- Email: [email protected]
How to detect trap cluster systems?
Radiation Measurements 43 (2008) 167 – 170 www.elsevier.com/locate/radmeas
How to detect trap cluster systems? Arkadiusz Mandowski Institute of Physics, Jan Długosz University, ul. Armii Krajowej 13/15, PL-42200 CzeR stochowa, Poland
Abstract Spatially correlated traps and recombination centres (trap-recombination centre pairs and larger clusters) are responsible for many anomalous phenomena that are difficult to explain in the framework of both classical models, i.e. model of localized transitions (LT) and the simple trap model (STM), even with a number of discrete energy levels. However, these ‘anomalous’ effects may provide a good platform for identifying trap cluster systems. This paper considers selected cluster-type effects, mainly relating to an anomalous dependence of TL on absorbed dose in the system of isolated clusters (ICs). Some consequences for interacting cluster (IAC) systems, involving both localized and delocalized transitions occurring simultaneously, are also discussed. © 2007 Elsevier Ltd. All rights reserved. Keywords: Thermoluminescence (TL); Trap clusters; Thermal bleaching; Recombination
1. Introduction Classical models for trapping and recombination of charge carriers in dielectrics relate to two analytical cases. The first one relates to uniform distribution of traps and recombination centres (RCs), where all transitions of charge carriers from traps to RCs go through delocalized transport bands. This is the simple trap model (STM—cf. Chen and McKeever, 1997). The second one relates to pairs of traps and RCs (T–RC) located close to each other. The model of localized transitions (LT), for the first time formulated by Halperin and Braner (1960) and then modified by Land (1969), assumes that recombination of trapped carriers proceeds through a local excited level. This system of T–RC pairs may be considered as a system of smallest clusters. Results of many experiments and theoretical considerations gave evidence that thermoluminescence (TL) kinetic processes proceed in large-scale defects rather than in pointlike traps (e.g. the paper of Townsend and Rowlands, 1999). None of the standard TL kinetic models is able to describe this type of kinetics properly. All the types of non-standard kinetics we will term the spatially correlated systems (SCSs). The studies of TL kinetics in various SCSs were started some ´ R tek (1992). For this purpose years ago by Mandowski and Swia they used several Monte Carlo algorithms modelling TL, e.g. in E-mail address: [email protected]. 1350-4487/$ - see front matter © 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.radmeas.2007.09.018
1-D and 3-D systems under different external conditions (e.g. ´ R tek, 1997, 1998). It was found that TL in Mandowski and Swia SCSs shows many unexpected features that cannot be explained within the framework of LT and STM models. Examples include apparently composite structure of monoenergetic peaks, additional ‘displacement’ peaks and the dependence of TL on the external electric field. For review, see Mandowski (1998, 2001, 2006a). Recently, an analytical model was proposed for the isolated clusters (ICs) model, which is a special case of SCSs (Mandowski, 2002, 2006b). The theory is based on two trap structural functions (TSFs)—n and h , for electrons and holes, respectively. Extensive numerical calculations gave the evidence that these functions depend only on structural properties of the solid. TSFs do not depend e.g. on the heating rate and the activation energy of traps. In this paper, the properties of IC systems are studied with respect to dose dependence. Using Monte Carlo algorithms TSFs were calculated for various initial fillings of traps. It will be shown that TSFs depend on dose in a very specific way, quite different from that found in both classical models—STM and LT. In this way it is possible to distinguish between classical and cluster-type kinetics of charge carrier recombination. Other properties relate to SCSs with both localized and delocalized transitions occurring simultaneously. The simplest case of this kind, relating to T–RC pairs, was recently formulated analytically as the model of semi-localized transitions (SLT)
168
A. Mandowski / Radiation Measurements 43 (2008) 167 – 170
