Molecules and Solids. 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
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.
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.
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.
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.
Holds many organic molecules together.
Free valence electrons may be shared by a number of atoms.
L is quantized.
The energy levels are
Erot varies only as a function of the quantum number l.
There is the possibility that a vibrational energy mode will be excited.
Assume that the two atoms are point masses connected by a massless spring with simple harmonic motion.
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 κ.
If it is a purely ionic bond, we can compute κ by assuming that the force holding the masses together is Coulomb.
emission features due to vibrational transitions appear at regular intervals.
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.
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.
ΔEincreases linearly with l as in Eq. (10.8).
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
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.
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.
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.
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).
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→ ∞. Then and u(f) becomes very large.
the probability of stimulated emission is approximately equal to the probability of absorption.
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.
An acronym for “light amplification by the stimulated emission of radiation.”
Microwaves are used instead of visible light.
The first working laser by Theodore H. Maiman in 1960.
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.
Use a multilevel atomic system to see those problems.
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.
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.
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.
The red helium-neon laser uses transitions between energy levels in both helium and neon.
The emitted radiation wavelength can be adjusted as wide as 200 nm.
Semi conductor lasers are replacing dye 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.
A pellet of deuterium and tritium would be induced into fusion by an intense burst of laser light coming simultaneously from many directions.
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!
The reference beam is on the same side of the film as the object and the illuminating beam is on the opposite side.
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.
Two holograms of the same object produced at different times can be used to detect motion or growth that could not otherwise be seen.
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.
Ex: eye operations.
Ex. Bar code of packaged product.
CD and DVD players
The reflected light is then sampled and turned into electronic signals that produce a digital output.
Condensed matter physics:
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.
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.
where r is the nearest-neighbor distance.
where the x3 term is responsible for the anharmonicity of the oscillation.
The mean displacement using the Maxwell-Boltzmann distribution function:
where β = (kT)−1 and use a Taylor expansion for x3 term.
Only the even (x4) term survived from −∞ to ∞.
We are interested only in the first-order dependence on T,
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.
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
The constant ratio is
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 uF
Rewrite Eq. (10.28)
where R = NAk and EF = ½ muF2.
------ Quantum Lorenz number
Agrees with experimental results
Positive for paramagnets
Negative for diamagnets
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.
N+ moments aligned parallel
N−moments aligned antiparallel to the applied field.
--------- Curie law
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
It simply stated as χ= C / T, where C = μ0Nμ2 / k
Sample magnetization curves
Curie law breaks down at higher values of B, when the magnetization reaches a “saturation point”
It is characterized by two macroscopic features:
- Onnes achieved temperatures approaching 1 K with liquid helium.
- In a superconductor the resistivity drops abruptly to zero at critical (or transition) temperatureTc.
- Superconducting behavior tends to be similar within a given column of the periodic table.
Resistivity of a superconductor
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.
The Meissner effect works only to the point where the critical fieldBc 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.
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.
Between Bc1 and Bc2 (vortex state), there is a partial penetration of magnetic flux although the zero resistivity is not lost.
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.
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).
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.
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.
Magnetic flux through a superconducting ring.
History of transition temperature
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
Higher values of n correspond to more stacked layers of copper and oxygen.
Magnetic levitation of trains.