Germano Maioli Penello

Chapter 7

Magnetism in the localised electron model

Presentation based on the bookMagnetism: 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 1s2, 2s2, 2p2

1s2 2s2

2p2

The multiplicity is 15!

A configuration where all of the shells are full is non degenerate!

= 1 0 -1

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 1s2, 2s2, 2p2

1s2 2s2

= 1 0 -1

Solving...

2p2

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

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: