Kubo–Toyabe

The Kubo–Toyabe family describes depolarisation by a random static (or stochastically fluctuating) distribution of local fields, with the characteristic zero-field dip and \(1/3\) tail. The notation follows Chapter 5 of Blundell, De Renzi, Lancaster and Pratt: the Gaussian width is \(\Delta\) (μs⁻¹), the Lorentzian half-width \(a_L\) (μs⁻¹), the fluctuation (hop) rate \(\nu\) (MHz ≡ μs⁻¹), and the applied longitudinal field \(B_L\) (Gauss). Two universal limits bracket every dynamic member of the family:

• \(\nu \to 0\) recovers the static function (with its zero-field \(1/3\) tail);

• \(\nu \gg \Delta\) gives motional narrowing — exponential decay with rate \(2\Delta^2/\nu\) (Gaussian, zero field), washing the tail away.

A longitudinal field adds a Larmor term \(\omega_0 = \gamma_\mu B_L\) that decouples the muon (\(G \to 1\) as \(B_L \to \infty\)); recovery of the polarisation under a small decoupling field is the unambiguous experimental signature that the local field is static. For an end-to-end walk-through see LF decoupling and static-vs-dynamic field distributions.

StaticGKT_ZF

\[A(t) = A\left[\tfrac{1}{3} + \tfrac{2}{3}\left(1-\Delta^2 t^2\right)e^{-\Delta^2 t^2/2}\right]\]

The zero-field static Gaussian Kubo–Toyabe function: the foundational response of a host in which a static, isotropic, Gaussian-distributed local field (typically randomly oriented nuclear moments) dominates. The \(1/3\) tail has a geometric origin — one third of the ensemble has its spin parallel to the local field and is not depolarised, while the remaining two thirds precess and dephase into the dip at \(t = \sqrt{3}/\Delta\). The width

\[\Delta = \gamma_\mu\sqrt{\langle B^2\rangle}\]

sets both the dip position and the early-time Gaussian behaviour, \(1 - \Delta^2 t^2 + \dots\). For metals and diamagnetic hosts with only nuclear moments \(\Delta \sim 0.1{-}0.5\;\mu s^{-1}\); appreciably larger values imply an electronic contribution.

Name	Symbol	Unit	Description
A	\(A\)	%	Component asymmetry amplitude.
Delta	\(\Delta\)	μs⁻¹	Static Gaussian field-distribution width.

A good seed follows directly from the data: the dip sits at \(t_{\min} = \sqrt{3}/\Delta\). If the data window does not extend past the dip, the function is indistinguishable from a Gaussian with \(\sigma = \Delta/\sqrt{2}\) (Gaussian); if the data decay monotonically through the plateau value, the field is partly dynamic — use DynamicGaussianKT. Common composites are StaticGKT_ZF + Constant and StaticGKT_ZF * Exponential + Constant when an additional dynamic channel modulates the static recovery. A standalone StaticGKT_ZF model is available in the MODELS registry.

References

• R. Kubo and T. Toyabe, in Magnetic Resonance and Relaxation, edited by R. Blinc (North-Holland, Amsterdam, 1967), p. 810.

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

LongitudinalFieldKT

Synthetic Ag polycrystal LF series with Δ ≈ 0.39 μs⁻¹ and B_L = 0, 5, 10, 25, 50 G. The 0 G run shows the characteristic 1/3 tail; as B_L grows the muon spins decouple from the nuclear dipolar field and the polarisation recovers toward unity.

\[A(t) = A\left\{1 - \frac{2\Delta^2}{\omega_0^2}\left[1 - e^{-\Delta^2 t^2/2} \cos(\omega_0 t)\right] + \frac{2\Delta^4}{\omega_0^3}\int_0^t e^{-\Delta^2\tau^2/2} \sin(\omega_0\tau)\,d\tau\right\}, \qquad \omega_0 = \gamma_\mu B_L\]

The static Gaussian Kubo–Toyabe function in a longitudinal field — the workhorse for magnetically disordered hosts where the local field is static on the muon time scale (frozen spin systems, dilute nuclear-dipole hosts) and the experiment sweeps \(B_L\) through the decoupling crossover \(\gamma_\mu B_L \sim \Delta\) to extract \(\Delta\). The \(B_L \to 0\) limit recovers StaticGKT_ZF exactly; at large \(B_L\) the polarisation decouples toward unity. If the polarisation does not recover with field, the local field is dynamic — use DynamicGaussianKT or Keren.

Name	Symbol	Unit	Description
A	\(A\)	%	Component asymmetry amplitude.
Delta	\(\Delta\)	μs⁻¹	Static Gaussian field-distribution width.
B_L	\(B_L\)	G	Applied longitudinal field.

