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Kinetics of excitation in TL and OSL detectors
Optical Materials 32 (2010) 1568–1572
Contents lists available at ScienceDirect
Optical Materials journal homepage: www.elsevier.com/locate/optmat
Kinetics of excitation in TL and OSL detectors A. Mandowski a,*, J. Orzechowski b, E. Mandowska a a b
Institute of Physics, Jan Dlugosz University, ul. Armii Krajowej 13/15, 42-200 Cze˛stochowa, Poland Stanislaw Staszic High School, ul. Szkolna 12, 27-200 Starachowice, Poland
a r t i c l e
i n f o
Article history: Received 29 January 2010 Received in revised form 18 June 2010 Accepted 18 June 2010 Available online 23 July 2010 Keywords: Thermoluminescence (TL) Optically stimulated luminescence (OSL) Semi-localized transitions (SLT) Recombination Traps
a b s t r a c t Kinetic equations for the semi-localized transitions (SLT) model are presented describing charge carrier’s kinetics for the excitation and fast relaxation stages. The formulation allows for dose dependence studies of thermoluminescence (TL) and optically stimulated luminescence (OSL) detectors based on the SLT model. The sets of equations were solved numerically demonstrating temporal evolution of all variables of the SLT model during excitation and fast relaxation. The influence of the dose rate on excitation kinetics is shown. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Thermoluminescence (TL) and optically stimulated luminescence (OSL) are widely known techniques, frequently used for characterization of traps and recombination centers (RCs) in insulating materials. Commercial applications include dosimetry of ionizing radiation and dating of archeological and geological objects. Prior to the measurement a sample is excited at appropriately low temperature to fill traps and RCs with charge carriers. Then, the sample is heated, usually with a constant rate (in TL) or stimulated by a strong light (in OSL). During the stimulation emitted luminescence is recorded. The luminescence is related to radiative recombination of charge carriers, which were thermally or optically released from traps and then moved to RCs. Theoretical description of these processes is usually based on two models: the model of localized transitions (LT) [1,2] and the simple trap model (STM) [3]. The first one (the LT model) allows trapped charge carriers to recombine solely to adjacent RC. It is assumed that traps and RCs are spatially correlated forming traprecombination centre (T–RC) pairs. The second one (STM) assumes transition of charge carriers via conduction band. These transitions are delocalized. Nevertheless, these two classical kinetic models are not successful in explanation of some TL properties, e.g. of two most widely used luminescence detectors, i.e. LiF:Mg,Ti and LiF:Mg,Cu,P. Recently, Mandowski [4,5] suggested a more general model combining both localized and delocalized transitions. This is the * Corresponding author. Tel.: +48 34 361 49 19x262; fax: +48 34 366 82 52. E-mail address: [email protected] (A. Mandowski). 0925-3467/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optmat.2010.06.014
model of semi-localized transitions (SLT). Analytic equations for SLT are constructed using enumeration technique which transforms concentrations of carriers to concentrations of states of localized T–RC pairs. The model allows explaining some unusual properties observed in some TL materials – e.g. the occurrence of very high frequency factors [6]. Some properties of SLT were analyzed also by Pagonis [7] and Kumar et al. [8]. A similar model for dose dependence studies was also constructed by Horowitz et al. [9]. To study kinetic dose–response characteristics of TL and OSL, one has to consider the initial stage of these processes, i.e. sample excitation. Most previous dose–response