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2B24 Atomic and Molecular Physics

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

Chapter 1
Figures 1

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

Chapter 2
Figures 2

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)Sli

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 Dl = ±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 H2+ and H2
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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