Accuracy of the activation energy
calculated from a thermoluminescence glow-peak using a method that uses three points on the peak
F. O. Ogundare*, ** and M. L. Chithambo
Department of Physics, Rhodes University, P.O. Box 94, Grahamstown 6140, South Africa Received 13 June 2005, revised 28 October 2005, accepted 24 November 2005
Published online 10 January 2006 PACS 78.60.Kn
This paper shows that the limitations of the general-order and peak-shape methods of kinetic analysis in cases where the retrapping probability greatly exceeds the recombination probability may be avoided if the analysis method proposed by Rasheedy [6–8] is applied. Analysis of computer simulated glow-curves using the general-order, peak-shape, and Rasheedy’s protocol showed that Rasheedy’s method gave the best agreement between calculated and default values. The accuracy of the method improved for data se- lected well below 10% of the peak maximum in the initial-rise region of a glow-peak. The observed im- provement has been attributed to the fact that in the initial-rise region, the quasiequilibrium condition is better satisfied than at the higher temperature end of the glow-peak. In general, we find that when the re- trapping probability increases, the analysis for kinetic parameters is better carried out on data selected well below 10% of maximum peak intensity.
© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Introduction
The utility of thermoluminescence (TL) in detecting changes in defect concentration in insulators is well established [1–3]. The underlying premise leading to the emission of TL is that exposure of a material to ionizing radiation causes a redistribution of charge in defect centres within the material. When the mate- rial is heated at a controlled linear rate, the thermoluminescence is emitted as a temperature-dependent set of peaks collectively known as a glow-curve. The shape and intensity of each of the glow-peaks may be characterized by a set of parameters consisting of the activation energy E, the frequency factor s, the number of electrons n0 trapped in defect centres at the start of the heating, and the order of kinetics b. The order of kinetics b is an indication of the retrapping probability i.e. the probability that a free elec- tron from the conduction band will be retrapped rather than recombine with a hole at a recombination centre to produce thermoluminescence. Retrapping of electrons reduces the TL intensity at any particular temperature during the heating process. The physical mechanisms of TL associated with a given glow curve are unique and may be characterized by analysis of the glow-curve for the said kinetic parameters.
There are many methods for analysis of glow-peaks for kinetic parameters including ones based on the use of the peak-shape, the heating rate, as well as methods of computerised deconvolution of the glow- peak [1, 4]. The applicability of some of these methods under various experimental conditions has been investigated and discussed by several authors [1–4]. In particular, Braunlich [4] showed that in cases where the retrapping probability is much higher than the recombination probability and the traps are filled up close to saturation, the initial-rise method for determining the activation energy is unreliable.
* Corresponding author: e-mail: [email protected]
** Permanent address: Department of Physics, University of Ibadan, Ibadan, Nigeria
Sunta et al. [5] using simulated glow-curves also found that when retrapping was dominant, glow-curves could not be properly described using either general-order fitting or peak-shape methods.
Under certain conditions, such as when the retrapping probability is greater than the recombination probability, the shape of the glow-peak may be distorted [5]. Analysing such a distorted glow-peak for kinetic parameters using methods that require the use of the whole or a portion of the glow-peak will inevitably produce erroneous kinetic parameters. However not all parts of the glow-peak will suffer equal distortion. This means that any analysis method that allows for free choice of data points from a glow-peak if applied to a region of a glow-peak with little or no distortion is likely to produce reliable kinetic parameters. With this justification, Rasheedy’s proposed method [6 – 8] for calculating kinetic parameters uses three points each of which may be selected from any position on a glow-peak.
The aim of the present work is to apply the method proposed by Rasheedy [6 – 8] to calculate the acti- vation energy in conditions where the retrapping probability is greater than the recombination probability and to assess the reliability of the method.
2 Rasheedy’s activation energy expression and charge-transfer models 2.1 Rasheedy’s activation energy expression
We first deduce a set of equations that are independent of order of kinetics b. These equations, to be used in calculating the activation energy, are developed from the following three equations, Eqs. (1) – (3), given by Rasheedy [8]:
1
( )x b exp
x b
x
A E
I S
N - kT
Ê ˆ
= ÁË- ˜¯, (1)
1
( )y b exp
y b
y
A E
I S
N - kT
Ê ˆ
= ÁË- ˜¯, (2)
and
1
( )z b exp
z b
z
A E
I S
N - kT
Ê ˆ
= ÁË- ˜¯, (3)
where Ix, Iy and Iz are, respectively, the three intensity values selected on a glow-peak and Tx, Ty and Tz
are their corresponding temperatures.
