Kinetic simulation of the optical absorption dose response of LiF:Mg,Ti (TLD-100) incorporating spatially correlated electron and hole trapping centers

Nuclear Instruments and Methods in Physics Research B 407 (2017) 282–290

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Kinetic simulation of the optical absorption dose response of LiF:Mg,Ti (TLD-100) incorporating spatially correlated electron and hole trapping centers I. Eliyahu a,⇑, Y.S. Horowitz b, L. Oster c, S. Druzhyna d, S. Biderman e, D. Ginzburg f, G. Reshes c a

Soreq Nuclear Research Center, Yavne, Israel Physics Department, Ben Gurion University of the Negev, Beersheva, Israel Physics Unit, Sami Shamoon College of Engineering, Beersheva, Israel d Nuclear Engineering Unit, Ben Gurion University of the Negev, Beersheva, Israel e Nuclear Research Center Negev, Beersheva, Israel f Department of Biotechnology, Ben Gurion University of the Negev, Beersheva, Israel b c

a r t i c l e

i n f o

Article history: Received 14 May 2017 Received in revised form 24 June 2017 Accepted 21 July 2017

Keywords: Kinetic simulations Conduction band/valence band Optical absorption dose response

a b s t r a c t A conduction band/valence band (CB/VB) model incorporating a spatially-correlated trapping center/luminescent center (TC/LC) complex is described which is associated with composite glow peak 5 in the thermoluminescence (TL) glow curve of LiF:Mg,Ti (TLD-100). The evidence for such a complex observed in the TL characteristics of TLD-100 is reviewed. Additional experimental evidence from the optical absorption (OA) energy spectrum and dose response of the individual bands has recently been reported in which the 4.0 eV band is interpreted to arise from two sub-bands at 3.84 eV and 4.3 eV. The 3.84 eV band demonstrates the commonly observed linear/exponentially saturating dose response and is interpreted to arise from the e-only TC which is not spatially correlated with an LC. The 4.3 eV band is interpreted to arise from two sub-bands which are unresolved in the OA spectrum but which show two very different dose-response characteristics. The first arising from the TC/LC which has captured only an electron (resulting again in a linear/exponentially saturating dose response) and the other from an electronhole captured configuration which continues to increase in concentration up to the highest levels of dose investigated. The CB/VB model is used to simulate the kinetics of charge carrier transport and capture leading to the dose response of all the observed OA bands. Ó 2017 Elsevier B.V. All rights reserved.

1. Introduction Unraveling the details of the complex mechanisms controlling the production of TL in LiF:Mg,Ti and other materials continues to be a subject of substantial interest [1,2]. LiF:Mg,Ti is the most widely used of the TL dosimetric materials so the development of a robust theory of its response to ionizing radiation is a significant undertaking. Recently the importance of TC/LC spatial correlation leading to geminate electron-hole (e-h) recombination has been recognized due to the inability of the delocalized kinetic models to simulate crucially important characteristics of TL mechanisms. For example, the dependence of the supralinearity in the TL dose response on ionization density [3]. The effects of ionization density (ID) on TL characteristics are observed in many materials so that

⇑ Corresponding author. E-mail address: [email protected] (I. Eliyahu). http://dx.doi.org/10.1016/j.nimb.2017.07.018 0168-583X/Ó 2017 Elsevier B.V. All rights reserved.

the incorporation of TC/LC spatial correlation into kinetic modeling and comparison with experimental data is an important step forward in the theoretical understanding of TL mechanisms. The first step in the production of TL is determined by the population of the various TCs, LCs and competitive centers (CCs) during irradiation and many of the relevant TC and CC concentrations can be measured via optical absorption (OA). The kinetic simulation of the OA dose response of the various OA bands is the subject of this paper. 1.1. Spatial correlation of traps and centers in the LiF:Mg,Ti system The presence of spatially-correlated TC/LC pairs in LiF:Mg,Ti was first pointed out following measurements of the emission spectra of LiF:Mg,Ti during thermoluminescence [4]. Further evidence came from the observation and interpretation of the dependence of the TL properties of composite glow peak 5 on ID shown in Fig. 1 [5]. The shift in the maximum intensity of the main peak to

