Solid State Physics - Home Assignment 10 Badri Narayanan Narasimhan (466013)

1. Response of valence Electrons to external Magnetic Field:The valence electrons in a metal can contribute to paramagnetic and diamagnetic response of a material. The electrons follow fermi dirac statistics (indistinguishable particles) and therefore due to pauli's exclusion principle the number of electrons that are able to reverse their spin with the application of external magnetic field are reduced as most orbitals will be occupied with electrons that have the same spin orientation. The electrons that have energy of the order KbT of Fermi level are only excited andcan reverse their spin which in turn contribute to paramagnetic susceptibility. It is expected that Curie suceptibility will reduce by a factor T/Tf which is smaller than classical prediction (using boltzmann statistics and curie law is not observed) and is temperature independent. The density of states for electrons after external magnetic field application can be split into electrons that have spin parallel to external field B0 whose energies are lowered by amount -ubB and the eletrons that have spin anti parallel to external field B0 whose energies are raised by ubB. Therefore the relative energy difference is 2ubB. It is to be noted that the fermi levels in both distributions are same except for the orientations to B.Magnetization of electron gas M = n+u+ + n-u- where n+u+ are parallel and n-u- are antiparallelThe magnetization originates due to the imbalance of Ef.The total magnetization M + ub(n+-n-) =ub2B0D(Ef) and m,P= u0ub2D(Ef) is called the pauli paramagnetic susceptibility. For free electron gas D(Ef)= 3n/2Ef , the T/Tf factor arises when we compare this expression with the classical treatment. The magnitude of pauli paramagnetic susceptibility is 3 times that of landau diamagnetic susceptibility.

2. If the temperature is such that KbT is greater than interaction energy between magnetic moments paramagnetic state will be favoured due to thermal fluctuations. But at low temperatures, an ordering of magnetic moments in a solid is ensued for electric dipoles. The ferromagnetic ordering occurs when the interaction between the magnetic moments makes sure that all moments are parallel to each other due to which they all contribute equally to spontaneous magnetization even in zero field. An antiferromagnetic ordering occurs when interaction produces antiparallel arrangement of moments which gives zero magnetization. It is also important to note that antiparallel moments do not cancel out entirely giving rise to a small magnetization.

The interaction responsible for ordering is the exchange interaction which is effectively a spin dependent coloumbic interaction however it is not due to the magnetic dipolar interactions between moments as they are too weak to overcome thermal disordering effects.

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