Dividing Eq. (1) by Eq. (2) one obtains 1 1 exp
b
x x
y y y x
I A E
I A k T T
Ê ˆ Ï Ê ˆ¸
=ÁË ˜¯ ÌÓ ÁË - ˜¯˝˛. (4)
Dividing Eq. (1) by Eq. (3) gives 1 1 exp
b
x x
z z z x
I A E
I =ÊÁËA ˆ˜¯ Ï ÊÌÓk TÁË -T ˆ¸˜¯˝˛. (5)
Taking natural logarithm of Eqs. (4) and (5) gives the following two equations
ln x x y ln x
y x y y
I E T T b A
I k T T A
Ê ˆ- - = Ê ˆ
Á ˜ Á ˜
Ë ¯ Ë ¯ (6)
and
ln Ix E T Tx z bln Ax
I k T T A
Ê ˆ- - = Ê ˆ
Á ˜ Á ˜
Ë ¯ Ë ¯. (7)
Now dividing Eq. (6) by Eq. (7) to eliminate b and rearranging one finds
ln x ln x ln x ln x x y ln x x z ln x
y z z y x y z x z y
I A I A E T T A E T T A
I A I A k T T A k T T A
- -
- = - , (8)
from which the expression for E is
( )
( )ln ln ln ln
ln ln
x x x x
x y z
y z z y
x x
z x y y x z
z y
I A I A
kT T T
I A I A
E T T T A T T T A
A A
È ˘
Í - ˙
Î ˚
= - - -
. (9)
In Eq. (9), Tx, Ty and Tz are the temperatures corresponding to each of the three intensities Ix, Iy and Iz
selected on the glow-peak. For the present work, the intensities Ix, Iyand Iz are selected on the ascending part of the glow-peak, k is Boltzmann’s constant; Ax, Ay and Az represent the areas under the glow-peak from Tx to Tf, Ty to Tf and Tz to Tfrespectively where Tf is the final temperature of the glow-peak. Details about the method have been discussed by Rasheedy [6–8].
One of the key assumptions that led to the general-order TL equations of Rasheedy i.e. Eqs. (1) – (3) above, from which Eq. (9) was obtained, is that TL phosphors consist of one trap and one recombination centre. Equation (9) is thus valid for the one-trap one-recombination centre model. Methods based on the use of Eq. (1) have been successfully applied to analyse glow-peaks [5]. Moreover, Rasheedy [6–8] has verified that the method successfully describes glow-curves that consist of one or more glow-peaks.
Sunta et al. [5] showed that methods based on Eq. (1) are reliable except in cases where the retrapping probability is much greater than the recombination probability and the trap is filled up near to saturation.
It is in these circumstances that we investigate the reliability of Rasheedy’s method.
In this work, glow-peaks are simulated for two types of physical models used by Sunta et al. [5].
These are the one-trap one-recombination centre (OTOR) model and the interactive multi-trap system (IMTS) model. The IMTS model, discussed by Sunta et al. [5], assumes the presence of one active trap (AT), one thermally-disconnected deep-trap (TDDT) and one recombination centre (RC). The activation energy E associated with each of the two models (OTOR, IMTS) is then calculated using Eq. (9).
2.2 One-trap one-recombination centre model
In the one-trap one-recombination centre model, the charge transfer process during the heating stage is described by the following set of differential equations:
( )
d exp
d c n
n ns E n N n A
t = - ÊË-kTˆ¯+ - , (10)
( ) ( )
d exp
d
c c n c c h
n ns E n N n A n n n A
t = ÊË-kTˆ¯- - - + . (11)
The TL intensity I is obtained by
( )
c c h
I n n n A= + , (12)
where N is the concentration of available traps; n the concentrations of filled traps, and nc the concentra- tions of the free charge carriers. E and s are the activation energy and frequency factor of the traps re- spectively. An and Ah are the capture coefficients of the traps and the recombination centres respectively.
Nine glow-peaks were simulated using the same input parameters as used by Sunta et al. [5]. The pa- rameters are listed in Table 1.
Table 1 Input parameters for the numerical glow-peaks from OTOR and IMTS models. The parameters common to all the peaks are E = 1 eV, s = 1012 s–1, N = 1010 cm–3, An = 10–7 cm3 s–1 and heating rate of 1 K s–1. In addition to these, for the TDDT in the IMTS model additional input parameters are M = 1010 cm–3. n0 and m0 are the initial values of N and M before the start of TL read-out heating.
OTOR model IMTS model
peak no.
An/Ah n0/N Am/Ah An/Ah n0/N m0/M
1 10–2 1 1 10–2 1 1
2 10–1 1 1 10–1 1 1
3 1 1 10–1 1 1 6.32 × 10–1
4 10 1 1 10 0.5 6.7 × 10–2
5 102 1 10 102 1 6.32 × 10–1
6 103 1 10 102 1 9.93 × 10–1
7 102 5 × 10–1 10 102 0.5 6.7 × 10–2
8 102 10–1 10 102 10–1 1.05 × 10–2
9 102 10–2 10 102 10–2 10–3
2.3 Interactive multi-trap system model
The differential equations that describe the charge transfer processes in the interactive multi-trap system model are Eq. (10) as well as Eqs. (13) and (14) below
( )
d
dm n M m Ac m
t = - , (13)
( ) ( ) ( )
d exp
d
c c n c m c c h
n ns E n N n A n M m A n m n n A
t = ÊË-kTˆ¯- - - + + (14)
where M and m are the concentrations of thermally-disconnected deep-traps and the filled thermally- disconnected deep traps respectively. The parameter Am is the capture coefficient of the thermally dis- connected deep traps. The TL intensity I is calculated from
( )
c c h
I n m n n A= + + . (15)
Nine glow-peaks were also simulated with this model also using the input parameters of Sunta et al. [5].
The parameters used are also listed in Table 1.
3 Results and discussion
The kinetic analysis method proposed by Rasheedy [6 – 8] was used to calculate the activation energy of the simulated glow-peaks including when the retrapping probability is greater than the recombination probability and to assess the reliability of the method by checking how well the method reproduces de- fault values of activation energy used to generate the test glow curves.
The activation energy values of the simulated glow-peaks were calculated using Eq. (9) for intensities below 1% as well as below 10% of maximum glow-peak intensity (Im) respectively. Values of activation energy so calculated from OTOR-model glow-peaks are listed in Table 2 and from IMTS-model glow-peaks are listed in Table 3. For comparison, the values reported by Sunta et al. [5] from general- order peak fitting and peak-shape methods are also included in both Tables 2 and 3.
Tables 2 and 3 together show that as the ratio An/Ah becomes much larger than 1, that is, as retrapping increases, the E values calculated using the Rasheedy’s method become increasingly less than the true value (the true value being the default value used to generated the glow-peak). When compared with the
Table 2 Calculated values of E (in eV) in the OTOR model (E values for peak-shape and peak-fit meth- ods are taken from Sunta et al. [5].
Rasheedy’s method peak shape method peak fit method peak no.
E (using I < 10% Im) E (using I < 1% Im) Eτ Eδ Eω E
1 1.000 1.000 1.004 0.989 1.006 1.014
2 1.000 1.000 1.024 1.029 1.035 1.065
3 1.000 1.000 1.000 0.986 0.997 1.001
4 0.998 1.000 0.565 0.619 0.595 0.613
5 0.923 1.000 0.470 0.549 0.510 0.487
6 0.499 0.952 0.466 0.556 0.512 0.511
7 1.000 1.000 0.897 0.893 0.897 0.891
8 1.000 1.000 0.986 0.994 0.994 0.987
9 1.000 1.000 0.998 0.994 0.998 1.047
method is closest to the true value. In Rasheedy’s method, the discrepancy between calculated and true value worsens when An/Ah reaches 103, unlike in the case of the other two methods where the effect be- comes noticeable when An/Ah reaches 10.
A possible reason why the results from Rasheedy’s method are not affected as much as the other methods may be due to selection of data from only the initial-rise region of the glow-peaks. One problem with this argument, however, is the fact that the initial-rise method (which also uses the initial-rise re- gion) has been reported to fail when the retrapping probability is much higher than the recombination probability and the dose is near saturation [4]. On this problem, Sunta et al. [5] have pointed out that the initial-rise method fails because the underlying assumption that the intensity increases exponentially up to 10% of Imno longer applies (i.e. that in this region the intensity is proportional to exp(–E/kT)); in fact they [5] showed that the intensity in the initial rise region does not always increase exponentially up to 10% of Imespecially when n0/N > 0.1. The success of Rasheedy’s method [6 – 8] in comparison with the other two methods may therefore be due to the fact that it is not built on a similar assumption i.e. that the TL intensity from Eq. (1) is proportional toexp(–E/kT).
The failure of both the peak-shape and peak-fitting methods in conditions of high retrapping has been attributed to the effect that high retrapping probability and dose in the saturation range have on the shape of the glow-peak [5]. One factor responsible for the distortion in the shape of the glow-peak in cases of high retrapping probability is that the frequency factor (rate of release of charge from trap) measured at any given temperature may be less than the actual value because some of the charges emitted at this Table 3 Calculated values of E (in eV) in the IMTS model (E values for Peak-shape and peak-fit meth- ods are taken from Sunta et al. [5].
Rasheedy’s method peak shape method peak fit method peak no.
