coupled rate and transport equations modeling light yield, pulse shape and proportionality to
TRANSCRIPT
COUPLED RATE AND TRANSPORT EQUATIONS MODELING LIGHTYIELD, PULSE SHAPE AND PROPORTIONALITY TO ENERGY IN
ELECTRON TRACKS: A STUDY OF CSI AND CSI:TL SCINTILLATORS
BY
XINFU LU
A Dissertation Submitted to the Graduate Faculty of
WAKE FOREST UNIVERSITY GRADUATE SCHOOL OF ARTS AND SCIENCES
in Partial Fulfillment of the Requirements
for the Degree of
DOCTOR OF PHILOSOPHY
Physics
December 2016
Winston-Salem, North Carolina
Approved By:
Richard Williams, Ph.D., Advisor
Todd Torgersen, Ph.D., Chair
David Carroll, Ph.D.
Daniel Kim-Shapiro, Ph.D.
K. Burak Ucer, Ph.D.
Table of Contents
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
List of Tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter 2 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1 Setting up coupled rate and transport equations . . . . . . . . . . . . 4
2.2 Experimental proportionality data. . . . . . . . . . . . . . . . . . . . 13
2.3 Finite difference methods in solving equations . . . . . . . . . . . . . 16
Chapter 3 Calculation of nonproportionality, and time/space distribution . . . . 18
3.1 Undoped CsI at room temperature . . . . . . . . . . . . . . . . . . . 19
3.1.1 Normalization: transition from continuous tracks to separatedclusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.1.2 Population distributions and the luminescence mechanism . . 29
3.2 Thallium-doped CsI at room temperature . . . . . . . . . . . . . . . 36
3.2.1 Population distributions and the luminescence mechanism . . 42
Chapter 4 Temperature dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.1 Temperature dependence of parameters . . . . . . . . . . . . . . . . . 53
4.2 Undoped CsI at 100 K . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2.1 Population distributions and the luminescence mechanism . . 62
Chapter 5 Energy-dependent scintillation pulse shape and proportionality ofdecay components in CsI:Tl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.1 Pulse shape and its energy dependence . . . . . . . . . . . . . . . . . 66
5.1.1 Experimental data . . . . . . . . . . . . . . . . . . . . . . . . 66
5.1.2 Fitting rise and decay times . . . . . . . . . . . . . . . . . . . 69
5.2 Nonproportionality of each decay component – experimental data andmodel results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
ii
5.3 Origin of three decay components of scintillation in CsI:Tl . . . . . . 76
5.3.1 Recombination reactions resulting in Tl+∗ light emission in CsI:Tl 77
5.3.2 Time-dependent radial population and reaction rate plots . . 79
5.4 Origin of anticorrelated fast and tail proportionality trends at roomtemperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.5 The material input parameters . . . . . . . . . . . . . . . . . . . . . . 107
Chapter 6 High tnergy tlectron tracks: GEANT4 and NWEGRIM . . . . . . . . . . . 112
6.1 GEANT4 results, trajectories and carrier density distributions, of CsI 112
6.2 NWEGRIM Results, trajectories and carrier density distributions, ofCsI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Appendix A Input files. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
A.1 GEANT4 sample input file . . . . . . . . . . . . . . . . . . . . . . . . 129
A.2 Local light yield model sample input file . . . . . . . . . . . . . . . . 130
Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
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List of Figures
1.1 Chopped track. The electric field at the beginning of the track is muchstronger than the end of the track so the thermalized electron needmore time to be attracted back to the center at the beginning of thetrack than the end of the track. . . . . . . . . . . . . . . . . . . . . . 3
2.1 Combined plot of the three experiments and their model fits for un-doped CsI (295 K), undoped CsI (100 K), and 0.082 mole% CsI:Tl(295 K). The “Energy (keV)”axis represents electron energy in theCompton coincidence measurements for CsI (295 K) and CsI:Tl (295K) and gamma ray energy for CsI (100 K). . . . . . . . . . . . . . . . 13
2.2 Radioluminescence spectra excited with Am-241 gamma rays at roomtemperature in the undoped sample (SGC unmarked, noisy line) com-pared with similar data extracted from Moszynski et al. [1] for CsI(A)(solid circles) and CsI(B) (solid diamonds). . . . . . . . . . . . . . . . 15
3.1 The proportionality curve of electron response modeled by Equations (2.1)to (2.3) from the material parameters listed in Table 3.1 is shown bythe solid triangles, and is superimposed on the Compton-coincidencedata for undoped CsI (SG sample) at 295 K shown by open trian-gles. Also shown by open squares is the gamma response experimen-tal curve for undoped CsI at 100 K, to be compared to the model inthe next Chapter. The schematic electron track at the bottom (afterVasil’ev [2]), will be used in discussion. . . . . . . . . . . . . . . . . . 25
3.2 Undoped CsI at 295 K. Radial density distributions for low on-axisexcitation density, 1018 e-h/cm3 (lower frames), and 100 x higher on-axis excitation density of 1020 e-h/cm3 (upper frames). Plotted arethe azimuthally-integrated densities of conduction electrons rne(r, t),self-trapped holes, rnh(r, t), self-trapped excitons, rN(r, t), and theaccumulated electrons trapped as deep defects, rned. The time afterexcitation for each plot is labeled on the frame near the curve. Thevertical scales are in units of 1016 nm/cm3 . . . . . . . . . . . . . . . 31
iv
3.3 Solid diamonds plot the calculated proportionality curve (electronresponse) using the combined parameters of Tables 3.1 and 3.2 for0.082 mole% thallium-doped CsI at room temperature inserted in themodel of Equations (2.1) to (2.7). The model curve is overlaid onthe Compton-coincidence experimental proportionality curve of CsI:Tl(0.082 mole%) at 295 K shown by the open diamonds. The experi-mental data for undoped CsI (295K) are reproduced in this figure byopen triangles for comparison. . . . . . . . . . . . . . . . . . . . . . . 42
3.4 CsI:Tl 295 K. Radial density distributions for (a) the azimuthally-integrated conduction electron density rne(r, t) and (b) the thallium-trapped electron density rnet(r, t) both for an original on-axis excita-tion density of 1019 e-h/cm3. Times after the original excitation areshown in the plots. The vertical scales are in units of 1016 nm/cm3. . 44
3.5 CsI:Tl 295 K. Expanded view of the radial Tl-trapped electron densitydistributions rnet(r) shown first in Figure 3.4 but here shown from 0to 25 nm with curves divided into two groups, 0 to 15 ns in frame (a)and 15 ns to 10 µs in (b). This is the distribution of electrons trappedby Tl+ dopant to form Tl0. The vertical scales are in units of 1016
nm/cm3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.6 CsI:Tl 295 K. Radial density distributions for (a) rnh self-trappedholes, (b) rnht Tl-trapped holes, (c) rN self-trapped excitons, (d)Nt Tl-trapped excitons, (e) STH +Tl0 self-trapped holes combiningwith Tl that has already trapped and electron and (f) Tl0 + Tl++
Tl-trapped holes migrating to combine with Tl0 all for an originalexcitation density of 1019 e-h/cm3. The vertical scales are in units of1016 nm/cm3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.1 Electron mobility calculated from different ways. Upper two use Equa-tions (4.1) and (4.2). Lower two use empirical Debye temperaturemethods from [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2 The solid square points show the calculated proportionality curve(electron response) at 100 K using the-low temperature parametersof Table 4.1 along with balance of parameters kept unchanged fromTable 3.1 as discussed in the text. The model curve is overlaid onthe experimental gamma yield spectra (open squares) of proportion-ality in undoped high-purity CsI (sample B) at 100 K measured byMoszynski et al [1]. The data of Figure 3.1 for undoped CsI(SG) atroom temperature are shown as open triangles for comparison. . . . . 58
v
4.3 Undoped CsI at 100 K. Radial density distributions for low on-axisexcitation density, 1018 e-h/cm3 (lower frames), and 100 x higher on-axis excitation density of 1020 e-h/cm3 (upper frames). Plotted arethe azimuthally-integrated densities of conduction electrons rne(r, t),self-trapped holes, rnh(r, t), self-trapped excitons, rN(r, t), and theaccumulated electrons trapped as deep defects, rned. The time afterexcitation for each plot is labeled on the frame near the curve. Thevertical scales are in units of 1016 nm/cm3 . . . . . . . . . . . . . . . 63
5.1 Experimental pulse rise and decay over the full measured range 0 to40 µs in CsI:Tl from [4] is shown for 662 keV gamma excitation in thered trace and for 6 keV gamma excitation in the lower blue trace. . . 67
5.2 Experimental scintillation decay curve from [4] for 662 keV gammaexcitation shown in red trace with noise on (a) 0 to 5 µs time scaleand (b) 0 to 40 µs scale. In both cases the superimposed smoothblack line is the modeled light output for 662 keV excitation. Modelis normalized to experiment at the peak. . . . . . . . . . . . . . . . . 70
5.3 Reconstructions of measured scintillation decay curves for 6 gamma-ray energies in CsI:Tl(0.06%) based on the time constants and inte-grated amplitudes reported in [4]. Only the decay curves are rep-resented. The curves for 122, 320, and 662 keV overlap in the topcurve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.4 Decay curves calculated from the model for six electron energies of thesame values as the gamma energies of the reconstructed experimentaldecay curves in Figure 5.3. . . . . . . . . . . . . . . . . . . . . . . . . 72
5.5 (a) Experimental proportionality curves for the fast (0.73 µs) andtail (16 µs) decay components as well as the proportionality of to-tal emission (Fast + Slow (τ2 = 3 µs) + Tail) in CsI:Tl are plottedversus gamma ray energy. Reproduced from [5]. (b) Simulated pro-portionality curves for fast, total, and tail decay components in CsI:Tlcalculated with the same model and parameter set used for Figure 5.2and Figure 5.4. The integration gate intervals for Fast, Total, andTail are given in the legend. Model curves are normalized at 200 keVfor reasons discussed in [6]. . . . . . . . . . . . . . . . . . . . . . . . . 75
5.6 The initial hole concentration profile, [STH] = nh, is plotted togetherwith the thallium-trapped electron concentration, [Tl0] = net, at earlytimes up to completion of electron trapping on Tl shortly after 5 ps.The on-axis excitation density is 1020 cm−3. Two formats are pre-sented. In frame (a), the population concentrations are multiplied bythe radius to convey number of carriers vs. radius. In frame (b), theconcentrations are reported directly. . . . . . . . . . . . . . . . . . . . 80
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5.7 The local rate of reaction #2 versus radius is plotted at evaluationtimes shown in the left two frames from 5 ps up to 10 ns and continuingin the right two frames from 20 ns to 800 ns. Reaction #2 ceases by800 ns when the supply of STH has been consumed by this reactionand by the competing process of STH capture on Tl+ activator sitesto create Tl++. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.8 Semi-logarithmic plot of spatially integrated rate of reaction #2 versustime, for on-axis excitation density of 1020 eh/cm3. . . . . . . . . . . 84
5.9 Plots proportional to azimuthally integrated local density of STH(rnh), Tl0 trapped electrons (rnet), Tl++ trapped holes (rnht), andTl+∗ trapped excitons rNt are displayed as a function of radius at sixindicated times between 10 ns and 10 µs. . . . . . . . . . . . . . . . 87
5.10 The Tl+∗ excited state (Nt) concentration distribution resulting fromall reactions at on-axis excitation density of 1020 cm−3 is plotted versusradius at times sampled from 5 ps to 20 µs. Notice that the radialscale range and the vertical axis range both change as time goes on. . 90
5.11 Radially weighted profiles of rate of change of [Tl0] due to reaction #3occurring ”in place” (Recombination, dashed curves) and due only totransport by diffusion and electric current (Transport, solid curves) arecompared at the indicated times. After about 3 µs, the rate of loss of[Tl0] (and identically of [Tl++] ) approaches equality with the positivegain of [Tl0] due to transport, indicating onset of the transport-limitedregime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.12 Radially weighted profiles of R#3 reaction rate are plotted for times(a) 0.1 ns, 0.5, 1, 5, 10, 20, 30, 40, 60, 80, 100 ns, and (b) 0.1 µs,0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.6, 0.7, 0.8, 0.9, 1, 1.5, 2, 3,4, 5, 6, 7, 8, 9, 10, 20, 30 µs. The radial weighting factor r comesfrom azimuthal integration of the cylindrical track to assess the totalreaction rate versus radius. Mixed units of 109 nm s−1 cm−3 are usedas in [6] so that division by the radius in nm recovers the local reactionrate at that radius in units of s−1cm−3. . . . . . . . . . . . . . . . . . 95
5.13 The spatially integrated rate of reaction #3 (black curve) is plot-ted as a function of time on semi-log scale for excitation densities of(a) 1017, (b) 1018, (c) 1019, and (d) 1020 eh/cm3. This model resultrepresents the time-dependent rate of change of the number of Tl+∗
excited activators due solely to R#3. It is the main contributor tothe Tl+∗ emitting state population at times longer than 700 ns. Threeexponential decay components of 730 ns, 3.1 µs, and 16 µs found tocharacterize 662 keV scintillation decay [4] are fitted and displayedalong with their sum in the magenta curve that can be compared tothe model-calculated black curve. . . . . . . . . . . . . . . . . . . . . 98
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5.14 The rate of reaction #3 as a function of excitation density was weightedby the probability of occurrence of each excitation density in a 662 keVelectron track based on GEANT4 simulations and is displayed versustime in the blue curve. Three exponential decay components of 730ns, 3.1 µs, and 16 µs found to characterize 662 keV scintillation de-cay [4] are fitted and displayed along with their sum in the magentacurve that can be compared to the model-calculated black curve. . . . 99
5.15 The time- and space-integrated yields of the reactions #2 and #3 areplotted versus initial on-axis excitation density in the solid blue andred curves, respectively. The yield is integrated from zero to 40 µs. . 102
5.16 The yields of reaction #2 and reaction #3 evaluated after 40 µs areplotted versus initial electron energy. . . . . . . . . . . . . . . . . . . 105
6.1 GEANT4 carrier density distributions. . . . . . . . . . . . . . . . . . 113
6.2 NWEGRIM tracks at 20 keV and 100 keV. . . . . . . . . . . . . . . . 114
6.3 NWEGRIM carrier density distributions. . . . . . . . . . . . . . . . . 115
6.4 Nonproportionality: NWEGRIM v.s. GEANT4. . . . . . . . . . . . . 116
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List of Tables
2.1 Parameter Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1 Parameters (and their literature references or comments on methods)as used for the calculation of proportionality and light yield in undopedCsI at 295 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Additional rate constants and transport properties used in Equa-tions (2.4) to (2.6) when modeling CsI:Tl at 295K. . . . . . . . . . . 37
4.1 Parameters (and literature references or estimation methods) pro-jected to T = 100 K for use in Equations (2.1) to (2.3) to fit un-doped CsI proportionality and light yield at 100 K. All other param-eters needed for Equations (2.1) to (2.3) were kept at their room-temperature value listed in Table 3.1 . . . . . . . . . . . . . . . . . . 57
5.1 Parameters used for the host parameters in the CsI:Tl model of thepresent work. Except for the deep defect trapping rate constant K1e
discussed in text, all parameters in this list are the same as used forthe calculation of proportionality and light yield in undoped CsI at295 K in [6]. In Table I of Ref. [6], literature references for the valueswere listed where available and otherwise comments on estimationmethods were listed and explained in the text. See [6] for definitionsof the parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.2 Additional rate constants and transport parameters used in Equa-tions (2.4) to (2.6) when modeling CsI:Tl (0.06%) at 295 K in thepresent work. S1e is the value measured on CsI:Tl (nominal 0.08 mole%) [7]. See [6] for definitions of the parameters. . . . . . . . . . . . . 110
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List of Abbreviations
Acronyms
GEANT4 GEometry ANd Tracking
NWEGRIM NorthWest Electron and Gamma Ray Interaction in Matter
PDEs Partial differential equations
STE Self-trapped exciton
STH Self-trapped hole
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Abstract
Xinfu Lu
This dissertation reports on development and testing of a scintillation responsemodel of progressive comprehensiveness that computes emission intensity over timeand space in electron tracks by solving coupled rate and transport equations de-scribing both the movement and the linear and nonlinear interactions of the chargecarriers deposited along the ionization track. The tracks are initially very narrowbefore hot and thermalized carrier diffusion takes effect. This suggests that an ade-quate and computationally manageable representation may be obtained by modelingdiffusion in one dimension, the radius. The initial track resulting from Monte Carlosimulations by GEANT4 or NWEGRIM codes is numerically chopped into cells smallenough to approximate their ionization density as constant, and these form the in-dividual parts of a finite element model. The initial ionization density values varyfrom cell to cell along the length of the track with the variation in dE/dx and wecalculate the light yield for each local value of dE/dx. This intermediate quantitythat we call local light yield as a function of dE/dx cannot itself be directly measuredby experiments. The local light yields must be multiplied by the number of timesthe associated ionization density occurs in repeated Monte Carlo simulations for thegiven initial electron energy, and then the yields are summed to report the total lightyield. When this calculation is carried out over a range of energies the results givethe predicted electron energy response or proportionality curve as a function of ini-tial electron energy, for comparison to Compton-coincidence and K-dip experiments.This has been done in the present work for CsI at 295 K and 100K, and for CsI:Tlat 295 K.
Relatively recent experiments on the scintillation response of CsI:Tl have foundthat there are three main decay times of about 730 ns, 3 us, and 16 us, i.e. onemore principal decay component than had been previously reported; that the pulseshape depends on gamma ray energy; and that the proportionality curves of eachdecay component are different, with the energy dependent light yield of the 16 uscomponent appearing to be anticorrelated with that of the 0.73 us component atroom temperature. These observations can be explained by the described modelof carrier transport and recombination in a particle track. It takes into accountprocesses of hot and thermalized carrier diffusion, electric field transport, trapping,nonlinear quenching, and radiative recombination.
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Chapter 1: Introduction
Gamma rays deposit energy in radiation detectors along ionized tracks left by
energetic electrons or positrons from photoelectric, Compton scattering, and pair-
production interactions. If there exists a known relation between detector output
and energy of the primary particle, the detector is spectroscopic. In scintillation
detectors, the response is the number of detected photons resulting from stopping
of the primary particle. If the scintillator’s intrinsic response is not proportional to
the particle energy, this so-called intrinsic nonproportionality combined with random
fluctuations in electron energies produced by scattering of gamma rays contributes
to degradation of energy resolution in gamma detectors. As electrons slow down,
their energy deposited per unit length, dE/dx, rises toward a maximum near 50 eV.
This variable energy deposition along the length of the main track and any branches
has long been considered to be a factor in the observed nonproportionality between
light yield and radiation energy in scintillators. In the last ten years particularly,
effort to understand and control nonproportionality has increased with the objective
of improving gamma energy resolution for a variety of practical applications. [8–11]
The experimental tools brought to bear have become more sophisticated. The
accurate measurement of light-yield produced by internally generated electrons over
a wide range of energies by Compton-coincidence [12, 13] and K-dip [14] methods
is one example. The use of pulsed lasers to measure transient behavior in the pi-
cosecond regime [7] and to induce specific ionization densities allowing measurement
of nonlinear processes at carrier densities found in the gamma ray induced electron
tracks [15] is another. There has been commensurate progress in the theoretical un-
derstanding of energy deposition and subsequent transport and recombination along
1
the ionized tracks. [16–24]
Since about 2010 the Wake Forest University group has been developing and
testing a scintillation response model of progressive comprehensiveness that computes
emission intensity over time and space in electron tracks by solving coupled rate and
transport equations describing both the movement and the linear and nonlinear
interactions of the charge carriers deposited along the ionized track. [25–27] The
tracks are initially very narrow before hot and thermalized carrier diffusion takes
effect, with a radius estimated as about 3 nm in NaI from hole thermalization range
[16], experiments on nonlinear quenching rate [15], and Monte Carlo simulations. A
similar size of the initial radius is indicated in other scintillators [28]. Even after hot
and thermalized carrier diffusion, the radius is much less than the track length of
several µm for 20 keV up to nearly a millimeter for 662 keV, which suggests that
a good representation can be obtained by modeling diffusion in one dimension, the
radius. The track is numerically chopped into cells small enough to approximate
their ionization density as constant and these form the individual parts of a finite
element model. The initial ionization density values vary from cell to cell along
the length of the track with the variation in dE/dx and we calculate the light yield
for each local value of dE/dx. This intermediate quantity that we call local light
yield as a function of dE/dx cannot itself be directly measured by experiments. The
local light yields must be multiplied by the number of times the associated ionization
density occurs in repeated simulations (e.g. using Geant4 [29,30]) for the given initial
electron energy, and then the yields are summed to report the total light yield. When
this calculation is carried out over a range of energies the results give the predicted
electron energy response or proportionality curve as a function of initial electron
energy, for comparison to Compton-coincidence and K-dip experiments. We are not
restricting the electron tracks modeled to be single linear tracks. Delta rays and high
2
energy Auger spurs are represented within the Geant4 simulations which determine
the weighting of each part of our modeled local light yield function (light yield versus
excitation density) in the final tally of electron response. The computation of local
light yield takes account of initially hot electrons and their thermalization; hole self-
trapping if it occurs in the material; electron, hole, and exciton diffusion; electrostatic
attraction of electrons and holes if there is charge separation; 2nd and 3rd order non-
linear quenching when ionization density is high enough; and carrier trapping with
and without luminescence.
Figure 1.1: Chopped track. The electric field at the beginning of the track is muchstronger than the end of the track so the thermalized electron need more time to beattracted back to the center at the beginning of the track than the end of the track.
3
Chapter 2: Equations
2.1 Setting up coupled rate and transport equations
The rate equations we are solving are expressed in terms of excitation density n.
A radial dimension is needed together with length x along the track in order to
convert a given dE/dx to an initial excitation density profile. We assume a Gaussian
cylindrical distribution of excitations, and the Gaussian radius is used to convert
dE/dx (eV/cm) to volume-normalized local initial excitation density n(r, t = 0)
(excitations/cm3). Calculation of local light yield in terms of volume-normalized
excitation density n rather than linear energy deposition dE/dx is an important
characteristic of this model. Volume-normalized density can be dramatically altered
by diffusion as time progresses during development of the light pulse. Rate terms
dependent on products of local electron and hole volumetric densities such as exciton
formation and Auger decay will be curtailed at lower densities after diffusion, or even
terminated to the extent that charge separation occurs by hot-electron diffusion
against hole self-trapping.
The calculation of response vs. electron energy has two parts: (1) solution of
coupled diffusion-limited rate equations in a spatial track geometry approximated as
cylindrical for one given on-axis excitation density, evaluating radiative and nonra-
diative recombination events and trapping in each cell and time step. The time- and
space-integrated radiative recombination events are tabulated as a function of the
initial on-axis excitation density. When normalized by the total number of electron-
hole pairs produced at that excitation density, this quantity is what we have termed
local light yield. (2) Monte Carlo simulations of linear energy deposition rate dE/dx
during stopping of an electron of initial energy Ei using the Geant4 code [29] are
4
averaged over multiple simulations to calculate distributions of the probability that
an electron of initial energy Ei will produce each local energy deposition rate dE/dx.
We multiply the local light yield YL(n0) by the probability P (n0, Ei) of occurrence
of each initial on-axis local density n0 in the stopping of an electron of initial energy
Ei. Integration of YL(n0) P (n0, Ei) over all n0 yields the electron energy response or
integrated light yield as a function of the initial energy of an electron launched inter-
nally within the sample [31]. Experimental electron energy response of scintillation
is typically measured by the Compton-coincidence [12, 13] or K-dip [14] techniques.
By convention, experimental electron energy response is usually normalized to unity
at 662 keV.
The local light yield in our model for an undoped scintillator and one doped with
a single activator is calculated using Equations (2.1) to (2.7).
dne
dt= Ge +De∇2ne + µe∇ · ne
−→E − (K1e + S1e)ne −Bnenh −Bhtnenht
−K3nenenh −K3nenenht
(2.1)
dnh
dt= Gh +Dh∇2nh − µh∇ · nh
−→E − (K1h + S1h)nh −Bnenh −Betnetnh
−K3nenenh −K3nenetnh
(2.2)
dN
dt= GE +DE∇2N −R1EN − (S1E +K1E)N +Bnenh −K2EN
2 (2.3)
dnet
dt= Det∇2net +µet∇ ·net
−→E +S1ene−K1etnet−Betnetnh−Bttnetnht−K3nenetnh
(2.4)
dnht
dt= Dht∇2nht +µht∇·nht
−→E +S1hnh−K1htnht−Bhtnenht−Bttnetnht−K3nenenht
(2.5)
dNt
dt= S1EN +Bhtnenht +Betnetnh +Bttnetnht −R1EtNt −K2EtN
2t (2.6)
5
S1x =nT l+
n0T l+
S01x where nT l+ = n0
T l+ − net − nht −Nt (2.7)
We will now describe each of the terms in the above equations and mention their
significance in selected cases. The electron and hole Equations (2.1) and (2.2) for
carrier densities ne and nh are of identical form, so we discuss them together. The
first term in Equations (2.1) and (2.2), Ge,h, is the generation term specifying the
initial Gaussian radial profile at t = 0. The Gaussian track radius of 3 nm is discussed
along with its justification in Section 3.1.1, which includes parameter values. The
magnitude of Ge,h(r = 0) is the on-axis excitation density.
n0 =dE/dx
πr2trackβEgap
(2.8)
Next in Equations (2.1) and (2.2) are the carrier diffusion terms. In alkali halides
including CsI, the hole is self-trapped very quickly [16,32,33]. In quantum molecular
dynamics calculations for NaI, hole self-trapping is indicated to occur as fast as 50
femtoseconds at room temperature. In view of such rapid self-trapping, the hole
Equation (2.2) is simply written in terms of the density of self-trapped holes, nh,
diffusing with the hopping diffusion coefficient of self-trapped holes. This effectively
ignores the first 50 fs of hole evolution, except as it may be represented in the initial
Tl++ production Ght in thallium-doped CsI [34, 35]. However, we cannot ignore the
early evolution of the electrons. The cooling of hot electrons is rather slow in CsI
(∼ 4 ps mean thermalization time) [22] due to its low optical phonon frequency. De
is a function of electron temperature Te and therefore a function of time during the
electron cooling process, De(Te(t)). The determination of the hot electron diffusion
coefficient in this work relies on the calculations of Wang et al [22] on hot electron
range in CsI. The great difference in diffusion ranges of hot electrons and self-trapped
holes in alkali halides means that electrons and holes are quickly separated as will
6
be seen directly in the radial distributions as a function of time. This is a significant
factor affecting the various 2nd and 3rd order rate terms in Equations (2.1) to (2.6)
that depend on overlap of the electron and hole populations.
