Relaxation

Relaxation components describe the decay of the muon-spin polarisation without coherent oscillation. They are used standalone for zero- and longitudinal-field relaxation, and multiplicatively as damping envelopes on oscillating components (Oscillatory * Exponential, FmuF_Linear * Exponential, …). This page covers the simple envelopes (Exponential, Gaussian, StretchedExponential), the dynamic-crossover functions (Abragam, Keren), and the 1D-transport function RischKehr. Relaxation from static field distributions with the characteristic \(1/3\) tail lives under Kubo–Toyabe.

Exponential

\[A(t) = A\,e^{-\lambda t}\]

The workhorse depolarisation function for any system in which the muon experiences a rapidly fluctuating local field. It is the Redfield motional-narrowing limit of slower-relaxation forms: when the fluctuation rate \(\nu\) of the local field is large compared with its static second moment \(\Delta\), the depolarisation collapses from a Kubo–Toyabe shape onto a pure exponential with rate \(\lambda \simeq 2\Delta^2/\nu\). In practice this covers paramagnetic spin fluctuations, electronic relaxation in metals and semiconductors where coherent dynamics are absent, and the high-temperature regime of essentially every diffusive system once the correlation time has dropped below the muon precession scale. The same shape also arises from a static but dilute (Lorentzian) field distribution in transverse field.

Name	Symbol	Unit	Description
A	\(A\)	%	Component asymmetry amplitude.
Lambda	\(\lambda\)	μs⁻¹	Exponential relaxation rate.

Both parameters are constrained non-negative by default. For a paramagnetic salt at room temperature \(\lambda \sim 0.1{-}1\;\mu s^{-1}\) is typical; above roughly \(\lambda \sim 10\;\mu s^{-1}\) the relaxation occurs almost entirely inside the instrumental deadtime window and Lambda becomes ill-conditioned — in that regime the signal is better described as missing initial asymmetry than as a fast exponential. When \(\lambda\) is recovered alongside a slow Gaussian or Kubo–Toyabe channel, expect correlation between the two rates; LF-decoupling data on the same sample is usually the cleanest way to break the degeneracy.

A standalone ExponentialRelaxation model (with explicit baseline) is available in the Python MODELS registry for scripted single-channel fits.

References

• A. Yaouanc and P. Dalmas de Réotier, Muon Spin Rotation, Relaxation, and Resonance: Applications to Condensed Matter (Oxford University Press, Oxford, 2011).

Gaussian

\[A(t) = A\,e^{-(\sigma t)^2}\]

The natural relaxation envelope when the muon ensemble experiences a static Gaussian distribution of local fields and the experimental time window is short compared with \(1/\sigma\). Expanding the static Gaussian Kubo–Toyabe function (StaticGKT_ZF) for \(\Delta t \ll 1\) gives \(1 - \tfrac{1}{2}\Delta^2 t^2\), which matches \(e^{-(\sigma t)^2}\) to leading order with \(\sigma = \Delta/\sqrt{2}\) — note the convention, since rates quoted for the \(e^{-\Delta^2 t^2/2}\) form differ by \(\sqrt{2}\). The rate is set by the second moment of the field distribution at the muon site, \(\sigma = \gamma_\mu\sqrt{\langle B^2\rangle}\,/\sqrt{2}\) in this convention, and the same form describes the Gaussian damping envelope of a TF precession line.

Name	Symbol	Unit	Description
A	\(A\)	%	Component asymmetry amplitude.
sigma	\(\sigma\)	μs⁻¹	Gaussian relaxation rate.

For polycrystalline samples with only nuclear moments, typical values lie in the range \(\sigma \sim 0.1{-}0.5\;\mu s^{-1}\), up to about \(1\;\mu s^{-1}\) when light, high-moment nuclei such as ¹H or ¹⁹F are dense at the muon site. Values substantially larger than this almost always signal that a coupled F–μ–F (or similar) entangled state is being mis-fitted as a Gaussian envelope; use the dedicated components in Nuclear dipolar instead. If the data reach \(t \gtrsim 1/\sigma\) without flattening onto a \(1/3\) tail, the relaxation is not purely static-Gaussian — try StretchedExponential or the full StaticGKT_ZF.

A standalone GaussianRelaxation 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).

StretchedExponential

\[A(t) = A\,e^{-(\lambda t)^{\beta}}\]

The Kohlrausch–Williams–Watts form, interpolating continuously between a simple exponential (\(\beta = 1\)) and a Gaussian (\(\beta = 2\)). It is the standard phenomenological relaxation function for systems in which the muon ensemble samples a distribution of relaxation rates rather than a single \(\lambda\) — spin glasses near and below freezing, dilute and concentrated magnetic alloys with broad RKKY-coupling distributions, frustrated magnets with quenched disorder.