E (using I < 10% Im) E (using I < 1% Im) Eτ Eδ Eω E
1 1.000 1.000 0.940 0.900 0.940 0.990
2 1.000 1.000 0.990 0.990 1.000 0.990
3 1.000 1.000 0.970 0.990 0.990 0.980
4 1.000 1.000 0.860 0.890 0.880 0.850
5 0.979 1.000 0.470 0.500 0.480 0.500
6 0.968 1.000 0.450 0.510 0.470 0.470
7 1.000 1.000 0.810 0.850 0.830 0.890
8 1.000 1.000 0.990 0.990 1.010 0.990
9 1.000 1.000 0.990 0.990 1.000 0.990
temperature are retrapped. Calculation of the frequency factor relies on the intensity of TL at that par- ticular temperature. On the other hand, the TL intensity does not account for all the charges released at the temperature but rather only those that recombine with holes in the luminescence centre to produce luminescence. For this reason, one expects the frequency factor to decrease as temperature increases from the initial-rise region to the end of the glow-peak because the retrapping probability increases that way. In such a situation the activation energy determined will not accurately represent the actual value, since its determination includes TL intensities in parts of the glow-peak affected by high retrapping. It follows therefore that any method (for example, general-order fitting method, peak-shape method) that uses intensities and/or temperatures from part or the whole of a glow-peak distorted due to high retrap- ping effects will produce values of E that may differ significantly from the true ones.
In order to get a more reliable E value, it then requires one to use intensities and/or temperatures from a region of the glow-peak where retrapping effect is minimal. Generally, it is in the initial-rise region of the glow-peak that the retrapping effect is minimal. Retrapping decreases when density of empty traps decreases.
Rasheedy’s method which allows for three data points to be taken from anywhere on the glow-peak can then be applied to the region where there is least distortion. The peak-shape method used by Sunta et al. [5], which also uses three points, requires the use of the temperature at maximum intensity and two other temperatures at half-maximum intensity on both sides of the glow-peak. Therefore in the case of the Rasheedy’s method [6 – 8], the accuracy of E calculated can be improved by choosing the points in the initial-rise region of the glow-peak (where intensities are ≤10% Im) as shown in Tables 2 and 3. By using the Rasheedy’s method with points selected in this initial-rise region for peaks 4 and 5 of the OTOR-model glow-peaks (Table 2) as well as peaks 5 and 6 of the IMTS-model glow-peaks (Table 3), the activation energy values obtained are now very close to the true one. Tables 2 and 3 also suggest that when the retrapping probability increases relative to recombination probability, the three points should be selected further below 10% of Im in the initial-rise region. For example, as shown in Table 2, for the highest ratio of retrapping to recombination probability (103) considered in this work, it required the points to be chosen in the initial-rise region where intensities are ≤1% Im for reliable values to be ob- tained.
In experimental application of Rasheedy’s method, one therefore needs to control the background component of the measured TL well enough to enable the use of intensities down to 1% of Im in the ini- tial-rise region. In this way, the effect of glow-peak distortion on the accuracy of E can be totally avoided even for a trap with very high retrapping probability. One way of ensuring that the TL intensities in the initial-rise region are above the equipment background is to increase trap occupancy by using high doses.
The use of a glow-peak with high dose will also minimize the effect of retrapping in the initial-rise re- gion of the glow-peak.
Most methods of kinetic analysis assume that the quasiequilibrium condition is satisfied [9, 10]. For- tuitously for Rasheedy’s method [6 – 8], the quasiequilibrium condition is better satisfied in the initial- rise region of the glow-peak than at its higher temperature side [9, 10]. In fact, Sunta et al. [11] have shown that departures from the quasiequilibrium condition become serious when the ratio of retrapping probability to recombination probability increases, an incidence important on the higher temperature end of the glow-peak.
4 Conclusion
The reliability of the kinetic analysis method proposed by Rasheedy [6 – 8] has been compared to that of the general-order peak-fit and peak-shape methods [5]. Each of the three methods was used to calculate the activation energy from computer-simulated glow-peaks generated using the one-trap-one- recombination centre model and the interactive multi-trap system model [5]. The reliability of each me- thod was assessed by how well the method reproduced default values of activation energy used to gener- ate the test glow-peaks. We found that the limitations experienced by the general-order and the peak-
Rasheedy’s method. The accuracy of the method improves for data selected well below 10% of the peak maximum in the initial-rise region of a glow-peak because the quasiequilibrium condition is better satis- fied there than at the higher temperature end of the glow-peak.
Acknowledgement This work was carried out during F.O.Ogundare’s post doctoral stay at Rhodes University in South Africa. F. O. Ogundare therefore acknowledges the postdoctoral fellowship support from Claude Harris Leon foundation (2005 – 2006). The support received from Rhodes University is also acknowledged.
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