osl curves

Radiation Protection Dosimetry (2012), Vol. 149, No. 4, pp. 363– 370 Advance Access publication 15 July 2011

doi:10.1093/rpd/ncr315

D. Afouxenidis1,2,*, G. S. Polymeris1, N. C. Tsirliganis1 and G. Kitis2 1 Archaeometry Laboratory, Cultural and Educational Technology Institute (C.E.T.I.), R.C. ATHENA’, Tsimiski 58, 67100 Xanthi, Greece 2 Nuclear Physics Laboratory, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece *Corresponding author: [email protected] Received March 3 2011, revised May 12 2011, accepted June 17 2011 This paper exploits the possibility of using commercial software for thermoluminescence and optically stimulated luminescence curve deconvolution analysis. The widely used software package Microsoft Excel, with the Solver utility has been used to perform deconvolution analysis to both experimental and reference glow curves resulted from the GLOw Curve ANalysis INtercomparison project. The simple interface of this programme combined with the powerful Solver utility, allows the analysis of complex stimulated luminescence curves into their components and the evaluation of the associated luminescence parameters.

INTRODUCTION The computerised curve deconvolution (CCD) analysis of thermoluminescence (TL) glow curves and optically stimulated luminescence (OSL) decay curves into their individual glow peaks and components respectively have been recognised over the last 30 y to be of major importance(1 – 5). The information by the CCD concerning the trap energy depth (E), frequency factor (s), kinetic order (b), photoionisation cross section (s), etc, is very useful in order to understand the luminescence mechanism of materials. The capabilities of several computer codes used by the various research groups for CCD and their assessment of the glow curve parameters were tested by the GLOw Curve ANalysis INtercomparison (GLOCANIN) project. Participants in the GLOC ANIN project were asked to analyse the so-called reference glow curves using their computer codes. The results on synthetic as well as on experimentally measured glow curves can be found in references(6, 7). The application of the CCD consists essentially of two steps: (i) deciding on the mathematical model that describes a single TL/OSL peak and (ii) assessing the values of the curve parameters that minimise the sum of squares of the differences between the fitted model and the experimental curve(8). Concerning the first step, there is a number of well-established analytical expressions in the literature describing single TL/OSL peaks(6, 9, 10). Although a great amount of work exists on deriving analytical single peak expressions and despite the significant information that the CCD analysis yield, the latter is not widely adopted by the TL/OSL community as a basic tool, while the

majority of TL/OSL publications does not involve any CCD analysis of relevant data. In authors’ opinion, this could be partly attributed to the lack of available commercial software; so every researcher has to write his/her own program. The aim of the present work is to offer a solution to the aforementioned problem that will enable any researcher to analyse easily and accurately even the most complex TL/OSL curves consisting of many overlapping individual peaks, employing the commonly used spreadsheet software package Microsoft Excel, along with its Solver, add-in utility. SINGLE TL/OSL PEAK ANALYTICAL EXPRESSIONS The analytical expressions used in the present work are the well-known expressions for general and mixed order kinetics, named GOK and MOK, respectively hereafter(7, 11 – 14). The selection of these expressions was based on the following: (1) The GOK expression, although empirical is almost exclusively used in the literature(7). (2) The MOK expression, although rarely used in the literature, is physically meaningful(11). (3) The GOK expression for b ¼ 2 coincides with second-order kinetics, whereas for b ! 1 coincides with first-order kinetics. (4) Similarly, the MOK expression for a ! 1 coincides with second-order kinetics and for a ¼ 0 coincides with first-order kinetics. The advantage of those expressions is that only one expression accounts for both first- and second-order kinetics, and furthermore it can

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COMPUTERISED CURVE DECONVOLUTION OF TL/OSL CURVES USING A POPULAR SPREADSHEET PROGRAM

