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An approximation of phosphorescence decay kinetics of ideal phosphors by a general order kinetics model
Physica B 304 (2001) 309–318
An approximation of phosphorescence decay kinetics of ideal phosphors by a general order kinetics model Z. Vejnovic! a, M.B. Pavlovic! b, M. Davidovic! b,c,* b
a Institute of Security, Kraljice Ane b. b. 11000 Belgrade, Yugoslavia Vinca Institute of Nuclear Sciences, P.O. Box 522, 11001 Belgrade, Yugoslavia c Faculty of Electrical Engineering, P.O. Box 816, 11000 Belgrade, Yugoslavia
Received 27 June 2000; accepted 7 February 2001
Abstract The phosphorescence of ideal phosphors has been investigated using a general order kinetics model. The regions where the model describes the phosphorescence processes exactly and approximately have been identified. In order to characterize these processes, the mean value of the kinetics order has been defined for the part of the curve of immediate interest. It has been shown that the value of the order of kinetics for ideal phosphors in the transition region cannot have a unique physical interpretation. The general order kinetics model enables theoretical explanation of some important physical characteristics of TL processes for the ideal phosphors. For the theoretical curves, obtained by the band model, the parameters of phosphorescence curves have been determined using the general order kinetics model. The results differ from the exact values by 3%. We conclude that the general order kinetics model enables a better understanding of the physical processes, and an approximate determination of the phosphorescence decay parameters. This points out the theoretical and practical value of the model. # 2001 Elsevier Science B.V. All rights reserved. PACS: 78.60.K Keywords: Thermoluminescence; Phosphorescence processes; The general order kinetics model; Retrapping factor
1. Introduction The simplest band model (BM) when explains the mechanism of TL relaxation must include the luminescence centers and traps. If in an ideal case all luminescence centers and traps are assumed to be identical, then the model describes a system *Corresponding author. Vinca Institute of Nuclear Sciences, P.O. Box. 522, 11001 Belgrade, Yugoslavia. Tel.: +381-11-458222; fax: +381-11-344-0100. E-mail address: [email protected] (M. Davidovic´).
known as ideal phosphor [1]. The general theory of luminescence decay process, explained by the energy band model, is mathematically complicated and cannot be solved for the general case. Starting from the band model [2] and taking that n þ nc ¼ pl , a general equation can be obtained [3]: 1 d2 pl 1 dpl 2 gt þ 1 þ pl dt2 p2l dt gl 1 dpl gl gt pl þ gt N þ S þ gl Spl ¼ 0; ð1Þ þ pl dt where pl is the concentration of holes occupying
0921-4526/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 0 4 9 1 - 4
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the luminescence centers, nc is the concentration of electrons in the conduction band, n is the concentration of electrons occupying the traps, N is the trap density, gl and gt are capture probabilities for electrons by luminescence centers and traps, respectively. Here S ¼ s expðE=kTÞ where s is the frequency factor, E is the trap activation energy, T is the material temperature and k is the Boltzmann constant. Klasens and Wise [3] have solved this equation for a special case of the second order kinetics when gl ¼ gt ¼ g. In this case, Eq. (1) becomes considerably simpler and the solution is
Taking S ¼ 0, an equation is obtained with solutions describing the first step of the decay process which is temperature independent. Solutions are exact and shown in parametric form: p gt =gl p nc l l ¼ C1 ð5Þ þ 1; N N N and Z dpl ¼ C2 gl Nt: ð6Þ gt =gl pl C1 pl =N þpl =N 1 The integration constants C1 and C2 depend on the initial conditions of the density of electrons nc0
gL þ g nc0 =pl0 L exp½ðgN þ S Þt I ¼
2 ; 1=pl0 þ gLt þ g nc0 =pl0 L =gN þ S ð1 exp½ðgN þ S ÞtÞ
where S ; L¼ gN þ S
ð3Þ
while nc0 and pl0 are the initial concentrations of electrons in the conduction band and empty luminescence centers, respectively. Klasens and Wise have noticed that the decay curves, obtained for various values of excitation, consist of two parts. The first part of the curve is temperature independent and results from the concentration of electrons in the conduction band which is higher than the quasistationary value characterizing the phosphorescence process. This step of the process is very short and takes place immediately after the end of excitation. Its duration varies from 108 up to 102 s [1,3]. The second part of the curve obtained by neglecting exponential terms is strongly temperature dependent. This step of the decay process is known as phosphorescence. Starting from these facts, Adirovich [1] has found the solutions for the first and the second step of luminescence decay process. After a rearrangement, Eq. (1) can be expressed as follows: dnc g t g t nc N 1 1 ¼1 þ þ þS : ð4Þ g l n c g l pl dpl g l g l pl pl
ð2Þ
in the conduction band and density of empty centers pl0 . For the second step of the process Adirovitch, has obtained the exact solutions also shown in parametric form: dn gl S n2 ; I ¼ ¼ dt gl n þ gt ðN nÞ and t¼
1 g n0 g N 1 1 1 t ln þ t : S gl n g l S n n0
ð7Þ
ð8Þ
Phosphorescence intensity is obtained by substituting the value of concentration n from the first into the second equation. Adirovitch denoted the resulting solutions the quasistationary ones [1] and the corresponding region the quasistationary region.
2. General order kinetics model In the general case, where neither of the two basic processes (luminescence and retrapping) is dominant, the models of the first and the second order kinetics cannot be applied. For the general case of isothermal TL relaxation (see discussion in Ref. [4]), Klasens and Wise have suggested an
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empirical differential equation instead of Eq. (7) to describe the process of phosphorescence decay in accordance with the ‘‘multimolecular’’ mechanism. Later on, May and Partridge [5] have used the model of general order kinetics expressed by such a differential equation to explain the TL glow curves which cannot be described by the models of the first and second order of kinetics. In order to give a better physical interpretation, Rasheedy [6] proposed a correction of the empirical differential equation, which can be expressed as: I ¼
dn S l ¼ n: dt N l1
ð9Þ
As shown in Ref. [7], this is an empirical interpolation equation which enables the calculation of an analytical approximate solution for the cases which cannot be described by the models of the first [8] and the second order kinetics [9]. In Eq. (9), the order of kinetics l is constant. In this equation, the constant r is substituted for l so that the differential equation has an analytical solution. In reality, however, l is not constant and depends on the initial conditions as has been shown for the curves close to the first and second order kinetics [1,10]. This is particularly prominent in the general order kinetics model, since, during the process of thermal relaxation there is a significant change in the ratio of parameters r to n0 , and hence in l as well. Therefore, the constant value of the parameter l in the model of general order kinetics is in fact an average value of this parameter for the part of the phosphorescence curve being examined. Solving the differential Eq. (9) and substituting the value of n0 for t ¼ 0, a generalized empirical Becqurel formula for the isothermal TL emission Ir ¼ dnr =dts at a given moment ts ¼ St is obtained: l=l1 Ir ¼ ½ðl 1Þts þ n1l ; r0
ð10Þ
where nr ¼ n=N and nr0 ¼ n0 =N. This equation, according to the general order kinetics model approximates the isothermal curve described by Eqs. (7) and (8). By analyzing particular solution (2), for the second order kinetics model, one can see that for different values of nc0 and pl0 , all the curves merge into one. It means that all the curves tend to one
Fig. 1. The luminescence decay curves for different retrapping factor values r ¼ gt =gl , parameter relations gl N=S ¼ 50, and the initial conditions nc0 =N ¼ 0:1 and pl0 =N ¼ 1. Broken lines represent the curves of the second order kinetics model.
limiting shape. This third part of the phosphorescence curve, which appears after sufficiently long time, characterizes emptiness of traps, i.e. n N, and can be called the limiting or asymptotic region. The quasistationary range in that case can be termed a transition range. For the well known models of phosphors: ideal phosphors [1,11], phosphors with deep inactive traps [11], phosphors with order of kinetics 1.5 [5], one obtains the concentration of electrons in the conduction band nc pK l , where K is a rational positive number or zero. Therefore, the third part of the curve can be presented by the model of general order kinetics, or in other words, the phosphorescence process can be shown in a simple way and directly. One can also see that the general order kinetics model describes the limiting case of phosphorescence process, a part of the general solution of this process (Fig. 1).