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lower temperatures by 4–5 °C following alpha particle irradiation coupled with the absence of the shoulder on the low temperature side of the main peak was demonstrated by Tm
Tstop analysis [6] to be due to the enhanced presence of a low temperature satellite (peak 5a) as shown in the deconvoluted glow curves. Theoretical analyses of spatially correlated systems soon followed [7–9]. Glow peak 5a in LiF:Mg,Ti (TLD-100) is interpreted as arising from geminate recombination of an electron-hole populated TC/ LC structure. Glow peaks 4 and 5 were further correlated with the hole-only and electron-only populated structures respectively. This description of composite glow peak 5 and glow peak 4 was used to explain the following TL characteristics: (i) The linear/supralinear dose response of composite peak 5 and its dependence on ID) (photon/electron energy) as shown in Fig. 2 [10]. At low dose levels geminate recombination in the e-h configuration (tunneling without involvement of the conduction band) leads to the TL linear dose response. At higher dose-levels electrons released from both e-occupied configurations can migrate to more distant holepopulated LCs due to the increased population (deactivation) of the competitive centers and decreased distances between the TCs and LCs [11]. The dependence of f (D) on photon energy is due to the increased population of the e-h configuration at higher ionization density which leads to lower values of the maximum supralinearity, f (D)max. (ii) the greatly increased relative intensity of glow peak 5a following high ionization density (heavy charged particle) irradiation due to the cumulative statistics required for the trapping of both charge carriers, [12]. (iii) The increased intensity of glow peak 4 following postirradiation annealing at 170 °C followed by photon bleaching at 4 eV (Fig. 3) which is explained as due to the direct removal of the electron from the e-h configuration (Fig. 4) [13]. The increased intensity of peak 4 is not observed following a post-irradiation anneal at 184 °C which totally depopulates the e-h configuration responsible for peak 5a.

Fig. 1. Deconvoluted glow curve following alpha particle irradiation (3  109 cm
2) (top) and beta irradiation at 10 Gy (bottom) in slow-cooled material. The ratio of glow peak 5a to 5 is approximately 0.8 and 0.3 respectively. Slow-cooled material enhances the ratio of peaks 5a/5.

Implementation of the ratio of peak 5a/5 to separate the components in mixed radiation fields has been hampered by the uncertainties in the deconvolution procedure leading to difficulties in reproducibility. Recently, however, significant improvement was reported in mixed a-c radiation fields using improved temperature resolution during glow curve readout [14].

Fig. 2. f(D) as a function of dose for various values of f(D)max corresponding to photon energies decreasing from approximately 100 keV (and above) to approximately 2 keV. Reproduced from Eliyahu et al. [10].

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2. The optical absorption energy spectrum and dose response: New interpretation 2.1. Experimental details

Fig. 3. Glow curve illustrating the effect of optical bleaching at 4 eV (310 nm) following a post-irradiation anneal of 170 °C/6 s. Note the increased absolute intensity of peak 4 following the bleach (reproduced from Horowitz et al. [12]).

The details of the OA measurements have been previously published [15] but are repeated herein for the convenience of the reader. The samples of TLD-100 (ThermoFisher Scientific) consisted of LiF doped with approximately 170 mol ppm Mg and 10 mol ppm Ti of geometry 3  3  0.9 mm3. Pre-irradiation annealing was applied before all measurements and consisted of a 400 °C anneal for 1 h in a dry atmosphere followed by natural cooling (average cooling rate approximately 75 °C min
1 to room temperature. The irradiations were carried out with a 90Sr/90Y beta source at room temperature and at a dose rate of approximately 0.13 Gy min
1 and covered the dose-range from 100 Gy to 2500 Gy. For higher dose levels, up to 105 Gy, a 60Co gamma source at a dose rate of about 1 Gy min
1, was used. The OA spectra were measured with a Genesis-5 UV/Visible wavelength spectrophotometer (Milton Roy Inc.). The OA spectra were deconvoluted with Gaussian peak shapes using a commercial ‘‘Peak-Fit” non-linear curve fitting program from Jandel Scientific. The Gaussian bands are characterized by the two parameters of band energy and width and these were kept constant at all levels of dose except for slight changes to obtain optimum fits to the spectra. Starting values of the parameters were adapted from the literature except for the 3.8 eV and 4.3 eV bands since these replaced the 4.0 eV band. The average parameters of the OA bands used in the deconvolution analysis are shown in Table 1. Two types of background subtraction were employed: (i) subtraction of the absorption spectrum of unirradiated material carried out on the same sample and (ii) estimation of the spectrum background in the irradiated material (following the subtraction of the OA spectrum of unirradiated material) which was estimated by linear interpolation from the

Fig. 4. Schematic representation of the spatially correlated TC/LC model showing the four possible configurations following irradiation. Each large oval represents a spatially correlated TC/LC complex (Reproduced from Horowitz [6]).

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Table 1 Parameters of the OA bands. Energy (eV)

FWHM (eV)

2.78 3.24 3.84 4.3 4.77 5.08 5.45 6.2

0.164 0.353 0.588 0.612 0.447 0.635 0.753 0.942

low and high energy regions of the spectrum. The latter estimation of the background gives rise to the main source of uncertainty and was the dominant contributor to the experimental uncertainties of 15% (1 SD). 2.2. OA results and interpretation A deconvoluted OA spectrum of LiF:Mg,Ti (TLD-100) following irradiation to a dose-level of 2500 Gy is shown in Fig. 5 [15]. The dominant OA band at 5.08 eV is due to F center excitation. The breadth of the F band is attributed to interaction with vibrational modes of the lattice and even at low temperature the F band retains appreciable half-widths [16]. In the range of energies studied herein it is usually assumed that all the observed bands are approximately Gaussian in shape and are due to electron occupied TCs. Gaussian broadening of the purely electronic absorption cross section due to phonon-impurity coupling has been discussed by Noras [17]. The dose response of all the OA bands has been previously reported to be linear/exponentially saturating at levels of dose up to 4000 Gy with no dependence of the dose-filling constants on photon energy [18] (see Fig. 6) The 4.0 eV band has been invariably associated to a Mg2+-Livac trimer (the trapping center –TC) coupled to Ti-OH (the luminescent center-LC) and associated with composite glow peak 5 in the TL glow curve. The only discord in this interpretation of the OA spectrum was the presence of a weak band at 4.45 eV with irregular properties which defied correlation with known defects and was inserted somewhat arbitrarily between the 4.0 eV and 4.77 eV bands by various authors in order to improve the goodness of fit in the deconvolution analysis [19–21]. An alternative interpretation of the OA energy spectrum has recently been proposed [15] based on the change in shape of the

Fig. 5. Deconvoluted OA energy spectrum of gamma irradiated LiF:Mg,Ti (TLD-100) illustrating a fit to the experimental data using OA bands at 3.84 eV and 4.3 eV (Reproduced from S. Druzhyna et al. [15]).

Fig. 6. Dose response of the 3.84 eV and 4.3 eV OA bands in TLD-100. (Reproduced from S. Druzhyna et al. [15]).