D. AFOUXENIDIS ET AL.

advantage of the I (Im, Tm, E, b or a, T ) representation is not easily seen in the cases of experimental curves with one single peak. However, it becomes clear in cases of experimental curves consisting of many overlapped peaks. It must be noted that the proposed expressions involving Im, Tm result as transformations of the original expression involving n0 and s. The only approximation in these expressions is the usual approximation of the exponential integral appearing in TL theory. The analytical TL expressions are derived using two terms of the asymptotic series approximation (ASA) of the exponential integral. The transformed equations can include more terms of the ASA through the term Dm, (see later text), which should be simply replaced by the ASA expressions above the third term. Details about the ASA approximation can be found in reference(15) SPREADSHEET PREPARATION The steps needed to perform a CCD analysis with an Excel spreadsheet are shown in Table 1. The example used is that of a complex TL glow curve consisting of two individual peaks using the GOK expression. In the first step, one ascribes the temperature and TL intensity of the experimental glow curve to the columns A and B. In the second step, the subsequent columns C and D are ascribed to each one of the individual peaks 1 and 2, respectively.

Table 1. Calculations for TL data under GOK. A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 a

B

C

D

E

F

S I(T)fit 0.0003 0.0004 0.0005

jI(T)exp – I(T)fitj 3.6E206 4.2E206 4.5E206

Heat rate (K/s) 1 Imax Tmax (K) E (eV) b M.F.

Temp 301.01 302.02 303.02

Peak #1 0.04 417 1.3 1.0001 1000a Integral 11 098 Freq factor (s – 1) 7Eþ15

FOM (%)

1.2

I(T)exp 0.0003 0.0004 0.0005

Peak 1 0.0003 0.0004 0.0005

Peak #2 0.05 456 1.5 1.0001 1000a 16 904 2.9Eþ15

Peak 2 4.8E206 5.9E206 7.1E206

The order of magnitude of maximum intensity Imax.

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additionally account for the intermediate orders. Concerning the decision between GOK and MOK, one has to know that their greater difference, as it was shown by Kitis et al.(12), is observed at about b¼1.6 and this difference is ,4 %. Moreover their difference is mainly at the high temperature range of the TL glow-peak. Taking into account that these differences were obtained using numerically generated peaks, one can conclude that these differences are rather difficult to be seen in an experimental TL peak. Therefore, in most practical experimental cases yielding an error higher than 1 %, there is no real dilemma in choosing between GOK and MOK, since they will both produce similar results. Another feature of the selected analytical expressions is that they are of the form of I (Im, Tm, E, b or a, T )(7, 11) and I (Im, tm, b or a, t)(13, 14), for TL and linearly modulated (LM)-OSL, respectively, instead of the form I (n0, s, E, b or a, T ) and I (n0, l, b or a, t) of the original equations(9, 10), where variable T is the temperature, Tm is the temperature at maximum TL intensity, Im is the maximum intensity, n0 is the initial concentration of trapped electrons, tm is the time at maximum LM-OSL intensity and l is the LM-OSL stimulation wavelength. The advantage of the proposed presentation is that the quantities Im, Tm and tm can be evaluated directly and accurately from the experimental TL/LM-OSL curves, whereas it is not possible to have any knowledge about the probable values of n0, s and E appearing in the original form of the equations. The

COMPUTERISED CURVE DECONVOLUTION

b=b1

IðTÞ ¼Im b

E T Tm exp kT Tm

T2 ðb 1Þ ð1 DÞ 2 Tm b=b1 E T Tm þ Zm exp kT Tm where D¼

2kT 2kTm ; Dm ¼ ; Zm ¼ 1 þ ðb 1Þ Dm E E

has to be written in Excel format in cell C19. It will be used to reproduce the first (lower Tmax) peak in column C by dragging it to the entire column C. Therefore a peak for the entire temperature region is calculated, according to the temperature T(K ) and its four trapping parameters that were preselected. This procedure is repeated for the next glow peak in column D, by copying and pasting the same expressions to cell D19 and dragging it to the entire corresponding column. In that way, two arbitrary glow peaks are created without any other limitation. However, the sum of these peaks should be similar to the total experimental curve. This similarity is usually checked by the linear regression coefficient (x 2), which however does not provide an immediate clue regarding the goodness of fit. In order to circumvent this problem, another mathematical index was selected, which is termed the FOM(16) and is defined as P p TLexp TLfit P FOMð%Þ ¼ 100 p TLfit

The FOM index value provides a measure for the goodness of fit; the lowest its value, the best fit. Therefore, every fitting attempt should result in minimising the FOM index value, which is achieved by changing the set of the parameter values of each glow peak. This is achieved by using certain optimisation software packages, the Solver, the power full add in of Excel. A full description for the latter is presented in the following section. Returning to spreadsheet preparation procedure, two more columns are required, for instance E and F, in order for the sum of the fitted (calculated) glow peaks and the absolute value of the difference between each experimental and calculated data point to be respectively presented. The main objective is to minimise the values in column F. Finally, showing the FOM index value among the initial cells of the spreadsheet is suggested. The spreadsheet of Table 1 described earlier stands as an example for the deconvolution of the glow curve into its two overlapping glow peaks, in order for the parameters associated with the individual peaks to be estimated. Only the first 21 rows are shown for the sake of brevity. Columns (A19:A21) and (B19:B21) contain the experimental data points for the TL glow curve, while columns (C19:C21) and (D19:D21) contain the fitted data points using the GOK model. Row nine holds the value of a multiplier factor that is necessary for further analysis and represents the order of peak height intensity. The heating rate used throughout the measurement is given in Cell B2 since it is a common parameter for all the glow peaks. Furthermore, the TL integral of each glow peak (sum of the data in column below row 19) is given in row 11 of the corresponding column, while in row 13, the corresponding frequency factors are shown. The frequency factor values are calculated from the respective equation(7) using the values of E, Tm and b obtained from the curve fitting and using the heating rate given in Cell B2.

ABOUT SOLVER The Solver(17, 18) is an Excel Add-in, a software program that could be found in the Tools menu; it can be installed by checking the Solver Add-in following the path: Tools !Add-Ins. Solver is a general-purpose optimisation package that is used in order to find a maximum, minimum or specified value of the target Cell. The Solver code is a product of Frontline Systems Inc. The Solver can be used to minimise the sum of squares of residuals (differences between yobsd and ycalc) and thus perform a least- square fitting. It can be used to

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For the fitting procedure, one has to insert initial, arbitrary but meaningful values for the parameters Im, Tm, E and b for each single glow-peak. As can be seen in Table 1, these values are inserted in the rows 5– 8 respectively in column C for the first peak and column D for the second peak. When the curve fitting procedure is completed, it gives the net values of Im, Tm, E and b. However, the evaluation of additional quantities such as the integral of each glow-peak, the frequency factor and figure-of merit (FOM) values (see later text) is also feasible. In the next step, the raw experimental data are inserted in the same spreadsheet in a way that the first data point appears in the 19th row of columns A and B, corresponding to the temperature and signal respectively. Following that, the GOK analytical expression(7)

D. AFOUXENIDIS ET AL.

Figure 1. Solver parameter dialog box.

in the set target cell (on the Solver parameter dialog box) be minimised.