(Mandowski, 2005, 2007). The model is able to explain many unexpected features observed in typical solid state detectors, e.g. extremely high-frequency factors (Mandowski, 2006a). It seems that these and some other features relate also to much complex systems consisting of large interacting clusters. Qualitative discussion of the dose dependence in interactive cluster (IAC) systems will be provided. 2. TL kinetics of isolated clusters Commonly accepted explanation of long-lasting phosphorescence and TL phenomena is based on the assumption of metastable levels (traps and RCs) situated within the energy gap. Although direct transition from trap to a recombination centre is possible, most of the transitions take place through excited states. In the model of ICs all trapping and recombination processes occur only within the clusters, i.e. no interaction between clusters is possible. TL is produced in the same way as in the LT model—a trapped charge carrier is thermally excited to a local excited level and then it may be retrapped or may recombine with an opposite charge carrier trapped at the RC level. For the simplest case of a single type of active traps, deep traps and RCs the following set of kinetic equations was proposed (Mandowski, 2002): −E −n˙ = n exp − n (n)ne , (1a) kT (t) −h˙ = h (h)ne ,
(1b)
h = n + ne + M,
(1c)
where E stands for the activation energy and is the frequency factor for active traps. n, ne and h denote the total concentrations of electrons trapped in active traps, electrons in the excited levels and holes trapped in RCs. M stands for the concentration of electrons in the thermally disconnected traps (deep traps), i.e. traps that are not emptied during the experiment. n (n) and h (h) denote two TSFs for trapping and recombination, respectively. It is interesting that the set of Eqs. (1) is valid also for both classical cases: LT and STM. For these two limiting cases one has to assume appropriate form of TSFs ¯ (LT) = A, n (LT)
h
= B¯
(2a) (2b)
and (STM) = A(N − n), n (STM)
h
= Bh,
(3a) (3b)
where A, B, A¯ and B¯ are constants and N is the total concentration of traps (in the case of STM, ne has the meaning of the concentration of carriers in the conduction band). 3. Dose dependence of TSFs To calculate TSFs for an arbitrary IC system one has to perform Monte Carlo calculations. The algorithm is performed
by considering elementary transitions—ID , IT and IR for detrapping, trapping and recombination, respectively: −E ID (t) = exp , (4a) kT (t) ¯ N¯ − n(t)], IT (t) = A[ ¯
(4b)
¯ IR (t) = B¯ h(t).
(4c)
Here, the dashed values denote variables and parameters relating to a single cluster of the system. N¯ denotes the number of trap levels, n¯ e is the number of electrons in the local excited level, n¯ is the number of electrons in traps and h¯ denotes the number of holes in RCs. A¯ and B¯ denote coefficients for trapping and recombination, respectively. A detailed method of the simulation as well as the scaling properties defining the relationship between the microscopic and the macroscopic parameters was given in some previous papers (e.g. Mandowski and ´ R tek, 1992, 1997). Swia For a given IC system TSFs can be calculated by performing a two-stage Monte Carlo simulation. In the first stage the clusters are randomly populated with electrons and holes. It is assumed that each hole–electron pair is trapped within the same cluster. Then, the TL process is simulated according to Eq. (4). Finally, n and h functions are calculated from Eqs. (1a) and (1b), respectively, i.e. −h˙ , ne −E 1 n (n) = n exp + n˙ . ne kT (t) h (h) =
(5a) (5b)
Now, it is interesting to study what is the dependence of these functions on dose. First of all let us look at both standard models where TSFs are given by analytical equations (2a), (2b) and (3a), (3b). These functions are plotted for various initial concentrations of trapped charge carriers 0 ≡ n0 /N = 0.1, 0.5 and 1.0 in Fig. 1. The common feature of the two models is that TSFs for partially filled systems (0 = 0.1 and 0 = 0.5) reproduce perfectly a part of the whole TSFs calculated for full initial filling (0 = 1). Such behaviour seems to be quite obvious. Surprisingly, for all other cases—including clusters of various sizes, the dependence is quite different. As an example, we calculated TL kinetics in two cluster systems consisting of clusters containing N¯ = 2 traps and RCs as well as N¯ = 5 traps and RCs. These systems were initially populated to the relative concentration of 0 =0.1, 0.5 and 1.0. Results of this simulations are shown in Fig. 2A with respect to h functions. It is clearly seen that the calculated functions do not overlap for various initial populations. Another interesting puzzle is that the shape of these curves is somewhat similar (within preciseness of the Monte Carlo calculations) and applying simple scaling on the horizontal axis, according to the equations n n n → n , (6a) N N h h h → h , (6b) h0 h0
A. Mandowski / Radiation Measurements 43 (2008) 167 – 170
Fig. 1. n and h functions calculated for standard LT and STM models (using Eqs. (2a), (2b) and (3a), (3b)) with various relative initial fillings of traps, i.e. 0 = 0.1 (䊉 for LT and for STM), 0 = 0.5 ( for LT and ♦ for STM) and 0 = 1.0 (solid lines for both models).
