.. _fit-relaxation: 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 :math:`1/3` tail lives under :doc:`kubo_toyabe`. .. _fit-exponential: Exponential ----------- .. math:: 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 :math:`\nu` of the local field is large compared with its static second moment :math:`\Delta`, the depolarisation collapses from a Kubo–Toyabe shape onto a pure exponential with rate :math:`\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`` :math:`A` % Component asymmetry amplitude. ``Lambda`` :math:`\lambda` μs⁻¹ Exponential relaxation rate. ========== =============== ===== =========================================== Both parameters are constrained non-negative by default. For a paramagnetic salt at room temperature :math:`\lambda \sim 0.1{-}1\;\mu s^{-1}` is typical; above roughly :math:`\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 :math:`\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). .. _fit-gaussian: Gaussian -------- .. math:: 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 :math:`1/\sigma`. Expanding the static Gaussian Kubo–Toyabe function (:ref:`fit-static-gkt-zf`) for :math:`\Delta t \ll 1` gives :math:`1 - \tfrac{1}{2}\Delta^2 t^2`, which matches :math:`e^{-(\sigma t)^2}` to leading order with :math:`\sigma = \Delta/\sqrt{2}` — note the convention, since rates quoted for the :math:`e^{-\Delta^2 t^2/2}` form differ by :math:`\sqrt{2}`. The rate is set by the second moment of the field distribution at the muon site, :math:`\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`` :math:`A` % Component asymmetry amplitude. ``sigma`` :math:`\sigma` μs⁻¹ Gaussian relaxation rate. ========= =============== ===== =========================================== For polycrystalline samples with only nuclear moments, typical values lie in the range :math:`\sigma \sim 0.1{-}0.5\;\mu s^{-1}`, up to about :math:`1\;\mu s^{-1}` when light, high-moment nuclei such as :sup:`1`\ H or :sup:`19`\ F 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 :doc:`nuclear_dipolar` instead. If the data reach :math:`t \gtrsim 1/\sigma` without flattening onto a :math:`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). .. _fit-stretched-exponential: StretchedExponential -------------------- .. math:: A(t) = A\,e^{-(\lambda t)^{\beta}} The Kohlrausch–Williams–Watts form, interpolating continuously between a simple exponential (:math:`\beta = 1`) and a Gaussian (:math:`\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 :math:`\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 :math:`\beta` is the physically informative parameter. :math:`\beta` near 1 indicates a narrow distribution of dynamic rates; :math:`\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 :math:`\beta` drifting from 1 toward :math:`\approx 1/3` on cooling through a transition is one of the canonical μSR signatures of glassy freezing. ========== =============== ===== ========================================== Name Symbol Unit Description ========== =============== ===== ========================================== ``A`` :math:`A` % Component asymmetry amplitude. ``Lambda`` :math:`\lambda` μs⁻¹ Relaxation rate scale. ``beta`` :math:`\beta` — Stretching exponent. ========== =============== ===== ========================================== Constrain :math:`\beta` away from 0 (``min=0.1`` or similar) and cap it at 2. :math:`\lambda` and :math:`\beta` are strongly correlated: changing :math:`\beta` rescales the effective rate, so the marginal uncertainty on :math:`\lambda` from the Hessian usually understates the truth. Where this matters, fix :math:`\beta` at a physically motivated value or quote the joint covariance. A standalone ``StretchedExponential`` model is available in the ``MODELS`` registry. **References** - R. E. 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). - I. A. Campbell *et al.*, Phys. Rev. Lett. **72**, 1291 (1994). .. _fit-abragam: Abragam ------- .. math:: 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 :math:`\Delta` fluctuating at rate :math:`\nu`, with the limits :math:`\nu \to 0:\ \exp(-\Delta^2 t^2/2)` (static Gaussian) and :math:`\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`` :math:`A` % Component asymmetry amplitude. ``Delta`` :math:`\Delta` μs⁻¹ Static Gaussian field-distribution width. ``nu`` :math:`\nu` MHz Fluctuation (hop) rate. ========= =============== ===== =========================================== **References** - A. Abragam, *The Principles of Nuclear Magnetism* (Oxford University Press, Oxford, 1961), Ch. X. .. _fit-keren: Keren ----- .. math:: 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 :math:`\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 (:math:`\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 :math:`B_L = 0`. Prefer the full :ref:`fit-dynamic-gaussian-kt` when fluctuations are slow (:math:`\nu \lesssim \Delta`) or the static :math:`1/3` tail matters. Evaluated in closed form (machine precision). ========= =============== ===== =========================================== Name Symbol Unit Description ========= =============== ===== =========================================== ``A`` :math:`A` % Component asymmetry amplitude. ``Delta`` :math:`\Delta` μs⁻¹ Static Gaussian field-distribution width. ``nu`` :math:`\nu` MHz Fluctuation (hop) rate. ``B_L`` :math:`B_L` G Applied longitudinal field. ========= =============== ===== =========================================== **References** - A. Keren, Phys. Rev. B **50**, 10039 (1994). .. _fit-risch-kehr: RischKehr --------- .. math:: 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 :math:`(\pi\Gamma t)^{-1/2}` long-time tail. A stretched-exponential fit drifting toward :math:`\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 :math:`\Gamma` then has a microscopic interpretation in terms of the intrachain diffusion rate and hyperfine coupling. ========= =============== ===== =========================================== Name Symbol Unit Description ========= =============== ===== =========================================== ``A`` :math:`A` % Component asymmetry amplitude. ``Gamma`` :math:`\Gamma` μs⁻¹ Risch–Kehr relaxation rate. ========= =============== ===== =========================================== :math:`\Gamma` is constrained non-negative. The implementation evaluates the scaled complementary error function (``erfcx``), which is numerically stable for all :math:`\Gamma t` — there is no asymptotic-branch switch (WiMDA changes form at :math:`\Gamma t = 20`), and WiMDA's mirrored branch for negative rates is intentionally not reproduced. **References** - R. Risch and K. W. Kehr, Phys. Rev. B **46**, 5246 (1992).