Nuclear dipolar

Synthetic PbF₂ ZF dataset with r:sub:`μF` = 1.17 Å and a λ = 0.3 μs⁻¹ relaxation envelope (literature-plausible for F–μ–F powders; see make_pbf2_fmuf in docs/screenshots/data/archetypes.py for the reasoning), fitted with the wizard-recommended FmuF_Linear * Exponential + Constant composite (converged, r_μF = 1.17 Å, λ = 0.30 μs⁻¹, \(\chi^2_\nu = 0.98\)). The plot is shown over the first 10 μs, where both the beats and their decay are resolved (the underlying dataset spans 20 μs). PbF₂ is a particularly clean F–μ–F host: the heavy Pb nuclei carry no significant nuclear moment, so the analytical FmuF_Linear component captures the full polarisation.

When the implanted muon stops close to one or a few nuclei with substantial moments, the muon and nuclear spins evolve as an entangled few-spin system under the magnetic dipole–dipole interaction, producing characteristic slow beats in the zero-field (or weak-LF) asymmetry. These beats are the textbook signature of a well-defined stopping site, and their envelope encodes the site geometry directly through the muon–nucleus distances. All components on this page derive from the dipolar Hamiltonian

\[H_{ij} = \omega_{ij}\left[\mathbf{S}_i\cdot\mathbf{S}_j - 3\left(\mathbf{S}_i\cdot\hat{\mathbf{r}}_{ij}\right) \left(\mathbf{S}_j\cdot\hat{\mathbf{r}}_{ij}\right)\right], \qquad \omega_{ij} = \frac{\mu_0}{4\pi}\,\gamma_i\gamma_j\hbar\,r_{ij}^{-3},\]

with distances entered in Å. They are polarisation building blocks, not full asymmetry models: the expected workflow is to multiply by a relaxation envelope and add a background, e.g. FmuF_Linear * Exponential + Constant. Distinguishing an F–μ–F beat from simple two-frequency precession is usually fastest in the frequency domain — F–μ–F gives a three-line (collinear) or multi-line (general) pattern with characteristic spacing, while precession gives one line per site (Fourier analysis).

Choosing a model: use MuF/ProtonDipole/ElectronDipole when one spin-½ partner dominates (or DipolarPairField to fit the dipolar field directly); FmuF_Linear for the classic symmetric linear centre; FmuF_General for bent or asymmetric two-fluorine geometries; FmuF_Triangle when a third fluorine matters; DynamicFmuF when the F–μ–F signal is washed out by muon hopping; and DipolarSpinJ for a single quadrupolar (\(J > 1/2\)) partner such as Cu or Nb.

MuF

\[D_z(t) = \frac{1}{6}\left[1 + 2\cos\left(\frac{\omega_d t}{2}\right) + \cos(\omega_d t) + 2\cos\left(\frac{3\omega_d t}{2}\right)\right]\]

The entangled two-spin μ–F pair: a muon strongly coupled to a single dominant ¹⁹F nucleus (\(I = 1/2\), 100 % abundant, no quadrupole), giving the characteristic three-frequency pattern. This is Case I of the molecular-magnet analysis of Lancaster et al. — the relevant model when a symmetric site between two fluorines is chemically disfavoured, as in CuF₂(H₂O)₂(pyz).

Name	Symbol	Unit	Description
A	\(A\)	%	Component asymmetry amplitude.
r_muF	\(r_{\mu F}\)	Å	Muon–fluorine distance.

Not intended for cases where two fluorines contribute comparably, or where an extra nearby nucleus (e.g. a proton) materially affects the spectrum.

References

• 20. Lancaster et al., Phys. Rev. Lett. 99, 267601 (2007).

ProtonDipole

The same two-spin form as MuF with the proton gyromagnetic ratio and an optional transverse damping \(\lambda_T\) applied to the oscillating \(5/6\) part only (the non-oscillating \(1/6\) term arises from field components parallel to the muon spin, which do not dephase):

\[A(t) = \frac{A}{6}\left[1 + e^{-\lambda_T t}\left( 2\cos\tfrac{\omega_d t}{2} + \cos\omega_d t + 2\cos\tfrac{3\omega_d t}{2}\right)\right]\]

Use for stopping sites adjacent to a single dominant proton — hydroxyl groups, hydrides, water of crystallisation. The fitted \(r_{\mu H}\) is the muon–proton distance through the \(r^{-3}\) coupling; \(\lambda_T\) absorbs weaker couplings to more distant nuclei. Proton moments are roughly ten times weaker than ¹⁹F at the same distance, so resolvable oscillations require a close, well-defined μ–H pair.

