- Email: [email protected]
Anomalous fading correction for dosimetry at small time scales
Radiation Physics and Chemistry 158 (2019) 137–142
Contents lists available at ScienceDirect
Radiation Physics and Chemistry journal homepage: www.elsevier.com/locate/radphyschem
Anomalous fading correction for dosimetry at small time scales Manish K. Sahai a b
a,b,⁎
, A.K. Bakshi
a,b
T
a,b
, D. Datta
Homi Bhabha National Institute,Bhabha Atomic Research Centre, Mumbai 400085, India Radiological Physics & Advisory Division,Bhabha Atomic Research Centre, Mumbai 400085, India
A R T I C LE I N FO Keywords: Afterglow Power law Computer simulation TL/OSL Fading correction
A B S T R A C T
In several cases after glow from an exposed sample shows deviation from power law decay of luminescence, I α twhere I is luminescence intensity, t is time of observation and k is a constant. It is shown that the non linearity is the general feature of afterglow and linear graphs observed in experiments or simulation are, in general, an artefact of time scale of observation. Present study also suggests curve fitting functions for afterglow. Study of afterglow is important for applying fading corrections in various domains of dosimetry.
k
1. Introduction Thermoluminescence (TL) is the phenomenon of emission of luminescence when thermal stimulation is given to a pre irradiated sample. Its optical counterpart is optically stimulated luminescence (OSL) where the stimulant is light. Historically the phenomenon of TL and OSL were explained on the basis of non localised models. When radiation is incident on a phosphor charge carriers are generated. Small fraction of charge carriers are trapped in defect centres (traps). They are excited to conduction band when stimulated and subsequently recombine at recombination/luminescence centres to produce luminescence (Chen and Mckeever, 1997; Chen and Pagonis, 2011; Mckeever 1988; Pagonis et al., 2006). Localised transition models (Bull, 1989; Mandowski, 2005) were also suggested to explain the behaviour of luminescence signal since the pioneering work of Halperin and Braner (Halperin and Braner, 1960). Localised transition models (Templer, 1986; Visocekas, 1985; Visocekas et al., 1994) could explain the loss of TL signal during storage. It has been established by various authors that loss of signal during storage at low temperature is mainly due to tunnelling from ground state of trap to recombination centre (anomalous fading) and the luminescence decay follow a power law with time (Huntley, 2006; Pagonis et al., 2013). This phenomenon can be represented by plotting intensity of luminescence (I) as a function of time (t) on a log-log scale. In practice, instead of plotting log(I) vs log(t), the plot is made between log (I) and log (st), where the time is scaled by the pre exponential factor (s) also known as attempt to escape frequency, as the numerical values of time are very small. Experimentally the behaviour of luminescence decay is known to generate both linear and non linear log (I) vs log(st) graphs (Jonscher and de Polignac, 1984). The
⁎
linear graphs have a slope (k) ranging from 0.5 to 2.0 (Jonscher and de Polignac, 1984). Huntley could explain partial range of k i.e. 0.95–2.0. Recent study (Sahai et al., 2017) could explain the full range of slope by modifying the model proposed by Huntley. However the non linear nature of log(I) vs log(st) graph observed experimentally (Jonscher and de Polignac, 1984) has not been addressed exhaustively in literature. This paper reports the possible reason for non linearity in log(I)-log(st) graph and various factors that can affect the advent of non linearity. This paper also reports the better curve fitting methodology for fading correction at small time scales than straight line. 2. Mathematical formulation In order to explain the full range of slope (k = 0.5–2.0) Sahai et al. had proposed (Sahai et al., 2017) the potential for one trap and one recombination centre system as given below.
RegionI ⎧− V V(r) = V2 = C0 + D0 r b Region II ⎨ RegionIII ⎩− V
(1)
where region I is trap, region III is recombination centre and region II is tunnelling region between trap and recombination centre. C0 , D0 and b are parameters which varies the shape of potential barrier in the region II. r is radial distance from trap centre. For this potential, the luminescence intensity of afterglow is given as
I (t ) = n (0)
∫0
∞
where
Corresponding author. E-mail address: [email protected] (M.K. Sahai).
https://doi.org/10.1016/j.radphyschem.2019.01.018 Received 11 June 2018; Received in revised form 22 January 2019; Accepted 29 January 2019 Available online 30 January 2019 0969-806X/ © 2019 Elsevier Ltd. All rights reserved.