3. The order of kinetics l in the general order kinetics model Comparing Eqs. (7) and (9) one can see that they are related by [12] nr þ rð1 nr Þ ¼ n2l r :
ð11Þ
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Taking the logarithm of (11) we get: ln½nr ð1 rÞ þ r ¼ 2 f ðnr ; rÞ: l ¼2 ln nr
ð12Þ
From this expression it is clear that l is not constant since it depends on the variable nr . An essential approximation in the model of general order kinetics is that l is independent of nr , that is, of r. For the phosphorescence process, a constant value of l can be obtained for the range of ts values by finding its average value: Rt %l ¼ R0 l dts ¼ 2 f%ðnr ; rÞ: ð13Þ t 0 dts From Eq. (12), it is clear that the average value of l requires the finding of an average value of the function f ðnr ; rÞ: R ts 0 ln½nr ð1 rÞ þ r=ln nr dts % f ðnr ; rÞ ¼ R ts 0 dts R nr nr0 ln½nr ð1 rÞ þ r=ln nr dts =dnr dnr R nr ¼ : nr0 dts =dnr dnr ð14Þ When the distribution function dts =dnr is found from Eq. (8) and substituted into Eq. (14) one obtains the final expression for the calculation of the average value of f ðnr ; rÞ for the interval of values of nr in which the phosphorescence is observed.
f%ðnr ; rÞ ¼
R nr nr0
always be found is of practical interest. The changes in function f ðnr ; rÞ are generally small within the nr range. The exception is that part of the curve where nr ! 0. This part of the curve, however, corresponds to the case where traps are almost depleted and therefore the intensities of phosphorescence in this region are very low. If this part of the curve is outside the observed range or if its influence is neglected, then we can take that f%ðnr ; rÞ f ðC; rÞ. In this expression C has a constant value in the range of r from 0 to 1, for the initial conditions of the observed region determined by nr0 , and by the final value nr , i.e. by the time interval in which the phosphorescence is observed. We conclude that C does not correspond to the average value of nr since, as can be seen from the above expression, the average value of nr does depend on r. An increase in the retrapping factor r leads to a slowdown of phosphorescence and to a decrease in the average value of nr . However, by calculating the average value of nr , it can be shown that this is a slowly changing function in the interval of r from 0 to 1. Therefore C is in fact an average value of nr which gives the best description of phosphorescence in the interval of r from 0 to 1, for the given initial condition nr0 and specified time interval. The value of C can be determined as follows. The differential Eq. (7) is solved for some values of the parameters r and nr0 . If this is done for a large
ln½nr ð1 rÞ þ r=ln nr 1=n2r þ ð1 rÞ=nr dnr R nr : 2 nr0 1=nr þ ð1 rÞ=nr dnr
The integrals in Eq. (15) for the arbitrary value of r in the interval from nr0 to 0 cannot be found analytically. Difficulties arise from the upper limits of the integral since the functions go to infinity when nr ! 0. Physically, this corresponds to the fact that the traps cannot become completely depleted during the isothermal relaxation. With the depleted traps, however, the probability of retrapping increases with respect to the probability of luminescent recombination. Therefore, only the range of phosphorescence within the limits from nr0 up to nr , for which numerical solutions can
ð15Þ
number of various values of r, a set of isothermal TL relaxation curves is obtained, which can be fitted by using expression (10). The value of nr0 is taken to be constant when fitting the parameter l. In this way a dependence of l on r is obtained (Fig. 2). By fitting the obtained curve to the function: l ¼2
ln½Cð1 rÞ þ r ¼ 2 f ðC; rÞ; ln C
ð16Þ
the value of parameter C is found. A constant value of l is always obtained by applying this
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Fig. 2. The dependence of the order of kinetics l on the retrapping factor r. The solid line is obtained by fitting the order of kinetics l using Eq. (10), with numerical solutions of the differential Eq. (7) for the exact value of nr0 ¼ 0:1. The parameter C is obtained by fitting the resulting curve to Eq. (16) (dotted line).
procedure for any constant value of nr0 , as a function of r. In the example shown in Fig. 2 we find C ¼ 0:032. The fitting was performed using the chi square method without weight factor when summing the squares. In this case, the sum of squares is computed using the expression Sðyi fi Þ2 , where yi is the experimental value of the intensity of phosphorescence at some point i, while fi is the value of the fitting function. It means that the highest TL emission values have the strongest influence on the fitting process of the experimental or simulated curve to the assumed function of the model. In practice this means that, in the case of a decrease in the TL emission by two orders of magnitude, any further increase in the range of nr , where the TL relaxation is observed, has a negligible influence on the change of the obtained values of l. Assuming that close to the end of the curve the phosphorescence proceeds by second order hyperbola, we may introduce the weighting factor in the process of chi square fitting for each point si ¼ yi , and get the sum of squares Sððyi fi Þ=si Þ2 . When using this method, all the points are fitted with the same relative error and for l we get the values closest to those computed by Eqs. (13) and (15). Thus we show that the parameter C depends
313
Fig. 3. Interpolation function (broken line) and exact values of the expression in the denominator of Eq. (7) (solid line). The parameter nr0 ¼ 0:1.
strongly on the method of fitting and on the interval of time for which the curve is observed, as shown by the theoretical expressions. Substituting Eq. (16) into Eq. (11) we get the form of the interpolation function which approximates the expression in the denominator of Eq. (7): nr ð1 rÞ þ r nrf ðC; rÞ :
ð17Þ
For the values of parameters of the phosphorescence curves, given as an example in Fig. 2, the ratio of these two functions is graphically presented for a few characteristic values of the parameter r (Fig. 3). The figure shows, how the interpolation is made by the model of general order kinetics, using the values of the function f ð0:032; rÞ, with the parameter r in the range from 0 to 1 when nr0 ¼ 0:1.
