The Genetic Algorithm and its Application in Calculating the Kinetic Parameters of the Thermoluminescence Curve

Abstract

This chapter explores the use of genetic algorithms as a tool for calculating the kinetic parameters of the thermoluminescence curve. Genetic algorithm is a search algorithm inspired by the process of natural selection, and it has proven to be effective in solving optimization problems in various fields. Author used genetic algorithm to estimate the activation energy and frequency factor of the thermoluminescence curve, which are important parameters in determining the dosimetric properties of materials. The results showed that genetic algorithm could accurately estimate the kinetic parameters of the thermoluminescence curve with high precision and efficiency compared to conventional methods. This approach can also handle noisy data and reduce the impact of outliers on the estimation process. Furthermore, author demonstrated that genetic algorithm can be generalized to different types of the thermoluminescence curves, such as those generated by different materials or under different experimental conditions. The proposed method is fast, accurate, and robust, making it useful for researchers in the field of dosimetry who require precise estimations of these parameters.

1. Introduction

The study of thermoluminescence (TL) has been an important area of research for many years due to its numerous applications in dating, dosimetry, and material characterization. The TL curve is a graph of the light emitted by a material as it is heated, and it provides valuable information about the energy states of the material. In order to extract useful information from the TL curve, it is necessary to analyze its kinetic parameters, such as activation energy and frequency factor. One powerful method for calculating these parameters is the genetic algorithm, which is a computational technique based on the principles of natural selection and evolution. Genetic algorithm is based on the simulation of genetic processes in living organisms and the principle of natural evolution. The experimentally obtained TL spectra are curves that are fitted according to different kinetic models [1, 2, 3]. The algorithm works with a biological population, each of which represents the ability to adapt to the explore space through synchronous combinations of evolutionary processes such as selection, crossover, and mutation [4].

Python provides a powerful platform for TL analysis, allowing for complex data analysis and visualization for accurate interpretation of experimental results. Python can be used for TL analysis by performing data analysis on experimental data obtained from TL measurements.

Here are some steps that can be followed: Import necessary modules: The first step is to import the necessary modules that will be required for data analysis. Some commonly used modules include NumPy, Pandas, Matplotlib, and Seaborn. Load experimental data: Load the experimental data into Python using the Pandas module. The data should be stored in a CSV or TXT file format that can be easily loaded into Python. Data processing: The next step is to clean and process the data. This involves removing any noise or unwanted signals present in the data. This can be done using NumPy’s signal processing functions such as filtering and smoothing. Plotting data: Visualizing the processed data is an important step in TL analysis. Using Matplotlib and Seaborn, create different types of plots such as scatter plots, line plots, histograms. Curve fitting: The next step is to fit a curve to the data. This helps in determining the kinetic parameters of the material being studied. Python provides various curve fitting algorithms such as least squares fitting and nonlinear regression. Analyzing results: Finally, interpret the results obtained from curve fitting and apply them to the material being studied.

2. Genetic algorithm approach to the TL curves

2.1 Theoretical basis

Genetic algorithm (GA) is based on the simulation of genetic processes in living organisms and the principle of natural evolution. The algorithm works with a biological population, each of which represents the ability to adapt to the explore space through synchronous combinations of evolutionary processes such as selection, crossover, and mutation [4]. The GA tuning process includes the following steps: (i) Setting initial chromosome and its encoding; (ii) Calculate GOK model distributions for each variable from individuals of a population and evaluate the fit of the fitness function; (iii) Select randomly parents and go to the next step; (iv) Crossover and mutation, go to the next step; (v) Select randomly individuals and go to the next step; (vi) Accept the results if there is better fitness value than the worst explore in the population and go to the next step, reject the worst explore and return step (iii); (vii) If the number of pre-determined steps (stopping condition) is reached and go to the next step; and (viii) Print results for best explore in population and GA finish. The block diagram of the GA is shown in Figure 1.

Figure 1.

Flowchart of the genetic algorithm.

