Fit functions

This chapter documents every fit function (component) available in the fit-function builder. The pages mirror the submenus of the builder’s component picker exactly, so the section a function is documented under is the submenu it lives under in the GUI:

Components are building blocks: each evaluates a normalised polarisation or relaxation shape scaled by its amplitude, and a fittable model is assembled by combining components with +, -, *, / and fraction groups in the builder (see Composite models). The two canonical patterns are an additive background,

StaticGKT_ZF + Constant

and a multiplicative relaxation envelope on an oscillating or entangled-state signal,

FmuF_Linear * Exponential + Constant

A small number of single-channel models (with an explicit baseline parameter) are also exposed through the Python MODELS registry for scripted fits; these are noted on the relevant component sections.

Conventions

Time is measured in μs, frequencies in MHz, fields in Gauss, distances in Å, and relaxation rates in μs⁻¹; phases are in radians. The muon gyromagnetic ratio is \(\gamma_\mu/2\pi = 135.539\) MHz/T. Component notation is kept consistent with Blundell, De Renzi, Lancaster and Pratt, Muon Spectroscopy: An Introduction (Oxford University Press, Oxford, 2022); each function’s section cites the original literature for its functional form.

The builder only offers components belonging to the analysis domain being fitted: time-domain representations see the time-domain categories (Relaxation, Oscillation, Kubo–Toyabe, Muonium, Nuclear dipolar, Background), while Fourier spectra are fitted with the Frequency domain components.

Documentation policy

Every fit component must be documented in the page of this chapter that corresponds to its category in the component picker, in the same pedagogical style: when the model is physically relevant, the mathematical form (rendered with LaTeX), parameter table, practical fitting guidance, and APS-style references to the original literature. This placement is enforced by tests/test_fit_function_docs.py — adding a new component without documenting it in the matching page fails the test suite.

Migrating from WiMDA

Asymmetry’s time-domain catalogue covers everything WiMDA’s fitting menu offers (built-in oscillation/relaxation grid plus the muonium and dipolar user-function libraries; see docs/porting/wimda-fit-function-parity/). A few WiMDA conveniences are deliberately not separate components because parameter constraints already express them:

Scaled frequency rotation (otScaledFRotation) — a cosine at frequency × scale. Use Oscillatory and tie the frequency with an affine tie (Parameter ties (links and equal spacing)), e.g. for a component locked to 1.2× the first component’s frequency:

from asymmetry.core.fitting import AffineTie, Parameter

Parameter("frequency_2", value=..., tie=AffineTie(main="frequency_1", scale=1.2))

or use a link group when the ratio is exactly 1.

Frequency-normalised stretched exponential (rtFstr) — a stretched exponential whose rate scales with the component’s precession frequency (Lambda = 2π·c·frequency). This couples two fitted parameters multiplicatively, so it is not an affine tie; it needs the general expression constraint (Parameter.expr), which is reserved but not yet evaluated by the engine. Until then, fit Lambda directly, or fix frequency and use an affine tie AffineTie(main="frequency", scale=2*pi*c) for a known c.

Gaussian variants (rtGau2, rtSig2) — reparameterisations of Gaussian \(e^{-(\sigma t)^2}\): WiMDA’s Gau2 \(e^{-(\sigma' t)^2/2}\) corresponds to \(\sigma = \sigma'/\sqrt{2}\) (also the mapping to the textbook’s \(e^{-\Delta^2 t^2/2}\) convention), and Sig2 fits \(s_2 = \sigma^2\) directly.

RIKEN BeCu pressure cell (BeCu ZF) — exactly the composite:

StaticGKT_ZF + Exponential

with the amplitude split between the two terms. The companion empirical BeCu LF 110G calibration curve (a polynomial λ(T) for one cell at one field) is instrument calibration data rather than a fit function and was not ported.