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Impact of non-fulfillment of the super position principle on the analysis of thermoluminescence glow-curve
Accepted Manuscript Impact of non-fulfillment of the super position principle on the analysis of thermoluminescence glow-curve A.M. Sadek, G. Kitis PII:
S1350-4487(18)30247-6
DOI:
10.1016/j.radmeas.2018.06.016
Reference:
RM 5938
To appear in:
Radiation Measurements
Received Date: 3 April 2018 Revised Date:
23 May 2018
Accepted Date: 16 June 2018
Please cite this article as: Sadek, A.M., Kitis, G., Impact of non-fulfillment of the super position principle on the analysis of thermoluminescence glow-curve, Radiation Measurements (2018), doi: 10.1016/ j.radmeas.2018.06.016. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Impact of non-fulfillment of the super position principle on the analysis of thermoluminescence glow-curve A.M. Sadeka* and G. Kitisb a
Ionizing Radiation Metrology Department, National Institute for Standards, El-Haram, Giza, Egypt Nuclear Physics and Elementary Particles Physics Section, Physics Department, Aristotle University of Thessaloniki, 54124 Thessaloniki, Makedonia, Greece
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b
Abstract
The analysis of complex thermoluminescence (TL) glow curves in its individual peaks is achieved by a
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computerized glow curve deconvolution (CGCD). The correct application of the CGCD requires the fulfillment of the superposition principle (SP), which postulates that the TL peaks have to be independent of each other. In
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the present work we simulate the application of the CGCD using interactive phenomenological models in which the SP is not fulfilled. The thermoluminescence (TL) processes of four active traps and one thermally disconnected deep trap (TDDT) was simulated over a wide range of doses and under different competition cases. The TL glow-curves were analyzed by the CGCD algorithm and the accuracy of the kinetic parameters and the dose dependence of each trap were investigated over the doses. In most of the simulated cases, the nonfulfillment of the superposition principles has no significant effect on the accuracy of the computed kinetics
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parameters. Different behaviors of dose response curve were discussed including sub-linear and supra-linear dose dependence. It has been also found that in the case of presence of competitions among the active traps, an almost linear dose response curve can be observed for the last glow-peak in a series of overlapping glow-peaks.
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Keywords
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Theory of thermoluminescence; Superposition principle; Glow-curve analysis.
Corresponding author
[email protected] (Amr M. Sadek)
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ACCEPTED MANUSCRIPT 1. Introduction The analysis of complex experimental thermoluminescence (TL) glow curves in their individual peaks can be achieved by a computerized glow curve deconvolution (CGCD). The CGCD analysis can provide the
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integrals of the individual peaks for dosimetric characterization of materials and the kinetic parameters of electron traps for energy level characterization (Horowitz & Yossian, 1995). The correct application of the CGCD requires the fulfillment of the superposition principle (SP), which postulates that every TL peak has to
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be independent of the other peaks.
In physics, the superposition principle states that for all linear systems, the net response at a given place and
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time caused by two or more stimuli is the sum of the response that would have been caused by each stimulus individually (Wolfson, 2016). Mathematically, for input
and response
= ( ), the superposition of the
input should yield the superposition of the respective responses, i.e., (
+
+⋯
) = ( ) + ( ) + ⋯+ (
)#1
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The energy band theory in solids is used to create phenomenological models to describe the TL processes. These phenomenological models consist of set of differential equations, which during irradiation
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stage describe the traffic of electrons form conduction band to traps and of holes from valence band to luminescence centers. During heating stage, they describe the thermal release of electrons to the conduction
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band and their re-trapping or recombination. In both stages the electron traps compete with each other in trapping electrons from the conduction band and altogether compete with the recombination to luminescence centers. All these competitions in the models act against the fulfillment of the SP defined above. Therefore, it is a clear theoretical prediction that under competition effects among traps, the SP is not fulfilled. In TL research, many researches adopted the applications of CGCD in different radiation dosimetry fields, which then was generalized and grown up very fast. However, others continue to oppose to the CGCD due to the non-fulfillment of the SP. Of special interest is an intermediate case of researchers who accept the 2
ACCEPTED MANUSCRIPT CGCD only if the TL peaks of TL glow curve obey first order kinetics and reject it if the peaks obey second order kinetics (e.g., (Bull et al., 1986). This case is a source of confusion for the following reasons; i.
It is not clear what is meant with first and second order kinetics, which are model dependent. In order to clarify this, let’s consider the two most common TL analytical expressions and the
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phenomenological models behind them. The most common and widely used analytical expressions is the empirical general order kinetics equation (May & Partridge, 1964) corresponding to one trap one recombination center (OTOR) phenomenological model. The second TL analytical expression in
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broad use is the physically meaningful mixed order kinetics (Chen et al., 1981) which corresponds to non-interactive multi trap system (NMTS) phenomenological model. In the case of OTOR model,
(i.e., negligible competition) with competition) with
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the boundary conditions of the first- and second-order kinetics are tending to negligible re-trapping → 1 and equal recombination- re-trapping (i.e., strong
= 2, respectively. However, in the mixed-order kinetics (MOK) model, the
boundary conditions are completely different. The mixed-order parameter in the MOK is defined as /(
+ ), where
is the trapped-electron concentration in the active trap and
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=
is a
constant representing the concentration of the thermally disconnected deep trap. In contrast with the GOK, in case of the first-order kinetic, the mixed-order parameter
→ 0 as
≫
(i.e., strong
ii.
→ 1 as
(i.e., weak competition).
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≪
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competition). While, in case of the second-order kinetic, the mixed-order parameter
Observing the characteristics of the first-order peak does not necessarily imply negligible re-trapping during the TL process. It has been shown that even with equally re-trapping and recombination probabilities, the TL glow-peak may have the characteristics of a first-order peak as a result of competition (Pagonis & Kitis, 2012). In other words, the first-order glow-peak in experimental glowcurve may not obey the definition introduced by (Randall & Wilkins, 1945).
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ACCEPTED MANUSCRIPT Recently, Kitis et al. (Kitis et al., 2017) investigated the effect of the competition among four overlapped active traps and one thermally disconnected deep trap (TDDT) on the accuracy of the activation energy determined by the initial rise (IR) method. They concluded that in the presence of high competition of TDDT, the superposition principle holds and the active traps behave independently of the other. On the other hand, in
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the absence of strong competition of TDDT, the IR method is valid for all active trap except for the active trap that plays the role of the competitor.
In their study, the attention was given only to the estimated trap parameters and the dose-saturation case was
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not considered. However, in TL dosimetry, the dose-response curve calculated using the CGCD algorithm is a
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crucial subject.
The main aim of the present work is to simulate, following the lines by Kitis et al. (Kitis et al., 2017), the application of the CGCD on TL glow curves derived from models containing strong competition so that the SP is not fulfilled at all. The specific goals will be:
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1. To simulate composite TL glow curve as a function of dose.
2. To investigate the goodness-of-fit achieved by the CGCD method regardless the accuracy of the obtained kinetic parameters.
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3. To investigate the impact of the non-fulfillment of the SP principles on the accuracy of the kinetics parameter obtained by the CGCD method.
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4. To investigate the behavior of the individual TL peak integrals (or peak heights) extracted from the CGCD, as a function of dose and examine its applicability for radiation dosimetry.
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ACCEPTED MANUSCRIPT 2. Phenomenological model
2.1.Computational model A phenomenological model based on energy band theory of solids which will be used for simulation in this
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work consists of three stages (Chen et al., 1996). The first stage is the irradiation stage. The electron transitions among the traps during this stage are described by the following set of differential equations:
− (
−
)
− "#
&
)
−
!
#1
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=
− (
− % "& ('& − &
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!
=
= "& ('& −
&) !
#2
& ) ! #3
(in *
Where
-
+, +
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= "# ! #4
) denotes the rate of production of electron-hole pairs by the excitation dose, which is
proportional to the dose rate, and hence, the excitation dose will be . = , where !
in *
respectively. +,
denote the concentrations of free holes in electrons in the conduction band and valence bad,
in *
+,
is the recombination center concentration with instantaneous occupancy
is the electron trap concentrations of the trap / with instantaneous occupancies of
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*
+,
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and
probability coefficient "& (in * (*
-
is the time of irradiation.
, +
-
, +
&(
( ), '& in
) and trapping
). The probability coefficient for holes to get trapped in the center is
) and the electron recombination probability coefficient is "# (in *
-
, +
).
The second stage is the relaxation stage, which could be simulated via solving the same set of differential equations eqs.(1-4) for an additional period of time when the excitation is switched off ( = 0).
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ACCEPTED MANUSCRIPT The third stage is the heating stage. The electrons transition among the traps during the heating stage can be described by the following set of differential equation:
!
= − & -& exp 3−
= % 9 & -& exp 3− &
4& 7+ 56
4& 7− 56
! "#
−
& )"& #5
−
& )"& :
+
#7
! "#
#6
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=−
! ('&
! ('&
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&
Where 4& (=>) and -& (- + ) are the activation energy and frequency factor of trap /, 5 (=> ? + ) is Boltzmann & -&
exp @−
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constant. If the trap / is considered non-interactive trap, then the term
AB
CD
E becomes zero.
Following the work of Kitis et al. (Kitis et al., 2017), glow-curves were simulated using the IMTS model given in section 2.1 with kinetic parameters values discussed in section 2.2 for different doses. In each simulation
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case, the shape of the glow-curve, the dose-response curves computed using the CGCD, and the trap kinetic parameters were investigated over the dose range.
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2.2.Selection of the kinetic parameters
In the current study, four active traps and one TDDT were considered. A schematic representation for the
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model used in the current study is presented in Figure 1.
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Figure 1. The electrons transitions durring (a) the irradiation stage, and (b) heating stage in a model of four active traps and one thermally disconnected deep trap (TDDT) used in the current study
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In literature, an enormous number of glow curves with different shapes were simulated. In order to have
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recognizable glow-curve shape, the kinetic parameters in the current work have been selected so that the shape of the simulated glow-curve is similar to the shape of the LiF:Mg,Ti glow-curve. It should be clear that this is only a reference shape and not a simulation study to LiF:Mg,Ti dosimeter. The parameters were selected from the GLOCANIN project (Bos et al., 1994). Therefore, the peak’s numbering starts from peak 2 to peak 5 as in
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the usual LiF:Mg,Ti glow-curve. The parameters presented in Table 1 are fixed in all simulations. Table 1. Fixed kinetic parameters values used in the current study (taken from (Bos et al., 1994)). These values are constant in all cases.
