Contact Hyperfine Fields - Some notes

Muesr assumes that hyperfine fields are isotropic. The contact term is obtained by specifying two terms: 1) the number of nearest neighbors magnetic atoms to the muon 2) a rcont parameters which governs the maximum radius of the interaction.

Finally, the LocalField object has a ACont property which is an effective hyperfine contact coupling term. The total field is therefore obtained as

\[\mathbf{B_T} = \mathbf{B_D} + \mathbf{B_L} + ACont \cdot \frac{2 \mu _0}{3} \sum _i ^N \frac{r _i ^{-3} }{\sum _i ^N r_i ^{-3}} \mathbf{m}_i\]

The value of N can strongly impact on the results. The best approximation strongly depend on the simulated system and must be considered case by case.

From CGS to SI

Hyperfine couplings are often reported in mol/emu while Muesr uses \({\buildrel _{\circ} \over {\mathrm{A}}}^{-3}\). Here’s how to convert the former into the latter.

\[\begin{split}B_c = -\frac{8}{3} \pi |\Psi (0)|^2 \boldsymbol{\mu}_e \qquad \mbox{(c.g.i)} \\ B_c = -\frac{2}{3} \mu_0 |\Psi (0)|^2 \boldsymbol{\mu}_e \qquad \mbox{(S.I.)}\end{split}\]

Assuming the following formula for the hyperfine contact field produced by the nearest neighboring magnetic atom

\[B_c = N_A A_{cont} \boldsymbol{\mu}_e\]

where \(A_{cont}\) is expressed in mol/emu and \(\boldsymbol{\mu}_e\) is in emu. As a consequence:

\[A_{cont} = \frac{8 \pi |\Psi (0)|^2}{3 N_A}\]

Assuming 1 mol/emu, the value for A in \({\buildrel _{\circ} \over {\mathrm{A}}}^{-3}\) is

\[A_{cont} = \frac{1 \mathrm{mol/emu} * 3 N_A}{8 \pi} = 7.188E22 cm^{-3} = 0.071884019 {\buildrel _{\circ} \over {\mathrm{A}}}^{-3}\]