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Chapter 7

Chapter 7. Magnetism in the localised electron model. Germano Maioli Penello. Presentation based on the book Magnetism: Fundamentals, Damien Gignoux & Michel Schlenker. Magnetism of a free atom or ion. A SINGLE ELECTRON Orbital magnetic moment.

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Chapter 7

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

  2. Magnetism of a free atom or ion • A SINGLE ELECTRON • Orbital magnetic moment

  3. 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.

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

  5. Spin magnetic moment 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. !!!

  6. 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!

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

  8. 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

  9. 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.

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

  11. 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.

  12. Example = 1 0 -1 Solving... Carbon atom. Configuration 1s2, 2s2, 2p2 1s2 2s2 The multiplicity is 15! 2p2 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.

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

  14. 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.

  15. Example Solving... Carbon atom. = 1 0 -1 Configuration 1s2, 2s2, 2p2 1s2 2s2 2p2

  16. Spin orbit coupling is negative for a shell less than half full, and positive for the opposite case. 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.

  17. 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.

  18. Example

  19. 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.

  20. Points to remember Hydrogen-like atoms or Many electron atoms

  21. 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

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

  23. 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:

  24. Points to remember:

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