Δℓ = ±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
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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
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Four-level system
1)
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.
2)
3)
4)
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Stimulated Emission and Lasers
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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
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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
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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.
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Pulsed lasers are used in thin-film deposition to study the
electronic properties of different materials.
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The use of lasers in fusion research.
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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
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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
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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.
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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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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.
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Quantum Entanglement, Teleportation, and
Information
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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.
1)
2)
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
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Used in surgery to make precise incisions.
Ex: eye operations.
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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
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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
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Each ion must experience a net attractive potential energy.
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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.
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There is a repulsive potential due to the Pauli exclusion principle.
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The value e−r /ρ diminishes rapidly for r > ρ.
ρ is roughly regarded as the range of the repulsive force.
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Structural Properties of Solids
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The net potential energy is
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At the equilibrium position (r = r0), F = −dV / dr = 0.
therefore,
and
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The ratio ρ / r0 is much less than 1 and must be less than 1.
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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.
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Thermal Expansion

The mean displacement using the Maxwell-Boltzmann
distribution function:
where β = (kT)−1 and use a Taylor expansion for x3 term.
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Only the even (x4) term survived from −∞ to ∞.
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We are interested only in the first-order dependence on T,
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Thermal Expansion
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Combining Eq. (10.24) and (10.25),
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Thermal expansion is nearly linear with temperature in the
classical limit. Eq. (10.26) vanishes as T → 0.
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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.
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The flow of heat per unit time along the rod is proportional to A
and to the temperature gradient dT / dx.
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The thermal conductivity K is the proportionality constant.
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Thermal Conductivity
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In classical theory the thermal conductivity of an ideal free electron
gas is
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Classically
, so
.
Compare the thermal and electrical conductivities:
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From classical thermodynamics the mean speed is
Therefore
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The constant ratio is
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Thermal Conductivity
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Eq. (10.32) is called the Wiedemann-Franz law, and the constant
L is the Lorenz number.
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Experiments show that K / σt has numerical value about 2.5 times
higher than predicted by Eq. (10.32).
We should replace
Fermi speed uF
quantum-mechanical result
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Rewrite Eq. (10.28)
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where R = NAk and EF = ½ muF2.
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Thermal Conductivity
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Now,
------ Quantum Lorenz number
Agrees with experimental results
48
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
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Diamagnetism
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The magnetization opposes the applied field.
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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
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Diamagnetism
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For a magnetic field from 0 to B, directed out of the page, the
angular moment changes by an amount
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This results in a magnetic moment changed by
which has a magnitude
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The change in magnetic moment is opposite to the applied field.
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Paramagnetism
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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 parallel
N− moments aligned antiparallel to the applied field.
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By Maxwell-Boltzmann statistics,
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where A is a normalization constant and β ≡ (kT)−1.
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Paramagnetism
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Net magnetic moment is
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Eliminate A by considering the mean magnetic moment per atom

It is only valid for T >> 0.
In the classical limit
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:
--------- Curie law
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It simply stated as χ = C / T, where C = μ0Nμ2 / k ----Curie constant
53
Paramagnetism
Sample magnetization curves

Curie law breaks down at higher values of B, when the
magnetization reaches a “saturation point”
54
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.
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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
Antiferromagnetic:

Adjacent magnet moments have opposing
directions.
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The net effect is zero magnetization below the
Neel temperature TN.
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Above TN, antiferromagnetic → paramagnetic.
Ferrimagnetic:
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A similar antiparallel alignment occurs, except
that there are two different kinds of positive ions
present.

The antiparallel moments leave a small net
magnetization.
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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 Tc.
- Superconducting behavior tends to be similar within a given
column of the periodic table.
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Superconductivity
Resistivity of a superconductor
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.
58
Superconductivity

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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 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.
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.
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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):
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
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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 electronphonon resonance allows the electron to move without resistance.

The complete BCS theory predicts other observed phenomena.
1)
2)
An isotope effect with an exponent very close to 0.5.
It gives a critical field.
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Superconductivity
Quantum 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
66
The Search for a Higher Tc

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
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
67
The Search for a Higher Tc

Higher values of n correspond to more stacked layers of copper
and oxygen.
thallium-based superconductor
68
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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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.
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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.
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Generation and Transmission of
Electricity

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
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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