# New designed software to deconvolute the thermoluminescence glow-curves

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## Abstract

Within the framework of the one trap-one recombination (OTOR) level model, a new software was designed to deconvolute the thermoluminescence glow-curves. The computer code was written considering deconvolution equations obtained, without any approximations, from the (OTOR) model with free parameters TM, IM, and ε. The code can also use the original equations of the OTOR model with free parameters s, ε, n_{o}, and β. The new software was applied to deconvolute glow curves measured from irradiated in-house prepared Al_{2}O_{3} doped with Cr_{2}O_{3} dosimeters. The consequences of the deconvolution analyses of the reference glow curves agree with those reported in the literature.

## Keywords

Glow curve OTOR model Thermoluminescence Deconvolution Kinetic parameters Al_{2}O

_{3}dosimeters

## 1 Introduction

Thermoluminescence (TL) is the thermally stimulated emission of light from insulator or semiconductor materials that absorb energy from radiation [1]. TL phenomenon requires the thermoluminescent material to possess impurities or defects. The simplest model to describe the TL phenomenon is the OTOR model. OTOR model assumes the existence of two kinds of localized states. One of the two states functions as an electron trap and the other as a hole trap (or a recombination center) [e.g. 2, 3]. In TL, thermal energy is used to release trapped electrons from their traps to the conduction band. Upon minimizing the internal energy the free electrons migrate from the conduction band to lower energy states. This transition may be radiative which causes the glow observed from the material. According to the OTOR model, this radiative transition results in the formation of a glow curve composing of a single peak. However, the glow curve measured from TL materials looks complicated due to the existence of multi-electron and hole traps. In the OTOR model, the measured glow curve is considered to be produced by summing up individual glow peaks. Each glow peak is a result of a certain electron trap. To analyze the obtained glow curve, computerized glow-curve deconvolution (CGCD) is the most used method [4].

CGCD analysis is used to find the separate glow-peaks of a composite TL glow-curve and the kinetic parameters (trap depth, frequency factor, and kinetic order) of each peak. Based on several models and approximations, many computer programs have been developed for TL glow-curve analysis [5, 6].

The aim of the current study is to consider mathematical deconvolution equations, obtained without approximations, from the OTOR model. A new software was designed to deconvolute TL glow curves using the obtained deconvolution equations. To test the new deconvolution equations and the developed code, it was, firstly, examined by analyzing the reference glow curves of GLOCANIN program [7, 8]. Secondly, the code was used to deconvolute measured TL glow curves of proposed dosimeters, which are made of Al_{2}O_{3} samples doped with Cr_{2}O_{3} of different concentrations [9].

## 2 Method

^{−1}), n

_{0}is the initial concentration of trapped electrons (cm

^{−3}), ϵ is the activation energy (eV), k is Boltzmann constant (J K

^{−1}), T is the absolute temperature (K), β is the heating rate and T

_{0}is the initial temperature of the heating profile. For the second and general order kinetics, the intensity of photons produced from the TL material is written as [11, 12]

### 2.1 First order kinetics model

_{0}, of a first order TL peak and it is a function of s, ϵ, T, β, and T

_{0}. In CGCD analysis it is desired to obtain a form of the intensity I as a function of maximum TL intensity and its corresponding temperature (i.e. I

_{M}and T

_{M}respectively) [16] as well as the energy ϵ. To achieve this goal in accordance with the last sentence, the conditions at maximum intensity are used [16, 17] and they lead to:

Equations 8 and 9 are used to calculate the frequency factor and the initial concentration of trapped electrons \(n_{0}\) of a first order glow peak.

### 2.2 Second- and general-order kinetics model

_{0}, ϵ, T, β, and T

_{0}being the same parameters defined earlier. The quantity s′ is the pre-exponential factor (cm

^{3}s

^{−1}). The conditions at maximum intensity, when they are used, lead to

_{0}. The two functions F(T,ϵ) and F(T

_{M},ϵ) are given by the two Eqs. (6) and (7), respectively. For the second order kinetics,

In a similar way to first order kinetics, Eqs. (12) and (13) are used to calculate the pre-exponential factor \(s'\) and \(n_{0}\) of a second order glow peak.