we are able to overlap these functions with acceptable accuracy. The constant is chosen numerically. Results are shown in Fig. 2B. A possible explanation for this divergence of TSFs may be explained by a difference in a time sequence (or time order) in which the clusters are populated (randomly) and depopulated (thermally). Thermal bleaching is not a uniform random
169
Fig. 2. Diagram A—h function calculated for two IC systems (N¯ = 2 and N¯ = 5) using Monte Carlo calculations and then Eqs. (5a) and (5b) with various relative initial fillings of traps 0 = 0.1 (䊉), 0 = 0.5 () and 0 = 1.0 (solid line). Diagram B represents the same function after scaling according to Eq. (6).
process. The highest probability of detrapping (4a) and recombination (4c) relates to charge carriers occupying these clusters which are populated at most. For these clusters the rate of retrapping (4b) is also much smaller. Therefore, these charge carriers will recombine in first order. These differences are not significant for LT model consisting of T–RC pairs (each cluster may consist of a maximum of one electron and one hole), as
170
A. Mandowski / Radiation Measurements 43 (2008) 167 – 170
well as the STM model for which the whole solid may be considered as a single very large cluster. The mathematical origin of the scaling property (6), however, well confirmed numerically, is not clear. Most likely it is an approximate feature. Possibly the results would be slightly different if one would apply a more sophisticated excitation scheme. However, it requires more additional assumptions and such cases will be considered later. 4. Experimental consequences As a consequence of the differences mentioned above, let us note that also TL curves measured for the same doses may be different. This property is characteristic exclusively for cluster systems ( = 1) and does not hold for any of the standard model (i.e. STM and LT where = 1). Therefore, by applying appropriate measurements we can distinguish between cluster and non-cluster systems. For these purposes we may suggest the following experimental procedure: Experiment 1: • Excite the sample up to a high dose (e.g. up to saturation 0 = 1). • Thermally bleach the traps, decreasing the relative population down to 1 . • Measure TL1 (determining 1 ). Experiment 2: • Excite the sample up to 1 . • Measure TL2 . In isolated cluster systems one may expect that TL1 = TL2 (different shapes of the two glow curves), but naturally T2 T2 T1 JT L1 (T ) dT = T1 JT L2 (T ) dT because the doses are the same. However, one should realize that in practical applications it may be difficult to differentiate between ICs and multiple trap levels. Therefore, most likely, such an analysis will require some elaborate calculations to compare the results with a multi-level model prediction. 5. Conclusions The results presented here apply mainly to isolated cluster systems, i.e. a set of non-interacting groups of traps and RCs where recombination takes place only within these clusters. Another assumption relates to the stability of trap levels. In the framework of IC model the trap levels are stable—it means that successive recombinations do not change the activation energies and other trap parameters for the remaining charge carriers. Fulfilling all of these conditions one may expect, that performing several simple measurements (as those suggested above) one is able to determine whether the system under consideration is cluster-like or not. Nevertheless, it is believed that many real solid-state detectors undergo much more complex type of TL kinetics. In particular, it was found that simultaneous localized and delocalized
transitions involving occupation-dependent energy levels produce first-order TL peaks characterized by very high-frequency factors (Mandowski, 2006a, 2007). This is a common situation in many popular TL detectors (e.g. Bilski, 2002). The result was analytically derived for rather simple case of interacting T–RC pairs; however, most likely it applies also to much more complex systems of interacting clusters (IACs). Dose-dependent TL kinetics of IAC may exhibit many other interesting features. Particularly, due to the cascade detrapping (CD) phenomenon one may expect the occurrence of very narrow peaks with giant frequency factors. However, as the dose increases, the clusters become larger. Due to simple electrostatic arguments one may suspect that trapped charge carriers are bounded stronger within traps. The higher activation energy results in shifting of TL peaks to higher