ABSTRACT
The aim of this study is to give some expressions able to give a criteria of acceptance for the activation energy values determined through the so called peak shape methods. The expressions are derived for both first and second order kinetics. Furthermore, using approximations concerning the peak temperature parameters, the reliability expressions are simplified for getting a more quickly criteria of acceptance. A table lists several data, from literature, concerning the activation energy determined for various Thermoluminescence materials; the experimental values are then tested using the criteria of acceptance showing the goodness of the method here presented in this research.
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How to cite this article
DOI: 10.3923/jas.2005.1340.1344
URL: https://scialert.net/abstract/?doi=jas.2005.1340.1344
INTRODUCTION
An important and widely used method for investigating the trapping levels in crystals is based on the Thermoluminescence (TL) technique. One of the most convenient way, among the various TL methods introduced during the years, is based on the geometrical characterization of a TL glow peak, i.e. the very well known Peak Shape (PS) methods. In fact, for calculating the activation energy of the trapping level corresponding to a peak in the glow curve, one needs to measure three temperature values on the peak itself: the temperature at the maximum, TM and the first and second half temperatures, T1 and T2, respectively on the ascending and descending parts of the peak.
The formulas proposed for finding the activation energy usually include the following factors:
In the following is given a list of the various expressions, for both first and second order of kinetics, allowing for the activation energy determination. All the expressions have been modified by Chen[1] for getting a better accuracy in the E values.
Grosswiener[2]:
(1) |
(2) |
The coefficient 1.41 in Eq.1 was computed by Dussel and Bube[3].
Lushchik[4]:
(3) |
(4) |
Halperin and Braner[5]:
(5) |
(6) |
Where, ΔM = 2kTM/E.
Both Eq. 5 and 6 require an iterative process owing the presence of the quantity ΔM. To avoid this difficulty, Chen modified them as following:
(7) |
(8) |
Chen also gave two more expressions based on the w factor:
Chen's additional expressions:
(9) |
(10) |
Balarin: Balarin[6] gave expressions based on the three temperatures T1, T2 and TM:
(11) |
(12) |
Chen's expressions (general): The previous methods were summed up by Chen[7], who considered a general order kinetics, l, ranging from 1 to 2, then giving the possibility of non-integer values for the kinetics order. The general expression is
(13) |
Where, αis t, δ or w. The values of cα and bα are summarized as:
with
being μ = 0.42 for a first order kinetics and μ = 0.52 for a second order.
The previous general expression, developed just for the 1st and the 2nd orders, gives:
1st Order
(14) |
(15) |
(16) |
2nd Order
(17) |
(18) |
(19) |
Comparison among the peak shape expressions: The variability of the E values determined using the various peak shape expressions was checked at first.
Synthetic glow curves following a first or a second order kinetics were generated for various values of E, keeping s constant and then for various values of s, keeping then E constant. After that, to the so generated glow curves the various peak shape methods have been applied.
It was observed that the E values calculated by the H-B expressions are always bigger than the other values. Furthermore, values more closed to the assigned E are obtained by the G, L and Cδ expressions.
Reliability expressions: From a theoretical point of view the various PS method do not give the same values for E because, mainly, the expressions were obtained according to different theoretical procedures as well as different approximations; furthermore, they are based on specific characteristics of the glow peak. Therefore, it seems to be correct to compare the E values obtained from expressions based on the same peak parameters, so that the relations obtained in this way could act as a test of acceptance for the E values.
The expressions so far given have been handled for getting a criteria of reliability of the E values obtained using the PS methods. In most of the cases the Chen's expressions have been used as reference because they have a more general meaning respect to the others and also give more accurate values of E.
1st Order
(20) |
(21) |
Because the presence of the term ΔM in the last expression, we need to calculate its limits of validity :
and then
(22) |
In this case also it is necessary to calculate the limits of validity of the expression. Then, we obtain:
and then
(23) |
2nd Order
(24) |
(25) |
Limits:
so that
(26) |
limits:
so that
(27) |
The expressions containing the parameter w are based on the three temperature values T1, T2 and TM. For getting the reliability expressions in this case, it is better to have only two temperature values, as in the previous expressions. For this purpose, one can consider the asymmetrical and symmetrical properties of a glow peak following a first or a second order kinetics respectively. Then, using the synthetic glow-peaks previously generated, some relations among the three temperatures T1, T2 and TM can be obtained. As a convenient first approximation, one may take 0.95 TM for T1 in both first and second order peaks, T2 = 1.036 TM and T2 = 1.054 TM, respectively for first and a second order peaks.
The following expressions can be then obtained:
1st order
from which, using the approximation T1 = 0.95 TM, the following expression, as a function of TM and T2, is obtained:
(28) |
Using the approximation T2 = 1.036 TM, the following expression is also obtained:
(29) |
2nd order
from which, using the approximation T1 = 0.95 TM, we get
(30) |
As before, using now the approximation T2 = 1.054 TM, the expression as a function of TM and T1 is obtained:
(31) |
Approximated expressions: Using the approximations of T1 and T2 as a function of TM, the previous expressions of reliability can be furthermore simplified.
1st order
(32) |
(33) |
(34) |
(35) |
2nd order
(36) |
(37) |
(38) |
(39) |
Table 1: | The goodness of the experimental values of E, determined using the peak shape methods, is tested according the reliability expressions given in the text. The columns calculated limits and approximated limits give the range of acceptance for the expected values of the activation energy. The column experimental ratios lists the ratios calculated using the experimental values of E |
Reliability of the method and conclusions: Table 1 lists some activation energy values from literature as well as the results obtained using the present method. The tenth column lists the limits of reliability as obtained inserting the glow peak temperatures in the various expressions. The experimental ratios are obtained as the ratios of the calculated E values. For instance, it can be observed that the (EL)δ value, calculated for the 5th peak in CaF2: Tm, is much lower compared to the other E values for the same peak. The experimental ratio (EL/EC)δ is very low, about 15% less than the calculated limit using the peak temperatures data. In this case, it seems to be correct to reject the activation energy data obtained using the Lushchik expression. In general, the E values calculated for the different materials listed in Table 1 are in good agreement among them, according to both calculated limits and experimental ratios.
The column of approximated limits shows the limits obtained using the values of T1 and T2 as function of TM. Also in this case it can be observed a quite good agreement between the experimental ratios and the approximated limits.
The method here presented gives a good way for testing the reliability of the experimental activation energy findings.