Van Vleck paramagnetism

In condensed matter and atomic physics, Van Vleck paramagnetism refers to a positive and temperature-independent contribution to the magnetic susceptibility of a material, derived from second order corrections to the Zeeman interaction. The quantum mechanical theory was developed by John Hasbrouck Van Vleck between the 1920s and the 1930s to explain the magnetic response of gaseous nitric oxide (NO) and of rare-earth salts. Alongside other magnetic effects like Paul Langevin's formulas for paramagnetism (Curie's law) and diamagnetism, Van Vleck discovered an additional paramagnetic contribution of the same order as Langevin's diamagnetism. Van Vleck contribution is usually important for systems with one electron short of being half filled and this contribution vanishes for elements with closed shells. The magnetization of a material under an external small magnetic field is approximately described by where is the magnetic susceptibility. When a magnetic field is applied to a paramagnetic material, its magnetization is parallel to the magnetic field and . For a diamagnetic material, the magnetization opposes the field, and . Experimental measurements show that most non-magnetic materials have a susceptibility that behaves in the following way: where is the absolute temperature; are constant, and , while can be positive, negative or null. Van Vleck paramagnetism often refers to systems where and . The Hamiltonian for an electron in a static homogeneous magnetic field in an atom is usually composed of three terms where is the vacuum permeability, is the Bohr magneton, is the g-factor, is the elementary charge, is the electron mass, is the orbital angular momentum operator, the spin and is the component of the position operator orthogonal to the magnetic field. The Hamiltonian has three terms, the first one is the unperturbed Hamiltonian without the magnetic field, the second one is proportional to , and the third one is proportional to . In order to obtain the ground state of the system, one can treat exactly, and treat the magnetic field dependent terms using perturbation theory.