Z dr
(2)
Radiation Physics and Chemistry 158 (2019) 137–142
M.K. Sahai, et al. t
k1 and k3 are wave vectors in Region I and III respectively. In case of rectangular potential barrier between trap and recombination centre (D0 = 0.0 , which makes potential V2 = C0) ζ has dimension of wave vector in Region II. E is energy of charge carrier. M is mass of charge carrier. ρ is recombination centre concentration. Further details of the parameters are available elsewhere (Sahai et al., 2017). We intent to explore the temporal behaviour of afterglow intensity which is spatial integration of parameter Z (Eq. (2)). As Z can be easily separated into time independent and time dependent parts, it is interesting to study their spatial and temporal behaviour. For this, Z can be written as
(− τ ) 1 4 Z = ⎛ ⎞ ρ 4πr 2exp{ − ⎛ ⎞ πr 3 ρ}e ⎝τ⎠ ⎝3⎠
−1 −1 τ = ηNR s e
F
b +1 ⎤ [ r+ A r 2 b +1 ⎥ ⎦
where
2 F = ⎛ ⎞ [2M C0− ⎝ℏ⎠
2ME ]
A = D0 /(C0 − E )
ηNR = 4
k1 (
k33 +
k12)
ζ2
Z = G1 x G 2
,
where G1 represents the time independent component and G2 is the time dependent component of Z and are given as
where
k1 =
2m (E − V ) ℏ2
G1
1 4 = ⎛ ⎞ ρ 4πr 2exp ⎧ −⎛ ⎞ πr 3 ρ ⎫ ⎨ ⎬ ⎝τ⎠ ⎩ ⎝3⎠ ⎭
k3 =
2m (E − V ) ℏ2
G2
= e (− τ )
1 ⎡d ζ = ⎢ ⎛⎜exp ⎛− dr ℏ ⎝ ⎣ ⎝
∫r
r
1
2m
(E −
t
By exploring the spatial and temporal behaviour of G1 and G2 (Section 3.1) one can understand the variation of luminescence intensity (I) with time. In literature (Jonscher and de Polignac, 1984) we observe various shapes of log(I)-log(st) graphs, hence, it is necessary to obtain the general shape of this graph. Section 3.2 explores this issue and establishes that non linearity is the general nature of log(I)-log(st) graph. Section 3.3 discusses about various factors that can affect the advent of non linearity. After arriving at the general shape of log (I)-log
⎤ k2 (r ) dr ⎞ ) ⎥ |r = r 1 ⎠ ⎦
where
k2 =
(3)
V2 (r ))
Fig. 1. Plot of normalised time independent (G1) and time dependent (G2) of expression Z (Eq. (2)) and their normalised product, Z. (A) Plot of normalised G1 (1) and G2 (2) with radial distance from trap. Parameters considered for the plot are well depth (V) = -1.0 eV, energy (E) = -0.5 eV, C0 = 0.5 (eV), D0 = −10−6 (eVm−b ), b = -0.5, initial time = 0.01 femto seconds, recombination centre concentration 1023 m−3, F = 1.6 × 109 m−1, pre exponential factor, s = 3 × 1015 s−1. (B) Plot of normalised Z with radial distance from trap. Values of all parameters are same as shown for (A). 138
Radiation Physics and Chemistry 158 (2019) 137–142
M.K. Sahai, et al.
Fig. 2. Plot of normalised G1, G2 and Z as a function of radial distance from trap for two different times. For plot (A and C) t = 0.40 femto seconds whereas for (B and D) t = 655.00 femto seconds. (E) and (F) depicts log(I) –log(st) graph up to above mentioned snapshots of times. All other parameters used for the plots are same as of Fig. 1. Plots are normalised with respect to their maximum value for better comparison in A, B, E and F. Plots of C and D are normalised with maximum value of C for comparison in C and D.
(st) graph in Sections (3.1) and (3.2), various curve fitting functions are proposed and their relative accuracy is compared in Section 3.4. Section 3.5 discusses about possible applications and extensions in model.
when maximum of G2 is after maximum of G1. In the former case full area of G1 contributes to log(I) – log(st) graph and is almost constant with time (Fig. 2(E)). In the latter case only a part of G1 contributes to log(I) – log(st) graph which leads to the decrease in intensity, thereby giving rise to the fall in log(I)-log(st) graph (Fig. 2(F)). So overall log(I) – log(st) graph can be considered to have two regions, the horizontal part and the falling part. Such decays had been observed experimentally (Jonscher and de Polignac, 1984).
3. Results and discussions 3.1. Cause for non linearity of log(I)-log(st) graph Fig. 1A illustrates a typical plot of time independent (G1), time dependent (G2) components and Z, (Eqs. (2) and (3)), as a function of radial distance from trap (r ). By observing the shape of G1 and G2, it is possible to explain the observed shape of Z. Initially G2≃0.0 and G1 ≠ 0.0 up to r = 25 angstroms and at r > 85 angstroms G2 ≠ 0.0 and G1≃0.0. So in both the cases (r > 85 and r < 25) the normalised value of the product Z is considerably small whereas in between (25 < r < 85) Z is significant (Fig. 1B). Fig. 2 shows the variation of G1, G2 and Z with time. Fig. 2(A) shows G1 and G2 at time t = 0.40 femto seconds. Fig. 2(B) shows the respective curves at time t = 655 femto seconds. It can be seen that both the components retain their shapes with time but, G2 traverses rightwards whereas G1 remains stationary. Fig. 2(C) and (D) shows the variation of Z with time at time t = 0.40 and t = 655 femto seconds respectively. It can be seen from the figures that the area under the curve decreases with time. This is due to lower contribution of G1 to Z. As the area under Z decreases with time, log(I) – log(st) graph is expected to fall with time. Fig. 2(E) and (F) show the variation of log(I) with respect to log (st) from 0.01 femto second to t = 0.40 femto seconds and t = 655 femto seconds respectively. Let us consider two different situations. (i) When maximum of traversing temporal graph (G2) is before the maximum of G1, and (ii)
3.2. Effect of time at which observation is initiated and period of observation It may be noted that observed nature of log(I)-log(st) graph depends on the time at which the observation was started after irradiation and the period of observation. The dependence on starting time of observation is shown in Fig. 3. Graph (A) of Fig. 3 shows the intensity for a duration of 10 nano seconds if the observation was initiated from 10 femto seconds after irradiation. A straight line (linear graph) was observed. However the nature of the graph differs if the observation was started at 0.01 femto seconds after irradiation (Graph (B)). The dependence on the period of observation is as shown in Fig. 4. Graph that was apparently linear becomes non linear for longer observation period. 3.3. Factors which affect the advent of non linearity Fig. 5 illustrates the shape of log(I)-log(st) curve for different recombination centre concentration. It can be seen from the figure that the curvature (non linearity) increases with the increase in recombination centre concentration. Hence, recombination centre 139
Radiation Physics and Chemistry 158 (2019) 137–142
M.K. Sahai, et al.
