Recommended Textbooks for the course:

Physics of Atoms and Molecules, B. H. Bransden & C. J. Joachain

Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles, R. Eisberg & R. Resnick

Introduction to the Structure of Matter, J. J. Brehm & W. J. Mullin

Introduction to atomic structure (3 lectures)

Early evidence for the existence of atoms

Thomson's measurement of e/m

Millikan's measurement of e

Rutherford scattering

Total and differential cross-sections

Elastic and inelastic cross-sections

Quantum scattering

the collisional cross section is a measure of the interaction probability of a projectile and a target

it is related, but not equivalent to the size of the particles

The Beer-Lambert law describes attenuation due to scattering

Scattering can be elastic or inelastic

Particle scattering is a powerful tool in physics

The Franck-Hertz experiment uses inelastic scattering to demonstrate quantization of energy levels

Rutherford scattering demonstrates the concentration of mass and positive charge in the nucleus of the atom

the total cross section (sigma)*T* does not contain information about the direction of scattering, for this we need the differential cross section, *ds*/*dW*

These are related by (sigma)*T* = ? [*ds* (q, f)/*dW* ]*dW*

The wave nature of the scattering particles is important, leading to a quantum treatment being necessary

A quantum description of scattering is made using incoming plane waves and outgoing

Bohr model of the atom

Hydrogen Spectra - Lyman, Balmer and Paschen Series

Correspondence Principle

Reduced Mass

Atomic units and Spectroscopic units

Review of quantum treatment of angular momentum and spherical harmonics

Chapter 3

Lecture10 - Allowed terms for equivalent electrons

Figures 3

The Bohr model is a simple model of hydrogen and hydrogen-like atoms

It assumes circular electron motion around a stationary nucleus

Nevertheless, it produces surprisingly good agreement with experimental data

Electrons are only allowed to occupy certain discrete states

In hydrogen, the energies of these states scale as E µ n-2

Transitions between these states involve the exchange of a photon, the energy of which is found from the Rydberg formula

The Correspondence Principle: in the limit of large quantum numbers *n* -> infinity, QM results should tend to the classical results

An atom with more than one electron is too complicated to solve analytically

Simple models can help us understand the fundamental physics without a great deal of mathematical complexity

The independent particle model neglects e- - e- interactions but produces poor solutions

The central field model includes the averaged effect e- - e- interactions as a screened potential

It produces reasonable solutions, especially for alkali-metal-like atoms

Modifying the potential lifts the degeneracy with respect to *l*

The quantum defect characterises the departure from hydrogenic behaviour

It is especially useful for atoms with a single optically active electron

A screened charge may be used to parameterize a non-hydrogenic atom

The Pauli Exclusion Principle states that the wavefunction of a system of identical fermions must be anti-symmetric with respect to exchange

An alternative formulation is that the electrons must have different sets of quantum numbers

The Helium atom wavefunction consists of a space part and a spin part

The total wavefunction must be antisymmetric overall with respect to exchange

For a non-vanishing wavefunction we then require the space part to be symmetric and the spin part to be antisymmetric

The result is a singlet state

Exchange is a quantum mechanical phenomenon for which there is no classical analogue

It arises from the spin-dependent properties of the wavefunction

As a result of the Pauli Exclusion Principle space and spin states are coupled

The exchange interaction modifies the energies of the states according to the symmetry of the wavefunction

It splits the energies of the singlet and triplet states

Excited helium may exist in two states - parahelium and orthohelium - with different energies

Electronic structure can be described in terms of filled and unfilled shells and subshells

The filling of the shells determines physical properties

Spectroscopic terms give the allowed angular momentum states of an electron configuration

Hund's Rules provide guidelines for the ordering (in energy) of terms

Terms are split into levels by the spin-orbit interaction

A magnetic moment *mX* is associated with an angular momentum *X*

Both spin and orbital angular momenta give rise to magnetic moments

The spin-orbit interaction arises from the interaction between the electron spin magnetic moment and the magnetic field due to the apparent motion of the nuclear charge

