| In metals, magnetism is due to | delocalised valence electrons |
| Magnetic susceptibility for Pauli paramagnet | independent of temperature |
| Why are Fe, Co, Ni ferromagnetic? | Large density of states at Fermi level due to narrow energy 3-d band |
| For ferromagnets, spontaneous splitting of spin up and spin down bands can take place due to the internal molecular field Bmf. What are the two competing energy contributions? | (i) increase in kinetic energy and (ii) decreases in potential |
| (Ferromagnetism) decrease in potential energy must be larger than any increase in kinetic energy when spins split, leading to...... | Stoner criterion U(Ef) >1. This Predicts that ferromagnetism is favoured by large Coulomb inter-action and/or high density of states at Fermi level (this is likely in transition metals). |
| J relation for more/less than half-filled shell | J = L+S for more
J = |L-S| for less |
| Hund's rules | (i) First maximise S by filling the positive ms side, (ii) maximise L by filling from positive biggest l downwards
(iii) selecting J that gives the smallest spin orbit interaction energy (this is J = |L - S| if the band is less than half-full and J = L + S for a band more than half-full). |
| Order of shells in increasing l value | s p d f g h i |
| Energy of magnetic dipole in B field | E = -µ_s . B |
| µ for an electron | - γ l where l is the angular momentum and γ is the gyromagnetic ratio (=e/2m) |
| µ for spin/J angular momentum | µ_(j/s) = -g_(j/s)γ * j/s
In this equation it is not j divided by s it is j or s |
| For a nearly free electron solid what can you assume? | Close to zero, electrons are free because far from the bragg condition (k+G)^2 = k^2 and because of 2nd order pertubation theory. This means E = (hbar k)^2/2m* and v_g = 1/hbar dE/dk |
| Relation between r^2 and <r^2> in 2D | r^2 = 2/3 <r^2> |
| Bohr magneton | µB = γ hbar |
| Diamagnetic susceptibility equation | X = M/H
Used to classify weakly magnetic materials (paramagnets positive, diamagnets
negative) |
| Magnetisation | Net dipole moment per unit volume
Equal to change in µ * electrons/atom *atoms/volume |
| H for weakly magnetic solids | B = µ0 (H+M) = µ0 H |
| Free electron theory limitations | predicts spherical Fermi surface, when really there is distortion at boundaries from Bragg diffraction off nuclei (interference between incident and scattered) |
| Origin of magnetocrystalline anistropy: | - crystal field effects arise due to anisotropic shape of electron orbitals
- results in orbital quenching where net ang mom L = 0
- orbital quenching modifies SO interaction so Hamiltonian now dependent on crystal direction |
| GL theory: why does free energy only depends on even powers? | - reversing direction of magnetisation vector must leave free energy unchanged
- only even powers do this |
| Explain, using classical electromagnetic theory, the origin of diamagnetism. | Electrical charges shield an interior of a body from an applied B field. When the flux through an electric circuit changings, an induced current is set up to oppose the flux change. This current persists as long as the field is present. The B field of the induced current is opposite to applied field. |
