Kinetic models — physics and model selection¶

Every TL peak in tldecpy is described by a kinetic model — a mathematical expression that relates TL emission intensity \(I(T)\) to a set of physical parameters. All models share the same three core parameters.

Shared parameters¶

Symbol	Parameter name	Unit	Physical meaning
\(T_m\)	`Tm`	K	Temperature at peak maximum
\(I_m\)	`Im`	counts	Intensity at peak maximum
\(E\)	`E`	eV	Trap activation energy (depth)

\(T_m\) and \(I_m\) are phenomenological anchors — the model is normalised so that \(I(T_m) = I_m\) regardless of the underlying kinetics. \(E\) is a genuine physical quantity that characterises the trap.

First-order kinetics (FO)¶

Assumption: negligible retrapping (\(A_n \ll A_m\)). After detrapping, electrons recombine immediately without being recaptured.

The Randall-Wilkins master equation in the FO limit:

\[ I(T) = n_0 \, s \exp\!\left(-\frac{E}{kT}\right) \exp\!\left(-\frac{s}{\beta}\int_{T_0}^{T} e^{-E/(kT')} \mathrm{d}T'\right) \]

The integral \(\int e^{-E/kT'}\mathrm{d}T'\) has no closed form; all four fo_* variants differ only in their numerical approximation:

Key	Backend for \(\int e^{-E/kT'}\mathrm{d}T'\)	Accuracy
`fo_rq`	Bos rational polynomial	~0.1 %
`fo_rb`	AAA barycentric rational (from stored \(Q(z)\) data)	~0.01 %
`fo_ka`	Classical asymptotic (Chen & McKeever 1997)	~1 %
`fo_wp`	Weibull empirical approximation	~2 %

Recommendation: use fo_rq or fo_rb for all routine fits.

Peak shape: asymmetric with geometric factor \(\mu_g \approx 0.42\).

Second-order kinetics (SO)¶

Assumption: dominant retrapping (\(A_n \gg A_m\)). Most detrapped electrons are recaptured before recombining.

\[ I(T) \propto \frac{n^2}{N} \]

Peak shape: symmetric (\(\mu_g \approx 0.52\)).

Key	Notes
`so_ks`	Garlick-Gibson (1948) standard form. Recommended.
`so_la`	Logistic asymmetric approximation

General-order kinetics (GO)¶

Assumption: phenomenological interpolation between FO and SO via kinetic order parameter \(b \in (1, 2)\).

\[ I(T) = I_m \, b^{b/(b-1)} \, \exp\!\left(\frac{E}{kT_m}\cdot\frac{T-T_m}{T_m}\right) \left[ (b-1)(1-\delta)\frac{T^2}{T_m^2}\exp\!\left(\frac{E}{kT_m}\cdot\frac{T-T_m}{T_m}\right) + 1 + (b-1)\delta_m \right]^{-b/(b-1)} \]

where \(\delta = 2kT/E\) and \(\delta_m = 2kT_m/E\).

Key	Approximation	Notes
`go_kg`	Kitis et al. (1998)	Most accurate. Default GO.
`go_rq`	Rational-quadratic	Faster but less accurate near \(b=2\)
`go_ge`	Exponentialized	Simplest form

Extra parameter: b ∈ (1.001, 2.0). b = 1 is equivalent to FO; b = 2 to SO. Allow the optimizer to find b freely (e.g. bounds (1.001, 2.0)).

Mixed-order kinetics (MO)¶

Assumption: explicit mixture of FO and SO recombination via mixing parameter \(\alpha \in (0, 1)\).

\(\alpha \to 0\) → FO behavior; \(\alpha \to 1\) → SO behavior.

Key	Reference
`mo_kitis`	Kitis & Gomez-Ros (2000). Recommended.
`mo_quad`	Quadratic Gomez-Ros variant
`mo_vej`	Vejnovic mixed-order variant

Extra parameter: alpha ∈ (0.001, 0.999).

OTOR — One Trap One Recombination center¶

Assumption: single trap level with a single recombination centre. The recombination–retrapping competition is parameterised by:

\[ R = \frac{A_n}{A_m} \]

where \(A_n\) is the retrapping coefficient (cm³/s) and \(A_m\) the recombination coefficient (cm³/s).

\(R\)	Physical meaning
\(R \to 0\)	No retrapping → pure FO
\(R \approx 0.5\)	Equal retrapping and recombination
\(R \to 1\)	Dominant retrapping → near-SO

The OTOR intensity uses the analytical Lambert W solution (Kitis & Vlachos 2013):

\[ I(T) \propto \frac{W(T_m) + W(T_m)^2}{W(T) + W(T)^2} \]

where \(W\) is the Lambert W function evaluated on different branches for \(R < 1\) and \(R > 1\).

Numerical singularity

otor_lw exhibits a singularity near \(R \approx 0.96\) on the \(R < 1\) branch. The library clamps \(R \leq 0.95\) internally on that branch. Always set bounds["R"] = (1e-6, 0.90) unless you have evidence for \(R > 1\).

Key	Implementation
`otor_lw`	Lambert W exact (recommended)
`otor_wo`	Wright Omega fast approximation

Extra parameter: R ∈ (1e-6, 0.90) for \(R < 1\) branch.

Continuous trap distributions¶

For materials with a broad distribution of trap depths, no single discrete level is appropriate. See Continuous trap distributions.

Model selection guide¶

Narrow, asymmetric peak (μg < 0.45)?
  → fo_rq  (or fo_rb for higher precision)

Broad, symmetric peak (μg ≈ 0.52)?
  → so_ks

Peak shape between FO and SO?
  → go_kg  (fit b freely, 1.001–2.0)
  → mo_kitis  (if explicit FO/SO mixture is needed)

Material known to have simple two-level trap physics (LiF, etc.)?
  → otor_lw  (physically rigorous)

Broad, structureless hump?
  → cont_gauss  or  cont_exp

When uncertain, fit with go_kg (freeing b) as a flexible baseline; then compare AIC/BIC against FO or SO to determine whether the extra parameter is justified.

References¶

Randall, J. T., & Wilkins, M. H. F. (1945). Phosphorescence and electron traps. Proc. R. Soc. London A 184, 366.
Garlick, G. F. J., & Gibson, A. F. (1948). The electron trap mechanism of luminescence. Proc. Phys. Soc. 60, 574.
Kitis, G., Gomez-Ros, J. M., & Tuyn, J. W. N. (1998). Thermoluminescence glow-curve deconvolution functions for first, second and general orders of kinetics. J. Phys. D 31, 2636.
Kitis, G., & Vlachos, N. D. (2013). General semi-analytical expressions for TL, OSL and other luminescence stimulation modes. Radiat. Meas. 48, 47.
Chen, R., & McKeever, S. W. S. (1997). Theory of Thermoluminescence and Related Phenomena. World Scientific.