4. Discussion For the transition region, relationship (12) is only an initial one in the process of defining the order of kinetics parameter using the general order kinetics model, the value of the order is taken to be constant during the TL process. We consider the method by which the constant l is obtained to be essential for the physical interpretation of this
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Fig. 4. The dependence of the kinetics order l on the initial condition nr .
parameter. The constant value of l can be defined as an average value of the function (12), which varies during phosphorescence process. This is an approximation as one can easily see. Leaving its quality aside for the moment, we have focussed our attention to the physical essence of the parameter l in the model of general order kinetics. Therefore, although we can conditionally take that the function (12) describes the value of the kinetics order, l is constant (Eq. (16)) throughout the phosphorescence process. Using Eq. (16) the functional dependence of l on r can be found in the entire region of l which is of practical interest (0.7, 2.5) [13]. Fig. 4 shows the curves for various values of the initial condition nr0 . If we examine these curves and look at the left and right hand side from the point r ¼ 1, the following can be concluded with a decrease in nr . The parameter l approaches the value of 2, which is its limiting value, the order of kinetics l > 2 exists only if r > 1, and kinetics order l52 appears for r51. On the left, as well as on the right hand side of the point r ¼ 1, depletion of traps leads to an absolute domination of the retrapping process but also to a less influence of the retrapping factor r and to an increase in depleted trap influence on the retrapping process. On the left hand side this means an increase of the retrapping probability, and on the right hand side a decrease of the retrapping probability with respect to its initial
value. This leads to asymptotic characteristics of phosphorescence processes [10]. Following the excitation with different intensities, the decay curves approach each other as the time goes on. For an ideal phosphor the relaxation processes are close to the process described by the second order kinetics. We can define the stable kinetics of the phosphorescence process, which should be understood as a process where l is constant and as a process to which all the other processes of phosphorescence decay asymptotically approach. For the ideal phosphors, the process flowing according to the second order kinetics model can be defined as a stable one. Phosphorescence process in these systems which is described by the model of the second order kinetics tends to go on unchanged. Those processes having the order of kinetics different from two tend to change into those, which can be described by the model of second order kinetics. For phosphorescence curves of ideal phosphors the region outside the second order kinetics represents the region of transition regime. The facts discussed above have been proved experimentally for some phosphors [10]. Assumed quasistationary electron density in the conduction band leads to a stable phosphorescence process. Only in two cases, the first and the second order kinetics, the process of phosphorescence can be observed to follow from the beginning to the end an unchanged law of decay. For the first order kinetics, one can see that the electron density in the conduction band nc ¼ S=gl is constant throughout the phosphorescence. The stable phosphorescence is a consequence of the constancy of nc . In the case of phosphors there is always a finite number of occupied traps, so it is clear that at the end of phosphorescence nc cannot remain constant. A detailed analysis is given in Ref. [1]. We conclude that at the end of phosphorescence process, taking place according to the first order kinetics model, the quasistationary electron density nc in the conduction band no longer holds. Therefore, in this case, we have a quasistable phosphorescence process. For the second order kinetics, the process proceeds in a stable manner due to the constant density ratio of electrons in the conduction band ðnc ¼ Sn=gt NÞ and in traps. This means that we
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have a stable process obeying the same law to its very end. From Eq. (16), we now conclude that the constant, having the value of 2, determines the stable character of the TL relaxation, and the function f ðC; rÞ is an average value of the deviation from the stable process for the initial condition nr0 . Therefore, the kinetics order l is the measure of the stability of TL relaxation process in respect to the initial condition nr0 , or in respect to the retrapping factor r in the TL relaxation region. From Eq. (16) derived above and Fig. 4, we can see that the conditions for the first and the second order kinetics can be simply expressed by r ¼ 0 and 1, respectively. It may be assumed that the given models can be approximately used in the neighborhood of these points. The condition r nr , defining the first order kinetics process, can simply be shown to be a close proximity of the point r ¼ 0. Since N > n we must have gl gt and this is r 0. The same holds for the condition r nr , which defines the second order kinetics process. However, since this is the region of stable processes, where even greater changes in the parameter r do not necessarily change the order of kinetics l, it may happen that for larger deviations from the point r ¼ 1 the phosphorescence process proceeds according to the second order kinetics. The condition r nr may hold even for some point r1 which is far from r ¼ 1, and this means that at this point the phosphorescence process can also be described by the model of the second order kinetics. As one can see from Fig. 4, if the phosphorescence can be approximately described by the second order kinetics at the point r1 , for a given initial condition nr0 , then this must be valid at any point 1 > r > r1 if r1 51 or 15r5r1 if r1 > 1. Consequently, the condition r ¼ 1 is the limiting value of the condition r nr . Therefore, the conditions r ¼ 1 and r nr define the range of r in which the phosphorescence processes can be described by the second order kinetics model, and they are interdependent. For a given band model with two types of centers: traps and luminescence centers we get using an approximation with the general order kinetics that l ¼ 1 represents a limit of physically real phosphorescence processes. For l51 we get
315
r51 and it means that gt 50 or gl 50, which is physically impossible. Therefore, the general order kinetics model cannot explain kinetics orders less than one, and an extrapolation of the curve (16), based on these values, for a given band model has no physical meaning. However, since the region for r ¼ 0 is the region of great instability of phosphorescence, it can be observed that a very small increase in both the luminescence centers, not followed by an increase in electron density in traps, and electrons in the conduction band or on excited levels, not arising from the traps, leads to the phosphorescence with the order of kinetics smaller than one. For ideal phosphors, expression (9) can be used to fit the phosphorescence curves with shapes ranging from exponential ðl ¼ 1Þ to hyperbolic decay ðl 1Þ [4]. In order to take account of all these cases of decay, there has to be some interdependence of the parameters in the exponent of relation (9) and the parameters next to the term ts . In the cases of the first and second order kinetics, the exponent in Eq. (10) is constant, hence this exponent and the coefficient ts are independent quantities. Therefore, they can be obtained by a simple fitting of the experimental curves to function (10). For those cases, where the phosphorescence decay is described by the expression having the order of kinetics l ¼ 1 and 2 each part of that curve can be described by Eq. (10). If various parts of that curve are fitted to expression (10), a different value of the parameter l will be obtained for each part. If we now make an approximation that l is constant throughout the phosphorescence decay, then by fitting the experimental curves to function (10) an error appears. This error is minimal in the process of fitting the parameter l and the coefficient ts . All this leads to a deviation of the parameter l and the coefficient ts from their exact values. This deviation depends on how much the parameter l deviates from l ¼ 1 or 2. The errors are larger near l ¼ 1, since we deal with an unstable process. Similar errors are present near l ¼ 2, because here we deal with a stable process. Assuming that the occupancy of traps nr is known, from the fitting of an experimental curve to Eq. (10) and from the value obtained for the
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parameter l, we conclude that there is an agreement between the theoretical procedure, based on general order kinetics, and the simple band model characterizing the ideal phosphor. Furthermore, we conclude that the simple band model of phosphor can explain the general empirical Becquerel law. It can also point out the complexity of the phenomenon of phosphorescence decay when other effects which can influence the phosphorescence are present (luminescence quenching, the presence of traps with various depths, etc). Expression (15) can be used to explain the fact that in the model of general order kinetics, for the processes of TL relaxation stimulated by the heating function R which is a simple hyperbolic function [11], we got analytical solutions. In the region where the distribution function tends to infinity, R tends to infinity as well, and this leads, at least theoretically, to the total depletion of traps.
