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

vi

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

viii

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

x

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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