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Molecular Bonding and Spectra Stimulated Emission and Lasers Structural Properties of Solids Thermal and Magnetic Properties of Solids Superconductivity

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Molecules andSolids

- 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

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.

- 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.

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.

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.

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.

Molecular spectroscopy:

- We can learn about molecules by studying how molecules absorb, emit, and scatter electromagnetic radiation.
- 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).
- In a purely rotational system, the kinetic energy is expressed in terms of the angular momentum L and rotational inertia I.

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.

- 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.

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.

and

- 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.

- 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.

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.

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.

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).

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 samewavelength and phase traveling in the same direction.

Because the incoming photon just triggers emission of the second photon.

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).

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.

helium-neon laser

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.

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.

Three-level system

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.

Four-level system

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.

Tunable laser:

The emitted radiation wavelength can be adjusted as wide as 200 nm.

Semi conductor lasers are replacing dye lasers.

Free-electron laser:

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.

- 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.

- Inertial confinement:

- 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.

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!

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.

- 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.

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.

- 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.

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.

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.

- 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 /ρ diminishes rapidly for r > ρ.
- ρ is roughly regarded as the range of the repulsive force.

- The net potential energy is
- At the equilibrium position (r = r0), F = −dV / dr = 0.
therefore,

and

- The ratio ρ / r0 is much less than 1 and must be less than 1.

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.

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.

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

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

quantum-mechanical result

Rewrite Eq. (10.28)

where R = NAk and EF = ½ muF2.

------ Quantum Lorenz number

Now,

Agrees with experimental results

- 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

- 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

- The magnetization opposes the applied field.

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.

- There exist unpaired magnetic moments that can be aligned by an external field.
- The paramagnetic susceptibility χis 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.

--------- Curie law

----Curie constant

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”

- 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 temperatureTC a ferromagnet changes to a paramagnet.

Antiferromagnetic:

- Adjacent magnet moments have opposing directions.
- The net effect is zero magnetization below the Neel temperatureTN.
- 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.

- 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:

- zero resistivity
- 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

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.

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.

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.

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):

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.

- 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.
- An isotope effect with an exponent very close to 0.5.
- It gives a critical field.

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.

Quantum fluxoid:

Magnetic flux through a superconducting ring.

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

thallium-based superconductor

- 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.

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.

- 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.

Maglev:

Magnetic levitation of trains.

- 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.