calculations were performed assuming the simple trap model (STM) with band-to-band transitions. This paper presents formulation and exemplary application of the SLT model to the analysis of excitation processes. A set of differential equations allows calculating populations of all states after irradiation of the solid with a given dose rate. After the excitation the system rapidly goes to a metastable quasi-eqilibrium state, which is termed the fast relaxation stage. Temporal behaviour of all variables characterizing the system during these stages will be given. The formulation allows for subsequent analysis of the SLT system with a standard set of SLT heating stage equations [4] yielding the final TL and OSL output. 2. Theory 2.1. Definitions We assume the typical SLT system as set of trap-recombination centre (T–RC) pairs, which was previously formulated for the
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heating stage of TL [4]. According to the theory we introduce the following variables characterizing concentrations of the allowed states of T–RC pair:
8 9 8 9 8 9 8 9 > > > > <0> <0> <1> <1> = = = = H00 0 H01 1 H10 0 H11 1 > > > > : > : > : > : > ; ; ; ; 1 1 1 1 8 9 8 9 8 9 8 9 > > > > <0> <0> <1> <1> = = = = E01 1 E10 0 E11 1 E00 0 > > > > > > > : ; : ; : ; : > ; 0 0 0 0
ð1Þ
where the numbers inside brackets
8 9 e > > : > ; h
ð2Þ
e is the numdenote occupation of states in a single T–RC unit, i.e. n is the number of elecber of electrons in the local excited level, n denotes the number of holes in RC level. trons in trap level and h The numbers may have value 0 or 1. The symbol (2) has the meaning of time-dependent variable denoting concentration (in cm3) of charge carriers in the respective e , n and h all T–RC units having n trap levels. Therefore, the variables Hnm ðtÞ and Enm ðtÞ denote the concentrations of states (i.e. corresponding T–RC units) with full and empty RCs, respectively. Due to small probability of occurrence the states H11 and E11 are not taken into account. Luminescence may arise from both localized (within each pair) and delocalized recombination. It could be measured during stimulation stage (i.e. heating in TL or optical in OSL). Next, we will show the formulation of SLT for the two basic stages of TL and OSL measurements, i.e. excitation done by high-energy radiation and the stage of fast relaxation in which the system comes to the metastable state. 2.2. Excitation stage High energy ionizing radiation produces hole-electron pairs in the material. We assume that the charge carries may freely travel through the solid within corresponding transport bands, i.e. the valence band and the conduction band, respectively. The luminescence detectors are dielectric materials, therefore, after some time the free carriers will eventually be trapped by corresponding traps. The diagram illustrating possible charge carriers transitions during the excitation stage is shown in Fig. 1. We denote the rate of creation of hole-electron pairs by G. Other transitions are described in the Fig. 1 caption. The figure does not contain thermally activated transitions as it is assumed that the temperature of the system is sufficiently low. Considering various possible states of T–RC pairs one may write the following diagram:
Fig. 1. Diagram illustrating possible charge carrier transitions for three T–RC pairs in SLT model during excitation stage. G is the rate of generation of hole-electron pairs by ionizing radiation; nc and hv are concentrations of electrons and holes, respectively, in transport bands; n, ne and h denote concentrations of electrons in B, C, K, Kv, Ke traps, electrons on the excited levels and holes in RCs, respectively; A, denote respective transitions probabilities. The same diagram applies to the fast relaxation stage except the rate of generation G.
