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On the general-order kinetics of the thermoluminescence glow peak and the calculation of parameters from glow curves
Journal of Luminescence 78 (1998) 279—287
On the general-order kinetics of the thermoluminescence glow peak and the calculation of parameters from glow curves Z. Vejnovic´!, M.B. Pavlovic´", D. Ristic´!, M. Davidovic´",#,* ! Institute of Security, Kraljice Ane b. b., 11000 Belgrade, Yugoslavia " Vinca Institute of Nuclear Sciences, P.O. Box. 522, 11001 Belgrade, Yugoslavia # Faculty of Electrical Engineering, P.O.Box 816, 11000 Belgrade, Yugoslavia Received 16 April 1997; received in revised form 30 December 1997; accepted 30 December 1997
Abstract It has been shown by a new approach that the equation which describes general-order kinetics is an interpolating function between analogous equations for first- and second-order kinetics. A new method for calculation of parameters is proposed. The method is based on the determination of the glow curve maximum, and the effective values of half-width and part of the half-width on the higher temperature side. The relation for the symmetry factor as a function of the corresponding order of kinetics l and the correction factor D is obtained. Approximate symmetry factor function is derived which enables analytical calculation of the parameters: activation energy E, order of kinetics l, and preexponential factor s(l). An iterative procedure is developed for more precise calculation of these parameters. The new method for calculation of these parameters is checked for some characteristic values of the parameters. ( 1998 Elsevier Science B.V. All rights reserved. PACS: 78.60.K Keywords: Thermoluminescence; Kinetics order; Glow curves
1. Introduction The first- and the second-order kinetics do not cover all possible cases of thermoluminescence (TL) kinetics found experimentally. May and Partridge [1] suggest an approximate relation to describe general order kinetics:
A B
dn E I"! "s@nl exp ! , dt k¹
(1)
is the pre-exponential factor, E (eV) is the energy depth of a single trap, ¹ (K) is the material temperature, and k (eV K~1) is the Boltzmann constant. Numerical solution of differential equation describing the TL emission, and analysis of so obtained TL curves as well as the extracted TL curves parameters, show that they are fairly well described by the general order kinetics model [2]. To highlight the physical meaning of Eq. (1), Rasheedy [3] suggested its rearrangement into
where at a time t (s), I (cm~3 s~1) is the thermoluminescence intensity, n (cm~3) is the electron trap concentration, l is kinetics order, s@ (cm3(l~1) s~1)
dn nl E I"! "s exp ! , dt Nl~1 k¹
* Corresponding author. Tel.: #381 11 444 0871; fax: #381 11 344 0100; e-mail: [email protected].
where N (cm~3) is the concentration of traps and the unit of s@ is now s~1. His aim was to avoid using
A B
0022-2313/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved PII S 0 0 2 2 - 2 3 1 3 ( 9 8 ) 0 0 0 0 6 - 4
(2)
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280
s@ the dimension of which changes with the order l. Eq. (2) describes the release of electrons from their traps and it is a basic relation for the calculation of the thermoluminescence intensity at constant heating rate R (Ks~1). From the above, Rasheedy has derived the following equation:
Only the first two terms of the expansion have been used and they are a good approximation for small values of D ((0.1).
I"nl s exp(!E/k¹) 0
We consider the phosphorescence process in the material which consists of trapping levels and luminescent centers both of a single kind [6]. General assumption about the phosphorescence process is that the concentration of conduction band electrons n is very low compared to the # concentration of trapped electrons n because the life time of electrons in the conduction band is much shorter than the life time of electrons in traps. Usually, N traps and Nl luminescent centers are partly occupied by electrons and holes, respectively, and due to the charge neutrality requirement, the concentration of trapped electrons n is equal to the concentration of empty luminescent centers pl (NAn). Initial increase in the TL glow curve is a part of exponential rise, dependent on activation energy E only. Using this part of glow curve the initial rise method has been developed to determine the activation energy. For the case n+N all traps are almost filled and the retrapping processes are negligible and the TL emission in the beginning can be neglected. As a first-order approximation one can take that the part of glow curve, preceding the maximum, is independent of the order of kinetics. This assumption is valid if all other parameters are constant, Fig. 1. Redistribution of glow curve emission as a function of temperature for different orders of kinetics appears after the maximum, which means that this simple model of TL emission describes quite well the different cases, as for instance n+N. The simple model is described by following equations:
A C
] Nl~1 1#M [s(l!1)(n /N)l~1]/RN 0
D B
T
]
P
exp(!E/k¹ @) d¹@
l@(l~1) ~1
,
(3)
T0 where n and ¹ are the initial values of trapped 0 0 electrons concentration and sample temperature, respectively. As shown for the second-order kinetics [4], the calculation of the n /N ratio is imposs0 ible. The same is true for general order kinetics. Because of that, it is more convenient to introduce, instead of the frequency factor s, which cannot be calculated except in the case of first-order kinetics, the pre-exponential factor s(-)"s(n /N)l~1 as has 0 been done by the above mentioned authors. I" n s(l) exp(!E/k¹) 0 . [1#M [s(l)(l!1)]/RN:T exp(!E/k¹@) d¹@]l@(l~1) T0 (4) To calculate the parameters of the TL glow curve one can use the equation given by Chen [5] which, with Rasheedy correction [3] and the pre-exponential factor s(-) introduced, has the form
A
B
RE E "s(l) exp ! [1#(l!1)D], (5) k¹2 k¹ . . where ¹ is the temperature corresponding to the . maximum intensity and D"2k¹ /E. This rela. tionship has been obtained by differentiating Eq. (4) and equating it to zero, and then using the approximation T
P A B
T0
A B A B
k¹ n E E exp ! d¹+¹ exp ! + E k¹ k¹ n/1
](!1)n~1n!.
2. Theoretical consideration
dpl ! "cln pl # dt
(7)
and
(6)
dn ! "!c n N#Sn, 5 # dt
(8)
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281
By assuming the existence of two kinds of centers (luminescent and trapping) and a very short life time of conduction band electrons, as noted above for a simple model, we can rewrite relation (9) (instead of pl we use n) as follows:
A
Sn" cl
Fig. 1. General order glow curves obtained according to Eq. (3) with s(l)"106 s~1, E"1.5 eV, R"0.5 K s~1, n "104 cm~3, 0 N"104 cm~3 and l are given as: (a) 1.9, (b) 1.5 and (c) 1.1.
where c and cl are the recombination probabilities 5 between the conduction band electrons and the traps or the luminescent centers, respectively, and S"s exp(!E/k¹). By assumptions which are valid for this simple model, Ddn /dtD@Ddn/dtD and # Ddn /dtD@Ddpl/dtD, we may write the following equa# tions: Sn n" # clpl#c N 5
(9)
and clS S dpl p2l " pl I"! " dt clpl#c N 1#rN/pl 5
(10)
where r"c /cl. If we assume that c N@clpl or 5 5 r"0, which is equivalent to the assumption that the concentration of traps is low compared with the concentration of luminescent centers, then these relations describe the first-order kinetics. But, if we take c NAclpl which is the same as the assumption 5 that the retrapping is dominant, these relations describe the second-order kinetics [7]. If for a given model we take c "cl, that is r"1, it is evident that 5 for this case Eq. (10) also describes the secondorder kinetics.