OA spectrum in the energy region between 4 and 4.5 eV as a function of dose. In the new understanding described herein, the 4.45 eV band is no longer necessary and the 4 eV band is composed of two sub-bands at 3.84 eV and 4.3 eV. The 3.84 eV band exhibits the usual linear/exponentially saturating dose response, however, the 4.3 eV band continues to increase at high levels of dose and does not reach saturation even at 105 Gy. The presence of two bands of different dose response explains the change in shape of the OA energy spectrum as a function of dose. The estimate of the centroid energies of the new bands is quite robust. With the values of 3.84 eV and 4.3 eV, a figure-of-merit (FOM) of 1–2% at all levels of dose is achievable with variations in the centroid energy of no more than 0.03 eV in either direction. Greater shifts in the centroid energies produce visually observable differences between the observed and fitted spectrum and increasing FOMs. Because of the cumulative statistics requiring capture of two charge carriers it follows that the high dose behavior of the 4.3 eV band may be associated with the e-h configuration in the TC/LC pair and correlated with glow peak 5a and the 3.84 eV with the e-only configuration associated with peak 5. The complex behavior of the 4.3 eV band dose response suggests that it may also be composed of two unresolved sub-bands and leads to the following possible interpretation of the LiF:Mg,Ti system involving composite glow peak 5 and peak 4. It is assumed that not all of the TCs and LCs are spatially correlated and that a certain fraction is uniformly distributed throughout the sample. This assumption is supported by the fact that the relative intensity of glow peak 5a to glow peak 5 is affected by the cooling rate following the 400 °C pre-irradiation anneal. Slower cooling rates allow enhanced formation of the TC/LC complex. The TCs which are not spatially correlated with an LC belong to the 3.84 eV band whose dose response is linear/exponentially saturating as is commonly observed for a single charge carrier trapping center. The linear low-dose response of the 4.3 eV band belongs to the spatially correlated TCs which have captured an electron-only. The component of the 4.3 eV band which increases with dose even beyond 104 Gy belongs to the e-h configuration. The continued increase at high dose arises naturally from the requirement of capture of two charge carriers. In the following a conduction band/valence band model is described which is used to simulate the dose response of all the experimentally observed OA bands and, as well, this interpretation of the dose response of the 3.84 eV and 4.3 eV bands.

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3. The valence band/ conduction band model The model used for the simulations is shown below in Fig. 7 and immediately below the differential equations (2)–(18) describing the traffic of the charge carriers. These were solved numerically using the second-order, modified Rosenbrock formula (Matlab Code 23 s). The excitation dose, D, is given by:

D ¼ ½W  X  t=m

ð1Þ

where t is the total length of excitation time and X is the production rate of the electron and holes. W is the average energy required to produce an electron-hole pair by low ionizing density radiation and was taken as 37 eV in LiF. Further details are available elsewhere [1,2,10]. The relaxation period following the excitation is simulated by setting X to zero and using the final values of the parameters at the end of excitation as the initial values for relaxation. In this model, the irradiation by ionizing radiation leads to the release of electrons (e) and holes (h) and a certain fraction of the charge carriers reach the conduction band (CB) and valence band (VB). These migrate in the CB and VB and are eventually captured by electron trapping centers (TCs) and hole recombination centers (RCs). The RCs can be further classified into luminescent centers (LCs) and non-luminescent centers (NLCs). The recombination coefficients in the kinetic simulations describe the probability of electron and hole capture into the trapping states respectively. The possible range of the values of the recombination coefficients extends over several orders of magnitude but can be constrained according to general guidelines as discussed in the following. 3.1. The electron trapping TCs The relative concentration as a function of dose, n(D), of many of the electron occupied states can be monitored by OA measurements. If the oscillator strength, f, of the optically excited transition is known the absolute concentration can be calculated. There are six trapping centers (TC1-TC6) corresponding to the bands observed in the OA energy spectrum at 3.24 eV, 3.84 eV, 4.3 eV, 4.77 eV, 5.08 eV and 5.45 eV. Eq. (7) describes the rate of change of creation or destruction of the e-h configuration. The first term represents destruction via transfer of an electron to the e-h populated TC,

annihilation of the hole, leaving an e-only configuration. The second term represents destruction via transfer of a hole to the e-h populated TC, annihilation of the electron, leaving a h-only configuration. The third term represents the creation of an e-h configuration via transfer of an electron to a hole-only configuration, ditto for the fourth term. Eq. (18) expresses the connection between the total number of available centers of the 3.8 eV and 4.3 eV bands and the occupied concentrations at all levels of dose: for example, before irradiation, Noo = noo. Similar considerations apply to the other equations describing the traffic of the charge carriers. The reader should note that the F band at 5.08 eV consists of two entities (TC5, TC50). Thus, Nvo, nF(vo) refers to vacancies present in the sample before irradiation, whereas Nvc, nF(vc) to vacancies created by the irradiation. In this model [1] it was demonstrated that the linear/exponentially saturating dose response of the F band (a surprising result for a center which is being created by irradiation) is due to the interplay of the dose dependent creation of vacancies and those initially present in the sample. 3.2. The hole trapping LCs A cation vacancy in the alkali halides is called a V center and is deeper in energy than the F center. The V3 center in LiF:Mg,Ti can be considered the analogue of the F2 two-electron center and consists of two holes trapped at three neighbouring halide ions. Less information is available on the hole trapping centers since these are not observed in the OA spectrum with the single exception of the V3 center which was observed at 11 eV by Mayhugh et al. [22] using a vacuum UV spectrophotometer. The V3 center following capture of an electron reverts to an unstable Vk center (a hole trapped by a pair of negative halide ions) which releases a hole into the valence band. The Vk center thus serves as an additional source of holes during irradiation. V3-Vk transformation was employed [15] to simulate the effects of photon bleaching on the OA spectrum of LiF:Mg,Ti (TLD-100). Finally, unlike the other recombination coefficients, Bm3 represents the value for simultaneous capture of two holes. This stipulation is necessary due to the instability of the one-hole Vk center at room temperature which requires that the V3 center be created by the near- simultaneous capture of two holes.

Fig. 7. Conduction band/valence band model used in the kinetic simulations. See the text for a detailed explanation.

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3.3. The differential equations describing charge carrier capture

dm1 ¼
Am1  m1  nc þ Bm1 ðM 1
m1 Þ  nv dt

ð2Þ

dm2 ¼
Am2  m2  nc þ Bm2 ðM 2
m2 Þ  nv dt

ð3Þ

dm3 ¼
Am3  m3  nc þ Bm3 ðM 3
m3 Þ  nv dt

ð4Þ

dn1 ¼ An1  ðN1
n1 Þ  nc
Bn1  n1  nv dt

ð5Þ

dn2 ¼ An2  ðN2
n2 Þ  nc
Bn2  n2  nv dt

ð6Þ

dneh ¼
Aeh  neh  nc
Beh  neh  nv þ ð1
kÞ  Ah  nh  nc dt þ ð1
kÞ  Be  ne  nv dne ¼
Be  ne  nv þ A00  ðN
ne
neh
nh Þ  nc þ Aeh  neh  nc dt

ð7Þ ð8Þ

ð10Þ

dn4 ¼ An4  ðN4
n4 Þ  nc
Bn4  n4  nv dt

ð11Þ

dnFðv cÞ ¼ An5  ðNv c
nFðv cÞ Þ  nc
Bn5  n5v c  nv dt

ð12Þ

dn6 ¼ An6  ðN6
n6 Þ  nc
Bn6  n6  nv 6 dt dn7 ¼ An7  ðN7
n7 Þ  nc
Bn7  n7  nv dt

ð13Þ

ð15Þ



2

ðn2 þ 2Þ

Description

Symbol

Value [m3 s
1]

LC – Luminescent hole center

Am1

5  10
24

Bm1

7  10
22

Am2

1  10
25

Bm2

1  10
24

Am3

2  10
24

Bm3

6  10
24

An1

2  10
23

Bn1

4  10
23

An2

2  10
22

Bn2

1  10
25

Ae-h

5  10
25

Be-h

6  10
22 –

V3 – Two hole center

Trapping center e-h (4.3 eV)

ð16Þ

ð17Þ ð18Þ

Trapping center e (4.3 eV)

– Be

Trapping center h

Ah

Trapping center oo

– A00

TC4 – Trapping center 4.77 eV TC5 – Trapping center 5.08 eV TC6 – Competitive center 5.45 eV

3.4. The trapping coefficients Current understanding of the values of these parameters considers them to be determined by the product of the cross-section for capture and the average velocity of the thermalized charge

ð19Þ

Table 2 Values of the Trapping Coefficients.

TC2 – Trapping center 3.8 eV


neh
nh Þ þ An4  ðN4
n4 Þ þ An5o  ðNv o
nFðv oÞ Þ

 amax  W

f is the oscillator strength of the transition, n is the index of refraction of the host crystal at the center of the absorption band, amax is the maximum absorption constant and W is the full width of the OA band at half maximum. Smakula [25] treated the F center as a classical damped oscillator embedded in the dielectric medium of the host crystal and acted on by the local field. It has been pointed out [26] that Eq. (19) should be regarded as rather approximate

TC1 – Trapping center 3.24 eV

dnc ¼ X
nc  ½Am1  m1 þ Am2  m2 þ Am3  m3 þ An1  ðN1 dt
n1 Þ þ An2  ðN2
n2 ÞAeh  neh þ Ah  nh þ A00  ðN
ne

NTC5 ¼ noo þ neh þ ne þ nh

N  f ¼ 0:87  10

!

n

NLC – Non luminescent center

þ Beh  neh ::: þ Be  ne þ B00  ðN
ne
neh
nh Þ þ Bn4  n4

þ An5  ðN5
n5 Þ þ An6  ðN6
n6 Þ þ An7  ðN7
n7 Þ

The values of the recombination coefficients used in the simulations are shown in Table 2 below. Center Concentrations The values of N (The TC and LC available concentrations) are estimated from the experimentally measured optical density (OD) at saturation of the dose response. The OD is a measure of the product N and f as shown below [25].

ð14Þ

dnv ¼ X þ Am3  m3  nc
nv  ½Bm1  ðM 1
m1 Þ dt þ Bm2 ðM 2
m2 Þ þ Bm3 ðM3
m3 Þ þ Bn1  n1 þ Bn2  n2 þ Bn5  n5v c þ Bn5  n5v 0 þ Bn6  n6 þ Bn7  n7 

(iii) Beh < Be and (iv) Be > Boo

17

ð9Þ

dnFð0Þ ¼ An5o  ðNv o
nFðv oÞ Þ  nc
Bn50  n5v 0  nv dt

(i) Aeh < Ah based on the expectation that e-h configuration has less overall positivity than hAonly configuration (ii) Ah > Aoo based on the expectation that the h-only configuration has greater positivity than the empty configuration. Similarly

dnh ¼
Ah  nh  nc þ B00  ðN
ne
neh
nh Þ  nv þ Beh  neh  nv dt dn00 ¼
A00  ðN
ne
neh
nh Þ  nc
B00  ðN
ne
neh dt
nh Þ  nv þ k  Be  ne  nv þ k  Ah  nh  nc

carrier. Although the average thermalized velocity of both charge carriers can be estimated with reasonable accuracy [23] the cross-sections are largely unknown. In addition, the possibility of capture of hot charge carriers has been raised in the literature [24]. In the current simulations, an attempt was made to comply with the following constraints:

TC7 – Catch all trapping center TC5.08 – Intrinsic TC 5.08 eV

4  10
22 1  10
23 – 2  10
24

B00

1  10
23

An4

1  10
23

Bn4

5  10
24

An5

2  10
24

Bn5

1  10
23

An6

1:4  10
23

Bn6

1  10
25

An7

7  10
25

Bn7

1  10
23

An5I

4  10
23

Bn5I

1  10
25

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due to the simplifying assumptions used in its derivation. It is generally assumed that the oscillator strengths obey the sum rule as shown in Eq. (2).

X

f ij ¼ 1

ð20Þ

There are exceptions to this rule if appreciable interaction with the core electrons occur [27]. Nevertheless, even if the oscillator strength is unknown it seems reasonable to assume that measurement of the OD determines N to within an order of magnitude corresponding to values of f between 0.1 and 1. Much lower values of f have not been reported in the Alkali Halide literature. The values of the available center concentrations used in the simulations are shown in Table 3. In LiF the value of f for the F band has been estimated by various authors between 0.5 and 0.9. In this work a value of f = 0.55 has been adopted which is appropriate for a Gaussian line shape [28]. For the other OA bands a value of f = 1 has been

adopted which yields minimum values of the concentrations. LiF: Mg,Ti (TLD-100) is doped with 100–200 ppm Mg so that the values of N for the combined 3.84 eV and 4.3 eV and 3.24 eV bands associated with Mg-Livac trimers and dipoles [10] respectively can be roughly estimated. In previous kinetic simulations we have also employed concentrations of N of approximately 1023 m
3 for these bands [1,2]. The concentration of the dominant F band is taken at