5. Theoretical and practical aspects of the general order kinetics model With respect to the asymptotic character of the phosphorescence process, observed long ago, the parameter l shows the characteristics of the stability curve in the physical sense. Therefore, this parameter shows the stability of phosphorescence or the distance from the asymptote, which defines stable conditions of the process for the real materials, which can be approximated by the model of general order kinetics. If we take the parameter l to be constant and get back to Eq. (9) and write it in another form, we get Ir ¼ expðljln nr jÞ:
ð18Þ
Here, we can define the parameter l as a constant of the velocity of emptying the traps with respect to variable ln nr for a given part of the phosphorescence curve. This means that the traps will empty
Table 1 r
ts
l
lf 1
rl1 (%)
nr0f 1
rn1 (%)
lf 2
rl2 (%)
nr0f 2
rn2 (%)
0 0.01 0.02 0.03 0.04 0.05 0.06 0.08 0.1 0.125 0.15 0.175 0.2 0.25 0.3 0.35 0.4 0.5 0.6 0.8 1 1.1 1.2 1.3 1.4 1.5
15 20 25 30 50 50 80 80 100 100 100 150 150 175 200 200 250 300 350 500 650 800 900 1000 1100 1200
1 1.0789 1.1307 1.1874 1.2291 1.2700 1.3015 1.3622 1.4076 1.4568 1.5012 1.5409 1.5746 1.6305 1.6781 1.7172 1.7534 1.8129 1.8618 1.9397 2 2.0255 2.0488 2.0702 2.0899 2.1083
1 1.0944 1.1643 1.2197 1.2685 1.3075 1.3453 1.4052 1.4515 1.4954 1.5378 1.5777 1.6072 1.6593 1.7027 1.7379 1.7718 1.8265 1.8715 1.9438 2 2.0238 2.0455 2.0654 2.0838 2.1009
0 1.44 2.97 2.72 3.21 2.96 3.37 3.16 3.12 2.65 2.44 2.39 2.07 1.75 1.47 1.21 1.05 0.75 0.52 0.21 0 0.09 0.16 0.23 0.29 0.35
0.1 0.0944 0.0915 0.0895 0.0873 0.0865 0.0849 0.0835 0.0836 0.0850 0.0848 0.0839 0.0848 0.0860 0.0871 0.0887 0.0894 0.0915 0.0935 0.0968 0.1 0.1025 0.1031 0.1047 0.1063 0.1079
0 5.52 8.47 10.50 12.62 12.62 15.05 16.44 16.39 14.98 15.15 16.01 15.11 13.98 12.86 11.22 10.53 8.45 6.48 3.16 0 2.59 3.18 4.76 6.34 7.90
1 1.0662 1.1171 1.1598 1.1981 1.2303 1.2621 1.3128 1.3576 1.4015 1.4438 1.4835 1.5177 1.5753 1.6256 1.6686 1.7086 1.7759 1.8328 1.9259 2 2.0319 2.0613 2.0884 2.1137 2.1373
0 1.18 1.20 2.32 2.52 3.13 3.03 3.62 3.55 3.79 3.83 3.72 3.66 3.39 3.13 2.83 2.56 2.04 1.55 0.71 0 0.32 0.61 1.12 1.14 1.38
0.1 0.0987 0.0982 0.0979 0.0975 0.0976 0.0972 0.0975 0.0976 0.0985 0.0987 0.0985 0.0987 0.0991 0.0993 0.0997 0.0997 0.0999 0.1000 0.1000 2 0.0999 0.0998 0.0998 0.0997 0.0997
0 1.25 1.79 2.07 2.44 2.37 2.76 2.47 2.37 1.46 1.30 1.49 1.27 0.88 0.63 0.29 0.26 0.03 0.08 0.09 0 0.05 0.11 0.17 0.24 0.30
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more slowly if the order of kinetics is higher. This is correct since in this case the retrapping factor is too large. As far as the calculation of phosphorescence parameters is concerned, only two parameters are of interest for some experimental curves, as one can see from Eq. (9), and they are nr 0 and r. Two cases will be examined. A direct fitting of the variables to the experimental curve (Table 1, lf 1 and nr0f 1 ) and a fitting with one parameter only if the exact value of the other one is known. Eq. (10) can be used for the purpose. If the value nr0 is known then the fitting l value can be taken to be the exact average value (Table 1, nr0 ¼ 0:1 and l). It comes out that the obtained values are not identical, except for the interpolation knots where the differential Eq. (7) is identical with differential Eq. (9) and this area can be described by the model of general order kinetics. These disagreements around knots are larger for that part of the curve where nr and r are of the same order of magnitude, particularly for the lower values of nr when traps are less occupied. The explanation of these discrepancies lies in the fact that the model of general order kinetics is an approximation of the phosphorescence process so that for some experimental phosphorescence curves this process is better described by close values than by the exact ones. However, one can see from Table 1 that these close values are questionable and that the deviations from the exact values do not influence the general physical picture and explanation of the phosphorescence process. If the general order kinetics model can approximately describe the phosphor specimen then it is possible to improve the method of calculation of the phosphorescence parameters. In the case where one parameter is fitted and the other is already known, the following relationship can be used to determine l:
Z 1
0 0
Ir ¼ exp l ln Ir ts dts
: ð19Þ ts
Here, one should know only Ir as a function of temperature ts and then nr0 is determine by fitting the function (7) to the experimental curve. This method of calculation gives more accurate values for nr0 and l (Table 1, lf 2 and nr0f 2 ) than the method of direct fitting (Table 1, rl1 , rn1 , rl2 and rn2 ).
317
6. Conclusions The phosphorescence, which for ideal phosphors is described by the band model with two types of competing capture centers (traps and luminescence centers), can be approximately described by the model of general order kinetics. According to the model, each phosphorescence curve can be approximated to a constant value of the kinetics order parameter l. It is shown that this constant value is the average value of l for the time interval in which the phosphorescence curve is observed. Based on such a definition of the order of kinetics, general Becquerel relation can be derived which approximates the phosphorescence curve. Its practical significance has been discussed in a series of papers [4,10,11]. The model of general order kinetics shows that all processes of phosphorescence decay tend to become stable, which is physically determined by domination of one type of capture centers, the traps. Stable flow of the phosphorescence process is defined by exact and constant value of parameter l at any moment. It is an asymptotic value too. Eq. (7) has an analytical solution in that point. Stable flow of the process, for ideal phosphors, is determined by the second order kinetics model and by the value l ¼ 2, which is the limiting case of the general order kinetics. The process stability is determined by the order of kinetics l via f ðnr ; rÞ function, and depends on the deviation from the value l ¼ 2. Therefore, l is a dynamic parameter that changes during the phosphorescence process, characterizing its stability. We may thus conclude that the model of general order kinetics is not only a mathematical formalism, but an explanation of some characteristics of the TL relaxation kinetics as well, and so it enables a better understanding of the kinetics of this process, and thus its practical and theoretical importance. The general order kinetics model, besides better understanding of the physical process enables better calculation of the parameters of the phosphorescence processes. In the general case with phosphors which besides traps and luminescence centers also contain inactive deep traps, phosphorescence pro-
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cesses described by the same order of kinetics as the processes in ideal phosphors have different physical interpretations [11]. These phosphors have the asymptotic value of the order of kinetics equal 1.
[2] [3] [4] [5] [6] [7]
Acknowledgements [8]
The authors are grateful to Professors D. $ ajkovski and Z. Ikoni!c for careful reading and C useful comments on the manuscript.
[9] [10] [11]
References [12] [1] E.I. Adirovitch, Nekotorie voprosi teorii lyminescencii kristallov, GITTL, Moskva, 1956, p. 148.
