This is a test of displaying math in html. The content is based on: W. A MacFarlane, “Hyperfine coupling in “kilogauss per µB” (Mar. 2007). The original document can be found here.
The hyperfine Hamiltonian is:
\[H_{\mathrm{hf}} = A_{12} \mathbf{S}_{1} \cdot \mathbf{S}_{2},\]where \(\mathbf{S}_{i}\) are the dimentionless spin operators (i.e., spin angular momentum operators divided by Planck’s constant), and \(A_{12}\) is the hyperfine coupling constant. Clearly, \(A_{12}\) has units of energy, so what the heck is the unit kG/µB (or kOe/µB) and why is it used?!
In a nuclear magnetic resonance (NMR) experiment, if you measure a shift \(K\) in the resonance that is proportional to a susceptibility \(\chi\), then the proportionality constant is also known as the hyperfine coupling \(A\), which is related to the \(A_{12}\) above. Practically then, \(K\) is a unitless quantity (e.g., in ppm), and so too is \(\chi\). This is easy to see since, for example,
\[M = χ H,\]and \(M\) and \(H\) have the same units. Of course, this is assuming we have the unitless “volume susceptibility” for \(\chi\) and not the per mole or per gram equivalents. If \(K\) and \(\chi\) are unitless in
\[K = A \chi,\]then so too must be the coupling \(A\). This, perhaps, is not helping clear things up, but it is good to explore the extent of the confusion!
To get the conventional hyperfine coupling in the units kG/μB, we use instead
\[K = \frac{A \chi}{N_{A} \mu_{B}},\]where \(N_{A}\) is Avogadro’s number and \(\mu_{B}\) is the Bohr magneton (in erg/G), which is about half the magnetic moment of the electron. The susceptibility can then be defined in the following way. The magnetic moment \(m\) of a sample (containing \(N\) moles of the compound) is measured. This moment is in units of emu (i.e., in the cgs system of units). Note that the emu (no relation to the large flightless bird) is a unit of magnetic moment equal to an erg per Gauss (or J/T in SI), so the “susceptibility” \(χ_\mathrm{total} = m / H\) has units of emu/G or erg/G/G (or J/T/T in SI). This is, of course, equal to units of volume, since the square of the magnetic field is just the magnetic energy density. The molar value is gotten just by dividing \(χ_\mathrm{total}\) by \(N\). Doing this we get the molar susceptibility in emu/mol (\(χ\)). Thus, \(A\) has units of (erg/G per erg/G/G) or just G (i.e., it is a magnetic field), but, confusingly, one quotes this as a certain number of G (e.g., kG, etc.) “per μB” because of the 1/μB in the expression for \(K\). The \(N_{A}\) in this definition makes the resulting coupling an atomic quantity (i.e., per atom whose nucleus is coupled to the \(χ\)). If the atom in question is hyperfine coupled to \(Z\) equivalent near neighbours, it is often useful to divide \(A\) further by \(Z\) to get a hyperfine coupling per neighbouring atom.