Table 3 Values of the available center concentrations. Description

Symbol

Value [m
3]

LC – Luminescent center

M1

2:5  1023

NLC –Non luminescent center

M2

8  1025

V3 – Two hole center

M3

1:6  1023

TC1 – Trapping center 3.24 eV

N1

1  1022

TC2 – Trapping center 3.8 eV

N2

1  1023

Noo

Noo

2:1  1023

TC4 – Trapping center 4.77 eV

N4

1:7  1023

TC5 – Trapping center 5.08 eV

N5

6  1023

TC6 – Competitive center 5.45 eV

N6

1:8  1023

TC7 – Catch all trapping center

N7

1:1  1026

TC5.08 – Intrinsic TC 5.08 eV

N50

1:6  1023

Fig. 9. Simulated dose response of the 3.84 eV OA band.

Fig. 8. Simulated dose response of the concentration ne-h and ne corresponding to the 4.3 eV OA band.

Fig. 10. Simulated dose response of the 4.77 eV OA band.

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Fig. 11. Simulated dose response of the 5.08 eV OA band.

289

Fig. 13. Simulated dose response of the V3 concentration, m3.

Fig. 12. Simulated dose response of the 5.45 eV OA band. Fig. 14. Simulated dose response of the hole center concentration, nh.

6  1023 m
3. Although these values of M and N are obviously approximate, they represent an effort to simulate the LiF:Mg,Ti (TLD-100) system as realistically as currently possible. It deserves mention that the results of the simulations are fairly insensitive to the absolute values of the concentrations, more so to their relative values. Although the model employs 19 coupled differential

equations and over fifty variable parameters, the results are required to accurately simulate a total of 70 experimentally measured optical densities. This places extreme constraints on the value of the parameters – only after a great many runs, well over a hundred, was a final set of parameter-values found which adequately simulated the dose response of all five OA bands.

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4. Results of the simulations The result of the simulation for the 4.3 eV OA band is shown below in Fig. 8. Two features of the dose response are especially worthy of mention. (i) the OA band of the e-only TC/LC configuration decreases at the highest levels of dose. At 105 Gy its concentration has decreased by an order of magnitude to its value at approximately 102 Gy. This is due to hole capture in the e-only configuration and occurs in parallel with the increase in the concentration of the e-h configuration. (ii) The OA band of the e-h configuration is supralinear up to a dose-level of approximately 1000 Gy and then continues to increase at a lower rate. The supralinear behavior arises from the increased probability of multiple charge carrier capture at high ionization density. The lower rate at very high levels of dose is due to the decreasing availability of unoccupied (noo) or e-only (ne) occupied centers. The results of the simulations for the others bands are shown in Figs. 9–12. It is interesting to note the supralinear dependence of both the V3 (two-hole center) and the hole-only configuration of the TC/LC configuration at approximately 1000 Gy before entry into saturation as shown in Figs. 13 and 14 respectively. Unfortunately, the hole-trapping centers are not observed in the OA spectrum so that this behavior cannot be substantiated experimentally. The behavior of the V3 center can be understood as due to the increased probability of multiple charge carrier trapping at high ionization density (dose). The supralinearity of the hole-only TC/LC configuration can also be understood as due to the combined effect of hole capture and recombination in the e-h configuration leaving behind a hole-only configuration as well as the main mechanism of hole capture in the empty TC/LC configuration. 5. Conclusions The CB/VB model described herein is used to investigate a new interpretation of the OA spectrum of LiF:Mg,Ti (TLD-100) and is shown to be capable of simulating the dose dependence of all the observed OA bands. The conduction band /valence band model incorporates a spatially correlated TC/LC configuration which gives rise to the 3.84 eV and 4.3 eV bands. The 3.84 eV band is postulated to arise from electron-occupied TCs not spatially correlated with an LC. The 4.3 eV band is composed of two sub-bands (not resolved in the OA spectrum) arising from an e-only and an e-h configuration. The former give rise to a linear/exponentially saturating dose response which decreases at high dose whereas the latter demonstrates a supralinear behavior at dose levels below approximately 1000 Gy followed by a slower rate of increase at the higher levels

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