Radiation Protection Dosimetry Advance Access published June 10, 2014 Radiation Protection Dosimetry (2014), pp. 1–11
doi:10.1093/rpd/ncu184
THE MODIFIED UNIFIED INTERACTION MODEL: INCORPORATION OF DOSE-DEPENDENT LOCALISED RECOMBINATION A. Lavon1, I. Eliyahu1,2, L. Oster3 and Y. S. Horowitz1,* 1 Ben Gurion University of the Negev, Beersheva 84105, Israel 2 Soreq Nuclear Research Center, Yavne 81800, Israel 3 Sami Shamoon College of Engineering, Beersheva 84100, Israel *Corresponding author: [email protected]
The unified interaction model (UNIM) was developed to simulate thermoluminescence (TL) linear/supralinear dose–response and the dependence of the supralinearity on ionisation density, i.e. particle type and energy. Before the development of the UNIM, this behaviour had eluded all types of TL modelling including conduction band/valence band (CB/VB) kinetic models. The dependence of the supralinearity on photon energy was explained in the UNIM as due to the increasing role of geminate (localised recombination) with decreasing photon/electron energy. Recently, the Ben Gurion University group has incorporated the concept of trapping centre/luminescent centre (TC/LC) spatially correlated complexes and localised/delocalised recombination into the CB/VB kinetic modelling of the LiF:Mg,Ti system. Track structure considerations are used to describe the relative population of the TC/LC complexes by an electron–hole or by an electron-only as a function of both photon/electron energy and dose. The latter dependence was not included in the original UNIM formulation, a significant over-simplification that is herein corrected. The modified version, the M-UNIM, is then applied to the simulation of the linear/supralinear dose–response characteristics of composite peak 5 in the TL glow curve of LiF:Mg,Ti at two representative average photon/electron energies of 500 and 8 keV.
INTRODUCTION The unified interaction model (UNIM) was developed at the Ben Gurion University (BGU) of the Negev by Horowitz et al.(1, 2) and Horowitz(3) in the late 1990s in order to simulate the linear/supralinear thermoluminescence (TL) dose –response observed for the glow peaks of LiF:Mg,Ti (TLD-100) and the dependence of the supralinearity on ionisation density, i.e. particle type and energy. TL supralinearity is a complex phenomenon dependent on many physical and experimental parameters(4) and no other TL models(5, 6) had previously been capable of modelling these characteristics of dose – response. For example, conventional conduction band (CB)/valence band (VB) kinetic models are blind to the effects of non-uniform ionisation density in particle tracks and the effects of overlapping tracks(7). The UNIM is based on the presence of spatially correlated/coupled trapping centres (TCs) and luminescent centres (LCs), which can lead to localised (geminate) recombination. Geminate recombination has been used to describe many luminescent phenomena(6 – 8). Recently, the BGU group has incorporated the concept of TC/LC spatially correlated complexes and localised/delocalised (LDL) recombination in CB /VB kinetic modelling of the LiF:Mg,Ti system(9). Track structure considerations are used to describe the relative concentration of the TC/LC complexes by an electron–hole ne – h, or
by an electron-only, ne, as a function of photon energy and dose. The same concept is introduced herein into a modified UNIM (M-UNIM). The dependence of ne – h/ne on dose was not employed in the original formulation of the UNIM and was a significant oversimplification. The M-UNIM is a multi-parameter model and many of the parameters (including those governing the relative e – h and e-only population of the TC/LC complex as a function of dose and energy) cannot be estimated from ab initio principles. However, as in the previous applications of the UNIM, an effort is made to restrict the range of allowed values of the parameters using ancillary measurements such as optical absorption, heavy charged particle (HCP) fluence response, reference to other known material characteristics, e.g. dopant levels, etc. The UNIM can also describe fluence response for HCPs (in the framework of the Extended Track Interaction Model)(10). In fact, two of the variable parameters in the UNIM applied to photons/electrons are estimated from low-energy alpha particle fluence response studies(11). The basic idea of the UNIM is that the linear response at low dose arises from geminate recombination in a localised entity. For gamma rays and electrons, the localised entity is the coupled/spatially correlated TC/LC. For HCPs, the localised entity is
# The Author 2014. Published by Oxford University Press. All rights reserved. For Permissions, please email: [email protected]
Downloaded from http://rpd.oxfordjournals.org/ at University of Regina on August 22, 2014
Received 6 April 2014; revised 4 May 2014; accepted 9 May 2014
A. LAVON ET AL.
standard deviations are too large to allow the reliable estimation of f (D)(15) max. SPATIALLY CORRELATED TCS AND LCS The presence of spatially correlated TCs and LCs in the TL mechanisms of LiF:Mg,Ti has been conclusively demonstrated by several means(16, 17); indeed, spatial correlation leading to ionisation density effects has a profound influence on the behaviour (shape) of composite peak 5 in the glow curve of this material. A typical glow curve of LiF:Mg,Ti (TLD-100) following 5-MeV alpha particle and 90Sr/90Y beta irradiation (500 keV average energy) is shown in Figure 1. The glow curve was obtained using a heating rate of 1.5 Ks21 and ‘natural’ cooling following the 4008C pre-irradiation anneal. Following photon/electron irradiation, glow peak 5a is present at a much lower intensity relative to peak 5 than in the alpha particle-induced glow curve due to lower ionisation density. Peak 5a has been demonstrated to arise from geminate recombination in an e–h occupied TC/LC complex, whereas peak 5 arises from a delocalised recombination originating from the
TL DOSE– RESPONSE The TL dose –response, F(D), is defined as the TL signal intensity per unit dose measured by the integral of the glow peak in question. The normalised TL dose –response, f (D) (actually a measure of the relative TL efficiency) is defined in Equation (1). In the linear dose–response region, f (D) ¼ 1; in the supralinear region, f (D) . 1; and at high levels of dose, the signal saturates and then decreases with increasing dose due to radiation damage effects. Thus, f (D) is given by f ðDÞ ¼
½FðDÞ=D ½F ðD Þ=D
ð1Þ
where D* is a low dose in the linear region of dose– response. The critical level of dose for composite peak 5 in the glow curve of LiF:Mg,Ti (TLD-100) above which supralinearity occurs is 1 Gy, and f (D)max reaches values of 3–4 at dose levels of 200 –400 Gy following irradiation by photons/electrons of energy .100 keV. The constancy of f (D)max above 100 keV is related to the ionisation density characteristics of single photon/electron tracks and to the levels of dose at which track overlap becomes important. There may be some variation in f (D)max at higher energies but this has never been demonstrated experimentally. On the other hand, f (D)max decreases to 1.7 for 8 keV (20 kVp) X-rays. Other measurements at energies between 100 and 8 keV confirm this behaviour(3, 6). Experiments at even lower photon energies (0.6 –1.5 keV) indicate some slight supralinearity; however, the
Figure 1. Glow curve of LiF:Mg,Ti (TLD-100) following alpha particle (5 MeV) and 90Sr/90Y (500 keV mean energy) electron irradiation. Note the lower relative intensity of peak 5a following electron irradiation.
Page 2 of 11
Downloaded from http://rpd.oxfordjournals.org/ at University of Regina on August 22, 2014
the isolated HCP track. Current investigations indicate that many TL-associated defects are based on complex molecules with internal localised transitions that may extend over many lattice sites. Particularly compelling is the evidence from TL emission spectra(12, 13) and the dependence of the glow peak 5 shape parameters on Ti concentration(14), which demonstrated that the TC and LC structures interact directly. The dependence on dose of the delocalised CB-mediated luminescence recombination gives rise to supralinearity. As the dose increases, the average of the distance distribution between occupied/active neighbouring TC/LC entities decreases, and the luminescence recombination efficiency increases due to the greater probability of charge carrier migration between neighbouring TC/LC complexes without interception by the competitive centres (CCs). In addition, if the CCs capture charge of the same sign as the TCs, their ‘competitive efficiency’ is decreased as the dose level increases since an occupied CC no longer serves as an active CC for the charge carrier liberated by the TC.
THE MODIFIED UNIFIED INTERACTION MODEL
available in the reader. This improved temperature resolution proves to be an important factor in reducing the temperature jitter in the appearance of Tmax (the temperature of maximum intensity of composite peak 5 in the glow curve) for a series of glow curves. Using these techniques, the deconvolution of the glow curve into component glow peaks and optimum readout techniques allow the measurement of peak 5a intensity to 5 % (1 SD) precision(20): 5a/5 relative intensities have been demonstrated to allow separation of alpha particle and beta components in a mixed radiation field to a precision of 15 % (1 SD)(21). Following irradiation, the TC/LC complex can exist in four different trapping configurations as shown in Figure 4. (1) The fraction of occupied TC/LC complexes that have simultaneously captured an e–h pair is given by s. (2) The fraction of occupied TC/LC complexes that have captured an e-only is given by 1 2 s. (3) The fraction of occupied TC/LC complexes that have captured a hole only is given by 1 2 s. (4) A certain fraction of the TC/LC complexes remains unoccupied. Two routes for recombination are also shown: (1) The electron recombines with the hole within the complex (solid arrow) giving rise to peak 5a. (2) Recombination of an electron from either electron occupied configuration with an external LC, with possibility of competition due to CB transport (dashed arrow) giving rise to peak 5.
Figure 2. Typical deconvoluted glow curve following beta particle irradiation with improved temperature resolution. The term mixed-batch refers to a sample population culled from several previously used batches of TLD-100 of unknown previous irradiation history.