[13]
A. Halperin, A.A. Braner, Phys. Rev. 117 (1960) 408. H.A. Klasens, M.E. Wise, Nature 158 (1948) 483. E.I. Adirovitch, J. Phys. Radiat. 17 (1956) 705. C.E. May, J.A. Partridge, J. Chem. Phys. 40 (1964) 1401. M.S. Rasheedy, J. Phys: Condens. Matter 5 (1993) 633. Z. Vejnovi!c, M. Pavlovi!c, D. Risti!c, M. Davidovi!c, J. Lumin. 78 (1998) 279. J.T. Randall, M.H.F. Wilkins, Proc. R. Soc. A 184 (1945) 356. G.F.J. Garlick, A.F. Gibson, Proc. R. Soc. A 188 (1947) 485. M.V. Fok, Vvedenie v kinetiku lyminescencii kristallofosforov, Nauka, Moskva, 1964, p. 40. R. Chen, Y. Kirsh, Analysis of Thermally Stimulated Process, Pergamon, Oxford, 1981. C.M. Sunta, R.N. Kulkalni, T.M. Piters, W.E. Feria Ayta, S. Watanabe, J. Phys. D 31 (1998) 2074. R. Chen, J. Electrochem. Soc. 116 (1969) 1254.
An approximation of phosphorescence decay kinetics of ideal phosphors by a general order kinetics model Z. Vejnovic! a, M.B. Pavlovic! b, M. Davidovic! b,c,* b
a Institute of Security, Kraljice Ane b. b. 11000 Belgrade, Yugoslavia Vinca Institute of Nuclear Sciences, P.O. Box 522, 11001 Belgrade, Yugoslavia c Faculty of Electrical Engineering, P.O. Box 816, 11000 Belgrade, Yugoslavia
Received 27 June 2000; accepted 7 February 2001
Abstract The phosphorescence of ideal phosphors has been investigated using a general order kinetics model. The regions where the model describes the phosphorescence processes exactly and approximately have been identified. In order to characterize these processes, the mean value of the kinetics order has been defined for the part of the curve of immediate interest. It has been shown that the value of the order of kinetics for ideal phosphors in the transition region cannot have a unique physical interpretation. The general order kinetics model enables theoretical explanation of some important physical characteristics of TL processes for the ideal phosphors. For the theoretical curves, obtained by the band model, the parameters of phosphorescence curves have been determined using the general order kinetics model. The results differ from the exact values by 3%. We conclude that the general order kinetics model enables a better understanding of the physical processes, and an approximate determination of the phosphorescence decay parameters. This points out the theoretical and practical value of the model. # 2001 Elsevier Science B.V. All rights reserved. PACS: 78.60.K Keywords: Thermoluminescence; Phosphorescence processes; The general order kinetics model; Retrapping factor
1. Introduction The simplest band model (BM) when explains the mechanism of TL relaxation must include the luminescence centers and traps. If in an ideal case all luminescence centers and traps are assumed to be identical, then the model describes a system *Corresponding author. Vinca Institute of Nuclear Sciences, P.O. Box. 522, 11001 Belgrade, Yugoslavia. Tel.: +381-11-458222; fax: +381-11-344-0100. E-mail address: [email protected] (M. Davidovic´).
known as ideal phosphor [1]. The general theory of luminescence decay process, explained by the energy band model, is mathematically complicated and cannot be solved for the general case. Starting from the band model [2] and taking that n þ nc ¼ pl , a general equation can be obtained [3]: 1 d2 pl 1 dpl 2 gt þ 1 þ pl dt2 p2l dt gl 1 dpl gl gt pl þ gt N þ S þ gl Spl ¼ 0; ð1Þ þ pl dt where pl is the concentration of holes occupying
0921-4526/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 0 4 9 1 - 4
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the luminescence centers, nc is the concentration of electrons in the conduction band, n is the concentration of electrons occupying the traps, N is the trap density, gl and gt are capture probabilities for electrons by luminescence centers and traps, respectively. Here S ¼ s expðE=kTÞ where s is the frequency factor, E is the trap activation energy, T is the material temperature and k is the Boltzmann constant. Klasens and Wise [3] have solved this equation for a special case of the second order kinetics when gl ¼ gt ¼ g. In this case, Eq. (1) becomes considerably simpler and the solution is
Taking S ¼ 0, an equation is obtained with solutions describing the first step of the decay process which is temperature independent. Solutions are exact and shown in parametric form: p gt =gl p nc l l ¼ C1 ð5Þ þ 1; N N N and Z dpl ¼ C2 gl Nt: ð6Þ gt =gl pl C1 pl =N þpl =N 1 The integration constants C1 and C2 depend on the initial conditions of the density of electrons nc0
gL þ g nc0 =pl0 L exp½ðgN þ S Þt I ¼
2 ; 1=pl0 þ gLt þ g nc0 =pl0 L =gN þ S ð1 exp½ðgN þ S ÞtÞ
where S ; L¼ gN þ S
ð3Þ
while nc0 and pl0 are the initial concentrations of electrons in the conduction band and empty luminescence centers, respectively. Klasens and Wise have noticed that the decay curves, obtained for various values of excitation, consist of two parts. The first part of the curve is temperature independent and results from the concentration of electrons in the conduction band which is higher than the quasistationary value characterizing the phosphorescence process. This step of the process is very short and takes place immediately after the end of excitation. Its duration varies from 108 up to 102 s [1,3]. The second part of the curve obtained by neglecting exponential terms is strongly temperature dependent. This step of the decay process is known as phosphorescence. Starting from these facts, Adirovich [1] has found the solutions for the first and the second step of luminescence decay process. After a rearrangement, Eq. (1) can be expressed as follows: dnc g t g t nc N 1 1 ¼1 þ þ þS : ð4Þ g l n c g l pl dpl g l g l pl pl
ð2Þ
in the conduction band and density of empty centers pl0 . For the second step of the process Adirovitch, has obtained the exact solutions also shown in parametric form: dn gl S n2 ; I ¼ ¼ dt gl n þ gt ðN nÞ and t¼
1 g n0 g N 1 1 1 t ln þ t : S gl n g l S n n0
ð7Þ
ð8Þ
Phosphorescence intensity is obtained by substituting the value of concentration n from the first into the second equation. Adirovitch denoted the resulting solutions the quasistationary ones [1] and the corresponding region the quasistationary region.
2. General order kinetics model In the general case, where neither of the two basic processes (luminescence and retrapping) is dominant, the models of the first and the second order kinetics cannot be applied. For the general case of isothermal TL relaxation (see discussion in Ref. [4]), Klasens and Wise have suggested an
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311
empirical differential equation instead of Eq. (7) to describe the process of phosphorescence decay in accordance with the ‘‘multimolecular’’ mechanism. Later on, May and Partridge [5] have used the model of general order kinetics expressed by such a differential equation to explain the TL glow curves which cannot be described by the models of the first and second order of kinetics. In order to give a better physical interpretation, Rasheedy [6] proposed a correction of the empirical differential equation, which can be expressed as: I ¼
dn S l ¼ n: dt N l1
ð9Þ
As shown in Ref. [7], this is an empirical interpolation equation which enables the calculation of an analytical approximate solution for the cases which cannot be described by the models of the first [8] and the second order kinetics [9]. In Eq. (9), the order of kinetics l is constant. In this equation, the constant r is substituted for l so that the differential equation has an analytical solution. In reality, however, l is not constant and depends on the initial conditions as has been shown for the curves close to the first and second order kinetics [1,10]. This is particularly prominent in the general order kinetics model, since, during the process of thermal relaxation there is a significant change in the ratio of parameters r to n0 , and hence in l as well. Therefore, the constant value of the parameter l in the model of general order kinetics is in fact an average value of this parameter for the part of the phosphorescence curve being examined. Solving the differential Eq. (9) and substituting the value of n0 for t ¼ 0, a generalized empirical Becqurel formula for the isothermal TL emission Ir ¼ dnr =dts at a given moment ts ¼ St is obtained: l=l1 Ir ¼ ½ðl 1Þts þ n1l ; r0
ð10Þ
where nr ¼ n=N and nr0 ¼ n0 =N. This equation, according to the general order kinetics model approximates the isothermal curve described by Eqs. (7) and (8). By analyzing particular solution (2), for the second order kinetics model, one can see that for different values of nc0 and pl0 , all the curves merge into one. It means that all the curves tend to one
Fig. 1. The luminescence decay curves for different retrapping factor values r ¼ gt =gl , parameter relations gl N=S ¼ 50, and the initial conditions nc0 =N ¼ 0:1 and pl0 =N ¼ 1. Broken lines represent the curves of the second order kinetics model.
