Mechanisms inherent in the thermoluminescence processes

Transcription

1 Indian Journal of Pure & Applied Physics Vol. 42, August 2004, pp Mechanisms inherent in the thermoluminescence processes J Prakash, S K Rai, P K Singh & H O Gupta Department of Physics, D D U Gorakhpur University, Gorakhpur Received 6 August 2003; revised14 May 2004; accepted 3 June 2004 Mechanisms inherent in the thermoluminescence (L) processes are re-investigated with an aim to establish a generalized approach. It has been observed that the extents of recombination and simultaneous retrapping decide the order of kinetics involved. First order kinetics has been found to be a recombination dominant process with negligible or no retrapping. Probabilities of recombination and retrapping are found to be equal in second order kinetics. With the increasing order of kinetics, the extent of recombination decreases alongwith a simultaneous increase in the retrapping. A generalized equation has been developed which is found to be capable of representing the intensities of L glow curves of different orders of kinetics including that of the first order. It has been found that irrespective of the order of kinetics involved, L intensity decays exponentially. [Keywords: hermoluminescence, Kinetics] IPC Code: F 21 K 2/04 1 Introduction An electron excited to the conduction band from the valence band returns to the valence band again. However, if the electron is trapped in the metastable state (or trap level), it needs an energy to be raised to the conduction band. When the system is heated, electrons are thermally released from their respective trap centres. Such released electrons may quickly recombine with an oppositely charged centre resulting in the appearance of a L glow curve. Let us suppose that m and n represent the density of holes and electrons in the respective recombination and trap centres at the time t. Recombination is the process through which m decreases. hus the L intensity, will be given by 1 i = - dm/dt = m n c A m (1) where n c is the density of electrons in the conduction band and A m the recombination probability. Rate of decrease of n i.e. (-dn/dt) will depend on the excitation of electrons into the conduction band and also on their retrapping. hus, - (dn/dt) = n s I exp (-E a /k) - (N - n) n c A n (2) he first term on the right hand side of Eq. (2) represents the rate of release of electrons from their respective traps and the second term represents the rate of retrapping. In Eq. (2), s, E a, N and A n represent exponential factor, activation energy, total number of electrons in the trap centre and retrapping probability respectively. Suffix 1 represents the parameters associated with first order kinetics. Apart from Eqs (1) and (2), charge neutrality requires m = n + n c (3) Eqs (1), (2) and (3) are known as Adirovitch 2 set of equations describing the mechanisms responsible for the appearance of a L glow curve. o have some meaningful information from Adirovitch set of equations, following two basic assumptions have been introduced 1,2 as; n c < < n and dn c /dt < < dn/dt In view of these assumptions, Eq. (3) gives; m = n and dm/dt = dn/dt (4) (5) Combination of Eqs (1), (2) and (5) results 1 into; i = -(dm/dt) = -(dn/dt) = n s I exp(-e a /k)x[ma m /{ma m + (N - n)a n }] (6)

2 566 INDIAN J PURE & APPL PHYS, VOL 42, AUGUS 2004 Eq. (6) has been used by various researchers to probe into the mechanisms inherent in the L processes. Randall and Wilkins 3 proposed that first order kinetics is a case of recombination dominant process with negligible retrapping. With reference to Eq. (6) it means that: ma m > > (N - n)a n (7) In view of Eqs (5) and (7), one gets from Eq. (6) i 1 = - (dm/dt) = -(dn/dt) = n s I exp (-E a /k) (8) When the experiment is recorded following a linear heating rate b according to: = 0 + bt (9) one gets for the intensity of a first order L glow curve: i 1 =n 0 s 1 exp[(-e a /k)-(s 1 /b) 0 exp (-E a /k/) d/] (10) where n 0 is the initial concentration of trapped electrons per unit volume, k the Boltzmann constant, 0 is the absolute temperature wherefrom the L glow curve starts to appear, an arbitrary temperature in the range 0 to. Garlick and Gibson 4 proposed that second order kinetics is a case of strong retrapping probability. In view of Eq. (6), a retrapping dominated process leads to (N - n) A n >> m A m (11) Eq. (11) means that electrons which are excited, retrap several times before recombining into the centre. If it is assumed that trap is far from saturation i.e., n << N and the probabilities of recombination and retrapping are equal, Eq. (6) in combination with Eqs. (5) and (11) gives: i 2 = n 2 s 2 exp (-E a /k) (12) When the experimental run is recorded following a linear heating rate b and the intensity of second order L glow curve is given by: i 2 = n 0 2 s 2 exp[(-e a /k)]x[1 + (n 0 s 2 /b) -2 exp(- Ea/ k' )d ' ]...(13) 0 For a general order L glow curve it has been shown by Chen 1,5 that Eq. (6) for L intensity is expressed as: i l = n l s l exp(-e a /k) (14) where s l is the pre-exponential factor having the dimension of m 3(l-1) s -1. he parameters s 1 and s l are related through the expression: s 1 = s l N (l-1) (15) It is obvious that for l = 2, parameters s 1 and s 2 are found to be related as: s 1 = Ns 2 (16) hus, the mechanisms responsible for the occurrence of a general order L glow curve are controlled by Eq. (14). When the experiment is recorded following a linear heating rate b, intensity of a general order L glow curve is represented by 5 : i l =(n 0 /N) (l-1) n 0 s 1 exp[-(e a /k)]x[1+ (n 0 /N) (l-1) {s 1 (l-1)/b} exp (-E a /k )d ] {-l/(l-1)} (17) 0 In the cases when all the available electron traps are filled initially i.e., when N = n 0, Eq. (17) changes to: i l = n 0 s 1 exp [-(E a /k)]x[1+{s 1 (l-1)/b} exp (-E a /k ) d ] ] {-l/(l - 1)} (18) 0 It has been reported by Chen and Winer 5 that the peak of a general order L glow curve appears at M such that: {1 + s 1 (l-1)/b} exp (-E a /k) d = 0 (ls 1 k M2 /be a ) exp(-e a /k M ) (19) Eq. (19) decides the location of the L glow peaks in the cases involving second and higher order kinetics including first order. In this reference, following observations are to be taken into account:

3 PRAKASH et al.: HERMOLUMINESCENCE PROCESSES 567 Eqs (8) and (12) dealing with first and second order kinetics, respectively are found to be the special cases of generalized Eq. (14); Eq. (17) representing the intensity of a general order L glow curve fails to represent the intensity of a first order L glow curve; Eq. (19) is found to be a general equation for deciding the location of the L glow peaks. o overcome these discrepancies, Prakash 6 modified Eq. (3) as m = x n + n c (20) where x is a dimensionless parameter expressed as: x = (n/n) (l-1) (21) With the help of Eqs (1), (2), (20) and (21), one can correlate L intensities of first and second order kinetics. he corresponding general equation when all the available electron traps are filled initially is given by i l = ln 0 s exp[(- E a /k) - (ls/b) exp (-E a /k ) d ] (22) 0 Although the anomalies mentioned above are removed through Eq. (22) following points should be taken into account: (i) For first order kinetics x = 1; (ii) Irrespective of the order of the kinetics involved, one gets x = 1 when n = N; (iii) x depends on n which itself is a function of time. hus, x happens to be a variable quantity. Consequently, x changes during an experimental run of a L glow curve; (iv) Since the order of the kinetics is exclusively expressed by l, it does not seem necessary to induct another parameter x for it. In order to satisfy these points Prakash and Prasad 7 suggested that charge neutrality requirement be expressed as: m l = n + n c (23) Substituting the value l in Eq. (23), one can analyse L glow curves of different order of kinetics. With the help of Eqs (1), (2) and (23) one can derive an equation for the intensity of a L glow curve involving general order kinetics as: i l = {1/(2l-1)} n 0 s l exp [-(E a l/k) {ls l /(2l-1) b} exp -(E al /k )}d ] (24) 0 While developing Eq. (24), it has been observed by Prakash and Prasad 7 that the probabilities of recombination and retrapping are related as: (l-1) m A m = (N - n) A n (25) Eq. (25) obviously suggests that the extents of recombination and simultaneous retrapping decide the order of the kinetics involved. Failure of Eq. (17) in representing the intensity of a general order L glow curve but the ability of Eq. (19) in representing the position of the peak of a general order L glow curve, led us to reinvestigate into the mechanisms inherent in the L processes. Further s l in Eq. (24) has the dimension m 3(l-1) s -1 which in accordance with Arrhenius 8 relation should have the dimension s -1. o meet these anomalies an attempt has, therefore, been made in this paper with an aim to establish a suitable model. Adirovitch set of equations has been modified and a generalized equation has been developed. 2 Suggested Mechanisms of hermoluminescence Although Eq. (24) seems to be a generalized equation to represent L intensities of different order of kinetics, even then following points still need reconsideration: (1) Charge neutrality condition ml = n + n c does not seem to be feasible on phenomenological grounds. he number of electrons trapped at the trap centres are excess negative charges which must be balanced by the opposite charges of recombination centres (i.e. holes) such that ml = n + n c with the condition that n c << n. As per Eq. (23) second order kinetics requires 2m = n which could be possible when each recombination centre has one positive charge and each trap level has two electrons. Similarly, third order kinetics requires 3m = n which means that each trap centre should have three electrons. (2) If we compare the cases of L and ionic thermocurrent 9 (IC), we find that n 0 of L is equivalent to Q 0 of IC. In IC measurements, Q 0 represents the total charge released in an IC run which is equal to the area enclosed in the IC

4 568 INDIAN J PURE & APPL PHYS, VOL 42, AUGUS 2004 spectrum. Consequently, the area enclosed in the L glow curve should be equal to n 0 instead of n 0 /l as proposed by Prakash and Prasad 7. (3) While deriving expression for L intensity from Eq. (6) for first order kinetics, it is assumed that first order kinetics is a recombination dominant process i.e. ma m >> (N-n) A n. his condition is fulfilled when either (i) n N or (ii) n = N or (iii) A n 0 or (iv) A n << A m or even in some other appropriate condition. L intensity of first order kinetics is also obtained when (N-n) A n = 0 which is an extended version of the condition ma m >> (N-n) A n. Now, (N-n)A n = 0 can be possible when either A n = 0 or (N-n) = 0. he first term A n cannot be zero and the second term N = n can only be possible just at the start of the experiment when all the available electron traps are filled initially. As soon as an electron is released from the trap, (N - n) will have some finite value at that time and hence (N - n) cannot be equal to zero during an experimental run. Hence, (N - n) A n can be set equal to zero for first order kinetics by expressing it as (l-1) (N - n) A n. (4) In Adirovitch set of equations, rates of recombination, retrapping and release of electrons from the trap centres are given by m n c A m, n c (N - n) A n and n s exp[-(e a /k)], respectively. For recombination dominant first order kinetics these rates lead to the condition ma m >> (N - n) A n. (5) m is the number of available recombination centres per unit volume at the time t. his effectively means that: m = m 0 - m F (26) where m 0 is the initial number of recombination centres per unit volume at t = 0 out of which m F is filled at time t. hese points further suggest a rigorous probe into the existing mechanisms of the L processes. It is obvious that the order of kinetics involved in the system should either be a function of the constitution of the specimen under investigation or it should depend on the mechanisms inherent in the L processes. his also supplements the argument proposed in point (1). If the first order kinetics is a recombination dominant process and if in the second order kinetics the probabilities of recombination and retrapping are equal, it obviously leads to the fact that the rate of recombination decreases with the increasing order of kinetics. hus, the first equation of the Adirovitch set of equations in the light of these arguments can be written as: i = - dm/dt = (1/l) m n c A m (27) It is obvious from Eq. (27) that the rate of recombination decreases with the increasing order of kinetics. he second equation of the Adirovitch set of equations in the light of the arguments proposed in point (3) can be expressed as: - dn/dt = n s exp [-E a /k]-(l-1)n c (N-n)A n (28) he first term on the right hand side of Eq. (28) represents the rate of release of the electrons from their respective traps and second term represents the rate of retrapping. It is obvious from Eq. (28) that the rate of retrapping increases with the increasing order of kinetics as wanted. Charge neutrality condition will however be expressed by Eq. (3). hus, Eqs (3), (27) and (28) are the modified Adirovitch set of equations. In view of two basic assumptions introduced through Eq. (4), one gets from Eqs (3), (27) and (28): i = - dm/dt = - dn/dt = n s exp [-E a /k] [(m n c A m /l)/{(m n c A m /l) + (l-1) n c (N - n) A n }] (29) In Eq. (29), (m n c A m /l) and (l-1) n c (N - n) A n represent rate of recombination and rate of retrapping respectively. From Eq. (29), information about L intensities of different order of kinetics can be obtained. 3 Expression for the Intensity of a General Order L Glow Curve It is now an established mechanism that first order kinetics is a recombination dominant process 3 whereas rates of recombination and retrapping are equal in second order kinetics 4. For a L glow curve involving lth order of kinetics, rates of recombination and retrapping are related as (l-1) (m n c A m /l) = (l-1) (N - n) n c A n (30) Obviously for lth order of kinetics, (l-1) x rate of recombination = rate of retrapping. Eq. (29), in the light of Eq. (30) results into

5 PRAKASH et al.: HERMOLUMINESCENCE PROCESSES 569 i = - dn/dt = (1/l) n s exp[-e a /k] Eq. (31) can be rewritten as i = - dn/dt = (1/l) (n/τ) (31) (32) where τ is the mean life time or the relaxation time at the temperature expressed 8 as: τ = (1/s) exp[e a /k] Eq. (32) can be solved to give: n = n 0 exp[- (1/l) (t/τ)] as: (33) (34) Non-isothermal form of Eq. (34) can be written n = n 0 exp [- (1/l) Eqs (32) and (35) give: t dt / τ ] (35) 0 t i = (1/l) (n 0 /τ) exp[(- 1/l) dt / τ ] (36) 0 proposed model. It is obvious from Eq. (38) that for l = 1 it gives: M 2 = (be a /k s) exp[e a /k M ] (39) Which is a condition for the occurrence of a L peak in first order kinetics 1. For second order kinetics i.e., for l = 2, Eq. (38) results into (1/2) M 2 = (be a /k s) exp[e a /k M ] (40) It is apparent from Eq. (40) that M is independent of n 0 and the location of L peak will depend on b similar to the case of first order kinetics. It is also obvious that the intensity of the L peak will increase proportionally with the initial concentration n 0. hese results are found to be in agreement with the experimental behaviour recorded in natural barite samples 10 and CaF 2 : Pr single crystals 11. he order of kinetics in these systems is reported to have a value more than 1. he location of two peaks observed in If the system is heated following a linear heating rate b as per Eq. (9), one then gets from Eq. (36): i = (n 0 s/l) exp[-(e a /k) - (s/l b) exp (- E a /k') d'] (37) 0 Eq. (37) is the generalized equation for expressing the intensity of a general order L glow curve. Expressions for the L intensity of different order of kinetics can be obtained from Eq. (37) after substituting corresponding values of l as 1, 2, 3, 4,... etc. It is obvious that Eq. (37) changes to Eq. (10) for l = 1. 4 Discussion In the proposed model, all the anomalies mentioned in sections (1) and (2) are removed. he location of the peak of a general order L glow curve can be ascertained with the help of Eq. (31). It is found that M depends on b, l, E a and s through the relation: (1/l) M 2 = (b E a /k s) exp [E a /k M ] (38) Eqs (19) and (38) can be compared for getting an idea about the changes introduced through the (k) Fig. 1 L glow curves of different orders of kinetics in a hypothetical system with E a = 0.55 ev, s = s -1, b = 0.1 Ks -1 & n 0 = m -3. Values of E a and s are assumed to be the same in different orders of kinetics. he number on the curves indicates the involved order of kinetics able 1 Location and intensity of L peak for different orders of kinetics in a hypothetical system with E a = 0.55 ev, b = 0.1 K s -1, n 0 = m -3 and s = s -1 Order of the kinetics (l) Location of the L peak M (K) Intensity of L peak I M (10 13 ) m -3 s

