Continuous trap distributions¶

Some TL materials — glasses, feldspars, natural minerals under variable irradiation history — show broad, structureless emission humps that cannot be decomposed into a small number of discrete kinetic peaks. The continuous trap-distribution models (cont_gauss, cont_exp) account for this by integrating the first-order kinetic emission over a distribution of activation energies.

Governing equation¶

Following Benavente et al. (2019), the TL intensity for a continuous distribution \(g(E)\) is:

\[ I(T) = I_N \, \frac{\displaystyle\int_0^\infty g(E)\, e^{-E/kT} \exp\!\left(-\frac{s}{\beta}\int_{T_0}^{T} e^{-E/kT'}\mathrm{d}T'\right)\mathrm{d}E} {\displaystyle\int_0^\infty g(E)\, e^{-E/kT_N} \exp\!\left(-\frac{s}{\beta}\int_{T_0}^{T_N} e^{-E/kT'}\mathrm{d}T'\right)\mathrm{d}E} \]

where the frequency factor is determined self-consistently from the characteristic temperature \(T_N\):

\[ \frac{s}{\beta} = \frac{E_0}{k T_N^2}\, e^{E_0/(kT_N)} \]

The temperature integral uses the \(E_2\) exponential integral approximation (Abramowitz & Stegun).

Parameters¶

Symbol	Parameter name	Unit	Physical meaning
\(T_N\)	`Tn`	K	Characteristic temperature (position of the distribution peak)
\(I_N\)	`In`	counts	Characteristic intensity at \(T_N\)
\(E_0\)	`E0`	eV	Centre (Gaussian) or lower bound (exponential) of the energy distribution
\(\sigma\)	`sigma`	eV	Width of the energy distribution

Legacy aliases

The discrete-peak aliases Tm, Im, E are accepted and mapped to Tn, In, E0 internally for backward compatibility.

Gaussian distribution (`cont_gauss`)¶

\[ g(E) = \frac{1}{\sigma\sqrt{2\pi}} \exp\!\left(-\frac{(E-E_0)^2}{2\sigma^2}\right) \]

The integration range spans \([E_0 - 8\sigma,\; E_0 + 8\sigma]\).

When to use: broad symmetric humps; overlapping peaks in heavily irradiated or natural samples.

import tldecpy as tl
import numpy as np

T = np.linspace(300.0, 700.0, 600)

peak_cg = tl.PeakSpec(
    name="CG1",
    model="cont_gauss",
    init={"Tn": 480.0, "In": 3000.0, "E0": 1.4, "sigma": 0.12},
    bounds={
        "Tn":    (350.0, 650.0),
        "In":    (0.0,   10000.0),
        "E0":    (0.5,   3.0),
        "sigma": (0.01,  0.5),
    },
)

Exponential distribution (`cont_exp`)¶

\[ g(E) = \frac{1}{\sigma}\, e^{-(E-E_0)/\sigma}, \quad E \geq E_0 \]

The integration range spans \([E_0,\; E_0 + 20\sigma]\).

When to use: one-sided trap distribution with a sharp lower energy cutoff; common in amorphous materials and feldspars.

peak_ce = tl.PeakSpec(
    name="CE1",
    model="cont_exp",
    init={"Tn": 480.0, "In": 3000.0, "E0": 1.2, "sigma": 0.10},
)

Performance note¶

The continuous models evaluate a numerical energy integral (\(n = 801\) nodes by default) at every optimizer iteration. They are 5–20× slower per evaluation than discrete kinetic models.

For large-scale fitting, reduce n_energy to 201–401:

# Direct model call with reduced quadrature
from tldecpy.models.continuous import continuous_gaussian

I = continuous_gaussian(T, Tn=480.0, In=3000.0, E0=1.4, sigma=0.12, n_energy=201)

When using fit_multi, the n_energy parameter is not currently exposed through PeakSpec; the default 801-node grid is used automatically.

References¶

Benavente, J. F., et al. (2019). Continuous trap-distribution models for TL glow-curve analysis. Radiat. Meas. 128, 106180.
Gomez-Ros, J. M., et al. (2006a). On the TL glow-curve deconvolution for continuous distributions of traps. Radiat. Meas. 41, 12.
Chen, R., & McKeever, S. W. S. (1997). Theory of Thermoluminescence and Related Phenomena. World Scientific.