27mm244 x 168

Thermally andOptically StimulatedLuminescence A Simulation Approach

Thermally and Optically Stimulated

Luminescence A Simulation Approach

Thermally and O

ptically Stimulated Lum

inescenceReuven Chen and Vasilis Pagonis

Chen Pagonis

Reuven ChenRaymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv, Israel

Vasilis PagonisMcDaniel College, Westminster, MD, USA

Thermoluminescence (TL) and optically stimulated luminescence (OSL) are two of the most important techniques used in radiation dosimetry. They have extensive practical applications in the monitoring of personnel radiation exposure, in medical dosimetry, environmental dosimetry, spacecraft, nuclear reactors, food irradiation etc., and in geological /archaeological dating.

Thermally and Optically Stimulated Luminescence: A Simulation Approach describes these phenomena, the relevant theoretical models and their prediction, using both approximations and numerical simulation. The authors concentrate on an alternative approach in which they simulate various experimental situations by numerically solving the relevant coupled differential equations for chosen sets of parameters.

Opening with a historical overview and background theory, other chapters cover experimental measurements, dose dependence, dating procedures, trapping parameters, applications, radiophotoluminescence, and effects of ionization density.

Designed for practitioners, researchers and graduate students in the fi eld of radiation dosimetry, Thermally and Optically Stimulated Luminescence provides an essential synthesis of the major developments in modeling and numerical simulations of thermally and optically stimulated processes.

Cover design: Gary Thompson

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Thermally and Optically StimulatedLuminescence

Thermally and OpticallyStimulated

LuminescenceA Simulation Approach

REUVEN CHEN

Raymond and Beverly Sackler School of Physics and Astronomy,Tel Aviv University, Tel Aviv, Israel

VASILIS PAGONIS

McDaniel College, Westminster, MD, USA

This edition first published 2011© 2011 John Wiley & Sons Ltd

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Library of Congress Cataloging-in-Publication Data

Chen, R. (Reuven)Thermally and optically stimulated luminescence : a simulation approach / Reuven Chen and Vasilis Pagonis.

p. cm.Includes bibliographical references and index.

ISBN 978-0-470-74927-2 (hardback)1. Thermoluminescence. 2. Thermoluminescence dosimetry. 3. Optically stimulated luminescence dating.

I. Pagonis, Vasilis. II. Title.QC479.C47 2011612′.014480287–dc22

2010053722

A catalogue record for this book is available from the British Library.

Print ISBN: 9780470749272ePDF ISBN: 9781119993773oBook ISBN: 9781119993766ePub ISBN: 9781119995760eMobi: 9781119995777

Set in 10/12pt Times Roman by Thomson Digital, Noida, IndiaPrinted in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire

Front cover image courtesy of Herb Yeates, http://superfluorescence.com/. Copyright (2007) with permission.

Contents

About the Authors ix

Preface xi

Acknowledgements xiii

1 Introduction 11.1 The Physical Mechanism of TL and OSL Phenomena 11.2 Historical Development of TL and OSL Dosimetry 21.3 Historical Development of Luminescence Models 5

2 Theoretical Basis of Luminescence Phenomena 72.1 Energy Bands and Energy Levels in Crystals 72.2 Trapping Parameters Associated with Impurities in Crystals 92.3 Capture Rate Constants 102.4 Thermal Equilibrium 122.5 Detailed Balance 142.6 Arrhenius Model 152.7 Rate Equations in the Theory of Luminescence 182.8 Radiative Emission and Absorption 192.9 Mechanisms of Thermal Quenching in Dosimetric Materials 202.10 A Kinetic Model for the Mott–Seitz Mechanism in Quartz 232.11 The Thermal Quenching Model for Alumina by Nikiforov et al. 26

3 Basic Experimental Measurements 293.1 General Approach to TL and OSL Phenomena 293.2 Excitation Spectra 323.3 Emission Spectra 343.4 Bleaching of TL and OSL 35

4 Thermoluminescence: The Equations Governing a TL Peak 394.1 Governing Equations 394.2 One Trap-One Recombination Center (OTOR) Model 444.3 General-order Kinetics 454.4 Mixed-order Kinetics 474.5 Q and P Functions 514.6 Localized Transitions 554.7 Semilocalized Transition (SLT) Models of TL 57

vi Contents

5 Basic Methods for Evaluating Trapping Parameters 635.1 The Initial-rise Method 635.2 Peak-shape Methods 655.3 Methods of Various Heating Rates 665.4 Curve Fitting 695.5 Developing Equations for Evaluating Glow Parameters 715.6 The Photoionization Cross Section 73

6 Additional Phenomena Associated with TL 796.1 Phosphorescence Decay 796.2 Isothermal Decay of TL Peaks 816.3 Anomalous Fading and Anomalous Trapping Parameters of TL 826.4 Competition Between Excitation and Bleaching of TL 946.5 A Model for Mid-term Fading in TL Dating; Continuum of Traps 1006.6 Photo-transferred Thermoluminescence (PTTL) 1096.7 TL Response of Al2O3:C to UV Illumination 1166.8 Dependence of the TL Excitation on Absorption Coefficient 1186.9 TL Versus Impurity Concentration; Concentration Quenching 1216.10 Creation and Stabilization of TL Traps During Irradiation 1296.11 Duplicitous TL Peak due to Release of Electrons and Holes 1316.12 Simulations of the Duplicitous TL Peak 140

7 Optically Stimulated Luminescence (OSL) 1437.1 Basic Concepts of OSL 1437.2 Dose Dependence of OSL; Basic Considerations 1447.3 Numerical Results of OSL Dose Dependence 1467.4 Simulation of the Dose-rate Dependence of OSL 1487.5 The Role of Retrapping in the Dose Dependence of POSL 1507.6 Linear-modulation OSL (LM-OSL) 1537.7 Unified Presentation of TL, Phosphorescence and LM-OSL 1567.8 The New Presentation of LM-OSL Within the OTOR Model 1587.9 TL-like Presentation of CW-OSL in the OTOR Model 1687.10 Dependence of Luminescence on Initial Occupancy; OTOR Model 1697.11 TL Expression Within the Unified Presentation 1717.12 Pseudo LM-OSL and OSL Signals under Various

Stimulation Modes 1737.13 OSL Decay and Stretched-exponential Behavior 1747.14 Optically Stimulated Exoelectron Emission 1807.15 Simulations of OSL Pulsed Annealing Techniques 186

