Barman, Supriya

Bhattacharya, Manabesh

Karmakar, Mahua

Mazumdar, P S

Sarkar, Bikash Kumar

Singh, S Dorendrajit

Singh, W Sambhunath

IACS Institutional Repository

Indian J. Phys. 82 (11), 1495-1502 (2008) Evaluation of kinetic parameters from thermoluminescence glow curves Mahua Karmakar1*, Bikash Kumar Sarkar2, Supriya Barman3, P S Mazumdar1, S Dorendrajit Singh4, W Sambhunath Singh5 and Manabesh Bhattacharya5 1 Department of Physics, A PC College, New Barrackpore, Kolkdta 700 131, West Bengal, India department of Physics, M D Mahavidyapith, Birati, Kolkata-700 051, West Bengal, India 3 Bhangar Balika Vidyalaya, Bhangar, 24 Parganas (South), West Bengal, India ^Department of Physics, Manipur University, Canchipur, Imphal 795 003, Manipur, India 5 Department of Physics, Manipur College, Singjamei Imphal-795 008, Manipur, India 0 Department of Physics, K N College, Berhampur 742 101, West Bengal, India L mail mahua, apc^yahoo com Received 12 Mav 2008, accepted 4 Julv 2008 Abstract In the present paper two methods of determination of trapping parameters from thermoluminescence glow curves are considered One is due to Kirsh and the other is suggested by Vejnovic et af By taking into account both the computer generated and the experimental TL peaks, It is established that the Kirsh method is superior to that of Vejnovic et at The possible application of Kirsh method to the case of temperature dependent frequency factor is also studied Keywords : Thermoluminescence, trapping parameters, temperature dependent frequency factor PACSNo. . 78.60 Kn 1. Introduction Thermoluminescence (TL) is often used for studies of trapping levels is solids, in particular for the evaluation of trap depth or the activation energy [1]. The study remains active because of its applications in the field of dosimetry and dating [1,2]. A number of methods [1] are currently been applied for the evaluation of the trapping parameters from a TL glow peak i.e. activation energy (E), order of kinetics (b) and frequency factor (s). Vejnovic et al [3] proposed a method of determination of TL trapping parameters. In the present paper Corresponding Author ©2008IACS 1496 Mahua Karmakar et al the method of Vejnovic et al is denoted by the acronym VPRD VPRD method makes use of a single expression for activation energy The main difficulty of using such expression is requirement of prior knowledge of order of kinetics It seems that Vejnovic et al is not aware of this fact pointed out by Gartia et al [4] Moreover they have not taken note of the existence of a number of expressions of activation energy [5,6] Kirsh [7] suggested a method of evaluation of TL trapping parameters The method is simple and straight forward in the sense that unlike other conventional methods of determination of trapping parameters such as peak shape, initial rise and various heating rates [8], it permits simultaneous evaluation of order of kinetics b and activation energy E In the present paper a critical comparison of VPRD method and Kirsh method has been done Finally an attempt has been made to adjudge the validity of Kirsh method for the case of temperature dependent frequency factor [1,8] 2. Theory The expression for TL intensity I can be expressed as [7] 1- dn_ dt (D with dt ( n \ \no J snQ exp - £ kT (2) where T is the temperature at time f, n and n0 are respectively, the number of trapped carriers at time t and initial number of trapped carriers, k is Boltzmann constant, b is the order of kinetics For a linear heating profile given by the eq [8] T^T^pt (3) where p is the linear heating rate and 7"0 is the initial temperature Equation (2) can be recast as [4] •L = exp[(um-u) + F(u,um)] (/> = 1) (4) and — = exp(um-u) 1 - ^ F M j ] " " 1 ( 6 ,1 ) (5) lm is intensity corresponding to the peak temperature Tm F(^um) = u2mexp{um) E2(um) E2(u) u u„ (6) Evaluation of kinetic parameters from thermolummescence glow curves 1497 With u = E/kT and um = E/kTm E (u) is the second exponential integral [9] Following (2) one can write [7] n _ A(T) (7) A(T) is the area of the peak from temperature T upto its end point temperature Te and is given by A(T)^ji(r)dr T From eqs (2) and (3) for any two points of the TL curve [7] (8) Jln(/) A\n(n / nQ) E J ln ( / /7 ) k A\r\(n I n0) (9) where J represents the difference between any two points of the TL curve n/n0 is calculated from eq (7) From eq (9) it follows that a plot of A\n(l)/A\n(nin0) against Jln(1 T)/Aln(n/n0) will give a straight line of slope - (Elk) and intercept b Thus Kirsh method allows simultaneous evaluation of b and E Knowing b and E the frequency factor s can be obtained from the equation [8] -A*£ exp kT. m / •ib-*£°»{w)dT