B
n #c n N. 5 # N
(11)
All electrons excited into the conduction band recombine with empty trap levels, with the recombination probability c@"cl (n/N)#c . For the 5 first-order kinetics c@"cl (n/N), since c N@clpl, 5 and for the second-order kinetics c@"c , since 5 NAn as well as c "cl. In other words, for the first5 and the second-order kinetics c@"cl (n/N)1 and c@"cl (n/N)0, respectively. The value of c@ can be interpolated between l"1 and l"2 by an empirical interpolation function c@"cl (n/N)m. One can see from Eq. (10), that the influence of the ratio N/pl on the shape of the curve increases with an increase of the factor r from 0 to 1, which at the same time means that it influences the order of kinetics. Since r can take any value between 0 and 1, the interpolating function must be continuous in this range. Therefore, one concludes that the change of the order of kinetics depends on N/pl or, according to this model, N/n ratio. By inspecting the expressions for the first- and second-order kinetics, one can see that this means a change from linear to quadratic function. Since the power function is increasing, it may be taken as the interpolation function. A constant multiplier for this power function must have the dimension of recombination probability. It is practical to take as a multiplier the recombination probability appearing in the interpolating points, i.e., the points describing the first- and second-order kinetics. This value is cl. For a small variation in l one can assume the linear dependence of the exponent m"a l#a . 0 1 Taking for these two points l "1, m "1 and 1 1 l "2, m "0, after solving the linear equation we 2 2 find that a "!1 and a "2. These solutions give 0 1 c@"cl (N/n)l~2. By inserting c@ in relations (9) and (10), we obtain the following relations: S n" nl~1 # clNl~1
(12)
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282
For k we obtain g
and
A B
dn nl E dpl exp ! , I"! "! "s Nl~1 dt dt k¹
(13)
which corresponds to the Rasheedy [3] suggestion (2).
3. Calculation of parameters Halperin and Braner [8] introduce a symmetry factor which depends on the thermal peak and is defined as follows: n k " ., g n 0
(14)
where n is the electron concentration in the traps . at the maximum intensity. For any single peak we have `= `= Rn " I d¹"I i d¹ 0 . ~= ~=
P
P
(15)
and `= `= Rn " I d¹"I i d¹. . . T. T.
P
P
(16)
Here i is a normalized function of TL curve. By applying these relations we can introduce the effective values of half-width u and effective values of %&& that part of the half-width which is on the higher temperature side d , and get the following rela%&& tions: `= Rn u " 0" i d¹ %&& I . ~=
P
(17)
and `= Rn i d¹. d " ." %&& I . T.
P
(18)
d k " %&& . g u %&&
(19)
Solving the differential equation (2) we get
C
D
~1@(l~1) T s(l)(l!1) E n"n 1# exp ! d¹@ 0 R k¹@ T0 (20)
P A
B
and for k g
C
D
~1@(l~1) T. . s(l)(l!1) E exp ! d¹@ k " 1# g R k¹@ T0 (21)
P A
B
Differentiating Eq. (4) and equating the derivative to zero we find T. s(l)(l!1) E 1# exp ! d¹@ R k¹@ T0 E s(l)lk¹2 . exp ! . " RE k¹ .
P A
B
A
B
(22)
This equation is the same as the one obtained by Chen [5], except that the pre-exponential factor has a different form. For k we derived the followg ing expression after substituting Eq. (22) into Eq. (21)
C
k" g
A
s(l)lk¹2 E . exp ! RE k¹
BD
~1@(l~1) .
(23)
.
By substituting Eq. (5) into Eq. (23), one obtains
C
k" g
D
1#(l!1)D 1@(l~1) . l
(24)
Eq. (24) has been derived using the first two terms of the approximation relation given by Eq. (6). In order to increase the accuracy of l, the number of power terms in Eq. (6) can be increased, which is important for higher values of l (over 1.3) and D (over 0.1). In order to obtain more accurate
Z. Vejnovic& et al. / Journal of Luminescence 78 (1998) 279—287
values of the symmetry factor, up to the third decimal place, it is necessary to add another two terms. After these modifications, k takes the following g form:
C
k" g
D
1#(l!1)(D!3D2#3D3) 1@(l~1) 2 . l
(25)
An expression for calculating the order of kinetics l and the activation energy E of the trap can be obtained from Eq. (4) by substituting the value for the peak maximum. I " . n s(l) exp(!E/k¹ ) 0 . . [1#[s(l)(l!1)/R] :T. exp(!E/k¹@) d¹@]l@(l~1) T0 (26) Substituting Eq. (22) into Eq. (26) we get n s(l)exp(!E/k¹ ) 0 . I " . m [(s(l)lk¹2 /RE) exp(!E/k¹ )]1`1@(l~1) . .
(27)
and, after substituting Eq. (23) into Eq. (27) and some rearrangements, n RE I " 0 k . k¹2 l g .
(28)
2k¹ 2d ." %&&. D" E ¹ l .