The third terms in Equations (2.1) and (2.2) represent the electric field driven
currents. The tendency for separation of charge between hot electrons and relatively
immobile self-trapped holes in the alkali halides means that large radial electric fields
can arise and will tend to drive corresponding radial currents. The displacement
of a given electron imparted by any reasonable space-charge electric field between
electron-phonon scattering events is much smaller than the displacement due to ki-
netic energy of a hot (e.g. 3 eV) electron between the same scattering events. So
initially the hot electrons run outward to a radial distribution peak shown to be
about 50 nm in CsI (with a tail extending as far as 200 nm) [22], leaving behind self-
trapped holes (STH) in a cylinder with radius about 3 nm [15–17]. As the electrons
cool to thermalized energy near the conduction band minimum, the hot diffusion
coefficient drops toward the smaller thermalized diffusion coefficient De and the elec-
tric field term can finally assert itself as stronger than the diffusion term. At that
point the direction of electron current reverses from outward to inward as thermal-
ized conduction electrons in the undoped pure material are collected back toward
the line charge of STH where recombination can occur. In an activated scintillator
such as CsI:Tl, a similar process occurs but on a much slower time scale set by the
hopping diffusion of electrons trapped on thallium as they are drawn back toward a
charged core of Tl++ ions, and as the STH diffuse out to find Tl0.
The fourth terms in Equations (2.1) and (2.2) represent carrier capture on deep
defect traps with rate constants K1e,h and on the activator dopant with rate constants
S1e,h. The symbol K was chosen for representing killing of the radiative probability
when a carrier is caught on a deep defect trap. The symbol S was chosen to rep-
7
resent the concept that trapping on the activator represents storage of the carrier
for possible radiative emission through the activator-trapped exciton Equation (2.6).
The first order rate constants for capture are proportional to the respective trap con-
centration, so for example if there is no activator, the rate constants S1e,h coupling
free carriers into the trapped-carrier and trapped-exciton Equations (2.4) to (2.6)
vanish, and the model automatically reduces just to Equations (2.1) to (2.3) for a
pure material.
The fifth terms in Equations (2.1) and (2.2) are the bimolecular exciton formation
rates characterized by rate constant B and proportional to the product of electron
and hole densities at a given location and time. This term can vanish due to charge
separation of hot electrons from STH, but the bimolecular rate of exciton formation
will come into play later as thermalized carriers are united in their mutual space-
charge field. The exciton formation term, −Bnenh, is a loss term for Equations (2.1)
and (2.2) but it is the main source term in Equation (2.3) governing exciton density
N .
The sixth terms in Equations (2.1) and (2.2) are the bimolecular rates of forming
trapped excitons from capture of one free carrier on a trap (activator in the case
considered) already occupied by the other carrier. Similar to the commentary im-
mediately above, this is a loss term for the free carrier density but a source term in
Equation (2.6) for trapped excitons on the activator at density Nt.
The seventh terms in Equations (2.1) and (2.2) are the third order Auger recom-
bination rates of free carriers. McAllister et al [36] found that in NaI, the valence
band structure does not have states to receive the excited spectator hole in an nenhnh
Auger process. The valence band structure of CsI seems to support the same con-
clusion. Therefore we retain only the Auger rate term of the form K3nenhne in this
work. The Auger rate constant in CsI has been measured by interband z-scan ex-
8
periments [15]. The excitation density gradient and consequent charge separation
experienced in the laser z-scan experiment are significantly less than in an electron
track. This renders K3 more readily measurable by laser z-scan, whereas the charge
separation phenomenon in an alkali halide can act to severely limit the importance
of free-carrier Auger recombination in tracks excited by high-energy electrons. The
eighth terms in Equations (2.1) and (2.2) are similar Auger terms in which one of
the carriers already occupies the activator dopant.
There are other rate terms that could be included in the free carrier equations.
Examples would be source terms due to thermal ionization of shallow and deep traps.
Thermal ionization of deep traps is omitted if the time for release is longer than usual
scintillator gate times of the order of 5 microseconds, because study of afterglow is
beyond what we want to tackle during first tests of this model. Ionization from known
shallow traps, specifically electron release from Tl0 in CsI:Tl, is included effectively
in this system of equations in a way that will be discussed during description of the
trapped carrier Equations (2.4) to (2.6) below.
Equation (2.3) for the density of excitons, N , has no source term other than the
bimolecular exciton formation transferring population from the free carrier equations,
because of our setting GE = 0 for reasons discussed in relation to Table ?. Similar
to our earlier discussion regarding the holes as immediately self-trapped in an alkali
halide, the excitons represented by N in Equation (2.3) are regarded as self-trapped
excitons (STE) when alkali halides are the materials of interest. Their diffusion, by
thermally activated hopping/reorientation [18, 37, 38], is represented by the second
term. The third term in Equation (2.3) is the radiative decay rate. This is the only
rate term that produces light in the first 3 equations for a pure material. In the case of
pure CsI, R1E is the reciprocal of the radiative lifetime of the 3.7-eV Type II STE at
100 K, and of the 4.1-eV luminescence of the equilibrated Type I & II STEs at room
9
temperature identified by Nishimura et al. [39] The fourth term in Equation (2.3) is
a linear loss term from the exciton population involving two rate constants. The S1E
rate constant represents linear trapping of STEs on Tl+ activator (if present) and
is therefore an energy storage term that can contribute ultimately to Tl+∗ activator
luminescence via Equation (2.6) while subtracting from intrinsic STE luminescence in
Equation (2.3). The K1E linear loss rate constant is used to describe the dominant
path of quenching STE luminescence at room temperature, which in many alkali
halides is nonradiative thermally-activated crossing to the ground state. The fifth
term in Equation (2.3) is the bimolecular source term due to exciton formation from
free carriers. The final term represents second-order dipole-dipole quenching of STEs.
Equations (2.4) to (2.6) describe populations of trapped electrons, holes, and
excitons respectively on the activator dopant, in this case Tl+ substituting for Cs+.
Without going through all the rate and transport coefficient symbol names again,
we comment generally that the carrier/exciton densities, the rate constants, and
the transport coefficients carry the same symbols as in Equations (2.1) to (2.3) to
indicate corresponding physical quantities, except now the fact of being trapped
on the activator is indicated by the subscript “t”on all such terms. To illustrate
the meaning of a few of the trapped carrier/exciton terms, for example, +S1ene is
the rate of trapping thermalized electrons on the activator dopant, −Betnetnh is
the bimolecular rate of converting trapped electron population, net, into trapped
excitons feeding the corresponding source term in Equation (2.6), and −K3nenetnh
is the rate of Auger recombination involving a trapped electron and free electrons and
holes. −R1EtNt is the first order radiative rate of decay of dopant-trapped excitons
at density Nt, and −K2EtN2t is the rate of second-order dipole-dipole quenching of
two trapped excitons.
The terms in Equation (2.4) representing diffusion of trapped electrons, Det∇2net,
10
current in a field, µet∇ · net
−→E , and implied motion of the trapped electrons in bi-
molecular recombination with immobile holes trapped as Tl++ in the term Bttnetnht
deserve special comment. These terms account for thermal untrapping from shal-
low traps even though no explicit untrapping rate is represented in Equations (2.1)
to (2.6). The following two equations would replace Equations (2.1) and (2.4) above
if we were to explicitly represent untrapping and re-trapping of electrons on thallium
in CsI:Tl.
dne
dt= Ge + Uetnet +De∇2ne − µe∇ · ne
−→E − · · · (1a)
dnet
dt= S1ene − Uetnet −Betnetnh −K3nenetnh (4a)
In these equations, the untrapping loss term−Uetnet in Equation (4a) for electrons
trapped as Tl0 is an added positive source term after Ge in Equation (1a) for free elec-
trons. There are now no diffusion or electric current terms in Equation (4a) because
all of the transport occurs while the electrons are free and therefore accounted for in
Equation (1a). Likewise the term −Bttnetnht introduced in Equation (2.4) to repre-
sent bimolecular recombination of thallium-trapped electrons with thallium-trapped
holes is absent in Equation (4a) because such processes formally take place through
the term −Bhtnenht in the free-electron equation during the time that the electron
is untrapped from Tl0. In such a description, Equations (2.5) and (2.6) would also
be without the Btt terms. But although the equations themselves are simpler in
this physically realistic formulation, their numerical solution spanning time scales
from femtoseconds to microseconds presents computational difficulties. In fact we
coded the model first for the equation set with Equations (1a) and (4a) in place of
Equations (2.1) and (2.4), as well as the modification of Equations (2.5) and (2.6),
and ran the first calculation of local light yield. It took an unacceptably long time.
The Equation (2.7) keeps track of the change in concentration of available dopant
11
traps in their initial charge state of Tl+. For example, a lattice-neutral Tl+ ion that
trapped a free electron to become Tl0 is not available to trap another electron in
the same way until subsequent events return that dopant ion to its original Tl+
charge state. The coupling rate constants S1e, S1h, S1E = S1x for electrons, holes
and excitons on Tl+ are themselves proportional to the dopant concentration in
the available charge state Tl+, so the capture rates that they govern will decrease
as the local concentration of Tl+ gets “used up”temporarily. This saturation can
have the effect of contributing as one factor to the “roll-off”of local light yield at
high excitation density (low electron energy). Gwin and Murray concluded that the
activator concentration was not a dominant effect in their experiments on CsI:Tl [40].
In other scintillators than CsI:Tl, there have been observed experimental activator
concentration effects on proportionality, such as LSO:Ce [9] and YAP:Ce.
Table 2.1: Parameter Definitions.
Parameter Definition
rtrack track radius
τhot thermalization time
rhot (peak) thermalization distance
βEgap average energy to create a e-h pair
Gx(r = 0) carrier generation term
ε0 static dielectric constant
µx carrier mobility
Dx carrier diffusion coefficient
K1x deep defect trapping rate/STE thermal quenching rate
S1x activator trapping rate
Bxx bimolecular recombination rate
K3 3rd Auger recombination rate
K2x 2nd dipole-dipole quenching rate
R1x radiative rate
12
We applied the system of equations just described to calculate electron response
curves for comparison to three experimental measurements [6]: electron response in
undoped CsI and CsI:Tl at 295 K and gamma response of undoped CsI at 100 K [1].
The experimental data and superimposed model calculation of proportionality for
the three experimental conditions are summarized in Figure 2.1.
1 10 100 10000.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
Energy (keV)
Lig
ht
yiel
d
CsI(pure)−295KCsI(pure)−100KCsI(Tl)−295Kmodel: CsI(pure)−295Kmodel: CsI(pure)−100Kmodel: CsI(Tl)−295K
Figure 2.1: Combined plot of the three experiments and their model fits for undopedCsI (295 K), undoped CsI (100 K), and 0.082 mole% CsI:Tl (295 K). The “Energy(keV)”axis represents electron energy in the Compton coincidence measurements forCsI (295 K) and CsI:Tl (295 K) and gamma ray energy for CsI (100 K).
2.2 Experimental proportionality data.
The purpose of this section is to describe the experimental data used to develop and
verify the model. This is the experimental part of Figure 2.1 just presented. We
measured Compton-coincidence electron energy response for nominally pure CsI and
for CsI:Tl (0.082 mole%) in the same apparatus in order to have a matched pair of
13
data for the electron response of doped and undoped material at room temperature.
Data for gamma ray energy response in undoped CsI at about 100 K are available
from Moszynski et al. [1] The model in this study calculates electron response, strictly
speaking, so comparison with this 100 K gamma response data requires additional
attention.
To measure scintillation light output proportionality, a Compton coincidence sys-
tem [12] was set up by Menge et al at Saint Gobain according to the close-coupled
design of Ugorowski [41]. Each crystal sample was coupled to a Hamamatsu R1306
photomultiplier (PMT) with optical grease. A Zn-65 source (1115.5 keV) was used
to excite the crystals. An Ortec GMX-30200-P high purity germanium (HPGe) de-
tector was used to capture the Compton scattered gamma rays. Coincidence pulses
from the HPGe and PMT detectors were recorded for periods of 30 minutes. Then
un-gated pulses were recorded for both PMT and HPGe for 5 minutes in between
data acquisition in coincidence mode. The centroids of un-gated pulse height spectra
were continuously tracked to correct for drift of the gain in both detectors. Several
cycles were run to reduce statistical uncertainty. Results at room temperature for
doped and undoped CsI are shown in the upper two experimental curves of Figure 2.1
above.
Moszynski et al measured the gamma yield spectra of proportionality and the to-
tal light yield at 662 keV of two undoped CsI samples, CsI(A) and CsI(B), cooled to
low temperature [1]. The samples were close-coupled to a large area silicon avalanche
photodetector in a liquid nitrogen cryostat which cooled the detector/sample assem-
bly to a temperature characterized as about 100 K. Their sample B obtained from
a university group had the higher light yield, which was measured to have the ex-
traordinarily high value of 124,000 photons/MeV ± 12,000 at the temperature of 100
K. Sample B at 100 K also produced a flatter proportionality curve at high energy,
14
making it the interesting first target for comparisons to our model at low temper-
ature. Sample A from a commercial supplier had lower light yield and displayed
a more humped proportionality curve. Because 100 K data were only available as
gamma response we also measured the gamma response of our undoped and Tl doped
samples.
The differences between samples CsI(A) and CsI(B) in the Moszynski et al. [1]
work motivated characterization and comparison of our undoped CsI sample with
results shown in Figure 2.2.
200 300 400 500 600 7000
0.2
0.4
0.6
0.8
1
Wavelength (nm)
Nor
mal
ized
Res
pons
e (a
.u.)
SGCCsI(A)CsI(B)
Figure 2.2: Radioluminescence spectra excited with Am-241 gamma rays at roomtemperature in the undoped sample (SGC unmarked, noisy line) compared withsimilar data extracted from Moszynski et al. [1] for CsI(A) (solid circles) and CsI(B)(solid diamonds).
The figure shows that their samples and ours are similar in that they display a
dominant uv radioluminescence peak near 310 nm at room temperature associated
with fast emission but they differ in the amount of visible signal produced, in par-
ticular a band near 425 nm sometimes ascribed to vacancies [42, 43] and associated
15
with slow emission. CsI(B) had the least visible emission in this region, CsI(A) the
most, and our sample an intermediate amount. Sample SGC also displayed sub-
stantially more emission toward the red but measured in a system with greater red
sensitivity. Chemical analysis for 31 elements was performed by inductively coupled
plasma-mass spectrometry (ICP-MS) on a slice taken from one end of the sample.
Only iron at 0.003% was detected. Tl was < 0.0005%. Sodium was not tested. The
impurity analysis and optical absorption measurements establish that the 550 nm
emission is not due to Tl. The fast to total ratio for our sample was measured at
74%, a normal result for this parameter in CsI.
Recognizing that undoped samples have variable properties, presumably due to
variation in trace impurities and defects, we are assuming that these differences can
be accounted for in the model by variation of a single deep trapping parameter.
In the present work, the modeled light yield and proportionality of undoped CsI
refer only to the fast (15 ns) component. The slow component of scintillation can
be included in this model in future work as the defect(s), their rate constants and
radiative properties become better understood.
2.3 Finite difference methods in solving equations
We use the finite difference method to obtain an approximate solution for coupled
partial differential equations (PDEs), n(r,t), at a finite set of r and t. The discrete r
are uniformly spaced in the interval [0,L] in cylindrical coordinates. At time 0, we set
the initial electron/hole density distribution to 3 nm Gaussian distribution. We use
first order forward difference and second order central difference to approximate the
spatial derivatives and forward difference to approximate the time derivative. The
entire solution is contained in two loops: an outer loop over all time steps, and an
inner loop over all interior cells. [44]
16
Instead of using a classic fixed time step, we use dynamic time step from searching
the dt to satisfy the criteria, the maximum variance of all carrier densities in all cells
would not exceed c%. (5%, 10% and 20 % limits were tried with essentially the same
result.)
Calculating the outcomes of the free-carrier Equations (2.1) and (2.2) requires
time steps as short as 0.1 femtosecond in the finite difference method. Calculating
the outcomes of the trapped carrier Equations (2.4) to (2.6) on the other hand
requires calculations running out to microseconds. Calculations spanning such time
ranges were made manageable in terms of computer time by varying the time steps
progressively longer from beginning to end when thermal untrapping of carriers was
not included. It worked because as time went on, the free carriers were trapped or
were combined as self-trapped excitons, and then larger time steps could be used. But
if thermal release of trapped carriers is included directly, then continual re-injection
of free carriers occurs over long time scales, so that short time steps continue to be
needed to accurately determine the fate of the fresh free electrons via Equation (1a).
This leads to the unacceptably long computational times.
Coupled Continuous PDEs, n(r, t)
Coupled Discrete PDEs, n(r(i), t)
Search the Fastest dn/dtn
in All Cells at Time t
Determine dt at t from the critieria: max(dn/dtn
)× dt = c%
t = t+ dt
17
Chapter 3: Calculation of nonproportionality, and
time/space distribution
In this chapter we detail the application of the model of Chapter 2 to calculate
proportionality for comparison to the experimental data as already summarized in
Figure 2.1. For each sample at room temperature, tables of material parameters are
provided. These are mainly the rate constants and transport coefficients in Equa-
tions (2.1) to (2.7). We use parameter values found directly in the literature when
possible, or that can be scaled by quantitative physical arguments from parameters
known in a similar material. This is the case for 21 of the 23 parameters listed in
Table 3.1 for the well-studied case of undoped CsI at room temperature. The two
remaining values are undetermined for physical reasons and are thus appropriately
treated as fitting parameters: K1e, the electron capture rate on deep defects of un-
determined identity and concentration; and the incident electron energy at which
normalization is performed. The normalization energy is treated as a one-time fit-
ting parameter because the usual normalization energy of 662 keV turns out to be
outside the electron energy range in which the cylinder track approximation is valid.
The best-fit values of these two parameters will be examined later when there are
additional data on the traps and better understanding of how multiple clusters of
excitation in a line act together. In particular, the effect of spacing of excitation
clusters along the track on attracting dispersed electrons to the STH track core will
be described in Section 3.1.1.
18
3.1 Undoped CsI at room temperature
Table 3.1 lists the material parameters used in the model prediction of proportionality
in undoped CsI at 295 K. The first two parameters listed, the initial Gaussian track
radius rtrack, and the average energy invested per electron-hole pair created by an
energetic electron, βEgap, are needed to convert dE/dx to the volume-normalized den-
sity of excitation via Equation (2.8) in terms of which the rate and transport Equa-
tions (2.1) to (2.7) are written. Both the initial radius and the volume-normalized
on-axis excitation density are introduced into the equations via the electron and hole
generation terms (Gaussian spatial profiles) Ge and Gh in Equations (2.1) and (2.2).
The initially deposited track radius r0 = 3 nm was first estimated for NaI based
on consideration of hole thermalization range by Vasil’ev [16], then deduced exper-
imentally by equating expressions for observed nonlinear quenching in K-dip and
interband laser z-scan experiments on NaI [15]. The value r0 ≈ 3 nm was further
supported by calculation of the initial hole distribution in NaI using the NWEGRIM
Monte Carlo code at PNNL. We have assigned the same initial track radius in CsI
based on similarity of the two alkali iodides.
The value of βEgap adopted in Table 3.1 is required for consistency with the light
yield of the 124,000 photons/MeV ± 12,000 (@ 662 keV) in undoped CsI at 100 K
measured by Moszynski et al [1]. The listed βEgap = 8.9 eV is calculated based
on the lower end of the experimental uncertainty range, 112,000 photons/MeV. The
band gap of CsI at T = 20 K has been reported as 6.02 eV on the basis of two-photon
spectroscopy [45]. From this we may estimate the room-temperature band gap of
CsI as 5.8 eV [15] The CsI light yield at 100 K thus implies β ≈ 1.5 if the light
emission is 100 % efficient and if we use the 5.8 eV band gap. Values of β are around
2.5 in most materials [46] including most scintillators [47], so a value of β ≤ 1.5
implied for CsI at 100 K is remarkable. Just to assess the effect of adopting a more
19
conservative estimate of the light yield in CsI at 100 K, we ran the proportionality
calculation at 100 K for βEgap values corresponding to both 112,000 photons/MeV
(shown in Figure 2.1) and 90,000 photons/MeV (not shown). The low energy end
of the proportionality curve was raised about 5% relative to the plotted curve for
112,000 photons/MeV. The effect is not dramatic, and is in line with what will be
discussed about the effects of lower excitation density in the track on both nonlinear
quenching and on electric-field collection of dispersed electrons back to the core of
self-trapped holes.
The electron mobility in CsI has been measured by Aduev et al. [48] using a
picosecond electron pulse method. The thermalized conduction electron diffusion
coefficient De is given in terms of µe by the Einstein relation, D = µkT/e. During
the hot-electron phase, which has a duration in CsI of τhot ≈4 ps [22], the diffu-
sion coefficient De has an elevated value De(Te) relative to the thermalized electron
mobility basically because the hot electrons have higher velocity between scattering
events. This is an important factor in the early radial dispersal of the hot electrons.
Wang et al calculated the peak of the radial distribution of hot electrons in CsI
upon achieving thermalization to be rhot(peak) ≈ 50 nm [22]. For simplicity in this
model, we have assumed a step-wise time-dependent electron diffusion coefficient
such that De(t < τhot) has a constant value that reproduces the Wang et al. result of
rhot(peak) ≈ 50 nm in the solution of Equation (2.1) at the end of τhot ≈ 4 ps. (Wang
et al. also stated that the tail of the hot-electron radial distribution in CsI extends as
far as 200 nm and the tail of the thermalization times is as long as 7 ps.) [22] It will
be possible in future versions of this model to use a time-dependent De(Te(t)) that
tracks electron temperature on the picosecond time scale without making the step-
wise assumption. The thermalization time and mean radial range of hot electrons,
τhot, rhot(peak), are imbedded in the code because of their use to specify the elevated
20
electron diffusion coefficient De(t < τhot). The cooling time, τhot, is also imbedded
to enforce the 4-ps delay of capture of electrons on self-trapped holes which was
directly observed in picosecond absorption spectroscopy of CsI [7]. In the picosec-
ond measurements, the bimolecular capture rate constant B for exciton formation in
CsI was time dependent, remaining zero until after the electron thermalization time
τhot ≈ 4ps, whereupon it achieved the value of B(t > τhot) that is listed in Table 3.1.
The self-trapped hole diffusion coefficient and thus mobility at 295 K are known
from the literature on thermally activated hopping of self-trapped holes [37,38]. The
nonlinear quenching rate constants K2E and K3 were measured in undoped CsI at
room temperature by laser interband z-scan experiments [15].
The radiative rate R1E and nonradiative decay rate K1E of self-trapped excitons
listed in Table 3.1 are the room-temperature values of temperature-dependent func-
tions R1E(T ) and K1E(T ), where the function K1E(T ) is assumed to be a thermally
activated path to the ground state. In typical treatments of thermally quenched
simple excited states, the radiative rate is independent of temperature and can be
identified as the decay rate at low temperature. Nishimura et al [39] have shown that
the STE luminescence in CsI comes from on-center (Type I) and off-center (Type II)
lattice configurations that communicate over barriers and finally come into thermal
equilibrium as temperature is raised above 250 K. The total radiative rate of the
communicating STE configurations is thus temperature-dependent, which we write
R1E(T ). The temperature-dependent total light yield is then
LY (T ) =R1E(T )
R1E(T ) +K1E(T )(3.1)
At temperatures above 250K when luminescence bands of the two STE config-
urations are no longer distinguishable from one another, the single temperature-
dependent decay time of the 310 nm fast intrinsic luminescence band is given in
21
these terms by
1
τobs(T ) = R1E(T ) +K1E(T ) (3.2)
These two equations can be fitted to the data of Nishimura et al. [39] as well as
Amsler et al [49] and Mikhailik et al [50] to obtain the functions R1E(T ) and K1E(T )
from 100 K up to 295K. The following method has been used to obtain the values of
R1E(295K)andK1E(295K) listed in Table 3.1.
To determine K1E(295 K) from data other than proportionality, the model of
Equations (2.1) to (2.3) is first run with the sole objective of reproducing the total
light yield of pure CsI at room temperature, which is 2,000 photons/MeV for 662
keV gamma rays as published in the Saint-Gobain CsI data sheet [51]. This is also
expressible as a 1.8% photon yield per e-h pair produced (using βEgap = 8.9 eV). The
fitting uses K1E(295 K) as the variable fitting parameter for that single light-yield
data point, but it is not varied for fitting the proportionality curve shape. Then
Equation (3.2) for the reciprocal of the experimentally measured STE decay time at
room temperature, τobs = 15 ns [39], can be solved for R1E(295 K).