Fig. 3. Plots of log(I) – log(st) for two different starting time of observation. Except initial time all other parameters are same as in Fig. 1. Fig. 5. Plots of log(I) – log(st) for different recombination centre concentrations. Figure illustrates log(I)-log(st) graph becomes non linear (curves labelled 2, 3 and 4) from linear curve (1) with increase in recombination centre concentration. 1. 1022 m−3 2. 1023 m−3 3. 1023.2 m−3 4. 1023.3 m−3 All other parameters are same as of Fig. 1. Value of D0 = −10−4 (eVm−b) and initial time is 1 milli seconds. Curves are normalised with respect to their maximum value for better comparison.
Fig. 4. Plot of log(I) – log(st) for (1) 1 s (2) 3.6 × 107 years observation periods. Figure illustrates that for same set of parameters but for longer observation period apparently straight line graph (1) becomes curved (2). Curves are fitted with straight line for more clarity. All parameters except length of observation period are same as of Fig. 1 only value of D0 = -10−4 (eVm−b) and initial time is 1 milli seconds. Fig. 6. Plot of log(I) – log(st) for different shape of potential barrier between trap and recombination centre as defined in Eq. (1). For all the curves recombination centre concentration is 1023per m3. Rest all parameter are same as of Fig. 5 except D0. They are as below. 1. D0 = −1.9 × 10−4 (eVm−b) 2. D0 = 0.0 (eVm−b) 3. D0 = −10−5 (eVm−b) Curve 1 is apparently linear whereas 2 and 3 have larger curvature.
concentration is one of the factors that affect the advent of non linearity. Fig. 6 shows the dependence of log(I) vs log(st) graph with the shape of potential barrier between trap and recombination centre. It can be seen that curvature (non linearity) varies with D0, where D0 is one of the parameters governing the shape of potential barrier between trap and recombination centre. Hence, shape of potential barrier is also an influencing factor that can affect advent of non linearity in log(I)-log (st) graph.
y = −lx 2 + mx + n
where l, m and n are curve fitting parameters. y is log of luminescence data (log I) and x is log of product of pre exponential factor and time (log(st)). Fig. 8 illustrates fitting of data with inverted half parabola and Table 1 tabulates values of l, m and n for curves of Fig. 8. It is pertinent to mention that negative sign before l is crucial so as to fit the data with part (A) of parabola 2. With this half parabola equation we get a much better fitting than a straight line. Even for cases when afterglow data is apparently a straight line this equation gives a good fit. The data which were fitted to straight line and inverted half parabola was also fitted with circular arc. The general equation of a circle is
3.4. Curve fitting Fig. 7 shows basic parabolas with vertex at origin and their corresponding equations. By visual inspection it is obvious that luminescence afterglow (Fig. 3) has a similar shape as that of part A of parabola 2 in Fig. 7 (inverted half parabola). General equation (with variable vertex and latus rectum) of such a parabola is:
y − B = −4A (x − C )2
(x > 0)
(5)
(4)
A, B and C are fitting parameters. Eq. (4) can be rearranged as
(x − h)2 + (y − k )2 = R2 140
(6)
Radiation Physics and Chemistry 158 (2019) 137–142
M.K. Sahai, et al.
Fig. 9. Plot of log(I) – log(st) graph as a function of recombination centre concentration. Figure illustrates the fitting of various curves with function (x − h)2 + (y − k )2 = R2 . The curve fitting parameters R, h and k for all the three curves are tabulated in Table 2. Parameters (recombination centre concentration) values are same as of Fig. 8.
Fig. 7. Figure illustrates basic parabolas for the shape of their sides, orientation and equations. Equations are as below: 1. X = AY 2 2. Y = Ax 2 3. X = −AY 2 4. Y = −AX 2 By visual inspection part (A) of parabola 2 has the shape of our simulated after glow.
Table 2 Table illustrates curve fitting parameters for various curves of Fig. 9. Curves are fitted with function (x − h)2 + (y − k )2 = R2 . Curve No.
R
h
k
1
1.3050 × 103 123.1800 48.4143
− 212.5357
− 1.2842 × 103 − 121.7693 − 47.8002
2 3
− 5.1779 2.1755
Table 3 Table shows comparison of error estimated using, error= ∑ (Simulated data − fitting curve )2 , for curve fitted with various functions. Error in curve fitting with various functions
Fig. 8. Plot of log(I) – log(st) graph as a function of recombination centre concentration. Figure illustrates the fitting of various curve with function y = −lx 2 + mx + n. The curve fitting parameters l, m and n for all the three curves are tabulated in Table 1. Curve (1) is for a recombination centre concentration 1019 m−3, curve (2) for 1022.5 m−3 and curve (3) for 1023 m−3. Rest parameters are same as of Fig. 1.
-l
m
n
1 2 3
0.0004 0.0043 0.0121
− 0.1688 − 0.0354 0.0986
3.4259 1.2611 0.6539
Straight line (linear)
Parabola (Non linear)
Circular arc (Non linear)
1 2 3
0.4047 1.9025 5.5645
0.0515 0.1887 1.1106
0.0231 0.0776 0.1694
3.5. Possible applications and extensions in model In present work small time scales have been considered so that time scale of situation shown in Fig. 2(E) is comparable with the time scale of situation shown in Fig. 2(F). Only then the two situations can be represented in a graph and concept is visually clear. Present work may find application in pulsed systems where irradiation time is very small and fading time is of order of seconds or minutes. One example may be fading correction in situations like in vivo online dosimetry (Andersen et al., 2009). Given formulation can be easily extended to durations of time which are of interest in geological dating at expense of computational time. Situations of simultaneous exposure and fading can also be incorporated in present model. In present work a fixed value of initial charge carrier concentration is taken and fading is calculated with time. In each time step, the initial charge carrier concentration can be increased by an amount depending on exposure rate and value of time increment in each time step. Though exact nature of afterglow in this situation is expected to be dependent of exposure rate and fading rate (which depends on distance between trap and recombination centre i.e.
Table 1 Table illustrates curve fitting parameters for various curves of Fig. 9. Curves are fitted with function y = −lx 2 + mx + n . Curve No.
Curve No
where (h, k ) is coordinate of centre and R is radius. Fig. 9 illustrates the fitting of data with circular arcs and Table 2 gives values of radius (R) and centre coordinates (h and k). Table 3 gives comparison of error in curve fitting with straight line, inverted half parabola and circular arc for three sets of data. In the present study based on data from Table 3, it appears that circular arc gives a better fit. This fit can be used for correction for the fading at different times.
141
Radiation Physics and Chemistry 158 (2019) 137–142
M.K. Sahai, et al.
Funding
defect concentration), but we do not expect afterglow curve to be a straight line in general for any exposure and fading rate. Hence, our general result that non linearity is the general feature of afterglow remains invariant. Although the present study was conducted for one trap one recombination centre model, the complex shapes of log(I)-log(st) graphs due to presence of multiple traps and recombination centres (Jonscher and de Polignac, 1984) can also be explained by suitably modifying the model. Effect of band tails is not considered in present work, however, the same can be considered in full detail by taking the two cases of closely located traps (where energy miniband may be formed) and separated traps (where hopping is dominant mechanism) (Poolton et al., 2002). This effect can be considered by adding the luminescence intensity expression due to band tail effect (Poolton et al., 2002) with luminescence intensity expression due to tunnelling effect. The combined luminescence intensity will be the afterglow.
This project did not receive any specific grant from funding agencies in public, commercial or not-for- profit sectors. References Andersen, C.E., et al., 2009. Characterization of a fiber-coupled Al2O3: C luminescence dosimetry system for online in vivo dose verification during 192Ir brachytherapy. Med. Phys. 36, 708–718. Bull, R.K., 1989. Kinetics of localised transition model for thermoluminescence. J. Phys. D: Appl. Phys. 22, 1375–1379. Chen, R., McKeeverS, W.S., 1997. Theory of Themoluminescence and Related Phenomenon. World Scientific. Chen, R., Pagonis, V., 2011. Thermally and Optically Stimulated Luminescence A Simulation Approach. John Wiley & Sons Ltd, UK. Halperin, A., Braner, A.A., 1960. Evaluation of thermal activation energies from glow curves. Phys. Rev. 117, 408–415. Huntley, D.J., 2006. An explanation of the power law decay of luminescence. J. Phys. Cond. Matter 18, 1359–1365. Jonscher, A.K., de Polignac, A., 1984. The time dependence of luminescence in solids. J. Phys. C: Solid State Phys. 17, 6493–6519. Mckeever, S.W.S., 1988. Thermoluminescence of Solids. Cambridge University press. Mandowski, A., 2005. Semi- localized transition model for thermoluminescence. J. Phys. D: Appl. Phys. 38, 17–21. Pagonis, V., Kitis, G., Furetta, C., 2006. Numerical and Practical Exercises in Thermoluminescence. Springer Science + Business Media, Inc, USA. Pagonis, V., Phan, H., Ruth, D., Kitis, G., 2013. Further investigations of tunnelling recombination processes in random distribution of defects. Radiat. Meas. 58, 66–74. Poolton, N.R., Ozanyan, K.B., Wallinga, J., Murray, A.S., Botter-Jesen, L., 2002. Electrons in feldspar II: a consideration of the influence of band-tail states on luminescence processes. Phys. Chem. Mater. 29, 217–225. Sahai, M.K., Bakshi, A.K., Datta, D., 2017. Revisit to power law decay of luminescence. J. Lumin. 195, 240–246. Templer, R.H., 1986. The localised transition model of anomalous fading. Radiat. Prot. Dosim. 17, 493–497. Visocekas, R., 1985. Tunnelling radiative recombination in labradorite: its association with anomalous fading of thermoluminescence. Nucl. Tracks Radiat. Meas. 10, 521–529. Visocekas, R., Sponner, N.A., Zink, A., Blanc, P., 1994. Tunnel afterglow, fading and infrared emission in thermoluminescence of feldspars. Radiat. Meas. 23, 377–385.
4. Conclusions Non linearity is a general feature of afterglow. Linearity observed in log(I) vs log(st) graph of afterglow can be considered as an specific case in time scale within which experiment or simulation is performed. The present study shows that the non linearity of afterglow can be better fitted with an inverted half parabola or a circular arc than a straight line. The use of best fit helps in estimating accurate fading correction in various domains of dosimetry. Acknowledgement Authors are thankful to Dr Pradeep Kumar K S, Associate Director, Health, Safety and Environment Group, Bhabha Atomic Research Centre for his constant support and encouragement. Authors are thankful to Shri S. N. Menon, Scientific Officer (F), Bhabha Atomic Research Centre for fruitful discussions and inputs to modify the manuscript.
142
Contents lists available at ScienceDirect
Radiation Physics and Chemistry journal homepage: www.elsevier.com/locate/radphyschem
Anomalous fading correction for dosimetry at small time scales Manish K. Sahai a b
a,b,⁎
, A.K. Bakshi
a,b
T
a,b
, D. Datta
Homi Bhabha National Institute,Bhabha Atomic Research Centre, Mumbai 400085, India Radiological Physics & Advisory Division,Bhabha Atomic Research Centre, Mumbai 400085, India
A R T I C LE I N FO Keywords: Afterglow Power law Computer simulation TL/OSL Fading correction
A B S T R A C T
In several cases after glow from an exposed sample shows deviation from power law decay of luminescence, I α twhere I is luminescence intensity, t is time of observation and k is a constant. It is shown that the non linearity is the general feature of afterglow and linear graphs observed in experiments or simulation are, in general, an artefact of time scale of observation. Present study also suggests curve fitting functions for afterglow. Study of afterglow is important for applying fading corrections in various domains of dosimetry.
k
1. Introduction Thermoluminescence (TL) is the phenomenon of emission of luminescence when thermal stimulation is given to a pre irradiated sample. Its optical counterpart is optically stimulated luminescence (OSL) where the stimulant is light. Historically the phenomenon of TL and OSL were explained on the basis of non localised models. When radiation is incident on a phosphor charge carriers are generated. Small fraction of charge carriers are trapped in defect centres (traps). They are excited to conduction band when stimulated and subsequently recombine at recombination/luminescence centres to produce luminescence (Chen and Mckeever, 1997; Chen and Pagonis, 2011; Mckeever 1988; Pagonis et al., 2006). Localised transition models (Bull, 1989; Mandowski, 2005) were also suggested to explain the behaviour of luminescence signal since the pioneering work of Halperin and Braner (Halperin and Braner, 1960). Localised transition models (Templer, 1986; Visocekas, 1985; Visocekas et al., 1994) could explain the loss of TL signal during storage. It has been established by various authors that loss of signal during storage at low temperature is mainly due to tunnelling from ground state of trap to recombination centre (anomalous fading) and the luminescence decay follow a power law with time (Huntley, 2006; Pagonis et al., 2013). This phenomenon can be represented by plotting intensity of luminescence (I) as a function of time (t) on a log-log scale. In practice, instead of plotting log(I) vs log(t), the plot is made between log (I) and log (st), where the time is scaled by the pre exponential factor (s) also known as attempt to escape frequency, as the numerical values of time are very small. Experimentally the behaviour of luminescence decay is known to generate both linear and non linear log (I) vs log(st) graphs (Jonscher and de Polignac, 1984). The
⁎
linear graphs have a slope (k) ranging from 0.5 to 2.0 (Jonscher and de Polignac, 1984). Huntley could explain partial range of k i.e. 0.95–2.0. Recent study (Sahai et al., 2017) could explain the full range of slope by modifying the model proposed by Huntley. However the non linear nature of log(I) vs log(st) graph observed experimentally (Jonscher and de Polignac, 1984) has not been addressed exhaustively in literature. This paper reports the possible reason for non linearity in log(I)-log(st) graph and various factors that can affect the advent of non linearity. This paper also reports the better curve fitting methodology for fading correction at small time scales than straight line. 2. Mathematical formulation In order to explain the full range of slope (k = 0.5–2.0) Sahai et al. had proposed (Sahai et al., 2017) the potential for one trap and one recombination centre system as given below.
RegionI ⎧− V V(r) = V2 = C0 + D0 r b Region II ⎨ RegionIII ⎩− V
(1)
where region I is trap, region III is recombination centre and region II is tunnelling region between trap and recombination centre. C0 , D0 and b are parameters which varies the shape of potential barrier in the region II. r is radial distance from trap centre. For this potential, the luminescence intensity of afterglow is given as
I (t ) = n (0)
∫0
∞
where
Corresponding author. E-mail address: [email protected] (M.K. Sahai).
https://doi.org/10.1016/j.radphyschem.2019.01.018 Received 11 June 2018; Received in revised form 22 January 2019; Accepted 29 January 2019 Available online 30 January 2019 0969-806X/ © 2019 Elsevier Ltd. All rights reserved.