8 Analytical and Approximate Expressions of Dose Dependenceof TL and OSL 1938.1 General Considerations 1938.2 Competition During Excitation 1968.3 Competition During Heating 1988.4 The Predose (Sensitization) Effect 205

Contents vii

8.5 Sensitization and De-sensitization in Quartz 2078.6 Dose-rate Dependence 2108.7 Sublinear Dose Dependence of TL and LM-OSL in the

OTOR System 2118.8 Dose-dependence and Dose-rate Behaviors by Simulations 2228.9 Simulations of the Dose-rate Effect of TL 2258.10 Nonmonotonic Dose Dependence of TL and OSL 2298.11 Nonmonotonic Dose Dependence of TL; Simulations 2318.12 Nonmonotonic Effect of OSL; Results of Simulations 239

9 Simulations of TL and OSL in Dating Procedures 2459.1 The Predose Effect in Quartz 2459.2 Simulation of Thermal Activation Characteristics in Quartz 2479.3 The Bailey Model for Quartz 2569.4 Simulation of the Predose Dating Technique 2619.5 The Single Aliquot Regenerative Dose (SAR) Technique 2669.6 Thermally Transferred OSL (TT-OSL) 272

10 Advanced Methods for Evaluating Trapping Parameters 27910.1 Deconvolution 27910.2 Monte-Carlo Methods 28210.3 Genetic Algorithms 28710.4 Application of Differential Evolution to Fitting OSL Curves 291

11 Simultaneous TL and Other Types of Measurements 29711.1 Simultaneous TL and TSC Measurements; Experimental Results 29711.2 Theoretical Considerations 30111.3 Numerical Analysis of Simultaneous TL-TSC Measurements 31011.4 Thermoluminescence and Optical Absorption 31311.5 Simultaneous Measurements of TL and ESR (EPR) 31811.6 Simultaneous Measurements of TL and TSEE 323

12 Applications in Medical Physics 32712.1 Introduction 32712.2 Applications of Luminescence Detectors in Medical Physics 32812.3 Examples of in-vivo Dosimetric Applications 33212.4 Radioluminescence 334

13 Radiophotoluminescence 34113.1 Development and Use of RPL Materials 34113.2 The Simplest RPL Model 344

14 Effects of Ionization Density on TL response 34714.1 Modeling TL Supralinearity due to Heavy Charged Particles 34714.2 Defect Interaction Model 35114.3 The Unified Interaction Model 352

viii Contents

15 The Exponential Integral 35715.1 The Integral in TL Theory 35715.2 Asymptotic Series 35815.3 Other Methods 362

Previous Books and Review Papers 363

Appendix A Examples 365A.1 Simulation of OSL Experiments Using the OTOR Model 365A.2 Simulation of OSL Experiments Using the IMTS Model 366A.3 Simulation of TL Experiment Using the Bailey Model 369

References 371

Author Index 395

Subject Index 411

About the Authors

Reuven Chen

Professor Reuven Chen is a Professor Emeritus at Tel Aviv University. He has been workingon thermoluminescence, optically stimulated luminescence and other related topics over thelast 48 years. Professor Chen has published approximately 170 scientific papers and twobooks. He has been a Visiting Professor at several universities in the USA, UK, Canada,Australia, Brazil, France and Hong Kong. At present, he is an Associate Editor of RadiationMeasurements and referee for several international journals.

Vasilis Pagonis

Professor Vasilis Pagonis is a Professor of Physics at McDaniel College. His researchinvolves working on modeling properties of dosimetric materials and their applicationsin luminescence dating and radiation dosimetry. Professor Pagonis has published approx-imately 70 scientific papers, as well as the book Numerical and Practical Exercises inThermoluminescence, published in 2006. He currently holds the Kopp endowed chairin the physical sciences at McDaniel College.

Preface

Thermoluminescence (TL) and optically stimulated luminescence (OSL) are two of themost important techniques used in radiation dosimetry. Hundreds of papers are publishedevery year in the scientific literature on different aspects of TL and OSL. These cover awhole spectrum of subjects, from experimental papers describing various aspects of thesephenomena in different materials under different experimental conditions, many times hav-ing in mind the potential applications, to publications interested only in the dates reachedby these methods and to publications on dosimetry measurements in different environments(e.g., in spaceships). On the other side of the spectrum, one can find work on the physicalbasis of TL and OSL, in which researchers try to obtain better understanding of the under-lying processes. These include the dose dependence of the effects (which may be linear ornonlinear), possible dose-rate dependence, the stability of the effects at ambient temperature(which may include normal and anomalous fading), the dependence of these effects on therelevant defects and impurities, and the nature of the emission spectrum.

The theoretical work on TL and OSL consists, in most cases, of the study of the simultane-ous differential rate equations governing the transitions of charge carriers, usually electronsand holes, between the different trapping states associated with impurities and defects inthe studied sample, and the conduction and valence bands. These equations are not linearand therefore, in most cases, cannot be solved analytically. In many cases, approximationsconcerning the trapping parameters and functions are made to reduce the complication, andexplicit equations for simplified models such as the first- and second-order kinetics can bewritten and solved analytically. Here, general solutions are reached from which one canconsider how different phenomena may take place, e.g. how the signal fades with time atroom temperature under first- or second-order kinetics, and to what extent these predic-tions agree with specific luminescence experiments for a given material. Obviously, thisapproach has a strong limitation since one does not know whether the assumptions madehold all along the temperature and time range of a TL/OSL measurement.

This book concentrates on an alternative approach, in which we simulate various exper-imental situations by numerically solving the relevant coupled differential equations forchosen sets of parameters. Using this approach, several complex situations can be demon-strated such as superlinear and nonmonotonic dose dependencies, dose-rate effects, theoccurrence of abnormally high frequency factors and others. Obviously, the shortcomingof this approach is that it does not provide us with general solutions but rather with resultsassociated with specific sets of trapping parameters. However, this kind of demonstrationthat certain behaviors are commensurate with our understanding of the underlying pro-cesses is of great importance. With the present availability of strong computing powerand advanced numerical methods, this approach has become very popular during the past20 years. A second approach that is emphasized throughout this book is demonstrating thepossibility of obtaining analytical solutions of the systems of differential equations by usingthe quasi-equilibrium approximation. Numerous examples are given in which this approach

xii Preface

leads to exact analytical solutions which describe accurately the experimental results. Thisbook is designed for practitioners, researchers and graduate students in the field of radi-ation dosimetry. It is a synthesis of the major developments in modeling and numericalsimulations of thermally and optically stimulated processes during the past 50 years.

