Properties of the thermoluminescence glow peaks simulated by the interactive multiple-trap system (IMTS) model

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Properties of the thermoluminescenceglow peaks simulated by the interactivemultiple-trap system (IMTS) model

A. M. Sadek*,1, H. M. Eissa1, A. M. Basha2, and G. Kitis3

1 Ionizing Radiation Metrology Department, National Institute for Standards, El-Haram, Giza, Egypt2 Physics Department, Faculty of Science, Fayoum University, Fayoum, Egypt3 Nuclear Physics and Elementary Particles Physics Section, Physics Department, Aristotle University of Thessaloniki,54124 Thessaloniki, Makedonia Greece

Received 6 August 2014, revised 28 October 2014, accepted 4 November 2014Published online 8 January 2015

Keywords charge-carrier trapping, glow curve analysis, thermoluminescence

* Corresponding author: e-mail [email protected]; Phone: þ20 111 40 77 224

Thermoluminescence (TL) glow peaks were simulated over awide range of absorbed doses using the interactive multiple-trap system (IMTS) model. The absorbed dose range wasdivided into three regions; a region in which the measured TLsignal grows quadratically with the absorbed dose, a supra-quadratic dose-response region and dose-saturation region.The properties of the simulated glow peaks were investigatedfor each dose region. The different behaviors of the maximumpeak position (Tmax) with the absorbed doses were discussed.The applicability of applying the general-order kinetics(GOK), the mixed-order kinetics (MOK), and the developed

one trap–one recombination (OTOR) center expressions onthe IMTS glow peaks was investigated. The results showedthat, in general, the developed OTOR expressions are themost apt expressions to be used in the peak-fitting method forthe deconvolution of the experimental glow curves. Theaccuracy of the GOK and MOK expressions in describing theIMTS glow peaks is dependent on the value of the ratio ofthe trapping probability to the recombination probability(R¼A1/Am) and the amount of the absorbed dose. New TLexpressions based on the non-interactive multiple-trap system(NMTS) model were deduced.

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1 Introduction The most common model for ther-moluminescence (TL) in materials is the one trap–onerecombination (OTOR) level model. This model involvesonly one trap and one recombination center in which thethermal elevation of pretrapped electrons from the trap sitesinto the conduction band. Once in the conduction band,the electrons either are captured by the holes trapped inluminescence centers, or are retrapped in empty electrontraps [1]. The OTOR level model shows all the character-istics of the TL and accounts for this behavior of the glow-peak shape under variation of the dose and heating rate.However, there is no existing TL material known that isaccurately described by the simple model [2]. In fact, theOTORmodel failed to explain some observed phenomena insome TL materials. One of the most important phenomenathat the OTOR model failed to explain is the supralinearbehavior reported (e.g., [3–7]) for some important TLmaterials used in doismetry and in dating. Indeed, the OTOR

model can be applicable to the TL materials that exhibit onlyone glow peak or, if more than one glow peak occurs, it canbe applicable only if the peaks do not interact, i.e., if they arecompletely independent [8]. If more than one glow peak isobserved, it is likely that some interaction among the trapsmust occur.When interactions among different types of trapsoccur, i.e., when more than one glow peak is present andcharges released from one type of trap may be trapped onother types of traps, the TL kinetics can be described by theinteractive traps model. This does not imply that the simplemodel has no meaning. On the contrary, it can assist us inthe interpretation of many features that can be consideredvariations of the OTOR model.

The interactive multitrap system (IMTS) model wasdeveloped to describe the situations in which the interactionsbetween the traps occur. This model was employed [9, 10]for explaining the supralinearity of dose dependence of TL.In the IMTS model, not all the traps shall be active in the

Phys. Status Solidi B, 1–9 (2014) / DOI 10.1002/pssb.201451406

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� 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

temperature range in which the specimen is heated. Acompetitor, which is a thermally disconnected deep trap, isone that can be filled with electrons during irradiation.Unlike in the OTOR model, in the IMTS model not all theelectrons excited by the radiation dose are captured by theactive trap. The competitor captures some of these electronsduring the irradiation stage. However, when the competitorapproaches the dose saturation, the interaction between theactive trap and the competitor vanishes and hence, the IMTSmodel turns into the non-interactive multitrap system(NMTS) model.