Page 3 of 11
Downloaded from http://rpd.oxfordjournals.org/ at University of Regina on August 22, 2014
transfer of the electron to the CB. Although extraction of peak 5a intensity from the glow curve appears difficult (18), a deconvolution protocol has been developed at BGU, based on ancillary techniques which allow the determination of the kinetic peak shape parameters of peaks 4, 5a and 5(19). The material is bleached using 4 eV photons corresponding to the optical ionisation energy of the composite peak 5 trapping structures. Glow peak 4 is regrown in this procedure due to the release of electrons from the e– h populated TC/LC and peak 5 is strongly reduced in intensity, resulting in a glow curve whose main components are peaks 4 and 5a. A 5 eV bleach, on the other hand, following irradiation and 4008C high temperature annealing releases electrons from the remaining F centres which repopulate peak 5 TCs but peaks 4 and 5a are only very weakly populated due to the small concentration of residual holes not thermally detrapped by the 4008C anneal. This results in a single glow peak 5 in the glow curve. The peak shape parameters deduced from these procedures are then adopted as fixed parameters in the deconvolution protocol for pure and mixed radiation fields. In addition, improved temperature resolution(20) using a reading resolution of 0.4758C per channel instead of the reading temperature resolution of 1.58C per channel as shown in Figure 1 improves the precision of the deconvolution. The improved temperature resolution results in the glow curves shown in Figures 2 and 3 following beta particle and alpha particle irradiation, respectively, and is obtained by simply narrowing the temperature range of measurement while maintaining data collection over the 200 channels
A. LAVON ET AL.
Figure 4. Schematic representation of the spatially correlated TC/LC model showing the four possible configurations following irradiation and possible routes for recombination.
A probability distribution function exists for the likelihood of finding the trapped electron and hole at various distances of separation. At large distances (say, greater than several tens of nm) the complex acts essentially as two independent centres and e–h recombination mainly occurs via charge carrier migration in the
CB. On the other hand, as the distance between the trapped electron in the TC and a trapped hole in the LC decreases, localised recombination comes into play, which occurs via a quantum-mechanical tunnelling process or via short-range, semi-localised migration between nearest neighbour TCs and LCs.
Page 4 of 11
Downloaded from http://rpd.oxfordjournals.org/ at University of Regina on August 22, 2014
Figure 3. Typical deconvoluted glow curve following alpha particle irradiation with improved temperature resolution.
THE MODIFIED UNIFIED INTERACTION MODEL
The presence of semi-localised recombination via band-tail states has been invoked in recent LDL kinetic modelling in order to simulate the ratio of glow peaks 5a/5 as a function of photon dose(22). A major postulate of the UNIM was that s is photon/electron energy dependent due to the more localised pattern of microscopic ionisation density as the electron energy decreases. However, the effect of overlapping/intersecting electron tracks with increasing dose was ignored and thus s was assumed to be independent of dose. This omission has been corrected in the current modelling and is the main subject of this paper.
(6) ro is the radius of the TC/LC complex; (7) g(rh.Ri) is a 2D solid angle factor between two neighbouring TC/LC pairs given approximately by Pr2h =4PR2i ; (8) SLC ¼ Pr2h is the cross section for capture of an electron by the LC; (9) Ri is the distance between neighbouring TC/LC pairs; (10) l is the mean free path of the electrons between the TC/LC pairs, is an increasing function of dose (due to the filling (de-activation) of the competitors with increasing dose) and is given by
MATHEMATICAL FORMULATION OF THE UNIM
(11) lo is the mean free path for charge carrier diffusion in the inter-track unirradiated region and is given by (NCCSCC)21; (12) SCC is the capture cross section of the unoccupied competing centre.
The TL signal intensity, F(D), is given by FðDÞ ¼ ksne ðDÞ þ ð1 ksÞne
rð 3 max X i¼1
r0
ð3Þ
gðRÞ ð2Þ
eðR=lðDÞÞ Pi ðnh ; R; DÞdR The first term, ksne, determines the contribution of the e–h populated TC to the total TL intensity and results in a linear/exponentially saturating contribution to the dose –response. The second term estimates the increased TL intensity at higher dose levels (supralinearity) due to CB mediated recombination in the presence of dose-dependent competitive processes. The decrease in f (D)max with decreasing photon energy was achieved in the previous UNIM simulations by increasing the value of ks leading to geminate recombination from 0.07 to 0.17. This increase in the e –h populated TC/LC is expected due to increased ionisation density of single electron tracks with decreasing photon energy. The parameters in Equation (2) are defined as follows: (1) k is the relative probability of geminate/localised recombination in the TC/LC complex giving rise to luminescence; (2) s is the fraction of TC/LC complexes that have captured an e –h pair following irradiation; (3) ne, nLC (nh) and nCC represent the density of occupied TCs, LCs and competitive centres (CCs), respectively. Their dose –response is described by a linear/exponentially increasing function, Ni (1 2 exp[2biD)], as demonstrated by optical absorption dose –response experimental measurements(23); (4) bTC, bLC and bCC are the dose-filling constants of the TCs, LCs and CCs, respectively; (5) NTC, NLC and NCC represent the density of available TCs, LCs and CCs, respectively;
The nearest neighbour probability distribution functions, Pi (nLC, Ri)dRi, are a crucial element in the calculation of F(D) since it is over these distances that the electron must migrate in the CB before recombination with a hole trapped in an LC; F(D) is calculated using the sum of the first, second and third nearest neighbour interactions. The derivation of Pi ¼ 1 – 3 has been described in previous publications(3, 6) and is given by the following: (1) First nearest neighbour PDF: P1 dR ¼ nh 4pR2 dR eð4=3ÞpR
3
nh
ð4Þ
(2) Second nearest neighbour PDF: P2 dR ¼
16 2 2 5 3 n p R dR eð4=3ÞpR nh 3 h
ð5Þ
(3) Third nearest neighbour PDF: P3 dR ¼
64 3 3 8 3 n p R dR eð4=3ÞpR nh 9 h
ð6Þ
Figure 5 illustrates the first and second nearest neighbour probability distribution function for various values of the dose. As expected, these shift to lower values of R with increasing dose. For the values of nh shown in Table 1 the probabilities are non-zero over ˚ to a few thoutypical distances from a few tens of A ˚ sand A. The TL signal intensity in the UNIM is thus given by F ðDÞ ¼ ksne ðDÞ þ ð1 ksÞne
3 X i¼1
Page 5 of 11
Ii
ð7Þ
Downloaded from http://rpd.oxfordjournals.org/ at University of Regina on August 22, 2014
lðDÞ ¼ l0 ebcc D
A. LAVON ET AL.
Table 1. UNIM parameters. Parameter
Value
ks
0.17 (20 kVp) 0.12 (50 kVp) 0.07 (.100 keV) 6` 1022 (m23) 1.95` 1022 (m23) 5` 1024 (Gy21) 0.5` 1024 (Gy21) 1.5` 1023 (Gy21) 4.5` 1023 (Gy21) 2` 1029 (m) 9.6` 10216 (m2) 4.75` 1028 (m)
Ne NLC bTC bCC bRD bLC ro SLC lo ¼ (NCC` SCC)21
where Ii is the contribution of the ith nearest neighbour interaction. At low dose, in the linear part of the dose– response, the second term describing the increased efficiency due to nearest neighbour interactions is vanishingly small due to the competitive processes, so that FðD Þ ksne ðD Þ
e-only in a TC/LC spatially coupled complex. The relative e–h to e-only concentrations (ne – h/ne) as a function of photon/electron energy and dose are then adopted in the recombination stage as starting parameters in order to simulate the dependence of the TL dose –response on ionisation density. The total population of TCs, NTC, giving rise to peak 5, is composed of a certain fraction of TCs, Ne – h, which are spatially correlated/coupled to the LCs. The remaining fraction of TCs are denoted by Ne. The fraction Ne – h /NTC may be influenced by various factors including methods of crystal growth, thermal treatments such as cooling rate following the 4008C pre-irradiation anneal(24) and others. The fraction, ne – h/ne, of the density of e –h to e-only populated complexes following irradiation is given by neh ¼ a ð1 ebeh D Þ þ b ne
ð10Þ
The combined densities must, of course, yield the total density of occupied peak 5-TCs as follows:
ð8Þ
The normalised TL dose–response, f(D), or relative TL efficiency is then given by
3 ½ne ðDÞ=D ð1 ksÞ X f ðDÞ ¼ Ii 1 þ ½ne ðD Þ=D ks i¼1
ð9Þ
REFORMULATION OF THE UNIM (THE M-UNIM) The LDL kinetic model(9) incorporates in the irradiation stage dose-dependent capture of an e –h and
ntot ¼ NTC ð1 ebTC D Þ ¼ neh þ ne
ð11Þ
The parameters a, b and be – h are postulated to depend on photon/electron energy. These equations are employed to yield values of ne – h/ne which increase slowly up to a threshold dose of 1– 10 Gy where supralinearity begins to appear in the dose–response of peak 5. Above this dose, the ratio increases rapidly and is governed by be – h. The parameter ‘b’ determines the ratio of ne – h/ne at the lowest dose levels; the dose-filling factor be – h is expected to decrease with decreasing photon energy due to the onset of track interaction at higher levels of dose for the lower photon energies. The parameter ‘a’ determines the
Page 6 of 11
Downloaded from http://rpd.oxfordjournals.org/ at University of Regina on August 22, 2014
Figure 5. First and second nearest neighbour probability distribution functions as a function of dose from 1 mGy to 10 000 Gy. Note the shift in the maximum probability to higher distances from the first to the second PDFs.