temperatures. Then, undergoing the cascade detrapping mechanism a narrow delocalized TL peak is produced. For larger clusters the CD mechanism should occur much stronger, as each delocalized trapping event to a larger cluster releases more charge carriers from traps. Therefore, we should observe an increase in the apparent values of activation energy and frequency factor (for even higher doses one may also expect saturation of these values). This qualitative theoretical picture seems to be well confirmed by some experimental results. Recently, Bilski et al. (2007) studying LiF:Mg,Cu,P detectors at ultra-high dose range found very pronounced first-order peak. With increasing dose the peak shifts to higher temperatures having also increased values of activation energy and frequency factor. Qualitative account of this phenomenon as well as other features allowing the identification of ICs and IAC systems will be discussed in a separate paper. Acknowledgment This work was partly supported by research project from the Polish Ministry of Science over the years 2007–2010. References Bilski, P., 2002. Radiat. Prot. Dosim. 100, 199–206. Bilski, P., Obryk, B., Olko, P., Mandowska, E., Mandowski, A., Kim, J.L., 2007. Characteristics of LiF:Mg,Cu,P thermoluminescence at ultra-high dose range. Radiat. Meas., this issue, doi:10.1016/j.radmeas.2007.10.015. Chen, R., McKeever, S.W.S., 1997. Theory of Thermoluminescence and Related Phenomena. World Scientific, Singapore. Halperin, A., Braner, A.A., 1960. Phys. Rev. 117, 408–415. Land, P.L., 1969. J. Phys. Chem. Solids 30, 1693–1708. Mandowski, A., 1998. Radiat. Prot. Dosim. 84 (1–4), 21–24. Mandowski, A., 2001. Radiat. Measurements 33, 747–751. Mandowski, A., 2002. Radiat. Prot. Dosim. 100 (1–4), 115–118. Mandowski, A., 2005. J. Phys. D. Appl. Phys. 38, 17–21. Mandowski, A., 2006a. Radiat. Prot. Dosim. 119, 23–28. Mandowski, A., 2006b. Radiat. Prot. Dosim. 119, 85–88. Mandowski, A., 2007. Semi-localized transitions model—general formulation and classical limits. Radiat. Meas., this issue, doi:10.1016/j.radmeas. 2007.10.009. ´ R tek, J., 1992. Phil. Magazine B 65, 729–732. Mandowski, A., Swia ´ R tek, J., 1997. J. Phys. III (France) 7, 2275–2280. Mandowski, A., Swia ´ R tek, J., 1998. Radiat Measurements 29, 415–419. Mandowski, A., Swia Townsend, P.D., Rowlands, A.P., 1999. Radiat. Prot. Dosim. 84, 7–12.
How to detect trap cluster systems? Arkadiusz Mandowski Institute of Physics, Jan Długosz University, ul. Armii Krajowej 13/15, PL-42200 CzeR stochowa, Poland
Abstract Spatially correlated traps and recombination centres (trap-recombination centre pairs and larger clusters) are responsible for many anomalous phenomena that are difficult to explain in the framework of both classical models, i.e. model of localized transitions (LT) and the simple trap model (STM), even with a number of discrete energy levels. However, these ‘anomalous’ effects may provide a good platform for identifying trap cluster systems. This paper considers selected cluster-type effects, mainly relating to an anomalous dependence of TL on absorbed dose in the system of isolated clusters (ICs). Some consequences for interacting cluster (IAC) systems, involving both localized and delocalized transitions occurring simultaneously, are also discussed. © 2007 Elsevier Ltd. All rights reserved. Keywords: Thermoluminescence (TL); Trap clusters; Thermal bleaching; Recombination
1. Introduction Classical models for trapping and recombination of charge carriers in dielectrics relate to two analytical cases. The first one relates to uniform distribution of traps and recombination centres (RCs), where all transitions of charge carriers from traps to RCs go through delocalized transport bands. This is the simple trap model (STM—cf. Chen and McKeever, 1997). The second one relates to pairs of traps and RCs (T–RC) located close to each other. The model of localized transitions (LT), for the first time formulated by Halperin and Braner (1960) and then modified by Land (1969), assumes that recombination of trapped carriers proceeds through a local excited level. This system of T–RC pairs may be considered as a system of smallest clusters. Results of many experiments and theoretical considerations gave evidence that thermoluminescence (TL) kinetic processes proceed in large-scale defects rather than in pointlike traps (e.g. the paper of Townsend and Rowlands, 1999). None of the standard TL kinetic models is able to describe this type of kinetics properly. All the types of non-standard kinetics we will term the spatially correlated systems (SCSs). The studies of TL kinetics in various SCSs were started some ´ R tek (1992). For this purpose years ago by Mandowski and Swia they used several Monte Carlo algorithms modelling TL, e.g. in E-mail address: [email protected]. 1350-4487/$ - see front matter © 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.radmeas.2007.09.018
1-D and 3-D systems under different external conditions (e.g. ´ R tek, 1997, 1998). It was found that TL in Mandowski and Swia SCSs shows many unexpected features that cannot be explained within the framework of LT and STM models. Examples include apparently composite structure of monoenergetic peaks, additional ‘displacement’ peaks and the dependence of TL on the external electric field. For review, see Mandowski (1998, 2001, 2006a). Recently, an analytical model was proposed for the isolated clusters (ICs) model, which is a special case of SCSs (Mandowski, 2002, 2006b). The theory is based on two trap structural functions (TSFs)—n and h , for electrons and holes, respectively. Extensive numerical calculations gave the evidence that these functions depend only on structural properties of the solid. TSFs do not depend e.g. on the heating rate and the activation energy of traps. In this paper, the properties of IC systems are studied with respect to dose dependence. Using Monte Carlo algorithms TSFs were calculated for various initial fillings of traps. It will be shown that TSFs depend on dose in a very specific way, quite different from that found in both classical models—STM and LT. In this way it is possible to distinguish between classical and cluster-type kinetics of charge carrier recombination. Other properties relate to SCSs with both localized and delocalized transitions occurring simultaneously. The simplest case of this kind, relating to T–RC pairs, was recently formulated analytically as the model of semi-localized transitions (SLT)
168
A. Mandowski / Radiation Measurements 43 (2008) 167 – 170
(Mandowski, 2005, 2007). The model is able to explain many unexpected features observed in typical solid state detectors, e.g. extremely high-frequency factors (Mandowski, 2006a). It seems that these and some other features relate also to much complex systems consisting of large interacting clusters. Qualitative discussion of the dose dependence in interactive cluster (IAC) systems will be provided. 2. TL kinetics of isolated clusters Commonly accepted explanation of long-lasting phosphorescence and TL phenomena is based on the assumption of metastable levels (traps and RCs) situated within the energy gap. Although direct transition from trap to a recombination centre is possible, most of the transitions take place through excited states. In the model of ICs all trapping and recombination processes occur only within the clusters, i.e. no interaction between clusters is possible. TL is produced in the same way as in the LT model—a trapped charge carrier is thermally excited to a local excited level and then it may be retrapped or may recombine with an opposite charge carrier trapped at the RC level. For the simplest case of a single type of active traps, deep traps and RCs the following set of kinetic equations was proposed (Mandowski, 2002): −E −n˙ = n exp − n (n)ne , (1a) kT (t) −h˙ = h (h)ne ,
(1b)
h = n + ne + M,
(1c)
where E stands for the activation energy and is the frequency factor for active traps. n, ne and h denote the total concentrations of electrons trapped in active traps, electrons in the excited levels and holes trapped in RCs. M stands for the concentration of electrons in the thermally disconnected traps (deep traps), i.e. traps that are not emptied during the experiment. n (n) and h (h) denote two TSFs for trapping and recombination, respectively. It is interesting that the set of Eqs. (1) is valid also for both classical cases: LT and STM. For these two limiting cases one has to assume appropriate form of TSFs ¯ (LT) = A, n (LT)
h
= B¯
(2a) (2b)
and (STM) = A(N − n), n (STM)
h
= Bh,
(3a) (3b)
where A, B, A¯ and B¯ are constants and N is the total concentration of traps (in the case of STM, ne has the meaning of the concentration of carriers in the conduction band). 3. Dose dependence of TSFs To calculate TSFs for an arbitrary IC system one has to perform Monte Carlo calculations. The algorithm is performed
by considering elementary transitions—ID , IT and IR for detrapping, trapping and recombination, respectively: −E ID (t) = exp , (4a) kT (t) ¯ N¯ − n(t)], IT (t) = A[ ¯
(4b)
¯ IR (t) = B¯ h(t).