CURVE FITTING EXAMPLES AND DISCUSSION TL case The synthetic glow curve REFGLOW.002 as well as the measured glow curve REFGLOW.009 of the GLOCANIN project (19) are used next as examples in order to estimate their parameters, using the earlier-mentioned procedure in Excel. The synthetic glow curve REFGLOW.002 is the sum of four glow peaks obtained by solving directly the differential equation with trapping parameters such as the glow curve structure of TLD-100, which describes the charge transport in the TL material according to the well-known Randall-Wilkins model with the Bulirsch-Stoer method. This glow curve is included in order to evaluate the accuracy in the determination of parameters in the case of overlapping peaks. Figure 2 shows the reference glow curve REFGLOW.002 in open cycles together with the calculated curve as a solid line passing through them, using the GOK model. Table 2 gives the calculated parameters for the peaks 2, 3, 4 and 5 for the GLOCANIN curve REFGLOW.002 as found by applying the solver Addin on the functions for both GOK and MOK(7, 11), using the Microsoft Excel spreadsheet. As can be seen from the low FOM values (0.0090 and 0.0093 % for the GOK model and the MOK model, respectively), the glow curve deconvolution

Figure 2. Curve fitting of REFGLOW.002 using excel spreadsheet and the Solver Add-in under GOK model. In the lower graph residual TL intensity versus temperature, after curve fitting is presented.

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perform either linear or non-linear least-square fitting. The Solver uses the Generalised Reduced Gradient non-linear optimisation code. For each of the changing cells, the Solver evaluates the partial derivative of the objective function (the target cell) with respect to the changing cell ai, by means of the finite-difference method. The Solver uses a matrix including the partial derivatives to determine the gradient of the response surface and thus decides how to change the values of the changing cells in order to approach the desired solution. The Solver Parameter dialog box is presented in Figure 1. In the Set Target Cell box the cell containing the quantity that is going to be minimised, here the FOM (%), must be typed. Because of the minimisation process, the Min button must be checked. In the By Changing Cells box, the Cells that contain the fitting parameter values must be typed—here the values of Imax, Tmax, E and b. With Solver, some physically imposed constrains can be applied to the solution. For example, in the case of GOK, the value of the kinetic parameter b is limited by the inequality 1,b 2 and in the case of MOK, the value of the kinetic parameter a is limited by the inequality 0 a , 1. So this constrains must be used for the above cases. Since the Solver operates by a search routine, a solution would be found most rapidly and efficiently if the initial estimates that have been provided are close to the final values. A chart of the data that displays both yobsd and ycalc is often useful to be created, and then a good set of initial parameters can be estimated by varying the parameters manually. It must be noticed that the initial values of the fitting parameters and the optimised parameter values are placed in the same cell because the Solver requires that the user should define the cells that have to be modified (by changing cells, on the Solver parameter dialog box) so that the quantity selected

COMPUTERISED CURVE DECONVOLUTION Table 2. Estimated TL parameters for both GOK and MOK using Microsoft Excel Add-in, Solver for REFGLOW.002 as well as the references values as reported in the CLOCANIN project.

Peak 2

Peak 3

Peak 4

General-order kinetic (FOM (%)¼0.008998) 456.5 484 Tmax (K) 417.1 E (eV) 1.383 1.483 1.584 b 1.0006 1.0009 1.0009 TL 11 100 16 914 27 382 3.9Eþ16 1.6Eþ16 2.1Eþ16 s (s21) Mixed-order kinetic (FOM (%)¼0.009339) 413.6 451.6 478.0 tmax (K) E (eV) 1.383 1.483 1.583 a 7.2E205 0 0 TL 11 098 16 896 27 397 –1 5.6Eþ16 2.5Eþ16 3.3Eþ16 s (s ) (19) CLOCANIN tmax (K) 417.3 456.8 484 E (eV) 1.383 1.483 1.583 –1 3.9 Eþ16 1.6Eþ16 2Eþ16 s (s ) TL 111 001 16 898 27 401

Peak 5

511.6 2.0038 1.0006 47 302 4Eþ19 503.7 2.0038 0 47309 8.5Eþ19 511.9 2.0038 4Eþ19 47 302

Note that Tmax for the CLOCANIN project is calculated according to the Randall-Wilkins model from the given heating rate, E and s values.