Parameter Peak 2 4(=>)
-(- + )
+,
)
3.9 × 10 0.2 'L
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' (*
1.38
Peak 3
J
1.48
2.0 × 10
0.25 'L
Peak 4 J
1.60
2.0 × 10
0.55 'L
Peak 5 J
2.01
4.0 × 10 10
K
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Different cases of the interactions among the traps were considered in the current study. These cases are presented in Table 2.
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ACCEPTED MANUSCRIPT Table 2. The kinetic parameters values used to simulate different competition cases.
Case
Strong competition
"& < "#
"S = 10+T 'S = 2'L
"& > "#
The values of the dose-rate was set to
Weak competition "S = 10+
*
'S = 0.1'L
= 10T *
-
+, +
-
, +
MN
MO & MQ
10+K *
10+T *
-
, +
-
, +
10+T * 10+V *
-
, +
-
, +
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Sub-case
, and the irradiation times were selected in each cases
in such a way that the dose-saturation level is approached.
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3. Analysis of glow-curve
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In the current study, the glow curves were analyzed using the CGCD algorithm. Instead of using the recently analytical equation derived from the OTOR model by (Kitis & Vlachos, 2013; Sadek et al., 2014; Sadek et al., 2015; Sadek et al., 2014), the general-order kinetics GOK equation of May and Partridge (May & Partridge, 1964) and transformed by Kitis et al. (Kitis et al., 1998) was used because it is the most commonly
With Δ =
CD
A
, Δ# =
4 6 − 6# 6 4 6 − 6# exp 3 7 Y ( − 1)(1 − Δ) exp 3 7 + [# \ 56 6# 56 6# 6#
CD^ A
X + X+
#8
and [# = 1 + ( − 1)Δ# .
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W = W#
X X+
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equation implemented in the CGCD algorithm and it is given by:
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Where W is the intensity of TL signal,
is the order of kinetics. W# and 6# are the peak maximum and peak
maximum position. Although, the GOK equation is an empirical equation, it has been shown that it can accurately describe the glow-peak generated by the OTOR model (Sadek et al., 2014). The goodness of fit was tested by the figure of merit (FOM) (Balian & Eddy, 1977) which is given by:
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ACCEPTED MANUSCRIPT cd
_` (%) = % cB
Where _`
|
&
− ( & )| × 100 #9 "
is the figure of merit, e& is the first channel in the region of interest, ef is the last channel in the
region of interest,
&
is the information content of channel e, ( & ) is the value of the fitting function in
4. Results and discussion 4.1.Case of MN < MO and strong competition effect
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channel e, and " is the integral of the fitted glow-peak in the region of interest.
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In this case, the glow-curves were simulated with trapping probability coefficients less than the recombination probability coefficient by two order of magnitudes, which is the first-order kinetics case. Horowtiz and Yossian (Horowitz & Yossian, 1995) reported that in many cases, where the kinetic order has been studied in dosimetric material, first order kinetics or glow peak very close to first order kinetics model has been reported. Therefore,
4.1.1. Glow curves
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one may expect that this is the most common case that describes the experimental glow-curve.
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1.
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The glow-curves simulated over different doses and an example of their CGCD analysis are presented in Figure
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Figure 2. a) Glow-curves simulated by NMTS model in cases of "& < "# and strong competition for different doses, b) The same glow-curves but normalized to W# and 6# of peak 5, and c) An example of CGCD analysis for one of glow-curves. The FOM was < 0.2%.
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The shapes of the simulated glow-curves are identical for all the doses, even when the dose approaches the saturation level. Kitis et al. (Kitis et al., 2017) concluded that this is in agreement with the experimental stability of LiF:Mg,Ti. In fact, this behavior will not be observed in the other cases simulated in the current study. In this case, there is a strong competition due to the TDDT during the irradiation stage. However, the competition
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among the active traps during the heating stage is negligible due to the high recombination probability coefficient. However, it was reported that the shape of the glow-curve of some dosimeters, especially for the LiF:Mg,Ti changes as the dose approaches the saturation level (Obryk et al., 2014; Farag et al., 2017). This, in
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fact, might be attributed to the trap creation during the irradiation process (McKeever, 1985) which is not
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consider in the current model.
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Figure 3. Dose-response curve of peak 5 using the area under the peak calculated by CGCD algorithm in case of "& < "# and strong competition.
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4.1.2. The TL dose-response curve
The dose-response curve using the area under the peak calculated from the CGCD analysis of the active peaks are approximately identical. Hence only the dose-response curve of the last peak (peak 5) is illustrated in Figure 3. Although the traps were linearly filled up by electrons during the irradiation stage, the dose-response curve shows a supra-linear (quadratic) dose dependence followed by a sub-linear dose dependence as the trap approaches the saturation level. In experimental glow-curve, the dose-response usually shows a linear, then supra-linear, then saturation (Horowitz, 2001). 11
ACCEPTED MANUSCRIPT In experimental work, a dose response curve similar to the dose response curve presented in Figure 3 was also reported. In low dose range., Halperin and Chen (Halperin & Chen, 1966) reported supra-linearity in semiconducting diamonds, Rodine and Land (Rodine & Land, 1971) reported a quadratic dose dependence in ThO2, and Chen et al. (Chen et al., 1988) reported a strong supra-linearity of the 110oC peak in g irradiated
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synthetic quartz. In most of these cases, the supra-linearity dose dependence was attributed to the competition among the traps during the irradiation stage accompanied by a reduction in the concentration of competing traps as the absorbed dose increases. Such mechanism has been used as an explanation to the supra-linearity
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phenomena (Horowitz, 1984; Horowitz, 1984; Horowitz, 1984). In fact, Mische and McKeever (Mische & McKeever, 1989) conclude that the supralinearity should not be viewed as over-response when the competition
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is weak, but rather as under-response at low doses when the competition is strong. However, it should be noted that this explanation is valid only when the competition among the active traps during the heating stage is
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negligible.
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4.1.3. The activation energy computed by the CGCD algorithm
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Figure 4.Activation energy values computed over the dose range by the CGCD algorithm in case of "& < "# and strong competition effect. The solid red lines are the input values used in the simulations process. The vertical dashed line indicates the dose-saturation level.
The activation energy values computed by the CGCD algorithm for each peak over the absorbed dose range are illustrated in Figure 4. From the figure it obvious the CGCD can accurately compute the activation energy of
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the all the deconvoluted peaks over the entire dose range.
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The conclusion of this case can be summarized as following. a) The shape of the glow-curves was stable over a wide dose range. This is a very important result, because in literature, the stability of the TL glow-curves over a wide range of doses was reported for different types of TL dosimeters.
b) All simulated glow curves were successfully fitted with FOM vales less than 0.2%. This means that in this case the CGCD acts as an excellent technique.
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ACCEPTED MANUSCRIPT c) The output values of activation energy of each TL peak is accurately reproduced. The integral of each TL peak was found to follow a clear functional dependence on dose, which is principal property required for dosimetry. The values of the kinetics order
was found to be almost ~1 indicating that the
TL peaks are of first-order kinetics. Note that the general order equation for b = 1.001 is identical with
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the usual first order kinetic equation to the sixth significant digit. Therefore, the results of the CGCD analysis have a clear physical meaning.
d) The effect of the strong TDDT competition dominates the effect of the competition among the active
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traps themselves. In other words, the electrons competed by the active traps are negligible compared with the electrons competed by the TDDT. This result was also reported by Kitis et al. (Kitis et al.,
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2017). In the current study, this case would be called “quasi-superposition principle” in which the results obtained by the CGCD method are reliable.
4.2.1. Glow curves
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4.2.Case of MN < MO and weak competition
The simulated glow-curves over different doses in case of "& < "# and weak competition are presented in
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Figure 5 and the geometrical properties of each glow-peak are illustrated in Figure 6.
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Figure 5. The IMTS model simulation results showing a) the glow-curves, b) normalized glow-curves, c) example of CGCD of glow-curve in case simulated at low-dose, and d) example of CGCD of glow-curve simulated at high dose. The simulations were performed considering the case of "& < "# and weak competition effect. The FOM was less than 0.25%.
A general conclusion can be drawn from figures 5 and 6 that the glow-peak 5 has different properties than the other glow-peaks. The position of peak 5 shifts toward the lower temperatures with increasing the 15
ACCEPTED MANUSCRIPT absorbed dose as it is shown in Figures 6. This behavior is known for the second-order kinetics peaks where "& = "# . This observation is in agreement with the conclusion made by Chen and Pagonis (Chen & Pagonis, 2013) that the last peak in a series obtained by a model of a single recombination center and multiple traps may be of second order. However, this is restricted only to the case of the absence competition due to TDDT. For the
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other glow-peaks, the geometrical properties tend to be stable over the absorbed dose range as the glow-peak is
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farther away from competitor peak, i.e., peak 5.
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Figure 6. The peak position and the kinetics order of the glow-peaks simulated by the IMTS model over different doses in case of "& < "# and weak comptition effect.
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4.2.2. TL dose response curve
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Figure 7. The dose-response curves of peaks 2 and 5 of glow-curves simulated by the IMTS model in case of "& < "# and weak competition..
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The dose-response curves of peaks 2 and 5 are presented in Figure 7. The behavior of the dose-response curves of peaks 3 and 4 are identical with the dose-response curve of peak 2, therefore they were not represented in this figure. As in case 4.1, a supra-linear (quadratic) dose dependent was observed for the peaks 2, 3 and 4 followed by a sub-linear region as the traps approach the dose-saturation level. However, a nonfamiliar behavior has been observed for peak 5. A sub-linear dose response for peak 5 was observed over the investigated dose range. The degree of the sub-linearity increases with increasing the absorbed dose.
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ACCEPTED MANUSCRIPT Usually, the sub-linear dose dependence is observed as the trap approaches the saturation level at high dose when the available traps are close to be filled. However, the case of the dose-response curve of peak 5 is different. In this case, the sub-linear dose dependence region is far from the dose-saturation region. Mische and McKeever (Mische & McKeever, 1989) suggested that the sub-linear dose dependence is may be caused by
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unknown radiation damage mechanisms or self-absorption of the TL light. On the other hand, Laweless et al. (Lawless et al., 2009) proved that the dose-dependence of the trap carriers may be significantly sublinear even when the traps are far from approaching the saturation level. They attributed this behavior to the competition
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between the trap and the recombination center during the irradiation stage.
Figure 8. The concentration of the electrons filled the traps 2 and 5 after the irradiation stage in case of " < "# and weak competition.
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ACCEPTED MANUSCRIPT It should be noted that the trap of peak 5 is the deepest trap. Therefore, during the heating stage, the trap of peak 5 acts as the competitor for the other traps. Thereby, it has different geometrical and dose-dependence behavior than the other traps. This competition occurs only during the heating stage. Therefore, it is anticipated that the behaviors of all the traps during the irradiation stage are identical. This is obvious in Figure 8 which presents
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the sub-linear dependence of the free carriers accumulated in the traps over different doses during the irradiation stage. The behaviors of the accumulated free carriers in the other traps are approximately the same. Figure 8 confirms that all the traps have the almost the same behavior during the irradiation stage and the dose-
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dependence behavior of peak 5 was different due to its competition effect occurring during the heating stage. Based on our knowledge, similar dose response curve like the one of peak 5 presented in Figure 8 was not
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observed in real experimental glow-curves. This implies that most of the experimental glow-curves are often subjected to strong competition effect either due to TDDT during the irradiation stage or competition among the
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active traps during the heating stage.
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4.2.3. The activation energy computed by CGCD
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Figure 9. The activation energy values estimated using CGCD algorithm in case of " < "# and weak competition. The simulation input 4 values are repsented by solid line for each glow-peak. The vertical dashed line indicates the dose-saturation level.
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The activation energy values for the deconvoluted glow-peaks are presented in Figure 9. The CGCD algorithm could compute the activation energy satisfactorily for the glow-peaks 2 and 3. However, at low dose level, an
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over estimation for the activation energy value of peak 4 was observed. As the dose increases, the accuracy of the estimated activation energy of peak 4 increases. It should be noted that this over estimation may not be due to the competition effect but rather because of the complex overlapping between this peak and its neighbor glow-peaks especially at the low dose level (Sadek, 2013). The conclusion of this case can be summarized as following;
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ACCEPTED MANUSCRIPT a) The instability of the shape of the glow-curves over the different doses would be a clear evidence that one of the active trap has acted as a competitor. In this case, the accuracy of the kinetic parameters estimated by the CGCD is satisfactory for all the peak except for the peak which has acted as a competitor.
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b) In this case, the quasi super-position state is very difficult to be established. However, even so, the
as a dosimeter. 4.3.Case of MN > MO and strong competition
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application of CGCD offer useful and reliable information about the suitability of a material to be used
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This case simulates the effects of strong competition due to the TDDT during the irradiation stage and also the strong competition among the active traps during the heating stage. Therefore, one may consider this case as
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extreme competition effects. The simulated glow-curves are presented in Figure 10.
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Figure 10.The IMTS model simulation results showing a) the glow-curves, b) normalized glow-curves, c) example of CGCD of glow-curve in case simulated at low-dose, and d) example of CGCD of glow-curve simulated at high dose. The simulations were performed consider "& > "# .The FOM was less than 0.45%.
4.3.1. Glow curves
Although in this case the simulations were performed assuming strong competition due to TDDT as in case discussed in section 4.1, the stability of the geometrical properties of the glow-curve of the different doses did not hold and the shape of the glow-curves changed as the traps approached the saturation level. It should be 22
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noted that in the current case "& > "# , and this is the only difference from case 4.1. In fact, setting "& > "# yields a strong competition between the active traps during the heating stage and the deepest traps will act as competitors. Therefore, the changes in the glow-curves are significant for peaks 4 and 5 as shown in Figure 10c. The width of these glow-peaks increased significantly and the goodness of the fitting decreases. It should be
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noted that the case discussed by Kitis et al. (Kitis et al., 2017) did not considered the high dose cases, and thereby the shape of the glow-curve presented in Figure 10c was not reported in their study.
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4.3.2. The dose response curve
Figure 11. The dose-response curves of peaks 2 and 5 of glow-curves simulated by the IMTS model in case of "& > "# and strong competition..
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ACCEPTED MANUSCRIPT The dose-response curves of peaks 2 and 5 are illustrated in Figure 11. The dose response of the other glowpeaks is almost identical with the dose response curves presented in this figure. The difference between the dose-response curves in the current case and the dose-response curves presented in case 4.1 is that in the current case, the quadratic dose dependence is followed by supra-quadratic dose dependence instead of sub-quadratic
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dose dependence. This behavior was reported for the 110oC TL peak of synthetic quartz (Chen et al., 1988). Later, Chen et al. (Chen et al., 1996) attributed this behavior to the competition effect during the heating stage. However, it should be noted that in their model, two recombination centers were considered in their simulation
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which is not the case of the simulation in the current work.
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4.3.3. The activation energy computed by the CGCD algorithm
Figure 12. Activation energy values computed by the CGCD algorithm in case of "& > "# and strong competition. The simulation input values are represented by dashed lines. The vertical line indicates to the dose-saturation level.
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ACCEPTED MANUSCRIPT The activation energy values computed by the CGCD algorithm over the absorbed dose range are presented in Figure 12. The CGCD algorithm using the GOK equation could compute the activation energies of peaks 2 and 3 all over the complete dose range with error < 10%. However, for peaks 4 and 5, the CGCD algorithm could accurately compute the activation energy as the traps are far from the saturation level. As the traps approach the
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saturation level, the activation energy values are underestimated by more than 30%. It should be noted that these results are different than the results reported by Kitis et al. (Kitis et al., 2017) using the initial region (IR) method. In their results, they reported an underestimation in the activation energy value of peak 5 by 20% in the
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evaluated with error ≤ 5% in the same dose region.
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absorbed dose region 10T : 10 K . However, in the current study, the activation energy of peak 5 could be
It worthwhile to mention that the underestimation of the activation energy of peak 5 was reported only at the very high dose in the extreme competition case. In other words, this underestimation should not have significant effect on the general accuracy of the CGCD algorithm because this case is rare to be found in experimental
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glow-curve.
The conclusion of this case can be summarized as following. a) The shapes of TL glow curves are very stable and show some differentiation only when approaching the
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saturation.
b) The fitting of the TL glow curves was excellent giving very low FOM values. The CGCD analysis
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appear in this case again as an excellent technique. c) The output parameters are successfully reproduced for peaks 2, 3 and 4 in whole dose region. For the last peak there is an underestimation appearing at onset of saturation. The behavior of individual peaks as a function of dose is easily discriminated and the practical dosimetry is applicable in a wide dose region up to the onset of saturation mainly far and not near the saturation. Therefore, the CGCD results could be physically meaningful.
25
ACCEPTED MANUSCRIPT d) In this case the TDDT is a strong competitor, which however, cannot eliminate the competition among the active traps. Due to the remnant competition among the active traps (see also (Kitis et al., 2017)) there is a weak violation of the SP, so that a quasi SP state can be poorly established. However, even so the CGCD analysis is correct and therefore derives meaningful results for each glow peak in the
4.4.Case of MN > MO and weak competition.
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greater part of their TL dose response curves.
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active traps during both the irradiation and heating stages.
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In this case, the competition due to TDDT is negligible, and thus, the competition occurs among the
4.4.1. Glow Curves
The glow-curves simulated by the IMTS model in case of "& > "# and weak competition are presented in
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Figure 13.
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Figure 13. The IMTS model simulation results showing a) the glow-curves, b) normalized glow-curves, c) example of CGCD of glow-curve in case simulated at low-dose, and d) example of CGCD of glow-curve simulated at high dose. The simulations were performed considering "& > "# and weak competition. The FOM is less than 1.2%.
In the current case the shape of the glow-curves changes with increasing the absorbed dose. At very low dose, the glow-peaks 2, 3 and 4 are hardly observable. Therefore, the deconvolution analysis process was difficult to be performed for these glow-curves, especially for peak 4 because of its complex overlapping with peaks 3 and 5. Moreover, it is obvious that the geometrical properties of the peak 5 changes with increasing the absorbed dose. The peak position decreases and the peak width increases with increasing the absorbed dose. 27
ACCEPTED MANUSCRIPT Therefore, a general conclusion can be made that the last glow-peak in a series of overlapped peaks acts as a second-order peak only in the absence of competition of TDDT during the irradiation stage or in the presence of strong competition among the active traps during the heating stage. Similar to case 4.2, the geometrical
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properties of the other glow peaks are almost stable over the absorbed dose range (Figure 14).
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Figure 14. The geometrical properties showing a) the peak position and b) peak width of the glow-peaks simulated with IMTS models over different doses in case of "& > "# and strong competition.
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4.4.2. The dose response curve
In fact, the changes in the geometrical properties of peak 5 confirms that this glow-peak plays the role of the
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competitor during the heating stage. In all the previous cases, the dose response curves were either supra-linear or sub-linear dose dependence for all the deconvoluted peaks. In the current case, a complete different situation was observed. The dose-response curves of peaks 2, 3 and 4 showed a supra-linear dose dependence. While, the dose-response curve of peak 5 showed an almost linear dose-dependence as it is illustrated in Figure 15.
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Figure 15. Dose response curves for a) peak 5 and b) peak 2 of glow-curves simulated using IMST model in case of "& > "# and weak competition.
An almost linear dose-dependence was observed for peak 5, while a quadratic dose-dependence was observed for the other glow-peaks. The results presented in Figure 15 indicate that even in the presence of competition
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among active traps, a linear dose-dependence can be observed for the last glow-peak. However, this occurs only in the absence of the competition of the TDDT. It should also be noted that the linear region is followed by sublinear region prior to approaching the dose-saturation level, i.e., no supra-linear dose dependence was observed.
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On the other hand, the dose-response curve of the other glow-peaks starts with quadratic then supra-quadratic
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dose-dependence before approaching the dose-saturation level.
4.4.3. The activation energy computed by the CGCD The activation energy values obtained by the CGCD over the different doses are presented in Figure 16. The activation energy values of peaks 2 and 5 have almost the same trend over the absorbed dose range. The CGCD method can obtain the activation energy values of these peaks with a 5% error level as the traps are far from the saturation level. However, as these traps approach the saturation level, the computed activation energy values are underestimated. On the other hand, significant deviations from the true value were observed for the 29
ACCEPTED MANUSCRIPT activation energy values computed for peaks 3 and 4. This case is similar to case 4.3 in which the complexity of the glow-curve, especially at the low dose affects the efficiency of the CGCD algorithm. The conclusion of this case can be summarized as following.
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a) The shapes of TL glow curves vary strongly as a function of dose. b) Even in this extreme case the CGCD analysis appears as a satisfactory analytical technique. c) The output parameters do not reproduce the input values for all peaks. Therefore, the CGCD analysis
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despite the good fits give results which maybe seems physical but they are not necessary correct. d) The behavior of individual peaks as a function of dose is easily discriminated. This means that even in
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this extreme case the practical dosimetry is possible.
e) The competition among the active traps themselves becomes dominant when the TDDT competition is negligible. This case is different from the case that was previously discussed in section 4.2. In the current case, most of the electrons are captured by the last active trap because of its high trapping probability. As a consequence, the electrons concentration in the other low-temperature trap, and thereby
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their TL intensity are decreased. Therefore, at low doses, the low-temperature glow-peaks were not visible compared to the last glow-peak of the highest trapping probability.
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f) The results of the cases are characteristic of strong violation of SP.
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Figure 16.Activation energy values computed by the CGCD algorithm in case of "& > "# and weak competition. The simulation input values are represented by solid lines.
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ACCEPTED MANUSCRIPT 5. General discussion In the current study, it has been shown that in many cases, the activation energy values obtained by the CGCD algorithm are close to the input values even with lack of the superposition principle. This does not mean
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the CGCD algorithm always gives accurate activation energy value as it has been also shown in the current study. However, one should distinguish between the accuracy and the applicability of the CGCD algorithm. In other words, not obtaining accurate kinetic parameters because of the complexity of the glow-curve is not the same as not obtaining accurate kinetics parameters because the model used in the CGCD failed to describe the
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experimental peak.
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In literature, many investigators examined the accuracy of the peak fitting algorithm via simulations. Recently, Sadek and Kitis (Sadek & Kitis, 2017) have concluded that two main reasons may lead to unrealistic simulated peaks; the improper selections of the simulation inputs, and performing the TL simulation considering the heating stage only, i.e., without considering the irradiation stage in the simulation process.
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Nevertheless, this does not mean that the results obtained by the CGCD algorithm are always correct. In some cases, especially for complex glow-curves in which infinite analysis CGCD solutions are available, it would be useful to validate the results obtained by the CGCD via using another analytical method, e.g., the
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different heating rate method (Hoogenstraaten, 1958), peak shape method (Kitis & Pagonis, 2007), Tl − Tmnop
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method (McKeever, 1980), or the isothermal decay method (Moharil, 1981)
6. Conclusions
The basic and general conclusion of the present work can be summarized as following. •
Regardless the validity of the SP principles, the CGCD algorithm could be used to investigate the TL dose-response function of the individual glow-peaks. Thus, in dosimetric applications, using the peak
32
ACCEPTED MANUSCRIPT area calculated by the CGCD algorithm would be more appropriate than using the area under a certain portion of the glow-curve. •
The quality of fit by CGCD analysis was excellent in all cases. Therefore, the present simulation showed clearly that the use of CGCD to analyze composite TL glow curves using anyone of the available TL
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expressions and not only the one used in the present work, is completely independent of the violation or not of the SP. •
The CGCD results concerning the trapping parameters needs more attention in their interpretation. It
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was found that the output values of activation energy are physically meaningful in case Ar < Al and strong TDDT competitor, physically meaningful up to the onset of dose saturation in cases Ar < Al
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weak TDDT competitor and Ar < Al strong TDDT competitor. The output values of activation energy
were erroneous in the case of Ar > Al weak TDDT competitor. •
Finally, the use of the CGCD as an analysis technique can be applied with confidence in experimental work. However, special care is need for the interpretation of its results concerning the trapping
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parameters, which however, can be evaluated using many of the methods available in TL theory.
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ACCEPTED MANUSCRIPT References Balian, H.G. & Eddy, W., 1977. Figure-of-merit (FOM), an improved criterion over the normalized Chi-squared test for assessing goodness-of-fit of Gamma-ray spectral peaks. Nucl. Instrum. Methods, 2389-395. Bos, A.J.J., Piters, T.M., Gomez-Ros, J.M. & Delgado, A., 1994. An intercomparison of glow curve analysis computer programs: II. measured glow curves. Radiat. Prot. Dosim., 2257-264.
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Bull, R.K. et al., 1986. Thermoluminescnce kinetics for multiple glow peak curves produced by the release of electrons and holes. J. Phys. D. Appl. Phys, 19, pp.1321-34. Chen, R., Fogel, G. & Lee, C.K., 1996. A new look at the models of the superlinear dose dependence of thermoluminescence. Radiat. Prot. Dosim., 65, pp.63-68.
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Chen, R., Kristianpoller, N., Davidson, Z. & Visocekas, R.J., 1981. Mixed first and second order kinetics in thermally stimulated process. J. Lumin., 2293-303.
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Chen, R. & Pagonis, V., 2013. On the expected order of kinetics in a series of thermoluminescence (TL) and thermally stimulated conductivity (TSC) peaks. Nucl. Instrum. Method Phys. Res., B 312, pp.60-69. Chen, R., Yang, X.H. & McKeever, S.W.S., 1988. The strongly superlinear dose dependence of thermoluminescence in synthetic quartz. J. Phys. D: Appl. Phys., 21, pp.1452-57. Chen, R., Yang, X.H. & McKeever, S.W.S., 1988. The strongly superlinear dose dependence of thermoluminescence in synthetic quartz. J. Phys. D: Appl. Phys., 21, pp.1452-57.
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Farag, M.A. et al., 2017. Radiation damage and sensitization effects on thermoluminescence of LiF:Mg,Ti (TLD-700). Nucl. Intrum. Methods, B 407, pp.180-90. Halperin, A. & Chen, R., 1966. Thermoluminescence of semiconducting diamands. Phys. Rev., 148, pp.839-45. Hoogenstraaten, W., 1958. Electron traps in zinc sulphide phosphors. Phlips Res. Rep., 13, p.515.
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Horowitz, Y.S., 1984. General characteristics of thermoluminescent materials. In Y.S. Horowitz, ed. Thermoluminscence and thermoluminescent dosimetry. Boca Raton: CRC Press. pp.89-172. Horowitz, Y.S., 1984. Recent models for TL supralinearity. Radiat. Prot. Dosim., 6, pp.17-20.
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Horowitz, Y.S., 2001. Theory of thermoluminescence gamma dose response: The unified interaction model. Nucl. Instrum. Methods, B 184, pp.68-84. Horowitz, Y.S. & Yossian, D., 1995. Computerized glow curve deconvolution: Application to thermoluminescence dosimetry. Radiat. Prot. Dosim., 21-110. Kitis, G., Gomez-Ros, J.M. & Tuyn, J.W.N., 1998. Thermoluminescence glow-curve deconvolution functions for first, second and general orders of kinetics. J. Phys. D: Appl. Phys., 31, pp.2636-41. Kitis, G. & Pagonis, V., 2007. Peak shape methods for general order thermoluminescence glow-peaks: A reappraisal. Nucl. Instrum. Methods , B 262, pp.313-22. Kitis, G., Pagonis, V. & Tzamaris, S.E., 2017. The influence of competition effects on the initial rise method during thermal stimulation of luminescence: A simulation study. Radiat. Meas., 100, pp.27-36. 34
ACCEPTED MANUSCRIPT Kitis, G. & Vlachos, N.D., 2013. General semi-analytical expressions for TL, OSL and other luminescence stimulation modes derived fron the OTOR model using the Lambert W-function. Radiat. Meas., 247-54. Lawless, J.L., Chen, R. & Pagonis, V., 2009. Sublinear dose dependence of thermoluminescence and optically stimulated luminescence prior to the approach to saturation level. Radiat. Meas., 44, pp.606-10. May, C.E. & Partridge, J.A., 1964. Thermoluminescent kinerics of alpha-irradiated alkali halides. J. Chem. Phys., 21401-1409.
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McKeever, S.W.S., 1980. On the analysis of complex thermoluminescence glow-curves: Resolution into individual peaks. Phys. Status Solidi, a 62, pp.331-40. McKeever, S.W.S., 1985. Thermoluminescence of solids. Cambridge: Cambridge University Press.
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Mische, E.F. & McKeever, S.W.S., 1989. Mechanisms of supralinearity in lithium fluoride thermoluminescence dosemeters. Radiat. Prot. Dosim., 29, pp.159-75. Moharil, S.V., 1981. Trapping parameters from iso-thermal decay of TL. J. Phys. D. Appl. Phys., 14, p.1677.
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Obryk, B. et al., 2014. On LiF:Mg,Cu,P and LiF:Mg,Ti phosphors high & ultra-high dose features. Radiat. Meas., 71, pp.25-30. Pagonis, V. & Kitis, G., 2012. Prevalence of first-order kinetics in thermoluminescence material: An explanation based on multiple competition process. Phys. Status Solidi B, 21590-1601. Randall, J.T. & Wilkins, M.H.F., 1945. Phosphorescence and electron traps. I. The study of trap distribution. In Proc. R. Soc. London, 1945.
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Rodine, E.T. & Land, P.L., 1971. Electronic defect structure of single crystal ThO2 by thermoluminescence. Phys. Rev., B4, pp.2701-24. Sadek, A.M., 2013. Test of the accuracy of the computerized glow curve deconvolution algorithm for the analysis of thermoluminescence glow curves. Nucl. Instrum. Methods A, 712, pp.56-61.
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Sadek, A.M. et al., 2015. The deconvolution of thermoluminescence glow-curves using general expressions derived from the one trap-one recombination (OTOR) level model. Appl. Radiat. Isot., 2214-221.
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Sadek, A.M., Eissa, H.M., Basha, A.M. & Kitis, G., 2014. Properties of the thermoluminescence glow peaks simulated by the interactive multiple-trap system (IMTS) model. Phys. Status Solidi B, 2721-729. Sadek, A.M., Eissa, H.M., Basha, A.M. & Kitis, G., 2014. Resolving the limitation of the peak fitting and peak shape methods in the determination of the activation energy of thermoluminescence glow peaks. J. Lumin, 2418-423. Sadek, A.M. & Kitis, G., 2017. A critical look at the kinetic parameter values used in simulating the thermoluminescence glow-curve. Journal of Luminescence, 183, pp.533-41. Wolfson, R., 2016. Facts101:Essebtial University Physics. Cram101 Textbook Reviews.
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ACCEPTED MANUSCRIPT
Highlights: The impact of non-fulfillment of the SP theorem on the CGCD analysis was discussed. TL glow curves were simulated over a wide range of doses and different conditions. In general, the non-fulfillment of the SP theorem has no effect on the CGCD method.
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• • •
S1350-4487(18)30247-6
DOI:
10.1016/j.radmeas.2018.06.016
Reference:
RM 5938
To appear in:
Radiation Measurements
Received Date: 3 April 2018 Revised Date:
23 May 2018
Accepted Date: 16 June 2018
Please cite this article as: Sadek, A.M., Kitis, G., Impact of non-fulfillment of the super position principle on the analysis of thermoluminescence glow-curve, Radiation Measurements (2018), doi: 10.1016/ j.radmeas.2018.06.016. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Impact of non-fulfillment of the super position principle on the analysis of thermoluminescence glow-curve A.M. Sadeka* and G. Kitisb a
Ionizing Radiation Metrology Department, National Institute for Standards, El-Haram, Giza, Egypt Nuclear Physics and Elementary Particles Physics Section, Physics Department, Aristotle University of Thessaloniki, 54124 Thessaloniki, Makedonia, Greece
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b
Abstract
The analysis of complex thermoluminescence (TL) glow curves in its individual peaks is achieved by a
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computerized glow curve deconvolution (CGCD). The correct application of the CGCD requires the fulfillment of the superposition principle (SP), which postulates that the TL peaks have to be independent of each other. In
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the present work we simulate the application of the CGCD using interactive phenomenological models in which the SP is not fulfilled. The thermoluminescence (TL) processes of four active traps and one thermally disconnected deep trap (TDDT) was simulated over a wide range of doses and under different competition cases. The TL glow-curves were analyzed by the CGCD algorithm and the accuracy of the kinetic parameters and the dose dependence of each trap were investigated over the doses. In most of the simulated cases, the nonfulfillment of the superposition principles has no significant effect on the accuracy of the computed kinetics
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parameters. Different behaviors of dose response curve were discussed including sub-linear and supra-linear dose dependence. It has been also found that in the case of presence of competitions among the active traps, an almost linear dose response curve can be observed for the last glow-peak in a series of overlapping glow-peaks.
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Keywords
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Theory of thermoluminescence; Superposition principle; Glow-curve analysis.
Corresponding author
[email protected] (Amr M. Sadek)
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ACCEPTED MANUSCRIPT 1. Introduction The analysis of complex experimental thermoluminescence (TL) glow curves in their individual peaks can be achieved by a computerized glow curve deconvolution (CGCD). The CGCD analysis can provide the
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integrals of the individual peaks for dosimetric characterization of materials and the kinetic parameters of electron traps for energy level characterization (Horowitz & Yossian, 1995). The correct application of the CGCD requires the fulfillment of the superposition principle (SP), which postulates that every TL peak has to
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be independent of the other peaks.
In physics, the superposition principle states that for all linear systems, the net response at a given place and
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time caused by two or more stimuli is the sum of the response that would have been caused by each stimulus individually (Wolfson, 2016). Mathematically, for input
and response
= ( ), the superposition of the
input should yield the superposition of the respective responses, i.e., (
+
+⋯
) = ( ) + ( ) + ⋯+ (
)#1
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The energy band theory in solids is used to create phenomenological models to describe the TL processes. These phenomenological models consist of set of differential equations, which during irradiation
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stage describe the traffic of electrons form conduction band to traps and of holes from valence band to luminescence centers. During heating stage, they describe the thermal release of electrons to the conduction
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band and their re-trapping or recombination. In both stages the electron traps compete with each other in trapping electrons from the conduction band and altogether compete with the recombination to luminescence centers. All these competitions in the models act against the fulfillment of the SP defined above. Therefore, it is a clear theoretical prediction that under competition effects among traps, the SP is not fulfilled. In TL research, many researches adopted the applications of CGCD in different radiation dosimetry fields, which then was generalized and grown up very fast. However, others continue to oppose to the CGCD due to the non-fulfillment of the SP. Of special interest is an intermediate case of researchers who accept the 2
ACCEPTED MANUSCRIPT CGCD only if the TL peaks of TL glow curve obey first order kinetics and reject it if the peaks obey second order kinetics (e.g., (Bull et al., 1986). This case is a source of confusion for the following reasons; i.
It is not clear what is meant with first and second order kinetics, which are model dependent. In order to clarify this, let’s consider the two most common TL analytical expressions and the
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phenomenological models behind them. The most common and widely used analytical expressions is the empirical general order kinetics equation (May & Partridge, 1964) corresponding to one trap one recombination center (OTOR) phenomenological model. The second TL analytical expression in
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broad use is the physically meaningful mixed order kinetics (Chen et al., 1981) which corresponds to non-interactive multi trap system (NMTS) phenomenological model. In the case of OTOR model,
(i.e., negligible competition) with competition) with
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the boundary conditions of the first- and second-order kinetics are tending to negligible re-trapping → 1 and equal recombination- re-trapping (i.e., strong
= 2, respectively. However, in the mixed-order kinetics (MOK) model, the
boundary conditions are completely different. The mixed-order parameter in the MOK is defined as /(
+ ), where
is the trapped-electron concentration in the active trap and
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=
is a
constant representing the concentration of the thermally disconnected deep trap. In contrast with the GOK, in case of the first-order kinetic, the mixed-order parameter
→ 0 as
≫
(i.e., strong
ii.
→ 1 as
(i.e., weak competition).
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≪
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competition). While, in case of the second-order kinetic, the mixed-order parameter
Observing the characteristics of the first-order peak does not necessarily imply negligible re-trapping during the TL process. It has been shown that even with equally re-trapping and recombination probabilities, the TL glow-peak may have the characteristics of a first-order peak as a result of competition (Pagonis & Kitis, 2012). In other words, the first-order glow-peak in experimental glowcurve may not obey the definition introduced by (Randall & Wilkins, 1945).
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ACCEPTED MANUSCRIPT Recently, Kitis et al. (Kitis et al., 2017) investigated the effect of the competition among four overlapped active traps and one thermally disconnected deep trap (TDDT) on the accuracy of the activation energy determined by the initial rise (IR) method. They concluded that in the presence of high competition of TDDT, the superposition principle holds and the active traps behave independently of the other. On the other hand, in
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the absence of strong competition of TDDT, the IR method is valid for all active trap except for the active trap that plays the role of the competitor.
In their study, the attention was given only to the estimated trap parameters and the dose-saturation case was
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not considered. However, in TL dosimetry, the dose-response curve calculated using the CGCD algorithm is a
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crucial subject.
The main aim of the present work is to simulate, following the lines by Kitis et al. (Kitis et al., 2017), the application of the CGCD on TL glow curves derived from models containing strong competition so that the SP is not fulfilled at all. The specific goals will be:
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1. To simulate composite TL glow curve as a function of dose.
2. To investigate the goodness-of-fit achieved by the CGCD method regardless the accuracy of the obtained kinetic parameters.
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3. To investigate the impact of the non-fulfillment of the SP principles on the accuracy of the kinetics parameter obtained by the CGCD method.
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4. To investigate the behavior of the individual TL peak integrals (or peak heights) extracted from the CGCD, as a function of dose and examine its applicability for radiation dosimetry.
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ACCEPTED MANUSCRIPT 2. Phenomenological model
2.1.Computational model A phenomenological model based on energy band theory of solids which will be used for simulation in this
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work consists of three stages (Chen et al., 1996). The first stage is the irradiation stage. The electron transitions among the traps during this stage are described by the following set of differential equations:
− (
−
)
− "#
&
)
−
!
#1
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=
− (
− % "& ('& − &
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!
=
= "& ('& −
&) !
#2
& ) ! #3
(in *
Where
-
+, +
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= "# ! #4
) denotes the rate of production of electron-hole pairs by the excitation dose, which is
proportional to the dose rate, and hence, the excitation dose will be . = , where !
in *
respectively. +,
denote the concentrations of free holes in electrons in the conduction band and valence bad,
in *
+,
is the recombination center concentration with instantaneous occupancy
is the electron trap concentrations of the trap / with instantaneous occupancies of
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*
+,
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and
probability coefficient "& (in * (*
-
is the time of irradiation.
, +
-
, +
&(
( ), '& in
) and trapping
). The probability coefficient for holes to get trapped in the center is
) and the electron recombination probability coefficient is "# (in *
-
, +
).
The second stage is the relaxation stage, which could be simulated via solving the same set of differential equations eqs.(1-4) for an additional period of time when the excitation is switched off ( = 0).
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ACCEPTED MANUSCRIPT The third stage is the heating stage. The electrons transition among the traps during the heating stage can be described by the following set of differential equation:
!
= − & -& exp 3−
= % 9 & -& exp 3− &
4& 7+ 56
4& 7− 56
! "#
−
& )"& #5
−
& )"& :
+
#7
! "#
#6
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=−
! ('&
! ('&
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&
Where 4& (=>) and -& (- + ) are the activation energy and frequency factor of trap /, 5 (=> ? + ) is Boltzmann & -&
exp @−
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constant. If the trap / is considered non-interactive trap, then the term
AB
CD
E becomes zero.
Following the work of Kitis et al. (Kitis et al., 2017), glow-curves were simulated using the IMTS model given in section 2.1 with kinetic parameters values discussed in section 2.2 for different doses. In each simulation
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case, the shape of the glow-curve, the dose-response curves computed using the CGCD, and the trap kinetic parameters were investigated over the dose range.
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2.2.Selection of the kinetic parameters
In the current study, four active traps and one TDDT were considered. A schematic representation for the
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model used in the current study is presented in Figure 1.
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Figure 1. The electrons transitions durring (a) the irradiation stage, and (b) heating stage in a model of four active traps and one thermally disconnected deep trap (TDDT) used in the current study
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In literature, an enormous number of glow curves with different shapes were simulated. In order to have
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recognizable glow-curve shape, the kinetic parameters in the current work have been selected so that the shape of the simulated glow-curve is similar to the shape of the LiF:Mg,Ti glow-curve. It should be clear that this is only a reference shape and not a simulation study to LiF:Mg,Ti dosimeter. The parameters were selected from the GLOCANIN project (Bos et al., 1994). Therefore, the peak’s numbering starts from peak 2 to peak 5 as in
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the usual LiF:Mg,Ti glow-curve. The parameters presented in Table 1 are fixed in all simulations. Table 1. Fixed kinetic parameters values used in the current study (taken from (Bos et al., 1994)). These values are constant in all cases.