level transforms the state to E01 or, alternatively, by trapping a hole from the valence band to RC (the KV transition) it may transform the state to H10 and so on. Further, we assume that the high-energy radiation G generates hole-electron pairs directly to valence and conduction band, respectively. Its influence on already trapped and B denote charge carriers will be neglected. The bar variables A 1 probability densities (in s ) of localized transitions for trapping and recombination, respectively. Similarly, K, C, KV and Ke denote probability densities (in cm3 s1) of delocalized transitions. The C transition denotes recombination of an electron from the conduction band and Ke transition denotes recombination of a hole from the valence band. Temperature-assisted transitions D1, D2 and V1, V2 from the diagram (3) will be neglected, because we assume, that temperature of the sample during excitation is sufficiently low. Considering the diagram, together with the generation rate for electrons and holes, allows us to write the corresponding set of kinetic equations:
1 þ K V hV E0 H_ 01 ¼ ðCnc þ K e hV ÞH01 þ AH 0 1 þB þ Cnc ÞH1 þ Knc H0 þ K V hv E1 H_ 10 ¼ ðA 0 0 0 H_ 0 ¼ K e hv H0 ðC þ KÞnc H0 þ K V hv E0 0
1
0
ð4aÞ ð4bÞ ð4cÞ
0
1 E_ 01 ¼ Cnc H01 ðK V þ K e Þhv E01 þ AE 0 1 1 1 þ K V hv ÞE þ Knc E0 E_ 0 ¼ Cnc H0 ðA 0 0 1 þ Cnc H0 þ K e hv E0 ðK V hv þ Knc ÞE0 E_ 00 ¼ BH 0 0 1 0
ð4dÞ ð4eÞ ð4fÞ
n_ c ¼ G Cnc H01 Cnc H10 ðC þ KÞnc H00 Knc E00 h_ v ¼ G K e hv H0 ðK V þ K e Þhv E0 K V hv E1 K V hv E0 1
1
0
0
ð4gÞ ð4hÞ
Here nc and hv denote concentrations of free electrons and holes, respectively. The dot above variables denotes, as usual, time derivative. The set of Eqs. (4) is highly nonlinear and most likely it has no general analytical solutions. 2.3. Fast relaxation stage
ð3Þ
It may be interpreted as follows: trapping of an electron from the conduction band to the excited level (the K transition) of empty T–RC pair E00 (the lower right corner of Eq. (3) transforms this state to E10 . Subsequent trapping of the electron to the lowest trapping
When the excitation is ceased the solid rapidly goes to a metastable state in which it may remain for a very long time (even many years). In this state the concentration of charge carriers in transport bands, as well as on the local excited levels is almost zero. The time of constituting the metastable state is usually very short, but depending mostly on the generation rate G (dose rate). When the dose is deposited very slowly, the material is all the time very close to the quasi-equilibrium, therefore, in this case, it is not necessary to calculate its time evolution during the fast relaxation stage. However, for high generation rates the relaxation may significantly adjust the final concentrations of SLT variables. In
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Fig. 2. Numerical solution of SLT equations during the excitation stage (4) ¼ 1010 s1 , B ¼ 1010 s1 , calculated for the following parameter values: A K = 1014 cm3 s1, C = 1014 cm3 s1, KV = 1015 cm3 s1, Ke = 1015 cm3 s1 and the rate of excitation G = 1015 cm3 s1. Initially, the system consists solely of empty pairs E00 with concentration N = 1016cm3. All other variables of the model, i.e. Enm , Hnm , nc and hv are initially set to 0.
general, the fast relaxation stage is governed by the same transition diagrams as the excitation stage (Fig. 1 and Eq. (3)) however, the generation rate G should be put to zero. 3. Temporal evolution of SLT variables Numerical calculations were performed for the excitation and fast relaxation stages. As previously argued, at low generation rates G the fast relaxation stage does not play significant role. For that reason, as an example, we will present results calculated for very high generation rate, where the time of saturation is comparable to the time of fast relaxation. The calculations were performed ¼ 1010 s1 , B for the following set of parameter values: A 10 1 14 3 1 14 3 1 15 3 1 ¼ 10 s , K = 10 cm s , C = 10 cm s , KV = 10 cm s ,
Fig. 3. Numerical solution of SLT during fast relaxation stage. Parameter values are the same as for Fig. 2 except G = 0.
Fig. 4. Numerical solution of SLT during fast relaxation stage. Results for empty states E00 , ‘doublets’ H01 and ‘singlets’ – E01 , H00 . Parameter values the same as for Fig. 3.