analysis process was very accurate. The similar values of peak temperature (Tmax), activation energy (E) and Integral, for each individual glow curve presented, as well as the same order of magnitude of frequency factor (s), indicate the agreement between the used models. These values are in great agreement with those reported in the GLOCANIN project (19), indicating the computational power of Solver utility. The value of the calculated parameter (b) that stands for kinetic order according to the GOK has been calculated as b¼1.00065 for peak 2, very close to unity (b¼1) that stands for the first-order kinetics model that has been used in order to obtain the REFGLOW.002, according to the Randall-Wilkins model. On the other hand, the value of the calculated parameter (a) that stands for kinetic order according to the MOK has been calculated as a ¼7.261025. For this value (a¼0), the MOK model is equivalent with the first-order kinetic model as has been described by Kitis et al.(12). Reference glow curve REFGLOW.009 stems for a highly irradiated (D¼600 Gy) TLD 700 sample. Apart from the well-known glow peaks 2– 5, it features a complex group of high temperature peaks. These overlapping peaks make the fitting procedure much more complicated. Figure 3. Curve fitting of REFGLOW.009 using excel spreadsheet and the Solver Add-in under General-Order Kinetics model. In the lower graph

Figure 3. Shows the reference glow curve REFGLOW.009 in open cycles together with the calculated curve as a solid line passing through them, using the GOK model.

residual TL intensity versus temperature, after curve fitting is presented. Table 3 gives the calculated parameters for the peaks 1, 2, 3, 4, 5, 5a, 6, 7, 8, 9, 10 and 11 of the GLOCANIN curve REFGLOW.009 as they were estimated by applying the Solver add-in utility on the function of the GOK(7), in the Microsoft Excel spreadsheet. These values are in great agreement with those reported in the GLOCANIN project (19), indicating the computational power of Solver utility.

OSL case As opposed to the case of TL, there is no project such as the GLOCANIN for OSL. So, there are not any reference LM-OSL curves with their corresponding parameters in order to evaluate the capabilities of computer codes for fitting in LM-OSL cases. Due to the lack of any reference LM-OSL curves, simulation was necessary in order to obtain data for reliable curve fitting. The latter was performed using the equation(10) sg b=1b I ðtÞOSL ¼ n0 sgt 1 þ ðb 1Þ t2 2 where n0 is the initial concentration of electrons in traps, s the photoionisation cross section, g the stimulation increase rate and b the kinetic order parameter. To obtain numerical values for I(t) OSL, the values for n0 ¼100 000, s.g¼0.0001 and b¼1.00000001 were used. The resulting data were used to perform a curve fitting using the equations for GOK and MOK of LM-OSL. Figure 4 shows the simulated LM-OSL curve as open cycles and the calculated curve as a solid line passing through them under the GOK model.

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TL parameters

D. AFOUXENIDIS ET AL. Table 3. Estimated TL parameters for General-Order Kinetics using Microsoft Excel Add-in, Solver for REFGLOW.009 as well as the reference values as reported in the CLOCANIN project. Peak 1

Peak 2

Peak 3

Peak 4

Peak 5a

Peak 6

Peak 7

General-order kinetic (FOM¼0.78 %) E (eV) 0.95 1.23 1.30 1.56 2.08 s (s – 1) 11013 11015 21014 11016 51020

1.77 91016

1.46 1.37 1.86 1.97 11013 91011 71015 11016

1.75 31013

1.41 11010

GLOCANIN(19) E (eV) — 1.25 1.3 1.6 2.02 11015 21014 81016 81019 s (s – 1) —

— —

— —

— —

— —

— —

Peak 8

— —

Peak 9

— —

Peak 10 Peak 11

Table 5. Estimated parameters for both GOK and MOK using Microsoft Excel Add-in, Solver for Al2O3:C LM-OSL curve structure. Al2O3:C

Figure 4. Curve fitting of simulated LM-OSL curve using Microsoft Excel spreadsheet and the Solver Add-in according GOK model. In the lower graph values for residual LM-OSL intensity versus stimulation time, after curve fitting are presented.