Z dr
(2)
Radiation Physics and Chemistry 158 (2019) 137–142
M.K. Sahai, et al. t
k1 and k3 are wave vectors in Region I and III respectively. In case of rectangular potential barrier between trap and recombination centre (D0 = 0.0 , which makes potential V2 = C0) ζ has dimension of wave vector in Region II. E is energy of charge carrier. M is mass of charge carrier. ρ is recombination centre concentration. Further details of the parameters are available elsewhere (Sahai et al., 2017). We intent to explore the temporal behaviour of afterglow intensity which is spatial integration of parameter Z (Eq. (2)). As Z can be easily separated into time independent and time dependent parts, it is interesting to study their spatial and temporal behaviour. For this, Z can be written as
(− τ ) 1 4 Z = ⎛ ⎞ ρ 4πr 2exp{ − ⎛ ⎞ πr 3 ρ}e ⎝τ⎠ ⎝3⎠
−1 −1 τ = ηNR s e
F
b +1 ⎤ [ r+ A r 2 b +1 ⎥ ⎦
where
2 F = ⎛ ⎞ [2M C0− ⎝ℏ⎠
2ME ]
A = D0 /(C0 − E )
ηNR = 4
k1 (
k33 +
k12)
ζ2
Z = G1 x G 2
,
where G1 represents the time independent component and G2 is the time dependent component of Z and are given as
where
k1 =
2m (E − V ) ℏ2
G1
1 4 = ⎛ ⎞ ρ 4πr 2exp ⎧ −⎛ ⎞ πr 3 ρ ⎫ ⎨ ⎬ ⎝τ⎠ ⎩ ⎝3⎠ ⎭
k3 =
2m (E − V ) ℏ2
G2
= e (− τ )
1 ⎡d ζ = ⎢ ⎛⎜exp ⎛− dr ℏ ⎝ ⎣ ⎝
∫r
r
1
2m
(E −
t
By exploring the spatial and temporal behaviour of G1 and G2 (Section 3.1) one can understand the variation of luminescence intensity (I) with time. In literature (Jonscher and de Polignac, 1984) we observe various shapes of log(I)-log(st) graphs, hence, it is necessary to obtain the general shape of this graph. Section 3.2 explores this issue and establishes that non linearity is the general nature of log(I)-log(st) graph. Section 3.3 discusses about various factors that can affect the advent of non linearity. After arriving at the general shape of log (I)-log
⎤ k2 (r ) dr ⎞ ) ⎥ |r = r 1 ⎠ ⎦
where
k2 =
(3)
V2 (r ))
Fig. 1. Plot of normalised time independent (G1) and time dependent (G2) of expression Z (Eq. (2)) and their normalised product, Z. (A) Plot of normalised G1 (1) and G2 (2) with radial distance from trap. Parameters considered for the plot are well depth (V) = -1.0 eV, energy (E) = -0.5 eV, C0 = 0.5 (eV), D0 = −10−6 (eVm−b ), b = -0.5, initial time = 0.01 femto seconds, recombination centre concentration 1023 m−3, F = 1.6 × 109 m−1, pre exponential factor, s = 3 × 1015 s−1. (B) Plot of normalised Z with radial distance from trap. Values of all parameters are same as shown for (A). 138
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Fig. 2. Plot of normalised G1, G2 and Z as a function of radial distance from trap for two different times. For plot (A and C) t = 0.40 femto seconds whereas for (B and D) t = 655.00 femto seconds. (E) and (F) depicts log(I) –log(st) graph up to above mentioned snapshots of times. All other parameters used for the plots are same as of Fig. 1. Plots are normalised with respect to their maximum value for better comparison in A, B, E and F. Plots of C and D are normalised with maximum value of C for comparison in C and D.
(st) graph in Sections (3.1) and (3.2), various curve fitting functions are proposed and their relative accuracy is compared in Section 3.4. Section 3.5 discusses about possible applications and extensions in model.
when maximum of G2 is after maximum of G1. In the former case full area of G1 contributes to log(I) – log(st) graph and is almost constant with time (Fig. 2(E)). In the latter case only a part of G1 contributes to log(I) – log(st) graph which leads to the decrease in intensity, thereby giving rise to the fall in log(I)-log(st) graph (Fig. 2(F)). So overall log(I) – log(st) graph can be considered to have two regions, the horizontal part and the falling part. Such decays had been observed experimentally (Jonscher and de Polignac, 1984).
3. Results and discussions 3.1. Cause for non linearity of log(I)-log(st) graph Fig. 1A illustrates a typical plot of time independent (G1), time dependent (G2) components and Z, (Eqs. (2) and (3)), as a function of radial distance from trap (r ). By observing the shape of G1 and G2, it is possible to explain the observed shape of Z. Initially G2≃0.0 and G1 ≠ 0.0 up to r = 25 angstroms and at r > 85 angstroms G2 ≠ 0.0 and G1≃0.0. So in both the cases (r > 85 and r < 25) the normalised value of the product Z is considerably small whereas in between (25 < r < 85) Z is significant (Fig. 1B). Fig. 2 shows the variation of G1, G2 and Z with time. Fig. 2(A) shows G1 and G2 at time t = 0.40 femto seconds. Fig. 2(B) shows the respective curves at time t = 655 femto seconds. It can be seen that both the components retain their shapes with time but, G2 traverses rightwards whereas G1 remains stationary. Fig. 2(C) and (D) shows the variation of Z with time at time t = 0.40 and t = 655 femto seconds respectively. It can be seen from the figures that the area under the curve decreases with time. This is due to lower contribution of G1 to Z. As the area under Z decreases with time, log(I) – log(st) graph is expected to fall with time. Fig. 2(E) and (F) show the variation of log(I) with respect to log (st) from 0.01 femto second to t = 0.40 femto seconds and t = 655 femto seconds respectively. Let us consider two different situations. (i) When maximum of traversing temporal graph (G2) is before the maximum of G1, and (ii)
3.2. Effect of time at which observation is initiated and period of observation It may be noted that observed nature of log(I)-log(st) graph depends on the time at which the observation was started after irradiation and the period of observation. The dependence on starting time of observation is shown in Fig. 3. Graph (A) of Fig. 3 shows the intensity for a duration of 10 nano seconds if the observation was initiated from 10 femto seconds after irradiation. A straight line (linear graph) was observed. However the nature of the graph differs if the observation was started at 0.01 femto seconds after irradiation (Graph (B)). The dependence on the period of observation is as shown in Fig. 4. Graph that was apparently linear becomes non linear for longer observation period. 3.3. Factors which affect the advent of non linearity Fig. 5 illustrates the shape of log(I)-log(st) curve for different recombination centre concentration. It can be seen from the figure that the curvature (non linearity) increases with the increase in recombination centre concentration. Hence, recombination centre 139
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Fig. 3. Plots of log(I) – log(st) for two different starting time of observation. Except initial time all other parameters are same as in Fig. 1. Fig. 5. Plots of log(I) – log(st) for different recombination centre concentrations. Figure illustrates log(I)-log(st) graph becomes non linear (curves labelled 2, 3 and 4) from linear curve (1) with increase in recombination centre concentration. 1. 1022 m−3 2. 1023 m−3 3. 1023.2 m−3 4. 1023.3 m−3 All other parameters are same as of Fig. 1. Value of D0 = −10−4 (eVm−b) and initial time is 1 milli seconds. Curves are normalised with respect to their maximum value for better comparison.
Fig. 4. Plot of log(I) – log(st) for (1) 1 s (2) 3.6 × 107 years observation periods. Figure illustrates that for same set of parameters but for longer observation period apparently straight line graph (1) becomes curved (2). Curves are fitted with straight line for more clarity. All parameters except length of observation period are same as of Fig. 1 only value of D0 = -10−4 (eVm−b) and initial time is 1 milli seconds. Fig. 6. Plot of log(I) – log(st) for different shape of potential barrier between trap and recombination centre as defined in Eq. (1). For all the curves recombination centre concentration is 1023per m3. Rest all parameter are same as of Fig. 5 except D0. They are as below. 1. D0 = −1.9 × 10−4 (eVm−b) 2. D0 = 0.0 (eVm−b) 3. D0 = −10−5 (eVm−b) Curve 1 is apparently linear whereas 2 and 3 have larger curvature.
concentration is one of the factors that affect the advent of non linearity. Fig. 6 shows the dependence of log(I) vs log(st) graph with the shape of potential barrier between trap and recombination centre. It can be seen that curvature (non linearity) varies with D0, where D0 is one of the parameters governing the shape of potential barrier between trap and recombination centre. Hence, shape of potential barrier is also an influencing factor that can affect advent of non linearity in log(I)-log (st) graph.
y = −lx 2 + mx + n
where l, m and n are curve fitting parameters. y is log of luminescence data (log I) and x is log of product of pre exponential factor and time (log(st)). Fig. 8 illustrates fitting of data with inverted half parabola and Table 1 tabulates values of l, m and n for curves of Fig. 8. It is pertinent to mention that negative sign before l is crucial so as to fit the data with part (A) of parabola 2. With this half parabola equation we get a much better fitting than a straight line. Even for cases when afterglow data is apparently a straight line this equation gives a good fit. The data which were fitted to straight line and inverted half parabola was also fitted with circular arc. The general equation of a circle is
3.4. Curve fitting Fig. 7 shows basic parabolas with vertex at origin and their corresponding equations. By visual inspection it is obvious that luminescence afterglow (Fig. 3) has a similar shape as that of part A of parabola 2 in Fig. 7 (inverted half parabola). General equation (with variable vertex and latus rectum) of such a parabola is:
y − B = −4A (x − C )2
(x > 0)
(5)
(4)
A, B and C are fitting parameters. Eq. (4) can be rearranged as
(x − h)2 + (y − k )2 = R2 140
(6)
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Fig. 9. Plot of log(I) – log(st) graph as a function of recombination centre concentration. Figure illustrates the fitting of various curves with function (x − h)2 + (y − k )2 = R2 . The curve fitting parameters R, h and k for all the three curves are tabulated in Table 2. Parameters (recombination centre concentration) values are same as of Fig. 8.
Fig. 7. Figure illustrates basic parabolas for the shape of their sides, orientation and equations. Equations are as below: 1. X = AY 2 2. Y = Ax 2 3. X = −AY 2 4. Y = −AX 2 By visual inspection part (A) of parabola 2 has the shape of our simulated after glow.
Table 2 Table illustrates curve fitting parameters for various curves of Fig. 9. Curves are fitted with function (x − h)2 + (y − k )2 = R2 . Curve No.
R
h
k
1
1.3050 × 103 123.1800 48.4143
− 212.5357
− 1.2842 × 103 − 121.7693 − 47.8002
2 3
− 5.1779 2.1755
Table 3 Table shows comparison of error estimated using, error= ∑ (Simulated data − fitting curve )2 , for curve fitted with various functions. Error in curve fitting with various functions
Fig. 8. Plot of log(I) – log(st) graph as a function of recombination centre concentration. Figure illustrates the fitting of various curve with function y = −lx 2 + mx + n. The curve fitting parameters l, m and n for all the three curves are tabulated in Table 1. Curve (1) is for a recombination centre concentration 1019 m−3, curve (2) for 1022.5 m−3 and curve (3) for 1023 m−3. Rest parameters are same as of Fig. 1.