When a dataset’s metadata carries a known applied field, the fit panel initialises B_L from it; fix B_L whenever it is not the quantity of interest. \(\Delta\) is partially degenerate with the amplitude in a single run — pin it with a decoupling field sweep, or a global fit sharing \(\Delta\) across runs (Global fit wizard). The oscillatory integral is evaluated for all requested times at once by cumulative trapezoidal integration on a shared fine grid (accurate to better than 10⁻⁶); for zero-field-only data use the cheaper StaticGKT_ZF directly. A standalone LFKuboToyabe model is available in the MODELS registry.

References

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

• 1. 4. Hillier and R. Cywinski, Appl. Magn. Reson. 13, 95 (1997).

DynamicGaussianKT

The strong-collision (Markovian) dynamic generalisation of the Gaussian Kubo–Toyabe function: the static field of width \(\Delta\) reorients stochastically at rate \(\nu\) — muon hopping, ionic motion, or thermally fluctuating moments. The dynamic polarisation is obtained from the static function \(G^{\mathrm{s}}(t)\) by the strong-collision relation

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

solved on a uniform grid (trapezoidal rule) with the step chosen so the result is grid-independent to better than 0.5 %; solutions are cached per \((\Delta, \nu, B_L, t_{\max})\). This is the standard model for extracting a hop/fluctuation rate and its activation energy in metals (Cu) and ionic conductors. \(\nu \to 0\) recovers the static (LF) function; \(\nu \gg \Delta\) approaches exponential decay at rate \(2\Delta^2/\nu\).

Name	Symbol	Unit	Description
A	\(A\)	%	Component asymmetry amplitude.
Delta	\(\Delta\)	μs⁻¹	Static Gaussian field-distribution width.
nu	\(\nu\)	MHz	Fluctuation (hop) rate.
B_L	\(B_L\)	G	Applied longitudinal field.

In the fast/intermediate regime the analytic Keren function is an excellent and cheaper alternative. A standalone DynamicGaussianKT model is available in the MODELS registry.

References

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

DynamicLorentzianKT

The Lorentzian analogue of DynamicGaussianKT, for dilute or randomly diluted moments (spin glasses, dilute-spin systems) whose local-field distribution is Lorentzian with half-width \(a_L\) rather than Gaussian. The zero-field static limit is the analytic Lorentzian Kubo–Toyabe,

\[G^{\mathrm{s}}(t) = \tfrac{1}{3} + \tfrac{2}{3}(1 - a_L t)\,e^{-a_L t},\]

and dynamics are added with the same strong-collision solver. The longitudinal-field static line shape has no closed form; it is computed from the stochastic field average over an isotropic Lorentzian distribution, with the angular and precession integrals done analytically so only a single smooth 1-D quadrature remains. The result is accurate to ≈ 0.2 % for \(B_L \gtrsim 20\) G (≈ 0.3–0.5 % near 5 G) — well below the statistical scatter of typical data; the line shape is cached, but it remains the most expensive member of the family, so fix \(B_L\) from the known applied field where possible.

Name	Symbol	Unit	Description
A	\(A\)	%	Component asymmetry amplitude.
a_L	\(a_L\)	μs⁻¹	Lorentzian field-distribution half-width.
nu	\(\nu\)	MHz	Fluctuation (hop) rate.
B_L	\(B_L\)	G	Applied longitudinal field.

A standalone DynamicLorentzianKT model is available in the MODELS registry.

References

• Y. J. Uemura, T. Yamazaki, D. R. Harshman, M. Senba, and E. J. Ansaldo, Phys. Rev. B 31, 546 (1985).

GaussianBroadenedKT

\[A(t) = A\int d\Delta'\,p(\Delta')\, G^{\mathrm{LF}}_{\mathrm{KT}}(t;\Delta',B_L), \qquad p = \mathcal{N}\!\left(\Delta,\,(w_\Delta\Delta)^2\right)\]

The static (longitudinal-field) Gaussian Kubo–Toyabe averaged over a Gaussian distribution of widths \(\Delta\) — for disordered hosts where a single-width KT fit is qualitatively right but the dip is too sharp and the \(1/3\)-tail recovery too pronounced: structurally disordered systems, dilute magnetic alloys, or several inequivalent muon sites. The fractional standard deviation \(w_\Delta\) is the broadening parameter (\(w_\Delta = 0\) reduces exactly and continuously to LongitudinalFieldKT); the average is evaluated by Gauss–Hermite quadrature directly at the requested times, vectorised over the nodes. WiMDA’s Gau broad KT “rel width” equals \(w_\Delta\sqrt{2}\).

Name	Symbol	Unit	Description
A	\(A\)	%	Component asymmetry amplitude.
Delta	\(\Delta\)	μs⁻¹	Mean Gaussian field-distribution width.
B_L	\(B_L\)	G	Applied longitudinal field.
w_rel	\(w_\Delta\)	—	Fractional standard deviation of Δ.

Beware a fundamental ambiguity: width broadening and dynamics both fill in the dip and soften the tail, and a single spectrum rarely distinguishes them — vary temperature or field before preferring this model over DynamicGaussianKT.

References

• 4. 18. Noakes and G. M. Kalvius, Phys. Rev. B 56, 2352 (1997).