Mixed-order kinetics [FOM (%)¼1.389] Peak 1 Peak 2 93.552 415.368 tmax (s) a 0.950 0.909 Integral 1 932 742 4 516 100

Peak 3 1316.279 0.914 3 318 872

General-order kinetics [FOM (%)¼1.410] tmax (s) 93.471 415.367 b 2.000 2.000 Integral 1 967 657 4 554 843

1316.277 1.916 3 245 255

Deviations (%) tmax (s) 0.086 Integral 21.806

0.000 2.218

0.000 20.858

Table 4. Estimated LM-OSL parameters for both GOK and MOK using Microsoft Excel Add-in, Solver for simulated LM-OSL curve. LM-OSL parameters GOK tmax (s) b Integral FOM (%)

100.000013 1.000000001 99998.75394 0.00004929

MOK

a

99.99991708 0.000002696 99998.80336 0.000054287

Table 4 gives the peak parameters for the simulated curve as found by applying the Solver add-in utility on the function of the GOK(13) and on the function of the MOK(14), using Microsoft Excel spreadsheet. As can be seen from Table 4, the accuracy of the fitting procedure using Microsoft Excel Add-in, Solver, is extremely good, as described by the FOM

Figure 5. Shows an experimental LM-OSL curve of the widely used Al2O3:C dosemeter (open cycles) together with the fitted curve (solid line) using the MOK model. The individual components of the Al2O3:C LM-OSL curve are also shown in the same figure.

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Peak 5

COMPUTERISED CURVE DECONVOLUTION

CONCLUSIONS The deconvolution process for TL as well as linear modulated optically stimulated luminescence curves using a common Microsoft Excel spreadsheet was outlined in this paper. Both GOK and MOK functions can be used for deconvolution. It should be noted that the same process can be used for analysing the sum of a number of exponentials and is not limited to TL and LM-OSL cases presented explicitly in this paper. For instance, CW-OSL curves can be analysed in a similar way by applying the transformations proposed by Bulur. The process uses the Solver utility, a Microsoft Excel add-in, that is also briefly presented here; Specific examples of both TL and OSL computerised curve deconvolution analysis using the spreadsheet along with the Solver add-in, are given. These examples show that the luminescence trapping parameters can be estimated with a high accuracy with a simple procedure without the need of complicated and specialised computer codes. The accuracy of the results was further confirmed in the case of TL by comparison with the results of the GLOCANIN project. This work shows the power of using a simple interface commercial software such as Microsoft Excel, in order to carry out complex scientific problems, such as computerised curve deconvolution analysis. It should be noted that the authors strongly recommend the usage of the Solver utility because it is used within the familiar Excel environment; so new commands and procedures do not have to be learnt.