-l
m
n
1 2 3
0.0004 0.0043 0.0121
− 0.1688 − 0.0354 0.0986
3.4259 1.2611 0.6539
Straight line (linear)
Parabola (Non linear)
Circular arc (Non linear)
1 2 3
0.4047 1.9025 5.5645
0.0515 0.1887 1.1106
0.0231 0.0776 0.1694
3.5. Possible applications and extensions in model In present work small time scales have been considered so that time scale of situation shown in Fig. 2(E) is comparable with the time scale of situation shown in Fig. 2(F). Only then the two situations can be represented in a graph and concept is visually clear. Present work may find application in pulsed systems where irradiation time is very small and fading time is of order of seconds or minutes. One example may be fading correction in situations like in vivo online dosimetry (Andersen et al., 2009). Given formulation can be easily extended to durations of time which are of interest in geological dating at expense of computational time. Situations of simultaneous exposure and fading can also be incorporated in present model. In present work a fixed value of initial charge carrier concentration is taken and fading is calculated with time. In each time step, the initial charge carrier concentration can be increased by an amount depending on exposure rate and value of time increment in each time step. Though exact nature of afterglow in this situation is expected to be dependent of exposure rate and fading rate (which depends on distance between trap and recombination centre i.e.
Table 1 Table illustrates curve fitting parameters for various curves of Fig. 9. Curves are fitted with function y = −lx 2 + mx + n . Curve No.
Curve No
where (h, k ) is coordinate of centre and R is radius. Fig. 9 illustrates the fitting of data with circular arcs and Table 2 gives values of radius (R) and centre coordinates (h and k). Table 3 gives comparison of error in curve fitting with straight line, inverted half parabola and circular arc for three sets of data. In the present study based on data from Table 3, it appears that circular arc gives a better fit. This fit can be used for correction for the fading at different times.
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Funding
defect concentration), but we do not expect afterglow curve to be a straight line in general for any exposure and fading rate. Hence, our general result that non linearity is the general feature of afterglow remains invariant. Although the present study was conducted for one trap one recombination centre model, the complex shapes of log(I)-log(st) graphs due to presence of multiple traps and recombination centres (Jonscher and de Polignac, 1984) can also be explained by suitably modifying the model. Effect of band tails is not considered in present work, however, the same can be considered in full detail by taking the two cases of closely located traps (where energy miniband may be formed) and separated traps (where hopping is dominant mechanism) (Poolton et al., 2002). This effect can be considered by adding the luminescence intensity expression due to band tail effect (Poolton et al., 2002) with luminescence intensity expression due to tunnelling effect. The combined luminescence intensity will be the afterglow.
This project did not receive any specific grant from funding agencies in public, commercial or not-for- profit sectors. References Andersen, C.E., et al., 2009. Characterization of a fiber-coupled Al2O3: C luminescence dosimetry system for online in vivo dose verification during 192Ir brachytherapy. Med. Phys. 36, 708–718. Bull, R.K., 1989. Kinetics of localised transition model for thermoluminescence. J. Phys. D: Appl. Phys. 22, 1375–1379. Chen, R., McKeeverS, W.S., 1997. Theory of Themoluminescence and Related Phenomenon. World Scientific. Chen, R., Pagonis, V., 2011. Thermally and Optically Stimulated Luminescence A Simulation Approach. John Wiley & Sons Ltd, UK. Halperin, A., Braner, A.A., 1960. Evaluation of thermal activation energies from glow curves. Phys. Rev. 117, 408–415. Huntley, D.J., 2006. An explanation of the power law decay of luminescence. J. Phys. Cond. Matter 18, 1359–1365. Jonscher, A.K., de Polignac, A., 1984. The time dependence of luminescence in solids. J. Phys. C: Solid State Phys. 17, 6493–6519. Mckeever, S.W.S., 1988. Thermoluminescence of Solids. Cambridge University press. Mandowski, A., 2005. Semi- localized transition model for thermoluminescence. J. Phys. D: Appl. Phys. 38, 17–21. Pagonis, V., Kitis, G., Furetta, C., 2006. Numerical and Practical Exercises in Thermoluminescence. Springer Science + Business Media, Inc, USA. Pagonis, V., Phan, H., Ruth, D., Kitis, G., 2013. Further investigations of tunnelling recombination processes in random distribution of defects. Radiat. Meas. 58, 66–74. Poolton, N.R., Ozanyan, K.B., Wallinga, J., Murray, A.S., Botter-Jesen, L., 2002. Electrons in feldspar II: a consideration of the influence of band-tail states on luminescence processes. Phys. Chem. Mater. 29, 217–225. Sahai, M.K., Bakshi, A.K., Datta, D., 2017. Revisit to power law decay of luminescence. J. Lumin. 195, 240–246. Templer, R.H., 1986. The localised transition model of anomalous fading. Radiat. Prot. Dosim. 17, 493–497. Visocekas, R., 1985. Tunnelling radiative recombination in labradorite: its association with anomalous fading of thermoluminescence. Nucl. Tracks Radiat. Meas. 10, 521–529. Visocekas, R., Sponner, N.A., Zink, A., Blanc, P., 1994. Tunnel afterglow, fading and infrared emission in thermoluminescence of feldspars. Radiat. Meas. 23, 377–385.
4. Conclusions Non linearity is a general feature of afterglow. Linearity observed in log(I) vs log(st) graph of afterglow can be considered as an specific case in time scale within which experiment or simulation is performed. The present study shows that the non linearity of afterglow can be better fitted with an inverted half parabola or a circular arc than a straight line. The use of best fit helps in estimating accurate fading correction in various domains of dosimetry. Acknowledgement Authors are thankful to Dr Pradeep Kumar K S, Associate Director, Health, Safety and Environment Group, Bhabha Atomic Research Centre for his constant support and encouragement. Authors are thankful to Shri S. N. Menon, Scientific Officer (F), Bhabha Atomic Research Centre for fruitful discussions and inputs to modify the manuscript.
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