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 10Molecules, Lasers and Solids

The experiment left no doubt that, as far as accuracy of measurement went, the resistance disappeared. At the same time, however, somethingunexpected occurred. . . . The mercury at 4.2K has entered a new state, which, owing to its particular electrical properties, can be called the state of superconductivity.

Heike Kamerlingh Onnes, Nobel Lecture (1913)

What happens when individual atoms join together to form molecules and solids?

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

How are atoms held together to form molecules?

An approximation of the potential of one atom in the vicinity of another atom is

: A and B are positive constants that depend on the types of atoms

: n and m are small, positive numbers. If n and m are equal to 1, the forces are Coulomb. But, because of the complicated shielding effects

of the various electron shells, n and m may not be equal to 1.

(Repulsive potential)

(Attractive potential)

Molecular Bonding and Spectra

The shape of the curve depends on the parameters A, B, n, and m.

n > m is required to produce a potential well.

The potential well provides a stable equilibrium for total energy E < 0.

(Repulsive potential)

(Attractive potential)

E

Vibrations/rotations are excited thermally, so the exact level of E depends on temperature.

The corresponding value of r at the minimum value is an equilibrium separation.

The amount of energy to separate the two atoms completely is the binding energy Eb which is roughly equal to the depth of the potential well.Eb

Molecular BondsIonic bonds: The simplest bonding mechanisms. (Ex) NaCl molecule

Sodium (1s22s22p63s1) readily gives up its 3s electron to become Na+, Chlorine (1s22s22p63s23p5) readily gains an electron to become Cl−. Finally, Na+ and Cl− have filled electron shells That forms the NaCl molecule.

Covalent bonds: The atoms are not as easily ionized. (Ex) Diatomic molecules - H2, N2, O2

The combination of two identical atoms (referred to as homopolar molecules) tend to be covalent. Neither atom is more likely than the other to gain or lose an electron.

Larger molecules (like organic molecules) are formed principally with covalent bonds.

Molecular BondsVan der Waals bond: Relatively weak bond found mostly in liquids and solids at low

temperature (Ex) 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.

Induced dipole

Polar molecule

Nonpolar molecule

Nonpolar Nonpolar

Fluctuation ofInduced dipoles

Molecular BondsHydrogen bond: It is also a kind of Van der Waals bond, but much stronger than normal ones. It holds many organic molecules together by the attractive force

between a hydrogen atom and an electronegative atom, typically O, N, F Water is an excellent example of hydrogen bonding. Double helix in DNA

Metallic bond: Valence electrons easily escape from an atom and form electron cloud (gas). The electron cloud may be shared by a number of atoms positively charged.

Molecular Bonds

1. Rotational Energy States Let us begin by considering a simple two-atom molecule, such as N2. 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.

Molecular spectroscopy:We can learn about molecules by studying how molecules absorb, emit, and scatter electromagnetic radiation. This kind of study is referred to broadly as molecular spectroscopy.

Energy levels of a molecule due to rotational and vibrational motions How to determine rotational energy states and vibrational energy states

Erot varies only as a function of the quantum number l.

There is also the possibility that a vibrational energy mode will be excited. At ordinary temperature no thermal excitation of this mode is possible in a diatomic gas. But, 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:

2. Vibrational Energy States

The energy levels are those of a quantum-mechanical oscillator.

The frequency of a two-particle oscillator is

If it is a purely ionic bond, we can compute κ by assuming that the force holding the masses together is Coulomb.

(reduced mass)

Vibration and Rotation Combined When the rotational and vibrational modes are excited simultaneously. Total energy of the simple vibration-rotation system:

Characteristic of emission spectra Because the vibrational energies are spaced at regular intervals Emission features due to vibrational transitions appear at regular intervals

For the case of rotational energies, the photon produced by a transition from the state to the state will have an energy ( )

Emission features due to rotational transitions also appear at regular intervals Vibrational energies are greater than rotational energies. This energy difference results in the band spectrum

Evenly spaced vibrational spectrum with a more closely spaced rotational spectrum superimposed on each vibrational line.

Vibration and Rotation CombinedVibrational

band

Rotationallevels

Rotationallevels

Vibrationallevel (band)

Vibrationallevel (band)

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 ∆ℓ = ±1.

In the absorption spectrum of HCl, the regular 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:

Absorption spectra:Within ∆ℓ = ±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.

phE

Equipment and data reduction methods for studying molecular spectra.

Fourier transform infrared (FTIR) spectroscopy: 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 ∆E is absorbed by a molecule,

the excess energy may be released in the form ofa scattered photon of lower energy.

Examine the spectrum of Raman-scattered photons. The angular momentum selection rule becomes ∆ℓ = ±2

because of the second photon involved. Raman spectroscopy is used to study the vibrational properties

of liquids and solids. The quanta of vibration are called phonons.