The rate constants S1e, S1h, S1E for trapping of electrons, holes, and excitons on
thallium are proportional to thallium concentration and so are zero in Table 3.1 for
undoped CsI.
It was argued in [52,53] based on generalized oscillator strength and Monte Carlo
calculations by Vasil’ev for BaF2 [54] that the number of excitons created initially by
stopping of a high-energy electron should be no more than about 4% of the production
of electron-hole pairs in wide-gap solids quite generally. This was approximately con-
firmed for CsI by picosecond absorption spectroscopy [7] which tracked the initially
created (t < 1 ps) exciton and free-carrier spectra throughout the infrared and visible
ranges from 0.45 eV photon energy upward, including the Type I STE peak. The
22
spectra also revealed that the initially-created STE population is destroyed within
about 2 ps by impact ionization from the hot electrons. Creating hotter initial elec-
trons by exciting 3 eV above the band gap resulted in more complete destruction of
the initial STE population. Re-constitution of STEs from bimolecular recombination
of thermalized electrons and self-trapped holes did not commence until after a 4 ps
delay for thermalization [7]. This is the meaning of our notation 4%Ge → 0 in the
“published ”column for the parameter GE(r = 0). The value used in the model is
GE = 0, i.e. the excitons that exist beyond the first 4 picoseconds are those formed
later through the bimolecular recombination term Bnenh.
The last two rate constants listed govern the capture rates for electrons and holes
on deep defects, K1ene and K1hnh. The suspected most numerous deep electron traps
in pure CsI are iodine vacancies, either empty vacancies as F+ centers or having
trapped an electron to form F centers. We can roughly estimate relative magnitudes
of the rate constants K1e and K1h by reference to the equation that relates trapping
rate constant K to cross section σ,
K = σ[trap] < v > (3.3)
where [trap] is the concentration of the trap and < v > is the root mean square
velocity of the carriers approaching the trap. The rate constants for capture on
otherwise equivalent traps at the same concentrations would scale as the speed of
the carrier being trapped. We argue in the following that in alkali halides the contrast
between conduction electron and self-trapped hole velocities dominates in comparison
to smaller differences in cross sections. The speed of self-trapped holes, calculated
as jump rate times jump distance averaged over 90 and 180 degree jumps, is about
6 x 10−6 of the speed of conduction electrons in CsI at 295 K. The rate constant
K1h in Table 3.1 is thus listed approximately as 10−5K1e if electron and hole traps
may be presumed to have similar cross sections and concentrations. Therefore K1h
23
is neglected, leaving only one variable rate constant in Table 3.1, the deep defect
trapping rate K1e. The last entry in the table, Ei(norm), is a vertical scaling factor
stated as the energy at which the calculated curve is normalized to unity.
Table 3.1: Parameters (and their literature references or comments on methods) asused for the calculation of proportionality and light yield in undoped CsI at 295 K.
Parameter Value Units Publ/Est References and Notes
rtrack 3 nm 3,3,2.8 refs [15, 16] for NaI
βEgap 8.9 (eV/e-h)avg 8.9ref [1] CsI 100 K Light Yield112,000 ph/MeV
ε0 5.65 N/A 5.65 refs [55, 56]
µe 8 cm2/Vs 8 ref [48]
De 0.2 cm2/s 0.2 De = µekT/e
µh 10−4 cm2/Vs 10−4 Dh = µhkT/e, ref [18]
Dh 2.6 x 10−6 cm2/s 2.6 x 10−6 refs [18, 38]
DE 2.6 x 10−6 cm2/s 2.6 x 10−6 DSTE ≈ DSTH ref [57]
B(t > τhot) 2.5 x 10−7 cm3/s 2.5 x 10−7 ref [7]
K3 4.5 x 10−29 cm6/s 4.5 x 10−29 ref [15]
R1E 6.7 x 106 s−1 6.7 x 106 ref [39] and eqs. (3.1)and (3.2) for R1E & K1E
K1E 6 x 107 s−1 6 x 107 solve model 662 keV for 2ph/keV ref [51]
τhot 4 ps 4 ref [22]
K2E 0.8 x 10−15 t-1/2cm3s-1/2 0.8 x 10−15 ref [15]
rhot (peak) 50 nm 50 ref [22]
De(t < τhot) 3.1 cm2/s to reproduce rhot(peak) atτhot
S1e 0 s−1 0 zero in undoped host
S1h 0 s−1 0 zero in undoped host
S1E 0 s−1 0 zero in undoped host
GE(r = 0) 0 cm−3 4%Ge → 0 refs [52–54]
K1e 2.7 x 1010 s−1 fitting variable #1, CsI pro-portionality
K1h 0 s−1 10−5K1eratio to K1e based oneq. (3.3)
Ei(norm) 200 keV fitting variable #2 normal-ization
24
The comparison between model and experiment is shown in Figure 3.1 below,
for undoped CsI at room temperature. The open blue triangles (upper curve) are
the experimental Compton-coincidence electron energy response data measured as
described in Chapter 2. As is the custom, the experimental Compton-coincidence
data are normalized to unity at 662 keV. The solid triangular points are the calcu-
lated electron response (proportionality) using the parameters of Table 3.1 in the
Equations (2.1) to (2.7), which reduce to Equations (2.1) to (2.3) for pure CsI.
1 10 100 10000.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
Energy (keV)
Lig
ht
yiel
d
295 K
100 K (Moszynski et al. data)
Figure 3.1: The proportionality curve of electron response modeled by Equa-tions (2.1) to (2.3) from the material parameters listed in Table 3.1 is shown bythe solid triangles, and is superimposed on the Compton-coincidence data for un-doped CsI (SG sample) at 295 K shown by open triangles. Also shown by opensquares is the gamma response experimental curve for undoped CsI at 100 K, to becompared to the model in the next Chapter. The schematic electron track at thebottom (after Vasil’ev [2]), will be used in discussion.
25
3.1.1 Normalization: transition from continuous tracks to separated clus-
ters
The model is in respectable agreement with the experimental data in the energy
range below about 200 keV. In our opinion, the respectable agreement becomes
more impressive when one considers that the other curves shown in Figure 2.1 were
calculated by the same model with parameters that are fairly highly constrained as
we will show later. One also notices that with the choice of the normalization energy
Ei(norm) for good fit at energies below 200 keV, the calculated proportionality
curve slopes decidedly below the experiment at electron energies greater than 200
keV. In Figure 2.1 shown previously, the same is true for CsI (100 K) and CsI:Tl
(295 K), with 200 keV always the normalization energy defining the upper limit of
the range for good fit. A suggestion of what is responsible comes from noticing that
the experimental proportionality curve is nearly flat from 200 keV to 662 keV and
higher for all three CsI data sets and indeed for scintillator materials generally. If all
scintillators have proportionality curves nearly flat between 200 keV and the usual
662 keV normalization energy, then we may as well normalize to unity at 200 keV,
amounting to a concession that our present model based on a cylinder approximation
of the track has a systematic departure from accurate representation of experiment
above 200 keV. The schematic track representation in the lower part of Figure 3.1
illustrates the likely cause. Vasil’ev used a similar track schematic to introduce the
concept that energy deposition occurs in a series of e-h clusters at a spacing that
increases with particle energy, reaching about 100 nm around 662 keV in NaI. [2]
Using a generalized oscillator strength model of the deposition of energy from
a high-energy electron, Vasil’ev describes energy transfers during stopping of the
electron as producing electron-hole clusters of a size that varies somewhat with the
energy of the primary electron but whose mean size is relatively constant in the
26
range of 5 to 6 electron-hole pairs per cluster in the high-energy part of the electron
track [2,16,23]. Thus from cluster to cluster, the mean local excitation density within
a typical cluster is approximately constant over a considerable range of primary
electron energy, and the decreasing energy deposition rate dE/dx with increasing
primary electron energy is then mainly reflected as increasing distance between these
clusters along the track. When the clusters are far enough apart that each acts in
isolation to attract its own dispersed (formerly hot) electrons back to the positive
STH cluster core of their origin, the electron response should approach the ideal
horizontal line of perfect proportionality. As long as they are far enough apart to be
non-interacting, the total light yield of N clusters in a track segment should just be
N times the responses of individual clusters, i.e. proportional. The proportionality
curves for most scintillators, including alkali halides on which we are focusing, do
tend toward a horizontal line at high enough electron energy.
But moving toward lower energy of the primary particle and thus higher average
dE/dx, the spacing of such clusters along the track becomes smaller [2,16]. One can
expect that cooperative effects between the clusters will be manifested. The most
important cooperative effect is that of an emerging line charge of STH clusters, as can
be appreciated looking at the track schematic in Figure 3.1. The 50 nm mean radius
of dispersed hot electrons is illustrated quantitatively by the length of an arrow that
may be compared to the 3 nm radius of the STH distribution around the track, and to
the ∼ 100 nm spacing between clusters typical of 662 keV electron energy in NaI and
CsI. Consider a test charge at 50 nm from a line of positive charges (STH clusters).
If there are multiple positive point charges along a line segment of roughly 50 nm
length, they will all contribute significant radial components to the force on the test
charge. The positive charges (e.g. STH clusters) are then acting cooperatively like a
line charge segment. In classical electrostatics the familiar example is the logarithmic
27
(infinite range) potential of an infinite line charge and the extended range even of
a finite line charge segment compared to that of a point charge or sphere. Even
with screening by an equal number of dispersed electrons balancing the core charge,
Gauss’s law shows that an enhanced electric field of the line charge extends almost
as far as the outer bound of the screening electron distribution.
In the model we have used, significant computational economy was achieved by
neglecting clustering and assuming that with each increase in dE/dx there is an in-
creasing but uniform charge distribution that packs into each cylindrical segment
of constant radius. This was done knowing that at some elevated energy threshold,
separation into clusters will cause the assumption to be inaccurate. What determines
that threshold and what is its value? The touching or overlap of clusters has some-
times been regarded as the condition for the track to resemble a cylinder. That is
based on a visual concept, but not a physical electrostatic charge-collection criterion.
We have calculated the efficiency of electrostatic collection of dispersed electrons in a
radial Gaussian (50 nm mean distribution) toward a line containing an equal number
of immobile positive charges arrayed in clusters of variable spacing. The result was
that as energy decreases and clusters move closer together the collection efficiency
turns upward when the cluster spacing is about 50 nm. From this, we generalize
that when the spacing of immobile (e.g. self-trapped) hole clusters in a line becomes
closer than the mean radius of dispersed (formerly hot) electrons, they begin to act
cooperatively in attracting the thermalized electrons back toward recombination.
The subsequent electrostatic collection after thermalization of distant electrons is
essential for forming excitons in the pure material and obtaining radiative emission.
As will be further noted in discussing the radial distributions, there is a competition
during the electrostatic collection process between the rate of collection back to the
central core where the holes are located and the simultaneous rate of electron capture
28
on deep traps. The rate of electron collection is proportional to the density of holes
on axis (excitation density) and to the electron mobility µe while the trapping is
proportional to capture rate constant K1e and the density of electrons. These two
competing rates, one leading to luminescence within the scintillation gate width and
the other not, determine the slope of the high-energy side of the halide hump. That
slope underlies the whole modeled proportionality curve. A worthwhile experimental
test could be to introduce concentrations of intentional deep defects and measure
proportionality, looking for an effect on the slope.
What the local light yield model is unable to reproduce is the rather sharp con-
cave upward bend from a linear downward slope to a nearly flat high-energy region
as electron energy goes above about 200 keV. We conclude that the concave upward
bend lies outside the rate and transport model itself. It is the cross-over from coop-
erative electrostatics of multiple STH clusters in a line, to independent STH clusters
interacting only with their own electrons dispersed to about 50 nm mean distance.
In Figure 3.1, the model using the parameters in Table 3.1 produces a reasonable
match of the falling experimental electron response from 28 keV to about 200 keV.
Together with the decrease of yield at even lower electron energy due principally to
2nd order nonlinear quenching, the model displays the halide hump that is familiar
from CsI:Tl electron response. Because of the low light yield of undoped CsI at room
temperature the experimental data end before going over the top of the hump, so
that all we see and have available to fit is the slope on its high-energy side.
3.1.2 Population distributions and the luminescence mechanism
To understand what controls the slope of the proportionality plot below 200 keV it
is helpful to examine how the carriers move and interact with themselves and with
traps from the first picoseconds onward. Dependence of the light emission process
29
on excitation density is believed to be the root of intrinsic nonproportionality of
scintillator response, so observing the locations and trapping or recombination status
of carriers and excitons at low and high excitation density can be instructive. In
Figure 3.2, we plot conduction electron density ne, the self-trapped hole density nh,
the self-trapped exciton density N , and the accumulated density ned of electrons
trapped on the assumed deep defect. The time after excitation for each plot is
labeled near the curve. The plotted quantity in all of these radial distributions is
the product of radius r and the carrier density, such as rne(r), to take account of
integrating azimuthally around the track. The gradient along the length of the track
is so small relative to the radial gradient that we can assume no net diffusion along
the length of the track. Thus the integral over radius and azimuth, or area under
these radial plots, should be constant as long as there is no loss from the population
such as by exciton formation, trapping or Auger recombination. The vertical scale
units for rn(r) are expressed in mixed form (units of 1016 nm/cm3) on all of the
radial distribution plots so that division of the plotted rn(r) value by the radius in
nm gives the carrier density (cm−3) at that radius.
30
0 50 100 1500
500
1000
1500
0 50 100 1500
20
40
60
80
100
0 50 100 1500
5
10
15
0 5 10 150
50
100
150
0 5 10 150
1
2
3
4
0 50 100 1500
1
2
3
4
5
0 5 10 150.0
5.0k
10.0k
15.0k
0 5 10 150.0
2.0k
4.0k
6.0k
8.0k
10.0k(b)
20 ps10 ps
4 ps
1 ps
r ne
r (nm)
0 ps(h)
High excitation density
4 ps
r ned
r (nm)
0.05 - 1 ns
20 ps
10 ps
6 ps
10 ps
50 ps100 ps
20 ps
4 ps
1 ps
r ne
r (nm)
0 ps
Low excitation density
(a) (c)
20 ns
10 nsr nh
r (nm)
0 - 1 ns
10 ps
(e)
10 ns
1 ns200 ps
50 psr N
r (nm)
20 ns
(g)
20 ps
50 ps
6 ps10 ps
r ned
r (nm)
0.1 - 1 ns
(d)
20 ns10 ns1 ns200 ps20 ps10 ps
r nh
r (nm)
0 - 4 ps(f)
50 ps
20 ns10 ns
1 ns
200 ps
10 ps
r N
r (nm)
Figure 3.2: Undoped CsI at 295 K. Radial density distributions for low on-axisexcitation density, 1018 e-h/cm3 (lower frames), and 100 x higher on-axis excita-tion density of 1020 e-h/cm3 (upper frames). Plotted are the azimuthally-integrateddensities of conduction electrons rne(r, t), self-trapped holes, rnh(r, t), self-trappedexcitons, rN(r, t), and the accumulated electrons trapped as deep defects, rned. Thetime after excitation for each plot is labeled on the frame near the curve. The verticalscales are in units of 1016 nm/cm3
Figure 3.2 plots four paired figures showing various radial distributions around
the track center for selected times after excitation. The material is undoped CsI at
295 K modeled with the material parameters of Table 3.1. In each vertically arrayed
pair of figures, the lower figure is for on-axis excitation density of 1018 e-h/cm3, a
low value that is encountered in the cylinder track approximation at the beginning
of a high-energy electron track, e.g. 662 keV. The upper figure compares the same
selection of radial plot times for 100 x higher on-axis excitation density of 1020 e-
h/cm3, as encountered toward the end of an electron track where electron energy is
below about 1 keV. Notice that the vertical scale of azimuthally-integrated carrier
31
density (e.g. rne(r, t) and rnh) is generally 100 x larger at 100 x the excitation
density. Exceptions are made when the plotted species shows a big nonlinearity vs
excitation density, such as the excitons, rN , and defect trapped electrons, rned.
In Figure 3.2 frames (a) and (b), we can see the spatial distribution of hot con-
duction electrons expanding rapidly from creation in a 3-nm track at t = 0 ps, past
25-nm mean radius in about 1 ps, and on to the thermalized electron distribution
at 4 ps in agreement with the Monte Carlo simulation results of Wang et al [22]. It
is worth emphasizing at this point the pronounced outward and subsequent inward
migrations of electrons at high excitation density. The outward migration in about 4
ps (in CsI) is driven by excess kinetic energy of the initially formed hot electrons [22].
Its effect, highlighted by comparison of frames (a,b) for electrons with frames (c,d)
for self trapped holes in Figure 3.2, is to separate electrons and holes very rapidly,
suppressing exciton formation, free-carrier Auger decay, and 2nd-order quenching at
least temporarily. But as shown in frames (b) and (f) of the figure, the electrostatic
attraction of the conduction electrons toward the cylinder (∼ line charge) of posi-
tive self-trapped holes asserts itself as the dominant factor at high excitation density
after the hot electrons have lost their excess kinetic energy (Dhot → Dthermalized).
Electrons are then drawn back toward the ∼ 3-nm cylinder of STH where they can
form self-trapped excitons that ultimately emit light in pure CsI. At low excitation
density, on the other hand, deep trapping dominates. The plots in Figure 3.2 show
that for n0 = 1018 e-h/cm3 almost 90% of the original electrons are trapped and
only about 3% form excitons. We have not explicitly included shallow defect traps,
so the electrons that are not captured on the deep defects (rate constant K1e) are
regarded as thermalized conduction electrons. This is the same thermalized elec-
tron population that is labeled “stopped”in the Monte Carlo simulations of Wang et
al [22].
32
The return migration of dispersed conduction electrons back to the STH core is
fast. The peak in exciton formation is reached in 20 ps at high excitation density
when about 60% of the original electrons and holes have formed excitons as seen in
frames (b) and (f) of Figure 3.2. When the on-axis excitation density is high, there is
a denser line charge of positive STH in the 3 nm cylinder, thus a larger electric field
drawing dispersed electrons radially inward, and so faster collection via the third
term in Equation (2.1). The time of exposure of those electrons to deep trapping via
the K1ene term in Equation (2.1) is therefore smaller. Accordingly the total density
of deep-trapped electrons (ned) and the corresponding complement of self-trapped
holes left near the core at the end is a strong function of initial excitation density.
Along with the nonlinear quenching terms that also depend sensitively on these radial
distributions, this accounts for the main part of intrinsic nonproportionality in alkali
halide scintillators.
Self-trapped exciton formation via the Bnenh rate term in Equation (2.1) is set
to zero during the 4-ps hot-electron phase, based on direct observation by picosecond
spectroscopy of STE formation in CsI [7]. According to those experiments, electrons
do not begin to be captured on self-trapped holes until they have thermalized. A
4-ps step function delay was built into our model using the parameter τhot. On
the other hand, picosecond absorption spectroscopy showed that electron capture on
Tl+ commences even during the hot electron phase [7]. This too is contained in the
model for thallium-doped CsI, although a theoretical reason for the different rates
of hot electron capture on STH and Tl+ is not yet in hand. In the absence of ps
absorption spectroscopy on the deep defect electron capture, we have assumed that
it is zero until after electron thermalization similar to capture on STH. Thus because
of the way we have zeroed STE formation and deep defect capture during electron
thermalization, the areas under the electron and hole distribution curves in frames
33
(a) & (c) and (b) & (d) of Figure 3.2 are constant for the first 4 ps.
Going beyond 4 ps, capture on STH and defects commences and it can be seen
in frames (a,b) of Figure 3.2 that the conduction electron population decreases sig-
nificantly in 10 to 50 ps depending on excitation density. The loss of conduction
electrons in this model of undoped CsI is due to capture on STH to form STE and
capture on the deep defects. The capture of conduction electrons on STH, converting
them to STE, can be seen directly as the decrease of STH (nh) population in frames
(c,d) of Figure 3.2 and as the creation of STE (N) in frames (e,f). The comparison
of low and high excitation density in frame pairs (a,b), (c,d), and (e,f) of Figure 3.2
supports the conclusion that the recombination of electrons and holes to create (self-
trapped) excitons in undoped CsI occurs much faster at high excitation density than
at low. This has traditionally been attributed to the fact that the Bnenh rate term is
quadratic in excitation density of electron-hole pairs. As excitation density increases,
exciton formation should become increasingly favored over the linear rate of trapping
on defects as pointed out by Murray and Meyer [58]. But frames (a,b) and (c,d) of
Figure 3.2 show that the electrons are separated spatially from the self-trapped holes
in less than a picosecond during hot electron diffusion, before exciton formation can
occur. So there is more to it than just the bimolecular rate term in a uniform dis-
tribution of electrons and holes. The term that acts to restore overlap of electron
and self-trapped hole populations in undoped CsI is the current of thermalized con-
duction electrons drawn back toward the positive STH track core by the long-range
electric field of this approximate line charge of positive holes. The line charge is
screened by the widely dispersed electrons, but they lie mostly outside the STH core
so that there is a very strong electric field experienced by the electrons lying close
to the core, diminishing farther out as screening charge builds radially. One can see
this effect in Figure 3.2(b), where the distribution of conduction electrons at 10 ps
34
has its most probable radius shifted out from the 50 nm radius of thermalization
at 4 ps to about 80 nm at 10 ps. In the 6 picoseconds after thermalization, more
than half of the dispersed conduction electrons are drawn back to the STH core by
its electric field, and those come predominantly from the ∼ 50 nm range nearest the
core where the screening of the STH core is small. The electrons withdrawn to the
track make the remaining electron population asymmetric with a maximum near 80
nm and a lower population toward the track.
In Chapter 2 describing the rate equations, we commented qualitatively that if
exciton formation occurs at a higher rate for higher excitation density, then compet-
ing nonradiative rate terms in the equations such as trapping on deep defects will
have less time of exposure to the available conduction electrons and so the radiative
yield can be expected to rise. If so, we should see the fraction of conduction electrons
captured on deep traps decrease as excitation density rises. Frames (g) and (h) of
Figure 3.2 plotting the accumulated distribution of electrons on deep traps shows
this effect clearly. The fraction of created electrons that are cumulatively captured
on the deep traps is approximately 5x smaller for on-axis excitation density of 1020
e-h/cm3 than for 1018 e-h/cm3. This is a strong expression of rising light yield as
excitation density increases. At the same time the competing nonlinear quenching
term of dipole-dipole transfer will be quadratically stronger with increasing excita-
tion density.
Commenting further on the trapped electron distribution in frame (g,h) of Fig-
ure 3.2, we see that the radial distribution of trapped electrons for high excitation
density is asymmetric with a maximum near 80 nm and a lower population toward
the track, similar to the skewed conduction electron distribution and for the same
reason. A peak of defect-trapped electrons actually appears in the central track core
for high excitation density because the many electrons drawn there and awaiting
35
capture by STH are also subject to capture by defects in that region.
The distribution of self-trapped holes in frames (c,d) of Figure 3.2 deserves com-
ment on the effect of excitation density as well. We have already noted that the
STH population decreases quickly at high excitation density because their cumula-
tive electric field pulls back dispersed electrons which recombine with the STHs to
form STEs. But because some electrons are trapped on deep defects in undoped
CsI, there will be a corresponding residual population of STH that do not convert
to STE. We can see those in the populations remaining at 10 and 20 ns in frames
(c,d). The radial diffusion of self-trapped holes is significantly enhanced at the higher
excitation density. Because the hot electrons disperse quickly to much larger aver-
age radius than the self-trapped holes, there is a repulsive electric field pushing the
positive STH apart, and the effect can be seen in frame (d). This becomes an im-
portant factor in the nonproportionality of CsI:Tl. The STH diffuse outward and
recombine on a nanosecond with the electrons that were captured on thallium as Tl0
in the first picoseconds in that material. In the presently discussed case of undoped
CsI, a similar process may occur as the STH diffuse outward through the field of
electrons trapped on defects. If that is the source of the 425-nm defect luminescence
in undoped CsI, its lifetime and yield should be related to STH diffusion.
3.2 Thallium-doped CsI at room temperature
Table 3.2 displays the additional rate constants and transport parameters used in
Equations (2.4) to (2.6) for trapped electrons, trapped holes, and trapped excitons in
predicting the proportionality (electron response) of CsI:Tl (0.082 mole%) at room
temperature. All of the parameters used in Equations (2.1) to (2.3) retain their host
parameter values shown in Table 3.1 for the pure CsI host when the model is applied
to CsI:Tl. K1e is the exception because the defect concentrations can change upon
36
doping.
Table 3.2: Additional rate constants and transport properties used in Equations (2.4)to (2.6) when modeling CsI:Tl at 295K.