Chapter 1 is mostly a historical overview of the developments in TL and OSL dosimetryduring the past 50 years, followed in Chapter 2 by an overview of the theoretical basis andseveral quantum aspects of luminescence phenomena, which is based on the energy-bandmodel of solids. Chapter 3 deals with a number of basic experimental measurements relevantto the study of TL and OSL. In Chapter 4 we present the basic kinetic equations govern-ing the TL process, including simple kinetic models based on first- second- general- andmixed-order kinetics. In addition, some aspects of localized versus delocalized electronictransitions during the luminescence process are discussed. The basic methods of evaluat-ing kinetic parameters in TL and OSL experiments are the main topic of Chapter 5, andChapter 6 addresses a variety of physical phenomena commonly encountered during TL andOSL measurements. The basic theoretical aspects and experimental techniques used in OSLdosimetry are presented in Chapter 7, with a specific emphasis on the relationship betweenthe various models used in obtaining OSL data (LM-OSL, CW-OSL, pseudo LM-OSL, etc.).Chapter 8 addresses a topic of prime importance for radiation dosimetry researchers, namelythe dose dependence of TL/OSL signals. Different types of experimentally observed dosebehaviors are examined using both an analytical approach and approximate expressionsobtained using certain approximations. The topic of TL and OSL simulations for datingapplications is presented in Chapter 9, including simulations of recent major developmentsin TL/OSL dating protocols. Chapter 10 examines the use of several alternative methodsfor evaluating trapping parameters, based on a variety of advanced numerical methods likeMonte-Carlo techniques, genetic algorithms and advanced curve-fitting methods. Chap-ter 11 contains a more general approach to thermally stimulated phenomena, and severalmethods of analyzing simultaneous thermal measurements are presented. The processes ofthermally stimulated conductivity (TSC), thermally stimulated electron emission (TSEE),optical absorption (OA) and electron spin resonance (ESR) are briefly discussed, in par-ticular in cases where their simultaneous measurements with TL can produce additionalinformation and can be simulated along with the simulation of TL. Chapter 12 deals withapplications of luminescence in medical physics and Chapter 13 with the associated phe-nomenon of radiophotoluminescence. Chapter 14 summarizes theoretical developmentsand simulation results on the effects of ionization density on TL response, which is a topicof major interest in radiation dosimetry for particles of varying ionization density. FinallyChapter 15 presents various numerical approaches to the exponential integral which appearscommonly in TL applications. In addition to the comprehensive list of references, coveringall the subjects discussed in the book, we have also included a list of books and reviewarticles published in the literature from 1968 onwards. Finally, in Appendix A, we presentsome simple examples of computer code that simulate three important models which appearfrequently in the book, namely, the one trap-one recombination center (OTOR) model, theinteractive multiple trap system (IMTS), and the widely used Bailey model for quartz.

Acknowledgements

We thank our wives Shula Chen and Mary Jo Boylan for their patience, encouragementand sound advice during the years of writing this book. We would like to thank alsoDr John L. Lawless for his contributions to some of the analytical aspects discussed inthe book. Thanks are also due to Dr Doron Chen for important technical help in preparingthe manuscript.

Reuven Chen and Vasilis Pagonis

I would like to thank all my research collaborators, students and friends in the luminescencecommunity for their helpful contributions and stimulating discussions during the develop-ment of various luminescence models over the years. Special thanks among these are dueto Dr George Kitis of Aristotle University in Greece for his friendship and extensive col-laboration over the last 10 years; Dr Ann Wintle for teaching me the importance of seekingperfection in the preparation of manuscripts; to my colleagues Andrew Murray, Mayank Jainand Christina Ankjærgaard in Denmark for their hospitality and many stimulating conver-sations; and last but not least, special thanks to my co-author, good friend and luminescencementor Dr Reuven Chen, for teaching me the importance of accuracy and precision duringluminescence modeling work.

Vasilis Pagonis

1Introduction

In this introductory chapter we first provide an overview of the physical mechanism involvedin thermoluminescence (TL) and optically stimulated luminescence (OSL) phenomena,followed by a brief historical review of the development of TL and OSL dosimetry. Thisis followed by a section on the parallel development of luminescence models for TL/OSLphenomena during the past 50 years.

1.1 The Physical Mechanism of TL and OSL Phenomena

The phenomenon of phosphorescence seems to have been discovered first by VincenzoCasciarolo (see e.g., Arnold [1]), an amateur alchemist in Bologna in 1602 who discoveredthe “Bologna Phosphorus”, the mineral barium sulfide, which was glowing in the dark afterexposure to sunlight. An account was later published by Fortunio Liceti in “Litheosphorus,sive de lapide Bononiensi lucem”, Utino, 1640. In 1663, Robert Boyle gave the RoyalSociety one of the first accounts of TL. He described some experiments he had carried outon a diamond, saying “I also brought it to some kind of glimmering light, by taking it intobed with me, and holding it a good while upon a warm part of my naked body” (see e.g.Heckelsberg [2] ). The phenomenon of TL had been known since the 17th century, and hasbeen studied intensively since the first half of the 20th century. For example, in 1927, Wick[3] reported on the TL of X-irradiated fluorite and other materials. In 1931, she reported[4] on TL in calcium sulfate doped by manganese and fluorite, following their exposureto radium. She also described the effect of applying pressure on the TL properties of thesamples. A preliminary qualitative explanation of the occurrence of TL, based on the bandtheory of solids was given by Johnson [5] only in 1939. The first quantitative theoreticalaccount based on the model of energy bands in crystals, was given in 1945 in a seminal workby Randall and Wilkins [6]. Basically, TL consists of the excitation of an insulator, usuallyby ionizing radiation but sometimes by non-ionizing radiation or other means, followed bya “read-out” stage of heating the sample and measuring the light emitted in excess of the

Thermally and Optically Stimulated Luminescence: A Simulation Approach, First Edition. Reuven Chenand Vasilis Pagonis. © 2011 John Wiley & Sons, Ltd. Published 2011 by John Wiley & Sons, Ltd.

2 Thermally and Optically Stimulated Luminescence

“black-body radiation”. In the OSL method, discovered significantly later, the read-out stageconsists of releasing the charge carriers, previously excited by irradiation, by illuminationwith light of an appropriate wavelength; the incident light is capable of releasing trappedcharge carriers at the ambient temperature.

