Magnetic Hyperfine Splitting

Magnetic hyperfine splitting is caused by the dipole interaction between the nuclear spin moment and a magnetic field ie Zeeman splitting. The effective magnetic field experienced by the nucleus is a combination of fields from the atom itself, from the lattice through crystal field effects and from external applied fields. This can be considered for now as a single field, $\boldsymbol{H}$, whose direction specifies the principal $z$ axis.

The Hamiltonian for the magnetic hyperfine dipole interaction is given as

$$\mathcal{H} = -\boldsymbol{\mu}.\boldsymbol{H} = -g\mu_{N}\boldsymbol{I}.\boldsymbol{H}$$ (2.15)

where $\mu_{N}$ is the nuclear Bohr magneton, $\boldsymbol{\mu}$ is the nuclear magnetic moment, $\boldsymbol{I}$ is the nuclear spin and $g$ is the nuclear $g$-factor.[5]

This Hamiltonian yields eigenvalues of

$$E_{M} = -g\mu_{N}Hm_{I}$$ (2.16)

where $m_{I}$ is the magnetic quantum number representing the $z$ component of $I$ (ie $m_{I} = I, I-1, \ldots, -I$). The magnetic field splits the nuclear level of spin $I$ into $(2I+1)$ equispaced non-degenerate substates. This and the selection rule of $\Delta{}m_{I} = 0,\pm{}1$ produces splitting and a resultant spectrum as shown in Figure 2.5 for a $\nicefrac{3}{2} \to \nicefrac{1}{2}$ transition.

Figure 2.5: The effect of magnetic splitting on nuclear energy levels in the absence of quadrupole splitting. The magnitude of splitting is proportional to the total magnetic field at the nucleus.

This splitting is a combination of a constant nuclear term and a variable magnetic term, influenced by the electronic structure. The magnetic field at the nucleus has several terms associated with it. A general expression is

$$H = H_{0} - DM + \frac{4}{3}\pi{}M + H_{S} + H_{L} + H_{D}$$ (2.17)

where $H_{0}$ is the value of magnetic field at the nucleus due to an external magnetic field, $-DM$ is the demagnetising field, $\nicefrac{4}{3}\pi{}M$ is the Lorentz field, $H_{S}$ is the Fermi contact term, $H_{L}$ is the orbital magnetic term and $H_{D}$ is the dipolar term. The demagnetising field and Lorentz field are usually negligible compared to the other terms.

$H_{S}$ is produced by the polarisation of electrons whose wavefunctions overlap the nucleus, ie $s$-electrons. This polarisation is due to unpaired electrons in the $d$ or $f$ orbitals and gives an imbalance in spin density at the nucleus from the difference in interaction between the unpaired electron with $s$-electrons of parallel or antiparallel spin to its own. This can be expressed formally as

$$H_{S} = -\frac{8\pi}{3}\mu_{0}\mu_{B} \sum \{ \vert\psi_{s\uparrow}(0)\vert^{2} - \vert\psi_{s\downarrow}(0)\vert^2 \}$$ (2.18)

$H_{L}$ arises from the net orbital moment at the nucleus caused by the orbital motion of electrons in unfilled shells and given by

$$H_{L} = \frac{2\mu_{0}\mu_{B}}{4\pi} \langle r^{-3} \rangle \langle \boldsymbol{L} \rangle$$ (2.19)

In transition metals $\boldsymbol{L}$ is usually quenched by interactions with the crystal field, but it can be substantial in Rare Earth ions.

$H_{D}$ arises from the dipolar interaction between the nucleus and the spin moment of 3$d$ or 4$f$ electrons and can be expressed as

$$H_{D} = -2\mu_{B} \langle \boldsymbol{S} \rangle \langle r^{-3} \rangle \langle 3\cos^{2}\theta - 1 \rangle$$ (2.20)

In transition metal compounds with cubic symmetry this has zero magnitude but can be substantial in Rare Earths.