Parameter Peak 2 4(=>)
-(- + )
+,
)
3.9 × 10 0.2 'L
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' (*
1.38
Peak 3
J
1.48
2.0 × 10
0.25 'L
Peak 4 J
1.60
2.0 × 10
0.55 'L
Peak 5 J
2.01
4.0 × 10 10
K
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Different cases of the interactions among the traps were considered in the current study. These cases are presented in Table 2.
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ACCEPTED MANUSCRIPT Table 2. The kinetic parameters values used to simulate different competition cases.
Case
Strong competition
"& < "#
"S = 10+T 'S = 2'L
"& > "#
The values of the dose-rate was set to
Weak competition "S = 10+
*
'S = 0.1'L
= 10T *
-
+, +
-
, +
MN
MO & MQ
10+K *
10+T *
-
, +
-
, +
10+T * 10+V *
-
, +
-
, +
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Sub-case
, and the irradiation times were selected in each cases
in such a way that the dose-saturation level is approached.
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3. Analysis of glow-curve
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In the current study, the glow curves were analyzed using the CGCD algorithm. Instead of using the recently analytical equation derived from the OTOR model by (Kitis & Vlachos, 2013; Sadek et al., 2014; Sadek et al., 2015; Sadek et al., 2014), the general-order kinetics GOK equation of May and Partridge (May & Partridge, 1964) and transformed by Kitis et al. (Kitis et al., 1998) was used because it is the most commonly
With Δ =
CD
A
, Δ# =
4 6 − 6# 6 4 6 − 6# exp 3 7 Y ( − 1)(1 − Δ) exp 3 7 + [# \ 56 6# 56 6# 6#
CD^ A
X + X+
#8
and [# = 1 + ( − 1)Δ# .
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W = W#
X X+
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equation implemented in the CGCD algorithm and it is given by:
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Where W is the intensity of TL signal,
is the order of kinetics. W# and 6# are the peak maximum and peak
maximum position. Although, the GOK equation is an empirical equation, it has been shown that it can accurately describe the glow-peak generated by the OTOR model (Sadek et al., 2014). The goodness of fit was tested by the figure of merit (FOM) (Balian & Eddy, 1977) which is given by:
8
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_` (%) = % cB
Where _`
|
&
− ( & )| × 100 #9 "
is the figure of merit, e& is the first channel in the region of interest, ef is the last channel in the
region of interest,
&
is the information content of channel e, ( & ) is the value of the fitting function in
4. Results and discussion 4.1.Case of MN < MO and strong competition effect
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channel e, and " is the integral of the fitted glow-peak in the region of interest.
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In this case, the glow-curves were simulated with trapping probability coefficients less than the recombination probability coefficient by two order of magnitudes, which is the first-order kinetics case. Horowtiz and Yossian (Horowitz & Yossian, 1995) reported that in many cases, where the kinetic order has been studied in dosimetric material, first order kinetics or glow peak very close to first order kinetics model has been reported. Therefore,
4.1.1. Glow curves
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one may expect that this is the most common case that describes the experimental glow-curve.
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1.
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The glow-curves simulated over different doses and an example of their CGCD analysis are presented in Figure
9
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Figure 2. a) Glow-curves simulated by NMTS model in cases of "& < "# and strong competition for different doses, b) The same glow-curves but normalized to W# and 6# of peak 5, and c) An example of CGCD analysis for one of glow-curves. The FOM was < 0.2%.
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The shapes of the simulated glow-curves are identical for all the doses, even when the dose approaches the saturation level. Kitis et al. (Kitis et al., 2017) concluded that this is in agreement with the experimental stability of LiF:Mg,Ti. In fact, this behavior will not be observed in the other cases simulated in the current study. In this case, there is a strong competition due to the TDDT during the irradiation stage. However, the competition
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among the active traps during the heating stage is negligible due to the high recombination probability coefficient. However, it was reported that the shape of the glow-curve of some dosimeters, especially for the LiF:Mg,Ti changes as the dose approaches the saturation level (Obryk et al., 2014; Farag et al., 2017). This, in
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fact, might be attributed to the trap creation during the irradiation process (McKeever, 1985) which is not
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consider in the current model.
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Figure 3. Dose-response curve of peak 5 using the area under the peak calculated by CGCD algorithm in case of "& < "# and strong competition.
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4.1.2. The TL dose-response curve
The dose-response curve using the area under the peak calculated from the CGCD analysis of the active peaks are approximately identical. Hence only the dose-response curve of the last peak (peak 5) is illustrated in Figure 3. Although the traps were linearly filled up by electrons during the irradiation stage, the dose-response curve shows a supra-linear (quadratic) dose dependence followed by a sub-linear dose dependence as the trap approaches the saturation level. In experimental glow-curve, the dose-response usually shows a linear, then supra-linear, then saturation (Horowitz, 2001). 11
ACCEPTED MANUSCRIPT In experimental work, a dose response curve similar to the dose response curve presented in Figure 3 was also reported. In low dose range., Halperin and Chen (Halperin & Chen, 1966) reported supra-linearity in semiconducting diamonds, Rodine and Land (Rodine & Land, 1971) reported a quadratic dose dependence in ThO2, and Chen et al. (Chen et al., 1988) reported a strong supra-linearity of the 110oC peak in g irradiated
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synthetic quartz. In most of these cases, the supra-linearity dose dependence was attributed to the competition among the traps during the irradiation stage accompanied by a reduction in the concentration of competing traps as the absorbed dose increases. Such mechanism has been used as an explanation to the supra-linearity
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phenomena (Horowitz, 1984; Horowitz, 1984; Horowitz, 1984). In fact, Mische and McKeever (Mische & McKeever, 1989) conclude that the supralinearity should not be viewed as over-response when the competition
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is weak, but rather as under-response at low doses when the competition is strong. However, it should be noted that this explanation is valid only when the competition among the active traps during the heating stage is
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negligible.
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4.1.3. The activation energy computed by the CGCD algorithm
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Figure 4.Activation energy values computed over the dose range by the CGCD algorithm in case of "& < "# and strong competition effect. The solid red lines are the input values used in the simulations process. The vertical dashed line indicates the dose-saturation level.
The activation energy values computed by the CGCD algorithm for each peak over the absorbed dose range are illustrated in Figure 4. From the figure it obvious the CGCD can accurately compute the activation energy of
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the all the deconvoluted peaks over the entire dose range.
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The conclusion of this case can be summarized as following. a) The shape of the glow-curves was stable over a wide dose range. This is a very important result, because in literature, the stability of the TL glow-curves over a wide range of doses was reported for different types of TL dosimeters.
b) All simulated glow curves were successfully fitted with FOM vales less than 0.2%. This means that in this case the CGCD acts as an excellent technique.
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ACCEPTED MANUSCRIPT c) The output values of activation energy of each TL peak is accurately reproduced. The integral of each TL peak was found to follow a clear functional dependence on dose, which is principal property required for dosimetry. The values of the kinetics order
was found to be almost ~1 indicating that the
TL peaks are of first-order kinetics. Note that the general order equation for b = 1.001 is identical with
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the usual first order kinetic equation to the sixth significant digit. Therefore, the results of the CGCD analysis have a clear physical meaning.
d) The effect of the strong TDDT competition dominates the effect of the competition among the active
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traps themselves. In other words, the electrons competed by the active traps are negligible compared with the electrons competed by the TDDT. This result was also reported by Kitis et al. (Kitis et al.,
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2017). In the current study, this case would be called “quasi-superposition principle” in which the results obtained by the CGCD method are reliable.
4.2.1. Glow curves
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4.2.Case of MN < MO and weak competition
The simulated glow-curves over different doses in case of "& < "# and weak competition are presented in
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Figure 5 and the geometrical properties of each glow-peak are illustrated in Figure 6.
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Figure 5. The IMTS model simulation results showing a) the glow-curves, b) normalized glow-curves, c) example of CGCD of glow-curve in case simulated at low-dose, and d) example of CGCD of glow-curve simulated at high dose. The simulations were performed considering the case of "& < "# and weak competition effect. The FOM was less than 0.25%.
A general conclusion can be drawn from figures 5 and 6 that the glow-peak 5 has different properties than the other glow-peaks. The position of peak 5 shifts toward the lower temperatures with increasing the 15
ACCEPTED MANUSCRIPT absorbed dose as it is shown in Figures 6. This behavior is known for the second-order kinetics peaks where "& = "# . This observation is in agreement with the conclusion made by Chen and Pagonis (Chen & Pagonis, 2013) that the last peak in a series obtained by a model of a single recombination center and multiple traps may be of second order. However, this is restricted only to the case of the absence competition due to TDDT. For the
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other glow-peaks, the geometrical properties tend to be stable over the absorbed dose range as the glow-peak is
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farther away from competitor peak, i.e., peak 5.
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Figure 6. The peak position and the kinetics order of the glow-peaks simulated by the IMTS model over different doses in case of "& < "# and weak comptition effect.
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4.2.2. TL dose response curve
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Figure 7. The dose-response curves of peaks 2 and 5 of glow-curves simulated by the IMTS model in case of "& < "# and weak competition..
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The dose-response curves of peaks 2 and 5 are presented in Figure 7. The behavior of the dose-response curves of peaks 3 and 4 are identical with the dose-response curve of peak 2, therefore they were not represented in this figure. As in case 4.1, a supra-linear (quadratic) dose dependent was observed for the peaks 2, 3 and 4 followed by a sub-linear region as the traps approach the dose-saturation level. However, a nonfamiliar behavior has been observed for peak 5. A sub-linear dose response for peak 5 was observed over the investigated dose range. The degree of the sub-linearity increases with increasing the absorbed dose.
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ACCEPTED MANUSCRIPT Usually, the sub-linear dose dependence is observed as the trap approaches the saturation level at high dose when the available traps are close to be filled. However, the case of the dose-response curve of peak 5 is different. In this case, the sub-linear dose dependence region is far from the dose-saturation region. Mische and McKeever (Mische & McKeever, 1989) suggested that the sub-linear dose dependence is may be caused by
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unknown radiation damage mechanisms or self-absorption of the TL light. On the other hand, Laweless et al. (Lawless et al., 2009) proved that the dose-dependence of the trap carriers may be significantly sublinear even when the traps are far from approaching the saturation level. They attributed this behavior to the competition
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between the trap and the recombination center during the irradiation stage.
Figure 8. The concentration of the electrons filled the traps 2 and 5 after the irradiation stage in case of " < "# and weak competition.