Ke = 1015 cm3 s1, G = 1015 cm3 s1, N = 1016 cm3. Initially, the system consists solely of the empty pairs E00 with concentration N. All other variables of the model, i.e. Enm , Hnm , nc and hv are initially set to 0. Temporal evolution of all these variables, except the concentration of empty states E00 , during first 300 ms is shown in Fig. 2. All these variables are increasing and after 300 ms seem to be close to saturation (on the logarithmic scale). The concentration of empty states E00 is a slowly decreasing function. At the time t = 300 ms the excitation is ceased and the system goes to the fast relaxation stage. In Fig. 3 the same variables are plotted as in Fig. 2 for the first 100 ms of the fast relaxation stage. Four of the variables are rapidly decreasing. These are T–RC pairs with an excited electron, i.e. E10 and H10 , as well as the free charge carrier’s concentrations nc and hv. Effectively, these concentrations go to zero. The other variables tend toward higher values. These are: concentration of empty states E00 , concentration of fully occupied T–RC states
Fig. 5. Numerical solution of SLT during excitation stage on log–log diagram. Parameter values are the same as for Fig. 2 except G = 106 cm3 s1.
A. Mandowski et al. / Optical Materials 32 (2010) 1568–1572
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Fig. 6. Numerical solution of SLT during excitation stage on log–log diagram. Parameter values are the same as for Fig. 2 except G = 1013 cm3 s1.
Fig. 8. Numerical solution of SLT during excitation stage on log–log diagram. Parameter values are the same as for Fig. 2 except G = 1017 cm3 s1.
Fig. 7. Numerical solution of SLT during excitation stage on log–log diagram. Parameter values are the same as for Fig. 2.
Fig. 9. Numerical solution of SLT during excitation stage on log–log diagram. Parameter values are the same as for Fig. 2 except G = 1019 cm3 s1.
H01 and the concentration of partially occupied T–RC states – E01 (filled with electron) and H00 (filled with hole). We will call H01 as ‘doublets’ and E01 , H00 as ‘singlets’. Temporal behaviour of these stable configurations during 500 ms is plotted in Fig. 4. The concentrations of two ‘singlets’ tend to the same value as it results from sample neutrality condition. The final population of these states depends on various SLT parameter values. In this particular case the system consists mainly of ‘singlets’, however, many states are still unoccupied. Figs. 5–9 present SLT excitation kinetics for a very broad range of generation rates G from 106 to 1019 cm3 s1. For all presented figures the same dose was deposited. With increasing dose rate the relative concentration of all SLT variables changes. The higher increase relates to free charge carriers in conduction and valence bands. Therefore, for very high dose rates the ‘fast relaxation’ stage plays a more important role.
4. Conclusions Kinetic equations for the excitation and fast relaxation stages were formulated in the framework of the SLT model. The equations were used to illustrate temporal evolution of SLT variables in the case of high dose rate where both excitation and fast relaxation stages are of great importance. Using numerical simulation it may be shown, that after excitation the SLT system consists of ‘doublets’ (fully occupied T–RC pairs), ‘singlets’ (partly occupied T–RC pairs) and empty states with a concentration depending on various SLT parameter values. This information is of great importance for calculating dose–response in various TL and OSL materials, e.g. luminescent detectors of ionizing radiation. The occurrence of ‘singlets’ forces delocalized transitions as the recombination mechanism for TL or OSL processes. SLT is a complex kinetic model containing many parameters. Therefore, further extensive
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numerical calculations are required to study its properties under excitation. In particular, the most important seems to be relative population of ‘singlets’ vs. ‘doublets’ determining luminescence response of TL and OSL materials. In the case of very high dose rates the ‘fast relaxation’ stage must be taken into account. Acknowledgment This work was supported by research project from the Polish Ministry of Science and Higher Education over the years 2007–2010.
References [1] A. Halperin, A.A. Braner, Phys. Rev. 117 (1960) 408. [2] P.L. Land, J. Phys. Chem. Solids 30 (1969) 1693. [3] R. Chen, S.W.S. McKeever, Theory of Thermoluminescence and Related Phenomena, World Scientific Publishing Co. Pvt. Ltd., Singapore, 1997. [4] A. Mandowski, J. Phys. D: Appl. Phys. 38 (2005) 17. [5] A. Mandowski, Radiat. Measur. 43 (2008) 199. [6] A. Mandowski, Radiat. Prot. Dosimetry 119 (2006) 23. [7] V. Pagonis, J. Phys. D: Appl. Phys. 38 (2005) 2179. [8] M. Kumar, R.K. Kher, B.C. Bhatt, C.M. Sunta, J. Phys. D: Appl. Phys. 39 (2006) 2670. [9] Y.S. Horowitz, L. Oster, Y. Einav, S. Biderman, J. Phys. D: Appl. Phys. 36 (2003) 446.
Contents lists available at ScienceDirect
Optical Materials journal homepage: www.elsevier.com/locate/optmat
Kinetics of excitation in TL and OSL detectors A. Mandowski a,*, J. Orzechowski b, E. Mandowska a a b
Institute of Physics, Jan Dlugosz University, ul. Armii Krajowej 13/15, 42-200 Cze˛stochowa, Poland Stanislaw Staszic High School, ul. Szkolna 12, 27-200 Starachowice, Poland
a r t i c l e
i n f o
Article history: Received 29 January 2010 Received in revised form 18 June 2010 Accepted 18 June 2010 Available online 23 July 2010 Keywords: Thermoluminescence (TL) Optically stimulated luminescence (OSL) Semi-localized transitions (SLT) Recombination Traps
a b s t r a c t Kinetic equations for the semi-localized transitions (SLT) model are presented describing charge carrier’s kinetics for the excitation and fast relaxation stages. The formulation allows for dose dependence studies of thermoluminescence (TL) and optically stimulated luminescence (OSL) detectors based on the SLT model. The sets of equations were solved numerically demonstrating temporal evolution of all variables of the SLT model during excitation and fast relaxation. The influence of the dose rate on excitation kinetics is shown. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Thermoluminescence (TL) and optically stimulated luminescence (OSL) are widely known techniques, frequently used for characterization of traps and recombination centers (RCs) in insulating materials. Commercial applications include dosimetry of ionizing radiation and dating of archeological and geological objects. Prior to the measurement a sample is excited at appropriately low temperature to fill traps and RCs with charge carriers. Then, the sample is heated, usually with a constant rate (in TL) or stimulated by a strong light (in OSL). During the stimulation emitted luminescence is recorded. The luminescence is related to radiative recombination of charge carriers, which were thermally or optically released from traps and then moved to RCs. Theoretical description of these processes is usually based on two models: the model of localized transitions (LT) [1,2] and the simple trap model (STM) [3]. The first one (the LT model) allows trapped charge carriers to recombine solely to adjacent RC. It is assumed that traps and RCs are spatially correlated forming traprecombination centre (T–RC) pairs. The second one (STM) assumes transition of charge carriers via conduction band. These transitions are delocalized. Nevertheless, these two classical kinetic models are not successful in explanation of some TL properties, e.g. of two most widely used luminescence detectors, i.e. LiF:Mg,Ti and LiF:Mg,Cu,P. Recently, Mandowski [4,5] suggested a more general model combining both localized and delocalized transitions. This is the * Corresponding author. Tel.: +48 34 361 49 19x262; fax: +48 34 366 82 52. E-mail address: [email protected] (A. Mandowski). 0925-3467/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optmat.2010.06.014
model of semi-localized transitions (SLT). Analytic equations for SLT are constructed using enumeration technique which transforms concentrations of carriers to concentrations of states of localized T–RC pairs. The model allows explaining some unusual properties observed in some TL materials – e.g. the occurrence of very high frequency factors [6]. Some properties of SLT were analyzed also by Pagonis [7] and Kumar et al. [8]. A similar model for dose dependence studies was also constructed by Horowitz et al. [9]. To study kinetic dose–response characteristics of TL and OSL, one has to consider the initial stage of these processes, i.e. sample excitation. Most previous dose–response calculations were performed assuming the simple trap model (STM) with band-to-band transitions. This paper presents formulation and exemplary application of the SLT model to the analysis of excitation processes. A set of differential equations allows calculating populations of all states after irradiation of the solid with a given dose rate. After the excitation the system rapidly goes to a metastable quasi-eqilibrium state, which is termed the fast relaxation stage. Temporal behaviour of all variables characterizing the system during these stages will be given. The formulation allows for subsequent analysis of the SLT system with a standard set of SLT heating stage equations [4] yielding the final TL and OSL output. 2. Theory 2.1. Definitions We assume the typical SLT system as set of trap-recombination centre (T–RC) pairs, which was previously formulated for the
1569
A. Mandowski et al. / Optical Materials 32 (2010) 1568–1572
heating stage of TL [4]. According to the theory we introduce the following variables characterizing concentrations of the allowed states of T–RC pair:
8 9 8 9 8 9 8 9 > > > > <0> <0> <1> <1> = = = = H00 0 H01 1 H10 0 H11 1 > > > > : > : > : > : > ; ; ; ; 1 1 1 1 8 9 8 9 8 9 8 9 > > > > <0> <0> <1> <1> = = = = E01 1 E10 0 E11 1 E00 0 > > > > > > > : ; : ; : ; : > ; 0 0 0 0
ð1Þ
where the numbers inside brackets
8 9 e > >
ð2Þ
e is the numdenote occupation of states in a single T–RC unit, i.e. n is the number of elecber of electrons in the local excited level, n denotes the number of holes in RC level. trons in trap level and h The numbers may have value 0 or 1. The symbol (2) has the meaning of time-dependent variable denoting concentration (in cm3) of charge carriers in the respective e , n and h all T–RC units having n trap levels. Therefore, the variables Hnm ðtÞ and Enm ðtÞ denote the concentrations of states (i.e. corresponding T–RC units) with full and empty RCs, respectively. Due to small probability of occurrence the states H11 and E11 are not taken into account. Luminescence may arise from both localized (within each pair) and delocalized recombination. It could be measured during stimulation stage (i.e. heating in TL or optical in OSL). Next, we will show the formulation of SLT for the two basic stages of TL and OSL measurements, i.e. excitation done by high-energy radiation and the stage of fast relaxation in which the system comes to the metastable state. 2.2. Excitation stage High energy ionizing radiation produces hole-electron pairs in the material. We assume that the charge carries may freely travel through the solid within corresponding transport bands, i.e. the valence band and the conduction band, respectively. The luminescence detectors are dielectric materials, therefore, after some time the free carriers will eventually be trapped by corresponding traps. The diagram illustrating possible charge carriers transitions during the excitation stage is shown in Fig. 1. We denote the rate of creation of hole-electron pairs by G. Other transitions are described in the Fig. 1 caption. The figure does not contain thermally activated transitions as it is assumed that the temperature of the system is sufficiently low. Considering various possible states of T–RC pairs one may write the following diagram:
Fig. 1. Diagram illustrating possible charge carrier transitions for three T–RC pairs in SLT model during excitation stage. G is the rate of generation of hole-electron pairs by ionizing radiation; nc and hv are concentrations of electrons and holes, respectively, in transport bands; n, ne and h denote concentrations of electrons in B, C, K, Kv, Ke traps, electrons on the excited levels and holes in RCs, respectively; A, denote respective transitions probabilities. The same diagram applies to the fast relaxation stage except the rate of generation G.