^{3(b−1)}s

^{−1}. Upon using the conditions at maximum intensity, it can be obtained

For general order kinetics, Eqs. (16) and (17) can be used to calculate the pre-exponential factor \(s^{\prime\prime}\) and \(n_{0}\) of a general order glow peak.

Equations (15) and (17) are used to achieve the TL glow curve analysis in view of the general order kinetics, for T > T0. Equations (6) and (7), respectively, represent the two terms F(T,ϵ) and F(T_{M},ϵ).

### 2.3 Other forms

_{M}, E) given by Eqs. (6) and (7), respectively. In Eqs. (19) and (20) R = A

_{n}⁄A

_{m}where A

_{n}and A

_{m}are the trapping and recombination probability coefficients. In case of A

_{n}< A

_{m}, R in the OTOR model resembles the kinetic order b in the GOK model. And W(x) is the Lambert function. The value of s (OTOR model) is calculated by using the peak maximum condition and it is given by the equation:

### 2.4 MATLAB code

The developed MATLAB code employs Eqs. (4), (5), (10), (11), (14), (15) and (18) to be used in the process of TL glow curve deconvolution analysis. Therefore, the user chooses the desired equation for TL deconvolution in the first step. The MATLAB code utilizes Nelder-Mead, Levenberg–Marquardt, and Trust-Region algorithms. Any of these non-linear optimization algorithms are used to perform TL glow curve deconvolution with considerable efficiency, and they are accurate in determining the peak parameters for experimental overlapping TL glow curves. As a second step, the user chooses the fitting algorithm. It is well known that the non-linear fitting is hugely dependent on the choice of the initial guess of the fitting parameters. Thus, in the third step, the user determines the number of peaks. Then he/she ignites the algorithm with an initial guess of each peak parameters. As declared in almost all optimization algorithms, they stop when they reach specific accuracy by changing the optimization parameters regardless of the physical meaning of the resultant final parameters. Therefore, the user is given the privilege of fixing some parameters if the solver fails to get acceptable physical parameters.

## 3 Material

GLOCANIN represents standard reference glow curves that belong to the first order kinetics. It is a good candidate for testing the MATLAB code for TL glow curve deconvolution. GLOCANIN project [7, 8] includes ten reference glow curves (RGCs) ensuing first order kinetics model only. RGCs 1 and 2 are obtained from computer simulation [7]. Those RGCs, 1 and 2, were simulated to be composed of 1 single peak for RefGlow.001 and 4 summed peaks for RefGlow.002 so that RefGlow.002 imitates the glow curve structure of TLD-100 [7]. The left eight RGCs are measured from samples irradiated at low and high doses [8]. Most of those eight reference glow curves show, relatively, simple glow curves composing of three overlapping peaks for RGCs 7 & 8, 4 overlapping peaks for RGCs 3, 4, & 10 and 5 overlapping peaks for RGCs 5 & 6. The most complicated RGC is number 9 which is obtained from a highly irradiated TLD-700 sample. The deconvolution of this RGC differs, in literature, not only in the number of deconvoluted peaks but also in their position and shape [8].

Second, the current deconvolution expressions are employed to analyze the experimental glow curves of samples of Cr_{2}O_{3}–Al_{2}O_{3} nanoparticles doped with Cr_{2}O_{3}. A different series of Cr_{2}O_{3}–Al_{2}O_{3} nanoparticles containing several molar compositions of Cr_{2}O_{3} (0–5 wt%) are prepared by the sol–gel method. The samples are irradiated using beta rays with a dose of 3 Gy. Then the irradiated samples are read out at a heating rate of 4 K/s. The TL measurements were achieved by RISO-TL-DA-12 reader with a built-in 90Sr/90Y β-source with a rate of 3.33 mGy/s. The samples were read out at the Institute of Radiation Protection, Helmholtz research centre, Munich, Germany [9].