The understanding of the phenomenon is associated with the energy-band theory of solids,and has to do with the trapping of charge carriers in the forbidden gap states associated withimperfections in the crystalline material, be it impurities or defects. The trapping states areentities that can capture either electrons or holes during the excitation period and duringthe read-out stage which, in the TL process is the time when the sample is heated andmeasurable light is recorded. The energy absorbed during the excitation period causes theproduction of electrons and holes, which may move around the conduction and valencebands, respectively, and get trapped in electron and hole trapping states. Some of thesetraps may be rather close to their respective bands, electrons to the conduction band andholes to the valence band, so that within the temperature range of the subsequent heating,they may be thermally released into the band. These entities are usually called “traps”.

The trapping states which are farther from their respective bands, in which a recombina-tion of trapped charge carriers and mobile carriers of the opposite sign may take place areusually termed “recombination centers” or just “centers”. Thus, during the read-out stagecharge carriers, say electrons, may be thermally elevated into the conduction band, wherethey can move around before recombining with the opposite-sign carriers, say a hole, andemit at least part of the previously absorbed energy in the form of photons. However, someof these recombinations may be radiationless, meaning that the produced energy turns intophonons. It is also possible that recombinations produce photons in a spectral range whichis not measurable by the device being used, and for the purpose of our analysis of the results,may be considered as being radiationless.

Note that, although very often one discusses the TL/OSL process as being related to thethermal or optical release of trapped electrons and their subsequent recombination with holesin centers, the inverse situation in which the mobile entity is the positive hole which movesin the valence band and then recombines with a stationary electron in a luminescence centeris just as likely to occur. One should also mention the possibility of localized transitions, asituation where the hole and electron trapping states are located in close proximity to eachother, and the radiative process takes place by thermal or optical stimulation of one kind ofcarrier into an excited state which is not in the conduction/valence band, and its subsequentrecombination with its opposite-sign companion.

1.2 Historical Development of TL and OSL Dosimetry

The two most important applications of TL and OSL are in the broad fields of radia-tion dosimetry and geological/archaeological dating. In this section we present a briefoutline of the historical development of luminescence techniques in these two broadapplication areas.

Although the first theoretical work, by Randall and Wilkins and later by Garlick andGibson was published in the 1940s, the first practical applications of TL were suggested inthe 1950s. The applications of TL in radiation dosimetry were initiated in the early 1950s byDaniels [7, 8] who also suggested that natural TL from rocks is related to radioactivity from

Introduction 3

uranium, thorium and potassium in the material. Later, Kennedy and Knopf [9] discoverednatural TL emitted from samples of ancient pottery, which led the way to the work on TLdating of archaeological samples which was developed quickly in the 1960s, first in Oxfordby Aitken and his group [10] and later, in dozens of laboratories all over the world. Thepossible use of optical stimulation instead of thermal stimulation for evaluating the absorbeddose in a sample for dosimetry purposes was first suggested by Antonov-Romanovsiı [11]in the mid 1950s and mentioned later by a number of researchers who referred usually toinfra-red stimulated luminescence (IRSL). The use of OSL for archaeological and geolog-ical dating was suggested in 1985 by Huntley et al. [12], and it has been in use in manylaboratories since then.

Since the 1950s there has been a continuous extensive search for the “perfect” thermo-luminescent dosimetric (TLD) material that will exhibit the ideal linear response over thewidest possible range of doses, high sensitivity, excellent reproducibility and stability of theluminescence signal. The historical development, properties and uses of various TLD mate-rials have been summarized in some detail in the book by McKeever et al. [13]. The use ofTL as a radiation dosimetry technique was first suggested by Farrington Daniels and collab-orators at the University of Wisconsin (USA) during the 1950s. Daniels et al. [7, 8] first usedLiF for radiation dosimetry during atomic bomb testing, and they also studied and consid-ered CaSO4:Mn, sapphire, beryllium oxide and CaF2:Mn as possible TL dosimeters duringthe same decade. In the 1960s a variety of new materials were also studied, namely CaF2:Dy,CaSO4:Tm, CaSO4:Dy, CaF2 and LiF:Mg,Ti. The latter material eventually became oneof the most commonly used TLD materials. In the next 20 years various forms of Al2O3,CaF2 and LiF were developed and considered as TLD candidates. Other commonly usedand studied TLD materials are Al2O3:C and LiF:Mg,Cu,P. The most common applicationsof TLD materials are in monitoring of personnel radiation exposure, in medical dosimetry,environmental dosimetry, spacecraft, nuclear reactors, mineral prospecting, food irradiation,retrospective dosimetry, and in geological/archaeological dating.

Kortov [14] recently summarized the current status and future trends in the develop-ment of materials for TL dosimetry. This author listed the main requirements for practicaluse of TL dosimeters as: a wide linear dose response, high TL sensitivity per unit ofabsorbed dose, low signal dependence on the energy of the incident radiation, low sig-nal fading over time, the presence of simple TL curve, luminescence spectrum matchingphotomultiplier (PM) tube response and appropriate physical characteristics. The authorlisted the useful dose range and thermal fading properties of the following seven main prac-tical dosimetric materials: LiF:Mg,Ti (TLD-100), LiF:Mg,Cu,P (TLD-100H), 6LiF:Mg,Ti(TLD-600), 6LiF:Mg,Cu,P (TLD-600H), CaF2:Dy (TLD-200), CaF2:Mn (TLD-400), andAl2O3:C (TLD-500). Kortov [14] also discussed the intrinsic luminescence efficiency η ofTL materials; he specifically attributed the high sensitivity of several dosimetric materialsto the efficient trapping/detrapping/excitation mechanisms associated with the presence ofF-centers.

In a recent comprehensive review of luminescence dosimetry materials Olko [15] sum-marized the progress of luminescence detectors and dosimetry techniques for personaldosimetry and medical dosimetry. The author discussed traditional personal dosimetrybased on OSL, TL and radiophotoluminescence (RPL), and also reviewed more novelluminescence detectors used in clinical dosimetry applications such as radiotherapy, inten-sity modulated radiotherapy (IMRT) and ion beam radiotherapy. The major advantages of

4 Thermally and Optically Stimulated Luminescence

luminescence dosimeters were summarized as: high sensitivity measurement of very lowdoses, linear dose dependence, good energy response to X-rays, reusability, and sturdiness.However, the review also recognized the problem of decreased response with increasing ion-ization density of the radiation field. This problem may lead to underestimation of dose afterheavy charged particle irradiation. Personal dosimetry is also used widely in the medicalsector, with dosimetric films gradually being replaced by TLD, OSL and RPL materials.