l"1 we can write ln k "ln k #D. (31) g g0 The value of ln k is in fact the tangent of the angle g0 that a straight line passing trough l"1 and l"l makes with the x-axis. This straight line is at the same time a chord on the curve ln l with intersections at the above two points (Fig. 2). When lP1 one can assume that the curvature of the arc intersected by the chord is small and that the chord is parallel to a tangent passing through the midpoint of ln l line enclosed between l"1 and l"l. In that case ln k "!2/(l#1), and substituting this reg0 sult into Eq. (25) we get 2 ln k "! #D. g 1#l
(32)
The region of validity of this function is for l"1 and for very small values of D(D(0.1). Since the shape of the function, for the region of practical interest (0.7—2.5) and D(0.05—0.15), changes only slightly, it can be precisely determined in the whole relevant region by using a general approximate function which has the same shape as function (32). This expression can be generalized by replacing numerical values with parameters P and l as well 1 0 as D with the function f (D). l P P 1 #f (D), ln k " 0 1 #f (D)" g l #l 1#l/l 0 0
and it follows that
283
(33)
(29)
For all the experimental data obtained from the thermoluminescence [5], the parameter l lies between 0.7 and 2.5. To get numerical values of the parameter l it is necessary to approximate the obtained relations to avoid the singularity values of l near l "1. Logarithm of Eq. (24) gives the symmetry factor as a sum of two terms ln l ln[1#(l!1)D] ln k "! # . g l!1 l!1
(30)
The first term represents ln k for D"0 and we can g write it as ln k . It can be shown that the second g0 term tends to D when lP1. In the neighborhood of
Fig. 2. Geometrical presentation of approximate expression ln k . g0
Z. Vejnovic& et al. / Journal of Luminescence 78 (1998) 279—287
284
where P and l are the weakly changing para1 0 meters dependent on D, and f (D) is a linear function of D as given in Eq. (32). For the range of values of D(0.05—0.15), which is of practical interest, P and 1 l are constants. Therefore, the function (33) can be 0 approximated by a function
thus being the solution. This is !A #JA2!4A A 1 1 0 2. l" 2A 0
From Eqs. (29) and (5) we obtain values for E and s(l), respectively. In case the values of TL curve are known, ¹ , u and d can be determined by . %&& %&& numerical integration of this curve within certain limits. If an adjustment of the TL curve has been done, then u and d can be obtained by integra%&& %&& tion of the fitted curve within certain limits. The values of calculated parameters and accuracy obtained for some characteristic values are given in Table 1. Using Eq. (4), curves are obtained and from them ¹ . Values of u and d are obtained . %&& %&& by numerical integration. For the parameter R, the value of 0.5 K s~1 was used. Parameters E, l and s(l) chosen for the curve described by Eq. (4) to simulate an experimental curve are shown on the left-hand side of Table 1. For each point of the curve given by Eq. (4), numerical evaluation of the integral has been done using the Gauss—Legendre method with estimated accuracy of the order of 10~9. Table 1 also contains other three parameters ¹ , u , and d obtained from the glow curve and, . %&& %&& further to the right-hand side, calculated parameters E , l , and s(l) with the corresponding rela# # # tive errors r , r , and r . The accuracy with which E l s the parameters are obtained depends on the accuracy of the adjustment of the function ln k and on g the accuracy with which ¹ , u and d are deter. %&& %&& mined. Since the approximate values for correction
P 1 #P D#P . ln k " (34) g 1#l/l 2 3 0 By adjusting the function (34) to (25) using the method of least squares, we get the following approximate values P "!2.127, P "0.76, 1 2 P "!0.2, and l "0.592. Substituting Eq. (29) 3 0 into Eq. (34) we get the quadratic equation A l2#A l#A "0, 0 1 2 where
(35)
d d A "P !ln %&& "!0.2!ln %&& , 0 3 u u %&& %&& d 2d A "l P #P !ln %&& #P %&& 1 0 1 3 2¹ u %&& . 1.52d d %&&!0.592 ln %&& !1.378 " ¹ u . %&& and
A
(36)
B
2d d A " %&&l P "0.9 %&&. 2 ¹ 0 2 ¹ . . Two values are obtained for the parameter l and only one of them belongs to the interval (0.7, 2.5),
Table 1 D
E
l
s(l)
¹ .
u
0.0597 0.0597 0.0622 0.0805 0.0848 0.0896 0.0994 0.1220 0.131 0.142 0.142
1.6 1.6 0.4 1.6 0.4 0.1 0.1 1.6 0.4 0.1 0.1
2.5 1.9 1.5 1.5 1.5 1.5 0.7 0.7 1.9 1.9 2.5
1013 1013 1013 109 109 109 108 105 105 105 105
554.24 554.48 144.31 746.76 198.81 52.01 57.65 1132.2 303.48 82.41 82.16
72.68 61.01 14.35 94.80 26.27 7.31 6.01 142.36 69.53 20.35 24.10
%&&
d %&&
E #
l #
s(l) #
r (%) E
r (%) l
r (%) s
41.56 31.59 6.75 45.32 12.61 3.52 1.90 47.78 37.92 11.18 14.56
1.608 1.601 0.399 1.601 0.400 0.100 0.090 1.586 0.396 0.0987 0.0979
2.524 1.908 1.501 1.510 1.513 1.511 0.691 0.686 1.893 1.885 2.450
1.166]1013 1.017]1013 9.380]1012 1.022]109 1.032]109 1.017]109 8.904]107 8.524]104 8.679]104 8.259]104 7.269]104
0.48 0.043 !0.23 0.06 0.06 0.13 !0.88 !0.86 !0.90 !1.40 !2.10
0.97 0.43 0.01 0.66 0.78 0.75 !2.07 !2.00 !0.33 !0.89 !1.98
16.6 0.71 !6.26 2.45 2.60 4.00 !17.74 !14.54 !13.10 !18.37 !27.30
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285
Table 2 D
E
l
s(l)
¹ .
u
0.0597 0.131 0.142 0.142
1.6 0.4 0.1 0.1
2.5 1.9 1.9 2.5
1013 105 105 105
554.24 303.48 82.41 82.16
72.68 69.53 20.35 24.10
%&&
d %&&
E #
l #
s(l) #
r (%) E
r (%) l
r (%) s
41.56 37.92 11.18 14.56
1.600 0.398 0.0994 0.0993
2.513 1.903 1.899 2.488
9.942]1012 9.370]104 9.240]104 9.010]104
0.02 0.45 !0.57 !0.62
0.50 !0.14 !0.036 !0.50
!0.58 !6.34 !7.62 !9.31
factor D are also obtained, in calculation of these parameters, it is possible to use more accurate values for P , P , P and l for the second calcu1 2 3 0 lation. If we look at the first and last three values in Table 1, we see an increase in calculation precision. In Table 2, the calculated values are given for the range of D(0.05, 0.07) and D(0.13, 0.15) when P "0.856, l "0.6 and P "0.679, l "0.578, re2 0 2 0 spectively. Values of P and P are taken to be the 1 3 same since their changes are negligible. In this way, the calculation precision can be increased so that the accuracy of the results depends only on the accuracy in determination of ¹ , . u and d , i.e., it depends on the accuracy in %&& %&& obtaining the TL curves.
4. Discussion In order to show how Eq. (1) is derived, a simplified model with one kind of traps and one kind of luminescent centers has been used. Further simplification of the model assumes the traps to be partly filled which is mostly the case. It can be shown that this simplification of the model does not have any effect on the analysis carried out in Section 2, and that it is valid for the case n+N. In that case, the same expression is obtained for the first- and second-order kinetics. If the term c n n in 5 # Eq. (8) is not neglected, c@"(cl!c )(n/N)#c is 5 5 obtained. Applying the condition for the first-order kinetics, c N@clpl, we get c@"(cl!c )(n/N). Since 5 t n(N or n+N and c n+c N@clpl, we get for the 5 5 first-order kinetics c@"cl(n/N). Applying condition for the second order kinetics, c "cl [7], we get 5 c@"cl. Comparing Eq. (1) to Eq. (10) one can see that all the existing parameters are the same except r which
changes with l. This change is the result of the interpolation process. In fact this interpolation means a substitution of one constant r for another constant l in the way which enables analytical solution of the differential equation describing the TL kinetics. In this way, physical meaning of the parameter l is in direct connection with the parameter r which is in accordance with the physical meaning of parameters l"1 or l"2. In these cases the physical meaning of l is defined by the ratio of luminescent to retrapping probabilities. According to the relation valid for lth order of kinetics and taking into account the Halperin and Braner [8] definition, a symmetry factor relation was evaluated. The obtained relation, Eq. (31) can be written more transparently as k "k eD. Since g g0 the symmetry factor k characterizes the order of g kinetics of thermoluminescence process, this equation can be used for determining the criterion for characterizing the order of kinetics. For example, k "e~1 corresponds to the first-order kinetics, g0 while k "0.5 corresponds to the second-order g0 kinetics. If eD is expanded, retaining only the first two terms of the series, i.e., eD+1#D, then the criteria obtained are k "(1#D)/e and g k "(1#D)/2 for the first- and the second-order of g kinetics, respectively, as given by Halperin and Braner [8] and used by others [5,9—12]. For general order kinetics, one can obtain the relation k "k (1#D) which represents a criterion for deg g0 termining the order of kinetics in general form. In the case of hyperbolic heating function used instead of linear, used the integral in Eqs. (3) and (4) can be solved analytically [13]. The hyperbolic function has the simple form ¹"R /(t !t), where 0 k R (Ks~1) and t (s) are constants. With such as0 k sumption one can derive the expression for the
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286
symmetry factor [13].