Parameter Value Units Publ/Est References and Notes
S1e 3.3 x 1011 s−1 3.3 x 1011 ps absorption [7]
S1h 9.9 x 107 s−1 2 x 106 varied ratio scaled from S1eEquation (3.3)
S1E 9.9 x 107 s−1 assumed same as S1h
R1Et 1.7 x 106 s−1 1.7 x 106 Tl∗ lifetime 575 ns from [59]
Uet(eq. (1a)) 7.1 x 105 s−1 7.1 x 105 Tl0 decay time 1.4µs [60]
Det/De 2.2 x 10−6 2.2 x 10−6 scaled by UT l0/S1e factor
µet/µe 2.2 x 10−6 2.2 x 10−6 scaled by UT l0/S1e factor
Btt/B 2.2 x 10−6 2.2 x 10−6 scaled by UT l0/S1e factor
K1et/K1e 2.2 x 10−6 2.2 x 10−6 scaled by UT l0/S1e factor
K1e 2.7 x 1010 s−1 same as CsI
Bet 2.5 x 10−7 cm3/s assumed same as B
Bht 2.5 x 10−7 cm3/s assumed same as B
K2Et 1.4 x 10−14 t-1/2cm3s-1/2 1.7 x 10−15 variable, z scan [15]
The first 3 rates in Table 3.2 are the S1x-type energy storage rate constants
for electron, hole, and exciton capture on Tl+ at the measured concentration of
0.082 mole%. The electron capture rate S1e is the value measured by picosecond
absorption spectroscopy in CsI:Tl at 0.07 mole% (0.3 wt% in melt) [7]. According to
those experiments, electron capture on Tl starts from time zero and proceeds while
the electrons are hot, in contrast to electron capture on STH, which was shown
to exhibit a 4 ps delay of onset when the electron energy started 3 eV above the
conduction band minimum.
37
The capture rate S1h of self-trapped holes on Tl+ to form Tl++ was scaled from
S1e by the velocity ratio implied in Equation (3.3) relating capture cross section and
carrier velocity to the capture rate constant. In this case of electron and hole capture
on Tl+, the concentration of the trap is exactly the same in both cases. The capture
cross sections were hypothesized to be of the same order of magnitude or closer for
electrons and holes since Tl+ is a lattice-neutral trap for both carriers. Following
the earlier discussion of K1e, K1h, the relevant thermal velocities for S1h and S1e are
those of self-trapped holes and conduction electrons, respectively, which are in the
ratio of about 6 x 10−6 at room temperature in CsI. Thus the estimated value is S1h
= 2 x 106 s−1 by scaling from S1e. The value for best fit of proportionality data was
S1h = 9.9 x 107 s−1.
We have chosen S1E (capture of a self-trapped exciton at a Tl+ activator) equal
to S1h for capture of a self-trapped hole on Tl+. The thermal velocities estimated
from jump rate times average jump length for STH and STE in alkali iodides are
about the same. [57,61] The parameter S1E has little effect in fitting proportionality
because the population of STEs free of thallium is very small in Tl-doped CsI, mainly
due to the enormous trapping rate S1e of electrons on thallium as measured in ps
absorption experiments.
Continuing in Table 3.2, the radiative rate of an STE trapped on Tl+, R1Et, is
the reciprocal of the published T+∗ lifetime measured in CsI by direct uv excitation
of Tl [59]. The next 4 parameters in the table are the effective values for electron
diffusion coefficient, electron mobility, bimolecular recombination of electrons from
Tl0 with holes trapped as Tl++ and deep defect trapping of electrons while untrapped
from Tl0. As discussed in Chapter 2, the reason for using these effective time-
averaged transport and capture coefficients to represent the trapped electrons on Tl0
is to avoid the computational expense of handling short time steps for free electrons
38
simultaneously with long time steps for trapped electrons. The trapped-electron
coefficients subscripted with “t”are set as a ratio to the free-electron value of the
corresponding parameter: Det/De, µet/µe, Btt/Bt, K1et/K1e. All of the ratios have
the same value, because all four of the listed parameters with subscript t refer to time
averaged transport or capture of trapped electrons that are inactive and immobile
during most of their existence and can diffuse, move in electric fields, or be captured
on a different site only during the short fraction of time that the carrier is thermally
freed to the conduction band. By this reasoning, the four parameter ratios in the
second grouping in Table 3.2 are described by just one parameter, which is the ratio
of free to trapped electron lifetime in CsI:Tl (0.082 mole%), calculated as follows.
The value for Uet = 7.1 x 105 s−1 in Table 3.2 is the reciprocal of the Tl0 lifetime
at room temperature, 1.4 x 10−6 s, as given in the thesis of S. Gridin [60]. Gridin also
measured thermoluminescence data and deduced the activation energy for electron
release from Tl0 in CsI:Tl as EA = 0.28 eV and the attempt frequency as s = 3.3
x 1010 s−1. [60] These parameters yield a room-temperature untrapping rate Uet =
4.5 x 105 s−1. The rate constant Uet does not appear directly in the rate equations
for reasons discussed in Chapter 2, but the ratio Uet/K1e determines the fraction of
time that an electron trapped as Tl0 spends in the conduction band.
The last 4 parameters in Table 3.2 include the capture rate of conduction electrons
on deep defects, K1e. K1e in Table 3.1 was determined by fitting proportionality in
undoped CsI and is kept at this value in Table 3.2. Next within the last four pa-
rameters, the 2nd order rate constants for recombination of thallium-trapped carriers
with the partner untrapped carrier (Bet and Bht) were assumed to have the same
values as for the corresponding rate constants of free carriers and excitons. Dipole-
dipole quenching of STEs involved in energy transfer and of thallium-trapped STEs
in CsI:Tl are together described by the 2nd order quenching rate constant K2Et which
39
has been measured by laser interband z-scan experiments. [15]
In summary, the parameters in Table 3.2 that were allowed to vary for best fit
are S1h and K2Et. The values of Ei(norm) and K1e for the host crystal in Table 3.1
remain the same as pure CsI. In this sense there are two fitting parameters for CsI:Tl.
We have treated S1h as a fitting parameter, even though its expected value was
estimated based on the discussion of Equation (3.3), as listed in the “publ/est”column
of Table 3.2. The reason for allowing it to vary was that we were unable to obtain
a good fit with the estimated S1h, but noticed that a 30x larger S1h could give a
reasonable fit. The other parameters in Table 3.2 were held at their estimated values
(hence not regarded as fitting parameters), while only S1h and K2Et were allowed to
vary near their estimated / measured values, respectively, for good fit. Among the
parameters held fixed at estimated values was Bet, where the term Betnetnh is the
rate of recombination of STH (nh) with electrons trapped as Tl0 (net). This is one
of the two main fates for STH in CsI:Tl, the other being capture on Tl+ to create
Tl++ at the rate S1hnh. Capture of free electrons by STH is a main term in undoped
CsI, but in CsI:Tl the results in Figure 3.3 show it to be a relatively minor third
channel because there are very few free electrons in the presence of Tl doping. The
summary point is that Bet and S1h are the rate constants governing the main two
competing channels for capture of STH in CsI:Tl. If one of these rate constants is
varied without varying the other, the relative contributions of (STH + Tl0) and (Tl0
released electron + Tl++) will be strongly affected. If both are varied together, the
relative contributions of these two routes for STH capture and eventual Tl* emission
will at least remain in balance.
Figure 3.3 shows with solid diamond points the calculated proportionality curve
(electron response) using the combined parameters of Tables 3.1 and 3.2 in the Equa-
tions (2.1) to (2.7) for 0.082 mole% thallium-doped CsI at room temperature. The
40
model curve is overlaid on the Compton-coincidence experimental proportionality
curve of CsI:Tl (0.082 mole%) at 295 K shown by the open diamonds. The data
for undoped CsI are reproduced in this figure as open triangles for comparison. The
Compton-coincidence measurements for both the CsI and CsI:Tl samples were done
in close succession on the same apparatus as described in Chapter 2. All the model
curves are normalized to unity at 200 keV, for reasons already discussed regarding
the validity ranges of the cylinder and cluster track approximations. The room tem-
perature light yield of CsI:Tl at 662 keV is 54,000 ph/MeV from the Saint-Gobain
data sheet [51]. The model predicts lower absolute light yield of 26,000 ph/MeV at
662 keV (somewhat too low because of the cylinder approximation breakdown) and
about 28,000 ph/MeV at 200 keV.
Comparison of the experimental proportionality curves for undoped and Tl-doped
CsI in Figure 3.3 supports a general conclusion that Tl doping makes the response
more proportional. On the one hand a difference in proportionality should not be
surprising given what the model is demonstrating about the quite different recombi-
nation paths leading to STE emitters and Tl* emitters in the two systems, but on the
other hand finding that CsI:Tl in fact has the flatter proportionality suggests looking
again for a Tl concentration at which the proportionality might be optimized, and for
a careful study of modeled proportionality through the transition from “undoped”to
doped material. This model could be a tool for understanding detailed effects of
activator concentration.
41
1 10 100 10000.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
Energy (keV)
Lig
ht
yiel
d
Figure 3.3: Solid diamonds plot the calculated proportionality curve (electron re-sponse) using the combined parameters of Tables 3.1 and 3.2 for 0.082 mole% thalli-um-doped CsI at room temperature inserted in the model of Equations (2.1) to (2.7).The model curve is overlaid on the Compton-coincidence experimental proportion-ality curve of CsI:Tl (0.082 mole%) at 295 K shown by the open diamonds. Theexperimental data for undoped CsI (295K) are reproduced in this figure by opentriangles for comparison.
3.2.1 Population distributions and the luminescence mechanism
The radial distribution plots from modeling CsI:Tl at 295 K with the parameters
in Table 3.2 are shown in Figures 3.4 to 3.6. The plots that are shown were calcu-
lated for the on-axis excitation density of 1019 e-h/cm3, mid-way on a logarithmic
scale between the high and low densities compared above for undoped samples. Fig-
ure 3.4(a) displays the azimuthally-integrated conduction electron density rne. The
initial distribution in the 3-nm track is seen going off scale vertically on the left at
t = 0 ps. The next distribution at t = 1 ps catches the hot electrons in outward
flight as before. The curve for t = 4 ps is peaked near 50 nm as in the undoped
42
samples, but it is already greatly diminished in area because electron trapping on
the thallium dopant proceeds with a 3 ps time constant measured in the picosecond
absorption experiments discussed earlier. [7] By 10 ps, free conduction electrons are
no longer visible on this plot. Figure 3.4(b) plotting electrons trapped as Tl0 (rne)
shows where the electrons are going; they have been immobilized on this short time
scale in a distribution of Tl0 = net peaking at about 20 nm radius. In contrast, the
dispersed electrons in the undoped crystal remained free until they were pulled back
to the STH core to form STEs, or were trapped in deep defects. This establishes
an important difference between radially dispersed mobile conduction electrons and
radially dispersed trapped electrons on the Tl activator as effective starting points
for recombination in pure CsI versus CsI:Tl.
43
0 50 100 1500
20
40
60
80
100
0 50 100 1500
50
100
150
10 s1 s100 ns15 ns1 ns50 ps10 ps4 ps
(b)
r net
r (nm)
1 ps
(a)
4 ps
1 psr ne
r (nm)
0 ps
Figure 3.4: CsI:Tl 295 K. Radial density distributions for (a) the azimuthally-inte-grated conduction electron density rne(r, t) and (b) the thallium-trapped electrondensity rnet(r, t) both for an original on-axis excitation density of 1019 e-h/cm3.Times after the original excitation are shown in the plots. The vertical scales are inunits of 1016 nm/cm3.
44
0 5 10 15 20 250
20
40
60
80
100
0 5 10 15 20 250
20
40
60
80
100
100 ns
10 s
15 ns
1 sr net
r (nm)
(b)
15 ns
(a)
1 ns
50 ps
10 ps
4 ps1 ps
0 ps
r net
r (nm)
Figure 3.5: CsI:Tl 295 K. Expanded view of the radial Tl-trapped electron densitydistributions rnet(r) shown first in Figure 3.4 but here shown from 0 to 25 nm withcurves divided into two groups, 0 to 15 ns in frame (a) and 15 ns to 10 µs in (b).This is the distribution of electrons trapped by Tl+ dopant to form Tl0. The verticalscales are in units of 1016 nm/cm3.
The distribution of electrons trapped by Tl+ dopant as Tl0 (density net) is shown
in Figure 3.5 with an expanded scale focused on 0 to 25 nm and divided into two
time groups, 0 to 15 ns in frame (a) and 15 ns to 10 µs in (b). Theses plots show
clearly what was stated in the paragraph above, that the peak of the trapped elec-
tron distribution is at 20 nm in contrast to the peak of thermalized electrons at 50
45
nm in undoped CsI. This expresses what was built into the model on the basis of
the picosecond absorption experiments showing that hot electrons are captured on
Tl+, with an exponential trapping time of 3 ps, that is shorter than the conduction
electron thermalization time. Thus some of the hot electrons are captured by Tl+
“in flight”on their way out toward what would have been the 50 nm distance of
thermalization. This has the practical consequence of keeping Tl-trapped electrons
close to the STH core and thus affecting the probability of recombination by STH
diffusion and electron de-trapping from Tl+ in the plots below.
In Figure 3.5(a), one can see the captured electron distributions growing at 1
ps and 4 ps, both starting up from the origin with nonzero slope. Then proceeding
forward from 10 ps through 15 ns, one sees a steep initial slope eating into the
otherwise stable and extended distribution of trapped electrons (Tl0). In the central
core the STH population (nht) and the Tl-trapped electrons (net) intermingle in
the same space. Where the densities are highest trapped electrons and holes are
closest to one another and hole diffusion to create STEs on neighboring trapped
electrons takes less time. The result is that trapped electrons are eliminated first
in the middle and the time to elimination increases as the radius increases. We see
this effect as an increase in the point where the trapped electron distribution begins.
This transformation eliminates trapped electrons from the center outward through
about 1 ns even though the shape of the hole distribution changes little in this time
frame.
Figure 3.6 provides radial plots for additional entities. Comparing the thallium
trapped electrons just viewed to the plot of STH (nh) distribution in Figure 3.6(a)
confirms that to 1 ns the hole population is essentially stationary but between 1 and
15 ns the hole distribution has expanded radially outward and now this progress-
ing front of holes is consuming trapped electrons and converting them to trapped
46
excitons. At any given time in this range, the tail of the STH radial distribution co-
incides with the initial rise of the steep slope in the Tl0 distribution; i.e. the STH are
diffusing out and recombining with Tl0 to create the emitting centers, excited Tl*,
represented as trapped density Nt in Figure 3.6(d). This is a very graphic illustration
of how STH diffusion controls the rate of creation of excited thallium activator in
the nanosecond time range. This rate of creation and the 575 ns lifetime of the Tl*
excited state are expected to bring about the rising part of the fast 650 ns emission
in CsI:Tl. Within 100 ns, the STH population plotted as rnh in Figure 3.6(a) has
decreased to a level indistinguishable from the baseline, consumed both in recombi-
nation with Tl0 and in trapping as Tl++ (nht), shown in Figure 3.6(b).
47
0 5 10 150
25
50
75
0 5 10 150
100
200
300
0 5 10 150
5
10
15
20
25
0 5 10 150
25
50
75
0 5 10 150
500
1000
1500
0 5 10 150
250
500
750
(e)0.1 - 10 s
50 ps
10 ps
15 ns1 ns
(STH
+ T
l0 )r (nm)
0 - 4 ps
(f)10 s
1 s
(Tl0 +
Tl++
)
r (nm)
(c)
1 s
1 ns
r N
r (nm)
10 - 50 ps(d)
10 ps
10 s
50 ps
1 s
0.1 s1 ns
r Nt
r (nm)
15 ns
15 ns
1 ns
r nh
r (nm)
0 - 50 ps(a) (b)100 ns
1 ns
15 ns10 s
r nht
r (nm)
1 s
Figure 3.6: CsI:Tl 295 K. Radial density distributions for (a) rnh self-trapped holes,(b) rnht Tl-trapped holes, (c) rN self-trapped excitons, (d) Nt Tl-trapped excitons,(e) STH +Tl0 self-trapped holes combining with Tl that has already trapped andelectron and (f) Tl0 + Tl++ Tl-trapped holes migrating to combine with Tl0 all foran original excitation density of 1019 e-h/cm3. The vertical scales are in units of 1016
nm/cm3.
48
Once the diffusing STH are gone, the curves for times of 100 ns and longer take
on a different radial profile as seen in Figure 3.5(b). The steep edge softens as the
distribution of Tl0 shifts by de-trapping of electrons from Tl0, diffusing back toward
the origin, drawn there by attraction of the positive track core of Tl++. It can
be seen that the 1 µs curve of Tl-trapped electrons (Tl0) in Figure 3.5 develops
a peak near the track core due to this influence, and then decays substantially up
to 10 µs as recombination of the untrapped Tl0 electrons with the Tl++ occurs to
produce a later stage of Tl* and consequent light. This constitutes the 3-µs decay
time component of CsI:Tl. From the 10-µs curve, one can see that the collection of
de-trapped Tl0 electrons is accelerated for the close ones, where the electric field of
the track core is relatively unscreened, whereas the distant ones represent a growing
proportion of the radial distribution of electrons stored as Tl0. They will contribute
to afterglow or simply find defect traps at longer times. Again, this graphically
associates different radial portions of the hot dispersed and then trapped electrons
with different identified decay time components of the emission in CsI:Tl.
Figure 3.6(a) shows the distribution of self-trapped holes at density rnh. The
population remains constant out to 4 ps because the model prevents electron capture
on STH until after electron thermalization in keeping with picosecond absorption
experiments [7]. It then decreases only slowly for the following nanosecond, because
there are almost no free electrons for recombination, and the STH must diffuse to
find electrons trapped on Tl0 or to become trapped as Tl++. Only a small number
of STH remain at 15 ns and virtually none at 100 ns.
Figure 3.6(b) shows the evolving distribution of Tl++ (density nht). It grows
monotonically up to 100 ns by trapping of STH on Tl+. From the nh plot in frame (a)
we have just seen that the STH population is exhausted at 100 ns. From that time on,
the Tl++ population decreases slowly by recombination with electrons released from
49
Tl0 to produce light emitting Tl*, with about 2/3 of the maximum Tl++ population
remaining at 10 µs. Those will contribute to afterglow or be caught on deeper defect
traps at longer time.
Figure 3.6(c) shows the distribution of self-trapped excitons not associated with
activators. The main comment here is that they are confined to about 4 nm radius
and that the STE population is roughly 3 times smaller than that of Tl*. This does
not mean that STE emission will be 0.3 times the intensity of Tl* emission, because
the STE population is subject to strong thermal quenching. With quenching taken
into account, this STE population will hardly produce any observable luminescence
at room temperature, in agreement with observation.
Figure 3.6(d) shows the evolution in time of the distribution of excitons trapped
on thallium, or simply excited Tl* (density rNt). This is the state mainly responsible
for emission of light in CsI:Tl and can be formed by recombination of STH with Tl0,
capture of STE on Tl+, or capture of an electron (free or released from Tl0) on
T++. We see the earliest distribution of Tl* at 10 ps as a nearly symmetric peak
versus radius. This 10-ps distribution qualifies as “prompt”creation of Tl*. Going
forward in time to 50 ps and 1 ns, the peak becomes asymmetric as the frontier of
STH + Tl0 recombinations moves outward. This is the beginning of the observable
slow rise of the fast 650 ns component of Tl* emission in CsI:Tl. At 15 ns and 100
ns, the addition of Tl* population at the frontier of diffusing STH continues, but
radiative decay of the whole population has also started (with 575 ns lifetime [59]).
Up to this point the radial distribution of Nt (Tl*) results almost entirely from the
recombination of STH with Tl0 as clearly indicated by the shapes of the curves in
Figure 3.6(d,f). Going forward still more to 1 µs and 10 µs, the frontier stops moving
outward because the STH population is exhausted. A peak emerges near the track
core as electrons start to be untrapped from Tl0 on the µs time scale and are drawn
50
in by the positive charge of Tl++ near the track core. The rate of light emission
from Tl* is just R1EtNt, where R1Et = 1/(575 ns) is the radiative decay rate of Tl*.
We will discuss sources and losses of the Tl* (Nt) population below. Separating the
components in time, it is possible to deduce predictions of proportionality of each of
the spatial and decay-time components.
Figure 3.6(e,f) plot two time-integrated source terms contributing to the pop-
ulation Nt (Tl*), specifically the third and fourth terms of Equation (2.6) for the
thallium-trapped exciton population Nt. The solution of Equation (2.6) as plotted in
Figure 3.6(d) includes the effects of loss terms for radiative decay and dipole-dipole
quenching, but one can readily see qualitative correspondence between the radial
distributions in Figure 3.6(e,f) and the several humps in Figure 3.6(d) represent-
ing identifiable physical processes and their spatial locations contributing to the Tl*
excited state that emits light.
As we’ve mentioned, the plot in Figure 3.6(e) can be considered as a source of
the fast (650 ns) scintillation in CsI:Tl, while the graphs for 1 and 10 µs in Fig-
ure 3.6(d,f) indicate the source of the middle (3 µs) decay time component of scintil-
lation in CsI:Tl at room temperature, probably including components of afterglow.
Experiments are available that show for particular doping levels and experimental
conditions that the fast component accounts for about 75% of the light and the 3-µs
component accounts for about 25% of the light [62, 63]. Attempting to reproduce
these relative magnitudes of emission taking into account the Tl* formation rates
and the decay kinetics will constitute a more rigorous additional test of the model
and especially the material parameters that enter it.
The rate of recombination of free conduction electrons with Tl++ according to
the rate term Bhtnenht is found to be negligible because the Tl++ form slowly (see
Figure 3.6(b)) and the free electrons don’t last very long in the presence of Tl+. Also,
51
electron trapping on deep defects becomes nearly negligible when thallium is present,
because the thallium is at higher concentration than typical defects and is a very
good electron trap. The point is that deep electron traps are so overwhelmed by the
efficient and numerous thallium traps in CsI:Tl (0.082%), that they barely become
populated on the scintillation time scale of 10 µs. Their effect on proportionality is
greatly reduced in CsI:Tl compared to undoped CsI.
52
Chapter 4: Temperature dependence
4.1 Temperature dependence of parameters
The mobility of the electron, µe, equals 8 cm2/V s in CsI at room temperature [48]. To
calculate the temperature dependence, we use empirical formulas given by Ahrenkiel
and Brown [3]. They used two formulas to fit the observed Hall mobility µe=µ0
(exp(θ/T)) for KBr and µe=µ0 (exp(θ/T)-1) for KI. There is no clear way to choose
one instead of the other so we use the average of the results coming from these two
formulas. Also we can not directly use the Debye temperature of CsI found in refer-
ences [64, 65] since the Debye temperatures, θKBr=233 K and θKI=222 K [3], used
in fitting experimental results are quite different from values found in handbooks,
θKBr=174 K and θKI=132 K [64,65].
From theory, the main lattice scattering mechanism is due to the interaction of
carriers with the longitudinal-optical phonons, so we can also calculate the electron
mobility using the following equations [66].
µL =e
2αω0m∗(exp(
~ω0
kT− 1)) (4.1)
α = (1
ε∞− 1
ε)
√m∗EH
me~ω0
(4.2)
The parameters can be found in reference [22] and EH=13.6 eV is the first ion-
ization energy of the hydrogen atom.
53
T (K)100 150 200 250 300 350
µe (
cm2/V
s)
0
10
20
30
40
50
60
70CsI Electron Mobility
theorytheory scaled to room TDebye T=93.6 KDebye T=124 K
Figure 4.1: Electron mobility calculated from different ways. Upper two use Equa-tions (4.1) and (4.2). Lower two use empirical Debye temperature methods from [3].
The mobility of the hole, µh, equals 10−4 cm2/V s. [18] From Wang et al [61], the
STH hopping rate is described by an Arrhenius equation.
k = Aexp(−WkBT
) (4.3)
Where W=0.1500 from their previous paper. The hole mobility is proportional to
the hopping rate, so we can use the following equation to calculate hole mobility at
different temperature.
µTh = µT0
h
exp(−WkBT
)
exp( −WkBT0
)(4.4)
We can use Equation (4.3) to scale trapping parameters proportional to the hole
hopping rate like K1h and S1h.
For electron trapping rates, we also need to figure out the electron thermal ve-
54
locity temperature dependence.
mv2e ∝ kBT (4.5)
So we can use the following equation to calculate electron thermal velocity at different
temperature.
ve = v0e
√T
T0(4.6)
4.2 Undoped CsI at 100 K
Table 4.1 presents the parameter values used in the calculation that will be com-
pared to experimental proportionality and light yield data for undoped CsI at 100
K measured by Moszynski et al [1]. The low-temperature electron mobility in CsI is
scaled from the room-temperature mobility of Aduev et al [48] using the formulae of
Ahrenkiel and Brown for temperature-dependent mobility in KBr and KI. [3] In the
non-cryogenic temperature regime where phonon scattering dominates, Ahrenkiel
and Brown showed that the temperature dependence of their measured Hall mobil-
ity fit an exponential expression where the semi-log slope parameter is proportional
to the Debye temperature. Using Debye temperatures for CsI, KI, and KBr along
with the Aduev [48] room-temperature mobility of CsI to scale from the measured
KBr and KI temperature dependences, we estimate µe(100K) ≈ 31 cm2/Vs in CsI.
The thermalized electron diffusion coefficient De is then given in terms of µe by the
Einstein relation.
The diffusion coefficient Dh and mobility µh of self-trapped holes in CsI at 100 K
are calculated from the thermally activated hopping rate following the references and
procedure used for room-temperature values in [37,38,61]. As remarked in regard to
DE in Table 3.1 following [57,61] the hopping rate of the STE is about the same as
for the STH, so DE ≈ Dh = 1.9 x 10−13cm2/s in Table 4.1.