Now according to VPRD method expressions for trapping parameters are given by [3] (10) E = ^ H Jaff and fo = -VVA2 -4V*2 2A, (11) with 4>=-H 'eff - 0 2 w. eff . (12) 1498 Mahua Karmakar et a/ A, = -0 592 In "eft LWeffJ - 1 3 7 8 + 1 5 2 ^ f f (13) and A, = 0 9 'eft (14) where Se„ and weff are respectively, the areas of the falling side and the whole peak s has been found out by using the approximate equation S =^re x p kT„ U(b-1)^IsL (15) Normally in the analysis of the TL curves, the frequency factor is assumed to be temperature independent But this assumption needs modification in view of temperature dependence of capture cross-sections [1,7,8,10,11] As a results s varies as Ta ( -2 < a < 2) i e we can write s{T) = s0 Ta where s0 is a constant For the case of temperature dependent frequency factor the analogues of equations (4)~(6) are given by [12] i;=[ir) eM("™-")+F^M (b=:1> (16) t /„ exp{um-u) A & - 1 _y , 1— r - ^ . f / n ) b b-1 (6*1) (17) With F(u,um) = (u2m+aum)uamexp(um) Ea+2(Um) Ea+l(u) - a + 2 l u m ,a+1 a+2^ (a+1 Where En(u) is the n-th order exponential integral [9] 3. Results and discussion It is clear from eq (7) that in order to find nJn0 one has to compute the area of the peak which requires the computation of the integral (Te being the end point T temperature) Now considering the case of non first order kinetics (b * 1 ) [sqs (5) and (6)] the area of the portions of the peak from temperature T to the end point temperature T is given by Evaluation of kinetic parameters from thermoluminescence glow curves 1499 r b--\ F(u',um) dT' (19) Te is obtained by solving the equation l(Te) = 0 (Te > Tm) numerically by using Newton, McAuley method [13] Changing the independent variable from T to u (= ElkT) eq (19) can be expressed as A{U) = J JexPKi~a') 1 ~ W « J to 1 u'~2du' (20) Integral occurring in eq (20) has been evaluated by using Gauss Legendre quadrature method [14] with range (u, ue) being split into a number of subranges The integral over each subranges has been carried out by using 16-pomt Gauss Legendre quadrature The second exponential integral has been evaluated [15] by using its continued fraction representation [16] For the first order kinetics similar numerical technique has been followed As a check of our computer code we have reproduced the values of Seff and weff of numerically computed TL peaks reported by Vejnovic et al [3] The order of kinetics (b) and activation energy (E) have been found out by using the technique of standard least square regression [17] In Table 1 we present the trapping parameters of a number of computer generated TL peaks considered by Vejnovic et al Table 1. rate {[\ = E„(eV> 16 1 6 04 1 6 04 01 01 1 6 04 01 01 Trapping parameters of 0 5K/sec) K 2 5 1 9 1 5 1 5 1 5 1 5 0 7 0 7 1 9 1 9 2 5 Sin (sec"1) 1013 1013 1013 109 109 109 108 10s 105 105 105 [3] Em, Ek and Ev denote respectively the input some numencally computed TL peaks corresponding to the Ek (eV) 1 608 1 601 0 399 1 601 0 400 0 100 0 090 1 586 0 396 0 099 0 098 \ 2 524 1 908 1 501 1 510 1 513 1 511 0 691 0 686 1 893 1 885 2 450 sK(sec~1) 1 166*1013 1 017*1013 9 380x1012 1 022x109 1 032x109 1 0 1 7 X 1 0 9 8 904x107 8524x104 8679x104 8259x104 7 269x10 Ev (eV) 1 599 1 600 0 400 1 600 0 400 0 100 0 100 1 600 0 400 0 100 0 100 bv 2 493 1 896 1 498 1 498 1 498 1 498 0 700 0 700 1 894 1 894 2 485 linear heating sv (sec1) 9 86x1012 9 90X1012 9 93x1012 9 95x108 9 95x108 9 95x108 1 00x106 1 00x105 9 95x104 9 95x104 9 91X104 1500 Mahua Karmakaretal values of activation energy, activation energy as calculated by Kirsh method and activation energy as calculated by VPRD method bm, bk and bv% sm, sk and sv are the corresponding values of order of kinetics and frequency factor respectively A close inspection of Table 1 reveals that in most of the cases the values of the trapping parameters as calculated by the Kirsh method are more closer to the input values than those of VPRD method Encouraged by our findings in the case of numerically computed peaks we consider well analyzed experimental first order 165 5°C TL peak (/? = 25°C/min) [5] of Ca doped KCI, X-irradiated at room temperature followed by thermal cleaning upto 150°C and 320°C second order TL peak of bluish green microhne ( /?= 40°C/min) [18] In Table 2 we depict the results of application of Kirsh method and VPRD method to the aforesaid experimental TL peaks EcV bcf and Scf are the values of trapping parameter obtained by using the rigorous method of curve fitting It is evident from Table 2 that also in the case of experimental TL peaks the values of the trapping parameters as calculated by using Kirsh method show a better overall agreement with the curve fitted values than those by using VPRD method Table 2. Trapping parameters of some experimental TL peaks Ecf(eV) bcf S t f (sec-1) Ek (eV) bk sk(sec"1) Ev (eV) bv sv (sec"1) 1 36 1 42 1 0 2 0 1 46x1014 3 44x1010 1 36 1 43 1 00 2 01 1 53*1014 3 84X1010 1 36 1 42 1 00 2 00 1 46*1014 3 41x1010 Finally we apply Kirsh method to some computer generated TL peaks with the temperature dependent frequency factor as reported by Flemming [19] A perusal of Table 3 shows that in this case the error in the determination of the activation energy is of the order of 10% as in the case of other methods of analysis such as peak shape, various heating rates and initial rise [8] We get good agreement in the values of b This is because of the fact that the symmetry factor ng of a TL peak with temperature dependent frequency factor is not much sensitive to value of temperature exponent a [12, 19] According to Chen and Kirsh [8] the activation energy obtained in the case of temperature dependent frequency factor should be corrected by subtracting akTm from its calculated value In Table 3. Trapping parameters of some numerically computed TL peaks ith temperature dependent frequency factor All peaks correspond to a linear heating rate of 0 42 K/s a 2 2 2 2 T„(K) 595 595 580 580 E«(eV) 1 303 1 497 0 409 0 591 K 2 0 2 0 1 0 1 0 (sec-1) 1 27*1010 1 27*1010 158 63 158 63 Ek (eV) 1 395 1 404 0 489 0 510 \ 1 981 2 011 0 963 1 037 (s0Tma)k (sec*1) 1 27x1010 1 27x1010 151 82 158 97 ECK <eV) 1 292 1 507 0 389 0610 Evaluation of kinetic parameters from thermolummescence glow curves 1501 table 3 we also present the corrected value of Ek given by Eck = Ek - akTm as obtained by following the prescription of Chen and Kirsh They are in good agreement with input value of E 4. Conclusion In the present paper a comparative study of Kirsh method [7] and VPRD method [3] for the determination of TL trapping parameters have been made by considering both computer generated and experimental TL peaks In general, the prediction of Kirsh method is found to be better than those of VPRD method As mentioned by Vejnovic et al [3] themselves VPRD method does not give good results for b - 0 7 but Kirsh method gives good results even for b = 0 7 This is because of the fact that Kirsh method has a sound mathematical basis resulting directly from basic TL equations whereas the development of VPRD method requires some approximation Moreover as we have already noted that unlike other conventional method of determination of trapping parameters such as initial rise, various heating rates and peak shape Kirsh method admits of simultaneous evaluation of activation energy (E) and order of kinetics (b) It is gratifying to note that Kirsh method can also be applied to the case of temperature dependent frequency factor Acknowledgment Two of us (WSS and MB) are grateful to University Grants Commission, New Delhi for partial financial assistance References [1) R Chen and S W S M c Keever Theory of Thermolummescence and Related Phenomena (Singapore World Scientific) (1997) [2] M J Aitken thermolummescence Dating (London Academic) (1985) [3] Z Vejnovic, M B Paviovic, D Ristic and M Devidovic J Lumm 78 279 (1998) [4] R K Gartia, S J Singh and P S Majumdar Phys Stat Sol (a) 106 291 (1988) [5] P S Mazumdar, S J Singh and R K Gartia J Phys D21 815 (1988) [6] R K Gartia, S J Singh and P S Mazumdar Phys Stat Sol(a) 114 p407 (1989) [7] Y Kirsh Phys Sat Sol (a) 129 15 (1992) IB] R Chen and Y Kirsh Analysis of Thermally Stimulated Processes (Oxford Pergamon) (1981) [9] M Abramowitz and I A Stegun Handbook of Mathematical Functions (New York Dover) (1965) [10] S G Elkomoss, M Samimi, M H All and P Sifferet J Appl Phys 57 5331 (1985) [11] P N Keating Proc Phys Soc 78 p1408 (1961) [12] S D Singh, R K Gartia and P S Majumdar Phys Stat Sol (a) 146 825 (1994) [13] E V Knshnamurty and S K Sen Numencal Algonthms (New Delhi East West) (1994) [14] W H Press, S A Tenkolssky, W T Vellenng and B P Flannery Numencal Receipes m Fortran (Cambridge Cambridge University Press) (1992) [15] W G Devi Vidyasagar University Journal of Physical Sciences 3 18 (1997) 1502 Mahua Karmakar et a/ [16] A Erdeyli Higher Transcendental Functions Vol 2 Ch 9 (New York : McGraw Hill) (1988) [17] E J Dudewicz and S N Misra Modem Mathematical Statistics (New York : Wiley) (1988) [18] A B Ahmed and R K Gartia Phys Stat. Sol (a) 94 645 (1985) [19] R J Flemming J. Phys. D23 950 (1990) 

Evaluation of Kinetic Parameters from Thermoluminescence Glow Curves

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Evaluation of Kinetic Parameters from Thermoluminescence Glow Curves

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