AB
1 1@(l~1) . k" g l
(37)
One can see that this equation is identical to the relationship obtained for k defined by Eqs. (30) g0 and (31). Therefore, k is in fact a symmetry factor g0 of thermoluminescence curve of the lth order kinetics when the heating function is simple hyperbolic. Expressions (24), k "k eD and k "k (1#D) g g0 g g0 in the region l (0.7—2.5) are very similar, in contrast to expression (25), large discrepancies are visible with respect to the referred three curves particularly for values l'1.3 (Fig. 3). According to these, one can state that addition of another two terms into Eq. (6) increases the accuracy of calculation of TL curve parameters. This improvement in calculation of parameters is particularly pronounced for higher values of D and higher values of l. According to the derived relations it is possible to calculate analytically the parameters E, l and s(l) from the shape of the TL curve. For calculation of the above parameters in the ranges 0.7(l(1.3 and D(0.06, relations (29) and (32) can be used. Combining these relations one gets the quadratic equation identical in form to Eq. (35). Relative error bars for the calculated values of l and E are smaller than 2.5%, and for s(30%. Since the correction parameter D is very small, it can be assumed that DP0 as a first approximation. According to the assumption, Eqs. (29) and (31) can
Fig. 3. Different relations describing symmetry factor for D"0.14.
be applied within the range of the parameter l up to 1.5. Since in this case ln k "f (ln l,l), one has to use g iterative procedure in order to obtain the values for l. For exact calculation of l in the entire range which is significant from the point of view of the experiment, improved expressions Eqs. (25) and (29) have to be used. The obtained result is a higherorder polynomial. In order to get an accurate solution, an iterative procedure should be used. In order to avoid this tedious procedure an analytical expression was proposed for calculation of the above parameters, Eq. (25). The relative error for these calculations given in Tables 1 and 2 is less then 1% for parameters l and E and less than 10% for parameter s(l). This kind of calculation is obviously better for the parameter E and much better for parameters l and s(l), than the one obtained by Chen [5]. As one can see in Table 1, the only exemption are the values obtained for l"0.7. Principal reason for this is that Eq. (4) does not define the TL curve very well in all regions of interest for calculation of u and d . More precisely, for %&& %&& higher temperature values the second term in the denominator, which is negative, becomes smaller than !1. The reason is the fact that Eq. (4), in this region and order of kinetics, is an extrapolation function and does not completely describe the TL processes. Since the region of undefined TL curve, according to Eq. (4), is largest for the value l"0.7, the obtained values of relative errors are maximal. In reality, the undefined region of the TL curve does not exist and because of that one can assume that the calculation errors lie inside the abovementioned limits. The determination of parameters of an experimental TL curve by determining the values of u and d is certainly the simplest method. However, the curve is not uniquely determined by these two parameters. There are curves of various shapes having the same delta and omega values. Although these deviations are comparatively small for certain order of kinetics, as is well known [5], by using these values it is not possible to determine the parameter l of the order of kinetics with the same accuracy, as achieved for E. From theoretical point of view the calculation method using the integral of the TL curve and part of the glow curve is more general since it considers the shape of the curve.
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Therefore, this method should have a higher precision. The determination of the integral of the peak and of a part of the peak from the glow curve by simple numerical treatment of the peak most certainly leads to additional errors and the method of determination of TL curve parameters by using delta and omega values can have higher precision. However, if one wants more precise calculations by using any of these two methods it is necessary first to obtain, from the experimental glow curve, the wanted peak by some fitting method. In this case we think that the method of calculation based on the integral of the TL curve and a part of this curve will give better result.
5. Conclusions Using a simple model, it has been shown that the equations for general order kinetics, with correction proposed by Rasheedy, actually represent an interpolation between the first- and the secondorder kinetics and an extrapolation beyond these values, assuming the exponent of the ratio (n/N) changes linearly. The introduction of a pre-exponential factor s(l) is justified since this parameter influences the shape of the TL curve, and it is not possible to deduce from this curve neither the values of n and N nor their ratio. This factor is not 0 a material constant, since it depends on the absorbed dose, but it explains the dynamics of the TL process for each single curve. The proposed method for parameters calculation is quite general since it is based on evaluation of integral values of TL curves and these curves originate from the derived equations valid for general order kinetics. This means that calculations of parameters of the curve for general order kinetics by this method are not determined by limits D(0.05—0.15) which are of practical value, but instead they are valid outside these limits, as well. Increased accuracy of calculations, besides relying
287
on the use of integral values of the TL curve, is obtained by including another two power terms defined in Eq. (6), and it influences the calculation of symmetry factor, Eq. (25). Certain approximations made for the symmetry factor have been made only to enable analytical results and to avoid a singularity value lP1. Due to the corresponding simple iteration procedure the accuracy of the results practically does not depend on the approximation made, but rather on the accuracy of the experimental curve and evaluation of ¹ , d and . %&& u . By inspecting the results of calculations shown %&& in Tables 1 and 2 one may conclude that evaluation of the above parameters with higher precision improves the accuracy of the results for E , l , and # # s(l) given in the tables. # Acknowledgements The authors are grateful to Professor D. C[ ajkovski and Professor Z. Ikonic´ for reading and making useful comments on the manuscript. This paper was supported in part by the Serbian Research Fund. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
C.E. May, J.A. Partridge, J. Chem. Phys. 40 (5) (1964) 1401. Y. Kirsh, Phys. Stat. Sol. 129 (15) (1992) 15. M.S. Rasheedy, J. Phys. Condens. Matter 5 (1993) 633. M.S. Rasheedy, A.M.A. Amry, J. Lumin. 63 (1995) 149. R. Chen, J. Electrochem. Soc. 116 (9) (1969) 1254. V.V. Serdjuk, Ju.F. Vaksman, Luminiscencija poluprovodnikov, Visa skola, Kiev-Odesa, 1988, p. 120. R. Chen, Y. Kirsh, Analysis of Thermally Stimulated Process, Pergamon Press, Oxford, 1981, p. 32. A. Halperin, A.A. Braner, Phys. Rev. 117 (2) (1960) 408. H.J. Beaven, J. Phys. Chem. Solids 36 (1975) 949. R. Chen, J. Appl. Phys. 40 (2) (1969) 570. A. Halperin, A.A. Braner, A. Ben Zvi, N. Kristianpoller, Phys. Rev. 117 (2) (1960) 416. J.K. Zope, Indian J. Pure Appl. Phys. 20 (1982) 816. R. Chen, N. Kristianpoller, Radiat. Prot. Dosim. 17 (1986) 443.
On the general-order kinetics of the thermoluminescence glow peak and the calculation of parameters from glow curves Z. Vejnovic´!, M.B. Pavlovic´", D. Ristic´!, M. Davidovic´",#,* ! Institute of Security, Kraljice Ane b. b., 11000 Belgrade, Yugoslavia " Vinca Institute of Nuclear Sciences, P.O. Box. 522, 11001 Belgrade, Yugoslavia # Faculty of Electrical Engineering, P.O.Box 816, 11000 Belgrade, Yugoslavia Received 16 April 1997; received in revised form 30 December 1997; accepted 30 December 1997
Abstract It has been shown by a new approach that the equation which describes general-order kinetics is an interpolating function between analogous equations for first- and second-order kinetics. A new method for calculation of parameters is proposed. The method is based on the determination of the glow curve maximum, and the effective values of half-width and part of the half-width on the higher temperature side. The relation for the symmetry factor as a function of the corresponding order of kinetics l and the correction factor D is obtained. Approximate symmetry factor function is derived which enables analytical calculation of the parameters: activation energy E, order of kinetics l, and preexponential factor s(l). An iterative procedure is developed for more precise calculation of these parameters. The new method for calculation of these parameters is checked for some characteristic values of the parameters. ( 1998 Elsevier Science B.V. All rights reserved. PACS: 78.60.K Keywords: Thermoluminescence; Kinetics order; Glow curves
1. Introduction The first- and the second-order kinetics do not cover all possible cases of thermoluminescence (TL) kinetics found experimentally. May and Partridge [1] suggest an approximate relation to describe general order kinetics:
A B
dn E I"! "s@nl exp ! , dt k¹
(1)
is the pre-exponential factor, E (eV) is the energy depth of a single trap, ¹ (K) is the material temperature, and k (eV K~1) is the Boltzmann constant. Numerical solution of differential equation describing the TL emission, and analysis of so obtained TL curves as well as the extracted TL curves parameters, show that they are fairly well described by the general order kinetics model [2]. To highlight the physical meaning of Eq. (1), Rasheedy [3] suggested its rearrangement into
where at a time t (s), I (cm~3 s~1) is the thermoluminescence intensity, n (cm~3) is the electron trap concentration, l is kinetics order, s@ (cm3(l~1) s~1)
dn nl E I"! "s exp ! , dt Nl~1 k¹
* Corresponding author. Tel.: #381 11 444 0871; fax: #381 11 344 0100; e-mail: [email protected].
where N (cm~3) is the concentration of traps and the unit of s@ is now s~1. His aim was to avoid using
A B
0022-2313/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved PII S 0 0 2 2 - 2 3 1 3 ( 9 8 ) 0 0 0 0 6 - 4
(2)
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280
s@ the dimension of which changes with the order l. Eq. (2) describes the release of electrons from their traps and it is a basic relation for the calculation of the thermoluminescence intensity at constant heating rate R (Ks~1). From the above, Rasheedy has derived the following equation:
Only the first two terms of the expansion have been used and they are a good approximation for small values of D ((0.1).
I"nl s exp(!E/k¹) 0
We consider the phosphorescence process in the material which consists of trapping levels and luminescent centers both of a single kind [6]. General assumption about the phosphorescence process is that the concentration of conduction band electrons n is very low compared to the # concentration of trapped electrons n because the life time of electrons in the conduction band is much shorter than the life time of electrons in traps. Usually, N traps and Nl luminescent centers are partly occupied by electrons and holes, respectively, and due to the charge neutrality requirement, the concentration of trapped electrons n is equal to the concentration of empty luminescent centers pl (NAn). Initial increase in the TL glow curve is a part of exponential rise, dependent on activation energy E only. Using this part of glow curve the initial rise method has been developed to determine the activation energy. For the case n+N all traps are almost filled and the retrapping processes are negligible and the TL emission in the beginning can be neglected. As a first-order approximation one can take that the part of glow curve, preceding the maximum, is independent of the order of kinetics. This assumption is valid if all other parameters are constant, Fig. 1. Redistribution of glow curve emission as a function of temperature for different orders of kinetics appears after the maximum, which means that this simple model of TL emission describes quite well the different cases, as for instance n+N. The simple model is described by following equations:
A C
] Nl~1 1#M [s(l!1)(n /N)l~1]/RN 0
D B
T
]
P
exp(!E/k¹ @) d¹@
l@(l~1) ~1
,
(3)
T0 where n and ¹ are the initial values of trapped 0 0 electrons concentration and sample temperature, respectively. As shown for the second-order kinetics [4], the calculation of the n /N ratio is imposs0 ible. The same is true for general order kinetics. Because of that, it is more convenient to introduce, instead of the frequency factor s, which cannot be calculated except in the case of first-order kinetics, the pre-exponential factor s(-)"s(n /N)l~1 as has 0 been done by the above mentioned authors. I" n s(l) exp(!E/k¹) 0 . [1#M [s(l)(l!1)]/RN:T exp(!E/k¹@) d¹@]l@(l~1) T0 (4) To calculate the parameters of the TL glow curve one can use the equation given by Chen [5] which, with Rasheedy correction [3] and the pre-exponential factor s(-) introduced, has the form
A
B
RE E "s(l) exp ! [1#(l!1)D], (5) k¹2 k¹ . . where ¹ is the temperature corresponding to the . maximum intensity and D"2k¹ /E. This rela. tionship has been obtained by differentiating Eq. (4) and equating it to zero, and then using the approximation T
P A B
T0
A B A B
k¹ n E E exp ! d¹+¹ exp ! + E k¹ k¹ n/1
](!1)n~1n!.
2. Theoretical consideration
dpl ! "cln pl # dt
(7)
and
(6)
dn ! "!c n N#Sn, 5 # dt
(8)
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281
By assuming the existence of two kinds of centers (luminescent and trapping) and a very short life time of conduction band electrons, as noted above for a simple model, we can rewrite relation (9) (instead of pl we use n) as follows:
A
Sn" cl
Fig. 1. General order glow curves obtained according to Eq. (3) with s(l)"106 s~1, E"1.5 eV, R"0.5 K s~1, n "104 cm~3, 0 N"104 cm~3 and l are given as: (a) 1.9, (b) 1.5 and (c) 1.1.
where c and cl are the recombination probabilities 5 between the conduction band electrons and the traps or the luminescent centers, respectively, and S"s exp(!E/k¹). By assumptions which are valid for this simple model, Ddn /dtD@Ddn/dtD and # Ddn /dtD@Ddpl/dtD, we may write the following equa# tions: Sn n" # clpl#c N 5
(9)
and clS S dpl p2l " pl I"! " dt clpl#c N 1#rN/pl 5
(10)
where r"c /cl. If we assume that c N@clpl or 5 5 r"0, which is equivalent to the assumption that the concentration of traps is low compared with the concentration of luminescent centers, then these relations describe the first-order kinetics. But, if we take c NAclpl which is the same as the assumption 5 that the retrapping is dominant, these relations describe the second-order kinetics [7]. If for a given model we take c "cl, that is r"1, it is evident that 5 for this case Eq. (10) also describes the secondorder kinetics.