limiting shape. This third part of the phosphorescence curve, which appears after sufficiently long time, characterizes emptiness of traps, i.e. n N, and can be called the limiting or asymptotic region. The quasistationary range in that case can be termed a transition range. For the well known models of phosphors: ideal phosphors [1,11], phosphors with deep inactive traps [11], phosphors with order of kinetics 1.5 [5], one obtains the concentration of electrons in the conduction band nc pK l , where K is a rational positive number or zero. Therefore, the third part of the curve can be presented by the model of general order kinetics, or in other words, the phosphorescence process can be shown in a simple way and directly. One can also see that the general order kinetics model describes the limiting case of phosphorescence process, a part of the general solution of this process (Fig. 1).
3. The order of kinetics l in the general order kinetics model Comparing Eqs. (7) and (9) one can see that they are related by [12] nr þ rð1 nr Þ ¼ n2l r :
ð11Þ
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Taking the logarithm of (11) we get: ln½nr ð1 rÞ þ r ¼ 2 f ðnr ; rÞ: l ¼2 ln nr
ð12Þ
From this expression it is clear that l is not constant since it depends on the variable nr . An essential approximation in the model of general order kinetics is that l is independent of nr , that is, of r. For the phosphorescence process, a constant value of l can be obtained for the range of ts values by finding its average value: Rt %l ¼ R0 l dts ¼ 2 f%ðnr ; rÞ: ð13Þ t 0 dts From Eq. (12), it is clear that the average value of l requires the finding of an average value of the function f ðnr ; rÞ: R ts 0 ln½nr ð1 rÞ þ r=ln nr dts % f ðnr ; rÞ ¼ R ts 0 dts R nr nr0 ln½nr ð1 rÞ þ r=ln nr dts =dnr dnr R nr ¼ : nr0 dts =dnr dnr ð14Þ When the distribution function dts =dnr is found from Eq. (8) and substituted into Eq. (14) one obtains the final expression for the calculation of the average value of f ðnr ; rÞ for the interval of values of nr in which the phosphorescence is observed.
f%ðnr ; rÞ ¼
R nr nr0
always be found is of practical interest. The changes in function f ðnr ; rÞ are generally small within the nr range. The exception is that part of the curve where nr ! 0. This part of the curve, however, corresponds to the case where traps are almost depleted and therefore the intensities of phosphorescence in this region are very low. If this part of the curve is outside the observed range or if its influence is neglected, then we can take that f%ðnr ; rÞ f ðC; rÞ. In this expression C has a constant value in the range of r from 0 to 1, for the initial conditions of the observed region determined by nr0 , and by the final value nr , i.e. by the time interval in which the phosphorescence is observed. We conclude that C does not correspond to the average value of nr since, as can be seen from the above expression, the average value of nr does depend on r. An increase in the retrapping factor r leads to a slowdown of phosphorescence and to a decrease in the average value of nr . However, by calculating the average value of nr , it can be shown that this is a slowly changing function in the interval of r from 0 to 1. Therefore C is in fact an average value of nr which gives the best description of phosphorescence in the interval of r from 0 to 1, for the given initial condition nr0 and specified time interval. The value of C can be determined as follows. The differential Eq. (7) is solved for some values of the parameters r and nr0 . If this is done for a large
ln½nr ð1 rÞ þ r=ln nr 1=n2r þ ð1 rÞ=nr dnr R nr : 2 nr0 1=nr þ ð1 rÞ=nr dnr
The integrals in Eq. (15) for the arbitrary value of r in the interval from nr0 to 0 cannot be found analytically. Difficulties arise from the upper limits of the integral since the functions go to infinity when nr ! 0. Physically, this corresponds to the fact that the traps cannot become completely depleted during the isothermal relaxation. With the depleted traps, however, the probability of retrapping increases with respect to the probability of luminescent recombination. Therefore, only the range of phosphorescence within the limits from nr0 up to nr , for which numerical solutions can
ð15Þ
number of various values of r, a set of isothermal TL relaxation curves is obtained, which can be fitted by using expression (10). The value of nr0 is taken to be constant when fitting the parameter l. In this way a dependence of l on r is obtained (Fig. 2). By fitting the obtained curve to the function: l ¼2
ln½Cð1 rÞ þ r ¼ 2 f ðC; rÞ; ln C
ð16Þ
the value of parameter C is found. A constant value of l is always obtained by applying this
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Fig. 2. The dependence of the order of kinetics l on the retrapping factor r. The solid line is obtained by fitting the order of kinetics l using Eq. (10), with numerical solutions of the differential Eq. (7) for the exact value of nr0 ¼ 0:1. The parameter C is obtained by fitting the resulting curve to Eq. (16) (dotted line).
procedure for any constant value of nr0 , as a function of r. In the example shown in Fig. 2 we find C ¼ 0:032. The fitting was performed using the chi square method without weight factor when summing the squares. In this case, the sum of squares is computed using the expression Sðyi fi Þ2 , where yi is the experimental value of the intensity of phosphorescence at some point i, while fi is the value of the fitting function. It means that the highest TL emission values have the strongest influence on the fitting process of the experimental or simulated curve to the assumed function of the model. In practice this means that, in the case of a decrease in the TL emission by two orders of magnitude, any further increase in the range of nr , where the TL relaxation is observed, has a negligible influence on the change of the obtained values of l. Assuming that close to the end of the curve the phosphorescence proceeds by second order hyperbola, we may introduce the weighting factor in the process of chi square fitting for each point si ¼ yi , and get the sum of squares Sððyi fi Þ=si Þ2 . When using this method, all the points are fitted with the same relative error and for l we get the values closest to those computed by Eqs. (13) and (15). Thus we show that the parameter C depends
313
Fig. 3. Interpolation function (broken line) and exact values of the expression in the denominator of Eq. (7) (solid line). The parameter nr0 ¼ 0:1.
strongly on the method of fitting and on the interval of time for which the curve is observed, as shown by the theoretical expressions. Substituting Eq. (16) into Eq. (11) we get the form of the interpolation function which approximates the expression in the denominator of Eq. (7): nr ð1 rÞ þ r nrf ðC; rÞ :
ð17Þ
For the values of parameters of the phosphorescence curves, given as an example in Fig. 2, the ratio of these two functions is graphically presented for a few characteristic values of the parameter r (Fig. 3). The figure shows, how the interpolation is made by the model of general order kinetics, using the values of the function f ð0:032; rÞ, with the parameter r in the range from 0 to 1 when nr0 ¼ 0:1.