(4c)
Here, the dashed values denote variables and parameters relating to a single cluster of the system. N¯ denotes the number of trap levels, n¯ e is the number of electrons in the local excited level, n¯ is the number of electrons in traps and h¯ denotes the number of holes in RCs. A¯ and B¯ denote coefficients for trapping and recombination, respectively. A detailed method of the simulation as well as the scaling properties defining the relationship between the microscopic and the macroscopic parameters was given in some previous papers (e.g. Mandowski and ´ R tek, 1992, 1997). Swia For a given IC system TSFs can be calculated by performing a two-stage Monte Carlo simulation. In the first stage the clusters are randomly populated with electrons and holes. It is assumed that each hole–electron pair is trapped within the same cluster. Then, the TL process is simulated according to Eq. (4). Finally, n and h functions are calculated from Eqs. (1a) and (1b), respectively, i.e. −h˙ , ne −E 1 n (n) = n exp + n˙ . ne kT (t) h (h) =
(5a) (5b)
Now, it is interesting to study what is the dependence of these functions on dose. First of all let us look at both standard models where TSFs are given by analytical equations (2a), (2b) and (3a), (3b). These functions are plotted for various initial concentrations of trapped charge carriers 0 ≡ n0 /N = 0.1, 0.5 and 1.0 in Fig. 1. The common feature of the two models is that TSFs for partially filled systems (0 = 0.1 and 0 = 0.5) reproduce perfectly a part of the whole TSFs calculated for full initial filling (0 = 1). Such behaviour seems to be quite obvious. Surprisingly, for all other cases—including clusters of various sizes, the dependence is quite different. As an example, we calculated TL kinetics in two cluster systems consisting of clusters containing N¯ = 2 traps and RCs as well as N¯ = 5 traps and RCs. These systems were initially populated to the relative concentration of 0 =0.1, 0.5 and 1.0. Results of this simulations are shown in Fig. 2A with respect to h functions. It is clearly seen that the calculated functions do not overlap for various initial populations. Another interesting puzzle is that the shape of these curves is somewhat similar (within preciseness of the Monte Carlo calculations) and applying simple scaling on the horizontal axis, according to the equations n n n → n , (6a) N N h h h → h , (6b) h0 h0
A. Mandowski / Radiation Measurements 43 (2008) 167 – 170
Fig. 1. n and h functions calculated for standard LT and STM models (using Eqs. (2a), (2b) and (3a), (3b)) with various relative initial fillings of traps, i.e. 0 = 0.1 (䊉 for LT and for STM), 0 = 0.5 ( for LT and ♦ for STM) and 0 = 1.0 (solid lines for both models).
we are able to overlap these functions with acceptable accuracy. The constant is chosen numerically. Results are shown in Fig. 2B. A possible explanation for this divergence of TSFs may be explained by a difference in a time sequence (or time order) in which the clusters are populated (randomly) and depopulated (thermally). Thermal bleaching is not a uniform random
169
Fig. 2. Diagram A—h function calculated for two IC systems (N¯ = 2 and N¯ = 5) using Monte Carlo calculations and then Eqs. (5a) and (5b) with various relative initial fillings of traps 0 = 0.1 (䊉), 0 = 0.5 () and 0 = 1.0 (solid line). Diagram B represents the same function after scaling according to Eq. (6).
process. The highest probability of detrapping (4a) and recombination (4c) relates to charge carriers occupying these clusters which are populated at most. For these clusters the rate of retrapping (4b) is also much smaller. Therefore, these charge carriers will recombine in first order. These differences are not significant for LT model consisting of T–RC pairs (each cluster may consist of a maximum of one electron and one hole), as
170
A. Mandowski / Radiation Measurements 43 (2008) 167 – 170
well as the STM model for which the whole solid may be considered as a single very large cluster. The mathematical origin of the scaling property (6), however, well confirmed numerically, is not clear. Most likely it is an approximate feature. Possibly the results would be slightly different if one would apply a more sophisticated excitation scheme. However, it requires more additional assumptions and such cases will be considered later. 