Germano Maioli Penello Chapter 7 Magnetism in the localised electron model Presentation based on the book Magnetism: Fundamentals, Damien Gignoux & Michel Schlenker

Magnetism of a free atom or ion  A SINGLE ELECTRON Orbital magnetic moment

This general result shows that the orbital magnetic moment of a charged particle is proportional to its angular momentum. In order to develop this idea further, one needs to make use of quantum mechanics. The stationary states of an electron is characterised by 4 quantum numbers n,, m, and σ.

Remembering that spatial representation cannot take into account all of the subtleties of quantum mechanics! Classical Quantum mechanics Angular momentum Any direction, any length Length and projection an one axis can only take discrete, well defined values.

Stern and Gerlach showed experimentally that the electron also has a magnetic moment which has come to be known as spin. The electron can only have two spin states characterised by σ = ±1/2. The associated angular momentum is written as: In a similar way to the orbital magnetic moment, the spin magnetic moment is proportional to the angular momentum. !!! Spin magnetic moment

Total magnetic moment is not in general collinear with the total angular momentum. Remembering that spatial representation cannot take into account all of the subtleties of quantum mechanics!

States of individual electrons or hydrogen like atoms Stationary states of an electron, Operador For the hydrogen atom, Solution: with wave functions

For hydrogen,, For other atoms (or ions) depends on the atomic number of the element, and on the number of electrons considered in the central potential. total wave function: spin state

MANY ELECTRON ATOMS Hartree's method- The central field approximation: configurations is the spin-orbit coupling hamiltonian. Schrodinger's equation for such a system is impossible to solve directly.

re-writing where, is a fictitious potential >> perturbação, state of an atom: amongst the N electrons there is one in a state, another in a state, and so on. The energy of the atomic state: Such a state is called a configuration.

The configuration of lowest energy is found by successively filling the individual states of lowest energy Total orbital and spin angular momenta: For a full shell: A full shell is not magnetic, and thus does not carry an intrinsic magnetic moment.

Carbon atom. Configuration 1s 2, 2s 2, 2p 2 1s 2 2s 2 2p 2 The multiplicity is 15! A configuration where all of the shells are full is non degenerate! = The interesting cases to consider from the point of view of magnetism are the cases where there are unfilled shells. Solving... Example

Terms partially lifts the degeneracy of each configuration. Intra-atomic correlations Leads to energy levels known as "terms". Each term is characterised by and

the values of the individual spins are those which maximise S, and are compatible with the Pauli exclusion principle. the values of the individual orbital angular momenta are those which maximize, and are compatible with the first rule and the Pauli exclusion principle.

Example Carbon atom. Configuration 1s 2, 2s 2, 2p 2 1s 2 2s 2 = Solving... 2p 2

Spin orbit coupling The origin of this coupling is the following: in the referential of the electron, the motion of the nucleus produces a magnetic field which interacts with the spin magnetic moment. This perturbation leads to different terms. is negative for a shell less than half full, and positive for the opposite case.

Multiplets The degeneracy of each term is partially lifted by the spin-orbit interaction. Each new energy level, known as a "multiplet", is characterised by the quantum number The ground state multiplet is such that if the shell is more than half full, and when it is less than half full.

Example

At this stage, the further liftings of the degeneracy can only take place as a result of external perturbations such as a magnetic field (Zeeman effect), or the effects of neighbouring atoms when the atom is no longer free, and is part of a solid.

Hydrogen-like atoms or Many electron atoms Points to remember

Hartree’s method For an iron atom the atomic number is Z = 26, the ground state configuration is written as: Terms States of a given term are of the type

Multiplets Within a multiplet. there exists a basis of 2J + 1 states Points to remember

magnetism of free ions and atoms Using not necessary collinear with quantum mechanics reveals that within each multiplet the total magnetic moment and the total angular momentum can be considered to be collinear, and linked by the formula:

Points to remember: