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

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1 Adv. Studies Theor. Phys., Vol. 4, 200, no. 4, The Effect of the Activation Energy, Frequency Factor and the Initial Concentration of Filled Traps on the TL Glow Curves of Thermoluminescence Md. Shah Alam,2 and Sabar Bauk Physics Section, School of Distance Education Universiti Sains Malaysia, 00 Penang, Malaysia 2 Department of Physics Shahjalal University of Science and Technology Sylhet, Bangladesh Abstract We have represented different types of models for the explanation of thermoluminescence glow curves such as Randall-Wilkins model, Garlick-Gibson model and May-Partridge model. We have studied the effect of the activation energy, frequency factor and the initial concentration of filled traps on the glow curves using MATHEMATICA for the above mentioned models. PACS:.60.Kn Keywords: Thermoluminescence glow curves, activation energy, frequency factor, concentration of filled traps

2 666 Md. Shah Alam and S. Bauk. Introduction Thermoluminescence (TL) is the thermally stimulated emission of light following the previous absorption of energy from radiation. The typical glow curve contains one or more glow peaks. Each peak gives information about each trap level and its occupation state, etc in the TL material. The glow peak is 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 in 945 []. Instead, if the retrapping dominates, the recombination with the holes is suppressed and the curve has a wide peak explained by Garlick and Gibson in 94 [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 in 964 [3]. In the second section we will clearly explain these three models i.e., Randall-Wilkins model, Garlick-Gibson model and May- Partridge model. We explain the effect of the activation energy on the glow curves in section three, the effect of the frequency factor on the glow curves in section four and the effect of the initial concentration of filled traps on the glow curves in section five using MATHEMATICA for the above mentioned models. 2. Different Models of Thermoluminescence 2. Randall and Wilkins Model E Electron Trap, Tr CB R VB Figure : Randall Wilkins Model We have found the Randall and Wilkins model for the thermoluminescence in different books [4-6] and journals [-9]. In 945, Randall and Wilkins extensively used a mathematical representation for each peak in a glow curve, starting from studies on phosphorescence. Their mathematical treatment was based on the energy band model and yields the well-known first order expression. Figure

3 Thermoluminescence glow curves 66 shows the simple model used for the theoretical treatment. Between the delocalized bands, conduction band CB, and valence band VB, two localized levels (metastable states) are considered, one acting as a trap Tr, and the other acting as a recombination center R. The distance between the trap Tr and the bottom of the CB is called the activation energy or trap depth E. This energy is the energy required to liberate a charge, i.e. an electron, which is trapped in Tr. According to this model, the TL intensity I, at a constant temperature can be written as.. () where n (cm 3 ) is the electron concentration trapped at time t(s), k (evk ) is the Boltzmann s constant, S(sec ) is the frequency factor and T(K) is the temperature. If the sample is heated up so that the temperature rises at a linear heating rate (K s ), then exp.. (2) On integration of Eq. (2), one obtains exp / /.. (3) where n 0 (cm 3 ) is the concentration of traps populated at the starting heating temperature T 0 (K). The temperature dependence of the emitted TL is given by / 2.2 Garlick-Gibson Model /.. (4) Garlick and Gibson [2] considered the case of strong retrapping, which brought about the second-order kinetics. According to this model, the TL intensity, I, at a constant temperature can be written as.. (5) where N (cm 3 ) is the trap concentration. The equation describing the TL

4 66 Md. Shah Alam and S. Bauk intensity for second-order glow peak is given by [2] exp.. (6) 2.3 May-Partridge Model May and Partridge [3, ] and others have proposed an empirical equation to describe the TL glow peak when conditions for neither first-order nor secondorder are satisfied. This equation is known as the general-order kinetics and its final form has been suggested as [2] exp /.. () where b is the order of kinetics. Usually b is assumed to be between and 2, but sometimes can exceed this range [3]. The solution of Eq. () for b is given by [2] "/ " exp /.. () where " /... (9) 3. The Effect of The Activation Energy on The TL Glow Curves 3. The effect of the activation energy on the TL glow curves for the Randall and Wilkins model Using MATHEMATICA we can plot the TL glow curves for different activation energies. Here is an example of three Randall and Wilkins model (first order) TL curves, calculated using three different values of the activation energies E =.0 ev, E 2 =.5 ev and E 3 =2.0 ev. All other parameters are the same i.e., frequency factor, s =0 2 sec and Boltzmann s constant, k= evk and the initial concentration of filled traps, n 0 = 0 0 cm 3. We get the curves as shown in Figure 2.

5 Thermoluminescence glow curves 669 3ґ0 E =.0 ev TL glowcurves: E=.0, E2=.5, E3=2.0 E 2 =.5 ev 2.5 ґ0 E 3 = 2.0 ev 2ґ0.5 ґ0 ґ0 5ґ T Figure 2: Randall and Wilkins model TL glow curves for the activation energies.0 ev,.5ev and 2.0 ev The effect of the activation energy on the TL glow curves for the Garlick- Gibson model Here is an example of three Garlick-Gibson model (second order) TL curves, calculated using three different values of the activation energies E =.0 ev, E 2 =.5 ev and E 3 =2.0 ev. All other parameters are the same i.e., frequency factor, s =0 2 sec and Boltzmann s constant, k= evk and the initial concentration of filled traps, n 0 = 0 0 cm 3. We get the curves as shown in Figure 3..5 ґ0 E =.0 ev TL glow curves: E=.0, E2=.5, E3=2.0 E 2 =.5 ev E 3 = 2.0 ev 5 ґ T Figure 3: Garlick-Gibson model TL glow curves for the activation energies.0

6 60 Md. Shah Alam and S. Bauk ev,.5 ev and 2.0 ev. 3.3 The effect of the activation energy on the TL glow curves for the May- Partridge model Here is an example of three May-Partridge model (general order) TL curves, calculated using three different values of the activation energies E =.0 ev, E 2 =.5 ev and E 3 =2.0 ev. All other parameters are the same i.e., frequency factor, s =0 2 sec and Boltzmann s constant, k= evk and the initial concentration of filled traps, n 0 = 0 0 cm 3. We get the curves as shown in Figure 4. E =.0 ev ґ0 TL glow curves: E=.0, E2=.5, E3=2.0 E 2 =.05eV E 3 = 2.0 ev 6 ґ0 4 ґ T Figure 4: May-Partridge model TL glow curves for the activation energies.0 ev,.5 ev and 2.0 ev. Remarks: As the activation energy E is increased, the TL glow curve shifts towards higher temperatures, but the curve maintains its overall shape for these three models.

7 Thermoluminescence glow curves 6 4. The Effect of The Frequency Factor on The TL Glow Curves 4. The effect of the frequency factor on the TL glow curves for the Randall and Wilkins model Here is an example of three Randall and Wilkins model (first order) TL curves, calculated using three different values of the frequency factors, s = 0 sec, s 2 =0 0 sec and s 3 =0 2 sec. All other parameters are the same i.e., activation energy, E =.0 ev and Boltzmann s constant, k = evk and the initial concentration of filled traps, n 0 = 0 0 cm 3. We obtained the curves in Figure 5. 3 ґ0 2.5 ґ0 s 3 = 0 2 sec TL glow curves : s=0^, s2=0^0, s3=0^2 s 2 = 0 0 sec s = 0 sec.5 ґ0 5 ґ T Figure 5: Randall and Wilkins model TL glow curves for the frequency factors, s = 0 sec, s 2 =0 0 sec and s 3 =0 2 sec. 4.2 The effect of the frequency factor on the TL glow curves for the Garlick- Gibson model Here is an example of three Garlick-Gibson model (second order) TL curves, calculated using three different values of the frequency factors, s = 0 sec, s 2 =0 0 sec and s 3 =0 2 sec. All other parameters are the same i.e., activation energy, E =.0 ev and Boltzmann s constant, k= evk and the initial concentration of filled traps, n 0 = 0 0 cm 3. We get the curves as shown in Figure 6.