4. Experimental consequences As a consequence of the differences mentioned above, let us note that also TL curves measured for the same doses may be different. This property is characteristic exclusively for cluster systems ( = 1) and does not hold for any of the standard model (i.e. STM and LT where = 1). Therefore, by applying appropriate measurements we can distinguish between cluster and non-cluster systems. For these purposes we may suggest the following experimental procedure: Experiment 1: • Excite the sample up to a high dose (e.g. up to saturation 0 = 1). • Thermally bleach the traps, decreasing the relative population down to 1 . • Measure TL1 (determining 1 ). Experiment 2: • Excite the sample up to 1 . • Measure TL2 . In isolated cluster systems one may expect that TL1 = TL2 (different shapes of the two glow curves), but naturally T2 T2 T1 JT L1 (T ) dT = T1 JT L2 (T ) dT because the doses are the same. However, one should realize that in practical applications it may be difficult to differentiate between ICs and multiple trap levels. Therefore, most likely, such an analysis will require some elaborate calculations to compare the results with a multi-level model prediction. 5. Conclusions The results presented here apply mainly to isolated cluster systems, i.e. a set of non-interacting groups of traps and RCs where recombination takes place only within these clusters. Another assumption relates to the stability of trap levels. In the framework of IC model the trap levels are stable—it means that successive recombinations do not change the activation energies and other trap parameters for the remaining charge carriers. Fulfilling all of these conditions one may expect, that performing several simple measurements (as those suggested above) one is able to determine whether the system under consideration is cluster-like or not. Nevertheless, it is believed that many real solid-state detectors undergo much more complex type of TL kinetics. In particular, it was found that simultaneous localized and delocalized
transitions involving occupation-dependent energy levels produce first-order TL peaks characterized by very high-frequency factors (Mandowski, 2006a, 2007). This is a common situation in many popular TL detectors (e.g. Bilski, 2002). The result was analytically derived for rather simple case of interacting T–RC pairs; however, most likely it applies also to much more complex systems of interacting clusters (IACs). Dose-dependent TL kinetics of IAC may exhibit many other interesting features. Particularly, due to the cascade detrapping (CD) phenomenon one may expect the occurrence of very narrow peaks with giant frequency factors. However, as the dose increases, the clusters become larger. Due to simple electrostatic arguments one may suspect that trapped charge carriers are bounded stronger within traps. The higher activation energy results in shifting of TL peaks to higher temperatures. Then, undergoing the cascade detrapping mechanism a narrow delocalized TL peak is produced. For larger clusters the CD mechanism should occur much stronger, as each delocalized trapping event to a larger cluster releases more charge carriers from traps. Therefore, we should observe an increase in the apparent values of activation energy and frequency factor (for even higher doses one may also expect saturation of these values). This qualitative theoretical picture seems to be well confirmed by some experimental results. Recently, Bilski et al. (2007) studying LiF:Mg,Cu,P detectors at ultra-high dose range found very pronounced first-order peak. With increasing dose the peak shifts to higher temperatures having also increased values of activation energy and frequency factor. Qualitative account of this phenomenon as well as other features allowing the identification of ICs and IAC systems will be discussed in a separate paper. Acknowledgment This work was partly supported by research project from the Polish Ministry of Science over the years 2007–2010. References Bilski, P., 2002. Radiat. Prot. Dosim. 100, 199–206. Bilski, P., Obryk, B., Olko, P., Mandowska, E., Mandowski, A., Kim, J.L., 2007. Characteristics of LiF:Mg,Cu,P thermoluminescence at ultra-high dose range. Radiat. Meas., this issue, doi:10.1016/j.radmeas.2007.10.015. Chen, R., McKeever, S.W.S., 1997. Theory of Thermoluminescence and Related Phenomena. World Scientific, Singapore. Halperin, A., Braner, A.A., 1960. Phys. Rev. 117, 408–415. Land, P.L., 1969. J. Phys. Chem. Solids 30, 1693–1708. Mandowski, A., 1998. Radiat. Prot. Dosim. 84 (1–4), 21–24. Mandowski, A., 2001. Radiat. Measurements 33, 747–751. Mandowski, A., 2002. Radiat. Prot. Dosim. 100 (1–4), 115–118. Mandowski, A., 2005. J. Phys. D. Appl. Phys. 38, 17–21. Mandowski, A., 2006a. Radiat. Prot. Dosim. 119, 23–28. Mandowski, A., 2006b. Radiat. Prot. Dosim. 119, 85–88. Mandowski, A., 2007. Semi-localized transitions model—general formulation and classical limits. Radiat. Meas., this issue, doi:10.1016/j.radmeas. 2007.10.009. ´ R tek, J., 1992. Phil. Magazine B 65, 729–732. Mandowski, A., Swia ´ R tek, J., 1997. J. Phys. III (France) 7, 2275–2280. Mandowski, A., Swia ´ R tek, J., 1998. Radiat Measurements 29, 415–419. Mandowski, A., Swia Townsend, P.D., Rowlands, A.P., 1999. Radiat. Prot. Dosim. 84, 7–12.