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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.
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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.
Molecular Bonding and Spectra • 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.
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 (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.
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 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.
Rotational States L is quantized. The energy levels are Erot varies 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 = 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
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
Vibration and Rotation Combined 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 ΔEincreases linearly with l as in Eq. (10.8).
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.
Vibration and Rotation Combined 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.
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. • 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 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 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).
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 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.
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 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.
Stimulated Emission and Lasers 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.
Stimulated Emission and Lasers 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.
Stimulated Emission and Lasers 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.
Stimulated Emission and Lasers 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.
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. 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.
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.
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 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.
Structural Properties of Solids • 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.
Structural Properties of Solids • 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 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)−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,
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.
Thermal Conductivity 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 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.
Thermal Conductivity ------ Quantum Lorenz number Now, Agrees with experimental results
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 • 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.