55
The STE radiative emission rate constant R1E at 100 K can be read from the
temperature-dependence of STE decay time plotted by Nishimura et al [39]. Their
temperature dependent luminescence spectra also confirm that the STE emission
becomes nearly pure 3.7-eV band (Type II STE) from 100 K down to about 10
K, so R1E at 100 K is the reciprocal of the 900-ns pure Type II STE lifetime. The
competing nonradiative STE decay rate K1E must be small compared to R1E because
of the plateau in decay time [39] and also the plateau in light yield [39,49,50] that is
reached on cooling to 100 K. A detailed look at the Amsler et al plot of temperature-
dependent light yield [49] shows that the intensity at 100 K is about 3% below
the maximum at 80K. Temperature-dependent light loss can come from temperature
dependence of defect trapping (see below) and thermal diffusion of STEs to quenching
sites as well as thermal quenching of the STE itself (the latter two represented in
K1E). If we assume that all of the light loss is from the K1E rate, its upper limit
implied by the Amsler data is K1E ≈ 3.4 x 104 s−1 at 100 K. In some sense a more
stringent limit is placed by the very large absolute light yield measured at 100 K and
its implication (see discussion of Table 3.1 above) that β is pushed even lower than
1.53 by any light-loss channel. To accommodate the absolute light yield measurement
together with a reasonable β parameter, we have taken the “used”value as K1E = 0
in Table 4.1 for 100 K. The “published”value based on the Amsler et al [49] data is
small as well and makes no noticeable difference in the proportionality curve shape.
56
Table 4.1: Parameters (and literature references or estimation methods) projectedto T = 100 K for use in Equations (2.1) to (2.3) to fit undoped CsI proportionalityand light yield at 100 K. All other parameters needed for Equations (2.1) to (2.3)were kept at their room-temperature value listed in Table 3.1
Parameter Value Units Publ/Est References and Notes
µe 31 cm2/Vs 31 [3, 39]
De 0.27 cm2/s 0.27 D = µkT/e
µh 2.2 x 10−11 cm2/Vs 2.2 x 10−11 [37, 38] eval. 100 K
Dh 1.9 x 10−13 cm2/s 1.9 x 10−13 D = µkT/e
DE 1.9 x 10−13 cm2/s 1.9 x 10−13 DSTE ≈ DSTH [57, 61]
R1E 1.1 x 106 s−1 1.1 x 106 [39]
K1E 0 s−1 < 3 x 104 STE thermal quench 100 K[49]
K2E 1.3 x 10−16 t-1/2cm3s-1/2 1.3 x 10−16 scale from z scan [15],R1E(100)/R1E(295) = 0.164
K1e 1.3 x 109 s−1
Ei(norm) 200 keV cluster spatial distribution isthe same as at 295 K
K3 4.5 x 10−29 cm6/s charge separation limitsAuger recombination
57
1 10 100 10000.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
Energy (keV)
Lig
ht
yiel
d
Figure 4.2: The solid square points show the calculated proportionality curve (elec-tron response) at 100 K using the-low temperature parameters of Table 4.1 along withbalance of parameters kept unchanged from Table 3.1 as discussed in the text. Themodel curve is overlaid on the experimental gamma yield spectra (open squares)of proportionality in undoped high-purity CsI (sample B) at 100 K measured byMoszynski et al [1]. The data of Figure 3.1 for undoped CsI(SG) at room tempera-ture are shown as open triangles for comparison.
The dipole-dipole quenching rate constant K2E describes losses of self-trapped
excitons that interact for quite some time at close quarters in CsI at 100 K after
electrons thermalize and regather at the line of holes which persists on the track.
Because of the resulting importance of dipole-dipole quenching at low temperature,
it is prudent to understand if a temperature-dependent trend of K2E can be esti-
mated. The dipole-dipole quenching rate constant K2E depends on the square of
the dipole matrix element and the spectral overlap of the emission and absorption
line shapes [67]. We do not yet have complete enough data on the temperature-
dependent spectra of STE absorption and emission in CsI to calculate the overlap
variation, although ps absorption spectroscopy toward this goal is underway in our
58
laboratory. The other temperature dependent factor, change of the magnitude of
the dipole matrix element for emission, can be estimated for CsI from the temper-
ature dependence of the radiative emission rate R1E. The radiative rate constant
in simple excited states is usually independent of temperature. But Nishimura et
al [39] have shown that the STE in CsI has communicating populations of on-center
and off-center STE [33], that come into thermal equilibrium above 250 K with a sin-
gle effective lifetime dominated by the fast on-center STE radiative rate. The ratio
R1E(100 K) / R1E(295 K) = 0.164, is adopted as the approximate scaling factor for
K2E(100 K) relative to K2E(295 K).
The deep defect trapping rate constant K1e was a fitting variable for the 295
K data, but for the modeling of 100 K data we have scaled K1e(100 K) from the
room-temperature experiment as follows: Best fit of undoped CsI at 295 K yielded
the parameter value K1e ≈ 2.7 x 1010 s−1. There are two reasons that the parameter
K1e can be expected to decrease for the modeling of the Moszynski et al. [1] sample
B at 100 K. One reason is that the sample B seems to have had remarkably low
concentration of defects. If we only take into account the 425 nm defect luminescence
band in Figure 2.2 as an indicator of defect concentration in the SGC sample relative
to Moszynski sample B, the SGC sample has about 30% higher defect concentration.
The 550 nm band is harder to use for comparison because the photodetectors used in
the two studies had different red sensitivity. Based on a comparison of fast to total
ratios we have used an estimated factor of 2.4 more total deep defects in the SGC
sample relative to Moszynski sample B.
In addition, there can be a temperature dependence of the capture rate constant
K1e. The first-principles calculation of carrier capture rate versus temperature by
Alkauskas, Yan, and Van de Walle [68] was applied in wide-gap semiconductor sys-
tems such as hole capture on the negatively charged center CN in GaN. In that case
59
since the capturing center is coulombically attractive to the carrier, the calculated
capture rate attains a minimum value near 100 K and then rises by about a factor
4 as the temperature rises to 295 K. If the defect center is neutral instead, their
results for GaN indicated that the temperature dependence is larger, increasing by a
factor of approximately 9 from 100 K up to 295 K. Calculations of the temperature-
dependent capture rate of electrons on F and F+ iodine vacancy centers in CsI by
the same method, as well as capture on self-trapped holes and on Tl+ ions in CsI,
are in progress. However since definite results on CsI have not been obtained yet,
we have reasoned by analogy to the temperature dependence of carrier capture in
GaN that the upper range of expected reduction of K1e on going from SGC undoped
CsI (295 K) to Moszynski B undoped CsI (100 K) could be a factor 1/(2.4 x 9) =
1/21 allowing for both sample-dependence and temperature-dependence. With this
scaling factor, we estimate K1e = 1.3 x 109 s−1 for the electron capture rate on deep
defects in Table 4.1 for modeling Moszynski sample B at 100 K.
The last two parameters in Table 4.1 are assigned the same values used at 295
K. Most such parameters are not listed in Table 4.1, but these two are worthy of
comment on why they are left the same. As we have noted, experimental propor-
tionality curves are usually normalized to unity at 662 keV, whereas the energy at
which this model is normalized is another parameter, Ei(norm). If indeed Ei(norm)
marks the approximate transition from independent STH clusters to cooperative
STH clusters attracting electrons from their far-flung locations reached at the end
of thermalization, it should not change much with temperature or doping since the
spacing of energy deposition clusters should not be strongly dependent on either of
those variables. To test this hypothesis, we set Ei(norm) fixed at the same 200 keV
value already determined by fitting undoped CsI at 295 K.
Theoretically, the Auger rate constant K3 is also temperature dependent, gener-
60
ally decreasing at low temperature along with the occupation of phonons that can
participate in indirect Auger transitions. [36]. Considering the comparison of di-
rect and phonon-assisted Auger rates in the work of McAllister et al [36] on NaI,
and basing relative populations of zone-boundary phonons at 100 K and 295 K on
a Debye temperature model similar to that used in Ref. [3], we estimate from the
direct/indirect ratio of Ref. [36] that the Auger rate constant in NaI would decrease
by a factor 4 at most on changing temperature from 295 K to 100 K. As can be seen
in the radial distribution plots, the present transport model shows that the spatial
separation of hot free electrons from self-trapped holes is so rapid (<1 ps) that the
practical importance of free-carrier Auger recombination is very limited. To avoid
unnecessary complexity in this early testing of temperature dependence in the model,
and recognizing that there remains at present an order-of-magnitude disagreement
between theoretical [36] and experimental [15] values of the Auger rate constant in
NaI, we have not attempted to predict the change of K3 with temperature. In Table
4.1 we assign it the room-temperature value measured in [15].
Figure 4.2 compares the calculated proportionality curve (electron response) at
100 K, shown by solid square points, overlaid on the experimental gamma yield
spectra of proportionality in undoped high-purity CsI (sample B) at 100 K measured
by Moszynski et al [1] (open squares).
We can see that the model has shifted from approximate match of the upward
trending data at 295 K to a surprisingly good match with the downward trending data
at 100 K in the applicability range below 200 keV. All material parameters were either
scaled by physical arguments for temperature and the sample defect content relative
to the 295 K experiment & model, or were kept at the 295 K values in a hypothesis
that some parameters do not have a big impact by their temperature dependence.
The fit appears too good in the sense that perfect overlap of calculated electron
61
response and measured gamma response is not expected. It is generally found that
gamma response proportionality curves resemble the shape of corresponding electron
response curves for a given material and conditions, but the gamma response curve
appears as if shifted to higher energy on the horizontal scale of (logarithmic) energy.
The experimental data in Figure 4.2 make a nearly horizontal line above about
60 keV. The model actually introduces a downward slope beginning above about 100
keV and falling distinctly below the data above the 200 keV normalization point.
As noted previously, we believe that this is an artifact from applying the cylinder
approximation in the model at energies above 200 keV where the discontinuous de-
posits of excitation clusters begin to act independently of one another in regard to
long-range collection of electrons back toward self-trapped holes.
4.2.1 Population distributions and the luminescence mechanism
Using the same format established in the previous subsection, Figure 4.3 plots radial
distributions at specified times for paired low and high excitation densities of 1018 e-
h/cm3 and 1020 e-h/cm3 on-axis for modeled undoped CsI at 100 K with the material
parameters of Table 4.1.
62
0 50 100 1500
5
10
15
0 5 10 150
50
100
150
0 5 10 150
10
20
30
40
50
0 50 100 1500.0
0.2
0.4
0.6
0.8
1.0
0 50 100 1500
500
1000
1500
0 5 10 150.0
5.0k
10.0k
15.0k
0 5 10 150.0
5.0k
10.0k
15.0k
0 50 100 1500
1
2
3
4
5
200 ps50 ps
10 ps4 ps
1 ps
r ne
r (nm)
0 ps(a) (c)
1 - 20 ns200 ps
50 ps
r nh
r (nm)
0 - 20 ps(e)
100 ps
20 ps
1 ns
200 ps
50 ps
r N
r (nm)
0 - 10 ps
(g)
Low excitation density
100 ps
1 ns
200 ps
50 ps
r ned
r (nm)
0 - 20 ps
(b)
High excitation density
6 ps 20ps10 ps
4 ps
1 ps
r ne
r (nm)
0 ps(d)
20 ps10 ps6 ps
20 ns
r nh
r (nm)
0 - 4 ps(f)
10 ns
6 ps10 ps
1 ns200 ps50 ps
r N
r (nm)
0 - 4 ps
(h)
0.1 -1 ns20 ps10 ps6 ps
r ned
r (nm)
0 - 4 ps
Figure 4.3: Undoped CsI at 100 K. Radial density distributions for low on-axisexcitation density, 1018 e-h/cm3 (lower frames), and 100 x higher on-axis excita-tion density of 1020 e-h/cm3 (upper frames). Plotted are the azimuthally-integrateddensities of conduction electrons rne(r, t), self-trapped holes, rnh(r, t), self-trappedexcitons, rN(r, t), and the accumulated electrons trapped as deep defects, rned. Thetime after excitation for each plot is labeled on the frame near the curve. The verticalscales are in units of 1016 nm/cm3
The high density frame (b) for electrons in Figure 4.3 at 100 K looks much like
the corresponding high density frame (b) of Figure 4.3 at 295 K. The 10 ps curve for
conduction electron distribution is about a factor of two lower at 100 K and the small
peak of conduction electrons overlapping the STH core at 3 nm is about a factor of
two higher at 100 K, all suggesting qualitatively that there is faster collection of
conduction electrons in the field of the STH due to the higher electron mobility at
low temperature. The conduction electron population decreases due both to capture
on STH and capture on defects. Compare frames (g,h) of the two figures showing
cumulative distribution of electrons trapped on defects at 100 K with those at 295 K.
63
A dramatically smaller fraction of electrons is captured on deep defects at low tem-
perature for high density. (Notice that the vertical scales of (g) and (h) in Figure 4.3
are in a much different ratio than the factor of 100 that should be expected if defect
trapping were simply proportional to e-h excitation. The azimuthally-integrated
distributions of defect-trapped electrons are seen to be small fractions of the e-h
distributions at 100 K. The reason for less defect trapping is shared by the higher
mobility of electrons and the lower cross sections of defects as discussed in connection
with Table 4.1.) At low excitation density the light yield is correspondingly enhanced
at 100 K by this effect in addition to less thermal quenching of the STE. The rate
equations express competitions between various terms, and faster rates dominate the
yield.
Figure 4.3(a) for low density excitation at 100 K does not appear even qualita-
tively similar to the corresponding frame of Figure 3.2 at 295 K after about 10 ps.
With fewer positive STH on axis to pull in the dispersed electrons, the electrons have
more time to diffuse in an electric field of the STH core that is evidently of marginal
importance in influencing the direction of diffusion. Because of the higher diffusion
coefficient at 100 K, a substantial number of the electrons simply escape to larger
radius as shown in Figure 4.3(a), and become trapped there as seen in frame (g).
Noting the vertical scale factors, comparison of the number of trapped electrons in
frames (g) of Figure 4.3 and 3.2 shows that even at low excitation density the success
of exciton formation versus trapping on defects is improved at 100 K relative to 295
K in undoped CsI.
Comparison of frames (c,d) of Figure 4.3 and 3.2 for self-trapped holes at the two
temperatures shows as expected that STH diffusion and the electric field enhance-
ment of it at high excitation density are both negligible at 100 K. Comparison of
the 10 ps curves at low and high density in 4.3(e,f) confirms visually that the STH
64
convert more rapidly to STE at high density than at low by drawing in the dispersed
electrons more quickly and then having a higher bimolecular rate of electron-hole
recombination. Conversely, it is the lower rate of this electron attraction and con-
version to STE at low excitation density which allowed the diffusion of thermalized
electrons to large radius in Figure 4.3(a). At high excitation density in frame (f), the
STE population has reached its maximum at 10 ps and thereafter begins to decay.
This enhanced decay rate at high density may be attributed mainly to dipole-dipole
quenching in the 10-100 picosecond time range. As the figure shows, the recon-
structed STE are tightly confined to the initial track radius because that is where
the STH reside, and their diffusion is very slow at 100 K. Even though the elec-
trons were dispersed to large radius initially while hot electrons, they return very
quickly (10 ps) to reconstruct the original track as shown. Despite wide dispersal
of the hot electrons, the STE finally form at the STH which retain the memory of
the initial track. The high density of excitons in this quickly reconstructed track
leads to enhanced dipole-dipole quenching at low temperature, which one can see in
the experimental data and in the modeled proportionality. The enhancement of the
amount of dipole-dipole quenching relative to what should be expected at higher tem-
perature has at least two origins. The electrons at low temperature survive against
trapping better and so create a higher density of STEs when captured in the track
core, and the STEs live longer at low temperature and so experience more nonlinear
quenching.
65
Chapter 5: Energy-dependent scintillation pulse shape
and proportionality of decay components in CsI:Tl
5.1 Pulse shape and its energy dependence
5.1.1 Experimental data
Syntfeld-Kazuch et al [4] measured pulse shape of CsI:Tl (0.06%) excited at several
gamma-ray energies between 662 keV and 6 keV using a so called slow-slow single-
photon method described in [1] and [69], tailored to reduce the background of random
coincidences. With this method they were able to resolve a ”tail” decay component
of about 16 µs in addition to the ”fast” and ”slow” components of 730 ns and 3.1
µs reported in prior studies of CsI:Tl scintillation decay [59, 63]. The pulse shapes
for 662 keV and 6 keV gamma excitation measured in [4] are plotted in Figure 5.1.
The 16 µs tail accounts for about 22% of the integrated pulse for 662 keV excitation,
compared to 48% and 30% for the fast and slow components respectively [4]. Decay
curves of scintillation from 16.6, 60, 122, and 320 keV gamma rays were also measured
in [4], reported in terms of fitt ed exponentials and their amplitudes.
66
0 1 0 2 0 3 0 4 01
1 0
1 0 0
1 0 0 0
1 0 0 0 0
0 1 2 31 0 0
1 0 2
1 0 4 6 6 2 k e V 6 k e V
Coun
ts
T i m e ( µs )
Coun
ts
T i m e ( µs )
Figure 5.1: Experimental pulse rise and decay over the full measured range 0 to 40µs in CsI:Tl from [4] is shown for 662 keV gamma excitation in the red trace and for6 keV gamma excitation in the lower blue trace.
The decay times of 730 ±30 ns, 3.1 ±0.2 µs, and 16 ±1 µs were determined by
fitting the decay curve for 662 keV excitation [4]. The corresponding decay times
under 6 keV excitation were 670±20 ns, 3.1±0.3 µs and 14±3 µs, indicating that the
fast and tail decay times decrease slightly with decreasing gamma energy. Noting
that there is only a weak dependence of the fitted decay times upon gamma ray
energy, the authors of [4] presented data on how the relative amplitudes (integrated
intensities) of the fast, slow, and tail decay components changed in six gamma energy
steps from 662 keV to 6 keV. A main conclusion of their study was that the fast/tail
ratio increases as gamma excitation energy is lowered. Fitting the observed pulse
shape as a function of energy and understanding the physical origins of the decay
components are among our objectives in this work.
The data in Figure 5.1 are a multichannel analyzer record of times from start
to single photon stop events which samples the scintillation lifetime and is adjusted
with delays to put the start-time for the pulse on scale. This means that the data
67
records shown in Figure 5.1 show the rising portion of the curve within the stated
20-ns experimental resolution in [4]. However, the measurement method itself does
not specify a time zero. The rise to a peak and initial decay out to 2.5 µs are
shown on an expanded time scale in the inset of Figure 5.1. Syntfeld-Kazuch et al
normalized their data at the peak for display and we will follow their lead in making
comparisons to the model. The time zero in the model is definite, corresponding
to the initial energy deposition, and can be read from the model curve matched to
the experiment curve at its peak. The curves in both the main figure and the inset
of Figure 5.1 are normalized and presented with the peak intensities coinciding in
time and amplitude. The red and blue traces in the inset are for 662 keV and 6
keV excitation measured by Syntfeld-Kazuch et al in an experiment optimized for
weak signals at long times [1,4,69]. We have also examined 511 keV excitation data
measured by Valentine et al [63] in an experiment optimized for fast response at
expense of resolving slow, weak signals.
Many previous studies [35, 38, 59, 63] have associated the ∼700 ns fast decay
mainly with the reaction STH + Tl0, and the ∼3 µs slow decay with electrons
thermally released from Tl0 recombining with Tl++ that were formed by STH capture
at Tl+ dopants. This association of the two main physical recombination routes
involving STH and Tl++ respectively with the decay components seemed complete
when there were only two decay components known experimentally (other than what
was considered afterglow). With three identifiable decay components having roughly
similar integrated strengths now known [4], an assessment of the responsible physical
mechanisms seems in order.
Our underlying model accounts for trapping of both electrons and holes by Tl+
in the lattice. Carriers created by high-energy radiation are initially hot, with excess
kinetic energy. As a result, electrons created in CsI spread quickly to a mean radius
68
of about 50 nm (extending as far as 200 nm) [22], and are trapped with a 1/e time of
∼3 ps [6,7] by Tl+ to form Tl0. Self-trapping of the co-produced holes is commonly
presumed to localize them initially at the original track. The transport and recom-
bination kinetics of these self-trapped holes (STH) with the Tl-trapped electrons
(denoted Tl0) initially governs the formation of excited Tl+∗ that are responsible for
scintillation light. Together with the Tl+∗ radiative lifetime of 575 ns, these trans-
port and recombination kinetics determine the finite rise time and the 730-ns fast
decay time of scintillation in CsI:Tl. In parallel, the STH are competitively trapped
on Tl+, accumulating a population of Tl-trapped holes (Tl++) until all STH are ex-
hausted by these two channels. With appropriate proximity and thermal untrapping
of electrons from Tl0, this Tl++ population, considered deeply trapped and immo-
bile, recombines with the electrons released from Tl0 to produce Tl+∗ at longer times.
For these calculations we adopt the conventional assumption that it is the electron
that is untrapping from Tl0. An alternative suggestion that Tl0 is a deep electron
trap [70–72] and instead holes untrap thermally from Tl++ to recombine with the
static Tl0 has been made [72].
5.1.2 Fitting rise and decay times
The 662 keV pulse shape data reported in [4] is reproduced by the red trace with
noise in Figure 5.2(a,b). The smooth curve superimposed shows the simulation of
Tl+∗ emission calculated by Equations (2.1) to (2.7) with material input parameters
to be tabulated and discussed later.
69
0 1 2 3 4 51 0 0
1 0 0 0
1 0 0 0 0 6 6 2 k e V E x p e r i m e n t 6 6 2 k e V M o d e l e x p o n e n t i a l , τ = 5 7 5 n s
Coun
ts
T i m e ( µs )(a)
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01
1 0
1 0 0
1 0 0 0
1 0 0 0 0 6 6 2 k e V E x p e r i m e n t 6 6 2 k e V M o d e l
Coun
ts
T i m e ( µs )(b)
Figure 5.2: Experimental scintillation decay curve from [4] for 662 keV gamma exci-tation shown in red trace with noise on (a) 0 to 5 µs time scale and (b) 0 to 40 µsscale. In both cases the superimposed smooth black line is the modeled light outputfor 662 keV excitation. Model is normalized to experiment at the peak.
The experimental data and model calculation for 662 keV excitation are shown on
a log scale versus linear time out to 5 microseconds in Figure 5.2(a). Superimposed
70
is a 575 ns exponential representing the radiative decay time measured for uv-excited
Tl+∗ photoluminescence in CsI:Tl by Hamada et al [59]. The observed scintillation
decay is slower than 575 ns, as can be seen in the figure and as many others have
observed. As discussed by previous authors [35, 59, 63, 73], the main mechanism for
formation of Tl+∗ in the first 200 ns or so is the hopping diffusion and capture of
self-trapped holes (STH) on Tl0 sites formed much earlier by rapid electron capture
on Tl+. For the scintillation decay to be longer than the 575 ns radiative lifetime,
the excited Tl population should be fed while it also undergoes radiative decay. This
formation process also accounts for the initial rise characteristics.
In part (b) of Figure 5.2, the model calculation with the same parameters is
compared to the full range of the experimental 662 keV scintillation decay out to
40 µs. In [4], raw data with noise and the rise to a peak before decay were shown
only for the 662 and 6 keV gamma energies. However, decay data for six gamma
energies of 662, 350, 122, 60, 16.6, and 6 keV were reported as a set of three fitted
exponential decay times and the amplitude of each [4]. We have reconstructed the
decay curves shown in Figure 5.3 from the experimentally determined decay constants
and amplitudes reported in [4]. The experimental curves for 662, 350, and 122 keV
overlap, so what is seen in Figure 5.3 is a single curve labeled 122-662 keV at the
top, with three curves below it for 60, 16.6, and 6 keV respectively. There is no
representation of the early rise to a peak. These reconstructed decay curves are
normalized to a value 105 at t = 150 ns, which corresponds to the peak of the
intensity curve in the model results.
71
0 1 0 2 0 3 0 4 01 0 2
1 0 3
1 0 4
1 0 5 1 2 2 - 6 6 2 k e V 6 0 k e V 1 6 . 6 k e V 6 k e V
Inten
sity (
a.u.)
T i m e ( µs )
Figure 5.3: Reconstructions of measured scintillation decay curves for 6 gamma-rayenergies in CsI:Tl(0.06%) based on the time constants and integrated amplitudesreported in [4]. Only the decay curves are represented. The curves for 122, 320, and662 keV overlap in the top curve.
0 1 0 2 0 3 0 4 01 0 2
1 0 3
1 0 4
1 0 5
6 6 2 k e V 3 5 0 k e V 1 2 3 k e V 6 0 k e V 1 6 . 6 k e V 6 k e V
Inten
sity (
a.u.)
T i m e ( µs )
Figure 5.4: Decay curves calculated from the model for six electron energies of thesame values as the gamma energies of the reconstructed experimental decay curvesin Figure 5.3.
72
The modeled light output curves are shown in Figure 5.4 for gamma energies of
662, 350, 122, 60, 16.6, and 6 keV. The curves are normalized to 105 at 150 ns, the
peak of light output. We did not display all of the experimental and modeled curves
in a single figure because it would be hard to distinguish them. Except for the 6
keV curve, the pulse shape is similar in model and experiment. The common trend
of increasing peak/tail ratio with decreasing energy is displayed in both.
There is disagreement in the amount of tail amplitude change (relative to the
peak of the fast component amplitude) between model and experiment at the lowest
excitation energy, 6 keV. In the model, the depression of the relative tail amplitude
continues at a rate consistent with the trend at higher energies, but the tail amplitude
in experiment drops a great deal more from 16.6 to 6 keV than at any other energy
intervals, and thus differs from the model. One possible reason for the disagreement
between experiment and model at very low gamma energy is a known difficulty of
light yield and decay time studies when the excitation occurs near the surface. A 6-
keV gamma- or x-ray has an attenuation length of about 3.6 µm in CsI. This is within
the range of the surface in which quenching effects have often been reported [39,74].
Such effects could be expected to affect the long tail of excitation decay more severely
than the fast component because a longer time interval allows more diffusion toward
the surface and quenching to occur. The amplitude of the tail could be decreased and
its apparent decay time shortened because of the competing channel for de-excitation
presented by quenching centers near the surface.
The model curves shown in Figure 5.2 reach a peak at about 200 ns, matching the
measurements in [75] which used methods chosen to reduce random coincidences [69]
at some sacrifice of rise time resolution. Rise and peaking data for CsI:Tl were
reported by Valentine et al. with better time resolution [63]. Their data include
an ultrafast component that can be aligned with the rise of the calculated STE
73
emission in the model curve. Comparison on that basis indicates that the experiment
reaches its peak 50 to 100 ns sooner than the results fitted to the Syntfeld-Kazuch
et al measurement [4]. Hamada et al. report decay data that for samples with
Tl content from 10−6 to 10−2 and their rise time ranges from 100 to 185 ns. As
commented earlier, these alternate fast-time data sets do not include decay data to
long times (20-40 µs), so we have concentrated on fitting the full set of data from
Syntfeld-Kazuch et al [4]. The contribution of STE luminescence is not plotted in
Figure 5.2. The STE contribution to luminescence at room temperature is very small
compared to the Tl+∗ emission except in details of the initial rise that occur within
the 20-ns resolution of the experimental data now being compared. It is known
from experiment that the STE in CsI at room temperature is thermally quenched to
about 2% yield coming as 15 ns emission from an equilibrated Type I/Type II STE
configuration [39].
5.2 Nonproportionality of each decay component – experi-
mental data and model results
The proportionality curves for the fast, tail, and total decay components in CsI:Tl(0.06%)
were reported by Syntfeld-Kazuch et al [5] in 2014, following the method developed
in [76] and are replotted below in Figure 5.5(a). The experimental proportionality
curves for decay components in Figure 5.5(a) were determined by fitting the mea-
sured decay curve at each gamma energy to 730 ns, 3 µs, and 16 µs components
and plotting the integrated light yield in each component versus gamma energy,
normalized at 662 keV.
74
1 0 1 0 0 1 0 0 00 . 6
0 . 8
1 . 0
1 . 2
1 . 4
F a s t ( τ1 = 7 3 0 n s ) T o t a l T a i l ( τ3 = 1 6 µ s )
Norm
alized
Ligh
t Yield
E n e r g y ( k e V )(a)
1 0 1 0 0 1 0 0 00 . 6
0 . 8
1 . 0
1 . 2
1 . 4 F a s t ( 0 - 7 5 0 n s ) T o t a l ( 0 - 4 0 u s ) T a i l ( 3 - 4 0 u s )
Norm
alized
Ligh
t Yield
E n e r g y ( k e V )(b)
Figure 5.5: (a) Experimental proportionality curves for the fast (0.73 µs) and tail (16µs) decay components as well as the proportionality of total emission (Fast + Slow (τ2= 3 µs) + Tail) in CsI:Tl are plotted versus gamma ray energy. Reproduced from [5].(b) Simulated proportionality curves for fast, total, and tail decay components inCsI:Tl calculated with the same model and parameter set used for Figure 5.2 andFigure 5.4. The integration gate intervals for Fast, Total, and Tail are given in thelegend. Model curves are normalized at 200 keV for reasons discussed in [6].
75
Figure 5.5(b) plots the calculated proportionality curves for the fast (730 ns) and
tail (16 µs) components as well as the total pulse proportionality in CsI:Tl(0.06%) us-
ing the model of Equations (2.1) to (2.7) [6] and the same parameters that produced
the preceding fits of the pulse shape data. To emphasize, the calculated proportion-
ality curves in Figure 5.5(b) came directly out of the model with its parameter set
refined to give good fits to the rise and decay data, without any further fitting to
reproduce the proportionality curves of separate decay components.
The approximate proportionality curves for specified decay times in Figure 5.5(b)
were calculated from the model output in the following way. The integrated light
emission from 0 to 750 ns was plotted as ”fast” light yield versus electron energy;
from 750 ns to 3 µs as ”slow” light yield versus energy; from 3 µs to 40 µs as
”tail” light yield versus electron energy; and 0 to 40 µs as the total light yield
versus energy. We are comparing proportionalities of decay components calculated
by a simulated gating time method with experimental proportionalities of decay
components analyzed as integrated strengths of three fitted exponential components.
There should be qualitative and reasonable quantitative correspondence between the
two methods. Qualitative correspondence is what we are pointing out in Figure 5.5.
The modeled proportionality curves were normalized at 200 keV for reason discussed
in [6]. It is a consequence of the approximate energy range over which the cylinder
approximation of the track is valid.
5.3 Origin of three decay components of scintillation in CsI:Tl
Two particularly intriguing questions are posed by the experimental observations:
(a) To what can the three main decay components of 0.73 µs, 3.1 µs, and 16 µs in
CsI:Tl be attributed? (b) To what can the different proportionality curves for the
three decay components, particularly the anticorrelation of fast and tail components,
76
be attributed? We address the origin of the three decay times first.
5.3.1 Recombination reactions resulting in Tl+∗ light emission in CsI:Tl
Beginning with Dietrich and Murray [35] and in many works since [38,63], four main
recombination processes have been considered to contribute to CsI:Tl scintillation.
Reaction #1 comprises direct Tl+ excitation and/or prompt electron and free hole
capture on Tl+ to contribute a promptly rising signal that should decay at the 575
ns radiative lifetime of Tl+∗. However this pure 575 ns decay is rarely distinguish-
able against the stronger 730 ns ”fast component” of CsI:Tl scintillation commonly
attributed to Reaction #2 – self-trapped holes recombining with electrons on Tl0,
STH + T l0 → T l+∗. Reactions #1 and #2 together should contribute to the ob-
served fast scintillation decay, with R#2 dominant, because self-trapping of holes is
very fast, and hot electrons disperse and form Tl0 mostly separated from the STH
in the track core.
The generally accepted Reaction #3 in CsI:Tl is the thermal release of electrons
trapped early in the track formation as Tl0, followed by diffusion through repeated
release and recapture until recombination with a hole trapped as Tl++ [6, 35, 38, 58,
59,63]. For a shorthand label, we write this as reaction #3: T l0+T l++ → T l+∗. This
reaction has been considered responsible for the single observed slow component of
roughly 3 µs seen in works prior to [4], e.g. as reviewed by Valentine et al [63]. Kerisit
et al noted that the 3 µs time range brackets the decay time of Tl0 due to electron
release at room temperature, ∼1.8 µs [77], and associated the single time constant
τe (for electron release from Tl0) with the scintillation decay time of approximately
3 µs [38].
Reaction #4 involving Tl in alkali halide scintillators is STE + T l+ → T l+∗,
i.e. self-trapped excitons migrating to encounter substitutional Tl+ and transferring
77
their excitation to create Tl+∗. Murray and Meyer in 1961 [58] suggested this STE
reaction channel as having main responsibility for scintillation in NaI:Tl. However,
following subsequent time-resolved kinetic studies on KI:Tl with partial extension to
NaI:Tl, Dietrich et al concluded in 1973 that ” . . . nearly all (≈95%) of the energy
transport takes place by electron-hole diffusion.” [35]. The computational model used
herein takes into account very rapid spatial separation of hot electrons relative to self-
trapped holes [22] and the effect of rapid electron trapping on Tl+ directly measured
by ps spectroscopy [7]. These effects hinder the formation of STEs in Tl-doped CsI,
relative to undoped CsI where the line of holes draws free conduction electrons back
to the track after they thermalize [6]. STEs that form in CsI:Tl despite the fast
competing channels of carrier capture on Tl must then survive thermal quenching at
room temperature [39] in order to finally excite Tl+. The model calculations indicate
that in CsI:Tl(0.06 mole%) at room temperature, STE formation amounts to ≤ 10%
of all electron hole pairs created in a 662 keV electron track. It shows furthermore
that the fraction of all initial excitations in the track that eventually result in STE
capture at Tl+ to form excited Tl+∗ is ≤5%. This model result supports extension
to CsI:Tl of the conclusion of Dietrich et al [35] noted above, namely that about 95%
of energy transfer to Tl is by binary electron and hole transfer, with STE transfer
(reaction #4) only a small contributor at perhaps 5%.
In summary, the detailed model results to be presented below show that the
two main factors favoring binary electron-hole energy transfer over STE transfer
in Tl-doped alkali halides are the rapid spatial separation of hot electrons from self-
trapped holes [22], combined with the very large capture rate of conduction electrons
on Tl+ [6,7]. The capture rate of electrons on Tl+ (0.08%) was shown by picosecond
absorption spectroscopy [7] to be even larger than the capture rate of electrons on
self-trapped holes so that doping CsI with ∼0.08 mole% Tl (≈0.3 wt% in melt)
78
strongly inhibits STE formation. As a result of these findings, we will not consider
reaction #4 involving STE energy transfer when seeking an explanation of the main
3 scintillation decay times in CsI:Tl.
A particular puzzle that we seek to answer in the rest of this section can be
phrased as follows: Experiments have revealed three distinct decay components in
CsI:Tl of 730 ns, 3.1 µs, and 16 µs, but apparently only the reactions #2 and #3
are available to account for them. Therefore it seems that at least one of the two
main reactions (#2 or #3) must be contributing two distinct decay components of
the scintillation pulse. How can that be? We look to plots of the time-dependent
radial population and reaction rate from the model for insight.
5.3.2 Time-dependent radial population and reaction rate plots
A good way to visualize the progress of various parts of the recombination process
in the modeled track is to plot populations or reaction rates as a function of radius
at a sequence of times for a given on-axis excitation density. Figure 5.6(a,b) plots
the initial hole distribution along with Tl-trapped electron distributions (Tl0) in the
critical first 10 picoseconds when hot-electron diffusion drives radial dispersal of elec-
trons [22,56] that are trapped in picoseconds as Tl0 (measured rate constant 3 ×1011
s−1 for nominally 0.08 mole% Tl [6,7]). This freezes in a charge-separated starting dis-
tribution of trapped electrons at larger radii of 40 nm or more and self-trapped holes
close to the core in a radius of about 3 nm. As mentioned above, this electron-hole
separation together with electron trapping as Tl0 discourages self-trapped exciton
formation and is probably the main reason for the finding by Dietrich, et al [35]
cited earlier that energy transport in Tl-activated KI and NaI occurs dominantly
by binary electron and hole transport rather than STE transport. On longer time
scales, as we shall see below, the STH will diffuse outward and ultimately electrons
79
thermally released from Tl0 will diffuse inward. In both cases, the carrier diffusion
is assisted by the strong internal electric field set up by the early charge separation
seen in Figure 5.6 particularly at high excitation density.
0 2 0 4 0 6 0 8 0 1 0 00 . 0
0 . 5
1 . 0
1 . 5 r n h / 1 0 - 0 p s r n e t - 1 p s r n e t - 2 p s r n e t - 5 p s r n e t - 1 0 p s
rn h and r
n et (1016
nm cm
-3 )
R a d i u s ( n m )(a)
0 1 0 2 0 3 0 4 00
1
2
3
4 n h / 2 0 - 0 p s n e t - 1 p s n e t - 2 p s n e t - 6 p s n e t - 1 0 p s
n h and n
et (1015
cm-3 )
R a d i u s ( n m )(b)
Figure 5.6: The initial hole concentration profile, [STH] = nh, is plotted togetherwith the thallium-trapped electron concentration, [Tl0] = net, at early times up tocompletion of electron trapping on Tl shortly after 5 ps. The on-axis excitationdensity is 1020 cm−3. Two formats are presented. In frame (a), the populationconcentrations are multiplied by the radius to convey number of carriers vs. radius.In frame (b), the concentrations are reported directly.
80
In Figure 5.6(a), the population concentration is multiplied by the radius to
produce a result proportional to the number of carriers present at each radius. Due
to hot-electron dispersal, the number of trapped electrons peaks at about 25 nm when
thermalization and electron trapping on the Tl activator have ended. As described
in Ref. [6], we set the hot-electron diffusion coefficient of the undoped CsI host in
our model to reproduce the same radial distribution of electrons thermalized after 4
ps in CsI as calculated by Wang et al [22], a mean radius of about 50 nm with some
electrons dispersed as far as 200 nm. Upon including 0.08 mole% Tl in the modeled
CsI with its measured electron capture rate constant of about 3 × 1011s−1 [7], the 25-
nm mean radius of Tl0 population in Figure 5.6(a) is found. Appreciable numbers
of trapped electrons extend as far as 100 nm and beyond. The radially weighted
plotting format of Figure 5.6(a) was used in [6] with mixed units of nm/cm3, chosen
so that division by the radius in nm recovers the local population density at that
radius in cm−3. Figure 5.6(b) simply plots the carrier concentrations vs. radius.
The reaction rates depend directly on the concentrations. We will use both plotting
formats as appropriate in the analysis and discussions that follow. The narrow peaks
at small radius in both frames of Figure 5.6 are labeled as the initial hole population
but equally well represent the electron population at t=0. Their values are divided
by the factors 10 and 20 in (a) and (b) respectively to bring them on scale.
An immediate and striking conclusion to be drawn from Figure 5.6(a) is that the
great majority of electrons are trapped within a few picoseconds on the Tl activator
ions in CsI:Tl at radial locations that have little overlap with the self-trapped holes.
The overlapped STH and Tl0 populations inside a radius of about 6 nm are immedi-
ately subject to recombination producing Tl+∗ excited activators at a rate given by
the term Betnet(1-fe)nh in Equation (2.6). Following the terminology introduced in
earlier studies [35, 38, 58, 63], we have called this Reaction #2. In the finite-element
81
solution of our rate model, the local rate of R#2 will be non-zero only when there
are overlapping populations of Tl0 (local concentration net) and STH (concentration
nh) in the same cell. Thus the significant portion of Tl0 trapped electrons that do
not spatially overlap the STH distribution in Figure 5.6(a) cannot immediately con-
tribute to the Reaction #2 rate term. They become eligible if diffusion brings them
into overlap. Such diffusion is assisted by the internal electric field set up between
the separated trapped charges.
To illustrate, Figure 5.7 plots the time dependent radial distributions of reaction
#2 itself. The rate term that is responsible for reaction #2 is Betnet(1 − fe)nh ≈
Betnetnh, where net is the local density of Tl0 (electrons trapped on Tl+ activator),
nh is the local density of self-trapped holes (STH), and Bet is the bimolecular rate
constant for this recombination of electrons and holes. The displayed results were
calculated for an initial excitation density of 1020 eh/cm3 on-axis of the track.
82
0 2 0 4 0 6 0 8 0 1 0 00
1
2 5 p s 2 0 p s 5 0 p s 1 0 0 p s 4 0 0 p s 1 0 0 0 p s
radiall
y weig
hted
R#2 r
ate (1
013 nm
s-1 cm-3 )
R a d i u s ( n m )0 2 0 4 0 6 0 8 0 1 0 0
0
1
2
3
4 2 0 n s 4 0 n s 6 0 n s 8 0 n s 1 0 0 n s
radiall
y weig
hted
R#2 r
ate (1
011 nm
s-1 cm-3 )
R a d i u s ( n m )(a) (c)
0 2 0 4 0 6 0 8 0 1 0 00
1
2
3 2 n s 4 n s 6 n s 8 n s 1 0 n s
radiall
y weig
hted
R#2 r
ate (1
012 nm
s-1 cm-3 )
R a d i u s ( n m )0 2 0 4 0 6 0 8 0 1 0 0
0
1
2
3 3 0 0 n s 3 5 0 n s 4 0 0 n s 4 5 0 n s 5 0 0 n s 6 0 0 n s 8 0 0 n s
radiall
y weig
hted
R#2 r
ate (1
010 nm
s-1 cm-3 )
R a d i u s ( n m )(b) (d)
Figure 5.7: The local rate of reaction #2 versus radius is plotted at evaluation timesshown in the left two frames from 5 ps up to 10 ns and continuing in the right twoframes from 20 ns to 800 ns. Reaction #2 ceases by 800 ns when the supply of STHhas been consumed by this reaction and by the competing process of STH captureon Tl+ activator sites to create Tl++.
Figure 5.7(a) shows that for roughly the first 200 ps, the reaction #2 occurs only
within a radius of about 6 nm where the STH and some of the Tl0 overlap from the
beginning. Starting around 1 nanosecond, outward movement of the reaction zone
tracking the diffusion of STH to overlap additional Tl0 at longer radius can first be
seen. The occurrence of significantly slower STH diffusion rates evaluated at lower
excitation densities of 1019 and 1018 eh/cm3 (not plotted) demonstrates that Coulomb
repulsion of the positive self-trapped holes in the track core significantly assists the
STH transport outward. From about 4 ns onward, the outwardly advancing reaction
zone leaves no significant activity in its wake because the Tl0 population (at density
83
net) is fully depleted by reaction with the dense advancing front of STH present at
this excitation density and time range. Integrating the curve radially, we obtain the
total R#2 rate at each considered time. The quantity netnh proportional to the rate
of R#2 is plotted in Figure 5.8 on a semi-log scale. This is not a light decay curve,
but a plot proportional to the reaction #2 rate for creating Tl+∗ excited states for
an initial excitation density of 1020 cm−3 on axis of the track. The 1/e time for decay
of the main R#2 rate is about 110 ns, corresponding to the straight line overlaid.
Reaction #2 is itself a bimolecular recombination process. If the bimolecular rate
term were the controlling factor in the decay at times longer than about 50 ns, we
should expect a t−1 decay at long time rather than the exponential decay evident in
Figure 5.8. We regard the finding of first-order exponential decay kinetics for this
reaction at long time as partial evidence of transport-limited reaction of spatially
separated populations at longer times.
0 2 0 0 4 0 0 6 0 0 8 0 01 0 1 2
1 0 1 4
1 0 1 6 r B e t n h n e t e x p o n e n t i a l , τ = 1 1 0 n s
radiall
y weig
hted
R#2 r
ate (n
m s-1 cm
-3 )
T i m e ( n s )
Figure 5.8: Semi-logarithmic plot of spatially integrated rate of reaction #2 versustime, for on-axis excitation density of 1020 eh/cm3.
The curve in Figure 5.8 begins with a fast spike of about 1 ns duration. Consid-
84
ering the initial stationary reaction zone seen in Figure 5.7, we conclude that the fast
spike represents reaction #2 in the initially overlapping STH and Tl0 populations,
while the 110 ns decay component of the main part of the reaction #2 represents the
transport-limited reaction rate of STH moving to encounter new Tl0 population. At
the end of Figure 5.7, reaction #2 can be seen taking place out to 80 nm, far beyond
the initial zone of creation of STH, so STH diffusion out into the surrounding field
of less mobile Tl0 has obviously been important. The ∼1 ns decay time of the spike
of reactions consuming the initial overlapped populations and the 110 ns decay of
the STH transport-limited reaction rate are two different manifestations of a single
reaction which we and previous writers have termed Reaction #2 between STH and
Tl0.
Both the 1 ns and the 110 ns decays for reaction #2 to form Tl+∗ excited states
are faster than the radiative decay time of the excited Tl+∗ state itself (575 ns), so
the two components do not lead to observably different decay times of light emission,
but rather contribute different rise-time components to the so-called fast scintillation
component decaying with an approximate 730 ns time constant. The full model
calculation already demonstrated in Figure 5.2 that the formation rate of Tl+∗ excited
states with a time constant of about 110 ns together with the 575 ns radiative
lifetime of Tl+∗ gives a good match to the observed 730 ns decay time of scintillation
light. The prevailing view that reaction #2 is the main contributor to the 730 ns
component is confirmed in this model, although we shall see later that Reaction #3
also contributes a decay component on the order of 800 ns. The model results in
Figures 5.7 to 5.9 furthermore confirm that reaction #2 expires too early (because of
STH depletion) to be a contributor to either the 3 µs or the 16 µs scintillation decay
components at the excitation density of 1020 eh/cm3 on-axis that is illustrated here.
Figure 5.9 below shows the radial population distributions of STH (rnh), Tl0
85
(rnet), Tl++ (rnht), and excited Tl+∗ (rNt), at six successive times from 10 ns to 10
µs. The first population to take note of is STH. It can be seen on the radial axis
that the STH population diffuses outward noticeably at times longer than about 10
ns. On the vertical axis, the number of STH can be seen decreasing rapidly with
time in this early range as they encounter and combine with Tl0 to produce Tl+∗
excited states (reaction #2) and with Tl+ to produce Tl++ trapped holes (setting
up reaction #3). At 800 ns, virtually all STH have been consumed mainly by these
two channels (i.e. nh is written to zero when below 0.1% of its initial value shortly
after 700 ns). At that point, R#2 has effectively stopped.
86
0 2 0 4 0 6 0 8 0 1 0 00
1
2
3 r n h r n e t r h h t r N t
radiall
y weig
hted
Conc
entra
tion (
1016
cm-3 )
R a d i u s ( n m )
1 0 n s
0 2 0 4 0 6 0 8 0 1 0 00
1
2
3
4
5 1 µs r n h r n e t r h h t r N t
R a d i u s ( n m )
radiall
y weig
hted
Conc
entra
tion (
1015
cm-3 )
(a) (d)
0 2 0 4 0 6 0 8 0 1 0 00123456 1 0 0 n s
r n h r n e t r h h t r N t
radiall
y weig
hted
Conc
entra
tion (
1015
cm-3 )
R a d i u s ( n m )0 2 0 4 0 6 0 8 0 1 0 0
0
1
2
3
4 r n h r n e t r h h t r N t
5 µs
radiall
y weig
hted
Conc
entra
tion (
1015
cm-3 )
R a d i u s ( n m )(b) (e)
0 2 0 4 0 6 0 8 0 1 0 00
1
2
3
4
5 r n h r n e t r h h t r N t
5 0 0 n s
radiall
y weig
hted
Conc
entra
tion (
1015
cm-3 )
R a d i u s ( n m )0 2 0 4 0 6 0 8 0 1 0 0
0
1
2
3
4 1 0 µs r n h r n e t r h h t r N t
radiall
y weig
hted
Conc
entra
tion (
1015
cm-3 )
R a d i u s ( n m )(c) (f)
Figure 5.9: Plots proportional to azimuthally integrated local density of STH (rnh),Tl0 trapped electrons (rnet), Tl++ trapped holes (rnht), and Tl+∗ trapped excitonsrNt are displayed as a function of radius at six indicated times between 10 ns and 10µs.
As seen in the 500 ns frame of Figure 5.9, most of the STH are exhausted by this
time but a population of Tl+∗ excited activators (Nt) produced by R#2 remain and
are available for continued radiative decay. In addition, the partly overlapped, partly
87
separated distributions of Tl++ and Tl0 that will produce subsequent additions to
Tl+∗ are evident. The overlapped Tl++ and Tl0 are immediately subject to recom-
bination producing Tl+∗ at a rate given by the term Bttnetfenht in Equation (2.6),
which we have termed Reaction #3. The significant portion of Tl0 trapped elec-
trons that are not spatially overlapping the Tl++ distribution in Figure 5.9 cannot
immediately contribute to this rate term for R#3, but become eligible if diffusion
of electrons released from Tl0 and recaptured elsewhere as another Tl0 (assisted by
the internal electric field of the separated charges that are clearly seen in Figure 5.6
and Figure 5.9) brings them into overlap. We have paraphrased the preceding two
sentences from the discussion of R#2 illustrated in Figure 5.7 earlier, because the
phenomena relating to reaction-rate-limited and transport-limited components apply
in very analogous ways to both R#2 and R#3. In the case of R#3, the rate-limited
and transport-limited rates of creating excited Tl+∗ are both slower than the Tl+∗
radiative decay time, so it can be expected that both reaction rates of R#3 may be
observed as separate decay components of light emission.
Note that particularly in the 0.5 µs and 1 µs frames of Figure 5.9, the Tl++
trapped hole distribution develops a tail on its large-radius side extending unusually
far into the Tl0 trapped electron population. In fact the Tl++ tail extends all the
way to the peak of the Tl0 radial population distribution at about 65 nm. In the
5 µs and 10 µs frames of Figure 5.9, the extended tail of Tl++ has disappeared.
This behavior suggests qualitatively that in the time leading up to roughly 0.5 µs,
STH diffusion and capture on Tl+ created Tl++ overlapping Tl0 at a faster rate than
Reaction #3 could consume them. This resulted in storage of spatially overlapped
reactant populations. The tail of Tl++ population extending into the region of high
Tl0 population seems to be one manifestation of that. After about 0.8 µs when STH
are effectively exhausted, the overlapped populations of Tl++ and Tl0 should be
88
the first consumed by R#3 in the few-microsecond time range. We suggest that this
accounts for the 3 µs decay component of R#3. When the main part of the stored-up
overlapped reactant population is exhausted, as we might judge from disappearance
of the Tl++ tail in the 5 µs frame, subsequent decay of R#3 is governed by the
transport of released and recaptured Tl0 electrons from their main population at
large radius toward the reservoir of Tl++ at small radius. This transport-limited
portion of R#3 is suggested to be responsible for the 16 µs decay component. R#3
is itself bimolecular, and yet the 16 µs decay component is found experimentally (and
in this model calculation as well) to be approximately exponential, signifying first-
order kinetics. This is consistent with its being a transport-limited reaction between
spatially-separated reactants. Notice the parallel reasoning between this discussion
of the origin of 3 µs and 16 µs decay components of R#3 and the origin of the 1 ns
and 100 ns rise components of R#2. The difference is that R#3 is slower than the
575 ns radiative time of excited Tl+∗, while R#2 is faster than that radiative time.
