# 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

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