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ACCEPTED MANUSCRIPT It should be noted that the trap of peak 5 is the deepest trap. Therefore, during the heating stage, the trap of peak 5 acts as the competitor for the other traps. Thereby, it has different geometrical and dose-dependence behavior than the other traps. This competition occurs only during the heating stage. Therefore, it is anticipated that the behaviors of all the traps during the irradiation stage are identical. This is obvious in Figure 8 which presents
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the sub-linear dependence of the free carriers accumulated in the traps over different doses during the irradiation stage. The behaviors of the accumulated free carriers in the other traps are approximately the same. Figure 8 confirms that all the traps have the almost the same behavior during the irradiation stage and the dose-
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dependence behavior of peak 5 was different due to its competition effect occurring during the heating stage. Based on our knowledge, similar dose response curve like the one of peak 5 presented in Figure 8 was not
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observed in real experimental glow-curves. This implies that most of the experimental glow-curves are often subjected to strong competition effect either due to TDDT during the irradiation stage or competition among the
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active traps during the heating stage.
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4.2.3. The activation energy computed by CGCD
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Figure 9. The activation energy values estimated using CGCD algorithm in case of " < "# and weak competition. The simulation input 4 values are repsented by solid line for each glow-peak. The vertical dashed line indicates the dose-saturation level.
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The activation energy values for the deconvoluted glow-peaks are presented in Figure 9. The CGCD algorithm could compute the activation energy satisfactorily for the glow-peaks 2 and 3. However, at low dose level, an
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over estimation for the activation energy value of peak 4 was observed. As the dose increases, the accuracy of the estimated activation energy of peak 4 increases. It should be noted that this over estimation may not be due to the competition effect but rather because of the complex overlapping between this peak and its neighbor glow-peaks especially at the low dose level (Sadek, 2013). The conclusion of this case can be summarized as following;
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ACCEPTED MANUSCRIPT a) The instability of the shape of the glow-curves over the different doses would be a clear evidence that one of the active trap has acted as a competitor. In this case, the accuracy of the kinetic parameters estimated by the CGCD is satisfactory for all the peak except for the peak which has acted as a competitor.
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b) In this case, the quasi super-position state is very difficult to be established. However, even so, the
as a dosimeter. 4.3.Case of MN > MO and strong competition
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application of CGCD offer useful and reliable information about the suitability of a material to be used
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This case simulates the effects of strong competition due to the TDDT during the irradiation stage and also the strong competition among the active traps during the heating stage. Therefore, one may consider this case as
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extreme competition effects. The simulated glow-curves are presented in Figure 10.
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Figure 10.The IMTS model simulation results showing a) the glow-curves, b) normalized glow-curves, c) example of CGCD of glow-curve in case simulated at low-dose, and d) example of CGCD of glow-curve simulated at high dose. The simulations were performed consider "& > "# .The FOM was less than 0.45%.
4.3.1. Glow curves
Although in this case the simulations were performed assuming strong competition due to TDDT as in case discussed in section 4.1, the stability of the geometrical properties of the glow-curve of the different doses did not hold and the shape of the glow-curves changed as the traps approached the saturation level. It should be 22
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noted that in the current case "& > "# , and this is the only difference from case 4.1. In fact, setting "& > "# yields a strong competition between the active traps during the heating stage and the deepest traps will act as competitors. Therefore, the changes in the glow-curves are significant for peaks 4 and 5 as shown in Figure 10c. The width of these glow-peaks increased significantly and the goodness of the fitting decreases. It should be
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noted that the case discussed by Kitis et al. (Kitis et al., 2017) did not considered the high dose cases, and thereby the shape of the glow-curve presented in Figure 10c was not reported in their study.
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4.3.2. The dose response curve
Figure 11. The dose-response curves of peaks 2 and 5 of glow-curves simulated by the IMTS model in case of "& > "# and strong competition..
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ACCEPTED MANUSCRIPT The dose-response curves of peaks 2 and 5 are illustrated in Figure 11. The dose response of the other glowpeaks is almost identical with the dose response curves presented in this figure. The difference between the dose-response curves in the current case and the dose-response curves presented in case 4.1 is that in the current case, the quadratic dose dependence is followed by supra-quadratic dose dependence instead of sub-quadratic
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dose dependence. This behavior was reported for the 110oC TL peak of synthetic quartz (Chen et al., 1988). Later, Chen et al. (Chen et al., 1996) attributed this behavior to the competition effect during the heating stage. However, it should be noted that in their model, two recombination centers were considered in their simulation
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which is not the case of the simulation in the current work.
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4.3.3. The activation energy computed by the CGCD algorithm
Figure 12. Activation energy values computed by the CGCD algorithm in case of "& > "# and strong competition. The simulation input values are represented by dashed lines. The vertical line indicates to the dose-saturation level.
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ACCEPTED MANUSCRIPT The activation energy values computed by the CGCD algorithm over the absorbed dose range are presented in Figure 12. The CGCD algorithm using the GOK equation could compute the activation energies of peaks 2 and 3 all over the complete dose range with error < 10%. However, for peaks 4 and 5, the CGCD algorithm could accurately compute the activation energy as the traps are far from the saturation level. As the traps approach the
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saturation level, the activation energy values are underestimated by more than 30%. It should be noted that these results are different than the results reported by Kitis et al. (Kitis et al., 2017) using the initial region (IR) method. In their results, they reported an underestimation in the activation energy value of peak 5 by 20% in the
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evaluated with error ≤ 5% in the same dose region.
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absorbed dose region 10T : 10 K . However, in the current study, the activation energy of peak 5 could be
It worthwhile to mention that the underestimation of the activation energy of peak 5 was reported only at the very high dose in the extreme competition case. In other words, this underestimation should not have significant effect on the general accuracy of the CGCD algorithm because this case is rare to be found in experimental
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glow-curve.
The conclusion of this case can be summarized as following. a) The shapes of TL glow curves are very stable and show some differentiation only when approaching the
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saturation.
b) The fitting of the TL glow curves was excellent giving very low FOM values. The CGCD analysis
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appear in this case again as an excellent technique. c) The output parameters are successfully reproduced for peaks 2, 3 and 4 in whole dose region. For the last peak there is an underestimation appearing at onset of saturation. The behavior of individual peaks as a function of dose is easily discriminated and the practical dosimetry is applicable in a wide dose region up to the onset of saturation mainly far and not near the saturation. Therefore, the CGCD results could be physically meaningful.
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ACCEPTED MANUSCRIPT d) In this case the TDDT is a strong competitor, which however, cannot eliminate the competition among the active traps. Due to the remnant competition among the active traps (see also (Kitis et al., 2017)) there is a weak violation of the SP, so that a quasi SP state can be poorly established. However, even so the CGCD analysis is correct and therefore derives meaningful results for each glow peak in the
4.4.Case of MN > MO and weak competition.
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greater part of their TL dose response curves.
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active traps during both the irradiation and heating stages.
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In this case, the competition due to TDDT is negligible, and thus, the competition occurs among the
4.4.1. Glow Curves
The glow-curves simulated by the IMTS model in case of "& > "# and weak competition are presented in
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Figure 13.
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Figure 13. The IMTS model simulation results showing a) the glow-curves, b) normalized glow-curves, c) example of CGCD of glow-curve in case simulated at low-dose, and d) example of CGCD of glow-curve simulated at high dose. The simulations were performed considering "& > "# and weak competition. The FOM is less than 1.2%.
In the current case the shape of the glow-curves changes with increasing the absorbed dose. At very low dose, the glow-peaks 2, 3 and 4 are hardly observable. Therefore, the deconvolution analysis process was difficult to be performed for these glow-curves, especially for peak 4 because of its complex overlapping with peaks 3 and 5. Moreover, it is obvious that the geometrical properties of the peak 5 changes with increasing the absorbed dose. The peak position decreases and the peak width increases with increasing the absorbed dose. 27
ACCEPTED MANUSCRIPT Therefore, a general conclusion can be made that the last glow-peak in a series of overlapped peaks acts as a second-order peak only in the absence of competition of TDDT during the irradiation stage or in the presence of strong competition among the active traps during the heating stage. Similar to case 4.2, the geometrical
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properties of the other glow peaks are almost stable over the absorbed dose range (Figure 14).
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Figure 14. The geometrical properties showing a) the peak position and b) peak width of the glow-peaks simulated with IMTS models over different doses in case of "& > "# and strong competition.
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4.4.2. The dose response curve
In fact, the changes in the geometrical properties of peak 5 confirms that this glow-peak plays the role of the
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competitor during the heating stage. In all the previous cases, the dose response curves were either supra-linear or sub-linear dose dependence for all the deconvoluted peaks. In the current case, a complete different situation was observed. The dose-response curves of peaks 2, 3 and 4 showed a supra-linear dose dependence. While, the dose-response curve of peak 5 showed an almost linear dose-dependence as it is illustrated in Figure 15.
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Figure 15. Dose response curves for a) peak 5 and b) peak 2 of glow-curves simulated using IMST model in case of "& > "# and weak competition.
An almost linear dose-dependence was observed for peak 5, while a quadratic dose-dependence was observed for the other glow-peaks. The results presented in Figure 15 indicate that even in the presence of competition
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among active traps, a linear dose-dependence can be observed for the last glow-peak. However, this occurs only in the absence of the competition of the TDDT. It should also be noted that the linear region is followed by sublinear region prior to approaching the dose-saturation level, i.e., no supra-linear dose dependence was observed.
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On the other hand, the dose-response curve of the other glow-peaks starts with quadratic then supra-quadratic
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dose-dependence before approaching the dose-saturation level.
4.4.3. The activation energy computed by the CGCD The activation energy values obtained by the CGCD over the different doses are presented in Figure 16. The activation energy values of peaks 2 and 5 have almost the same trend over the absorbed dose range. The CGCD method can obtain the activation energy values of these peaks with a 5% error level as the traps are far from the saturation level. However, as these traps approach the saturation level, the computed activation energy values are underestimated. On the other hand, significant deviations from the true value were observed for the 29
ACCEPTED MANUSCRIPT activation energy values computed for peaks 3 and 4. This case is similar to case 4.3 in which the complexity of the glow-curve, especially at the low dose affects the efficiency of the CGCD algorithm. The conclusion of this case can be summarized as following.