THE MODIFIED UNIFIED INTERACTION MODEL Table 2. Values of e– h filling parameters—LDL kinetic model.
ne – h/ne
Symbol
f(D)max ¼ 1.7, photon energy 8 keV
f (D)max ¼ 3.3, photon energy .100 keV
a b be – h(Gy21)
0.63 0.30 4` 1023
0.80 0.15 11` 1023
s¼
neh a ð1 ebeh D Þ þ b ¼ 1 þ a ð1 ebeh D Þ þ b ntot
Figures 6 and 7 show values of f(D) summed over the first to third nearest neighbour interactions at two photon energies of 100 keV and 20 kVp (8 keV) for the UNIM and M-UNIM using the parameters shown in Tables 1 and 2. The UNIM simulation has been previously fitted to experimental results. As can be seen, the values of f(D)max are significantly greater in the M-UNIM calculation due to the behaviour of ne – h/ne. This is demonstrated in Figures 8 and 9, which explicitly show the effects of ne – h/ne increasing with dose. In the following section, the influence of a, b and be – h on f(D) is demonstrated, which allowed the appropriate choice of parameter values to correct the overestimate of f(D)max.
ð12Þ
Therefore at low dose sj
D,0:1 b,,0:01
¼
b 1þb
ð13Þ
and at high dose sjDb.10 ¼
aþb 1þaþb
ð14Þ
f(D) is now given by f ðDÞ ¼
3 ½ne ðDÞ=D ksðDÞ ð1 ksðD ÞÞ X Ii þ ½ne ðD Þ=D ksðD Þ ksðD Þ i¼1
!
Figure 6. Comparison of UNIM and M-UNIM simulations for photon/electron energies .100 keV (solid line MUNIM; dashed line UNIM).
ð15Þ ksðDÞ ksðD Þ ¼
ða ð1 ebeh D Þ þ bÞ=ð1 þ a ð1 ebeh D Þ þ bÞ b=ð1 þ bÞ ð16Þ
with 1ksðDÞ ksðD Þ ¼
1kððað1ebeh D ÞþbÞ=ð1það1ebeh D ÞþbÞÞ kðb=ð1þbÞÞ ð17Þ
Figure 7. Comparison of UNIM and M-UNIM simulations for 8 keV (20 kVp) photons (solid line M-UNIM; dashed line UNIM).
Page 7 of 11
Downloaded from http://rpd.oxfordjournals.org/ at University of Regina on August 22, 2014
relative importance of the two terms. Table 2 gives the values of the parameters adopted by Eliyahu et al. (9) to simulate f (D) following photon irradiation of 100 keV ( f (D)max ¼ 3.3) and 8 keV ( f (D)max ¼ 1.7). Incorporation of ne – h/ne into the UNIM requires the calculation of s ¼ ne – h/ntot. It can be easily shown that
COMPARISON OF UNIM AND M-UNIM SIMULATIONS
A. LAVON ET AL.
Figure 10. The dependence of f(D) on the value of the parameter ‘a’ (circle, 0.1; square, 0.3; triangle, 0.5; square, 0.7; inverted triangle, 0.9).
Figure 11. The dependence of f(D) on the value of the parameter ‘b’ (circle, 0.005; plus, 0.01; triangle, 0.05; square, 0.1; inverted triangle, 0.15).
Figure 9. M-UNIM simulation of f(D) for 8 keV photons showing the contribution of ne – h/ne and the first, second and third nearest neighbour interactions (circle, total; plus, ne – h; triangle, first nearest neighbour; inverted triangle, second nearest neighbour; square, third nearest neighbour).
THE DEPENDENCE OF f(D) ON THE PARAMETERS OF ne – h/ne The dependence of f (D) on a, b and be – h is shown below in Figures 10–12. The basis set of parameters are the ones listed in Tables 1 and 2. Variation in the value of ‘a’ leads to a moderate decrease of f (D)max from 3.3 to 2.8, but Dmax is unaffected. Variations in ‘b’, however, lead to large changes in f (D)max, reaching values as high as 75. These very high values reaching even 200 have been measured
Figure 12. The dependence of f(D) on the value of the parameter ‘be – h’ ( plus, 0.4 Gy21; inverted triangle, 0.04 Gy21; circle, 0.004 Gy21; square, 0.0004 Gy21; triangle, 0.00004 Gy21).
Page 8 of 11
Downloaded from http://rpd.oxfordjournals.org/ at University of Regina on August 22, 2014
Figure 8. M-UNIM simulation of f (D) for 100 keV photons showing the contribution of ne – h/ne and the first, second and third nearest neighbour interactions. The contribution of ne – h/ne is dominant; the contribution of the third nearest neighbour interaction is negligible (circle, total; plus, ne – h; triangle, first nearest neighbour; inverted triangle, second nearest neighbour; square, third nearest neighbour).
THE MODIFIED UNIFIED INTERACTION MODEL
experimentally for the high temperature peaks in LiF:Mg,Ti (TLD-100)(25, 26). Very low values of ‘b’ reflect very low values of ne – h/ne at low dose, which implies a TL signal based almost completely on delocalised recombination. This leads to very high values of f(D)max due to the signal dependence on the product of ne and nh, i.e. to n 2. In this case, again, Dmax is unaffected. Figure 12 shows the behaviour of f(D) when be – h is varied. The very high values of be – h give rise to very broad (double-peaked) values of f (D) which have never been observed experimentally. Figure 13 emphasises the unusual behaviour of f (D) with high values of be – h showing explicitly the contribution of the first term to the supralinearity for low and high values of be – h. Of the other UNIM parameters, the only ones that significantly influence the shape of f (D) (Dmax and the dose range for which f (D) . 1) are bTC and b(3) LC. SIMULATIONS OF f(D) AT PHOTON ENERGIES .100 AND AT 8 keV The variable parameters in the M-UNIM are: k, a, be – h, b, bTC, bLC, bCC, bRD, lO, SLC, NLC and ro, 12 parameters in all. An additional parameter, NTC, cancels out since f(D) is normalised. Only a, be – h and b are expected to depend on photon energy; bRD is not explicitly shown in Equation (2) and is introduced to simulate radiation-induced damage to the LCs(26), resulting in reduced available concentration of LCs, NLC*, via NLC ¼ NLC ebRD D
ð18Þ
Radiation damage has been experimentally measured in the TL response to TLD-100. A dose level of
Figure 14. M-UNIM simulation of f (D) for 100 keV photons.
Figure 15. M-UNIM simulation of f (D) for 8 keV photons.
1000 Gy results in a decrease of the TL intensity of composite peak 5 of 40 %(6); at 10 000 Gy the TL intensity decreases by a factor 2 from its maximum value(27). Unlike the dose-filling parameters bTC, bCC, which have been demonstrated to be independent of photon energy(28), bRD might also be somewhat dependent on photon energy; however, there are no experimental data supporting this possibility. The choice of LCs in the radiation damage mechanism is dictated by the results of optical absorption measurements which show no decrease in the optical density of the 4 eV (TC) or 5.45 eV (CC) bands at high levels of dose, reaching 30 000 Gy(29). There are certain constraints on the values of the M-UNIM parameters. The a, be – h, b parameters have been adopted from LDL kinetic modelling; however, there is no compelling reason to expect that these values are appropriate to the M-UNIM since the two models simulate supralinearity in two very different theoretical frameworks. Some of the other
Page 9 of 11
Downloaded from http://rpd.oxfordjournals.org/ at University of Regina on August 22, 2014
Figure 13. The dependence of f(D) for 100 keV photons showing the contribution of ne – h/ne (with be – h ¼ 4` 1025 Gy21) and the first (triangle), second (left-pointing triangle), third (square) nearest neighbourinteractions and total (circle) of all contributions.
A. LAVON ET AL. Table 3. M-UNIM Values of e– h filling parameters.
ne – h/ne
Symbol
f(D)max ¼ 1.7, photon energy 8 keV
f (D)max ¼ 3.3, photon energy .100 keV
a b be – h(Gy21) k
0.15 0.80 1` 1024 0.42
0.50 0.30 40` 1024 0.42
CONCLUSIONS In this work, the UNIM has been modified (MUNIM) to include the dependence of ne – h/ne, on dose. The increase in the e –h population of the TC/ LC complex with increase in ionisation density (dose) is expected based on statistical and nanodosimetric considerations of electron/photon track structure. The functional form of the dose behaviour of ne – h/ ne has been adopted from recent LDL modelling which simulated the dependence of f (D) on photon energy. The M-UNIM is used to simulate the dependence
REFERENCES 1. Horowitz, Y. S., Mahajna, S. and Rosenkrantz, M. Unified approach to gamma and heavy charged particle supralinearity: the track defect interaction model. Radiat. Prot. Dosim. 65, 7– 12 (1996). 2. Horowitz, Y. S., Mahajna, S., Oster, L., Weizman, Y., Satinger, D. and Yossian, D. The Unified Interaction Model applied to the gamma induced supralinearity and sensitisation of peaks 4 and 5 in LiF:Mg,Ti (TLD-100). Radiat. Prot. Dosim. 78, 169 –193 (1998). 3. Horowitz, Y. S. Theory of thermoluminescence gamma dose response: the unified interaction model. Nucl. Instrum. Methods B184, 68–84 (2001). 4. Horowitz, Y. S. The theoretical and microdosimetric basis of thermoluminescence and applications to dosimetry. Phys. Med. Biol. 26, 765–824 (1981). 5. Horowitz, Y. S. Recent models for TL supralinearity. Radiat. Prot. Dosim. 6, 17–20 (1984). 6. Horowitz, Y. S. Ch. 2. A unified and comprehensive theory of the TL dose response of thermoluminescent systems applied to LiF:Mg,Ti. In: Microdosimetric Response of Physical and Biological Systems to Lowand High-LET Radiations: Theory and Applications to Dosimetry. Horowitz,, Y. S. Ed. Elsevier (2006). 7. Chen, R. and Pagonis, V. Thermally and Optically Stimulated Luminescence: A Simulation Approach. Wiley (2011). 8. McKeever, S. W. S. and Horowitz, Y. S. Charge trapping mechanisms and microdosimetric processes in lithium fluoride. Radiat. Phys. Chem. 36, 35– 46 (1990). 9. Eliyahu, I, Horowitz, Y. S., Oster,, L. and Mardor, I. A kinetic model incorporating both localized and delocalized recombination: application to the dependence of the TL dose response on photon energy. J. Lumin. 145, 600– 607 (2014). 10. Horowitz, Y. S., Avila, O. and Rodriguez-Villafuerte, M. Theory of heavy charged particle response (efficiency and supralinearity) in TL materials. Nucl. Instrum. Methods B184, 85– 112 (2001).