level transforms the state to E01 or, alternatively, by trapping a hole from the valence band to RC (the KV transition) it may transform the state to H10 and so on. Further, we assume that the high-energy radiation G generates hole-electron pairs directly to valence and conduction band, respectively. Its influence on already trapped and B denote charge carriers will be neglected. The bar variables A 1 probability densities (in s ) of localized transitions for trapping and recombination, respectively. Similarly, K, C, KV and Ke denote probability densities (in cm3 s1) of delocalized transitions. The C transition denotes recombination of an electron from the conduction band and Ke transition denotes recombination of a hole from the valence band. Temperature-assisted transitions D1, D2 and V1, V2 from the diagram (3) will be neglected, because we assume, that temperature of the sample during excitation is sufficiently low. Considering the diagram, together with the generation rate for electrons and holes, allows us to write the corresponding set of kinetic equations:
1 þ K V hV E0 H_ 01 ¼ ðCnc þ K e hV ÞH01 þ AH 0 1 þB þ Cnc ÞH1 þ Knc H0 þ K V hv E1 H_ 10 ¼ ðA 0 0 0 H_ 0 ¼ K e hv H0 ðC þ KÞnc H0 þ K V hv E0 0
1
0
ð4aÞ ð4bÞ ð4cÞ
0
1 E_ 01 ¼ Cnc H01 ðK V þ K e Þhv E01 þ AE 0 1 1 1 þ K V hv ÞE þ Knc E0 E_ 0 ¼ Cnc H0 ðA 0 0 1 þ Cnc H0 þ K e hv E0 ðK V hv þ Knc ÞE0 E_ 00 ¼ BH 0 0 1 0
ð4dÞ ð4eÞ ð4fÞ
n_ c ¼ G Cnc H01 Cnc H10 ðC þ KÞnc H00 Knc E00 h_ v ¼ G K e hv H0 ðK V þ K e Þhv E0 K V hv E1 K V hv E0 1
1
0
0
ð4gÞ ð4hÞ
Here nc and hv denote concentrations of free electrons and holes, respectively. The dot above variables denotes, as usual, time derivative. The set of Eqs. (4) is highly nonlinear and most likely it has no general analytical solutions. 2.3. Fast relaxation stage
ð3Þ
It may be interpreted as follows: trapping of an electron from the conduction band to the excited level (the K transition) of empty T–RC pair E00 (the lower right corner of Eq. (3) transforms this state to E10 . Subsequent trapping of the electron to the lowest trapping
When the excitation is ceased the solid rapidly goes to a metastable state in which it may remain for a very long time (even many years). In this state the concentration of charge carriers in transport bands, as well as on the local excited levels is almost zero. The time of constituting the metastable state is usually very short, but depending mostly on the generation rate G (dose rate). When the dose is deposited very slowly, the material is all the time very close to the quasi-equilibrium, therefore, in this case, it is not necessary to calculate its time evolution during the fast relaxation stage. However, for high generation rates the relaxation may significantly adjust the final concentrations of SLT variables. In
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Fig. 2. Numerical solution of SLT equations during the excitation stage (4) ¼ 1010 s1 , B ¼ 1010 s1 , calculated for the following parameter values: A K = 1014 cm3 s1, C = 1014 cm3 s1, KV = 1015 cm3 s1, Ke = 1015 cm3 s1 and the rate of excitation G = 1015 cm3 s1. Initially, the system consists solely of empty pairs E00 with concentration N = 1016cm3. All other variables of the model, i.e. Enm , Hnm , nc and hv are initially set to 0.
general, the fast relaxation stage is governed by the same transition diagrams as the excitation stage (Fig. 1 and Eq. (3)) however, the generation rate G should be put to zero. 3. Temporal evolution of SLT variables Numerical calculations were performed for the excitation and fast relaxation stages. As previously argued, at low generation rates G the fast relaxation stage does not play significant role. For that reason, as an example, we will present results calculated for very high generation rate, where the time of saturation is comparable to the time of fast relaxation. The calculations were performed ¼ 1010 s1 , B for the following set of parameter values: A 10 1 14 3 1 14 3 1 15 3 1 ¼ 10 s , K = 10 cm s , C = 10 cm s , KV = 10 cm s ,
Fig. 3. Numerical solution of SLT during fast relaxation stage. Parameter values are the same as for Fig. 2 except G = 0.