The pros and cons of using OSL versus TLD dosimeters have been summarized inMcKeever and Moscovitch [16]. Some of the advantages of OSL dosimeters are high effi-ciency and stable sensitivity, better precision and accuracy, fast read-out, and no thermalannealing steps. However, TL dosimeters have the advantages of high sensitivity, no lightsensitivity, simple automated read-out, possibility of neutron dosimetry, and flat photonenergy response.

Olko [15] also summarized some newer developments in luminescence detectors: devel-opment of a personal neutron dosimeter based on OSL [17], laser-scanned RPL glassesused to measure the dose from fast neutrons by counting tracks of charged recoil particles[18], and fluorescent nuclear track detectors (FNTDs) which allow imaging of individualtracks of heavy charged particles [19, 20]. Oster et al. [21] suggested the possibility ofusing standard LiF:Mg,Ti (TLD-100) and a combined TL/OSL signal to increase the effi-ciency of detecting high linear energy transfer (LET) particles. Additional novel techniquesinclude the development of a laser-scanned OSL system and TLD systems with a charge-coupled device (CCD) camera [22–24]. Olko [15] identified three active areas for researchin new luminescence detectors, namely developing new materials for the medical field, formaterials to be used in dosimetry of high LET radiation, and for materials mimicking theradiation response of biological systems. However, this author also identified the absenceof luminescence detectors for neutron dosimetry as a major gap in luminescence dosimetry.

The second broad area where TL and OSL dosimetry have found extensive practical appli-cations is in the field of geological and archaeological dating. In a comprehensive reviewarticle, Wintle [25] reviewed the historical and technological developments in the field ofluminescence dating. During the time period 1957–1979, TL techniques were applied toheated materials, while in the time period 1979–1985 TL dating was extended to oldersedimentary samples. The historical developments in the use of TL during this time periodinclude the fine-grain and coarse-grain TL dating techniques, improvements in the calcula-tion and measurement of natural dose rates, applications of TL dating to pottery and firedclay, and authenticity testing of ceramics using predose dating. During these early years,two major problems were identified which hindered successful application of TL dating:the problems of anomalous fading exhibited, e.g., by feldspars; and the phenomenon ofsupralinearity during dose response measurements. However, there were many attempts toextend the use of TL signals in the study of other materials, such as heated stones, calcitedeposits and burnt flint. In many of these areas, TL continues to be a valuable dating tool.Starting in 1979, researchers began exploring the possibility of using TL dating techniquesfor determining the time of deposition of quartz and feldspar grains. The exploration of newluminescence signals during the period 1979–1985 for the dating of sediment deposition ledto the next major phase in luminescence dating, which continues today. During the last 25years, research in luminescence dating has undergone a dramatic shift, due to the discoveryof new luminescence signals which could be zeroed by exposure to sunlight. These newsignals led to the development of OSL dating techniques. In 2008, Wintle [25] identified

Introduction 5

1999 as the seminal year in which the single aliquot regenerative (SAR) dating procedurewas developed; this technique has revolutionized luminescence dating, by providing anaccurate and precise tool for routine measurement of equivalent doses. Furthermore, theSAR protocol allows for a completely automated measurement process, resulting in majorimprovements in the speed of data acquisition and analysis. As a result of these majordevelopments during the past 25 years, OSL has become arguably the most accurate andprecise luminescence dating tool in Quaternary geology, as well as a valuable archaeologicaltool [26].

1.3 Historical Development of Luminescence Models

In this section we present a historical overview of the development of luminescence models,which took place in parallel to the historical development of experimental TL and OSLtechniques described in the previous section.

Randall and Wilkins [6] wrote a differential equation governing the TL process anddiscussed the properties of its solution, by assuming that retrapping is negligible and that therate of change of trapped carriers is proportional to the concentration of these trapped carriers(first-order kinetics). Garlick and Gibson [27] showed that under different relations betweenthe retrapping and recombination probabilities, the rate of change of the concentrationof trapped carriers is proportional to the square of this concentration, i.e. the kinetics is ofsecond order. They wrote the relevant differential equation and studied the properties ofits solution. Following a previous suggestion by Hill and Schwed [28], May and Partridge[29] extended this treatment to “general-order” kinetics, namely, cases in which the rateof change of the concentration of trapped carriers is proportional to a non-integer powerof their concentration. Although heuristic in nature, the approach has been rather popularin the study of TL. A milestone in the development of luminescence models is the workby Halperin and Braner [30], who introduced a more realistic presentation of a single TLpeak. They wrote three simultaneous differential equations governing the traffic of carriersbetween a trapping state, the conduction band and a recombination center. Since theseequations cannot be solved analytically, Halperin and Braner [30], Levy [31] and otherauthors made some simplifying assumptions, which enabled the solution of the problem ina relatively easy way for some specific circumstances. It is obvious, however, that the onlyroute to follow more complicated cases is by solving numerically the relevant simultaneousdifferential equations.

During the past 50 years numerous kinetic models have been published which attemptto explain various experimentally observed behaviors in luminescence phenomena. Per-haps the best overview of these models is the paper by McKeever and Chen [32] and thetextbook by Chen and McKeever [33]. The approach used in the majority of publishedTL/OSL papers is to solve numerically the relevant simultaneous differential equations.With modern available software, this is a relatively easy task. One can use reasonable setsof trapping parameters and find how the TL, as well as OSL, signals behave. The obviousdisadvantage is that it is usually very hard to draw general conclusions from the simula-tion. It is possible, however, to demonstrate that certain effects are compatible with specificassumptions concerning the relevant trapping states. For example, nonlinear dose depen-dencies of TL and OSL have been reported in some materials; even within the one trap-one

6 Thermally and Optically Stimulated Luminescence

recombination center (OTOR) model, called by Levy [31] General One Trap (GOT), nonlin-ear dose dependence can be expected under certain conditions. In addition, different kindsof such nonlinearity can be explained by taking into consideration the occurrence of com-petitors, the transitions into which are nonradiative. In some extreme cases, this behaviorcan be shown analytically, but the variety of nonlinear dose dependencies can be demon-strated by simulation through numerical solution of the relevant equations. The simulationshould be performed for the excitation stage and for the read-out stage, and properties ofthe solution can be compared with the experimental results. A comprehensive approachshould, however, include both the excitation and read-out stages, with a certain relaxationperiod in between.

The review article by McKeever and Chen [32] addressed several important questions onthe usefulness and need for modeling and numerical simulations of luminescence phenom-ena. These authors emphasized that one of the most important purposes of modeling is toprovide researchers with “a feeling of security”; the use of models can indeed improve ourbasic understanding of the physical processes being studied. In another familiar example,modeling can provide fundamental answers about the validity of the complex modern pro-tocols used during luminescence dating. In the same review paper, the authors provided acritique of modeling efforts and emphasized the need to test the actual behavior of the pro-posed models, in order to ascertain what behaviors are possible (or not) within the model.They also pointed out that often, modeling efforts lead to the development of ad hoc models,without regard to how well the model can describe other behaviors observed in the samematerial. It is our belief that to some extent these two criticisms of modeling efforts havebeen addressed during the past 20 years, with the development of comprehensive models fora variety of dosimetric materials. As an example of such comprehensive modeling efforts,we mention the recent development of comprehensive models for quartz by several authors[34–36]. Such models have proved to be very useful indeed for explaining a wide variety ofexperimental behaviors in quartz. As a second example of a comprehensive model, we men-tion the various models developed to explain the TL and OSL properties of the widely useddosimetric material Al2O3:C. Several of these comprehensive models have been shown tobe able to describe simultaneously a wide variety of TL/OSL phenomena in this importantdosimetric material [37–39].

2Theoretical Basis of Luminescence

Phenomena

Throughout this book, we will be examining how different combinations of rate constantsyield different luminescent behaviors. In this chapter, we examine what physics tells us aboutthese constants and their magnitude. We start with a discussion of electron and hole capturerate constants and discuss how the type of trap determines how large the rate constants areexpected to be. We then examine thermal equilibrium. Whether or not a TL material everreaches equilibrium, the theory allows us to relate the magnitude of a rate constant to thatof its reverse. Following that, we derive thermal detrapping rate constants from capture rateconstants. We will then consider Arrhenius’ theory of rate constants and how it relates toboth capture and detrapping rate constants. We also consider the role of rate equations inthe theory of TL and OSL and their origin in quantum statistics and relaxation theory. Wecontinue by considering emission and absorption of radiation, how emission and absorptionrates are related through detailed balance, and discuss estimates for their values. We brieflydiscuss the theoretical work found in the literature on the association of trapping parameterswith certain impurities embedded in given crystals. Finally, we give an account of a numberof aspects of possible mechanisms leading to thermal quenching of luminescence.

2.1 Energy Bands and Energy Levels in Crystals

We start with a very brief explanation of the basic properties of a crystal which enable therich variety of conductivity and luminescence properties. The basic theory of all kinds ofluminescence in solids, including TL and OSL has to do with the energy band of solids.The solution of the Schrodinger equation for electrons in a periodic potential yields allowedbands separated by forbidden bands (see e.g. Kittel [40] and Ibach and Luth [41]). In a pureinsulating and semiconducting crystal at absolute zero (0 K), all the bands up to the onecalled the valence band are full of electrons. The next allowed band, called the conduction

Thermally and Optically Stimulated Luminescence: A Simulation Approach, First Edition. Reuven Chenand Vasilis Pagonis. © 2011 John Wiley & Sons, Ltd. Published 2011 by John Wiley & Sons, Ltd.

8 Thermally and Optically Stimulated Luminescence

band, is empty of electrons and so are the higher allowed bands. The forbidden band betweenthe valence band and the conduction band is called the forbidden band or the gap. Electronicconduction in the crystal can take place only if electrons from the valence band are givenenough energy to reach the conduction band. Once an electron is in the conduction band, itcan contribute to the electrical conductivity. Moreover, the missing electron in the valenceband can be considered as a positive charge carrier, a “hole”, and it can move in the crystal,thus contributing to the conduction. At finite temperatures, electrons can be thermally raisedfrom the valence band into the conduction band. However, this may take place at relativelylow temperatures in semiconductors which have a relatively narrow band gap, and hardlyoccurs at all in insulators which are the main subject of the present book, due to their broadband gap. For both perfect semiconductors and insulators with a band gap Eg, opticalabsorption only takes place for light with photon energies larger than Eg, namely withfrequencies above Eg/h where h is the Planck constant.

All real crystals are not ideal in the sense that they always include imperfections, namelydefects and impurities. This causes a local change in the otherwise periodical system, andnew energy levels are thus produced in the forbidden gap, which makes it possible forelectrons and holes to get “trapped”. This means that these carriers may possess ener-gies that are forbidden in the ideal crystal. The occurrence of these traps or centers maycause additional optical absorption of light with photon energies significantly lower thanthe band-to-band energy. Thus, new absorption bands may be observed, which may changethe visible color of the crystal. The occurrence of the trapping states in the forbidden gapchanges in many cases very drastically the conductivity properties as well as the lumi-nescence features, and practically all the effects discussed in the present book have to dowith transitions between such energy levels and the valence and conduction bands. Thenature of these trapping states depends on the host material. In the case of impurities,the properties of these point defects also depend on the foreign atoms and ions presentin the material and in their location in the host material. As for the defects, the proper-ties of the relevant energy levels depend on the specific defect. The allowed energy levelsin the forbidden gap may be discrete or distributed depending on the host lattice and thespecific imperfections.

A well known defect type is the Frenkel defects which are interstitial atoms, ions ormolecules normally located on the lattice site, which have moved out of their originalplace. The corresponding vacancies are called Schottky defects. The latter may be the resultof a diffusion of the host ions to the surface of the crystal. In some cases high energyradiation may produce a pair of vacancy–interstitial located in rather close proximity toeach other, thus forming a defect of a different nature. Another cause for a disturbancein the periodicity of the perfect crystal is the presence of the surface. This can result intrapping levels in the periodic potential, thus yielding usually shallow trapping levels in thesurface region.

Low-energy radiation and sometimes even high-energy radiation applied to a samplemay not produce new defects, but in most cases play a crucial role in the filling of traps andcenters associated with existing impurities and defects. On the other hand, the absorptionof photons may photostimulate previously trapped charge carriers into the conduction orvalence band, and this leads to a reduction of an expected TL or OSL effect. Practically allthe phenomena discussed in this book are related to transitions of this kind. The connectionof the effects of excitation and de-excitation to the luminescence phenomena which are the

Theoretical Basis of Luminescence Phenomena 9

subject matter of the book, are elaborated upon starting with Chapter 4 which describes themost basic way of producing TL and OSL (see in particular Figure 4.1).