### 3.1 Test of the computer code

The deviation of the activation energies values, which are obtained by the CGCD using the exact TL expressions, from those values obtained in references (15, 16). The table also shows the values of FOM (%) for each glow curve

File name | Deviation in E (eV) | FOM (%) | ||||
---|---|---|---|---|---|---|

Peak 1 | Peak 2 | Peak 3 | Peak 4 | Peak 5 | ||

RefGlow.001 | 0 | – | – | – | – | 0.0099 |

RefGlow.002 | – | 0.001 | 0.001 | 0.001 | 0 | 0.0099 |

RefGlow.003 | – | − 0.036 | − 0.019 | − 0.016 | 0.0154 | 0.9541 |

RefGlow.004 | – | − 0.034 | − 0.011 | − 0.037 | − 0.05 | 0.9815 |

RefGlow.005 | – | − 0.012 | 0.004 | − 0.057 | 0.023 | 0.6793 |

RefGlow.006 | – | − 0.034 | − 0.018 | − 0.033 | − 0.028 | 0.8723 |

RefGlow.007 | – | – | − 0.066 | 0.003 | − 0.046 | 1.0348 |

RefGlow.008 | – | – | − 0.042 | − 0.014 | − 0.015 | 0.9048 |

RefGlow.009 | – | − 0.029 | 0.005 | − 0.045 | 0.042 | 0.8453 |

RefGlow.010 | – | − 0.065 | 0.057 | − 0.062 | − 0.014 | 1.2874 |

There exist no standard results or projects to test the second and general order equations. Therefore, synthetic second and general order glow curves were generated by convoluting peaks using the proper equations (Eqs. 10 and 2) of the second and general order kinetic equations. Then, we used the MATLAB code to deconvolute the synthetic glow curves. The FOM values obtained in the case of the currently proposed functions were of the order of 10^{−5}%.

### 3.2 Deconvolution of TL glow curves of Al_{2}O_{3} doped with Cr_{2}O_{3}

_{2}O

_{3}–Al

_{2}O

_{3}nanoparticles were deconvoluted using the designed software. The glow-curve analysis of Al

_{2}O

_{3}nanoparticles containing a molar composition of Cr

_{2}O

_{3}of 1.5 wt% , between 297 and 685 K, is shown in Fig. 3, which reveals a deconvolution of 6 peaks. The obtained trap depths of the 6 peaks were found to be 0.625, 0.691, 0.801, 0.949, 1.006, and 1.071 eV, respectively. The corresponding orders of kinetic parameters of those peaks were 1.57, 1.39, 1.69, 1.53, 1.52, and 1.28. Figure 4 shows the deconvolution of Al

_{2}O

_{3}nanoparticles containing a molar composition of Cr

_{2}O

_{3}of 5.0 wt%, between 236 and 685 K. The obtained trap depths of these 5 peaks were found to be 0.735, 0.773, 0.822, 0.937, and 0.942 eV, respectively. The orders of kinetic parameters of those peaks were 1.41, 1.29, 1.54, 1.55, and 1.45. A comparison between Figs. 3 and 4 indicates that Al

_{2}O

_{3}nanoparticles with compositions of 1.5% Cr

_{2}O

_{3}show a higher response for beta radiation than the other compositions.

## 4 Conclusions

Within the frame of the OTOR level model, the exact equations of the intensities of a TL glow peaks as functions of temperature for different kinetics orders were considered. A new software was developed to, computationally, deconvolute TL glow curves. The CGCD analysis, using the equations of the first order kinetics, is achieved for the GLOCANIN reference glow-curves of references [7, 8]. The capability of the current TL deconvolution expressions to get accurate analyses of the TL glow-curves with low values of FOM is shown. The deconvolution equations of the second and general order kinetics were tested using pre-constructed glow curves (synthetics), and the obtained results reveal excellent agreements with low FOM values.

The current deconvolution equation of the general kinetic order was employed to deconvolute the TL glow curves of Al_{2}O_{3} doped with different concentrations of Cr_{2}O_{3}. It was found that Al_{2}O_{3}–Cr_{2}O_{3} nanoparticles with compositions of 1.5% Cr_{2}O_{3} exhibit the highest response to beta radiation. Al_{2}O_{3}–Cr_{2}O_{3} nanoparticles material with a concentration of 1.5% of Cr_{2}O_{3} represented a promising dosimeter for beta radiation. It also exemplifies a promising dosimeter for gamma radiation at low doses. More details about Al_{2}O_{3}–Cr_{2}O_{3} nanoparticles material with a concentration of 1.5% of Cr_{2}O_{3} need more studies when it is irradiated with gamma radiation.

## Notes

### Acknowledgement

The authors are thankful to professor G. Kitis for providing the reference glow curves of GLOCANIN project.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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