The interaction energy depends on the relative orientation of *L* and *S*

The spin-orbit interaction splits terms into energy levels according to their total angular momentum

In multi-electron atoms the two extreme regimes are *LS*-coupling and *jj*-coupling, depending on the size of the interaction

The energy levels can be ordered using a third Hund's Rule

The separation of the energy levels can be found from the Landé interval rule

The parity of an N electron wavefunction is (-1)*S*_{l}*i*

Chapter 4

Lecture14 - Spherical harmonics

Lecture18 - Proof of Eqn. 40

Figures 4

Atoms in radiation: dipole allowed and forbidden transitions

Einstein A & B coefficients, metastable levels

Laser operation, population inversion & application to atom cooling

X-rays & inner shell transitions

Atoms in magnetic fields: normal and anomalous Zeeman effect

Hyperfine splitting

Atoms in electric fields: linear and quadratic Stark effect

Transitions between states are governed by selection rules

Selection rules for a transition may be obtained from Fermi's Golden Rule
The properties of the spherical harmonics determine the selection rules

For an electric dipole transition D_{l} = ±1, Dm = 0, ±1, Ds = 0, and the states must be of opposite parity. Also Dj = 0, ±1 (but not j = 0 ® j'=0) if spin-orbit interaction is significant

An atom interacts with a radiation field via absorption, spontaneous emission and stimulated emission of a photon

The Einstein relations may be derived by considering the equilibrium of the atom with the radiation field

The excited state has a finite lifetime and finite energy (frequency) width

An electric dipole forbidden transition has a longer lifetime, producing a metastable state

Photons emitted in stimulated emission are coherent, which is the basis of the laser
To achieve laser action needs a population inversion

This is not possible in a two-level system, so usually a three- or four-level system is used

Lasers can be used to cool atoms by carefully choosing their frequency to exert a velocity dependent force

X-rays are emitted when a target of heavy metal atoms is bombarded by energetic electrons

The typical spectrum consists of a continuous background of Bremmstrahlung with characteristic peaks superimposed

The Bremmstrahlung has a short wavelength cut-off determined by the energy of the electrons, independent of the target material

The peaks are caused by the removal of an electron from an inner shell and higher energy

Electrons making a transition to the inner shell by emitting an X-ray photon

The wavelength of the peaks is characteristic of the target and can be calculated from a Rydberg-type formula when a screening constant is included

The interaction between an atom and a magnetic field is called the Zeeman effect

The magnetic field interacts with the magnetic dipole moments of the atom that arise from the angular momenta (spin and orbital)

The interaction can be evaluated in the limit where it is a small perturbation

If the interaction is smaller than the spin-orbit interaction is is classed as 'weak'

The weak field Zeeman effect is classified as 'normal' or 'anomalous' depending on whether spin is needed to explain the effects

The Zeeman effect splits a transition into a number of spectral lines

In the limit of strong applied magnetic field spin and orbital angular momentum decouple and precess independently around the field direction

ml and ms are thus good quantum numbers

due to the electric dipole selection rules a single spectral line is split into three

The Stern-Gerlach Experiment provides evidence for the quantisation of angular momentum

The nucleus of an atom also possesses a nuclear spin I, and magnetic moment *mI*

mI interacts with the magnetic field of the orbiting electron to split energy levels into hyperfine sublevels depending on their value of F = I + J

Chapter 5

Lecture24 - Hydrogen Derivation

Figures 5

The Born-Oppenheimer approximation

Electronic spectra of H_{2}^{+} and H_{2}

Effects of symmetry and exchange: bonding and antibonding orbitals

Nuclear motion

Rotational, vibrational and electronic spectra for ideal molecules

Selection rules

Covalent and ionic bonds

The electron affinity is the energy required to add an electron to an atom

The ionisation energy is the energy required to remove an electron from an atom

The difference between these may be compensated by the Coulomb attraction to from a stable ionic molecule

To calculate the binding energy we must take account of the repulsion of the electron clouds

The electric dipole moment gives an indication of the fractional ionic character of the bond

2B24 Exam 2006 - Atomic and Molecular Physics

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