The stretching exponent \(\beta\) is the physically informative parameter. \(\beta\) near 1 indicates a narrow distribution of dynamic rates; \(\beta = 1/2\) is the Walstedt–Walker form expected for dilute, broadly distributed static moments in the fast-fluctuation limit; values approaching 2 indicate a static near-Gaussian distribution better fitted with a Kubo–Toyabe form. A fitted \(\beta\) drifting from 1 toward \(\approx 1/3\) on cooling through a transition is one of the canonical μSR signatures of glassy freezing.

Name	Symbol	Unit	Description
A	\(A\)	%	Component asymmetry amplitude.
Lambda	\(\lambda\)	μs⁻¹	Relaxation rate scale.
beta	\(\beta\)	—	Stretching exponent.

Constrain \(\beta\) away from 0 (min=0.1 or similar) and cap it at 2. \(\lambda\) and \(\beta\) are strongly correlated: changing \(\beta\) rescales the effective rate, so the marginal uncertainty on \(\lambda\) from the Hessian usually understates the truth. Where this matters, fix \(\beta\) at a physically motivated value or quote the joint covariance. A standalone StretchedExponential model is available in the MODELS registry.

References

• 18. 5. Walstedt and L. R. Walker, Phys. Rev. B 9, 4857 (1974).

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

• 1. 1. Campbell et al., Phys. Rev. Lett. 72, 1291 (1994).

Abragam

\[A(t) = A\,\exp\!\left[-\frac{\Delta^2}{\nu^2} \left(e^{-\nu t} - 1 + \nu t\right)\right]\]

The Gaussian-to-exponential crossover function: a Gaussian static width \(\Delta\) fluctuating at rate \(\nu\), with the limits \(\nu \to 0:\ \exp(-\Delta^2 t^2/2)\) (static Gaussian) and \(\nu \gg \Delta:\ \exp(-(\Delta^2/\nu)\,t)\) (motionally narrowed exponential). It is the classic model for extracting a hop or correlation rate from a transverse-field line shape — the textbook example being the Gaussian-to-Lorentzian change of the Cu line shape as muon diffusion sets in on warming. Evaluated in closed form (machine precision).

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

References

• A. Abragam, The Principles of Nuclear Magnetism (Oxford University Press, Oxford, 1961), Ch. X.

Keren

\[A(t) = A\,e^{-\Gamma(t)},\qquad \Gamma(t)=\frac{2\Delta^2}{(\omega_0^2+\nu^2)^2} \Big[(\omega_0^2+\nu^2)\,\nu t +(\omega_0^2-\nu^2)(1-e^{-\nu t}\cos\omega_0 t) -2\nu\omega_0 e^{-\nu t}\sin\omega_0 t\Big]\]

with \(\omega_0 = \gamma_\mu B_L\). Keren’s analytic generalisation of the Abragam function to a longitudinal field: an accurate strong-collision result in the fast/intermediate fluctuation regime (\(\nu \gtrsim \Delta\)) that avoids the numerical convolution of the full dynamic Kubo–Toyabe. It is the standard model for longitudinal-field decoupling analyses (e.g. ionic diffusion) and reduces to the Abragam function at \(B_L = 0\). Prefer the full DynamicGaussianKT when fluctuations are slow (\(\nu \lesssim \Delta\)) or the static \(1/3\) tail matters. Evaluated in closed form (machine precision).

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.

References

• 1. Keren, Phys. Rev. B 50, 10039 (1994).

RischKehr

\[A(t) = A\, e^{\Gamma t}\,\mathrm{erfc}\!\left(\sqrt{\Gamma t}\right)\]

Relaxation of the muon (or muonium) polarisation by a spin carrier diffusing in one dimension — a polaron moving along a conducting-polymer chain, or an excitation confined to a structural channel. The 1D random walk keeps returning the carrier to the muon, so instead of an exponential the polarisation acquires a \((\pi\Gamma t)^{-1/2}\) long-time tail. A stretched-exponential fit drifting toward \(\beta \approx 1/2\) at early times is the usual hint to try this form; prefer it over a stretched exponential whenever 1D transport is physically motivated, since \(\Gamma\) then has a microscopic interpretation in terms of the intrachain diffusion rate and hyperfine coupling.

Name	Symbol	Unit	Description
A	\(A\)	%	Component asymmetry amplitude.
Gamma	\(\Gamma\)	μs⁻¹	Risch–Kehr relaxation rate.

\(\Gamma\) is constrained non-negative. The implementation evaluates the scaled complementary error function (erfcx), which is numerically stable for all \(\Gamma t\) — there is no asymptotic-branch switch (WiMDA changes form at \(\Gamma t = 20\)), and WiMDA’s mirrored branch for negative rates is intentionally not reproduced.

References

• 18. Risch and K. W. Kehr, Phys. Rev. B 46, 5246 (1992).