8 62 Md. Shah Alam and S. Bauk s 3 = 0 2 sec TL glow curves : s=0^, s s2=0^0, s3=0^2 2 = 0 sec s = 0 sec.5 ґ0 5 ґ T Figure 6: Garlick-Gibson model TL glow curves for the frequency factors, s = 0 sec, s 2 =0 0 sec and s 3 =0 2 sec. 4.3 The effect of the frequency factor on the TL glow curves for the May- Partridge model Here is an example of three May-Partridge model (general order) TL curves, calculated using three different values of the frequency factors, s = 0 sec, s 2 =0 0 sec and s 3 =0 2 sec. All other parameters are same i.e., activation energy, E =.0 ev and Boltzmann s constant, k = evk and the initial concentration of filled traps, n 0 = 0 0 cm 3. We get the curves as shown in Figure.

9 Thermoluminescence glow curves 63 s 3 = 0 2 sec TL glowcurves : s=0^, s2=0^0, s3=0^2 s 2 = 0 0 sec ґ0 s = 0 sec 6 ґ0 4 ґ T Figure : May-Partridge model TL glow curves for the frequency factors, s = 0 sec, s 2 =0 0 sec and s 3 =0 2 sec. Remarks: As the frequency factor s is increased, the TL glow curve shifts towards lower temperatures, but the curve maintains its overall shape for these three models. 5. The Effect of the Initial Concentration of Filled Traps on the TL Glow Curves 5. The effect of the initial concentration of filled traps on the TL glow curves for the Randall and Wilkins model Here is an example of three Randall and Wilkins model (first order) TL curves, calculated using three different values of the initial concentration of filled traps, n 0 = N, 0.5 N and 0. N (where N = 0 0 cm 3 ). All other parameters are the same i.e., activation energy, E =.0 ev and Boltzmann s constant, k = evk and the frequency factor, s = 0 2 sec. We get the curves as shown in Figure.

10 64 Md. Shah Alam and S. Bauk 3 ґ0 2.5 ґ0 n 0 = 0 0 First order kinetics : no=* N, 0.5* N, 0.* N n 0 = ґ0 n 0 = ґ T Figure : Randall and Wilkins model TL glow curves for initial concentration of filled traps, n 0 = 0 0 cm 3, cm 3 and cm 3. Remarks: The initial concentration of filled traps (n o ) affects only the maximum height of the TL glow curve, and leaves the shape of the st order TL glow curve and the temperature of maximum TL intensity (T max ) unchanged. 5.2 The effect of the initial concentration of filled traps on the TL glow curves for the Garlick-Gibson model Here is an example of three Garlick-Gibson model (second order) TL curves, calculated using three different values of the initial concentration of filled traps, n 0 = N, 0.5 N and 0. N (where N = 0 0 cm 3 ). All other parameters are the same i.e. activation energy, E =.0 ev and Boltzmann s constant, k = evk and the frequency factor, s = 0 2 sec. We get the curves as shown in Figure 9.

11 Thermoluminescence glow curves 65 n 0 = 0 0 Second order kinetics : no=* N, 0.5* N, 0.* N.5 ґ0 n 0 = n 0 = ґ T Figure 9: Garlick-Gibson model TL glow curves for initial concentration of filled traps, n 0 = 0 0 cm 3, cm 3 and cm 3. Remarks: The initial concentration of filled traps (n o ) affects both the maximum height of the TL glow curve and the temperature of maximum TL intensity (T max ), but leaves the overall shape unchanged. 5.3 The effect of the initial concentration of filled traps on the TL glow curves for the May-Partridge model Here is an example of three May-Partridge model (general order) TL curves, calculated using three different values of the initial concentration of filled traps, n 0 = N, 0.5 N and 0. N (where N = 0 0 cm 3 ). All other parameters are the same i.e. activation energy, E =.0 ev and Boltzmann s constant, k = evk and the frequency factor, s = 0 2 sec. We get the curves as shown in Figure 0.