The electron transitions among the traps in both theirradiation and heating stages in the IMTS model arerepresented in Fig. 1. In the figure, N1 and N2 in cm�3

are the electron trap concentrations of the active trap(AT) and the competitor, respectively, with instantaneousoccupancies of n1(t) and n2(t), respectively. A1 and A2 incm3/s are the trapping probability coefficients of the twotraps, respectively. M in cm�3 is the recombination centerconcentration with instantaneous occupancy m(t). Theprobability coefficient for holes to get trapped in the centeris B in cm3/s and the electron recombination probabilitycoefficient is Am in cm3/s. nc(t) and nv(t) denote theconcentrations of free electrons and holes, respectively. X incm�3/s denotes the rate of production of electron–hole pairsby the excitation dose, which is proportional to the dose rate.The set of simultaneous differential equations governing the

process during irradiation stage is given by [9]:

dn1dt

¼ A1ðN1 � n1Þnc; ð1Þ

dn2dt


dmdt

¼ BðM � m1Þnv � Ammnc; ð3Þ

dnvdt

¼ X � BðM � mÞnv; ð4Þ

dncdt

¼ dmdt

þ dnvdt

� dn1dt

� dn2dt

: ð5Þ

The set of the simultaneous differential equationsgoverning the heating stage is given by:

dn1dt

¼ A1ðN2 � n1Þnc � s n1exp � E

kT

� �; ð6Þ

dn2dt


dmdt

¼ �Ammnc; ð8Þ

dncdt

¼ dmdt

� dn1dt

� dn2dt

; ð9Þ

where E in eV is the activation energy of the active trap, s ins�1 is the frequency factor. The aims of this work are to:

(i) investigate the geometrical properties of the glow peakssimulated by the IMTS model in different dose regions;

(ii) verify the possibility of applying methods not based onthe IMTSmodel to describe the glow peaks simulated bythe IMTS model over the different dose regions.

2 Theoretical and computational considerationsIn order to follow the experimental procedure, the simulationwas conducted on three stages. The first stage is the irradiationstage. In this stage, the set of the differential equationsgiven by Eqs. (1)–(5) is solved for a certain period of time(irradiation time), and thus, the dose applied is proportional toXt. The next stage of the simulation is the relaxation stage,solving the same set of equations for an additional period oftimewhen the excitation is switched off (X¼ 0). Thus, the freeelectrons and holes remaining in the conduction and valencebands, respectively, decay into the respective trapping states,thus contributing to the final concentrations. The final valuesof the instantaneous functions obtained in the irradiationstage were used as initial values in the relaxation stage. Thethird stage of the simulation is the thermal heating stage.In this stage, the set of the differential equations given by

Figure 1 Energy diagram of the electron transitions among thetraps during (a) the irradiation stage and (b) the heating stage in theIMTS model.

2 A. M. Sadek et al.: Properties of the TL glow peaks simulated by IMTS model

� 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-b.com

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Eqs. (6)–(9) is solved for a certain heating rate andtemperature range. The temperature as a function of time isT(t)¼T0þbt, where b is the linear heating rate and T0 is theinitial temperature. The final values of the instantaneousfunctions obtained in the relaxation stage were used as initialvalues in the heating stage. The sample is heated and theemitted TL signal is recorded as a function of temperature. Inall the simulations performed in this work, the sets of thedifferential equations were solved using the Fehlberg–Runge–Kutta method. The MATLAB ode15s built-in functionwas used to solve the sets of differential equations for certaininput parameters.

In this work, the glow peaks were simulated either withrandom values of irradiation doses or random values ofthe ratio of the trapping and recombination probabilitycoefficients of the active trap (R¼A1/Am). In the case ofdifferent irradiation doses, since the absorbed dose is assumedto produce X (in s/cm3) electron–hole pairs, the absorbed dosecan be considered to be Xt, where t is the total length ofirradiation time [11]. Thus, the random values of the absorbeddose were obtained by using a fixed value of X and randomvalues of the irradiation time twithin the chosen range.While,in the case of different values of R, the values of A1 and Am

were randomly selected from a certain chosen range.

3 The geometrical properties of the TL glowpeaks simulated by the IMTS model at the differentdose regions A set of glow peaks was simulated by theIMTS model at different doses. The behaviors of theconcentrations of the electrons filled up the active trap (n10)and those filled up the competitor (n20) after the relaxationstage, and the measured TL signal by the peak area and peakmaximum (Imax) over the absorbed dose range are illustratedin Fig. 2. The parameters employed in the simulation are

mentioned in the caption of the figure. The calculated TLsignal either by the peak area or Imax starts quadratically withthe absorbed dose in the dose region where the competitor isfar from saturation. It was concluded that in reality, the TLmaterials with two or more glow peaks, neither the peak areanor peak maximum varies linearly with dose, except inspecial situations [12]. This is because at low doses, both n1and n2 grow linearly. It means that the electron numbertrapped in the competitor increases with increasing absorbeddose. Because the capacitor of the competitor N2 is relativelysmall and it has a high A2 value, the competitor approachesthe saturation level. Thus, more electrons are released to theactive trap. When the competitor approaches saturation,the TL intensity, measured by the peak area or the peakmaximum, is thus necessarily supralinear with the dose [9].At very high doses, the active trap approaches saturationitself. In this region, there are no empty free states to captureany more electrons and thus the emitted TL signal shallremain constant over the doses in this region. It is alsoobserved that both the Imax and the peak area have the samebehavior over the entire dose range. It should be noted that asthe competitor approaches the dose-saturation level, theactive trap recovers from the impact of the competitor. Inother words, the effect of the competitor on the active trapdecreases as the dose-response curve starts to move into thesupraquadratic region. Therefore, it is observed from thefigure that both n10 and the measured peak area start tocoincide in this dose-response region. Thus, all the electronsreleased from the active trap, in this case, contribute inproducing the TL signal.