B
n #c n N. 5 # N
(11)
All electrons excited into the conduction band recombine with empty trap levels, with the recombination probability c@"cl (n/N)#c . For the 5 first-order kinetics c@"cl (n/N), since c N@clpl, 5 and for the second-order kinetics c@"c , since 5 NAn as well as c "cl. In other words, for the first5 and the second-order kinetics c@"cl (n/N)1 and c@"cl (n/N)0, respectively. The value of c@ can be interpolated between l"1 and l"2 by an empirical interpolation function c@"cl (n/N)m. One can see from Eq. (10), that the influence of the ratio N/pl on the shape of the curve increases with an increase of the factor r from 0 to 1, which at the same time means that it influences the order of kinetics. Since r can take any value between 0 and 1, the interpolating function must be continuous in this range. Therefore, one concludes that the change of the order of kinetics depends on N/pl or, according to this model, N/n ratio. By inspecting the expressions for the first- and second-order kinetics, one can see that this means a change from linear to quadratic function. Since the power function is increasing, it may be taken as the interpolation function. A constant multiplier for this power function must have the dimension of recombination probability. It is practical to take as a multiplier the recombination probability appearing in the interpolating points, i.e., the points describing the first- and second-order kinetics. This value is cl. For a small variation in l one can assume the linear dependence of the exponent m"a l#a . 0 1 Taking for these two points l "1, m "1 and 1 1 l "2, m "0, after solving the linear equation we 2 2 find that a "!1 and a "2. These solutions give 0 1 c@"cl (N/n)l~2. By inserting c@ in relations (9) and (10), we obtain the following relations: S n" nl~1 # clNl~1
(12)
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282
For k we obtain g
and
A B
dn nl E dpl exp ! , I"! "! "s Nl~1 dt dt k¹
(13)
which corresponds to the Rasheedy [3] suggestion (2).
3. Calculation of parameters Halperin and Braner [8] introduce a symmetry factor which depends on the thermal peak and is defined as follows: n k " ., g n 0
(14)
where n is the electron concentration in the traps . at the maximum intensity. For any single peak we have `= `= Rn " I d¹"I i d¹ 0 . ~= ~=
P
P
(15)
and `= `= Rn " I d¹"I i d¹. . . T. T.
P
P
(16)
Here i is a normalized function of TL curve. By applying these relations we can introduce the effective values of half-width u and effective values of %&& that part of the half-width which is on the higher temperature side d , and get the following rela%&& tions: `= Rn u " 0" i d¹ %&& I . ~=
P
(17)
and `= Rn i d¹. d " ." %&& I . T.
P
(18)
d k " %&& . g u %&&
(19)
Solving the differential equation (2) we get
C
D
~1@(l~1) T s(l)(l!1) E n"n 1# exp ! d¹@ 0 R k¹@ T0 (20)
P A
B
and for k g
C
D
~1@(l~1) T. . s(l)(l!1) E exp ! d¹@ k " 1# g R k¹@ T0 (21)
P A
B
Differentiating Eq. (4) and equating the derivative to zero we find T. s(l)(l!1) E 1# exp ! d¹@ R k¹@ T0 E s(l)lk¹2 . exp ! . " RE k¹ .
P A
B
A
B
(22)
This equation is the same as the one obtained by Chen [5], except that the pre-exponential factor has a different form. For k we derived the followg ing expression after substituting Eq. (22) into Eq. (21)
C
k" g
A
s(l)lk¹2 E . exp ! RE k¹
BD
~1@(l~1) .
(23)
.
By substituting Eq. (5) into Eq. (23), one obtains
C
k" g
D
1#(l!1)D 1@(l~1) . l
(24)
Eq. (24) has been derived using the first two terms of the approximation relation given by Eq. (6). In order to increase the accuracy of l, the number of power terms in Eq. (6) can be increased, which is important for higher values of l (over 1.3) and D (over 0.1). In order to obtain more accurate
Z. Vejnovic& et al. / Journal of Luminescence 78 (1998) 279—287
values of the symmetry factor, up to the third decimal place, it is necessary to add another two terms. After these modifications, k takes the following g form:
C
k" g
D
1#(l!1)(D!3D2#3D3) 1@(l~1) 2 . l
(25)
An expression for calculating the order of kinetics l and the activation energy E of the trap can be obtained from Eq. (4) by substituting the value for the peak maximum. I " . n s(l) exp(!E/k¹ ) 0 . . [1#[s(l)(l!1)/R] :T. exp(!E/k¹@) d¹@]l@(l~1) T0 (26) Substituting Eq. (22) into Eq. (26) we get n s(l)exp(!E/k¹ ) 0 . I " . m [(s(l)lk¹2 /RE) exp(!E/k¹ )]1`1@(l~1) . .
(27)
and, after substituting Eq. (23) into Eq. (27) and some rearrangements, n RE I " 0 k . k¹2 l g .
(28)
2k¹ 2d ." %&&. D" E ¹ l .