10.2 Stimulated Emission and LasersAccording to Einstein’s Theory of Radiation, the interaction of electromagnetic radiation with matter can be described in terms of 3 basic processes involving these 2 energy levels: Absorption, Spontaneous Emission, and Stimulated Emission.

absorption spontaneous emission stimulated emission

Spontaneous emission:

A molecule in an excited state will decay to a lower energy state and emit a photon spontaneously, without any stimulus from the outside.

As a consequence, the emitted photon has random phase and direction.

If a spectral line has a width ∆E, then Heisenberg’s uncertainty principle gives a lower-bound estimate of the lifetime of ∆t = ħ / (2 ∆E).

For example, suppose you observe an atomic state withE = 0.24 eV and E = 2.1 X 10-6 eV.

Then the lower-bound lifetime is

10.2 Stimulated Emission and LasersUsually the emission of photons by excited molecules occurs spontaneouslywithin very short time.

Stimulated emission: A photon incident upon a molecule in an excited state causes (stimulate)

the unstable system to decay to a lower state. The photon emitted tends to have the same phase and direction as the

stimulating radiation incident upon the molecule.

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 is not absorbedbut rather triggers emission of the second photon.

The two photons (of the same wavelength and phase) are then said to be coherent.

The emission of photons by molecules can also be triggered (stimulated) by an incident radiation.

A simple argument by Einstein “On the Quantum Theory of Radiation” (1917) Consider transitions between two molecular states with energies E1 and E2

(where E1 < E2). The photon energy and frequency of either emission or absorption: Eph = hf = E2 − E1.

If stimulated emission (that is, a process in which incoming radiation causes a transition from E2 to E1) occurs: the rate of emission transitions must be proportional to the number of molecules

in the higher state (call this N2) and the energy density of states, u(f). the rate at which stimulated transitions from E2 to E1 is B21N2u(f)

(where B21 is a proportional constant) Similarly, the absorption probability that a molecule at E1 will be B12N1u(f)

The rate of spontaneous emission is independent of u( f ), however,can be simply as AN2 (where A is a constant)

Einstein’s analysis on Stimulated emission

B21 A

Absorption

B12

B21 A

Absorption

B12

Einstein’s analysis on Stimulated emission

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 →∞; , and the energy density u(f) becomes very large, so the A term becomes insignificant.

The probability of stimulated emission is approximately equal to the probability of absorption.

That means that if the transition from E1 to E2 (absorption) can occur, then we should also expect that stimulated emission will occur.

This closely resembles the Planck radiation law, which can be expressed in terms of frequency:

The probability of stimulated emission (B) is proportional to the probability of spontaneous emission (A) in equilibrium.

In a process for which the probability of spontaneous emission is high, the probability of stimulated emission will also be high.

Solve for u(f)

Using

Stimulated emission is the fundamental physical process in a laser.

Laser: Light Amplification by the Stimulated Emission of Radiation

Masers: Microwaves are used instead of visible light.

The first working maser was made by Charles H. Townes in 1954. The first laser by a group led by Theodore H. Maiman in 1960.

helium-neon laser

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.

LaserLightAmplification byStimulatedEmission ofRadiation.

How does a laser work ?Light Amplification by Stimulated Emission of Radiation.

Stimulated Emission and Lasers

A laser is made up of 2 key parts: an optical Amplifier and an optical Resonator.

1. Amplifier

More atoms must be in the excited state than in the ground state. “population inversion.”

pumping “Metastable”level

Light Amplification by Stimulated Emission of Radiation.

No population inversionin 2-level system

The red helium-neon laser uses transitions between energy levels in both helium and neon.

2. Resonators

: Laser mode frequencies

Light Amplification by Stimulated Emission of Radiation.

Light Amplification by Stimulated Emission of Radiation.

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: 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

An 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

Photography Records intensity distribution of light .

Does not record direction. Two-dimensional image.

Holography = “whole + writing” Records intensity & direction of light.

Information in interference pattern. Reconstruct image by passing original light

through hologram. Need laser so that light interferes.

Dennis Gabor (1947) ••• Nobel Prize in Physics (1971)

Recording a Hologram Consider coherent laser light emitted by a reference source R. The light through a combination of mirrors and beamsplitter can be made to strike both

a photographic plate and an object. the image on the film will be an interference pattern “Hologram”

When the hologram is illuminated, a virtual (real) image can be observed.

Reconstructing the Hologram

Holographic 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.

Other Laser Applications Used in surgery to make precise incisions: Ex: eye operations

Scanning devices used by supermarkets and other retailersEx. Bar code of packaged product

Laser light is directed toward disk tracks in CD and DVD that contain encoded information. The reflected light is turned into electronic digital output.

Quantum entanglement Schrödinger used the term 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 quantum entanglement only , but

transmitting information using entangled systems in conjunction with classical information is possible.