Fit models

This page summarizes the analytical expressions implemented in TGAnalysis.model1, TGAnalysis.model2, and TGAnalysis.model3.

Notation

  • \(t\): time (ps).

  • \(t_0\): time-zero shift for the primary decay channel (ps).

  • \(t_{02}, t_{03}\): optional additional time shifts used by Models 2–3 for extra exponentials (ps).

  • \(\sigma\): Gaussian instrument-response width (fitted in ps; reported as fs in params_fit).

  • \(k, k_1, k_2, k_3\): decay rates in ps \(^{-1}\).

  • \(\tau = 1/k\): decay time (reported in fs in fit summaries).

  • \(\mathrm{erf}(\cdot)\): error function.

Model 1 (model1)

A mono-exponential decay convolved with a Gaussian response plus an offset step term:

\[f_1(t) = A_1 \exp\!\left(-(t-t_0)k\right) \exp\!\left(\frac{(k\sigma)^2}{2}\right) \left[1 + \mathrm{erf}\!\left(\frac{t-t_0-k\sigma^2}{\sqrt{2}\sigma}\right)\right] + A_{\mathrm{off}} \left[1 + \mathrm{erf}\!\left(\frac{t-t_0}{\sqrt{2}\sigma}\right)\right].\]

Model 2 (model2)

Model 1 plus a second exponential decay channel:

\[f_2(t) = f_{\mathrm{exp},1}(t) + f_{\mathrm{off}}(t) + f_{\mathrm{exp},2}(t),\]

where:

\[f_{\mathrm{exp},1}(t) = A_1 \exp\!\left(-(t-t_0)k_1\right) \exp\!\left(\frac{(k_1\sigma)^2}{2}\right) \left[1 + \mathrm{erf}\!\left(\frac{t-t_0-k_1\sigma^2}{\sqrt{2}\sigma}\right)\right],\]
\[f_{\mathrm{off}}(t) = A_{\mathrm{off}} \left[1 + \mathrm{erf}\!\left(\frac{t-t_0}{\sqrt{2}\sigma}\right)\right],\]
\[f_{\mathrm{exp},2}(t) = A_2 \exp\!\left[-k_2(t-t_{02})\right] \exp\!\left(-\frac{(k_2\sigma)^2}{2}\right) \left[1 + \mathrm{erf}\!\left(\frac{t-t_{02}-k_2\sigma^2}{\sqrt{2}\sigma}\right)\right].\]

Model 3 (model3)

A sum of three Gaussian-convolved exponential channels with independent time zeros:

\[f_3(t) = A_1 e^{-(t-t_0)k_1} e^{-\frac{(k_1\sigma)^2}{2}} \left[1+\mathrm{erf}\!\left(\frac{t-t_0-k_1\sigma^2}{\sqrt{2}\sigma}\right)\right] + A_2 e^{-(t-t_{02})k_2} e^{-\frac{(k_2\sigma)^2}{2}} \left[1+\mathrm{erf}\!\left(\frac{t-t_{02}-k_2\sigma^2}{\sqrt{2}\sigma}\right)\right] + A_3 e^{-(t-t_{03})k_3} e^{-\frac{(k_3\sigma)^2}{2}} \left[1+\mathrm{erf}\!\left(\frac{t-t_{03}-k_3\sigma^2}{\sqrt{2}\sigma}\right)\right].\]

Bounds and reported \(\tau\)

Fitted decay rates \(k\) (in ps \(^{-1}\)) are converted to \(\tau = 1000/k\) fs when filling "tau", "tau2", and "tau3" entries in params_fit. The Gaussian width sigma is stored in fs in params_fit. Model 1 and Model 2 also expose "ampoff" for the convolved offset term; for plotting compatibility across models, Model 3 stores "ampoff" as NaN.

Practical note

Use tg_analysis.TGAnalysis.get_fit_parameters() to fit all loaded scans. The method stores values and propagated errors in params_fit and reports reduced \(\chi^2\) and \(R^2\) for each scan.