Explanation of Figure 1: Initial population: create an initial population of candidate solutions randomly. Each solution represents a potential solution to the problem being solved. Selection: The selection process is typically based on a fitness function that evaluates each individual’s performance. Crossover: Two selected individuals are combined to produce offspring with characteristics from both parents. The crossover point is chosen randomly, and the resulting offspring replace their parents in the population. Mutation: To maintain diversity in the population, some individuals undergo random mutation, which introduces new characteristics not present in the parent population. Evaluation: The fitness function is used to evaluate the performance of each individual in the population, including the newly generated offspring. Termination: The algorithm terminates when either a termination criterion is met (e.g., a maximum number of generations) or the best solution has been found.

Code in Python:

The Python file FSG_GA.py at GitHub [5].

2.2 The thermoluminescence kinetic equation

The most commonly used models for analyzing such signals are the first order kinetics (FOK), second order kinetics (SOK), the general order kinetics (GOK) [6, 7].

The empirical equation describing the first-order TL curve has the form:

$$\mathrm{I}\left(\mathrm{T}\right)={\mathrm{I}}_{\mathrm{m}}exp\left[1+\frac{\mathrm{E}}{\mathrm{kT}}\frac{\mathrm{T}-{\mathrm{T}}_{\mathrm{m}}}{{\mathrm{T}}_{\mathrm{m}}}-\frac{{\mathrm{T}}^2}{{\mathrm{T}}_{\mathrm{m}}^2}exp\left(\frac{\mathrm{E}}{\mathrm{kT}}\frac{\mathrm{T}-{\mathrm{T}}_{\mathrm{m}}}{{\mathrm{T}}_{\mathrm{m}}}\right)\left(1-\frac{2\mathrm{kT}}{\mathrm{E}}\right)-\frac{2\mathrm{k}{\mathrm{T}}_{\mathrm{m}}}{\mathrm{E}}\right]$$E1

Code in Python:

Graph of first-order kinetic TL curve:

Explanation of Figure 2: Graph of the first-order kinetic model of the TL spectrum with a peak and symmetry.

Figure 2.

First-order glow peaks of TL curve.

The empirical equation describing the second-order TL curve has the form:

$$\mathrm{I}\left(\mathrm{T}\right)=4{\mathrm{I}}_{\mathrm{m}}exp\left(\frac{\mathrm{E}}{\mathrm{kT}}\frac{\mathrm{T}-{\mathrm{T}}_{\mathrm{m}}}{{\mathrm{T}}_{\mathrm{m}}}\right){\left[\frac{{\mathrm{T}}^2}{{\mathrm{T}}_{\mathrm{m}}^2}\left(1-\frac{2\mathrm{kT}}{\mathrm{E}}\right)exp\left(\frac{\mathrm{E}}{\mathrm{kT}}\frac{\mathrm{T}-{\mathrm{T}}_{\mathrm{m}}}{{\mathrm{T}}_{\mathrm{m}}}\right)+1+\frac{2\mathrm{k}{\mathrm{T}}_{\mathrm{m}}}{\mathrm{E}}\right]}^{-2}$$E2

Code in Python:

Graph of second-order kinetic TL curve:

Explanation of Figure 3: A graph of the second-order kinetic model of the TL spectrum with a peak and an asymmetrical shape, sloped forward of the peak.

Figure 3.

Second-order glow peaks of TL curve.

General-order glow peaks are produced in intermediate cases (neither of first-order nor of second-order). The four parameters describing a glow peak are I_m, E, T_m, and b.

The empirical equation describing the general-order TL curve has the form:

$$\begin{array}{ll}\mathrm{I}\left(\mathrm{T}\right)=& {\mathrm{I}}_{\mathrm{m}}{\mathrm{b}}^{\frac{\mathrm{b}}{\mathrm{b}-1}}exp\left(\frac{\mathrm{E}}{\mathrm{kT}}\frac{\mathrm{T}-{\mathrm{T}}_{\mathrm{m}}}{{\mathrm{T}}_{\mathrm{m}}}\right)[\left(\mathrm{b}-1\right)\left(1-\frac{2\mathrm{kT}}{\mathrm{E}}\right)\frac{{\mathrm{T}}^2}{{\mathrm{T}}_{\mathrm{m}}^2}exp\left(\frac{\mathrm{E}}{\mathrm{kT}}\frac{\mathrm{T}-{\mathrm{T}}_{\mathrm{m}}}{{\mathrm{T}}_{\mathrm{m}}}\right)+1\\ & {+\frac{2\mathrm{k}{\mathrm{T}}_{\mathrm{m}}\left(\mathrm{b}-1\right)}{\mathrm{E}}]}^{-\frac{\mathrm{b}}{\mathrm{b}-1}}\end{array}$$E3

Code in Python:

Graph of general-order kinetic TL curve:

Explanation of Figure 4: The graph of the general-order kinetic model of the TL spectrum has the form of an intermediate peak of a first- and second-order kinetics model, with a slightly sloping front peak.

