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.
[1][2][3][4] 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.
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
Van Vleck paramagnetism often refers to systems where
The Hamiltonian for an electron in a static homogeneous magnetic field
is the component of the position operator orthogonal to the magnetic field.
is the unperturbed Hamiltonian without the magnetic field, the second 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.
Note that for strong magnetic fields, Paschen-Back effect dominates.
First order perturbation theory on the second term of the Hamiltonian (proportional to
) for electrons bound to an atom, gives a positive correction to energy given by where
is the total angular momentum operator (see Wigner–Eckart theorem).
This correction leads to what is known as Langevin paramagnetism (the quantum theory is sometimes called Brillouin paramagnetism), that leads to a positive magnetic susceptibility.
If the ground state has no total angular momentum there is no Curie contribution and other terms dominate.
Note that Larmor susceptibility does not depend on the temperature.
While Curie and Larmor susceptibilities were well understood from experimental measurements, J.H.
term: where the sum goes over all excited degenerate states
Van Vleck called this term the "high frequency matrix elements".
[4] In this way, Van Vleck susceptibility comes from the second order energy correction, and can be written as where
are the projection of the spin and orbital angular momentum in the direction of the magnetic field, respectively.
Van Vleck summarizes the results of this formula in four cases, depending on the temperature:[3] While molecular oxygen O2 and nitric oxide NO are similar paramagnetic gases, O2 follows Curie law as in case (a), while NO, deviates slightly from it.
In 1927, Van Vleck considered NO to be in case (d) and obtained a more precise prediction of its susceptibility using the formula above.
The ground state of Eu3+ that has a total azimuthal quantum number
is very close to the ground state at 330 K and contributes through second order corrections as showed by Van Vleck.
A similar effect is observed in samarium salts (Sm3+ ions).
[7][6] In the actinides, Van Vleck paramagnetism is also important in Bk5+ and Cm4+ which have a localized 5f6 configuration.