Page 10 of 11
Downloaded from http://rpd.oxfordjournals.org/ at University of Regina on August 22, 2014
values of the M-UNIM parameters are severely constrained. For example, bTC and bCC have recently been re-measured to high levels of dose by the BGU group(29), and lo and bLC have been estimated from low-energy alpha particle fluence response data(11); NLC can be estimated approximately from the concentration of Ti in the LiF:Mg,Ti samples(30). The M-UNIM fits to the experimental data for 500 keV electrons and 8 keV photons (8 keV experimental data from Gamboa-deBuen et al.(31)) are shown in Figures 14 and 15 using the values of the parameters given in Table 3. A good fit to the experimental data could not be achieved without changing the e– h dose-filling rate parameters a, b and be – h previously employed in the LDL kinetic modelling. The behaviour of these parameters with photon/ electron energy deserves special comment. The value of s at low dose levels is given by the value of b/ (1þb). This would be expected to be higher for the lower photon/electron energies due to the higher ionisation density at lower energy and, as given in Table 3, these values are 0.44 and 0.23 for the 8 and 500 keV energies, respectively. Similarly, one expects that the value of be – h would be lower for the low energy photons due to the smaller extension of the low energy tracks requiring higher levels of dose to result in track overlap. This is also reflected in Table 3 with values of be – h decreasing from 0.004 Gy21 (500 keV electrons) to 0.0001 Gy21 (8 keV photons). At very high dose, the value of s approaches (a þ b)/(1 þ a þ b) and might be expected to be higher for the 8 keV photons—again the values are 0.7 and 0.45 for 8 and 500 keV, respectively.
of f(D) on photon/electron energies above 100 and at 8 keV. The simulations in both the UNIM and the M-UNIM employed the identical set of parameters at both energies except for the e–h TC/LC dosepopulation characteristics. The M-UNIM and the LDL kinetic model are currently the only existing models capable of predicting the linear/supralinear behaviour of f(D) and its dependence on photon energy. However, the values of the dose-filling characteristics a, b and be – h used in the two models are different. Obviously, further efforts are required in order to discover a single set of parameters that will yield fits to the experimentally measured f(D) in both model simulations. This is an imposing challenge considering their very different theoretical frameworks and may require certain refinements of both models. However, success would constitute the first-time-ever estimation of track structure ionisation-density-dependent effects in TL solid-state systems and perhaps in ionisation-densitysensitive solid-state systems in general.
THE MODIFIED UNIFIED INTERACTION MODEL 21. Sibony, D., Horowitz, Y. S., Oster, L., Wojcik, A. and Sollazzo, A. Combined measurement of dose and a/g radiation-field-components using the shape of composite peak 5 in the glow curve of LiF: Mg,Ti. Available on http:// dx.doi.org/10.1016/j.radmeas.2014.01.006. 22. Eliyahu, I., Horowitz, Y. S., Oster, L., Druzhyna, S. and Mardor, I. Nanodosimetric model incorporating localised and delocalised recombination: application to the prediction of the electron dose response of the peak 5a/5 ratio in the glow curve of LiF:Mg,Ti (TLD-100). Available on http://dx.doi.org/10.1016/j.radmeas.2014.01.015. 23. Issa, N., Oster, L. and Horowitz, Y. S. Optical absorption and sensitisation dose response in LiF:Mg,Ti:application to the unified interaction model predictions of thermoluminescence dose response. Radiat. Prot. Dosim. 100, 107–110 (2002). 24. Fuks, E., Horowitz, Y. S. and Oster, L. Investigation of the properties of composite glow peak 5 in slow-cooled TLD-100. Radiat. Meas. 43, 249– 253 (2008). 25. Massillon, J. L. G., Gamboa-Debuen, I. and Brandan, M. E. Onset of supralinear response in TLD-100 exposed to Co-60 gamma rays. J. Phys. D. Appl. Phys. 39, 262–268 (2006). 26. Livingstone, J., Horowitz, Y. S., Oster, L., Datz, H., Lerch, M., Rosenfeld,, A. and Horowitz, A. Experimental investigation of the 100 keV x-ray dose response of the high temperature thermoluminescence in LiF:Mg,Ti (TLD100):theoretical interpretation using the Unified Interaction Model. Radiat. Prot. Dosim. 138, 320–333 (2010). 27. Montano-Garcia, C. and Gamboa-deBuen, I. Measurements of the optical density and the thermoluminescent response of LiF:Mg,Ti exposed to high doses of 60Co gamma rays. Radiat. Prot. Dosim. 119, 230–232 (2006). 28. Biderman, S., Eliyahu, I., Horowitz, Y. S. and Oster, L. Dose response of F center optical absorption in LiF:Mg,Ti (TLD-100). Available on http://dx.doi.org/ 10.1016/j.radmeas.2014.02.001. 29. Nail, I. et al. Search for ionisation density effects in the radiation absorption stage in LiF:Mg,Ti”. Radiat. Prot. Dosim. 119, 180– 183 (2006). 30. Horowitz, Y. S. The average distance between Mg-based trapping structures in LiF:Mg,Ti and Li:Mg,Cu,P and the relevance to microdosimetry. Radiat. Prot. Dosim. 82, 51–54 (1999). 31. Gamboa-deBuen, I., Buenfil, A. E., Ruiz, C. G., Rodriguez-Villafuerte, M., Flores, A. and Brandan, M. E. Thermoluminescent response and relative efficiency of TLD-100 exposed to low-energy X rays. Phys. Med. Biol. 43, 2073–2083 (1998).
Downloaded from http://rpd.oxfordjournals.org/ at University of Regina on August 22, 2014
11. Rodriguez-Villafuerte, M., Buenfil, A. E., GamboadeBuen, I., Ruiz-Trejo, C., Brandan, M. E., Yossian, D., Satinger, D. and Horowitz, Y. S. Study of the TL response of LiF:Mg,Ti to 3 and 7.5 MeV Helium ions: measurements and interpretation in terms of the track interaction model. Nucl. Instrum. Methods B160, 377 –386 (2000). 12. Biderman, S., Horowitz, Y. S. and Oster, L. Investigation of the emission spectra of LiF:Mg,Ti (TLD-100) during thermoluminescence. Radiat. Prot. Dosim. 100, 369– 372 (2002). 13. Townsend, P. D., Ahmed, K., Chandler, P. J., McKeever, S. W. S. and Witlow, H. J. Measurements of the emission spectra of LiF during thermoluminescence. Radiat. Eff. 72, 245–257 (1983). 14. Weizman, Y., Horowitz, Y. S., Oster, L., Youssian, D. and Horowitz, A. Mixed-order kinetic analysis of the glow curve characteristics of single crystal LiF:Mg,Ti as a function of Ti concentration. Radiat. Prot. Dosim. 84, 47– 50 (1999). 15. Carrillo, R. E., Pearson, D. W., DeLuca, P. M. Jr and MacKay, J. F. Thermoluiminescent response of lithium fluoride (TLD-100) to monoenergetic photon beams of 600, 730, 900 and 1500 eV. Radiat. Prot. Dosim. 47, 519 –524 (1993). 16. Weizman, Y., Horowitz, Y. S. and Oster, L. Investigation of the composite structure of peak 5 in the thermoluminescent glow curve of LiF:Mg,Ti (TLD-100) using optical bleaching. J. Phys. D. Appl. Phys. 32, 2118–2127 (1999). 17. Horowitz, Y. S., Oster, L., Satinger, S., Biderman, S. and Einav, Y. The composite structure of peak 5 in the glow curve of LiF:Mg,Ti (TLD-100): confirmation of peak 5a arising from a locally trapped electron-hole configuration. Radiat. Prot. Dosim. 100, 123–126 (2002). 18. Horowitz, Y. S. and Moscovitch, M. Highlights and pitfalls of 20 years of application of computerised glow curve analysis to thermoluminescence research and dosimetry. Radiat. Prot. Dosim. 153, 1– 22 (2013). 19. Horowitz, Y. S., Fuks, E., Oster, L., Podpalov, L., Belaish, Y. and Ben Shachar, B. Advanced multistage deconvolution applied to composite glow peak 5 in LiF:Mg,Ti (TLD-100). Radiat. Prot. Dosim. 126, 322 –325 (2007). 20. Sibony, D., Horowitz, Y. S. and Oster, L. The effect of sample/planchet geometry and temperature resolution on the reproducibility of glow curve shapes and precision of dose measurement in LiF-TLD-100 thermoluminescent dosimetry. Available on http://dx.doi.org/10.1016/j. radmeas.2014.01.005.