Fig. 4. Numerical solution of SLT during fast relaxation stage. Results for empty states E00 , ‘doublets’ H01 and ‘singlets’ – E01 , H00 . Parameter values the same as for Fig. 3.
Ke = 1015 cm3 s1, G = 1015 cm3 s1, N = 1016 cm3. Initially, the system consists solely of the empty pairs E00 with concentration N. All other variables of the model, i.e. Enm , Hnm , nc and hv are initially set to 0. Temporal evolution of all these variables, except the concentration of empty states E00 , during first 300 ms is shown in Fig. 2. All these variables are increasing and after 300 ms seem to be close to saturation (on the logarithmic scale). The concentration of empty states E00 is a slowly decreasing function. At the time t = 300 ms the excitation is ceased and the system goes to the fast relaxation stage. In Fig. 3 the same variables are plotted as in Fig. 2 for the first 100 ms of the fast relaxation stage. Four of the variables are rapidly decreasing. These are T–RC pairs with an excited electron, i.e. E10 and H10 , as well as the free charge carrier’s concentrations nc and hv. Effectively, these concentrations go to zero. The other variables tend toward higher values. These are: concentration of empty states E00 , concentration of fully occupied T–RC states
Fig. 5. Numerical solution of SLT during excitation stage on log–log diagram. Parameter values are the same as for Fig. 2 except G = 106 cm3 s1.
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Fig. 6. Numerical solution of SLT during excitation stage on log–log diagram. Parameter values are the same as for Fig. 2 except G = 1013 cm3 s1.
Fig. 8. Numerical solution of SLT during excitation stage on log–log diagram. Parameter values are the same as for Fig. 2 except G = 1017 cm3 s1.
Fig. 7. Numerical solution of SLT during excitation stage on log–log diagram. Parameter values are the same as for Fig. 2.
Fig. 9. Numerical solution of SLT during excitation stage on log–log diagram. Parameter values are the same as for Fig. 2 except G = 1019 cm3 s1.
H01 and the concentration of partially occupied T–RC states – E01 (filled with electron) and H00 (filled with hole). We will call H01 as ‘doublets’ and E01 , H00 as ‘singlets’. Temporal behaviour of these stable configurations during 500 ms is plotted in Fig. 4. The concentrations of two ‘singlets’ tend to the same value as it results from sample neutrality condition. The final population of these states depends on various SLT parameter values. In this particular case the system consists mainly of ‘singlets’, however, many states are still unoccupied. Figs. 5–9 present SLT excitation kinetics for a very broad range of generation rates G from 106 to 1019 cm3 s1. For all presented figures the same dose was deposited. With increasing dose rate the relative concentration of all SLT variables changes. The higher increase relates to free charge carriers in conduction and valence bands. Therefore, for very high dose rates the ‘fast relaxation’ stage plays a more important role.
4. Conclusions Kinetic equations for the excitation and fast relaxation stages were formulated in the framework of the SLT model. The equations were used to illustrate temporal evolution of SLT variables in the case of high dose rate where both excitation and fast relaxation stages are of great importance. Using numerical simulation it may be shown, that after excitation the SLT system consists of ‘doublets’ (fully occupied T–RC pairs), ‘singlets’ (partly occupied T–RC pairs) and empty states with a concentration depending on various SLT parameter values. This information is of great importance for calculating dose–response in various TL and OSL materials, e.g. luminescent detectors of ionizing radiation. The occurrence of ‘singlets’ forces delocalized transitions as the recombination mechanism for TL or OSL processes. SLT is a complex kinetic model containing many parameters. Therefore, further extensive
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numerical calculations are required to study its properties under excitation. In particular, the most important seems to be relative population of ‘singlets’ vs. ‘doublets’ determining luminescence response of TL and OSL materials. In the case of very high dose rates the ‘fast relaxation’ stage must be taken into account. Acknowledgment This work was supported by research project from the Polish Ministry of Science and Higher Education over the years 2007–2010.
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