2.2 Trapping Parameters Associated with Impurities in Crystals

It is quite obvious that the properties of the traps and recombination centers are directlyderived from the nature of the host crystal and the imperfections, impurities and defects,embedded in the crystal. Thus, in principle, knowledge of which imperfections are involvedshould yield all the relevant parameters. This would mean that in addition to knowledgeabout the emission spectrum associated with the relevant recombination centers, one mightexpect to know the activation energies, frequency factors, as well as the recombination andretrapping-probability coefficients associated with the capture cross-sections for retrappingand recombination and the thermal velocity of the carriers. Note that another kind of param-eters, namely the total concentrations of traps and centers, is of a different nature since it hasto do only with the amounts of the relevant imperfections. One would expect that quantummechanical theory should yield the values of the trapping parameters. In the literature, thereare reports of such quantum mechanical treatments which yield predictions on lumines-cence properties as well as other related features of the materials with given imperfections.To the best of our knowledge, none of these is directly related to TL and OSL in the sensethat one cannot predict, for instance, the dosimetric behavior of a given crystal with givenamounts of certain impurities from first principles.

Some related theoretical works are to be mentioned here briefly. Williams [42, 43]describes the theory of luminescence of impurity-activated ionic crystals, using the absolutetheory of the absorption and emission of these crystals. The detailed atomic rearrangementsfollowing the optical transitions and the equilibrium among accessible atomic configu-rations of the activator system are determined quantitatively. The author shows that theabsorption and emission spectra of KCl:Tl at 298 K can be predicted on theoretical grounds.Norgett et al. [44] studied theoretically the electronic structure of the V− center in MgO,where a hole is trapped at a cation vacancy. Using a model for lattice relaxation calcu-lations with a Coulomb part and a short-range interaction which has overlap and van derWaals components, they can explain the energies of optical transitions of 1.5 and 2.3 eVwhich involve transitions occurring within an O− and hole hopping from one oxygenion to another, respectively. Lagos [45] describes a quantum theory of interstitial impu-rities and shows that the analytical calculation of a number of effects like phonon-assistedtunneling and optical absorption, which are valid for massive impurities and high concen-trations, follow as a direct application. Testa et al. [46] calculated the excitation energies ofVk-centers in NaCl by combining an unrestricted Hartree–Fock code with classical poten-tials to simulate the defect and the distorted lattice around it. In a textbook on solid-statetheory, Harrison [47] discusses impurity states in crystals. The tight-binding descriptionis considered, dealing with the situation in which one of the ions in an ionic crystal isreplaced by an impurity ion. Impurities, donors and acceptors are also studied in semi-conductors. The quantum theory of surface states and impurity states is elaborated upon.In a review paper and, in particular in an extensive book, Stoneham [48, 49] discussesthe theory of defects in solids, dealing with the electronic structure of defects in insu-lators, in particular ionic crystals, and semiconductors. Among other important subjects,

10 Thermally and Optically Stimulated Luminescence

he considers different forms of F-centers (F, F’, Ft), M-centers, R-centers, Vk-centers,H-centers, etc.

Extensive work by Dorenbos and co-workers [50–53] has been devoted to the evaluationof the trivalent and divalent lanthanide energy level location in different materials used forTL and OSL dosimetric materials such as CaSO4:Dy3+, SrAl2O4:Eu2+, YPO4:Ce3+ andmany others. Dorenbos [50] presented the systematic variation in the energy level positionsof divalent lanthanides in wide band gap ionic crystals. He concludes that the width of thecharge transfer (CT) band in spectra does not correlate with the width of the valence band.Also, the width of the CT luminescence in Yb3+-doped compounds is about the same asthe width of the CT absorption. Finally, there is no significant dependence of the width ofthe CT band on the type of lanthanide. By comparing the CT energies for different trivalentlanthanides in the same host, constant energy differences were revealed. This may help inpredicting the relevant energies. For instance, once the CT bands in Eu3+ is known, theenergies of other lanthanides can be evaluated. In subsequent papers, Dorenbos and Bos[51] and Bos et al. [52] use the same ideas to locate the energy levels in YPO4 doped bydifferent lanthanide ions and discuss the related TL phenomena. Bos et al. [53] state that thetrend of predicted trap depths agrees well with experimental results. However, the absoluteenergy-level positions show a systematic difference of ∼ 0.5 eV. They extend the researchby studying the excitation spectra of the OSL of a YPO4:Ce3+,Sm3+ sample in order toelucidate the discrepancy between the predicted value of 2.5 eV and experimental result of2.1 eV. From the OSL excitation spectra at different temperatures they deduce the value ofthe trap depth predicted by the Dorenbos model.

One should note, however, that as far as TL and OSL are concerned, the state of the art atpresent is such that in most cases one cannot calculate theoretically the important trappingparameters from first principles. Given a certain host crystal and a specific imperfection,be it a defect or impurity, one cannot predict in most cases the relevant values of frequencyfactors and recombination and retrapping-probability coefficients, and only in few casesthe activation energies can be determined using quantum-mechanical considerations. Fur-thermore, most crystals have different kinds of imperfections, and it is not always clearwhich one of them is directly (or indirectly) involved in the TL and OSL phenomena. Thus,although it is obvious that the imperfections are always the source of all the luminescenceeffects discussed in this book, bridging the gap between quantum theory and the theoret-ical work discussed in this book, which is based mainly on using the relevant sets of rateequations, is still a desirable goal to be dealt with in the future.

2.3 Capture Rate Constants

The lifetime for a free electron is typically of the form 1/A(N − n) where A is a capturerate constant and N − n is the concentration of available trapping sites. While both of thesequantities can vary by orders of magnitude from one material to another, the range is notunlimited. In practice, trap concentrations N are limited on both the high and low ends. Inpure materials, one expects trap concentrations to be small but, in even the more refinedsingle crystals, recombination centers are found with densities of 1012 cm−3 or more. At thehigh end, trap concentration is limited by the requirement that there be enough separationthat the wavefunctions do not overlap. Deep traps can be closely packed in high band gap

Theoretical Basis of Luminescence Phenomena 11

materials where values of N as high as 1019 cm−3 are observed. Next, the observed rangesfor capture rate constants will be discussed (see e.g. Rose [54]).