l"1 we can write ln k "ln k #D. (31) g g0 The value of ln k is in fact the tangent of the angle g0 that a straight line passing trough l"1 and l"l makes with the x-axis. This straight line is at the same time a chord on the curve ln l with intersections at the above two points (Fig. 2). When lP1 one can assume that the curvature of the arc intersected by the chord is small and that the chord is parallel to a tangent passing through the midpoint of ln l line enclosed between l"1 and l"l. In that case ln k "!2/(l#1), and substituting this reg0 sult into Eq. (25) we get 2 ln k "! #D. g 1#l
(32)
The region of validity of this function is for l"1 and for very small values of D(D(0.1). Since the shape of the function, for the region of practical interest (0.7—2.5) and D(0.05—0.15), changes only slightly, it can be precisely determined in the whole relevant region by using a general approximate function which has the same shape as function (32). This expression can be generalized by replacing numerical values with parameters P and l as well 1 0 as D with the function f (D). l P P 1 #f (D), ln k " 0 1 #f (D)" g l #l 1#l/l 0 0
and it follows that
283
(33)
(29)
For all the experimental data obtained from the thermoluminescence [5], the parameter l lies between 0.7 and 2.5. To get numerical values of the parameter l it is necessary to approximate the obtained relations to avoid the singularity values of l near l "1. Logarithm of Eq. (24) gives the symmetry factor as a sum of two terms ln l ln[1#(l!1)D] ln k "! # . g l!1 l!1
(30)
The first term represents ln k for D"0 and we can g write it as ln k . It can be shown that the second g0 term tends to D when lP1. In the neighborhood of
Fig. 2. Geometrical presentation of approximate expression ln k . g0
Z. Vejnovic& et al. / Journal of Luminescence 78 (1998) 279—287
284
where P and l are the weakly changing para1 0 meters dependent on D, and f (D) is a linear function of D as given in Eq. (32). For the range of values of D(0.05—0.15), which is of practical interest, P and 1 l are constants. Therefore, the function (33) can be 0 approximated by a function
thus being the solution. This is !A #JA2!4A A 1 1 0 2. l" 2A 0
From Eqs. (29) and (5) we obtain values for E and s(l), respectively. In case the values of TL curve are known, ¹ , u and d can be determined by . %&& %&& numerical integration of this curve within certain limits. If an adjustment of the TL curve has been done, then u and d can be obtained by integra%&& %&& tion of the fitted curve within certain limits. The values of calculated parameters and accuracy obtained for some characteristic values are given in Table 1. Using Eq. (4), curves are obtained and from them ¹ . Values of u and d are obtained . %&& %&& by numerical integration. For the parameter R, the value of 0.5 K s~1 was used. Parameters E, l and s(l) chosen for the curve described by Eq. (4) to simulate an experimental curve are shown on the left-hand side of Table 1. For each point of the curve given by Eq. (4), numerical evaluation of the integral has been done using the Gauss—Legendre method with estimated accuracy of the order of 10~9. Table 1 also contains other three parameters ¹ , u , and d obtained from the glow curve and, . %&& %&& further to the right-hand side, calculated parameters E , l , and s(l) with the corresponding rela# # # tive errors r , r , and r . The accuracy with which E l s the parameters are obtained depends on the accuracy of the adjustment of the function ln k and on g the accuracy with which ¹ , u and d are deter. %&& %&& mined. Since the approximate values for correction
P 1 #P D#P . ln k " (34) g 1#l/l 2 3 0 By adjusting the function (34) to (25) using the method of least squares, we get the following approximate values P "!2.127, P "0.76, 1 2 P "!0.2, and l "0.592. Substituting Eq. (29) 3 0 into Eq. (34) we get the quadratic equation A l2#A l#A "0, 0 1 2 where
(35)
d d A "P !ln %&& "!0.2!ln %&& , 0 3 u u %&& %&& d 2d A "l P #P !ln %&& #P %&& 1 0 1 3 2¹ u %&& . 1.52d d %&&!0.592 ln %&& !1.378 " ¹ u . %&& and
A
(36)
B
2d d A " %&&l P "0.9 %&&. 2 ¹ 0 2 ¹ . . Two values are obtained for the parameter l and only one of them belongs to the interval (0.7, 2.5),
Table 1 D
E
l
s(l)
¹ .
u
0.0597 0.0597 0.0622 0.0805 0.0848 0.0896 0.0994 0.1220 0.131 0.142 0.142
1.6 1.6 0.4 1.6 0.4 0.1 0.1 1.6 0.4 0.1 0.1
2.5 1.9 1.5 1.5 1.5 1.5 0.7 0.7 1.9 1.9 2.5
1013 1013 1013 109 109 109 108 105 105 105 105
554.24 554.48 144.31 746.76 198.81 52.01 57.65 1132.2 303.48 82.41 82.16
72.68 61.01 14.35 94.80 26.27 7.31 6.01 142.36 69.53 20.35 24.10
%&&
d %&&
E #
l #
s(l) #
r (%) E
r (%) l
r (%) s
41.56 31.59 6.75 45.32 12.61 3.52 1.90 47.78 37.92 11.18 14.56
1.608 1.601 0.399 1.601 0.400 0.100 0.090 1.586 0.396 0.0987 0.0979
2.524 1.908 1.501 1.510 1.513 1.511 0.691 0.686 1.893 1.885 2.450
1.166]1013 1.017]1013 9.380]1012 1.022]109 1.032]109 1.017]109 8.904]107 8.524]104 8.679]104 8.259]104 7.269]104
0.48 0.043 !0.23 0.06 0.06 0.13 !0.88 !0.86 !0.90 !1.40 !2.10
0.97 0.43 0.01 0.66 0.78 0.75 !2.07 !2.00 !0.33 !0.89 !1.98
16.6 0.71 !6.26 2.45 2.60 4.00 !17.74 !14.54 !13.10 !18.37 !27.30
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285
Table 2 D
E
l
s(l)
¹ .
u
0.0597 0.131 0.142 0.142
1.6 0.4 0.1 0.1
2.5 1.9 1.9 2.5
1013 105 105 105
554.24 303.48 82.41 82.16
72.68 69.53 20.35 24.10
%&&
d %&&
E #
l #
s(l) #
r (%) E
r (%) l
r (%) s
41.56 37.92 11.18 14.56
1.600 0.398 0.0994 0.0993
2.513 1.903 1.899 2.488
9.942]1012 9.370]104 9.240]104 9.010]104
0.02 0.45 !0.57 !0.62
0.50 !0.14 !0.036 !0.50
!0.58 !6.34 !7.62 !9.31
factor D are also obtained, in calculation of these parameters, it is possible to use more accurate values for P , P , P and l for the second calcu1 2 3 0 lation. If we look at the first and last three values in Table 1, we see an increase in calculation precision. In Table 2, the calculated values are given for the range of D(0.05, 0.07) and D(0.13, 0.15) when P "0.856, l "0.6 and P "0.679, l "0.578, re2 0 2 0 spectively. Values of P and P are taken to be the 1 3 same since their changes are negligible. In this way, the calculation precision can be increased so that the accuracy of the results depends only on the accuracy in determination of ¹ , . u and d , i.e., it depends on the accuracy in %&& %&& obtaining the TL curves.
4. Discussion In order to show how Eq. (1) is derived, a simplified model with one kind of traps and one kind of luminescent centers has been used. Further simplification of the model assumes the traps to be partly filled which is mostly the case. It can be shown that this simplification of the model does not have any effect on the analysis carried out in Section 2, and that it is valid for the case n+N. In that case, the same expression is obtained for the first- and second-order kinetics. If the term c n n in 5 # Eq. (8) is not neglected, c@"(cl!c )(n/N)#c is 5 5 obtained. Applying the condition for the first-order kinetics, c N@clpl, we get c@"(cl!c )(n/N). Since 5 t n(N or n+N and c n+c N@clpl, we get for the 5 5 first-order kinetics c@"cl(n/N). Applying condition for the second order kinetics, c "cl [7], we get 5 c@"cl. Comparing Eq. (1) to Eq. (10) one can see that all the existing parameters are the same except r which
changes with l. This change is the result of the interpolation process. In fact this interpolation means a substitution of one constant r for another constant l in the way which enables analytical solution of the differential equation describing the TL kinetics. In this way, physical meaning of the parameter l is in direct connection with the parameter r which is in accordance with the physical meaning of parameters l"1 or l"2. In these cases the physical meaning of l is defined by the ratio of luminescent to retrapping probabilities. According to the relation valid for lth order of kinetics and taking into account the Halperin and Braner [8] definition, a symmetry factor relation was evaluated. The obtained relation, Eq. (31) can be written more transparently as k "k eD. Since g g0 the symmetry factor k characterizes the order of g kinetics of thermoluminescence process, this equation can be used for determining the criterion for characterizing the order of kinetics. For example, k "e~1 corresponds to the first-order kinetics, g0 while k "0.5 corresponds to the second-order g0 kinetics. If eD is expanded, retaining only the first two terms of the series, i.e., eD+1#D, then the criteria obtained are k "(1#D)/e and g k "(1#D)/2 for the first- and the second-order of g kinetics, respectively, as given by Halperin and Braner [8] and used by others [5,9—12]. For general order kinetics, one can obtain the relation k "k (1#D) which represents a criterion for deg g0 termining the order of kinetics in general form. In the case of hyperbolic heating function used instead of linear, used the integral in Eqs. (3) and (4) can be solved analytically [13]. The hyperbolic function has the simple form ¹"R /(t !t), where 0 k R (Ks~1) and t (s) are constants. With such as0 k sumption one can derive the expression for the
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286
symmetry factor [13].