The GOK model introduced the kinetics order (b)parameter that goes from �1 to 2 as the shape of the glowpeaks gets out of the first-order case to the second-order case.While, the MOK model introduced the parameter a, whichdepends on the values of n10 and n20. As n10 � n20, the shapeof the glow peaks tends to the first-order peak shape.While, asn10 � n20, it tends to the second-order peak shape. For eachsimulated glow peak, the kinetics order, b, was calculatedusing the expression deduced byKitis and Pagonis [13] for thenumerically simulated glow peak, which is given as

nmn0

¼ b

1þ ðb� 1ÞDm

� �1=ðb�1Þ; ð10Þ

where Dm¼ 2kTmax/E, and k is the Boltzmann constant.According to Halperin and Braner [14], the quantity nm=n0,which represents the ratio of the high temperature half-integralof a glow peak over its total integral is the integral symmetryfactor of the glow peak (m0

g). In the case of the mixed-orderkinetics (MOK), the parameter a was calculated using

a ¼ n10n10 þ n20

: ð11Þ

Both b and a depend upon N, n, m, A1, and Am indifferent manners [15]. The behaviors of the kinetics order b,the mixed order a, and the integral symmetry factor m0

g for

Figure 2 Concentration of the electrons filling up the active trap(n10) and the competitor (n20), and the peak area and peakmaximum (Imax) for the glow peaks simulated by the IMTS modelover a certain dose range. The parameters used in the simulationwere: X¼ 108 cm�3/s, N1¼ 1012 cm�3, N2¼ 1010 cm�3, M¼ 1.01� 1012 cm�3, A1¼ 10�9 cm3/s, A2¼ 10�7 cm3/s, B¼ 10�9 cm3/s,Am¼ 10�9 cm3/s, E¼ 1 eV, s¼ 1012 s�1 and b¼ 1K/s.

Phys. Status Solidi B (2014) 3

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the glow peaks simulated by the IMTS model over theabsorbed dose range are illustrated in Fig. 3. In the caseof the OTOR model, Levy [12] concluded that over asufficiently large increasing absorbed dose range, the glowcurves would appear to change from the second- to the first-order kinetics. It is to be noted that this conclusion was basedon running a simulation using the general one-trap equationwith a very low value of the trapping coefficient. However, inthe case of the IMTSmodel, it is observed from Fig. 3 that boththe kinetics order, the mixed order and the integral symmetryfactor change from the first- to the second-order kineticswith increasing absorbed dose. In the quadratic dose-responseregion, the glow peaks are characterized by the first-orderkinetics with kinetics order values of �1.1 (m0

g� 0.42). Thisin fact was not anticipated, because the trapping probabilityused in the simulation was equal to the recombinationprobability (A1¼Am), which is a second-order kinetics case.The domination of the first-order kinetics on the glow peakssimulated at the quadratic dose-response region is caused bythe strong effect of the competitor in this dose region. This canbe observed from the values of a in this region, which are�0.5. This means that, in this dose region, the number ofelectrons captured by the competitor is almost the same as thenumber of the electrons captured by the active trap.

As the absorbed dose increases, the competitorapproaches to the saturation level and the number ofcaptured electrons decreases and thus the values of aincrease. In the supraquadratic region, the glow peaks arecharacterized by kinetics order values between �1.2 and�1.9 which indicates to general- and second-order kineticscases. It is to be noted that the kinetics order values in thesecases do not reflect the actual domination of the retrappingor the recombination during the TL process. This isexplained by the fact that the expressions derived fromthe OTOR model can be applied only when the assumptions

used to derive these expressions are strictly applicable [16].However, the kinetics order values in these cases describeonly the geometrical properties of the glow peaks regardlessof the mechanism of the TL process.

As the competitor approaches the saturation, the glowpeaks in this region are characterized by the second-orderkinetics model with b� 2 (m0

g� 0.52). It follows that thevalues of the kinetics order that reflect the actual mechanismof the TL process appear only in the dose-saturation region.In this region, the values of a are �1, which means that thecompetitor cannot capture any more electrons. Therefore, theeffect of the competitor on the active trap in this regionvanishes and the glow peaks are characterized by the second-order kinetics that are the real condition of the simulatedpeaks (A1¼Am). The model describing the glow peaks, inthis case, turns from the IMTS model into the NMTSmodel [17]. The NMTS model is similar to the IMTS modelexcept that the former assumes that the competitor isalways in the dose-saturation level (N2¼ n20). Thisimplies that the neutrality condition in the IMTS modelis m(t)¼ n1(t)þ n2(t)þ nc(t). While, in the NMTS modelit is m(t)¼ n1(t)þN2þ nc(t). In the OTOR model, whichassumes only one trap and only one recombination center,the neutrally condition is m(t)¼ n(t)þ nc(t). This may beconsidered the main difference between the three models.

Pagonis and Kitis [18] concluded that the strongerthe competition for the luminescence process, the lowerthe kinetic order b value. This is clearly observed in thequadratic dose-response region in which the competitor has astrong impact on the active trap. This in fact also accounts forthis prevalence of the first-order kinetics reported in theliterature. Since the majority of the TL glow curves consist ofmore than one glow peak, the natural mechanism of the TLprocess makes it unlikely that the interaction between thesetrap is zero [12, 19]. However, Chen and Pagonis [20]concluded that the last peak in a series obtained by a modelof a single recombination center and multiple traps may be ofsecond-order kinetics. In fact, the second-order kineticsmodel predicted a shift in the Tmax of the glow peak towardthe low temperatures as a function of increasing the absorbeddose, while the first-order kinetics model predicted nochange in the Tmax due to changes in the dose. Based on thesepredictions, the variable-dose (VD) method was adoptedand utilized [21, 22] to estimate the kinetics order of theexperimental glow peaks. However, this method usuallycould not observe any significant changes in the Tmax valuesof the glow peaks with increasing doses. This, in fact, is dueto the strong effect of the competitor that causes a remarkablestability of the glow-curve shapes and to the prevalence offirst-order kinetics. Therefore the changes in the Tmax withthe doses cannot be observed easily or accurately duringexperimental work. The behavior of the Tmax of the glowpeaks simulated over the absorbed dose range using theOTOR, the IMTS, and the NMTS models is presented inFig. 4. In the case of the IMTS model, the parameters used inthe simulation were the same as in Fig. 2. As the competitorapproaches the dose-saturation level (N2¼ n20), the IMTS

Figure 3 The kinetics order b, the mixed order a and the integralsymmetry factor m0

g for the glow peaks simulated by the IMTSmodel over the absorbed dose range. The dose range was dividedinto three regions: the quadratic region in which the TL signalgrows quadratically with the dose, the supra-quadratic region, andthe dose-saturation region. The parameters used in the simulationare the same as in Fig. 2.



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model turns into the NMTS model. In the case of the OTORmodel, N2¼ 0. It is observed that in the case of the IMTSmodel, no change in the Tmax values can be observed over thedose region where the competitor is filled up with electrons.This means that in this dose region, the effect of thecompetitor on the active trap holds the stability of the Tmax.This behavior was experimentally observed for peaks 1–6and 8 in CaF2:Dy at the dose range of 1.2–24Gy [21], for themain glow peak of CaF2:Mn at the dose range of �0.015–110Gy [22] and all the glow peaks except the last one in thenatural CaF2 at the dose range of 0.015 to �2 kGy of beta-rays from a 90Sr–90Y source [23].

In the case of the NMTS model, the competitor issaturated and its effect on active trap is removed. At thispoint, since A1¼Am, the glow peaks are characterized by thesecond-order kinetics. Therefore, abrupt change in the Tmax

is observed. At this point, the behavior of the Tmax predicatedby the NMTS model is the same as that predicted by theOTOR model. Therefore, it is observed from Fig. 4 that theTmax of the NMTS glow peaks coincides with the Tmax ofthe OTOR glow peaks. In both models, a decrease in the Tmax

values followed by a stable behavior was observed onincreasing the absorbed dose. However, it is important to notethat in the case of the OTOR model, only the stable behaviorof the Tmax occurs at the saturation dose level. While, in thecase of the NMTS model, both of the decreasing and thestable behaviors occur in the saturation dose level. This, infact, may explain the observation reported by Hsiang-ENet al. [24] of decreasing the Tmax of peak 2 of CaF2:Mn onincreasing the absorbed dose at the dose-saturation level(�80–1020Gy). The last authors also indicated a confusingincrease in the Tmax of peak 3 in the same dosimeter at thedose range of�80–200Gy. This increasing in the Tmax couldbe attributable to approaching the competitor to the saturationlevel. Therefore, if the glow peak was characterized by asecond-order kinetics, an increase in the Tmax would beobserved, as was discussed above.

4 The accuracy the activation energy obtainedby GOK andMOK expressions for the TL glow peakssimulated by the IMTS model over the absorbeddose range In this section, the activation energy (E) wascalculated for each glow peak simulated by the IMTS modelover the absorbed dose range using the peak-fitting methodand peak-shape method (PSM) based on the GOK and theMOK. The GOK- and MOK-PSM expressions are given,respectively, as [13, 25]:

EGOK ¼ Cvbb=ðb�1Þk

T2m

v� 2kTm; ð12Þ

EMOK ¼ Cv1m0

g

Fm þ a

Fm

kT2m

v; ð13Þ

where Cv is the respective triangle assumption pseudo-constant which is given by Cv ¼ vIm=bn0 [26], v the totalwidth of the peak, and k is the Boltzmann constant. Theparameters a and Fm can be calculated using the followingrelations [27]:

m0g ¼

1� a

Fm � a; ð14Þ

Fm ¼ 2:58226� 2:13911 aþ 0:55071 a2: ð15Þ

For the peak-fitting method, the following expressionswere used for the GOK and MOK, respectively, [28]:

IGOK ¼ s n0exp � E

kT

� �1þðb�1Þ s

b

ZT

T0

exp � E

kT

� �dT

8<:

9=;

b=ðb�1Þ

;

ð16Þ

IMOK ¼s0c2a exp � E

kT

� �exp

c s0

b

� �Z T

T0

exp � E

kT

� �dT

� �

expc s0

b

� �Z T

T0

exp � E

kT

� �dT � a

� �2 ;

ð17Þ

where n0 (cm�3) is the initial concentration of the trappedelectrons, s0 ¼ s=ðN þ cÞ, where c represents the number oftrapped electrons not taking part in the TL process in thetemperature range being considered due to their being indeep traps or in low-probability recombination centers. Theactivation energy calculated by the peak-fitting method usingthe GOK and MOK expressions, for the glow peakssimulated by the IMTS model over the absorbed dose rangeis shown in Fig. 5. It is observed that the MOK equationdescribes the glow peaks simulated by the IMTS better thanthe GOK equation over the absorbed dose range. Themaximum error in the E values obtained over the absorbeddose range using the MOK was �4% and the highest FOM

Figure 4 The Tmax of the glow peaks simulated at different dosesby the IMTS, NMTS and OTOR models. The n20 is the electronconcentration captured by the competitor. As n20 approaches thesaturation (N2¼ n20) the IMTS model turns into the NMTS model.In the case of the OTOR model, N2¼ 0.



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value was 0.08%, while, an error of �13% in the E valuesobtained by the GOK equation was found in the supra-quadratic dose-response region. The corresponding highestFOM value was 1.7%. This in fact was expected since itwas concluded [17] that the MOK expression is a superioralternative to GOK for the glow-peak characterization. Thisis because the mixed-order parameter a remains constant atall temperatures T in the glow curve, whereas the kineticsorder b, though assumed constant in the GOK expressions,changes with T when the GOK model is applied to anyphysically plausible model of TL emission, except whenb turns out to be �1 or 2. Due to this variation of b, thevalue found for E always has some error that increasessystematically as the b deviates further away from 1 to 2.Parallel to the error in E, the FOM of the fit also deteriorates.This explains the high error in the EGOK values and the highFOM values observed in the supraquadratic dose-responseregion. In this region, the kinetics order values of the glowpeaks are intermediate between the first- and the second-order kinetics cases. Therefore, it provokes a high error in theobtained E values and thus high FOM values. The samebehavior was noted when the activation energy of the glowpeaks was obtained by the PSM using the GOK and MOKexpressions given by Eqs. (12) and (13), respectively.

It is to be noteworthy that the results in Fig. 5 wereobtained for the glow peaks simulated by the IMTS modelwith R¼ 1. However, in the case of R¼ 103, an error of�53% in EGOK and EMOK values was found in the dose-saturation region. While, for the quadratic and supra-quadratic dose-response region, the error in the obtained Edid not exceed 5%. It was concluded that both the GOKand MOK expressions failed to describe the glow peakssimulated with R � 1 and the sample dose in the saturationregion [17, 18, 29–33]. Therefore, it was suggested [30] touse minimal sample doses for analyzing the glow peaks.However, in the next section, new TL expressions describing

the TL glow peaks simulated by NMTS model with R � 1have been deduced.

5 New TL analytical expressions describing asingle glow peak simulated by the NMTS model Infact, some attempts have been made in order to derive ananalytical equation based on the IMTS model that can fitthe IMTS glow peak. Unfortunately, these attempts havefailed. However, this was attempted for only the NMTSmodel. In the case of the dose saturation, the IMTS modelturns into NMTS model in which N2¼ n20. The analyticalexpression of the TL intensity for the NMTS mode is givenby [17]:

IðTÞ ¼ pn1 þ n2ð Þn1Am

b N1 � n1ð ÞAn þ n1 þ n2ð ÞAm½ � ; ð18Þ

where

p ¼ s exp � E

kT

� �; ð19Þ

mðtÞ ¼ n1ðtÞ þ N2 þ nc: ð20Þ

Introducing the so called quasiequilibrium (QE)approximations, which means assuming nc � n1 þN2,Eq. (20) can be written as

mðtÞ ffi n1ðtÞ þ N2: ð21Þ

Then, Eq. (18) can be written as

IðTÞ ¼ pmn1

N1 � n1ð ÞRþ m: ð22Þ

Introducing the function f as

f ¼ n1ðTÞmðTÞ ; ð23Þ

the values of f change from�0 to 1. It was observed that thevalues of f function are �1 in the low-temperature part ofthe glow peak, while in the high-temperature part of the glowpeak, the values of f decreases as the temperature increases.In this region, the value of f depends on the value of R andalso on the competitor components, i.e., N2 and A2. In orderto deduce TL expressions based on the NMTS model, it wasassumed that this function has an effective value feff that isconstant all over the temperature range of the glow peak.Following the same lines of Kitis and Vlachos [34] and aftersome algebra, one can get:

Zm

m0

N1Rþ mð1� feffRÞm2

dm ¼ �feff s

b

ZT

T0

exp � E

kT

� �dT

ð24Þ

Figure 5 Activation energy obtained by the peak-fitting methodusing the GOK and MOK expressions for the glow peak simulatedby the IMTS model over the absorbed dose range and thecorresponding FOM (%) values. The kinetics parameters used insimulating the glow peaks were the same as in Fig. 2.



ph

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i b

��!yields 1m0

� 1mþ 1� feffR

N1Rln

m

m0

� �

¼ � feffs

N1Rb

ZT

T0

exp � E

kT

� �dT

ð25Þ

��!yields m0

m eþ ln

m0

m e

� �¼ 1

e� lnðeÞ

þ feffs

1� feffR

ZT

T0

exp � E

kT

� �dT ;

ð26Þ

where

e ¼ m0ð1� feffRÞN1R

: ð27Þ

The solution of Eq. (25) for R< 1 is given by

m ¼ m0

eW0½expðz1Þ� ; ð28Þ

where W0[exp(z1)] is the principal branch of the Lambert-Wfunction for the exp(z1) function, and

z1 ¼ 1e� lnðeÞ þ feffs

1� feffR

ZT

T0

exp � E

kT

� �dT : ð29Þ

However, in the case of R> 1, the e defined by Eq. (27)will have negative value. In other words, e¼�|e| and thusthe solution of Eq. (26) in this case will be given by

m ¼ �m0

eW ½�1;�expð�z2Þ� ; ð30Þ

where W[�1, �exp(�z2)] is the second branch of theLambert-W function for the exp(�z2) function, and

z2 ¼ 1jej þ lnðjejÞ þ feffs

j1� feffRjZT

T0

exp � E

kT

� �dT :

ð31Þ

Then, the analytical expressions for the TL intensity ofthe NMTS model will be given by:

for R < 1:

IðTÞ ¼ N1Rfeff

bð1� feffRÞ2s exp �E=kTð Þ

W0½expðz1Þ� þW0½expðz1Þ�2;

ð32Þ

for R < 1:

IðTÞ ¼ N1Rfeff

bð1� feffRÞ2s exp �E=kTð Þ

W ½�1; expð�z2Þ� þW ½�1;�expðz2Þ�2:

ð33Þ

The peak-fitting method using the MOK and thedeveloped NMTS expressions was used to determine theactivation energy of each glow peak simulated with differentvalues of R by the NMTS model. The results are shown inFig. 6. It can be observed that the developed TL expressionsgiven by Eqs. (32) and (33) can accurately describe the glowpeaks simulated by the NMTSmodel over the full range of R.The error in the calculated E did not exceed 5% and theFOM did not exceed 0.4%. This means that the error due tousing a constant effective value, feff, is acceptable. It is alsoobserved that the values of feff at almost the full range of Rare �0.5, which is the average value of the function f(T)over the temperature range of the glow peak. However, itdecreases to �0.3 at very high values of R. The accuracy ofthe expressions given by Eqs. (32) and (33) decreases as theNMTS model turns into IMTS model. This in fact wasexpected. Since these expressions were derived under thecondition of N2¼ n20. As the absorbed dose decreases andthe competitor is far from the saturation and this conditiondoes not hold.

6 Fitting the IMTS glow peaks using analyticalexpressions derived from the one trap–onerecombination (OTOR) center level model Kitis andVlachos [34] developed new TL analytical expressions todescribe the TL glow peak simulated with linear heatingfunction by the OTOR model. These developed expressionssuccessfully described the TL glow peaks simulated by theOTOR model with R � 1 and the sample dose in the

Figure 6 EMOK, ENMTS, and fNMTSeff obtained by the peak-fitting

method using the MOK and the developed NMTS expressions forthe glow peaks simulated by the NMTS model with differentR¼A1/Am. The values of A1 and Am were randomly selected fromthe interval of 10�10 to 10�7. The other kinetics parameters were thesame as in Fig. 2.



Original

Paper

saturation level. The error in the activation energy obtainedby the peak-fitting method using these developed expres-sions for the numerically simulated glow peaks did notexceed 0.5% [27]. For the linear heating rate, theseexpressions are given as [34]:

for R < 1:

IðTÞ ¼ n0cð1� RÞ

s exp �E=kTð ÞW0½expðz01Þ� þW0½expðz01Þ�2

;ð34Þ

for R > 1:

IðTÞ ¼ n0cð1� RÞ

s exp �E=kTð ÞW0 �1;�expð�z01Þ½ � þW0 �1;�expð�z02Þ½ �2 ;

ð35Þ

where W0[exp(z01)] and W[–1,–exp(–z02)] are the principaland the second branches of the Lambert-W function for thefunctions exp(z01) and exp(–z02), respectively, and

z01 ¼ 1c� lnðcÞ þ s

R TT0exp �E=kTð ÞdTð1� RÞb ; ð36Þ

z02 ¼ 1jcj þ lnðjcjÞ þ s

R TT0exp �E=kTð ÞdTj1� Rjb ; ð37Þ

c ¼ n0ð1� RÞNR

: ð38Þ

The accuracy of the activation energy obtained by thepeak-fitting method using these TL expressions for the glowpeaks simulated with R¼ 100 by the IMTS model over theabsorbed dose range was tested. The results are shown inFig. 7. It is observed that the accuracy of the activation

energy obtained by the peak-fitting method using theseexpressions decreases and high FOM values are observed inthe supraquadratic dose-response region as it was observedin the case of using the GOK expressions. However, on theother hand, the values of the obtained E for the glow peakssimulated at the dose-saturation level are very accurate. Itis to be noted that although all the glow peaks weresimulated with R¼ 100, Eq. (35) was used to fit only thepeaks simulated at the dose saturation range. Otherwise,Eq. (34) was used. This means that the OTOR expressionsare peak-shape dependent regardless of the real mecha-nism of the TL process. Therefore, although, the peak-fitting method using these expressions can provide anaccurate value for the activation energy, the values of Robtained from the fitting do not reflect the actual ratio ofthe trapping and recombination probability. This meansthat the TL expressions given by Eqs. (34) and (35) can beused in the deconvolution of the experimental glow curveeven in very high doses (saturation case) and with thepossibility of R � 1. However, the value of R obtainedfrom the fitting process cannot be a reliable physicalparameter to describe whether retrapping or recombinationdominates.

The general conclusion is that choosing certain TLexpressions based on a certain TL model to be used in thedeconvolution of the experimental glow curve depends onthe experimental conditions including the irradiation doselevel. Except for the supraquadratic dose-response region,the developed OTOR TL expressions [34] are the mostapt expressions to be used in the deconvolution of theexperimental glow curve. This is because the peak fittingmethod using these expressions can provide accurate valuesfor the activation energy of the glow peaks whetherretrapping or recombination is dominates. However, thevalue of R obtained from the fitting process by usingthese expressions is not a reliable physical parameter todescribe whether retrapping or recombination dominates.

7 Conclusions The existence of the competitor mayaffect the geometrical properties, the peak position and themeasured TL signal of the glow peaks.

For the glow peaks simulated with A1¼Am, in thequadratic dose-response region, the glow peaks arecharacterized by the first-order kinetics. In the supra-quadratic region, the glow peaks are characterized by thegeneral-order kinetics, while in the dose-saturation region,the glow peaks are characterized by the second-orderkinetics.

The stability of shape of the glow peaks is held as thecompetitor is far away from the saturation. This accounts forthis prevalence of the first-order kinetics for the glow peaksreported in the literature.

When the competitor is saturated, its impact on the activetrap vanishes and the geometrical shape of the glow peaksreflects the real mechanism of the TL process.

At the dose-saturation level, the behavior of the Tmax

predicted by the NMTS model shows a decrease followed

Figure 7 Activation energy obtained by the peak-fitting methodusing Eqs. (34) and (35) and the corresponding FOM (%) valuesfor glow peaks simulated by IMTS model with R¼ 100 over theabsorbed dose range. The other kinetics parameters were the sameas in Fig. 2.



ph

ysic

a ssp stat

us

solid

i b

by stable behavior on increasing the absorbed dose, while theOTOR model predicted only stable behavior.

Both of the GOK and MOK models have almost thesame accuracy in describing the glow peaks simulated bythe IMTS model in both the quadratic dose-response and thedose-saturation regions, while in the supraquadratic dose-response region the accuracy of the MOK model is muchbetter than the accuracy of the GOK model.

An error of more than 50% in the E values obtained byboth the GOK and MOK models is observed for the glowpeaks simulated by the NMTS model with R � 1. However,for the quadratic dose-response region, the error did notexceed 5%.

Newly developed TL expressions based on the NMTSmodel were derived using the Lambert-W function. Theseexpressions can accurately describe the NMTS glow peaksin the cases in which the GOK and MOK expressions failed.