Recall that we ruled out reaction #4 (STE energy transport) as the source of any
of the three main decay components because of the implications of extreme charge
separation and electron trapping in Figure 5.6. By elimination of the alternatives,
attention is now focused on reaction #3 to understand from additional model per-
spectives how both the medium and tail decay components can arise from it.
Figure 5.10(a-d) plots the radial dependence of the concentration of Tl+∗ excited
states resulting from all reactions calculated for on-axis excitation density of 1020
cm−3, sampled at times from 5 ps to 10 µs as labeled in the legends. The time
sequence increases going down the left column and then going down the right column,
ending at 20 µs. Notice that the radial scale range and the vertical axis range both
change as time goes on. For the first 100 ps, the Tl+∗ excited states are formed at
increasing rate ”in place” defined by the initial STH distribution overlapping some
89
Tl0 formed near the axis. The total number of excited states (integrated azimuthally
and radially) is small in this stage. At times longer than about 200 ps, a shoulder
progressing out to larger radius indicates the onset of significant STH diffusion,
resulting in overlap with additional Tl0 to sustain the reaction #2. This continues
out to 800 ns in frame (c), at which time the supply of STH is exhausted as we saw
previously in Figure 5.7 and Figure 5.9. By this time, an underlying contribution
from reaction #3 has developed, so Tl+∗ formation is maintained going forward
beyond 800 ns.
0 5 1 0 1 5 2 00
1
2 5 p s 2 0 p s 4 0 p s 8 0 p s 1 5 0 p s 3 0 0 p s 1 0 0 0 p s
[Tl+ *] (
1015
cm-3 )
R a d i u s ( n m )0 2 0 4 0 6 0 8 0
0
1
2
3
4
5 1 5 0 n s 2 0 0 n s 3 0 0 n s 4 0 0 n s 6 0 0 n s 8 0 0 n s 1 0 0 0 n s[Tl
+ *] (10
14 cm
-3 )
R a d i u s ( n m )(a) (c)
0 1 0 2 0 3 0 4 00
1
2 2 n s 4 n s 8 n s 1 5 n s 3 0 n s 6 0 n s 1 0 0 n s
[Tl+ *] (
1015
cm-3 )
R a d i u s ( n m )0 2 0 4 0 6 0 8 0
0
1
2
3
4 1 . 5 u s 2 . 0 u s 2 . 5 u s 3 . 0 u s 4 . 0 u s 6 . 0 u s 1 0 u s 2 0 u s
[Tl+ *] (
1013
cm-3 )
R a d i u s ( n m )(b) (d)
Figure 5.10: The Tl+∗ excited state (Nt) concentration distribution resulting fromall reactions at on-axis excitation density of 1020 cm−3 is plotted versus radius attimes sampled from 5 ps to 20 µs. Notice that the radial scale range and the verticalaxis range both change as time goes on.
90
These plots of Tl+∗ at various times give additional clues to the origin of the two
distinct decay times found in the range longer than 730 ns, where Reaction #3 is the
only substantial recombination reaction still taking place. Indeed, the last frame in
Figure 5.10 showing times from 1.5 µs to 20 µs displays a distinct change in height,
width, and shift of radial position versus time for the peak in Tl-trapped exciton
population, starting after 3 µs.
The dominant radial distribution for times from 1.5 to 3 µs is a peak in Tl+∗
fixed at about 36 nm, overlying a background that slopes downward with increasing
radius. The background falls away due to Tl+∗ radiative decay over the interval
from 0.8 to 1.5 µs, revealing the stationary peak at 36 nm quite clearly as a main
contributor to the rate of Tl excited state production during this few-microsecond
range. Considering the 575 ns decay time of Tl+∗, the time interval of dominance
of this 36-nm peak in radial distribution of reaction #3 production of Tl+∗ lines
up with the experimental 3 µs decay component of light emission. Recall that in
Figure 5.9 we could see a tail of the Tl++ distribution penetrating deep into the Tl0
population during the few-microsecond period, suggesting that overlapped reactants
were being stored in the radial range from 30 to 60 nm during the first 0.5 µs, and
that afterward they were being consumed by R#3. We may conclude that the peak
of the radial reaction zone producing Tl+∗ in Figure 5.10 from roughly 1 to 3 µs
remains stationary because it is running mainly on the overlapped populations that
were stored previously. When those stored overlapped populations are exhausted,
the R#3 reaction zone begins to shift inward toward small radius, as the continued
R#3 depends on diffusion of electrons untrapped from Tl0 to find Tl++ at smaller
radius.
Starting at about 3 µs in Figure 5.10(d), the formerly stationary radial peak in
Tl+∗ population shifts toward smaller radius as just noted. It assumes a smaller
91
width and gradually decreasing height out to 20 µs. Empirically, it seems natural to
associate this radially shifting and slowly decreasing zone of R#3 with the 16 µs decay
component. This strongly suggests that the 3 µs and 16 µs decay components both
come from R#3 (thermally ionized Tl0 electrons reacting with stored Tl++ trapped
holes), with the distinct decay times rooted in different spatial distributions of the
reactants, calling into temporary dominance different rate terms in Equations (2.4)
to (2.6).
When the transport arrival rate of carriers increases the product of reactants
faster than the bimolecular reaction rate governed by B0ttnhtnetfe can decrease it,
the R#3 rate producing Tl+∗ is not transport limited. Also if the product of local
reactant densities built up from previous trapping added to the transport into that
location supports a bimolecular recombination rate faster than the arrival of new
overlapped populations, once again the R#3 rate producing Tl+∗ is not transport
limited. The rate in these cases will be set by the bimolecular rate constant B0tt mul-
tiplying the local product of densities ”in place”. As time goes on, the bimolecular
rate of ”in place” reactions will consume the stored excess population of overlapped
carriers. We propose that such reaction ”in place” combined with radiative decay
is happening in Figure 5.10(d) between about 1.5 and 3 µs. The bimolecular rate
falls until it equals the rate of increase of the reactant density product due to trans-
port processes (1) and (2) listed above. When the reaction rate becomes equal to
the transport rate, the rate of the bimolecular reaction is transport limited, and
should be determined partly by the concentration gradient and electric field that
drive directional transport. In contrast, the rate term B0ttnhtnetfe for bimolecular
recombination does not depend directly on concentration gradients or electric fields.
In this way, two distinct decay components of the R#3 can arise.
Figure 5.2 showing calculated light emission as a function of time confirms that
92
the rates of Tl+∗ production which are dissected in Fig. 5.10 do indeed produce
scintillation decay times corresponding to the observed values of 730 ns, 3.1 µs, and
16 µs. We conclude again that the latter two are due, respectively, to non-transport-
limited and to transport-limited bimolecular reaction #3 between Tl++ and Tl0.
Figure 5.11-Figure 5.13 below provide further support for this conclusion. Fig-
ure 5.11 illustrates the transition from ”in-place” consumption of local stored densi-
ties of reactants to transport-limited reaction at a slower rate. The radially weighted
rate of change of local density of Tl0 due only to transport is plotted in the solid
curves, while the radially weighted rate of change of local density of Tl++ at corre-
sponding times is plotted in the dashed curves. The latter are fully negative because
production of any new Tl++ ceased shortly after 700 ns, the first curve shown in this
figure. Tl++ are assumed not to diffuse on time scales of interest in scintillation, so
the reason for their population to decrease in this model is R#3, in which a Tl++ and
a Tl0 are annihilated as a pair with production of Tl+∗. Thus the dashed curves also
represent identical loss of Tl0 by R#3 recombination. In these terms, Figure 5.11
may be considered to compare radially weighted profiles of rate of change of [Tl0]
due to reaction #3 occurring ”in place” (Recombination, dashed curves) and due
only to transport by diffusion and electric current (Transport, solid curves) at the
indicated times.
93
0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0- 1 0
- 5
0
5
1 0 0 . 7 µ s 0 . 8 µ s 0 . 9 µ s 1 . 0 µ s 1 . 5 µ s 2 . 0 µ s 2 . 5 µ s 3 . 0 µ s
Reco
mbina
tion a
nd Tr
ansp
ort ra
tes of
chan
ge(10
8 nm s-1 cm
-3 )
R a d i u s ( n m )
Figure 5.11: Radially weighted profiles of rate of change of [Tl0] due to reaction #3occurring ”in place” (Recombination, dashed curves) and due only to transport bydiffusion and electric current (Transport, solid curves) are compared at the indicatedtimes. After about 3 µs, the rate of loss of [Tl0] (and identically of [Tl++] ) approachesequality with the positive gain of [Tl0] due to transport, indicating onset of thetransport-limited regime.
When the transport-limited regime is attained, every Tl0 arriving in the reaction
zone by diffusion and electric current transport of the thermally released electron
should correspond pairwise with the loss of a Tl++ from the reaction zone. The
positive peak of the Tl0 transport curve should come to have the same height, width,
and radial position as the inverted peak of the Tl0 and Tl++ pairwise consumption
curve. It can be seen in Figure 5.11 that this occurs for times of approximately 3
µs and greater. At earlier times, the dashed curve exceeds the transport peak of Tl0
arrivals in height and width, consistent with the consumption of locally stored Tl++
and Tl0 populations to feed part of the bimolecular recombination via reaction #3
94
during the 1-3 µs interval of the middle decay component.
Figure 5.12 presents a time sequence from 100 ps through 30 µs for the R#3 rate
term B0ttnhtnetfe, where nht is the Tl++ density, netfe is the local density of Tl0 that
are thermally ionized in equilibrium, and B0tt is the bimolecular rate constant for
reaction #3.
0 2 0 4 0 6 0 8 00
2
4
6rad
ially w
eighte
d R#
3 rate
(109 nm
s-1 cm-3 )
R a d i u s ( n m )
1 0 0 n s0 . 1 n s
(a)
0 2 0 4 0 6 0 8 00
1
2
3
4
3 0 µs
0 . 1 µs
3 . 0 µsradiall
y weig
hted
R#3 r
ate (1
09 nm s-1 cm
-3 )
R a d i u s ( n m )
0 . 5 µs
(b)
Figure 5.12: Radially weighted profiles of R#3 reaction rate are plotted for times(a) 0.1 ns, 0.5, 1, 5, 10, 20, 30, 40, 60, 80, 100 ns, and (b) 0.1 µs, 0.15, 0.2, 0.25, 0.3,0.35, 0.4, 0.45, 0.5, 0.6, 0.7, 0.8, 0.9, 1, 1.5, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30 µs. Theradial weighting factor r comes from azimuthal integration of the cylindrical trackto assess the total reaction rate versus radius. Mixed units of 109 nm s−1 cm−3 areused as in [6] so that division by the radius in nm recovers the local reaction rate atthat radius in units of s−1cm−3.
95
Figure 5.12 confirms much of what was seen in earlier radial representations of
different data, particularly Figure 5.10. The rate of R#3 increases rapidly at small
radius over the first 1000 ps. In the overview of the earlier times in Figure 5.12(a),
we can see that the reaction rate for R#3 initially grows versus time, inside 10 nm
radius, for about the first 1000 ps. Since the density profile for Tl0-trapped electrons,
net, was established in the first 10 ps, the growth in height of this B0ttnhtnetfe reaction
peak is due to increase of nht, the density of Tl++, by capture of STH from the intense
peak at small radius seen in Figure 5.6(a). This is governed by the rate term S1hnh
appearing in Equation (2.2) as a loss and in Equation (2.5) as a source term. The
evolution from about 10 ns to 0.5 µs is mainly that of a radially translating reaction
zone tracking the STH diffusion front as it creates new Tl++ overlapping existing
Tl0.
From 0.5 µs to about 3 µs, the width of the zone decreases rapidly as its peak
shifts inward toward smaller radius. This coincides in time with the evidence we have
noted earlier for consumption of stored overlapping Tl++ and Tl0 populations around
40 to 60 nm radius that were accumulated faster than the R#3 reaction could occur
in the preceding 0.5 µs. Thereafter until the last plot at 30 µs, a narrow reaction
zone moves inward as a transport-limited reaction fueled by arrival of Tl0 (diffusing
by electron release and recapture) from the reservoir at larger radius. This is the
same sequence that was evident in Figure 5.10, told this time from the perspective
of the R#3 rate term.
The profile of R#3 represented in Figure 5.12 was integrated over the radial coor-
dinate to obtain the total reaction #3 rate as a function of time. The resulting time
dependence of the R#3 rate is plotted in Figure 5.13(d) for excitation density 1020
eh/cm3 on axis, the same value used for the illustrations in Figure 5.6-Figure 5.12.
In Figure 5.13, we also show results for 1017, 1018, and 1019 eh/cm3 on axis and in all
96
the frames we have attempted to reconstruct the decay curve in terms of the three
exponential decay times, 730 ns, 3.1 µs, and 16 µs found to fit the experimental scin-
tillation decay data (662 keV) [4]. As shown earlier, reaction #2 (STH+T l0 → T l+∗)
is mainly responsible for the 730 ns scintillation decay, and that reaction is not rep-
resented in Figure 5.13. Nevertheless, R#3 turns out to exhibit a fast component of
the reaction rate decay in the range of 700 ns as well, so the 730 ns decay time was
included in the analysis. The main interest driving this analysis was in the 3.1 µs
and 16 µs components of R#3. We have seen that R#2 goes to completion within
800 ns at 1020 eh/cm3 and 1.4 µs at 1017 eh/cm3, so the two longer decay components
of scintillation should arise mainly from R#3. Furthermore, since these longer decay
times significantly exceed the 575 ns radiative lifetime of Tl+∗, the longer components
of scintillation decay can be expected to track the decay of the total R#3 reaction
rate producing Tl+∗.
97
0 1 0 2 0 3 0 4 01 0 8
1 0 9
1 0 1 0
1 0 1 1
T o t a l R # 3 r a t e e x p o n e n t i a l , τ1 = 0 . 7 3 µ s e x p o n e n t i a l , τ2 = 3 . 1 µ s e x p o n e n t i a l , τ3 = 1 6 µ s S u m
Rate
of R#
3 (s-1 cm
-3 )
T i m e ( µs )
1 0 1 7 e h / c m 3
0 1 0 2 0 3 0 4 01 0 1 0
1 0 1 1
1 0 1 2
1 0 1 3
Rate
of R#
3 (s-1 cm
-3 )
T i m e ( µs )
T o t a l R # 3 r a t e e x p o n e n t i a l , τ1 = 0 . 7 3 µ s e x p o n e n t i a l , τ2 = 3 . 1 µ s e x p o n e n t i a l , τ3 = 1 6 µ s S u m
1 0 1 9 e h / c m 3
(a) (c)
0 1 0 2 0 3 0 4 01 0 9
1 0 1 0
1 0 1 1
1 0 1 2
Rate
of R#
3 (s-1 cm
-3 )
T i m e ( µs )
T o t a l R # 3 r a t e e x p o n e n t i a l , τ1 = 0 . 7 3 µ s e x p o n e n t i a l , τ2 = 3 . 1 µ s e x p o n e n t i a l , τ3 = 1 6 µ s S u m
1 0 1 8 e h / c m 3
0 1 0 2 0 3 0 4 01 0 1 1
1 0 1 2
1 0 1 3
1 0 1 4
Rate
of R#
3 (s-1 cm
-3 )
T i m e ( µs )
T o t a l R # 3 r a t e e x p o n e n t i a l , τ1 = 0 . 7 3 µ s e x p o n e n t i a l , τ2 = 3 . 1 µ s e x p o n e n t i a l , τ3 = 1 6 µ s S u m
1 0 2 0 e h / c m 3
(b) (d)
Figure 5.13: The spatially integrated rate of reaction #3 (black curve) is plotted asa function of time on semi-log scale for excitation densities of (a) 1017, (b) 1018, (c)1019, and (d) 1020 eh/cm3. This model result represents the time-dependent rate ofchange of the number of Tl+∗ excited activators due solely to R#3. It is the maincontributor to the Tl+∗ emitting state population at times longer than 700 ns. Threeexponential decay components of 730 ns, 3.1 µs, and 16 µs found to characterize662 keV scintillation decay [4] are fitted and displayed along with their sum in themagenta curve that can be compared to the model-calculated black curve.
The three-component analysis in Figure 5.13 shows reasonably good fits at 1017,
1018 and 1019 eh/cm3 but a substantial under-representation at 1020 eh/cm3 from
about 5 µs to 30 µs. This is a reminder that the scintillation is a weighted sum over
many contributing local excitation densities. Furthermore, the analysis indicates a
reduction of the 730 ns component as the excitation density is lowered, trending
toward a mostly two-component sum of 3.1 µs and 16 µs decay for the R#3 curve
at 1017 eh/cm3.
98
The collective effect of all the excitation densities to R#3 can be calculated
by weighting each according to its frequency of occurrence in a 662 keV electron
deposition using GEANT4. This procedure is analogous to the method for weighting
local light yield in our full scintillation model. The result for weighted R#3 is shown
in Figure 5.14. The model-calculated R#3 decay curve in blue is matched fairly
well by the sum of 730 ns, 3.1 µs, and 16 µs components in orange. Two small
discrepancies around 4 µs and 16 µs remain and are similar to the full model fits of
scintillation decay in Figure 5.2. If the 3.1 µs decay time is replaced by a 4 µs decay
time, nearly exact matching of the calculated R#3 curve is obtained, but we will
stay with the set of fixed decay times from the experimental study [4].
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 01 0 4
1 0 5
1 0 6
1 0 7
1 0 8
6 6 2 k e V m o d e l e x p o n e n t i a l , τ1 = 0 . 7 3 µ s e x p o n e n t i a l , τ2 = 3 . 1 µ s e x p o n e n t i a l , τ3 = 1 6 µ s S u m
Rate
of R#
3 (s-1 cm
-3 )
T i m e ( µs )
Figure 5.14: The rate of reaction #3 as a function of excitation density was weightedby the probability of occurrence of each excitation density in a 662 keV electron trackbased on GEANT4 simulations and is displayed versus time in the blue curve. Threeexponential decay components of 730 ns, 3.1 µs, and 16 µs found to characterize662 keV scintillation decay [4] are fitted and displayed along with their sum in themagenta curve that can be compared to the model-calculated black curve.
We regard Figure 5.14 as confirmation that the R#3 reaction rate for the weighted
99
sum of excitation densities in a 662 keV track can be well represented by the three
decay time components of scintillation [4], even though that representation fails to
some degree at high excitation density around 1020 eh/cm3. Two notable features
have emerged at 1020 eh/cm3 in Figure 5.13(d). The 3.1 µs component has yielded
strength to a 730 ns component in the fitting at high density. Effectively, what
was the faster ( 3 µs) of two main slow components of R#3 at lower excitation
densities has become faster still at high density and will contribute light in the same
general time range as the main 730 ns fast component of scintillation due to R#2.
In experimental observations of scintillation pulse shape versus gamma energy, this
will appear as an increase in the ratio of fast compared to slow and tail components
at low gamma energy, i.e. high excitation density. This is the observed experimental
trend. Part of the reason is identified with increasing contribution of R#3 in the
same time range as the fast component of mainly R#2 light emission. Other reasons
for this energy dependence of pulse shape can be found in the kinetics and spatial
dependence of R#2 itself as already discussed.
In addition, Figure 5.13 shows that the ”tail” decay time trends to a shorter
value than 16 µs at the higher excitation densities, especially 1020 eh/cm3. The
experimental trend found in [4] was that at low gamma energy the tail decay time
became slightly faster, e.g. 14 ± 3 µs at 6 keV.
Finally, Figure 5.13 shows that the empirical ”3 µs decay time” is not a single
identified process with that decay time, but is the weighted sum of multiple decay
times dependent on excitation density that vary through the roughly 10 µs to 0.7 µs
range as excitation density encountered in a track spans the corresponding densities.
100
5.4 Origin of anticorrelated fast and tail proportionality
trends at room temperature
As could be seen in Figure 5.5, the model predicts proportionality curves of the fast
and tail components of scintillation showing the same remarkable anticorrelation of
trends for these two components as was found in experiment [4]. The measured fast
component falls as energy increases from 16 keV to 662 keV, while the tail component
rises over the same increasing energy interval. To get at the physical mechanisms
behind this anticorrelated behavior of fast and tail proportionality curves, we make
use of the results in the previous section confirming that the fast decay component
(730 ns) is mainly due to reaction #2, and the tail component (16 µs) is due to
the transport-limited part of reaction #3. As has been discussed, the rate term in
Equation (2.6) that is responsible for reaction #2 is Betnet(1 − fe)nh, and the rate
term responsible for reaction #3 is B0ttnetfenht. These are not light outputs, but they
both feed the Tl+∗ population from which light is emitted.
In Figure 5.15, the excitation density dependences of the yields of R#2 and R#3
producing Tl+∗ excited states are plotted in blue and red, respectively. We could call
this the ”local yield of reactions #2 and #3” in analogy to what we have previously
called local light yield as a function of excitation density. The dashed black curve
and grey horizontal line labeled ”Murray-Meyer” will be discussed later.
101
1 0 1 6 1 0 1 7 1 0 1 8 1 0 1 9 1 0 2 0 1 0 2 10
2 0
4 0
6 0
8 0
r e a c t i o n # 2 r e a c t i o n # 3 M u r r a y - M e y e r
Norm
alized
yield
(%)
E x c i t a t i o n d e n s i t y ( c m - 3 )
Figure 5.15: The time- and space-integrated yields of the reactions #2 and #3 areplotted versus initial on-axis excitation density in the solid blue and red curves,respectively. The yield is integrated from zero to 40 µs.
Look first at the blue solid curve for the yield of reaction #2. It starts near zero
at very low excitation density and rises with a concave upward curvature consistent
with the fact that reaction #2 is bimolecular in populations whose initial values scale
roughly with excitation density. We say ”roughly” because it should be apparent from
the above discussion and radial plots that the product of overlapping STH and Tl0
densities varies dramatically in time and space as a result of hot-electron diffusion
in the beginning followed by electric-field driven diffusion re-uniting free carriers and
trapped carriers over time. But in sweeping terms, the supply of reactant populations
that can participate in transport and recombination is roughly proportional to the
initial excitation density, so we should not be surprised that the R#2 yield (i.e. the
blue curve) looks roughly quadratic up to an excitation density around 7 × 1019
eh/cm3. Above that density it starts to bend over and eventually turns downward.
That trend is understandable first because the supply of reacting carriers is limited,
102
so it must be a saturating yield that finally bends toward a finite value if there are
no losses. The limit of the saturating yield is lowered by 2nd order quenching and
the sharp turn-down above 4 × 1020 eh/cm3 can be attributed to 3rd order Auger
quenching.
Bearing in mind that Figure 5.15 plots local reaction yield as a function of exci-
tation density, not light yield as a function of gamma energy, one nevertheless can
see that the fast reaction #2 curve has a shape consistent with the proportionality
curve of the fast 730 ns scintillation component shown in the experimental results
of Figure 5.5(a). In making this comparison, we qualitatively associate high gamma
energy with predominantly low excitation density and low gamma energy with pre-
dominantly high excitation density.
Now focus on the solid red curve plotting the local yield of the slower R#3. For
excitation densities above 1018 eh/cm3 that comprise most of the energy deposition
in high energy electron tracks, the curve of R#3 yield is flat or turns downward
with increasing excitation density. This trend is anticorrelated with the increasing
yield of R#2 versus excitation density, just as seen in the experimental fast and
tail proportionality versus gamma energy (Fig. 5.5(a)). The essential reason for
this anticorrelation is quite basic, namely the two processes compete for the same
STH supply in two different kinetic orders. Reaction #2 (STH + T l0 → T l+∗)
is bimolecular in excitation products and therefore wins at high excitation density
over the first-order process of Tl++ formation (STH + T l+ → T l++). The latter
process obeys first-order kinetics because Tl+ is a crystal dopant, not an excitation
product. Since Tl++ is a reactant for R#3, we see the result of the competition
as a decrease in R#3 at high density in Figure 5.15. The reaction #3 rate term is
proportional to the product of two trapped carrier populations, both of which are
essentially the ”leftovers” after completion of the faster reaction #2. By about 3 µs
103
when we can first clearly identify the tail component, R#2 has run to completion
and has consumed 54% of the starting STH and Tl0 at 1020 eh/cm3 versus 9.5% of
the starting STH and Tl0 at 1018 eh/cm3, according to the model results. Most of
the STH not used in R#2 were converted by capture into Tl++ and will serve as
one reactant for R#3. Most of the Tl0 not used in R#2 will be used as the other
reactant in R#3. The yield of reaction #3 scales approximately as the product of
two nearly equal populations that are both ”leftovers” after completion of reaction
#2. In those general terms, the yield of R#3 in this system must be anticorrelated
with the yield of R#2 versus excitation density and therefore versus gamma or
electron energy in the reversed sense of how particle energy and effective excitation
density are approximately related. The anticorrelation displayed in the experimental
measurements of Figure 5.5(a) is a direct consequence. The ratio of carriers being
used in R#2 or left over for R#3 is influenced by the electric field assisted transport
of STH in the first phase, which is dependent on excitation density.
Reaction #3 is itself bimolecular since the Tl++ and Tl0 reactants (in the sense
of the corresponding rate term in Equation (2.6)) are both excitation products. This
accounts for the rising slope of R#3 with excitation density at low densities in
Figure 5.15, i.e. before R#2 begins to deplete the STH supply in 2nd order faster
than the 1st-order STH + Tl+ capture can use them. The fact that R#3 starts
out larger than R#2 at low carrier densities is also understandable because a large
fraction of STH are converted to Tl++ at low excitation density where first-order
capture on numerous activators competes well with bimolecular recombination.
Proceeding from reaction yields versus excitation density to reaction yields versus
electron energy, Figure 5.16 plots the result of weighting the reaction yields at various
densities in Figure 5.15 by the probability of each density occurring in the Geant4
simulations for a given initial electron energy. Repeating for various electron energies
104
produced the curves in Figure 5.16 giving the reaction yield (R#2 or R#3) for that
initial energy. Note that the energy dependences of the yields for R#2 and R#3 have
the same general form as the proportionality curves of fast and tail decay components
in Figure 5.5.
1 1 0 1 0 0 1 0 0 00 . 6
0 . 8
1 . 0
1 . 2
1 . 4
1 . 6No
rmaliz
ed Yi
eld
E n e r g y ( k e V )
R e a c t i o n # 3 R e a c t i o n # 2
Figure 5.16: The yields of reaction #2 and reaction #3 evaluated after 40 µs areplotted versus initial electron energy.
Before leaving this topic, it is worthwhile to try to connect the results with
the approximate treatment by Murray and Meyer of competing bimolecular exciton
formation and defect trapping in a line track [58]. They postulated a system in
which the free electrons and holes were created pairwise in linear number density
n = nh = ne along a line of deposition. They considered ”. . . that the electron can
suffer two events, either recombining with a hole in the wake of the incident particle,
or trapping at an unspecified site in the lattice”, the latter according to a first-order
trapping rate Kn. The productive bimolecular rate of electron-hole recombination
to form the excitons that they suggested were responsible for Tl+∗ emission can be
105
written as the second order term Bn2. The productive rate divided by the sum of
all rates defined a yield written as
Y =Bn2
Kn+Bn2=
αn
1 + αn(5.1)
where α = B/K. The expression on the right-hand side of this equation is the
Murray-Meyer statement of expected radiative yield in this system. This describes a
yield decreasing as the excitation number n decreases (in an assumed line deposition).
The expression starts from near zero at small n, rises at first quadratically, and
approaches a saturating constant value of unity at large n. It does not turn down
at large n, but saturates at unity because Murray and Meyer did not include either
2nd order (dipole-dipole) or 3rd order (Auger) nonlinear quenching. Equation (5.1)
is plotted in Figure 5.15 with the dashed black curve, and its saturation asymptote
with a solid grey line.
As noted above (and as well by Murray and Meyer [58]), this simple formula can
give only a qualitative illustration of what goes on in a real particle track. For one
reason, as we have seen in Figure 5.6 and the surrounding discussion, roughly 90%
of the electrons and holes are separated by hot electron diffusion and are trapped or
self-trapped in different radial zones early in the pulse evolution [6, 56]. There are
important electric field effects and trapping at play in their eventual recombination.
As we have already noted, there are also nonlinear quenching terms not included in
the Murray-Meyer formula. A solution of the full model description of transport,
trapping, and recombination is necessary to make quantitative predictions of the
light yield versus particle energy, i.e. results such as are shown in Figure 5.5(b).
Nevertheless, comparison of the Murray-Meyer curve and the reaction #2 curve in
Figure 5.15 reveals considerable similarity. This confirms that fundamentally, re-
action #2 is bimolecular in excitation density when sufficient time is allowed for
106
transport and recombination of dispersed trapped-carrier populations. The main
linear trapping channel that competes with R#2 in the Murray-Meyer sense is actu-
ally STH capture on Tl+ to make Tl++. Although this trapped-hole species will later
produce light in R#3, it is a dark defect trap with respect to the ”fast” scintillation
of R#2. Electron trapping on deep defects in this model is also a competitor with
R#2, but on a smaller scale than linear hole trapping to form Tl++, simply because
the concentration of Tl+ dopant exceeds defect concentration in most cases.
5.5 The material input parameters
As described in [6], the model of scintillation that we have constructed tries to take
into account the important physical processes of carrier generation, transport, re-
combination, nonlinear quenching, and capture on dopants and defects that seem
logically required for physical description of the events in a particle track from which
light yield and proportionality are determined. The number of material parameters
necessary to specify those terms in a system of equations for free and trapped elec-
trons, holes, and excitons is large, as was enumerated in Tables I and III of [6] for
undoped and Tl-doped CsI. The good news about this circumstance is that for a
model to yield information about effects caused by variation of material composition
(concentration and species of doping, co-doping, defects, . . .) the coefficients and rate
constants of all those components should be in the model or no specific information
on material engineering by their variation can be obtained. The bad news is that
good values for all of the parameters must be supplied whenever a new material
system is modeled. There is a time investment for each new material. Over time, a
library of tested parameter sets for important scintillator systems of interest should
be built up. We believe that the material input parameters for CsI and CsI:Tl are
approaching a reasonably well-tested status by virtue of the fitting and predictions of
107
pulse shapes, energy dependence, absolute light yield, and proportionality (both to-
tal and by decay component) in the present work, evolving from the initial set in [6].
Undoubtedly there will be some further refinement of the material parameter values
following direct experimental measurements and theoretical work in the future. But
over time, validated parameter sets for a number of important scintillator systems
should emerge from continuing work on CsI:Tl, and then on other materials as well.
Table 5.1 lists the parameters of CsI, all of which except for the deep defect
trapping rate constant K1e have remained unchanged in the present work relative to
the values used for the calculation of proportionality and light yield in undoped CsI
at 295 K in [6]. Note that the material parameters of undoped CsI are also used to
describe the host when modeling CsI:Tl.
Table 5.2 lists the additional parameters needed to model CsI:Tl (0.06-0.08%),
some of which did change in the process of fitting the wider array of data (i.e.
pulse shape) in the present paper. The modeling in this study was done with the
rate constant S1e (for electron capture rate to form Tl0) at the value measured for
nominal 0.08 mole % Tl in CsI [7], the same as the CsI:Tl fitted in [6]. Although the
sample measured by Syntfeld-Kazuch et al [4,5] contained 0.06% mole % Tl, we are
not sure that falls outside the uncertainty of nominal 0.08% estimated from weight
% Tl in the melt for the sample used in the picosecond measurements of S1e [7].
When listing the Equations (2.1) to (2.7), we introduced the ”free electron frac-
tion”, fe = Uet/S1e, of Tl0 that are ionized in equilibrium, so that the free-electron
values of De, µe, and K1e could be used in Equations (2.4) to (2.6) rather than define
new parameters Det, µet, and K1et scaled by the same factor as done in [6]. The
defect trapping rate constant K1e is proportional to the concentration of the respon-
sible deep defects, which is sample dependent. Thus a determination of K1e was
made in this work from fitting the decay curve of the CsI:Tl (0.06%) sample studied
108
Table 5.1: Parameters used for the host parameters in the CsI:Tl model of the presentwork. Except for the deep defect trapping rate constant K1e discussed in text, allparameters in this list are the same as used for the calculation of proportionalityand light yield in undoped CsI at 295 K in [6]. In Table I of Ref. [6], literaturereferences for the values were listed where available and otherwise comments onestimation methods were listed and explained in the text. See [6] for definitions ofthe parameters.
Parameter Value Unitsrtrack 3 nmβEgap 8.9 (eV/e-h)avgε0 5.65 N/Aµe 8 cm2/VsDe 0.2 cm2/sµh 10−4 cm2/VsDh 2.6 x 10−6 cm2/sDE 2.6 x 10−6 cm2/sB(t > τhot) 2.5 x 10−7 cm3/sK3 4.5 x 10−29 cm6/sK2E 0.8 x 10−15 t−1/2cm3s−1/2
R1E 6.7 x 106 s−1
K1E 6 x 107 s−1
τhot 4 psrhot (peak) 50 nmDe(t < τhot) 3.1 cm2/sS1e 0 s−1
S1h 0 s−1
S1E 0 s−1
GE(r = 0) 0 cm−3
K1e 2.7 x 1010 s−1
K1h 10−5K1e s−1
Ei(norm) 200 keV
109
Table 5.2: Additional rate constants and transport parameters used in Equa-tions (2.4) to (2.6) when modeling CsI:Tl (0.06%) at 295 K in the present work.S1e is the value measured on CsI:Tl (nominal 0.08 mole %) [7]. See [6] for definitionsof the parameters.
Parameter Value UnitsS1e 3.3 x 1011 s−1
S1h 5.0 x 106 s−1
S1E 5.0 x 106 s−1
[T l] 0.06 mole % in sampleR1Et 1.7 x 106 s−1
UEt 5.4 x 105 s−1
Btt 2.5 x 10−7 cm3/sBet 1.3 x 10−10 cm3/sBht 2.5 x 10−7 cm3/sK2Et 1.7 x 10−15 t−1/2cm3s−1/2
by Syntfeld-Kazuch et al [4].
All of the changes in parameter values relative to [6] can be considered small or
modest except the two bimolecular rate constants involving thallium: Bet for capture
of STH on Tl0, and Btt (= B0ttfe) for capture of an electron released from Tl0 on Tl++,
including the effect of release and recapture. The 1st-order capture rate constants
that had not been directly measured were estimated in [6] as the product of a cross
section, the concentration of the capturing defect, and the mean velocity of approach
of the mobile carrier. When the approaching mobile carrier is a self-trapped hole, the
velocity of approach is quite low, governed by the STH hopping rate. For capture
on neutral traps (including substitutional Tl+ in the CsI lattice), a geometrical cross
section can usually be assumed without large error. In this way, the value for S1h,
the first-order rate constant for capture of STH on Tl+, was estimated. For the 2nd-
order capture of STH on Tl0 governed by Bet, the cross section was assumed to be the
same as found from ps absorption measurements of STH+e→ STE. During fitting
of pulse shape in the present work, the STH population was found to be vanishing
110
too quickly to support the observed rise time to peak. The need for a reduced value
of Bet became clear, and there was recognition that the estimated value should have
taken into account the low velocity of the approaching carrier (STH) in the case of
STH+T l0 → T l+∗. This is the largest correction in Table 5.2. Because the Betnetnh
rate term and the S1hnh rate term divide the available STH population as discussed
earlier, reduction in the value of Bet required a balancing decrease in the value of
S1h, returning it close to the value of S1h originally estimated in [6]. The fitting of
pulse shape in the present work required a significant increase in the value of Btt
relative to the estimate of this 2nd-order rate constant made in [6]. This underscores
a conclusion we reached in this study, that fitting the proportionality alone can be
fairly forgiving on some of the parameter choices. Fitting additional data with more
structure, such as multiple rise and decay components of the pulse shape, can be
used to refine parameters before undertaking calculation of proportionality, as done
in the present study.
111
Chapter 6: High tnergy tlectron tracks: GEANT4 and
NWEGRIM
6.1 GEANT4 results, trajectories and carrier density distri-
butions, of CsI
We use GEANT4 to do Monte Carlo (MC) simulation to get the carrier density
distribution of CsI. The sample input file is shown in Appendix A.1. Typically the
output file contains “Step#, X, Y, Z, KineE, dEStep, StepLeng, TrakLeng, Volume,
Process”. We use Equation (2.8) to calculate the carrier density.
dE/dx =dEStep
StepLeng(6.1)
After we calculate the carrier density between each energy deposit position to the
next energy deposit position along the track, we can do the weighted histogram to
get the carrier density distribution for this input energy. After repeating the same
process for different energies, we can get the following carrier density distributions
for different energies.
112
electron density (/cm3)
1017 1018 1019 1020 1021
prob
abili
ty
×10-19
0
0.5
1
1.5
2
2.5
3GEANT4
400 keV100 keV20 keV10 keV5 keV2 keV
Figure 6.1: GEANT4 carrier density distributions.
We can see that as the input primary electron energy increases the peak (density
lower than 1020 /cm3) moves toward left. The spike (density higher than 1020 /cm3)
stays the same. The reason for the spike probably comes from the cutoff 200 eV used
in the simulation Appendix A.1.
6.2 NWEGRIM Results, trajectories and carrier density dis-
tributions, of CsI
GEANT4 is not the only program can simulate the electron tracks. From Sebastien
Kerisit and his group at PNNL we also get the simulation results generated by the
MC code NWEGRIM (NorthWest Electron and Gamma Ray Interaction in Matter).
113
2000
1000
x (nm)
NWEGRIM, 20 keV
0
-1000-500
0
500
y (nm)
1000
×104
6
3
4
5
7
1500
z (n
m)
track 1track 2
20001000
x (nm)
0
NWEGRIM, 100 keV
-1000-20002000
3000
4000
y (nm)
5000
×106
1.475
1.474
1.473
1.476
1.477
6000
z (n
m)
track 300track 301track 302track 303
Figure 6.2: NWEGRIM tracks at 20 keV and 100 keV.
We can use the simulation results to calculate the carrier density distributions.
There are two parameters need to use in this calculation: 1. track cutoff; 2. cylinder
cutoff. Firstly, we need to separate each track which means we need to find the track
end. As shown in Figure 6.2, we assume if the distance between two created electrons
114
is larger than the track cutoff (300 nm in use) the first electron will be the end of
the track and the second electron will be the start of the new track. Using this track
cutoff, we will not get very unreasonable low carrier densities. Secondly, in local
light model, we need to chop each track into small cylinders. We use 15 nm which
is 5 times of the track radius as the cylinder cutoff. We still can use Equation (2.8)
to calculate the carrier density. The carrier density distributions from NWEGRIM
are as following.
electron density (/cm3)
1017 1018 1019 1020 1021
prob
abili
ty
×10-20
0
1
2
3
4
5
6
7
8
9NWEGRIM
400 keV100 keV20 keV10 keV5 keV2 keV
Figure 6.3: NWEGRIM carrier density distributions.
We can see in both GEANT4 and NWEGRIM the peak positions for different
energies are close and for higher input energy the peak (density lower than 1020 /cm3)
moves toward left. But different from GEANT4, in NWEGRIM there is no spike at
density higher than 1020 /cm3. We think this is because NWEGRIM follows electron
energy all the way down and does not dump everything below threshold.
Using the same set of parameters in fitting CsI:Tl(0.06%) pulse shape spectrum,
we can get the nonproportionality curves.
115
energy (keV)100 101 102 103
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4NWEGRIM vs GEANT4
NWEGRIMGEANT4
Figure 6.4: Nonproportionality: NWEGRIM v.s. GEANT4.
We can see there is a hump at 10 keV for combination of local light yield model
with NWEGRIM results which is similar with experimental results shown in Fig-
ure 2.1. But there is no such hump for GEANT4 results. The reason probably is
there is no spike at high density in NWEGRIM results which will lower the low
energies side on nonproportionality curve.
From this comparison, we can see that GEANT4 and NWEGRIM results are
different. GEANT4 give us positions of the energy deposition events sequentially
along the track. NWEGRIM give us the positions of electron/hole created at the
end of tracks which are exact what we want to combine with the local light yield
model results. Also from Figure 6.4 we can see there will be a way to match both
nonproportionality and pulse shape spectrum experiments from the same set of pa-
rameters.
116
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Appendix A: Input files
A.1 GEANT4 sample input file
/control/verbose 2 /run/verbose 2 /testem/det/setAbsMat CsI /testem/det/setAbsThick 2 cm /testem/det/setAbsYZ 2 cm /testem/phys/addPhysics empenelope /testem/phys/setECut 3 nm /testem/phys/setPCut 1 km /testem/phys/setGCut 1 km /cuts/setLowEdge 200 eV /run/initialize /process/eLoss/fluct true /testem/gun/setDefault /gun/position 0 /gun/particle e- /gun/energy 662 keV /tracking/verbose 1 /run/beamOn 500
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A.2 Local light yield model sample input file
//CsI //SI units //================= double _dt0=10.0*ns; // decay dt double _gatetime=40.0*us; // gate time double _er=6.6; // static dielectric constant double _nTl0=0.3E25; // Tl concentration double _nCo0=0.3E25; // Codopant concentration double _h0=(atof(argv[1])*1.0E6<3.0E26) ? 1.0 : ((atof(argv[1])*1.0E6<8.0E26) ? 0.05 : 0.025); // space step double _L=250.0; // max radius double _sigmae=3.0*nm; // electron track radius double _sigmah=3.0*nm; // hole track radius double _T0=4.0*ps; // thermalization time double _R0=50.0*nm; // thermalization distance double _T=295.0; // termperature double _pp=0.0; // prompt Tl-E //================= double _W=0.2; double _ratiocs=1.0; // cross section ratio double _ratiove=sqrt(_T/295.0); // electron thermal velocity ratio double _ratiovh=exp(-_W/kB/_T)/exp(-_W/kB/295.0); // hole thermal velocity ratio double _ratioveh=1.0E-5; // e-h thermal velocity ratio // double _ue=8.0*1.0E-4; // eletron mobility double _ue=8.0*1.0E-4*(exp(125.0/_T)/exp(125.0/295.0)+(exp(157.0/_T)-1)/(exp(157.0/295.0)-1))/2.0; // eletron mobility double _uh=1.0*1.0E-4*1.0E-4*_ratiovh/(_T/295.0); // hole mobility double _K1e=0.8*1.0/(30.0*ps)*_ratiocs*_ratiove; // electron trapped on defect rate double _S1e0=1.0*1.0/(3.0*ps)*_ratiocs*_ratiove; // electron trapped on Tl rate double _S1ec0=1.0*1.0/(3.0*ps)*_ratiocs*_ratiove; // electron trapped on codopant rate double _K1h=0.0*1.0/(30.0*ps)*_ratioveh*_ratiocs*_ratiovh; // hole trapped on defect rate double _S1h0=1.0*_S1e0*_ratioveh*_ratiovh/_ratiove; // hole trapped on Tl rate double _S1hc0=1.0*_S1ec0*_ratioveh*_ratiovh/_ratiove; // hole trapped on codopant rate double _B=1.0/(3.26*ps*1.2E18*1.0E6)*_ratiocs*_ratiove; // e-h recombination rate double _K3=1.0*3.2E-29*1.0E-12; // 3rd-order quenching rate double _uE=_uh; // exciton mobility double _K1E=0.9*1.0/(20.0*ns); // exciton trapped on defect rate double _S1E0=1.0*_S1h0; // exciton trapped on Tl rate double _S1Ec0=1.0*_S1hc0; // exciton trapped on codopant rate double _R1E=1.0/(20.0*ns)-_K1E; // exciton radiation rate double _K2E=1.0*0.8E-15*1.0E-6; // exciton 2nd-order quenching rate (a function of time) //================= // double _Uet=1.0/(1.4E-6); // electron escaping from Tl rate double _Uet=1.0*1.3E10*exp(-0.2/kB/_T); // electron escaping from Tl rate double _Uht=0.0*1.3E10*exp(-0.2/kB/_T); // electron escaping from Tl rate double _ratioet=1.0*_Uet/_S1e0; // electron escaping from Tl ratio double _ratioht=1.0*_Uht/_S1h0; // hole escaping from Tl ratio double _uet=1.0*_ratioet*_ue; // Tl-e mobility
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double _uht=1.0*_ratioht*_uh; // Tl-h mobility double _K1et=1.0*_ratioet*_K1e; // Tl-e trapped on defect rate double _K1ht=1.0*_ratioht*_K1h; // Tl-h trapped on defect rate double _Bet=25.0*_B*_ratioveh*_ratiovh/_ratiove; // Tl-e and hole recombination rate double _Bht=_B; // electron and Tl-h recmobination rate double _Btt=_ratioet*_Bht+_ratioht*_Bet; // Tl-e and Tl-h recombination rate double _R1Et=1.0/(560.0*ns); // Tl-E radiation rate double _K2Et=1.0*1.7E-15*1.0E-6; // Tl-E 2nd-order quenching rate (a function of time) //================= double _Uetc=1.0*1.3E10*exp(-0.2/kB/_T); // electron escaping from codopant rate double _Uhtc=0.0*1.3E10*exp(-0.2/kB/_T); // electron escaping from codopant rate double _ratioetc=1.0*_Uetc/_S1ec0; // electron escaping from codopant ratio double _ratiohtc=1.0*_Uhtc/_S1hc0; // hole escaping from codopant ratio double _uetc=1.0*_ratioetc*_ue; // Codopant-e mobility double _uhtc=1.0*_ratiohtc*_uh; // Codopant-h mobility double _K1etc=1.0*_ratioetc*_K1e; // Codopant-e trapped on defect rate double _K1htc=1.0*_ratiohtc*_K1h; // Codopant-h trapped on defect rate double _Betc=25.0*_B*_ratioveh*_ratiovh/_ratiove; // Codopant-e and hole recombination rate double _Bhtc=_B; // electron and Codopant-h recmobination rate double _Bttec=_ratioetc*_Bht; // Codopant-e and Codopant-h recombination rate double _Btthc=_ratiohtc*_Bet; // Codopant-e and Codopant-h recombination rate double _R1Etc=1.0/(560.0*ns); // Codopant-E radiation rate double _K2Etc=1.0*1.7E-15*1.0E-6; // Codopant-E 2nd-order quenching rate (a function of time) //================= double _M=0.10; // max variation //================= int _verbose=0; // output format 0 //int _verbose=1; // output format 1 //int _verbose=2; // output format 2 //int _verbose=3; // output format 3 //================= //SI units
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Curriculum Vitae
EDUCATION
Wake Forest University (WFU), Winston Salem, NC, USA 2010–2016
Ph.D., Physics, Advisor: Prof. R. T. Williams
Nanjing University (NJU), Nanjing, Jiangsu, China 2006–2010
B.S., Physics, Advisor: Prof. Jun Wang
COMPUTER SKILLS
C/C++; Matlab; Maple; Shell script; Python; Gnuplot; Geant4; LaTeX; Microsoft
Word, PowerPoint, Excel;
EXPERIENCE
Research Assistant as Modeler and Programmer, WFU 2013–2016
Developed and coded a model to solve seven coupled partial differential equations
describing scintillation in radiation detection materials.
• Combined the solution set with data provided via GEANT4/NWEGRIM Monte
Carlo simulation to describe CsI and CsI(Tl) performance, light yield, propor-
tionality and pulse time characteristics based on 20 fundamental properties of
the material.
• Used finite element methods and optimized performance by improving the
time-step algorithm and data structures. The computation provided time-
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and space-resolved distributions of the carrier & trap populations and recom-
bination events.
• Developed a user-friendly program (toolkit) to run the model written mainly
in C++ and Bash script.
• Projects are in cooperation with Lawrence Berkeley National Lab, Pacific
Northwest National Lab, National Centre for Nuclear Research in Poland and
Saint Gobain Crystals.
Graduate Student Researcher, WFU 2011–2013
Worked in Prof. F. R. Salsbury, Jr.’s biophysics group analyzing protein conforma-
tional change by statistical methods using Matlab.
Undergraduate Thesis Researcher, NJU 2009–2010
Worked in Prof. Jun Wang’s biophysics group. Wrote C codes to simulate protein
folding and search for protein native state.
Undergraduate Innovation Program Researcher, NJU 2008–2010
Designed two optical methods to measure the magnetostriction coefficient using a
Michelson interferometer and a double grating.
Teaching Assistant, WFU, 2011–2013
Tutored 50 students each semester. The tasks included teaching the labs, grading
homework, lab reports and exam papers.
PUBLICATIONS
Selected Peer-Reviewed Journal Articles and Conference Proceedings
X. Lu, S. Gridin, R. T. Williams, M. R. Mayhugh, A. Gektin, A. Syntfeld-Kazuch,
L. Swiderski, and M. Moszynski, “Energy-dependent scintillation pulse shape and
133
proportionality of decay components for CsI:Tl: Modeling with transport and rate
equations”, Phys. Rev. Applied (in press).
H. Huang, Q. Li, X. Lu, Y. Qian, Y. Wu, and R. T. Williams, “Role of hot electron
transport in scintillators: A theoretical study”, Phys. Status Solidi RRL, 10: 762768
(2016).
X. Lu, Q. Li, G. A. Bizarri, M. R. Mayhugh, K. Yang, P. R. Menge, and R. T.
Williams, “Coupled rate and transport equations modeling proportionality of light
yield in high-energy electron tracks: CsI at 295 K and 100 K; CsI:Tl at 295 K”,
Phys. Rev. B 92, 115207 (2015).
Q. Li, X. Lu, and R. T. Williams, “Toward a user’s toolkit for modeling scintillator
non-proportionality and light yield”, Proc. SPIE 9213, 92130K (2014).
CONFERENCE PRESENTATIONS
Selected First-Author Poster/Oral Presentations
2016 IEEE Symposium on Radiation Measurements and Applications (SORMA XVI,
Berkeley, May 2016), “Energy-dependent scintillation pulse shape and proportional-
ity of decay components in CsI:Tl modeling with transport and rate equations”.
2015 IEEE Nuclear Science Symposium & Medical Imaging Conference (NSS/MIC,
San Diego, Nov 2015), “Transport and rate equation modeling of experiments on
proportionality of decay time components in CsI:Tl”.
13th International Conference on Inorganic Scintillators and their Applications (SCINT,
Berkeley, Jun 2015), “Numerical Modeling of Scintillator Proportionality and Light
Yield from Experimentally and Computationally Determined Host and Dopant Pa-
rameters”.
2014 IEEE Symposium on Radiation Measurements and Applications (SORMA XV,
134
Ann Arbor, Jun 2014), “Numerical model of electron energy response in CsI:Tl and
SrI2:Eu with diffusion coefficient and rate constants dependent on electron temper-
ature and thus time”.
AWARDS
WFU Graduate School Alumni Student Travel Award 2014-2016
WFU Dean’s Fellowship For Top Graduate Student Admitted 2010
NJU People’s Scholarship For Undergraduates 2006-2010
8th Jiangsu Undergraduate Physics Innovation Contest 2nd Prize 2009
EXTRA-CURRICULAR ACTIVITIES
WFU Intramural Basketball League
North East Chinese Basketball League
NJU Undergraduate Volleyball League
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