12 66 Md. Shah Alam and S. Bauk n 0 = 0 0 Generalorder kinetics : no=* N, 0.5* N, 0.* N ґ0 6 ґ0 4 ґ0 n 0 = n 0 = T Figure 0: May-Partridge model TL glow curves for initial concentration of filled traps, n 0 = 0 0 cm 3, cm 3 and cm 3. Remarks: The initial concentration of filled traps (n o ) affects both the maximum height of the TL glow curve and the temperature of maximum TL intensity (T max ), but leaves the overall shape unchanged. 6. Conclusion We have studied the effect of the activation energy, frequency factor and the initial concentration of filled traps on the glow curves using Randall-Wilkins model, Garlick-Gibson model and May-Partridge model. We have observed the following results: (i) As the activation energy E is increased, the TL glow curve shifts towards higher temperatures, but the curve maintains its overall shape for these three models. (ii) As the frequency factor s is increased, the TL glow curve shifts towards lower temperatures, but the curve maintains its overall shape for these three models. (iii) The initial concentration of filled traps (n o ) affects only the maximum height of the TL glow curve, and leaves the shape TL glow curve and the temperature of maximum TL intensity (T max ) unchanged in the case of the Randall-Wilkins model. (iv) The initial concentration of filled traps (n o ) affects both the maximum height of the TL glow curve and the temperature of maximum TL intensity (T max ), but

13 Thermoluminescence glow curves 6 leaves the overall shape unchanged in the case of the Garlick-Gibson model and May-Partridge model. References [] Randall, J. T. and Wilkins, M. H. F. (945). Phosphorescence and electron traps. I. The study of trap distributions. Proc. Roy. Soc. Lond. A 4, 366. [2] Garlick, G. F. J. and Gibson, A. F. (94). The electron trap mechanism of luminescence in sulphide and silicate phosphors. Proc. Phys. Soc. 60, 54. [3] May, C. E. and Partridge, J. A. (964). Thermoluminescence kinetics of alpha irradiated alkali halides. J. Chem. Phys. 40, 40. [4] Chen, R. and McKeever, S.W.S. (99). Theory of thermoluminescence and related phenomena. World Scientific, Singapore, Chapters and 2. [5] Aitken, M.J. (95), Thermoluminescence Dating (London: Academic) [6] McKeever, S.W.S. (95), Thermoluminescence of solids, Cambridge University press. [] Furetta, C. and Kitis, G. (2004). Review models in thermoluminescence, Journal of Matherial Science 39, [] McKeever, S.W.S., and Chen, R. (99), Luminescence models, Radiat. Meas. 2, p [9] Sunta, C. M., Ayta, W. E. F., Kulkami, R. N., Piters, T. M. and Watanabe,S. (999). Theoretical models of thermoluminescence and their relevance in experimental work, Radiat. Prot. Dosim. 4, p.25-. [0] Rasheedy, M. S. (2005). A Modification of the Kinetic Equations Used for Describing the Thermoluminescence Phenomenon, Journal of Fluorescence Vol. 5, No. 4. [] Partridge, J. A. and May, C. E. (965). Anomalous thermoluminescence kinetics of irradiated alkali halides. J. Chem. Phys. 42, 9. [2] Rasheedy, M. S. (993). On the general-order kinetics of the thermoluminescence glow peak. J. Phys.: Condens. Matter 5, 633.

14 6 Md. Shah Alam and S. Bauk [3] Chen, R. (969). Glow curves with general order kinetics. J. Electrochem. Soc. 6, 254. Received: May, 200

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

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