Name	Symbol	Unit	Description
A	\(A\)	%	Component asymmetry amplitude.
r_muH	\(r_{\mu H}\)	Å	Muon–proton distance.
lambda_T	\(\lambda_T\)	μs⁻¹	Transverse damping of the oscillation.

References

• 16. 6. Meier, Hyperfine Interact. 18, 427 (1984).

ElectronDipole

As ProtonDipole with the electron gyromagnetic ratio: a muon coupled by the dipolar interaction to a single localised electronic moment at distance \(r_{\mu e}\), static on the muon time scale — a dilute paramagnetic defect or rare-earth ion adjacent to the muon site. Frequencies are about three orders of magnitude higher than the nuclear pairs at the same distance, so \(r_{\mu e}\) of several Å still gives MHz-scale oscillations. Not appropriate for muonium (contact hyperfine dominates — use the Muonium components) or for dense magnets (use Oscillatory or Bessel with an internal field). Parameters: A (%), r_mue (Å), lambda_T (μs⁻¹).

References

• 16. 6. Meier, Hyperfine Interact. 18, 427 (1984).

DipolarPairField

The same spin-½ pair polarisation parameterised by the dipolar field at the muon, \(\omega_d = \gamma_\mu B_{\mathrm{dip}}\), for when it is preferable to fit the field directly rather than assume a nucleus and distance — e.g. when the coupled nucleus is unknown, or when comparing with dipolar-field calculations of candidate sites. A fitted \(B_{\mathrm{dip}}\) converts to a distance through \(B_{\mathrm{dip}} = \mu_0\hbar\gamma_j/(4\pi r^3)\) once the partner nucleus is identified. Parameters: A (%), B_dip (G), lambda_T (μs⁻¹).

References

• 16. 6. Meier, Hyperfine Interact. 18, 427 (1984).

DipolarSpinJ

Zero-field polycrystalline precession of a muon coupled to one nucleus of spin \(J > 1/2\) with both dipolar and quadrupolar interactions. The implanted μ⁺ produces an electric field gradient that quadrupole-splits the neighbouring nucleus, so the two-spin spectrum depends on the quadrupolar splitting \(f_{\mathrm{quad}}\) (sign-sensitive) as well as the dipolar coupling \(f_{\mathrm{dip}}\). The component implements the closed-form eigen-solution of Celio and Meier, averaged as \((P_z + 2P_x)/3\) for a polycrystal. Typical applications are μ⁺–⁶³Cu (\(J = 3/2\)) and μ⁺–⁹³Nb (\(J = 9/2\)) pairs in metals.

Name	Symbol	Unit	Description
A	\(A\)	%	Component asymmetry amplitude.
f_dip	\(f_{\mathrm{dip}}\)	MHz	Dipolar coupling frequency.
f_quad	\(f_{\mathrm{quad}}\)	MHz	Quadrupolar splitting.
J_spin	\(J\)	—	Nuclear spin (hold fixed).

\(J\) is fixed by default (the model is piecewise-constant in it); set it to the known nuclear spin. For \(J = 1/2\) the quadrupole is inactive and the function reduces exactly to the spin-½ pair. For more than one strongly coupled nucleus use the F–μ–F family or a dedicated multi-spin model. Note that the implementation uses the signed block mixing angle, verified against exact diagonalisation; WiMDA’s Dip gen ZF PCR drops the sign and is wrong for every \(J > 1/2\), so fitted parameters are not comparable with WiMDA for those spins.

References

• 13. Celio and P. F. Meier, Hyperfine Interact. 18, 435 (1984).

• 15. Hartmann, Phys. Rev. Lett. 39, 832 (1977).

FmuF_Linear

\[G_{F\mu F}(t)=\frac{1}{6}\left[3 + \cos(\sqrt{3}\,\omega_d t) + \left(1-\frac{1}{\sqrt{3}}\right) \cos\left(\frac{3-\sqrt{3}}{2}\,\omega_d t\right) + \left(1+\frac{1}{\sqrt{3}}\right) \cos\left(\frac{3+\sqrt{3}}{2}\,\omega_d t\right)\right]\]

The classic collinear three-spin F–μ–F centre of ionic fluorides: the muon pulls two fluorines together into a hydrogen-bond-like linear configuration and sits midway between them. This closed form (which neglects the weak F–F coupling) is the correct starting point for LiF, NaF, CaF₂, BaF₂ and similar hosts; Brewer et al. extracted typical μ–F distances of about 1.17 Å (F–F separation ≈ 2.34–2.38 Å). Parameters: A (%), r_muF (Å). Do not use for inequivalent fluorines or bent geometries — use FmuF_General.

