Analysis of Thermoluminescence Glow-Curves for. General-Order Kinetics Using Mathematica

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1 Adv. Studies Theor. Phys., Vol. 4, 2010, no. 1, 3-53 Analysis of Thermoluminescence Glow-Curves for General-Order Kinetics Using Mathematica Md. Shah Alam 1,2 and Sabar Bauk 1 1 Physics Section, School of Distance Education Universiti Sains Malaysia, 1100 Penang, Malaysia 2 Department of Physics Shahjalal University of Science and Technology Sylhet, Bangladesh shahalam032003@yahoo.com Abstract We have discussed and presented the mathematical expression of the general order kinetics of the thermoluminescence glow curves. We plotted the glow curves of general order kinetics for the different values of the order of kinetics, b using MATHEMATICA. We analyzed these glow curves and discussed the results obtained from the analysis. Keywords: Thermoluminescence, general order kinetics, glow curve PACS:.60.Kn 1. INTRODUCTION Thermoluminescence (TL) is the thermally stimulated emission of light following the previous absorption of energy from radiation. If TL emission is detected and plotted as a function of time during readout using a linear time-temperature heating profile, a curve is obtained named Glow Curve. The glow peak is

2 3 Md. Shah Alam and S. Bauk analyzed by an empirical method in which a parameter called the order of kinetics is introduced. When the trapped electrons jump up to the conduction band by the thermal energy, they have two kinds of chances to jump down. One is the retrapping process returning to the same kind of traps and another is the recombination with the hole accompanied by the emission of TL light. When the probability of being retrapped is negligible, the glow curve has a narrow peak shape by a rapid recombination process explained by Randall and Wilkins [1]. Instead, if retrapping dominates, the recombination with the holes is suppressed and the curve has a wide peak explained by Garlick and Gibson [2]. These two descriptions are called the first order kinetics and the second order kinetics respectively. Between these two types, the general order kinetics is introduced for providing a proper analytic continuation from the discrete two types of kinetics explained by May and Partridge [3]. In the second section of this paper we present the mathematical expression of the general order kinetics. In the third section we plot the glow curves using MATHEMATICA and the analysis of the shape of the glow curves for the general order kinetics. 2. MATHEMATICAL EXPRESSION OF GENERAL ORDER KINETICS May and Partridge [3, 4] had proposed an empirical equation to describe the TL glow peak when conditions for neither first-order nor second-order are satisfied. This equation is known as the general-order kinetics. Let us assume that the number n of charge carriers present in a single energy level is proportional to n b, usually b is ranging in the interval between 1 and 2. Then, the probability rate of escape is: exp /.. (1) Where, is the pre-exponential factor. n = concentration of the filled electron traps in the crystal. E = activation energy of the electron traps. S = frequency factor of the electron trap. T = temperature k = Boltzmann s constant Equation (1) is the so called general order kinetics relation. From this equation we can write exp.. (2) Integrating equation (2) we get 1 1 exp.. (3) Putting in equation (3) we get 1 1 exp.. (4)

3 Analysis of thermoluminescence glow-curves 39 The intensity I(t) can be written as /.. (5) From equation (5) and (1) we get exp /.. (6) Putting the value of n from equation (4) into equation (6) we get exp 1 1 exp..... () Putting in equation () we get exp 1 1 exp.. () Assuming the linear heating rate β the formula for intensity can be written as [5] 1 1 "/ " exp /.. (9) 3. ANALYSIS OF GLOW CURVES We consider the values (E = 1.0 ev; S = S 1 ; k = evk 1 ; β = 1 C S 1 ; N = cm 3 ; no=1 N; b = 1.1, 1.2, 1.3, 1.4, 1.5, 1.5, 1.6, 1., 1., 1.9) and using MATHEMATICA to plot glow curves for different values of b. For the analysis of glow curves it is customary to define the quantities τ = Tm T1, δ = T2 Tm, ω = T2 T1 and μ = δ/ω [6]. Here Tm = temperature of maximum TL intensity and T1, T2 = temperatures at half the maximum TL intensity.

4 40 Md. Shah Alam and S. Bauk 4 ґ10 3 ґ10 2 ґ10 1 ґ T T1 Tm T2 Figure 1: Glow curve for b = 1.1 From figure 1 we get T1 = 625, Tm = 45 and T2 = 40. Therefore, τ = Tm T1 = 120, δ = T2 Tm = 95, ω = T2 T1 = 215 and μ = δ/ω = 0.44.

5 Analysis of thermoluminescence glow-curves 41 6 ґ10 5 ґ10 4 ґ10 3 ґ10 2 ґ10 1 ґ T T1 Tmax T2 Figure 2: Glow curve for b = 1.2 From figure 2 we get T1 = 505, Tm = 590 and T2 = 665. Therefore, τ = Tm T1 = 5, δ = T2 Tm = 5, ω = T2 T1 = 160 and μ = δ/ω = 0.46.

6 42 Md. Shah Alam and S. Bauk ґ 10 6ґ 10 5ґ 10 4ґ 10 3ґ 10 2ґ 10 1ґ 10 T T1 Tmax T2 Figure 3: Glow curve for b = 1.3 From figure 3 we get T1 = 416, Tm = 49 and T2 = 53. Therefore, τ = Tm T1 = 63, δ = T2 Tm = 59, ω = T2 T1 = 122 and μ = δ/ω = 0.4.

7 Analysis of thermoluminescence glow-curves 43 ґ10 6 ґ10 4 ґ10 2 ґ T T1 Tmax T2 Figure 4: Glow curve for b = 1.4 From figure 4 we get T1 = 340, Tm = 390 and T2 = 440. Therefore, τ = Tm T1 = 50, δ = T2 Tm = 50, ω = T2 T1 = 100 and μ = δ/ω = 0.50.

8 44 Md. Shah Alam and S. Bauk 1 ґ10 ґ10 6 ґ10 4 ґ10 2 ґ T T1 Tmax T2 Figure 5: Glow curve for b = 1.5 From figure 5 we get T1 = 20, Tm = 320 and T2 = 362. Therefore, τ = Tm T1 = 40, δ = T2 Tm = 42, ω = T2 T1 = 2 and μ = δ/ω = 0.51.

9 Analysis of thermoluminescence glow-curves ґ10 1 ґ10 ґ10 6 ґ10 4 ґ10 2 ґ T T1 Tmax T2 Figure 6: Glow curve for b = 1.6 From figure 6 we get T1 = 22, Tm = 261 and T2 = 29. Therefore, τ = Tm T1 = 33, δ = T2 Tm = 3, ω = T2 T1 = 0 and μ = δ/ω = 0.52.

10 46 Md. Shah Alam and S. Bauk 1.4 ґ ґ10 1 ґ10 ґ10 6 ґ10 4 ґ10 2 ґ T T1 Tmax T2 Figure : Glow curve for b = 1. From figure we get T1 = 1, Tm = 215 and T2 = 245. Therefore, τ = Tm T1 = 2, δ = T2 Tm = 30, ω = T2 T1 = 5 and μ = δ/ω = 0.52.

11 Analysis of thermoluminescence glow-curves ґ ґ10 1 ґ10.5 ґ10 5 ґ ґ T T1 Tmax T2 Figure : Glow curve for b = 1. From figure we get T1 = 14, Tm = 13 and T2 = 201. Therefore, τ = Tm T1 = 25, δ = T2 Tm = 2, ω = T2 T1 = 53 and μ = δ/ω = 0.52

12 4 Md. Shah Alam and S. Bauk 1.5 ґ ґ ґ10 1 ґ10.5 ґ10 5 ґ ґ10 T T1 Tmax T2 Figure 9: Glow curve for b = 1.9 From figure 9 we get T1 = 11, Tm = 140 and T2 = 165. Therefore, τ = Tm T1 = 22, δ = T2 Tm = 25, ω = T2 T1 = 4 and μ = δ/ω = RESULTS AND DISCUSSIONS We have given below the values of different parameters for the glow of different values of b. b Tm τ δ ω μ

13 Analysis of thermoluminescence glow-curves 49 We plotted Change of τ (= Tm T1) for the variation of b shown in figure 10, Change of temperature of maximum TL intensity for the variation of b shown in figure 11, Change of δ (= T2 Tm) for the variation of b shown in figure 12, Change of ω (= T2 T1) for the variation of b shown in figure 13 and finally Change of μ for the variation of b shown in figure 14. τ= Tm T b τ 0 0,2 0,4 0,6 0, 1 1,2 1,4 1,6 1, 2 Order of kinetics, b Figure 10: Change of τ (= Tm T1) for the variation of b. Temperatute of maximum TL intensity, Tm b Tm 0 0,2 0,4 0,6 0, 1 1,2 1,4 1,6 1, 2 Order of kinetics, b Figure 11: Change of temperature of maximum TL intensity for the variation of b.

14 50 Md. Shah Alam and S. Bauk 100 b δ 0 δ = T2 Tm ,2 0,4 0,6 0, 1 1,2 1,4 1,6 1, 2 Order of kinetics, b Figure 12: Change of δ (= T2 Tm) for the variation of b. 250 b ω 200 ω = T2 T ,2 0,4 0,6 0, 1 1,2 1,4 1,6 1, 2 Order of kinetics, b Figure 13: Change of ω (= T2 T1) for the variation of b.

15 Analysis of thermoluminescence glow-curves 51 0,6 b μ 0,5 μ = δ/ω 0,4 0,3 0,2 0, ,5 1 1,5 2 Order of kinetics, b Figure 14: Change of μ for the variation of b. From figure 10, 11, 12 and 13 we see that the value of τ (= Tm T1), the temperature of maximum TL intensity, the value of δ (= T2 Tm) and the value of ω (= T2 T1) all are exponentially decreases with the increase of the order of kinetics, b. From figure 14 we see that the value of μ increases with the increase of the order of kinetics, b.

16 52 Md. Shah Alam and S. Bauk REFERENCES 1. Randall, J. T. and Wilkins, M. H. F. Phosphorescence and electron traps. I. The study of trap distributions. Proc. Roy. Soc. Lond. A 14, 366 (1945). 2. Garlick, G. F. J. and Gibson, A. F. The electron trap mechanism of luminescence in sulphide and silicate phosphors. Proc. Phys. Soc. 60, 54 (194). 3. May, C. E. and Partridge, J. A. Thermoluminescence kinetics of alpha irradiated alkali halides. J. Chem. Phys. 40, 1401 (1964). 4. Partridge, J. A. and May, C. E. Anomalous thermoluminescence kinetics of irradiated alkali halides. J. Chem. Phys. 42, 9 (1965).. 5. Rasheedy, M. S. On the general-order kinetics of the thermoluminescence glow peak. J. Phys.: Condens. Matter 5, 633 (1993). 6. Pagonis, Vasilis, Kitis, George, Furetta, Claudio. Numerical and Practical Exercises in Thermoluminescence, SPRINGER (2006).

17 Analysis of thermoluminescence glow-curves 53 Received: August, 2010

The Effect of the Activation Energy, Frequency. Factor and the Initial Concentration of Filled. Traps on the TL Glow Curves of. Thermoluminescence

Adv. Studies Theor. Phys., Vol. 4, 200, no. 4, 665-6 The Effect of the Activation Energy, Frequency Factor and the Initial Concentration of Filled Traps on the TL Glow Curves of Thermoluminescence Md.