AB
1 1@(l~1) . k" g l
(37)
One can see that this equation is identical to the relationship obtained for k defined by Eqs. (30) g0 and (31). Therefore, k is in fact a symmetry factor g0 of thermoluminescence curve of the lth order kinetics when the heating function is simple hyperbolic. Expressions (24), k "k eD and k "k (1#D) g g0 g g0 in the region l (0.7—2.5) are very similar, in contrast to expression (25), large discrepancies are visible with respect to the referred three curves particularly for values l'1.3 (Fig. 3). According to these, one can state that addition of another two terms into Eq. (6) increases the accuracy of calculation of TL curve parameters. This improvement in calculation of parameters is particularly pronounced for higher values of D and higher values of l. According to the derived relations it is possible to calculate analytically the parameters E, l and s(l) from the shape of the TL curve. For calculation of the above parameters in the ranges 0.7(l(1.3 and D(0.06, relations (29) and (32) can be used. Combining these relations one gets the quadratic equation identical in form to Eq. (35). Relative error bars for the calculated values of l and E are smaller than 2.5%, and for s(30%. Since the correction parameter D is very small, it can be assumed that DP0 as a first approximation. According to the assumption, Eqs. (29) and (31) can
Fig. 3. Different relations describing symmetry factor for D"0.14.
be applied within the range of the parameter l up to 1.5. Since in this case ln k "f (ln l,l), one has to use g iterative procedure in order to obtain the values for l. For exact calculation of l in the entire range which is significant from the point of view of the experiment, improved expressions Eqs. (25) and (29) have to be used. The obtained result is a higherorder polynomial. In order to get an accurate solution, an iterative procedure should be used. In order to avoid this tedious procedure an analytical expression was proposed for calculation of the above parameters, Eq. (25). The relative error for these calculations given in Tables 1 and 2 is less then 1% for parameters l and E and less than 10% for parameter s(l). This kind of calculation is obviously better for the parameter E and much better for parameters l and s(l), than the one obtained by Chen [5]. As one can see in Table 1, the only exemption are the values obtained for l"0.7. Principal reason for this is that Eq. (4) does not define the TL curve very well in all regions of interest for calculation of u and d . More precisely, for %&& %&& higher temperature values the second term in the denominator, which is negative, becomes smaller than !1. The reason is the fact that Eq. (4), in this region and order of kinetics, is an extrapolation function and does not completely describe the TL processes. Since the region of undefined TL curve, according to Eq. (4), is largest for the value l"0.7, the obtained values of relative errors are maximal. In reality, the undefined region of the TL curve does not exist and because of that one can assume that the calculation errors lie inside the abovementioned limits. The determination of parameters of an experimental TL curve by determining the values of u and d is certainly the simplest method. However, the curve is not uniquely determined by these two parameters. There are curves of various shapes having the same delta and omega values. Although these deviations are comparatively small for certain order of kinetics, as is well known [5], by using these values it is not possible to determine the parameter l of the order of kinetics with the same accuracy, as achieved for E. From theoretical point of view the calculation method using the integral of the TL curve and part of the glow curve is more general since it considers the shape of the curve.
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Therefore, this method should have a higher precision. The determination of the integral of the peak and of a part of the peak from the glow curve by simple numerical treatment of the peak most certainly leads to additional errors and the method of determination of TL curve parameters by using delta and omega values can have higher precision. However, if one wants more precise calculations by using any of these two methods it is necessary first to obtain, from the experimental glow curve, the wanted peak by some fitting method. In this case we think that the method of calculation based on the integral of the TL curve and a part of this curve will give better result.
5. Conclusions Using a simple model, it has been shown that the equations for general order kinetics, with correction proposed by Rasheedy, actually represent an interpolation between the first- and the secondorder kinetics and an extrapolation beyond these values, assuming the exponent of the ratio (n/N) changes linearly. The introduction of a pre-exponential factor s(l) is justified since this parameter influences the shape of the TL curve, and it is not possible to deduce from this curve neither the values of n and N nor their ratio. This factor is not 0 a material constant, since it depends on the absorbed dose, but it explains the dynamics of the TL process for each single curve. The proposed method for parameters calculation is quite general since it is based on evaluation of integral values of TL curves and these curves originate from the derived equations valid for general order kinetics. This means that calculations of parameters of the curve for general order kinetics by this method are not determined by limits D(0.05—0.15) which are of practical value, but instead they are valid outside these limits, as well. Increased accuracy of calculations, besides relying
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on the use of integral values of the TL curve, is obtained by including another two power terms defined in Eq. (6), and it influences the calculation of symmetry factor, Eq. (25). Certain approximations made for the symmetry factor have been made only to enable analytical results and to avoid a singularity value lP1. Due to the corresponding simple iteration procedure the accuracy of the results practically does not depend on the approximation made, but rather on the accuracy of the experimental curve and evaluation of ¹ , d and . %&& u . By inspecting the results of calculations shown %&& in Tables 1 and 2 one may conclude that evaluation of the above parameters with higher precision improves the accuracy of the results for E , l , and # # s(l) given in the tables. # Acknowledgements The authors are grateful to Professor D. C[ ajkovski and Professor Z. Ikonic´ for reading and making useful comments on the manuscript. This paper was supported in part by the Serbian Research Fund. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
C.E. May, J.A. Partridge, J. Chem. Phys. 40 (5) (1964) 1401. Y. Kirsh, Phys. Stat. Sol. 129 (15) (1992) 15. M.S. Rasheedy, J. Phys. Condens. Matter 5 (1993) 633. M.S. Rasheedy, A.M.A. Amry, J. Lumin. 63 (1995) 149. R. Chen, J. Electrochem. Soc. 116 (9) (1969) 1254. V.V. Serdjuk, Ju.F. Vaksman, Luminiscencija poluprovodnikov, Visa skola, Kiev-Odesa, 1988, p. 120. R. Chen, Y. Kirsh, Analysis of Thermally Stimulated Process, Pergamon Press, Oxford, 1981, p. 32. A. Halperin, A.A. Braner, Phys. Rev. 117 (2) (1960) 408. H.J. Beaven, J. Phys. Chem. Solids 36 (1975) 949. R. Chen, J. Appl. Phys. 40 (2) (1969) 570. A. Halperin, A.A. Braner, A. Ben Zvi, N. Kristianpoller, Phys. Rev. 117 (2) (1960) 416. J.K. Zope, Indian J. Pure Appl. Phys. 20 (1982) 816. R. Chen, N. Kristianpoller, Radiat. Prot. Dosim. 17 (1986) 443.