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a) The shapes of TL glow curves vary strongly as a function of dose. b) Even in this extreme case the CGCD analysis appears as a satisfactory analytical technique. c) The output parameters do not reproduce the input values for all peaks. Therefore, the CGCD analysis
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despite the good fits give results which maybe seems physical but they are not necessary correct. d) The behavior of individual peaks as a function of dose is easily discriminated. This means that even in
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this extreme case the practical dosimetry is possible.
e) The competition among the active traps themselves becomes dominant when the TDDT competition is negligible. This case is different from the case that was previously discussed in section 4.2. In the current case, most of the electrons are captured by the last active trap because of its high trapping probability. As a consequence, the electrons concentration in the other low-temperature trap, and thereby
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their TL intensity are decreased. Therefore, at low doses, the low-temperature glow-peaks were not visible compared to the last glow-peak of the highest trapping probability.
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f) The results of the cases are characteristic of strong violation of SP.
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Figure 16.Activation energy values computed by the CGCD algorithm in case of "& > "# and weak competition. The simulation input values are represented by solid lines.
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ACCEPTED MANUSCRIPT 5. General discussion In the current study, it has been shown that in many cases, the activation energy values obtained by the CGCD algorithm are close to the input values even with lack of the superposition principle. This does not mean
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the CGCD algorithm always gives accurate activation energy value as it has been also shown in the current study. However, one should distinguish between the accuracy and the applicability of the CGCD algorithm. In other words, not obtaining accurate kinetic parameters because of the complexity of the glow-curve is not the same as not obtaining accurate kinetics parameters because the model used in the CGCD failed to describe the
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experimental peak.
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In literature, many investigators examined the accuracy of the peak fitting algorithm via simulations. Recently, Sadek and Kitis (Sadek & Kitis, 2017) have concluded that two main reasons may lead to unrealistic simulated peaks; the improper selections of the simulation inputs, and performing the TL simulation considering the heating stage only, i.e., without considering the irradiation stage in the simulation process.
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Nevertheless, this does not mean that the results obtained by the CGCD algorithm are always correct. In some cases, especially for complex glow-curves in which infinite analysis CGCD solutions are available, it would be useful to validate the results obtained by the CGCD via using another analytical method, e.g., the
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different heating rate method (Hoogenstraaten, 1958), peak shape method (Kitis & Pagonis, 2007), Tl − Tmnop
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method (McKeever, 1980), or the isothermal decay method (Moharil, 1981)
6. Conclusions
The basic and general conclusion of the present work can be summarized as following. •
Regardless the validity of the SP principles, the CGCD algorithm could be used to investigate the TL dose-response function of the individual glow-peaks. Thus, in dosimetric applications, using the peak
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ACCEPTED MANUSCRIPT area calculated by the CGCD algorithm would be more appropriate than using the area under a certain portion of the glow-curve. •
The quality of fit by CGCD analysis was excellent in all cases. Therefore, the present simulation showed clearly that the use of CGCD to analyze composite TL glow curves using anyone of the available TL
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expressions and not only the one used in the present work, is completely independent of the violation or not of the SP. •
The CGCD results concerning the trapping parameters needs more attention in their interpretation. It
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was found that the output values of activation energy are physically meaningful in case Ar < Al and strong TDDT competitor, physically meaningful up to the onset of dose saturation in cases Ar < Al
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weak TDDT competitor and Ar < Al strong TDDT competitor. The output values of activation energy
were erroneous in the case of Ar > Al weak TDDT competitor. •
Finally, the use of the CGCD as an analysis technique can be applied with confidence in experimental work. However, special care is need for the interpretation of its results concerning the trapping
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parameters, which however, can be evaluated using many of the methods available in TL theory.
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ACCEPTED MANUSCRIPT References Balian, H.G. & Eddy, W., 1977. Figure-of-merit (FOM), an improved criterion over the normalized Chi-squared test for assessing goodness-of-fit of Gamma-ray spectral peaks. Nucl. Instrum. Methods, 2389-395. Bos, A.J.J., Piters, T.M., Gomez-Ros, J.M. & Delgado, A., 1994. An intercomparison of glow curve analysis computer programs: II. measured glow curves. Radiat. Prot. Dosim., 2257-264.
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Bull, R.K. et al., 1986. Thermoluminescnce kinetics for multiple glow peak curves produced by the release of electrons and holes. J. Phys. D. Appl. Phys, 19, pp.1321-34. Chen, R., Fogel, G. & Lee, C.K., 1996. A new look at the models of the superlinear dose dependence of thermoluminescence. Radiat. Prot. Dosim., 65, pp.63-68.
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Chen, R., Kristianpoller, N., Davidson, Z. & Visocekas, R.J., 1981. Mixed first and second order kinetics in thermally stimulated process. J. Lumin., 2293-303.
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Chen, R. & Pagonis, V., 2013. On the expected order of kinetics in a series of thermoluminescence (TL) and thermally stimulated conductivity (TSC) peaks. Nucl. Instrum. Method Phys. Res., B 312, pp.60-69. Chen, R., Yang, X.H. & McKeever, S.W.S., 1988. The strongly superlinear dose dependence of thermoluminescence in synthetic quartz. J. Phys. D: Appl. Phys., 21, pp.1452-57. Chen, R., Yang, X.H. & McKeever, S.W.S., 1988. The strongly superlinear dose dependence of thermoluminescence in synthetic quartz. J. Phys. D: Appl. Phys., 21, pp.1452-57.
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Farag, M.A. et al., 2017. Radiation damage and sensitization effects on thermoluminescence of LiF:Mg,Ti (TLD-700). Nucl. Intrum. Methods, B 407, pp.180-90. Halperin, A. & Chen, R., 1966. Thermoluminescence of semiconducting diamands. Phys. Rev., 148, pp.839-45. Hoogenstraaten, W., 1958. Electron traps in zinc sulphide phosphors. Phlips Res. Rep., 13, p.515.
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Horowitz, Y.S., 1984. General characteristics of thermoluminescent materials. In Y.S. Horowitz, ed. Thermoluminscence and thermoluminescent dosimetry. Boca Raton: CRC Press. pp.89-172. Horowitz, Y.S., 1984. Recent models for TL supralinearity. Radiat. Prot. Dosim., 6, pp.17-20.
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Horowitz, Y.S., 2001. Theory of thermoluminescence gamma dose response: The unified interaction model. Nucl. Instrum. Methods, B 184, pp.68-84. Horowitz, Y.S. & Yossian, D., 1995. Computerized glow curve deconvolution: Application to thermoluminescence dosimetry. Radiat. Prot. Dosim., 21-110. Kitis, G., Gomez-Ros, J.M. & Tuyn, J.W.N., 1998. Thermoluminescence glow-curve deconvolution functions for first, second and general orders of kinetics. J. Phys. D: Appl. Phys., 31, pp.2636-41. Kitis, G. & Pagonis, V., 2007. Peak shape methods for general order thermoluminescence glow-peaks: A reappraisal. Nucl. Instrum. Methods , B 262, pp.313-22. Kitis, G., Pagonis, V. & Tzamaris, S.E., 2017. The influence of competition effects on the initial rise method during thermal stimulation of luminescence: A simulation study. Radiat. Meas., 100, pp.27-36. 34
ACCEPTED MANUSCRIPT Kitis, G. & Vlachos, N.D., 2013. General semi-analytical expressions for TL, OSL and other luminescence stimulation modes derived fron the OTOR model using the Lambert W-function. Radiat. Meas., 247-54. Lawless, J.L., Chen, R. & Pagonis, V., 2009. Sublinear dose dependence of thermoluminescence and optically stimulated luminescence prior to the approach to saturation level. Radiat. Meas., 44, pp.606-10. May, C.E. & Partridge, J.A., 1964. Thermoluminescent kinerics of alpha-irradiated alkali halides. J. Chem. Phys., 21401-1409.
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McKeever, S.W.S., 1980. On the analysis of complex thermoluminescence glow-curves: Resolution into individual peaks. Phys. Status Solidi, a 62, pp.331-40. McKeever, S.W.S., 1985. Thermoluminescence of solids. Cambridge: Cambridge University Press.
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Mische, E.F. & McKeever, S.W.S., 1989. Mechanisms of supralinearity in lithium fluoride thermoluminescence dosemeters. Radiat. Prot. Dosim., 29, pp.159-75. Moharil, S.V., 1981. Trapping parameters from iso-thermal decay of TL. J. Phys. D. Appl. Phys., 14, p.1677.
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Obryk, B. et al., 2014. On LiF:Mg,Cu,P and LiF:Mg,Ti phosphors high & ultra-high dose features. Radiat. Meas., 71, pp.25-30. Pagonis, V. & Kitis, G., 2012. Prevalence of first-order kinetics in thermoluminescence material: An explanation based on multiple competition process. Phys. Status Solidi B, 21590-1601. Randall, J.T. & Wilkins, M.H.F., 1945. Phosphorescence and electron traps. I. The study of trap distribution. In Proc. R. Soc. London, 1945.
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Rodine, E.T. & Land, P.L., 1971. Electronic defect structure of single crystal ThO2 by thermoluminescence. Phys. Rev., B4, pp.2701-24. Sadek, A.M., 2013. Test of the accuracy of the computerized glow curve deconvolution algorithm for the analysis of thermoluminescence glow curves. Nucl. Instrum. Methods A, 712, pp.56-61.
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Sadek, A.M. et al., 2015. The deconvolution of thermoluminescence glow-curves using general expressions derived from the one trap-one recombination (OTOR) level model. Appl. Radiat. Isot., 2214-221.
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Sadek, A.M., Eissa, H.M., Basha, A.M. & Kitis, G., 2014. Properties of the thermoluminescence glow peaks simulated by the interactive multiple-trap system (IMTS) model. Phys. Status Solidi B, 2721-729. Sadek, A.M., Eissa, H.M., Basha, A.M. & Kitis, G., 2014. Resolving the limitation of the peak fitting and peak shape methods in the determination of the activation energy of thermoluminescence glow peaks. J. Lumin, 2418-423. Sadek, A.M. & Kitis, G., 2017. A critical look at the kinetic parameter values used in simulating the thermoluminescence glow-curve. Journal of Luminescence, 183, pp.533-41. Wolfson, R., 2016. Facts101:Essebtial University Physics. Cram101 Textbook Reviews.
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Highlights: The impact of non-fulfillment of the SP theorem on the CGCD analysis was discussed. TL glow curves were simulated over a wide range of doses and different conditions. In general, the non-fulfillment of the SP theorem has no effect on the CGCD method.
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