Figure 4.

General-order glow peaks of TL curve.

In addition to the three kinetic models above, the TL spectrum is also matched according to the mixing model and the one-trap one-center recombination model. Each model has its own advantages and disadvantages in calculating TL trap parameters.

Details and full Python source code for TL kinetic models can be found on GitHub [5].

2.3 Fitting TL curves to estimate the energy value

2.3.1 Straight line fitting

The initial rise (IR) method introduced by Garlick and Gibson [8] is used to estimate the trapping energy value of the TL curve. The IR method works as follows: A sample is irradiated with a known dose of ionizing radiation. The sample is then heated at a constant rate, and the emitted light is measured as a function of temperature. This is called a TL curve. The TL curve is divided into equal temperature intervals, and the total integrated light output for each interval is calculated. The integrated light output is plotted against the natural logarithm of the heating rate for each interval. The slope of the resulting straight line is proportional to the activation energy required to release the trapped charges in the sample. This method is based on the principle that the intensity of TL increases initially with the temperature. The activation energy can be calculated using the Arrhenius Eq. (4)

$$\mathrm{I}\propto exp\left(-\frac{\mathrm{E}}{\mathrm{kT}}\right)$$E4

where E is the activation energy, k is the Boltzmann constant, and slope is the slope of the straight line. The IR method is repeated at different doses of ionizing radiation, and the activation energy is plotted against the dose. The trapping energy value can be estimated from the intercept of this plot with the x-axis (dose axis). In summary, the IR method involves measuring the total integrated light output as a function of temperature for a sample irradiated with a known dose of ionizing radiation [8, 9, 10, 11]. An example of an IR region of a glow peak is shown in Figure 5.

Figure 5.

Diagram of initial rise (IR).

Explanation of Figure 5: The low-temperature peak tail in this region increases up to a critical temperature T_C which is less than T_m. The values of E from the IR remain true for some critical values of temperature up to T_C, corresponding to a TL intensity I_C smaller than about 10% of the maximum TL intensity I_m [12].

Code in Python:

Linear spectral fitting graph:

Explanation of Figure 6: On the left is the TL1 sample matched by GA and applying the IR method, the kinetic parameters are also calculated, E = 0.72 eV.

Figure 6.

Fitting line-fit of the TL1 to estimate E.

2.3.2 Gaussian peak spectral fitting

The peak shape (PS) method is generally called as Chen’s [13] method, which is used to determine the kinetic parameters of the main glow peak of the TL materials. This method is mainly based on the temperatures T_m, T₁, and T₂, which are the peak temperatures, the temperatures at half of the maximum intensity, on the ascending and descending parts of the peak, respectively. Calculation of the activation energy by PS method is shown in Figure 7. The expression deduced by Chen [13] and valid for any kinetics is given by Eq. (9), where α stands for τ, δ, and ω, in which the low-temperature half width τ = T_m - T₁, the high-temperature half width δ = T₂ -T_m, and the total half intensity width ω = T₂ - T₁. The values of c_α and b_α are summarized as defined in Eq. (6).

Figure 7.

Diagram of peak shape (PS).

$${\mathrm{E}}_{\mathrm{\alpha }}={\mathrm{c}}_{\mathrm{\alpha }}\left(\frac{\mathrm{k}{{\mathrm{T}}_{\mathrm{m}}}^2}{\mathrm{\alpha }}\right)-{\mathrm{b}}_{\mathrm{\alpha }}\left(2\mathrm{k}{\mathrm{T}}_{\mathrm{m}}\right)$$E5