4. Discussion For the transition region, relationship (12) is only an initial one in the process of defining the order of kinetics parameter using the general order kinetics model, the value of the order is taken to be constant during the TL process. We consider the method by which the constant l is obtained to be essential for the physical interpretation of this
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Fig. 4. The dependence of the kinetics order l on the initial condition nr .
parameter. The constant value of l can be defined as an average value of the function (12), which varies during phosphorescence process. This is an approximation as one can easily see. Leaving its quality aside for the moment, we have focussed our attention to the physical essence of the parameter l in the model of general order kinetics. Therefore, although we can conditionally take that the function (12) describes the value of the kinetics order, l is constant (Eq. (16)) throughout the phosphorescence process. Using Eq. (16) the functional dependence of l on r can be found in the entire region of l which is of practical interest (0.7, 2.5) [13]. Fig. 4 shows the curves for various values of the initial condition nr0 . If we examine these curves and look at the left and right hand side from the point r ¼ 1, the following can be concluded with a decrease in nr . The parameter l approaches the value of 2, which is its limiting value, the order of kinetics l > 2 exists only if r > 1, and kinetics order l52 appears for r51. On the left, as well as on the right hand side of the point r ¼ 1, depletion of traps leads to an absolute domination of the retrapping process but also to a less influence of the retrapping factor r and to an increase in depleted trap influence on the retrapping process. On the left hand side this means an increase of the retrapping probability, and on the right hand side a decrease of the retrapping probability with respect to its initial
value. This leads to asymptotic characteristics of phosphorescence processes [10]. Following the excitation with different intensities, the decay curves approach each other as the time goes on. For an ideal phosphor the relaxation processes are close to the process described by the second order kinetics. We can define the stable kinetics of the phosphorescence process, which should be understood as a process where l is constant and as a process to which all the other processes of phosphorescence decay asymptotically approach. For the ideal phosphors, the process flowing according to the second order kinetics model can be defined as a stable one. Phosphorescence process in these systems which is described by the model of the second order kinetics tends to go on unchanged. Those processes having the order of kinetics different from two tend to change into those, which can be described by the model of second order kinetics. For phosphorescence curves of ideal phosphors the region outside the second order kinetics represents the region of transition regime. The facts discussed above have been proved experimentally for some phosphors [10]. Assumed quasistationary electron density in the conduction band leads to a stable phosphorescence process. Only in two cases, the first and the second order kinetics, the process of phosphorescence can be observed to follow from the beginning to the end an unchanged law of decay. For the first order kinetics, one can see that the electron density in the conduction band nc ¼ S=gl is constant throughout the phosphorescence. The stable phosphorescence is a consequence of the constancy of nc . In the case of phosphors there is always a finite number of occupied traps, so it is clear that at the end of phosphorescence nc cannot remain constant. A detailed analysis is given in Ref. [1]. We conclude that at the end of phosphorescence process, taking place according to the first order kinetics model, the quasistationary electron density nc in the conduction band no longer holds. Therefore, in this case, we have a quasistable phosphorescence process. For the second order kinetics, the process proceeds in a stable manner due to the constant density ratio of electrons in the conduction band ðnc ¼ Sn=gt NÞ and in traps. This means that we
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have a stable process obeying the same law to its very end. From Eq. (16), we now conclude that the constant, having the value of 2, determines the stable character of the TL relaxation, and the function f ðC; rÞ is an average value of the deviation from the stable process for the initial condition nr0 . Therefore, the kinetics order l is the measure of the stability of TL relaxation process in respect to the initial condition nr0 , or in respect to the retrapping factor r in the TL relaxation region. From Eq. (16) derived above and Fig. 4, we can see that the conditions for the first and the second order kinetics can be simply expressed by r ¼ 0 and 1, respectively. It may be assumed that the given models can be approximately used in the neighborhood of these points. The condition r nr , defining the first order kinetics process, can simply be shown to be a close proximity of the point r ¼ 0. Since N > n we must have gl gt and this is r 0. The same holds for the condition r nr , which defines the second order kinetics process. However, since this is the region of stable processes, where even greater changes in the parameter r do not necessarily change the order of kinetics l, it may happen that for larger deviations from the point r ¼ 1 the phosphorescence process proceeds according to the second order kinetics. The condition r nr may hold even for some point r1 which is far from r ¼ 1, and this means that at this point the phosphorescence process can also be described by the model of the second order kinetics. As one can see from Fig. 4, if the phosphorescence can be approximately described by the second order kinetics at the point r1 , for a given initial condition nr0 , then this must be valid at any point 1 > r > r1 if r1 51 or 15r5r1 if r1 > 1. Consequently, the condition r ¼ 1 is the limiting value of the condition r nr . Therefore, the conditions r ¼ 1 and r nr define the range of r in which the phosphorescence processes can be described by the second order kinetics model, and they are interdependent. For a given band model with two types of centers: traps and luminescence centers we get using an approximation with the general order kinetics that l ¼ 1 represents a limit of physically real phosphorescence processes. For l51 we get
315
r51 and it means that gt 50 or gl 50, which is physically impossible. Therefore, the general order kinetics model cannot explain kinetics orders less than one, and an extrapolation of the curve (16), based on these values, for a given band model has no physical meaning. However, since the region for r ¼ 0 is the region of great instability of phosphorescence, it can be observed that a very small increase in both the luminescence centers, not followed by an increase in electron density in traps, and electrons in the conduction band or on excited levels, not arising from the traps, leads to the phosphorescence with the order of kinetics smaller than one. For ideal phosphors, expression (9) can be used to fit the phosphorescence curves with shapes ranging from exponential ðl ¼ 1Þ to hyperbolic decay ðl 1Þ [4]. In order to take account of all these cases of decay, there has to be some interdependence of the parameters in the exponent of relation (9) and the parameters next to the term ts . In the cases of the first and second order kinetics, the exponent in Eq. (10) is constant, hence this exponent and the coefficient ts are independent quantities. Therefore, they can be obtained by a simple fitting of the experimental curves to function (10). For those cases, where the phosphorescence decay is described by the expression having the order of kinetics l ¼ 1 and 2 each part of that curve can be described by Eq. (10). If various parts of that curve are fitted to expression (10), a different value of the parameter l will be obtained for each part. If we now make an approximation that l is constant throughout the phosphorescence decay, then by fitting the experimental curves to function (10) an error appears. This error is minimal in the process of fitting the parameter l and the coefficient ts . All this leads to a deviation of the parameter l and the coefficient ts from their exact values. This deviation depends on how much the parameter l deviates from l ¼ 1 or 2. The errors are larger near l ¼ 1, since we deal with an unstable process. Similar errors are present near l ¼ 2, because here we deal with a stable process. Assuming that the occupancy of traps nr is known, from the fitting of an experimental curve to Eq. (10) and from the value obtained for the
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parameter l, we conclude that there is an agreement between the theoretical procedure, based on general order kinetics, and the simple band model characterizing the ideal phosphor. Furthermore, we conclude that the simple band model of phosphor can explain the general empirical Becquerel law. It can also point out the complexity of the phenomenon of phosphorescence decay when other effects which can influence the phosphorescence are present (luminescence quenching, the presence of traps with various depths, etc). Expression (15) can be used to explain the fact that in the model of general order kinetics, for the processes of TL relaxation stimulated by the heating function R which is a simple hyperbolic function [11], we got analytical solutions. In the region where the distribution function tends to infinity, R tends to infinity as well, and this leads, at least theoretically, to the total depletion of traps.
