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.[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.[5][6]
Description
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 .
Derivation
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. Note that for strong magnetic fields, Paschen-Back effect dominates.
First order perturbation theory
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 ground state, is the Landé g-factor of the ground state and 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. For sufficiently large temperatures , this contribution is described by Curie's law:
- ,
a susceptibility that is inversely proportional to the temperature , where is the material dependent Curie constant. If the ground state has no total angular momentum there is no Curie contribution and other terms dominate.
The first perturbation theory on the third term of the Hamiltonian (proportional to ), leads to a negative response (magnetization that opposes the magnetic field). Usually known as Larmor or Langenvin diamagnetism:
where is another constant proportional to the number of atoms per unit volume, and is the mean squared radius of the atom. Note that Larmor susceptibility does not depend on the temperature.
Second order: Van Vleck susceptibility
While Curie and Larmor susceptibilities were well understood from experimental measurements, J.H. Van Vleck noticed that the calculation above was incomplete. If is taken as the perturbation parameter, the calculation must include all orders of perturbation up to the same power of . As Larmor diamagnetism comes from first order perturbation of the , one must calculate second order perturbation of the term:
where the sum goes over all excited degenerate states , and are the energies of the excited states and the ground state, respectively, the sum excludes the state , where . Historically, J.H. 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 is the number density, and and are the projection of the spin and orbital angular momentum in the direction of the magnetic field, respectively.
In this way, , as the signs of Larmor and Van Vleck susceptibilities are opposite, the sign of depends on the specific properties of the material.
General formula and Van Vleck criteria
For a more general system (molecules, complex systems), the paramagnetic susceptibility for an ensemble of independent magnetic moments can be written as
where
- ,
- ,
and is the Landé g-factor of state i. Van Vleck summarizes the results of this formula in four cases, depending on the temperature:[3]
- If all , where is Boltzmann constant, the susceptibility follows Curie law: ,
- If all , the susceptibility is independent of the temperature
- If all is either or , the susceptibility has a mixed behavior and , where is a constant
- If all , there is no simple dependence on .
While molecular oxygen O
2 and nitric oxide NO are similar paramagnetic gases, O
2 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.[2][4]
Systems of interest
The standard example of Van Vleck paramagnetism are europium(III) oxide (Eu
2O
3) salts where there are six 4f electrons in trivalent europium ions. The ground state of Eu3+
that has a total azimuthal quantum number and Curie's contribution () vanishes, the first excited state with 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.[7]
References
- ^ Van Vleck, John Hasbrouck (1932). The Theory of Electric and Magnetic Susceptibilities. Clarendon Press.
- ^ a b Van Vleck, J. H. (1928-04-01). "On Dielectric Constants and Magnetic Susceptibilities in the New Quantum Mechanics Part III—Application to Dia- and Paramagnetism". Physical Review. 31 (4): 587–613. Bibcode:1928PhRv...31..587V. doi:10.1103/PhysRev.31.587. ISSN 0031-899X.
- ^ a b van Vleck, John H. (1977). "John H. van Vleck Nobel Lecture". Nobel Prize. Retrieved 2020-10-18.
- ^ a b c Anderson, Philip W. (1987). John Hasbrouck Van Vleck (PDF). Washington D.C: National Academy of Sciences.
- ^ Marder, Michael P. (2010-11-17). Condensed Matter Physics. John Wiley & Sons. ISBN 978-0-470-94994-8.
- ^ a b Nolting, Wolfgang; Ramakanth, Anupuru (2009-10-03). Quantum Theory of Magnetism. Springer Science & Business Media. ISBN 978-3-540-85416-6.
- ^ a b Coey, J. M. D. (2010). Magnetism and Magnetic Materials. Cambridge: Cambridge University Press. ISBN 978-0-521-81614-4.