$$\left\{\begin{array}{c}{\mathrm{c}}_{\mathrm{\tau }}=1,51+3,0\left({\mathrm{\mu }}_{\mathrm{g}}-0,42\right)\\ {\mathrm{b}}_{\mathrm{\tau }}=1,51+4,2\left({\mathrm{\mu }}_{\mathrm{g}}-0,42\right)\\ {\mathrm{c}}_{\mathrm{\delta }}=\mathrm{0,976}+7,3\left({\mathrm{\mu }}_{\mathrm{g}}-0,42\right)\\ {\mathrm{b}}_{\mathrm{\delta }}=0\\ {\mathrm{c}}_{\mathrm{\omega }}=2,52+10,2\left({\mathrm{\mu }}_{\mathrm{g}}-0,42\right)\\ {\mathrm{b}}_{\mathrm{\omega }}=1\end{array}\right.$$E6

where μ_g is the so-called geometrical shape or symmetry factor that determines the order of the kinetics. The order of the kinetics depends on the glow PS. The value of μ_g for first- and second-order kinetics is 0.42 and 0.52, respectively. The symmetry factor in GOK model can be evaluated from the following Eq. (7). The TL glow peaks corresponding to second-order kinetics are characterized by an almost symmetrical shape, whereas first-order peaks are asymmetrical [6].

$${\mathrm{\mu }}_{\mathrm{g}}=\frac{\mathrm{\delta }}{\mathrm{\omega }}=\left({\mathrm{T}}_2-{\mathrm{T}}_{\mathrm{m}}\right)/\left({\mathrm{T}}_2-{\mathrm{T}}_1\right)$$E7

Explanation of Figure 7: describes how to fit the Gaussian peak spectrum and the PS method to calculate the trap energy. The calculation results depend on the T₁, T₂, and T values of the TL peak.

Code in Python:

TL curves from R package tgcd [14] with 22 TL curves are measured from different materials provided by George Kitis. Among them is the TL curve denoted as R4, which is used to fit the Gaussian peak shape and calculate the kinematic parameters. The results are shown in Figure 8.

Figure 8.

The TL spectrum of the R4 sample is matched with four peaks.

Details and full Python source code for calculating the activation energy can be found on GitHub [5].

Explanation of Figure 8: spectrum of R4 consisting of four peaks matched and peaked by genetic algorithm. The kinetic parameters of the spectrum are calculated reasonably, and the obtained FOM coefficient is very small.

2.4 Calculation of the frequency factor

The intensity I(t) of the TL signal is measured at time t after the start of the experiment. This magnitude TL I(t) is proportional to the derivative −dn/dt, depending on the measurement conditions. In experimental research, experimentalists pay much attention to the frequency factor of the TL signal. Kitis et al. [7] obtain the following analytic equation for s with the GOK model:

$$\mathrm{s}=\left(\frac{\mathrm{\beta E}}{\mathrm{k}{{\mathrm{T}}_{\mathrm{m}}}^2}\right)exp\left(\frac{\mathrm{E}}{\mathrm{k}{\mathrm{T}}_{\mathrm{m}}}\right){\left[1+\left(\mathrm{b}-1\right)\frac{2\mathrm{k}{\mathrm{T}}_{\mathrm{m}}}{\mathrm{E}}\right]}^{-1}$$E8

Thus, kinematic parameters such as E and s of the TL curve will be estimated according to Eqs. (5) and (8). Each peak coordinate of the TL curve including two parameters of temperature and TL intensity will be recorded after each mouse click on the screen containing the TL curve. The kinematic order of the GOK model is also selected and changed from 1 to 2 until the FOM values of the TL curve reach the minimum condition.

Code in Python:

K2GdF5 materials doped with concentrations of 10%Tb are widely used in radiation dosimetry and materials science [15]. The spectrum of K2GdF5:Tb curve and results of calculating E and s values are shown in Figure 9.

Figure 9.

The spectrum of K2GdF5:Tb curve and results of calculating E and s values.

Explanation of Figure 9: depicts the result of spectral matching of sample K2GdF5:Tb with four peaks. The calculation results of sample K2GdF5:Tb in terms of E and s values are also calculated with high accuracy.

Details and full Python source code for calculating the frequency factor can be found on GitHub [5].

3. Conclusion

The application of GA to TL curve analysis provides an efficient and effective method for the estimation of kinetic parameters such as activation energy, frequency factor, and order of reaction. The algorithm can explore a large search space of candidate solutions and converge toward a solution that optimizes the fitness function. The use of GA in TL curve analysis has significant implications for the field of geochronology and archeological dating, as it provides a powerful tool for the accurate and precise determination of the age of materials.