The TL expressions deduced based on the OTOR modelusing the Lambert-W function can be used to fit the glowpeaks simulated by the IMTS and NMTS models even in thecase of R � 1. However, the fitted parameter R in this casecannot be a reliable physical parameter to describe whetherretrapping or recombination dominates.

References

[1] A. D. Franklin, W. F. Hornyak, and R. Chen, Radiat. Prot.Dosim. 47, 17–22 (1993).

[2] A. J. J. Bos, Nucl. Instrum. Methods B 184, 3–28 (2001).[3] A. Halperin and R. Chen, Phys. Rev. 148, 839–845 (1966).[4] J. R. Cameron, N. Suntharalingam, and G. N. Kenney,

Thermoluminescence Dosimetry (The University of Wiscon-sin Press, Madison, 1968), pp. 150–177.

[5] R. Chen, X. H. Yang, and S. W. S. McKeever, J. Phys. D,Appl. Phys. 21, 1452–1457 (1988).

[6] C. Charitidis, G. Kitis, C. Furetta, and S. Charalambous,Radiat. Prot. Dosim. 84, 95–98 (1999).

[7] C. Charitidis, G. Kitis, C. Furetta, and S. Charalambous, Nucl.Instrum. Methods Phys. B 168, 404–410 (2000).

[8] P. W. Levy, Nucl. Instrum. Methods B 1, 436–444 (1984).[9] R. Chen, G. Fogel, and C. K. Lee, Radiat. Prot. Dosim. 65,

63–68 (1996).

[10] N. Suntharalingam and J. R. Cameron, Phys. Med. Biol. 14,397–410 (1969).

[11] J. L. Lawless, R. Chen, and V. Pagonis, Radiat. Meas. 44,606–610 (2009).

[12] P. W. Levy, Nucl. Tracks 10, 547–556 (1985).[13] G. Kitis and V. Pagonis, Nucl. Instrum. Methods B 262, 313–

322 (2007).[14] A. Halperin and A. A. Braner, Phys. Rev. 117, 408–415 (1960).[15] D. Yossian and Y. S. Horowitz, Radiat. Meas. 27, 465–471

(1997).[16] P. W. Levy, Radiat. Effects 72, 259–264 (1983).[17] C. M. Sunta, W. E. F. Ayta, J. F. D. Chubaci, and S.

Watanabe, Radiat. Meas. 35, 47–57 (2002).[18] V. Pagonis and G. Kitis, Phys. Status Solidi B 249, 1590–

1601 (2012).[19] W. F. Hornyak, P. W. Levy, and J. A. Kierstead, Nucl. Tracks

10, 557–563 (1985).[20] R. Chen and V. Pagonis, Nucl. Instrum. Methods B 312, 60–

69 (2013).[21] A. N. Yazici, R. Chen, S. Solak, and Z. Yegingil, J. Phys. D,

Appl. Phys. 35, 2526–2535 (2002).[22] A. N. Yazici, M. Bedir, and A. Sibel Sokucu, Nucl. Instrum.

Methods B 259, 955–965 (2007).[23] M. Topaksu and A. N. Yazici, Nucl. Instrum. Methods B 264,

293–301 (2007).[24] H. Wang, S.-W. Lin, P.-S. Weng, and P.-C. Hsu, Appl.

Radiat. Isot. 46, 869–873 (1995).[25] G. Kitis, R. Chen, and V. Pagonis, Phys. Status Solidi A 205,

1181–1189 (2008).[26] R. Chen, J. Appl. Phys. 40, 570–585 (1969).[27] A. M. Sadek, H. M. Eissa, A. M. Basha, and G. Kitis, J.

Lumin. 146, 418–423 (2014).[28] R. Chen and Y. Kirsh, Analysis of Thermally Stimulated

Processes (Pergamon Press, Oxford, 1981).[29] T. M. Piters and A. J. J. Bos, Radiat. Prot. Dosim. 47, 41–47

(1993).[30] C. M. Sunta, A.W. E. Feria, T. M. Piters, and S. K. Watanabe,

Radiat. Meas. 30, 197–201 (1999).[31] C.M. Sunta, W. E. F. Ayta, J. F. D. Chubaci, and S.Watanabe,

J. Phys. D, Appl. Phys. 34, 2690–2698 (2001).[32] C.M. Sunta, W. E. F. Ayta, J. F. D. Chubaci, and S.Watanabe,

J. Phys. D, Appl. Phys. 39, 95–102 (2005).[33] A. M. Sadek, Nucl. Instrum. Methods A 712, 56–61 (2013).[34] G. Kitis and N. D. Vlachos, Radiat. Meas. 48, 47–54 (2013).



Original

Paper

Properties of the thermoluminescence glow peaks simulated by the interactive multiple-trap system (IMTS) model

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