References

• 10. 8. Brewer et al., Phys. Rev. B 33, 7813 (1986).

FmuF_General

For a bent or asymmetric two-fluorine geometry there is no compact closed form. The polarisation is computed numerically: the full three-spin dipolar Hamiltonian (including the F–F coupling) is diagonalised for each powder orientation and

\[D_z(t) = \frac{1}{N}\sum_{m,n} \left|\langle m|\sigma_z^{\mu}|n\rangle\right|^2 \cos\left[(\omega_m-\omega_n)t\right]\]

is averaged over orientations (Gauss–Legendre × uniform Euler-angle quadrature); the geometry-dependent eigenspectrum is cached. This is the model for distorted two-fluorine stopping states such as the [Cu(NO₃)(pyz)₂]PF₆ site of Lancaster et al. (\(r_1 = 0.106(3)\) nm, \(r_2 = 0.156(3)\) nm, \(\theta = 143(1)^\circ\)).

Name	Symbol	Unit	Description
A	\(A\)	%	Component asymmetry amplitude.
r1	\(r_1\)	Å	First muon–fluorine distance.
r2	\(r_2\)	Å	Second muon–fluorine distance.
theta	\(\theta\)	°	F–μ–F bond angle.

Assumes exactly three coupled spins; it does not cover configurations requiring an additional nucleus, such as the proton-coupled HF₂⁻ state (and for a third fluorine, use FmuF_Triangle).

References

• 20. Lancaster et al., Phys. Rev. Lett. 99, 267601 (2007).

• 10. 8. Brewer et al., Phys. Rev. B 33, 7813 (1986).

FmuF_Triangle

A collinear F–μ–F pair (both fluorines at \(r_{\mu F}\)) plus a third fluorine at distance \(r_3\) and angle \(\phi_3\) to the F–μ–F axis, solved exactly in the 16-dimensional four-spin space with all μ–F and F–F dipolar couplings and a full powder average. Use when second-neighbour fluorines visibly modify the F–μ–F beat pattern, as established for ionic fluorides by second-neighbour analyses. As \(r_3 \to \infty\) it approaches the collinear limit of FmuF_General (i.e. FmuF_Linear plus the F–F coupling).

Name	Symbol	Unit	Description
A	\(A\)	%	Component asymmetry amplitude.
r_muF	\(r_{\mu F}\)	Å	Muon–fluorine distance of the linear pair.
r3	\(r_3\)	Å	Distance to the third fluorine.
phi3	\(\phi_3\)	°	Angle of the third fluorine to the axis.

Unlike WiMDA’s F-u-F-F function this includes the F–F couplings and a proper powder average, so fitted distances are not directly comparable with WiMDA results. Evaluation is cached per geometry; fits are slower than the analytic F–μ–F forms.

References

• 10. 8. Brewer et al., Phys. Rev. B 33, 7813 (1986).

• 10. 13. Wilkinson and S. J. Blundell, Phys. Rev. Lett. 125, 087201 (2020).

DynamicFmuF

The collinear F–μ–F polarisation dynamicised by the strong-collision model at fluctuation rate \(\nu\):

\[G^{\mathrm{d}}(t) = G_{F\mu F}(t)\,e^{-\nu t} + \nu\int_0^t G^{\mathrm{d}}(t-t')\,G_{F\mu F}(t')\,e^{-\nu t'}\,dt'.\]

Use when an F–μ–F signal that is clear at low temperature progressively damps and loses its oscillations on warming because the muon hops away from the site (or the coupling fluctuates). \(\nu = 0\) recovers FmuF_Linear exactly; large \(\nu\) gives motional narrowing toward \(\exp(-2\omega_d^2 t/\nu)\) via an Abragam-form interpolation that keeps the model smooth in \(\nu\) across the solver crossover (seam below ~1 % at physical distances). Fitting a temperature series with shared \(r_{\mu F}\) and free \(\nu\) yields the hop rate and hence an activation energy for muon diffusion in the fluoride. Assumes the equal-distance collinear geometry of FmuF_Linear; the solver and caching follow DynamicGaussianKT.

Name	Symbol	Unit	Description
A	\(A\)	%	Component asymmetry amplitude.
r_muF	\(r_{\mu F}\)	Å	Muon–fluorine distance.
nu	\(\nu\)	MHz	Fluctuation (hop) rate.

References

• 10. 8. Brewer et al., Phys. Rev. B 33, 7813 (1986).

• R. S. Hayano, Y. J. Uemura, J. Imazato, N. Nishida, T. Yamazaki, and R. Kubo, Phys. Rev. B 20, 850 (1979).