doi:10.1093/rpd/ncu184
THE MODIFIED UNIFIED INTERACTION MODEL: INCORPORATION OF DOSE-DEPENDENT LOCALISED RECOMBINATION A. Lavon1, I. Eliyahu1,2, L. Oster3 and Y. S. Horowitz1,* 1 Ben Gurion University of the Negev, Beersheva 84105, Israel 2 Soreq Nuclear Research Center, Yavne 81800, Israel 3 Sami Shamoon College of Engineering, Beersheva 84100, Israel *Corresponding author: [email protected]
The unified interaction model (UNIM) was developed to simulate thermoluminescence (TL) linear/supralinear dose–response and the dependence of the supralinearity on ionisation density, i.e. particle type and energy. Before the development of the UNIM, this behaviour had eluded all types of TL modelling including conduction band/valence band (CB/VB) kinetic models. The dependence of the supralinearity on photon energy was explained in the UNIM as due to the increasing role of geminate (localised recombination) with decreasing photon/electron energy. Recently, the Ben Gurion University group has incorporated the concept of trapping centre/luminescent centre (TC/LC) spatially correlated complexes and localised/delocalised recombination into the CB/VB kinetic modelling of the LiF:Mg,Ti system. Track structure considerations are used to describe the relative population of the TC/LC complexes by an electron–hole or by an electron-only as a function of both photon/electron energy and dose. The latter dependence was not included in the original UNIM formulation, a significant over-simplification that is herein corrected. The modified version, the M-UNIM, is then applied to the simulation of the linear/supralinear dose–response characteristics of composite peak 5 in the TL glow curve of LiF:Mg,Ti at two representative average photon/electron energies of 500 and 8 keV.
INTRODUCTION The unified interaction model (UNIM) was developed at the Ben Gurion University (BGU) of the Negev by Horowitz et al.(1, 2) and Horowitz(3) in the late 1990s in order to simulate the linear/supralinear thermoluminescence (TL) dose –response observed for the glow peaks of LiF:Mg,Ti (TLD-100) and the dependence of the supralinearity on ionisation density, i.e. particle type and energy. TL supralinearity is a complex phenomenon dependent on many physical and experimental parameters(4) and no other TL models(5, 6) had previously been capable of modelling these characteristics of dose – response. For example, conventional conduction band (CB)/valence band (VB) kinetic models are blind to the effects of non-uniform ionisation density in particle tracks and the effects of overlapping tracks(7). The UNIM is based on the presence of spatially correlated/coupled trapping centres (TCs) and luminescent centres (LCs), which can lead to localised (geminate) recombination. Geminate recombination has been used to describe many luminescent phenomena(6 – 8). Recently, the BGU group has incorporated the concept of TC/LC spatially correlated complexes and localised/delocalised (LDL) recombination in CB /VB kinetic modelling of the LiF:Mg,Ti system(9). Track structure considerations are used to describe the relative concentration of the TC/LC complexes by an electron–hole ne – h, or
by an electron-only, ne, as a function of photon energy and dose. The same concept is introduced herein into a modified UNIM (M-UNIM). The dependence of ne – h/ne on dose was not employed in the original formulation of the UNIM and was a significant oversimplification. The M-UNIM is a multi-parameter model and many of the parameters (including those governing the relative e – h and e-only population of the TC/LC complex as a function of dose and energy) cannot be estimated from ab initio principles. However, as in the previous applications of the UNIM, an effort is made to restrict the range of allowed values of the parameters using ancillary measurements such as optical absorption, heavy charged particle (HCP) fluence response, reference to other known material characteristics, e.g. dopant levels, etc. The UNIM can also describe fluence response for HCPs (in the framework of the Extended Track Interaction Model)(10). In fact, two of the variable parameters in the UNIM applied to photons/electrons are estimated from low-energy alpha particle fluence response studies(11). The basic idea of the UNIM is that the linear response at low dose arises from geminate recombination in a localised entity. For gamma rays and electrons, the localised entity is the coupled/spatially correlated TC/LC. For HCPs, the localised entity is
# The Author 2014. Published by Oxford University Press. All rights reserved. For Permissions, please email: [email protected]
Downloaded from http://rpd.oxfordjournals.org/ at University of Regina on August 22, 2014
Received 6 April 2014; revised 4 May 2014; accepted 9 May 2014
A. LAVON ET AL.
standard deviations are too large to allow the reliable estimation of f (D)(15) max. SPATIALLY CORRELATED TCS AND LCS The presence of spatially correlated TCs and LCs in the TL mechanisms of LiF:Mg,Ti has been conclusively demonstrated by several means(16, 17); indeed, spatial correlation leading to ionisation density effects has a profound influence on the behaviour (shape) of composite peak 5 in the glow curve of this material. A typical glow curve of LiF:Mg,Ti (TLD-100) following 5-MeV alpha particle and 90Sr/90Y beta irradiation (500 keV average energy) is shown in Figure 1. The glow curve was obtained using a heating rate of 1.5 Ks21 and ‘natural’ cooling following the 4008C pre-irradiation anneal. Following photon/electron irradiation, glow peak 5a is present at a much lower intensity relative to peak 5 than in the alpha particle-induced glow curve due to lower ionisation density. Peak 5a has been demonstrated to arise from geminate recombination in an e–h occupied TC/LC complex, whereas peak 5 arises from a delocalised recombination originating from the
TL DOSE– RESPONSE The TL dose –response, F(D), is defined as the TL signal intensity per unit dose measured by the integral of the glow peak in question. The normalised TL dose –response, f (D) (actually a measure of the relative TL efficiency) is defined in Equation (1). In the linear dose–response region, f (D) ¼ 1; in the supralinear region, f (D) . 1; and at high levels of dose, the signal saturates and then decreases with increasing dose due to radiation damage effects. Thus, f (D) is given by f ðDÞ ¼
½FðDÞ=D ½F ðD Þ=D
ð1Þ
where D* is a low dose in the linear region of dose– response. The critical level of dose for composite peak 5 in the glow curve of LiF:Mg,Ti (TLD-100) above which supralinearity occurs is 1 Gy, and f (D)max reaches values of 3–4 at dose levels of 200 –400 Gy following irradiation by photons/electrons of energy .100 keV. The constancy of f (D)max above 100 keV is related to the ionisation density characteristics of single photon/electron tracks and to the levels of dose at which track overlap becomes important. There may be some variation in f (D)max at higher energies but this has never been demonstrated experimentally. On the other hand, f (D)max decreases to 1.7 for 8 keV (20 kVp) X-rays. Other measurements at energies between 100 and 8 keV confirm this behaviour(3, 6). Experiments at even lower photon energies (0.6 –1.5 keV) indicate some slight supralinearity; however, the
Figure 1. Glow curve of LiF:Mg,Ti (TLD-100) following alpha particle (5 MeV) and 90Sr/90Y (500 keV mean energy) electron irradiation. Note the lower relative intensity of peak 5a following electron irradiation.
Page 2 of 11
Downloaded from http://rpd.oxfordjournals.org/ at University of Regina on August 22, 2014
the isolated HCP track. Current investigations indicate that many TL-associated defects are based on complex molecules with internal localised transitions that may extend over many lattice sites. Particularly compelling is the evidence from TL emission spectra(12, 13) and the dependence of the glow peak 5 shape parameters on Ti concentration(14), which demonstrated that the TC and LC structures interact directly. The dependence on dose of the delocalised CB-mediated luminescence recombination gives rise to supralinearity. As the dose increases, the average of the distance distribution between occupied/active neighbouring TC/LC entities decreases, and the luminescence recombination efficiency increases due to the greater probability of charge carrier migration between neighbouring TC/LC complexes without interception by the competitive centres (CCs). In addition, if the CCs capture charge of the same sign as the TCs, their ‘competitive efficiency’ is decreased as the dose level increases since an occupied CC no longer serves as an active CC for the charge carrier liberated by the TC.
THE MODIFIED UNIFIED INTERACTION MODEL
available in the reader. This improved temperature resolution proves to be an important factor in reducing the temperature jitter in the appearance of Tmax (the temperature of maximum intensity of composite peak 5 in the glow curve) for a series of glow curves. Using these techniques, the deconvolution of the glow curve into component glow peaks and optimum readout techniques allow the measurement of peak 5a intensity to 5 % (1 SD) precision(20): 5a/5 relative intensities have been demonstrated to allow separation of alpha particle and beta components in a mixed radiation field to a precision of 15 % (1 SD)(21). Following irradiation, the TC/LC complex can exist in four different trapping configurations as shown in Figure 4. (1) The fraction of occupied TC/LC complexes that have simultaneously captured an e–h pair is given by s. (2) The fraction of occupied TC/LC complexes that have captured an e-only is given by 1 2 s. (3) The fraction of occupied TC/LC complexes that have captured a hole only is given by 1 2 s. (4) A certain fraction of the TC/LC complexes remains unoccupied. Two routes for recombination are also shown: (1) The electron recombines with the hole within the complex (solid arrow) giving rise to peak 5a. (2) Recombination of an electron from either electron occupied configuration with an external LC, with possibility of competition due to CB transport (dashed arrow) giving rise to peak 5.
Figure 2. Typical deconvoluted glow curve following beta particle irradiation with improved temperature resolution. The term mixed-batch refers to a sample population culled from several previously used batches of TLD-100 of unknown previous irradiation history.
Page 3 of 11
Downloaded from http://rpd.oxfordjournals.org/ at University of Regina on August 22, 2014
transfer of the electron to the CB. Although extraction of peak 5a intensity from the glow curve appears difficult (18), a deconvolution protocol has been developed at BGU, based on ancillary techniques which allow the determination of the kinetic peak shape parameters of peaks 4, 5a and 5(19). The material is bleached using 4 eV photons corresponding to the optical ionisation energy of the composite peak 5 trapping structures. Glow peak 4 is regrown in this procedure due to the release of electrons from the e– h populated TC/LC and peak 5 is strongly reduced in intensity, resulting in a glow curve whose main components are peaks 4 and 5a. A 5 eV bleach, on the other hand, following irradiation and 4008C high temperature annealing releases electrons from the remaining F centres which repopulate peak 5 TCs but peaks 4 and 5a are only very weakly populated due to the small concentration of residual holes not thermally detrapped by the 4008C anneal. This results in a single glow peak 5 in the glow curve. The peak shape parameters deduced from these procedures are then adopted as fixed parameters in the deconvolution protocol for pure and mixed radiation fields. In addition, improved temperature resolution(20) using a reading resolution of 0.4758C per channel instead of the reading temperature resolution of 1.58C per channel as shown in Figure 1 improves the precision of the deconvolution. The improved temperature resolution results in the glow curves shown in Figures 2 and 3 following beta particle and alpha particle irradiation, respectively, and is obtained by simply narrowing the temperature range of measurement while maintaining data collection over the 200 channels
A. LAVON ET AL.
Figure 4. Schematic representation of the spatially correlated TC/LC model showing the four possible configurations following irradiation and possible routes for recombination.
A probability distribution function exists for the likelihood of finding the trapped electron and hole at various distances of separation. At large distances (say, greater than several tens of nm) the complex acts essentially as two independent centres and e–h recombination mainly occurs via charge carrier migration in the
CB. On the other hand, as the distance between the trapped electron in the TC and a trapped hole in the LC decreases, localised recombination comes into play, which occurs via a quantum-mechanical tunnelling process or via short-range, semi-localised migration between nearest neighbour TCs and LCs.
Page 4 of 11
Downloaded from http://rpd.oxfordjournals.org/ at University of Regina on August 22, 2014
Figure 3. Typical deconvoluted glow curve following alpha particle irradiation with improved temperature resolution.
THE MODIFIED UNIFIED INTERACTION MODEL
The presence of semi-localised recombination via band-tail states has been invoked in recent LDL kinetic modelling in order to simulate the ratio of glow peaks 5a/5 as a function of photon dose(22). A major postulate of the UNIM was that s is photon/electron energy dependent due to the more localised pattern of microscopic ionisation density as the electron energy decreases. However, the effect of overlapping/intersecting electron tracks with increasing dose was ignored and thus s was assumed to be independent of dose. This omission has been corrected in the current modelling and is the main subject of this paper.
(6) ro is the radius of the TC/LC complex; (7) g(rh.Ri) is a 2D solid angle factor between two neighbouring TC/LC pairs given approximately by Pr2h =4PR2i ; (8) SLC ¼ Pr2h is the cross section for capture of an electron by the LC; (9) Ri is the distance between neighbouring TC/LC pairs; (10) l is the mean free path of the electrons between the TC/LC pairs, is an increasing function of dose (due to the filling (de-activation) of the competitors with increasing dose) and is given by
MATHEMATICAL FORMULATION OF THE UNIM
(11) lo is the mean free path for charge carrier diffusion in the inter-track unirradiated region and is given by (NCCSCC)21; (12) SCC is the capture cross section of the unoccupied competing centre.
The TL signal intensity, F(D), is given by FðDÞ ¼ ksne ðDÞ þ ð1 ksÞne
rð 3 max X i¼1
r0
ð3Þ
gðRÞ ð2Þ
eðR=lðDÞÞ Pi ðnh ; R; DÞdR The first term, ksne, determines the contribution of the e–h populated TC to the total TL intensity and results in a linear/exponentially saturating contribution to the dose –response. The second term estimates the increased TL intensity at higher dose levels (supralinearity) due to CB mediated recombination in the presence of dose-dependent competitive processes. The decrease in f (D)max with decreasing photon energy was achieved in the previous UNIM simulations by increasing the value of ks leading to geminate recombination from 0.07 to 0.17. This increase in the e –h populated TC/LC is expected due to increased ionisation density of single electron tracks with decreasing photon energy. The parameters in Equation (2) are defined as follows: (1) k is the relative probability of geminate/localised recombination in the TC/LC complex giving rise to luminescence; (2) s is the fraction of TC/LC complexes that have captured an e –h pair following irradiation; (3) ne, nLC (nh) and nCC represent the density of occupied TCs, LCs and competitive centres (CCs), respectively. Their dose –response is described by a linear/exponentially increasing function, Ni (1 2 exp[2biD)], as demonstrated by optical absorption dose –response experimental measurements(23); (4) bTC, bLC and bCC are the dose-filling constants of the TCs, LCs and CCs, respectively; (5) NTC, NLC and NCC represent the density of available TCs, LCs and CCs, respectively;
The nearest neighbour probability distribution functions, Pi (nLC, Ri)dRi, are a crucial element in the calculation of F(D) since it is over these distances that the electron must migrate in the CB before recombination with a hole trapped in an LC; F(D) is calculated using the sum of the first, second and third nearest neighbour interactions. The derivation of Pi ¼ 1 – 3 has been described in previous publications(3, 6) and is given by the following: (1) First nearest neighbour PDF: P1 dR ¼ nh 4pR2 dR eð4=3ÞpR
3
nh
ð4Þ
(2) Second nearest neighbour PDF: P2 dR ¼
16 2 2 5 3 n p R dR eð4=3ÞpR nh 3 h
ð5Þ
(3) Third nearest neighbour PDF: P3 dR ¼
64 3 3 8 3 n p R dR eð4=3ÞpR nh 9 h
ð6Þ
Figure 5 illustrates the first and second nearest neighbour probability distribution function for various values of the dose. As expected, these shift to lower values of R with increasing dose. For the values of nh shown in Table 1 the probabilities are non-zero over ˚ to a few thoutypical distances from a few tens of A ˚ sand A. The TL signal intensity in the UNIM is thus given by F ðDÞ ¼ ksne ðDÞ þ ð1 ksÞne
3 X i¼1
Page 5 of 11
Ii
ð7Þ
Downloaded from http://rpd.oxfordjournals.org/ at University of Regina on August 22, 2014
lðDÞ ¼ l0 ebcc D
A. LAVON ET AL.
Table 1. UNIM parameters. Parameter
Value
ks
0.17 (20 kVp) 0.12 (50 kVp) 0.07 (.100 keV) 6` 1022 (m23) 1.95` 1022 (m23) 5` 1024 (Gy21) 0.5` 1024 (Gy21) 1.5` 1023 (Gy21) 4.5` 1023 (Gy21) 2` 1029 (m) 9.6` 10216 (m2) 4.75` 1028 (m)
Ne NLC bTC bCC bRD bLC ro SLC lo ¼ (NCC` SCC)21
where Ii is the contribution of the ith nearest neighbour interaction. At low dose, in the linear part of the dose– response, the second term describing the increased efficiency due to nearest neighbour interactions is vanishingly small due to the competitive processes, so that FðD Þ ksne ðD Þ
e-only in a TC/LC spatially coupled complex. The relative e–h to e-only concentrations (ne – h/ne) as a function of photon/electron energy and dose are then adopted in the recombination stage as starting parameters in order to simulate the dependence of the TL dose –response on ionisation density. The total population of TCs, NTC, giving rise to peak 5, is composed of a certain fraction of TCs, Ne – h, which are spatially correlated/coupled to the LCs. The remaining fraction of TCs are denoted by Ne. The fraction Ne – h /NTC may be influenced by various factors including methods of crystal growth, thermal treatments such as cooling rate following the 4008C pre-irradiation anneal(24) and others. The fraction, ne – h/ne, of the density of e –h to e-only populated complexes following irradiation is given by neh ¼ a ð1 ebeh D Þ þ b ne
ð10Þ
The combined densities must, of course, yield the total density of occupied peak 5-TCs as follows:
ð8Þ
The normalised TL dose–response, f(D), or relative TL efficiency is then given by
3 ½ne ðDÞ=D ð1 ksÞ X f ðDÞ ¼ Ii 1 þ ½ne ðD Þ=D ks i¼1
ð9Þ
REFORMULATION OF THE UNIM (THE M-UNIM) The LDL kinetic model(9) incorporates in the irradiation stage dose-dependent capture of an e –h and
ntot ¼ NTC ð1 ebTC D Þ ¼ neh þ ne
ð11Þ
The parameters a, b and be – h are postulated to depend on photon/electron energy. These equations are employed to yield values of ne – h/ne which increase slowly up to a threshold dose of 1– 10 Gy where supralinearity begins to appear in the dose–response of peak 5. Above this dose, the ratio increases rapidly and is governed by be – h. The parameter ‘b’ determines the ratio of ne – h/ne at the lowest dose levels; the dose-filling factor be – h is expected to decrease with decreasing photon energy due to the onset of track interaction at higher levels of dose for the lower photon energies. The parameter ‘a’ determines the
Page 6 of 11
Downloaded from http://rpd.oxfordjournals.org/ at University of Regina on August 22, 2014
Figure 5. First and second nearest neighbour probability distribution functions as a function of dose from 1 mGy to 10 000 Gy. Note the shift in the maximum probability to higher distances from the first to the second PDFs.
