.. _fit-kubo-toyabe: 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 :math:`1/3` tail. The notation follows Chapter 5 of Blundell, De Renzi, Lancaster and Pratt: the Gaussian width is :math:`\Delta` (μs⁻¹), the Lorentzian half-width :math:`a_L` (μs⁻¹), the fluctuation (hop) rate :math:`\nu` (MHz ≡ μs⁻¹), and the applied longitudinal field :math:`B_L` (Gauss). Two universal limits bracket every dynamic member of the family: - :math:`\nu \to 0` recovers the static function (with its zero-field :math:`1/3` tail); - :math:`\nu \gg \Delta` gives **motional narrowing** — exponential decay with rate :math:`2\Delta^2/\nu` (Gaussian, zero field), washing the tail away. A longitudinal field adds a Larmor term :math:`\omega_0 = \gamma_\mu B_L` that **decouples** the muon (:math:`G \to 1` as :math:`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 :doc:`/workflows/lf_decoupling_dynamics`. .. _fit-static-gkt-zf: StaticGKT_ZF ------------ .. math:: 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 :math:`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 :math:`t = \sqrt{3}/\Delta`. The width .. math:: \Delta = \gamma_\mu\sqrt{\langle B^2\rangle} sets both the dip position and the early-time Gaussian behaviour, :math:`1 - \Delta^2 t^2 + \dots`. For metals and diamagnetic hosts with only nuclear moments :math:`\Delta \sim 0.1{-}0.5\;\mu s^{-1}`; appreciably larger values imply an electronic contribution. ========= =============== ===== =========================================== Name Symbol Unit Description ========= =============== ===== =========================================== ``A`` :math:`A` % Component asymmetry amplitude. ``Delta`` :math:`\Delta` μs⁻¹ Static Gaussian field-distribution width. ========= =============== ===== =========================================== A good seed follows directly from the data: the dip sits at :math:`t_{\min} = \sqrt{3}/\Delta`. If the data window does not extend past the dip, the function is indistinguishable from a Gaussian with :math:`\sigma = \Delta/\sqrt{2}` (:ref:`fit-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). .. _fit-lf-kubo-toyabe: LongitudinalFieldKT ------------------- .. image:: /_generated/screenshots/lf_kt_series_plot.png :alt: Overlay of five Ag LF Kubo–Toyabe runs spanning the decoupling regime :width: 100% *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.* .. math:: 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 :math:`B_L` through the decoupling crossover :math:`\gamma_\mu B_L \sim \Delta` to extract :math:`\Delta`. The :math:`B_L \to 0` limit recovers ``StaticGKT_ZF`` exactly; at large :math:`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`` :math:`A` % Component asymmetry amplitude. ``Delta`` :math:`\Delta` μs⁻¹ Static Gaussian field-distribution width. ``B_L`` :math:`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. :math:`\Delta` is partially degenerate with the amplitude in a single run — pin it with a decoupling field sweep, or a global fit sharing :math:`\Delta` across runs (:doc:`../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). - A. D. Hillier and R. Cywinski, Appl. Magn. Reson. **13**, 95 (1997). .. _fit-dynamic-gaussian-kt: DynamicGaussianKT ----------------- The strong-collision (Markovian) dynamic generalisation of the Gaussian Kubo–Toyabe function: the static field of width :math:`\Delta` reorients stochastically at rate :math:`\nu` — muon hopping, ionic motion, or thermally fluctuating moments. The dynamic polarisation is obtained from the static function :math:`G^{\mathrm{s}}(t)` by the strong-collision relation .. math:: 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 :math:`(\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. :math:`\nu \to 0` recovers the static (LF) function; :math:`\nu \gg \Delta` approaches exponential decay at rate :math:`2\Delta^2/\nu`. ========= =============== ===== =========================================== 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. ========= =============== ===== =========================================== In the fast/intermediate regime the analytic :ref:`fit-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). .. _fit-dynamic-lorentzian-kt: 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 :math:`a_L` rather than Gaussian. The zero-field static limit is the analytic Lorentzian Kubo–Toyabe, .. math:: 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 :math:`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 :math:`B_L` from the known applied field where possible. ========= =============== ===== =========================================== Name Symbol Unit Description ========= =============== ===== =========================================== ``A`` :math:`A` % Component asymmetry amplitude. ``a_L`` :math:`a_L` μs⁻¹ Lorentzian field-distribution half-width. ``nu`` :math:`\nu` MHz Fluctuation (hop) rate. ``B_L`` :math:`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). .. _fit-gaussian-broadened-kt: GaussianBroadenedKT ------------------- .. math:: 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** :math:`\Delta` — for disordered hosts where a single-width KT fit is qualitatively right but the dip is too sharp and the :math:`1/3`-tail recovery too pronounced: structurally disordered systems, dilute magnetic alloys, or several inequivalent muon sites. The fractional standard deviation :math:`w_\Delta` is the broadening parameter (:math:`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 :math:`w_\Delta\sqrt{2}`. ========= ================= ===== ========================================= Name Symbol Unit Description ========= ================= ===== ========================================= ``A`` :math:`A` % Component asymmetry amplitude. ``Delta`` :math:`\Delta` μs⁻¹ Mean Gaussian field-distribution width. ``B_L`` :math:`B_L` G Applied longitudinal field. ``w_rel`` :math:`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** - D. R. Noakes and G. M. Kalvius, Phys. Rev. B **56**, 2352 (1997).