5. Theoretical and practical aspects of the general order kinetics model With respect to the asymptotic character of the phosphorescence process, observed long ago, the parameter l shows the characteristics of the stability curve in the physical sense. Therefore, this parameter shows the stability of phosphorescence or the distance from the asymptote, which defines stable conditions of the process for the real materials, which can be approximated by the model of general order kinetics. If we take the parameter l to be constant and get back to Eq. (9) and write it in another form, we get Ir ¼ expðljln nr jÞ:
ð18Þ
Here, we can define the parameter l as a constant of the velocity of emptying the traps with respect to variable ln nr for a given part of the phosphorescence curve. This means that the traps will empty
Table 1 r
ts
l
lf 1
rl1 (%)
nr0f 1
rn1 (%)
lf 2
rl2 (%)
nr0f 2
rn2 (%)
0 0.01 0.02 0.03 0.04 0.05 0.06 0.08 0.1 0.125 0.15 0.175 0.2 0.25 0.3 0.35 0.4 0.5 0.6 0.8 1 1.1 1.2 1.3 1.4 1.5
15 20 25 30 50 50 80 80 100 100 100 150 150 175 200 200 250 300 350 500 650 800 900 1000 1100 1200
1 1.0789 1.1307 1.1874 1.2291 1.2700 1.3015 1.3622 1.4076 1.4568 1.5012 1.5409 1.5746 1.6305 1.6781 1.7172 1.7534 1.8129 1.8618 1.9397 2 2.0255 2.0488 2.0702 2.0899 2.1083
1 1.0944 1.1643 1.2197 1.2685 1.3075 1.3453 1.4052 1.4515 1.4954 1.5378 1.5777 1.6072 1.6593 1.7027 1.7379 1.7718 1.8265 1.8715 1.9438 2 2.0238 2.0455 2.0654 2.0838 2.1009
0 1.44 2.97 2.72 3.21 2.96 3.37 3.16 3.12 2.65 2.44 2.39 2.07 1.75 1.47 1.21 1.05 0.75 0.52 0.21 0 0.09 0.16 0.23 0.29 0.35
0.1 0.0944 0.0915 0.0895 0.0873 0.0865 0.0849 0.0835 0.0836 0.0850 0.0848 0.0839 0.0848 0.0860 0.0871 0.0887 0.0894 0.0915 0.0935 0.0968 0.1 0.1025 0.1031 0.1047 0.1063 0.1079
0 5.52 8.47 10.50 12.62 12.62 15.05 16.44 16.39 14.98 15.15 16.01 15.11 13.98 12.86 11.22 10.53 8.45 6.48 3.16 0 2.59 3.18 4.76 6.34 7.90
1 1.0662 1.1171 1.1598 1.1981 1.2303 1.2621 1.3128 1.3576 1.4015 1.4438 1.4835 1.5177 1.5753 1.6256 1.6686 1.7086 1.7759 1.8328 1.9259 2 2.0319 2.0613 2.0884 2.1137 2.1373
0 1.18 1.20 2.32 2.52 3.13 3.03 3.62 3.55 3.79 3.83 3.72 3.66 3.39 3.13 2.83 2.56 2.04 1.55 0.71 0 0.32 0.61 1.12 1.14 1.38
0.1 0.0987 0.0982 0.0979 0.0975 0.0976 0.0972 0.0975 0.0976 0.0985 0.0987 0.0985 0.0987 0.0991 0.0993 0.0997 0.0997 0.0999 0.1000 0.1000 2 0.0999 0.0998 0.0998 0.0997 0.0997
0 1.25 1.79 2.07 2.44 2.37 2.76 2.47 2.37 1.46 1.30 1.49 1.27 0.88 0.63 0.29 0.26 0.03 0.08 0.09 0 0.05 0.11 0.17 0.24 0.30
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more slowly if the order of kinetics is higher. This is correct since in this case the retrapping factor is too large. As far as the calculation of phosphorescence parameters is concerned, only two parameters are of interest for some experimental curves, as one can see from Eq. (9), and they are nr 0 and r. Two cases will be examined. A direct fitting of the variables to the experimental curve (Table 1, lf 1 and nr0f 1 ) and a fitting with one parameter only if the exact value of the other one is known. Eq. (10) can be used for the purpose. If the value nr0 is known then the fitting l value can be taken to be the exact average value (Table 1, nr0 ¼ 0:1 and l). It comes out that the obtained values are not identical, except for the interpolation knots where the differential Eq. (7) is identical with differential Eq. (9) and this area can be described by the model of general order kinetics. These disagreements around knots are larger for that part of the curve where nr and r are of the same order of magnitude, particularly for the lower values of nr when traps are less occupied. The explanation of these discrepancies lies in the fact that the model of general order kinetics is an approximation of the phosphorescence process so that for some experimental phosphorescence curves this process is better described by close values than by the exact ones. However, one can see from Table 1 that these close values are questionable and that the deviations from the exact values do not influence the general physical picture and explanation of the phosphorescence process. If the general order kinetics model can approximately describe the phosphor specimen then it is possible to improve the method of calculation of the phosphorescence parameters. In the case where one parameter is fitted and the other is already known, the following relationship can be used to determine l:
Z 1
0 0
Ir ¼ exp l ln Ir ts dts
: ð19Þ ts
Here, one should know only Ir as a function of temperature ts and then nr0 is determine by fitting the function (7) to the experimental curve. This method of calculation gives more accurate values for nr0 and l (Table 1, lf 2 and nr0f 2 ) than the method of direct fitting (Table 1, rl1 , rn1 , rl2 and rn2 ).
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6. Conclusions The phosphorescence, which for ideal phosphors is described by the band model with two types of competing capture centers (traps and luminescence centers), can be approximately described by the model of general order kinetics. According to the model, each phosphorescence curve can be approximated to a constant value of the kinetics order parameter l. It is shown that this constant value is the average value of l for the time interval in which the phosphorescence curve is observed. Based on such a definition of the order of kinetics, general Becquerel relation can be derived which approximates the phosphorescence curve. Its practical significance has been discussed in a series of papers [4,10,11]. The model of general order kinetics shows that all processes of phosphorescence decay tend to become stable, which is physically determined by domination of one type of capture centers, the traps. Stable flow of the phosphorescence process is defined by exact and constant value of parameter l at any moment. It is an asymptotic value too. Eq. (7) has an analytical solution in that point. Stable flow of the process, for ideal phosphors, is determined by the second order kinetics model and by the value l ¼ 2, which is the limiting case of the general order kinetics. The process stability is determined by the order of kinetics l via f ðnr ; rÞ function, and depends on the deviation from the value l ¼ 2. Therefore, l is a dynamic parameter that changes during the phosphorescence process, characterizing its stability. We may thus conclude that the model of general order kinetics is not only a mathematical formalism, but an explanation of some characteristics of the TL relaxation kinetics as well, and so it enables a better understanding of the kinetics of this process, and thus its practical and theoretical importance. The general order kinetics model, besides better understanding of the physical process enables better calculation of the parameters of the phosphorescence processes. In the general case with phosphors which besides traps and luminescence centers also contain inactive deep traps, phosphorescence pro-
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cesses described by the same order of kinetics as the processes in ideal phosphors have different physical interpretations [11]. These phosphors have the asymptotic value of the order of kinetics equal 1.
[2] [3] [4] [5] [6] [7]
Acknowledgements [8]
The authors are grateful to Professors D. $ ajkovski and Z. Ikoni!c for careful reading and C useful comments on the manuscript.
[9] [10] [11]
References [12] [1] E.I. Adirovitch, Nekotorie voprosi teorii lyminescencii kristallov, GITTL, Moskva, 1956, p. 148.
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