THE MODIFIED UNIFIED INTERACTION MODEL Table 2. Values of e– h filling parameters—LDL kinetic model.
ne – h/ne
Symbol
f(D)max ¼ 1.7, photon energy 8 keV
f (D)max ¼ 3.3, photon energy .100 keV
a b be – h(Gy21)
0.63 0.30 4` 1023
0.80 0.15 11` 1023
s¼
neh a ð1 ebeh D Þ þ b ¼ 1 þ a ð1 ebeh D Þ þ b ntot
Figures 6 and 7 show values of f(D) summed over the first to third nearest neighbour interactions at two photon energies of 100 keV and 20 kVp (8 keV) for the UNIM and M-UNIM using the parameters shown in Tables 1 and 2. The UNIM simulation has been previously fitted to experimental results. As can be seen, the values of f(D)max are significantly greater in the M-UNIM calculation due to the behaviour of ne – h/ne. This is demonstrated in Figures 8 and 9, which explicitly show the effects of ne – h/ne increasing with dose. In the following section, the influence of a, b and be – h on f(D) is demonstrated, which allowed the appropriate choice of parameter values to correct the overestimate of f(D)max.
ð12Þ
Therefore at low dose sj
D,0:1 b,,0:01
¼
b 1þb
ð13Þ
and at high dose sjDb.10 ¼
aþb 1þaþb
ð14Þ
f(D) is now given by f ðDÞ ¼
3 ½ne ðDÞ=D ksðDÞ ð1 ksðD ÞÞ X Ii þ ½ne ðD Þ=D ksðD Þ ksðD Þ i¼1
!
Figure 6. Comparison of UNIM and M-UNIM simulations for photon/electron energies .100 keV (solid line MUNIM; dashed line UNIM).
ð15Þ ksðDÞ ksðD Þ ¼
ða ð1 ebeh D Þ þ bÞ=ð1 þ a ð1 ebeh D Þ þ bÞ b=ð1 þ bÞ ð16Þ
with 1ksðDÞ ksðD Þ ¼
1kððað1ebeh D ÞþbÞ=ð1það1ebeh D ÞþbÞÞ kðb=ð1þbÞÞ ð17Þ
Figure 7. Comparison of UNIM and M-UNIM simulations for 8 keV (20 kVp) photons (solid line M-UNIM; dashed line UNIM).
Page 7 of 11
Downloaded from http://rpd.oxfordjournals.org/ at University of Regina on August 22, 2014
relative importance of the two terms. Table 2 gives the values of the parameters adopted by Eliyahu et al. (9) to simulate f (D) following photon irradiation of 100 keV ( f (D)max ¼ 3.3) and 8 keV ( f (D)max ¼ 1.7). Incorporation of ne – h/ne into the UNIM requires the calculation of s ¼ ne – h/ntot. It can be easily shown that
COMPARISON OF UNIM AND M-UNIM SIMULATIONS
A. LAVON ET AL.
Figure 10. The dependence of f(D) on the value of the parameter ‘a’ (circle, 0.1; square, 0.3; triangle, 0.5; square, 0.7; inverted triangle, 0.9).
Figure 11. The dependence of f(D) on the value of the parameter ‘b’ (circle, 0.005; plus, 0.01; triangle, 0.05; square, 0.1; inverted triangle, 0.15).
Figure 9. M-UNIM simulation of f(D) for 8 keV photons showing the contribution of ne – h/ne and the first, second and third nearest neighbour interactions (circle, total; plus, ne – h; triangle, first nearest neighbour; inverted triangle, second nearest neighbour; square, third nearest neighbour).
THE DEPENDENCE OF f(D) ON THE PARAMETERS OF ne – h/ne The dependence of f (D) on a, b and be – h is shown below in Figures 10–12. The basis set of parameters are the ones listed in Tables 1 and 2. Variation in the value of ‘a’ leads to a moderate decrease of f (D)max from 3.3 to 2.8, but Dmax is unaffected. Variations in ‘b’, however, lead to large changes in f (D)max, reaching values as high as 75. These very high values reaching even 200 have been measured
Figure 12. The dependence of f(D) on the value of the parameter ‘be – h’ ( plus, 0.4 Gy21; inverted triangle, 0.04 Gy21; circle, 0.004 Gy21; square, 0.0004 Gy21; triangle, 0.00004 Gy21).
Page 8 of 11
Downloaded from http://rpd.oxfordjournals.org/ at University of Regina on August 22, 2014
Figure 8. M-UNIM simulation of f (D) for 100 keV photons showing the contribution of ne – h/ne and the first, second and third nearest neighbour interactions. The contribution of ne – h/ne is dominant; the contribution of the third nearest neighbour interaction is negligible (circle, total; plus, ne – h; triangle, first nearest neighbour; inverted triangle, second nearest neighbour; square, third nearest neighbour).
THE MODIFIED UNIFIED INTERACTION MODEL
experimentally for the high temperature peaks in LiF:Mg,Ti (TLD-100)(25, 26). Very low values of ‘b’ reflect very low values of ne – h/ne at low dose, which implies a TL signal based almost completely on delocalised recombination. This leads to very high values of f(D)max due to the signal dependence on the product of ne and nh, i.e. to n 2. In this case, again, Dmax is unaffected. Figure 12 shows the behaviour of f(D) when be – h is varied. The very high values of be – h give rise to very broad (double-peaked) values of f (D) which have never been observed experimentally. Figure 13 emphasises the unusual behaviour of f (D) with high values of be – h showing explicitly the contribution of the first term to the supralinearity for low and high values of be – h. Of the other UNIM parameters, the only ones that significantly influence the shape of f (D) (Dmax and the dose range for which f (D) . 1) are bTC and b(3) LC. SIMULATIONS OF f(D) AT PHOTON ENERGIES .100 AND AT 8 keV The variable parameters in the M-UNIM are: k, a, be – h, b, bTC, bLC, bCC, bRD, lO, SLC, NLC and ro, 12 parameters in all. An additional parameter, NTC, cancels out since f(D) is normalised. Only a, be – h and b are expected to depend on photon energy; bRD is not explicitly shown in Equation (2) and is introduced to simulate radiation-induced damage to the LCs(26), resulting in reduced available concentration of LCs, NLC*, via NLC ¼ NLC ebRD D
ð18Þ
Radiation damage has been experimentally measured in the TL response to TLD-100. A dose level of
Figure 14. M-UNIM simulation of f (D) for 100 keV photons.
Figure 15. M-UNIM simulation of f (D) for 8 keV photons.
1000 Gy results in a decrease of the TL intensity of composite peak 5 of 40 %(6); at 10 000 Gy the TL intensity decreases by a factor 2 from its maximum value(27). Unlike the dose-filling parameters bTC, bCC, which have been demonstrated to be independent of photon energy(28), bRD might also be somewhat dependent on photon energy; however, there are no experimental data supporting this possibility. The choice of LCs in the radiation damage mechanism is dictated by the results of optical absorption measurements which show no decrease in the optical density of the 4 eV (TC) or 5.45 eV (CC) bands at high levels of dose, reaching 30 000 Gy(29). There are certain constraints on the values of the M-UNIM parameters. The a, be – h, b parameters have been adopted from LDL kinetic modelling; however, there is no compelling reason to expect that these values are appropriate to the M-UNIM since the two models simulate supralinearity in two very different theoretical frameworks. Some of the other
Page 9 of 11
Downloaded from http://rpd.oxfordjournals.org/ at University of Regina on August 22, 2014
Figure 13. The dependence of f(D) for 100 keV photons showing the contribution of ne – h/ne (with be – h ¼ 4` 1025 Gy21) and the first (triangle), second (left-pointing triangle), third (square) nearest neighbourinteractions and total (circle) of all contributions.
A. LAVON ET AL. Table 3. M-UNIM Values of e– h filling parameters.
ne – h/ne
Symbol
f(D)max ¼ 1.7, photon energy 8 keV
f (D)max ¼ 3.3, photon energy .100 keV
a b be – h(Gy21) k
0.15 0.80 1` 1024 0.42
0.50 0.30 40` 1024 0.42
CONCLUSIONS In this work, the UNIM has been modified (MUNIM) to include the dependence of ne – h/ne, on dose. The increase in the e –h population of the TC/ LC complex with increase in ionisation density (dose) is expected based on statistical and nanodosimetric considerations of electron/photon track structure. The functional form of the dose behaviour of ne – h/ ne has been adopted from recent LDL modelling which simulated the dependence of f (D) on photon energy. The M-UNIM is used to simulate the dependence
REFERENCES 1. Horowitz, Y. S., Mahajna, S. and Rosenkrantz, M. Unified approach to gamma and heavy charged particle supralinearity: the track defect interaction model. Radiat. Prot. Dosim. 65, 7– 12 (1996). 2. Horowitz, Y. S., Mahajna, S., Oster, L., Weizman, Y., Satinger, D. and Yossian, D. The Unified Interaction Model applied to the gamma induced supralinearity and sensitisation of peaks 4 and 5 in LiF:Mg,Ti (TLD-100). Radiat. Prot. Dosim. 78, 169 –193 (1998). 3. Horowitz, Y. S. Theory of thermoluminescence gamma dose response: the unified interaction model. Nucl. Instrum. Methods B184, 68–84 (2001). 4. Horowitz, Y. S. The theoretical and microdosimetric basis of thermoluminescence and applications to dosimetry. Phys. Med. Biol. 26, 765–824 (1981). 5. Horowitz, Y. S. Recent models for TL supralinearity. Radiat. Prot. Dosim. 6, 17–20 (1984). 6. Horowitz, Y. S. Ch. 2. A unified and comprehensive theory of the TL dose response of thermoluminescent systems applied to LiF:Mg,Ti. In: Microdosimetric Response of Physical and Biological Systems to Lowand High-LET Radiations: Theory and Applications to Dosimetry. Horowitz,, Y. S. Ed. Elsevier (2006). 7. Chen, R. and Pagonis, V. Thermally and Optically Stimulated Luminescence: A Simulation Approach. Wiley (2011). 8. McKeever, S. W. S. and Horowitz, Y. S. Charge trapping mechanisms and microdosimetric processes in lithium fluoride. Radiat. Phys. Chem. 36, 35– 46 (1990). 9. Eliyahu, I, Horowitz, Y. S., Oster,, L. and Mardor, I. A kinetic model incorporating both localized and delocalized recombination: application to the dependence of the TL dose response on photon energy. J. Lumin. 145, 600– 607 (2014). 10. Horowitz, Y. S., Avila, O. and Rodriguez-Villafuerte, M. Theory of heavy charged particle response (efficiency and supralinearity) in TL materials. Nucl. Instrum. Methods B184, 85– 112 (2001).
