The orbital magnetic moment of a finite system, such as a molecule, is given classically by^[1]

${\displaystyle \mathbf {m} _{\rm {orb}}={\frac {1}{2}}\int \mathbf {r} \times \mathbf {J} (\mathbf {r} )\ d^{3}\mathbf {r} }$ 

where J(r) is the current density at point r. (Here SI units are used; in Gaussian units, the prefactor would be 1/2c instead, where c is the speed of light.) In a quantum-mechanical context, this can also be written as

${\displaystyle \mathbf {m} _{\rm {orb}}={\frac {-e}{2m_{e}}}\langle \Psi \vert \mathbf {L} \vert \Psi \rangle }$ 

where −e and m_e are the charge and mass of the electron, Ψ is the ground-state wave function, and L is the angular momentum operator. The total magnetic moment is

${\displaystyle \mathbf {m} =\mathbf {m} _{\rm {orb}}+\mathbf {m} _{\rm {spin}}}$ 

where the spin contribution is intrinsically quantum-mechanical and is given by

${\displaystyle \mathbf {m} _{\rm {spin}}={\frac {-g_{s}\mu _{\rm {B}}}{\hbar }}\,\langle \Psi \vert \mathbf {S} \vert \Psi \rangle }$ 

where g_s is the electron spin g-factor, μ_B is the Bohr magneton, ħ is the reduced Planck constant, and S is the electron spin operator.

The orbital magnetization M is defined as the orbital moment density; i.e., orbital moment per unit volume. For a crystal of volume V composed of isolated entities (e.g., molecules) labelled by an index j having magnetic moments m_{orb, j}, this is

${\displaystyle \mathbf {M} _{\rm {orb}}={\frac {1}{V}}\sum _{j\in V}\mathbf {m} _{{\rm {orb}},j}\;.}$ 

However, real crystals are made up out of atomic or molecular constituents whose charge clouds overlap, so that the above formula cannot be taken as a fundamental definition of orbital magnetization.^[2] Only recently have theoretical developments led to a proper theory of orbital magnetization in crystals, as explained below.

Difficulties in the definition of orbital magnetization edit

For a magnetic crystal, it is tempting to try to define

${\displaystyle \mathbf {M} _{\rm {orb}}={\frac {1}{2V}}\int _{V}\mathbf {r} \times \mathbf {J} (\mathbf {r} )\ d^{3}\mathbf {r} }$ 

where the limit is taken as the volume V of the system becomes large. However, because of the factor of r in the integrand, the integral has contributions from surface currents that cannot be neglected, and as a result the above equation does not lead to a bulk definition of orbital magnetization.^[2]

Another way to see that there is a difficulty is to try to write down the quantum-mechanical expression for the orbital magnetization in terms of the occupied single-particle Bloch functions |ψ_{n k}⟩ of band n and crystal momentum k:

${\displaystyle \mathbf {M} _{\rm {orb}}={\frac {-e}{2m_{e}}}\sum _{n}\int _{\rm {BZ}}{\frac {1}{(2\pi )^{3}}}\,\langle \psi _{n\mathbf {k} }\vert \mathbf {r} \times \mathbf {p} \vert \psi _{n\mathbf {k} }\rangle \,d^{3}k\,,}$ 

where p is the momentum operator, L = r × p, and the integral is evaluated over the Brillouin zone (BZ). However, because the Bloch functions are extended, the matrix element of a quantity containing the r operator is ill-defined, and this formula is actually ill-defined.^[3]

Atomic sphere approximation edit

In practice, orbital magnetization is often computed by decomposing space into non-overlapping spheres centered on atoms (similar in spirit to the muffin-tin approximation), computing the integral of r × J(r) inside each sphere, and summing the contributions.^[4] This approximation neglects the contributions from currents in the interstitial regions between the atomic spheres. Nevertheless, it is often a good approximation because the orbital currents associated with partially filled d and f shells are typically strongly localized inside these atomic spheres. It remains, however, an approximate approach.

Modern theory of orbital magnetization edit

A general and exact formulation of the theory of orbital magnetization was developed in the mid-2000s by several authors, first based on a semiclassical approach,^[5] then on a derivation from the Wannier representation,^[6][7] and finally from a long-wavelength expansion.^[8] The resulting formula for the orbital magnetization, specialized to zero temperature, is

${\displaystyle \mathbf {M} _{\rm {orb}}={\frac {e}{2\hbar }}\sum _{n}\int _{\rm {BZ}}{\frac {1}{(2\pi )^{3}}}\,f_{n\mathbf {k} }\;\operatorname {Im} \;\left\langle {\frac {\partial u_{n\mathbf {k} }}{\partial {\mathbf {k} }}}\right|\times \left(H_{\mathbf {k} }+E_{n\mathbf {k} }-2\mu \right)\left|{\frac {\partial u_{n\mathbf {k} }}{\partial {\mathbf {k} }}}\right\rangle \,d^{3}k\,,}$ 

where f_{n k} is 0 or 1 respectively as the band energy E_{n k} falls above or below the Fermi energy μ,

${\displaystyle H_{\mathbf {k} }=e^{-i\mathbf {k} \cdot \mathbf {r} }He^{i\mathbf {k} \cdot \mathbf {r} }}$ 

is the effective Hamiltonian at wavevector k, and

${\displaystyle u_{n\mathbf {k} }(\mathbf {r} )=e^{-i\mathbf {k} \cdot \mathbf {r} }\psi _{n\mathbf {k} }(\mathbf {r} )}$ 

is the cell-periodic Bloch function satisfying

${\displaystyle H_{\mathbf {k} }\left|u_{n\mathbf {k} }\right\rangle =E_{n\mathbf {k} }\left|u_{n\mathbf {k} }\right\rangle \;.}$ 

A generalization to finite temperature is also available.^[3][8] Note that the term involving the band energy E_{n k} in this formula is really just an integral of the band energy times the Berry curvature. Results computed using the above formula have appeared in the literature.^[9] A recent review summarizes these developments.^[10]

The orbital magnetization of a material can be determined accurately by measuring the gyromagnetic ratio γ, i.e., the ratio between the magnetic dipole moment of a body and its angular momentum. The gyromagnetic ratio is related to the spin and orbital magnetization according to

${\displaystyle \gamma =1+{\frac {M_{\mathrm {orb} }}{(M_{\mathrm {spin} }+M_{\mathrm {orb} })}}}$ 

The two main experimental techniques are based either on the Barnett effect or the Einstein–de Haas effect. Experimental data for Fe, Co, Ni, and their alloys have been compiled.^[11]