6 570 INDIAN J PURE & APPL PHYS, VOL 42, AUGUS 2004 Order of the kinetics involved l able 2 Conditions for the appearance of different order L glow curves m & n Relationship inbetween Recombination and retrapping probabilities Extent of recombination Extent of simultaneous retrapping First 1 m = n no retrapping 100% 0% Second 2 m = n m n c A m = 2 (N - n) n c A m 50% 50% hird 3 m = n m n c A m = 3 (N - n) n c A m 33.3% 66.7% Fourth 4 m = n m n c A m = 4 (N - n) n c A m 25% 75% Fifth 5 m = n m n c A m = 5 (N - n) n c A m 20% 80% lth l m = n m n c A m = l (N - n) n c A m (100/l)% 100(l-1)/l % L glow curve of natural barite samples has been found to be independent of the extent of the irradiation dose i.e., independent of n 0 as expected. Similarly, the location of the L peak in CaF 2 : Pr single crystals is found to be independent of the initial concentration of Pr in CaF 2. he intensity of the L peak in both the cases has, however, been found to increase with n 0 as expected. In a hypothetical system with given values of b, E a, s and n 0, the location of the L peak has been found to change due to a change in the order of the kinetics as shown in Fig. 1. he location of the L peak shifts towards the higher temperature alongwith a simultaneous decrease in the intensity of the L peak as a result of increase in the value of l (Fig. 1). he extent of these changes is more pronounced when the order of kinetics changes from l = 1 to l = 2. An idea about the changes in M and I M can be had from able 1. On the basis of the theory proposed in section 3, one can tabulate different conditions required for the appearance of a L glow curve. It is obvious from able 2 that first order kinetics is a recombination dominant process. In this process, retrapping is practically negligible in comparison to recombination. his is the process, which is frequently observed and widely reported in the literature. In the second order kinetics, recombination and retrapping processes take Fig. 2 Exponential decay in a hypothetical system involving different orders of kinetics. Values of E a and s are assumed to be the same in different orders of kinetics. E a = 0.55 ev, s = s -1, n 0 = m -3 and = 300 K. he number on the curves indicates the involved order of kinetics place with equal probabilities. Irrespective of the order of the kinetics involved, the concentration of recombination centres and of trapped electrons are equal (able 2). he extent of recombination for third and fourth order kinetics is 33.3% and 25%,

7 PRAKASH et al.: HERMOLUMINESCENCE PROCESSES 571 respectively (able 2). hus, the appearance of a L glow curve can be expressed in terms of the extents of recombination and retrapping processes. It is obvious that with the increasing order of kinetics the extent of recombination decreases with a simultaneous increase in the extent of retrapping. It is obvious from Eq. (34) that irrespective of the order of the kinetics involved L intensity decays exponentially. he decay is the fastest in first order kinetics as shown in Fig. 2. he extent of exponential decay has been found to decrease with the increase in the order of the kinetics. Second order kinetics is found to be a slower decaying process than the first order kinetics, and the third order kinetics decays still slowly than the second order kinetics and so on. It is also obvious from Eq. (37) that irrespective of the order of kinetics involved, the escape frequency factor(s) has the dimension s -1 as expected and is in accordance with the requirement of Arrhenius relation. hus, the misleading proposition introduced through Eq. (15) is also removed. Acknowledgement he authors are thankful to Prof. R. Chen (el Aviv, Israel) for valuable suggestion. References 1 Chen R & Kirsh Y, Analysis of thermally stimulated processes (Pergamon Press, Oxford), Adirovitch E I, J Phys Rad, 17 (1956) Randall J & Wilkins M H F, Proc Roy Soc A, 184 (1945) Garlick G F J & Gibson A F, Proc Roy Soc, 60 (1948) Chen R & Winer S A A, J Appl Phys, 41 (1970) Prakash J, Solid State Commun, 85 (1993) Prakash J & Prasad D, Phys Status Solidi a, 142 (1994), Arrhenius S Z, Phys Chem, 226 (1889). 9 Bucci S, Fieschi R & Guidi G, Phys Rev, 148 (1966) Prokic M, J Phys Chem Solids, 18 (1977) Sinha R K & Mukherjee M I, Phys Status Solidi b, 105 (1981) 69.