Page 10 of 11
Downloaded from http://rpd.oxfordjournals.org/ at University of Regina on August 22, 2014
values of the M-UNIM parameters are severely constrained. For example, bTC and bCC have recently been re-measured to high levels of dose by the BGU group(29), and lo and bLC have been estimated from low-energy alpha particle fluence response data(11); NLC can be estimated approximately from the concentration of Ti in the LiF:Mg,Ti samples(30). The M-UNIM fits to the experimental data for 500 keV electrons and 8 keV photons (8 keV experimental data from Gamboa-deBuen et al.(31)) are shown in Figures 14 and 15 using the values of the parameters given in Table 3. A good fit to the experimental data could not be achieved without changing the e– h dose-filling rate parameters a, b and be – h previously employed in the LDL kinetic modelling. The behaviour of these parameters with photon/ electron energy deserves special comment. The value of s at low dose levels is given by the value of b/ (1þb). This would be expected to be higher for the lower photon/electron energies due to the higher ionisation density at lower energy and, as given in Table 3, these values are 0.44 and 0.23 for the 8 and 500 keV energies, respectively. Similarly, one expects that the value of be – h would be lower for the low energy photons due to the smaller extension of the low energy tracks requiring higher levels of dose to result in track overlap. This is also reflected in Table 3 with values of be – h decreasing from 0.004 Gy21 (500 keV electrons) to 0.0001 Gy21 (8 keV photons). At very high dose, the value of s approaches (a þ b)/(1 þ a þ b) and might be expected to be higher for the 8 keV photons—again the values are 0.7 and 0.45 for 8 and 500 keV, respectively.
of f(D) on photon/electron energies above 100 and at 8 keV. The simulations in both the UNIM and the M-UNIM employed the identical set of parameters at both energies except for the e–h TC/LC dosepopulation characteristics. The M-UNIM and the LDL kinetic model are currently the only existing models capable of predicting the linear/supralinear behaviour of f(D) and its dependence on photon energy. However, the values of the dose-filling characteristics a, b and be – h used in the two models are different. Obviously, further efforts are required in order to discover a single set of parameters that will yield fits to the experimentally measured f(D) in both model simulations. This is an imposing challenge considering their very different theoretical frameworks and may require certain refinements of both models. However, success would constitute the first-time-ever estimation of track structure ionisation-density-dependent effects in TL solid-state systems and perhaps in ionisation-densitysensitive solid-state systems in general.
THE MODIFIED UNIFIED INTERACTION MODEL 21. Sibony, D., Horowitz, Y. S., Oster, L., Wojcik, A. and Sollazzo, A. Combined measurement of dose and a/g radiation-field-components using the shape of composite peak 5 in the glow curve of LiF: Mg,Ti. Available on http:// dx.doi.org/10.1016/j.radmeas.2014.01.006. 22. Eliyahu, I., Horowitz, Y. S., Oster, L., Druzhyna, S. and Mardor, I. Nanodosimetric model incorporating localised and delocalised recombination: application to the prediction of the electron dose response of the peak 5a/5 ratio in the glow curve of LiF:Mg,Ti (TLD-100). Available on http://dx.doi.org/10.1016/j.radmeas.2014.01.015. 23. Issa, N., Oster, L. and Horowitz, Y. S. Optical absorption and sensitisation dose response in LiF:Mg,Ti:application to the unified interaction model predictions of thermoluminescence dose response. Radiat. Prot. Dosim. 100, 107–110 (2002). 24. Fuks, E., Horowitz, Y. S. and Oster, L. Investigation of the properties of composite glow peak 5 in slow-cooled TLD-100. Radiat. Meas. 43, 249– 253 (2008). 25. Massillon, J. L. G., Gamboa-Debuen, I. and Brandan, M. E. Onset of supralinear response in TLD-100 exposed to Co-60 gamma rays. J. Phys. D. Appl. Phys. 39, 262–268 (2006). 26. Livingstone, J., Horowitz, Y. S., Oster, L., Datz, H., Lerch, M., Rosenfeld,, A. and Horowitz, A. Experimental investigation of the 100 keV x-ray dose response of the high temperature thermoluminescence in LiF:Mg,Ti (TLD100):theoretical interpretation using the Unified Interaction Model. Radiat. Prot. Dosim. 138, 320–333 (2010). 27. Montano-Garcia, C. and Gamboa-deBuen, I. Measurements of the optical density and the thermoluminescent response of LiF:Mg,Ti exposed to high doses of 60Co gamma rays. Radiat. Prot. Dosim. 119, 230–232 (2006). 28. Biderman, S., Eliyahu, I., Horowitz, Y. S. and Oster, L. Dose response of F center optical absorption in LiF:Mg,Ti (TLD-100). Available on http://dx.doi.org/ 10.1016/j.radmeas.2014.02.001. 29. Nail, I. et al. Search for ionisation density effects in the radiation absorption stage in LiF:Mg,Ti”. Radiat. Prot. Dosim. 119, 180– 183 (2006). 30. Horowitz, Y. S. The average distance between Mg-based trapping structures in LiF:Mg,Ti and Li:Mg,Cu,P and the relevance to microdosimetry. Radiat. Prot. Dosim. 82, 51–54 (1999). 31. Gamboa-deBuen, I., Buenfil, A. E., Ruiz, C. G., Rodriguez-Villafuerte, M., Flores, A. and Brandan, M. E. Thermoluminescent response and relative efficiency of TLD-100 exposed to low-energy X rays. Phys. Med. Biol. 43, 2073–2083 (1998).
Downloaded from http://rpd.oxfordjournals.org/ at University of Regina on August 22, 2014
11. Rodriguez-Villafuerte, M., Buenfil, A. E., GamboadeBuen, I., Ruiz-Trejo, C., Brandan, M. E., Yossian, D., Satinger, D. and Horowitz, Y. S. Study of the TL response of LiF:Mg,Ti to 3 and 7.5 MeV Helium ions: measurements and interpretation in terms of the track interaction model. Nucl. Instrum. Methods B160, 377 –386 (2000). 12. Biderman, S., Horowitz, Y. S. and Oster, L. Investigation of the emission spectra of LiF:Mg,Ti (TLD-100) during thermoluminescence. Radiat. Prot. Dosim. 100, 369– 372 (2002). 13. Townsend, P. D., Ahmed, K., Chandler, P. J., McKeever, S. W. S. and Witlow, H. J. Measurements of the emission spectra of LiF during thermoluminescence. Radiat. Eff. 72, 245–257 (1983). 14. Weizman, Y., Horowitz, Y. S., Oster, L., Youssian, D. and Horowitz, A. Mixed-order kinetic analysis of the glow curve characteristics of single crystal LiF:Mg,Ti as a function of Ti concentration. Radiat. Prot. Dosim. 84, 47– 50 (1999). 15. Carrillo, R. E., Pearson, D. W., DeLuca, P. M. Jr and MacKay, J. F. Thermoluiminescent response of lithium fluoride (TLD-100) to monoenergetic photon beams of 600, 730, 900 and 1500 eV. Radiat. Prot. Dosim. 47, 519 –524 (1993). 16. Weizman, Y., Horowitz, Y. S. and Oster, L. Investigation of the composite structure of peak 5 in the thermoluminescent glow curve of LiF:Mg,Ti (TLD-100) using optical bleaching. J. Phys. D. Appl. Phys. 32, 2118–2127 (1999). 17. Horowitz, Y. S., Oster, L., Satinger, S., Biderman, S. and Einav, Y. The composite structure of peak 5 in the glow curve of LiF:Mg,Ti (TLD-100): confirmation of peak 5a arising from a locally trapped electron-hole configuration. Radiat. Prot. Dosim. 100, 123–126 (2002). 18. Horowitz, Y. S. and Moscovitch, M. Highlights and pitfalls of 20 years of application of computerised glow curve analysis to thermoluminescence research and dosimetry. Radiat. Prot. Dosim. 153, 1– 22 (2013). 19. Horowitz, Y. S., Fuks, E., Oster, L., Podpalov, L., Belaish, Y. and Ben Shachar, B. Advanced multistage deconvolution applied to composite glow peak 5 in LiF:Mg,Ti (TLD-100). Radiat. Prot. Dosim. 126, 322 –325 (2007). 20. Sibony, D., Horowitz, Y. S. and Oster, L. The effect of sample/planchet geometry and temperature resolution on the reproducibility of glow curve shapes and precision of dose measurement in LiF-TLD-100 thermoluminescent dosimetry. Available on http://dx.doi.org/10.1016/j. radmeas.2014.01.005.