Microsoft Excel spreadsheets for multi-component luminescence computerised curve deconvolution analysis can be obtained on-line from the web site (www.ipet.gr) of the Archaeometry Department, of the Cultural and Educational Technology Institute by filling a registration form with the contact information. The spreadsheets are offered free of charge, as long as they are not used for commercial purposes or as a part of a system for dose assessment (21). REFERENCES 1. Horowitz, Y. S. and Moscovitch, M. Computerized glow curve deconvolution applied to high dose (102– 105 Gy) TL dosimetry. Nucl. Instrum. Methods. Phys. Res. A 243(1), 207– 214 (1986). 2. Horowitz, Y. S, Moscovitch, M. and Wilt, M. Computerized glow curve deconvolution applied to ultralow dose LiF thermoluminescence dosimetry. Nucl. Instrum. Methods. Phys. Res. A 244(3), 556–564 (1986). 3. Horowitz, Y. S. and Yossian, D. Computerized glow curve deconvolution: the case of LiF TLD-100. J. Phys. D: Appl. Phys. 26(8), 1331– 1332 (1993). 4. Horowitz, Y.S. and Yossian, D. Computerized glow curve deconvolution: application to thermoluminescence dosimetry. Radiat. Prot. Dosim. 60(1), 1– 114 (1995). 5. Furetta, C., Kitis, G. and Kuo, C. -H. Kinetics parameters of CVD diamond by computerized glow-curve deconvolution (CGCD). Nucl. Instrum. Methods. Phys. Res. B: Beam Interact. Mater. At. 160(1), 65–72 (2000). 6. Pagonis, V., Furetta, C. and Kitis, G. Numerical and Practical Exercises in Thermoluminescence. Springer (2006). 7. Kitis, G., Gomes-Ros, J. M. and Tuyn, J. W. N. Thermoluminescence glow curve deconvolution functions for first, second and general orders of kinetics. J. Phys. D: Appl. Phys. 31 2636–2641 (1998). 8. Van Dijk, J. W. E. Thermoluminescence glow curve deconvolution and its statistical analysis using the flexibility of spreadsheet programs. Radiat. Prot. Dosim. 119, 332– 338 (2006). 9. Chen, R. and McKeever,, S. W. S. Theory of Thermoluminescence and Related Phenomena. World Scientific (1997). 10. Bøtter-Jensen, L., McKeever, S. W. S. and Wintle,, A. G. Optically Stimulated Luminescence Dosimetry. Elsevier (2003). 11. Kitis, G. and Gomes-Ros, J. M. Thermoluminescence glow curve deconvolution functions for mixed order of kinetics and continuous trap distribution. Nucl. Instrum. Methods Phys. Res. A 440, 224–231 (2000). 12. Kitis, G., Chen, R. and Pagonis, V. Thermoluminescence glow-peak shape methods based on mixed order kinetics. Physica Status Solidi (a) 205(5), 1181–1189 (2008). 13. Kitis, G. and Pagonis, V. Computerized curve deconvolution analysis for LM-OSL. Radiat. Meas. 43, 737– 741 (2008). 14. Kitis, G., Furetta, C. and Pagonis, V. Mixed order kinetics model for optically stimulated luminescence. Mod. Phys. Lett. B, 23(27), 3191–3207 (2009). 15. Kitis, G., Chen, R., Pagonis, V., Carinou, E. and Kamenopoulou,, V. Thermoluminescence under an

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index. The respective FOM values are 0.000049 % for the GOK model and 0.000054 % for the MOK model. It should be noted that the estimated values of the parameters in both cases are almost the same with very small deviations. In Table 5, the calculated peak parameters for the experimental LM-OSL curve of Al2O3:C are given, as they were estimated by applying the Solver add-in utility on the function of the GOKand on the function of the MOK, in the Microsoft Excel spreadsheet. Figure 5. LM-OSL Curve fitting of Al2O3:C using excel spreadsheet and the Solver Add-in under General-order kinetics model. In the lower graph values for residual LM-OSL intensity versus stimulation time, after curve fitting is presented. As can be seen from the low FOM values (1.410 and 1.389% for the GOK model and the MOK model, respectively), the curve deconvolution analysis process was very accurate. Deviations between the estimated values of each of the components presented, in stimulation time at peak height (tmax), in Integral, is also presented. These values are in great agreement with those reported in the literature(20).

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exponential heating function: I. Theory. J. Phys. D: Appl. Phys. 39, 1500, (2006). 16. Balian, H. G. and Eddy,, N. W. Figure-of-merit (FOM), an improved criterion over the normalized chisquared test for assessing goodness-of-fit of gamma-ray spectral peaks. Nucl. Instrum. Methods 145(2), 389– 395 (1977). 17. Billo, E. J. Excel for Scientists and Engineers. Wiley (2007). 18. Fylstra, D., Lasdon, L., Watson, L. and Waren, A. Design and use of the Microsoft Excel solver. Interfaces 28, 29– 55 (1998).