Van Vleck paramagnetism

The magnetization of a material under an external small magnetic field ${\displaystyle \mathbf {H} }$ is approximately described by

${\displaystyle \mathbf {M} =\chi \mathbf {H} }$

where ${\displaystyle \chi }$ is the magnetic susceptibility. When a magnetic field is applied to a paramagnetic material, its magnetization is parallel to the magnetic field and ${\displaystyle \chi >0}$. For a diamagnetic material, the magnetization opposes the field, and ${\displaystyle \chi <0}$.

Experimental measurements show that most non-magnetic materials have a susceptibility that behaves in the following way:

${\displaystyle \chi (T)\approx {\frac {C_{0}}{T}}+\chi _{0}}$,

where ${\displaystyle T}$ is the absolute temperature; ${\displaystyle C_{0},\chi _{0}}$ are constant, and ${\displaystyle C_{0}\geq 0}$, while ${\displaystyle \chi _{0}}$ can be positive, negative or null. Van Vleck paramagnetism often refers to systems where ${\displaystyle C_{0}\approx 0}$ and ${\displaystyle \chi _{0}>0}$.

The Hamiltonian for an electron in a static homogeneous magnetic field ${\displaystyle \mathbf {H} }$ in an atom is usually composed of three terms

${\displaystyle {\mathcal {H}}={\mathcal {H}}_{0}+\mu _{0}{\frac {\mu _{\rm {B}}}{\hbar }}(\mathbf {L} +g\mathbf {S} )\cdot \mathbf {H} +\mu _{0}^{2}{\frac {e^{2}}{8m_{\rm {e}}}}r_{\perp }^{2}H^{2}}$

where ${\displaystyle \mu _{0}}$ is the vacuum permeability, ${\displaystyle \mu _{\rm {B}}}$is the Bohr magneton, ${\displaystyle g}$ is the g-factor, ${\displaystyle e}$ is the elementary charge, ${\displaystyle m_{\rm {e}}}$ is the electron mass, ${\displaystyle \mathbf {L} }$ is the orbital angular momentum operator, ${\displaystyle \mathbf {S} }$ the spin and ${\displaystyle r_{\perp }}$ is the component of the position operator orthogonal to the magnetic field. The Hamiltonian has three terms, the first one ${\displaystyle {\mathcal {H}}_{0}}$ is the unperturbed Hamiltonian without the magnetic field, the second one is proportional to ${\displaystyle \mathbf {H} }$, and the third one is proportional to ${\displaystyle H^{2}}$. In order to obtain the ground state of the system, one can treat ${\displaystyle {\mathcal {H}}_{0}}$ exactly, and treat the magnetic field dependent terms using perturbation theory. Note that for strong magnetic fields, Paschen-Back effect dominates.

For a more general system (molecules, complex systems), the paramagnetic susceptibility for an ensemble of independent magnetic moments can be written as

${\displaystyle \chi _{\rm {para}}=\mu _{0}\mu _{\rm {B}}^{2}{\frac {n}{\sum _{i}p_{i}}}\sum _{i}p_{i}\left[{\frac {\left(W_{i}^{(1)}\right)^{2}}{kT}}-2W_{i}^{(2)}\right]\;;\;p_{i}=\exp \left(-{\frac {E_{i}^{(0)}}{kT}}\right)}$

where

${\displaystyle W_{i}^{(1)}=g_{J}^{(i)}\langle \mathrm {e} _{i}|J_{z}|\mathrm {e} _{i}\rangle /\hbar }$,

${\displaystyle W_{i}^{\rm {(2)}}={\frac {1}{\hbar ^{2}}}\sum _{k(k\neq i)}{\frac {|\langle \mathrm {e} _{i}|L_{z}+gS_{z}|\mathrm {e} _{k}\rangle |^{2}}{\delta E_{i,k}}}\;;\;\delta E_{i,k}=E_{\mathrm {e} ,i}^{(0)}-E_{\mathrm {e} ,k}^{(0)}}$,

and ${\displaystyle g_{J}^{(i)}}$ is the Landé g-factor of state i. Van Vleck summarizes the results of this formula in four cases, depending on the temperature:[3]

1. If all ${\displaystyle |\delta E_{i,k}|\ll k_{\rm {B}}T}$, where ${\displaystyle k_{\rm {B}}}$ is Boltzmann constant, the susceptibility follows Curie law: ${\displaystyle \chi _{\rm {para}}\propto 1/T}$,
2. If all ${\displaystyle |\delta E_{i,k}|\gg k_{\rm {B}}T}$, the susceptibility is independent of the temperature
3. If all ${\displaystyle |\delta E_{i,k}|}$ is either ${\displaystyle \gg k_{\rm {B}}T}$ or ${\displaystyle \ll k_{\rm {B}}T}$, the susceptibility has a mixed behavior and ${\displaystyle \chi _{\rm {para}}\propto 1/T+c}$, where ${\displaystyle c}$ is a constant
4. If all ${\displaystyle |\delta E_{i,k}|\approx k_{\rm {B}}T}$, there is no simple dependence on ${\displaystyle T}$.

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]

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 ${\displaystyle j=0}$ and Curie's contribution (${\displaystyle C_{0}/T}$) vanishes, the first excited state with ${\displaystyle j=1}$ 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]