2.3.1 General Considerations

Let us consider an electron (or hole) traveling in a solid at a speed of v. Let us supposethat there is some distance rc, such that if the electron comes within rc of a trap, it will becaptured. Over a time t, the electron travels a distance vt and it will have been capturedif there was a trap anywhere within the volume vt(πr2

c ). The probability of capture withintime t is then the probability that there was a trap within the volume vt(πr2

c ). If traps have adensity of N, then the expected number of traps within that volume is Nvt(πr2

c ). The capturerate, that is the probability of capture per unit time, is thus Nv(πr2

c ). Since we are interestedin free electrons with a velocity distribution characterized by a temperature T , we need toaverage that capture rate over all thermal speeds which yields Nv(πr2

c ) where v is the meanthermal speed. Elsewhere in this book, the capture rate for a free electron is written as AN

where A is the capture rate constant. It is thus clear that

A = v(πr2c ). (2.1)

Very often, the area πr2c is called a cross-section and denoted by σ. In this case, the rate

constant A can be written as

A = vσ. (2.2)

The mean thermal speed is given by

v =√

8kT

πm, (2.3)

where k is Boltzmann’s constant, T is temperature, and m is the effective mass of theelectron. Effective masses for electrons and holes vary with the crystal structure. Typicalvalues range from 0.05me to 2me where me = 9.1 × 10−31kg is the mass of an electron infree space. As a rough order of magnitude, 107cm s−1 is a typical mean thermal speed foreither electrons or holes.

To investigate the orders of magnitude involved, let us suppose that the capture radiusof a trap is its physical radius. If the trap is the size of an atom, then one might guess thatrc ∼ 2 × 10−8cm. It would follow that the cross-section is

σ = πr2c = 3.14 × (2 × 10−8cm)2 ≈ 10−15cm2. (2.4)

Using this σ, a typical capture rate constant would then be

A = vσ ∼ 107cm s−1 × 10−15cm2 ∼ 10−8cm3s−1. (2.5)

Experiments show that Equations (2.4) and (2.5) are actually typical magnitudes forcapture cross-sections and rate constants, respectively, if the trap is neutral. For exampleBemski [55] measured a cross-section of 5 × 10−16 cm2 for electrons being captured bya neutral Au trap in a silicon crystal at room temperature. Alekseeva et al. [56] measuredσ = 10−15 cm2 for capture of holes by neutral Bi traps in germanium crystals. Lax [57]

12 Thermally and Optically Stimulated Luminescence

surveyed experimental data for electron or hole capture by neutral traps and concludedthat typical cross-sections range from 10−17 to 10−15 cm2 (i.e., A ranging from ∼ 10−10 to10−8 cm3 s−1).

When traps have a net charge, very different cross-sections are observed. If the free par-ticle and the trap have charges of the same sign, there is a long-range electrostatic repulsionbetween them. Consequently, capture is unlikely. Experiments for this case show capturecross-sections of ∼10−21 cm2 (i.e., A ∼ 10−14 cm3 s−1) or smaller [57]. If the free particleand trap have, by contrast, opposite signs, then the free particle will be electrostaticallyattracted to the trap. In these cases, some very large cross-sections are observed. Values of10−15–10−12 cm2 are observed.

A trapped electron is at a considerably lower energy state than a free electron. Therefore,as part of the capture process, the electron must lose a significant amount of energy. It isbelieved that the energy is lost to lattice vibrations (i.e., phonons). Lax [57] has analyzedthis process.

In a later work, Mitonneau et al. [58] present a method which combines optical absorp-tion and electrical refilling of deep levels, allowing one to measure the majority-carriercapture cross-section for minority carrier traps. They report very large (10−15 cm2) and verysmall (10−21 cm2) electron capture cross-sections for two levels in GaAs:Cr, and suggest apossible capture mechanism for these cases.

The above considerations apply to capture of an electron or hole by a trap. The reverseprocess of thermal detrapping is closely related and the rate constants for detrapping can beestimated if the trapping rate constants are known. Before this can be discussed, we needto review thermodynamics, which we shall do in the next section.

2.4 Thermal Equilibrium

In a solid, there are many possible quantum states that an electron could occupy. Thesestates could be in a valence band, in a conduction band, or attached to traps in between.Because there are many such states, we will not deal with them individually but ratherconsider their distribution N(E) where N(E)dE is the number of such states per unit volumewith energies between E and E + dE. In thermodynamic equilibrium, the probability thatany electron state is occupied by an electron depends on the state’s energy E and is given bythe formula

f (E) = 1

1 + exp(E − Ef )/kT. (2.6)

Equation (2.6) is called the Fermi–Dirac distribution or, for short, the Fermi distribution. Tis the absolute temperature of the solid, measured in Kelvin. k is the Boltzmann constantand Ef is known as the Fermi energy.

Since we know that N(E)dE is the number of states with energy between E and E + dE

and f (E) is the probability that any such state is filled with an electron, it follows that theexpected number of electrons occupying a state between E and E + dE is f (E)N(E)dE.If we integrate this quantity over all energies, we can find the expected total number of

Theoretical Basis of Luminescence Phenomena 13

electrons in the solid, Ne

Ne =∫

N(E)f (E)dE. (2.7)

Under normal circumstances, solids are nearly electrically neutral. This means that, for everyproton in a nucleus, the solid also has an electron. If one rubs the solid with an electronegativematerial or an electropositive material, one can generate a charge imbalance in the solid(“static electricity”). Even when such imbalances are present, the difference between thenumber of electrons and the number of protons is typically much smaller than the populationof either alone, and can be ignored for our purposes. This means that the total number ofelectrons in the solid Ne should match the total number of positive charges in the nuclei.

Consider a solid held at a temperature of absolute zero: T = 0. In this case, the Fermidistribution, Equation (2.6) simplifies to

f (E) =

⎧⎪⎨⎪⎩

0 if E > Ef ,

1/2 if E = Ef ,

1 if E < Ef .

(2.8)

At room temperature, kT = 1/40 eV while we deal with materials with band gap energiesof 1–12 eV. Consequently, it is useful to consider approximations in which temperature issmall. For energies more than a few kT above the Fermi energy, the term exp[(E − Ef )/kT ]is much larger than one. On the other hand, for energies more than a few kT below theFermi energy, that exponential is much smaller than one. These observations lead to usefulapproximations for the Fermi–Dirac distribution

f (E) =

⎧⎪⎨⎪⎩

exp[−(Ef − E)/kT ] if kT � E − Ef ,

1/2 if E = Ef ,

1 − exp[(E − Ef )/kT ] if kT � Ef − E.

(2.9)

This indicates that traps with energies sufficiently below the Fermi level are nearly filledwith electrons. We will find it convenient to call these traps “hole-type”.

Let us consider a luminescent material with a valence band, a conduction band, and oneor more traps as shown in Figure 2.1. If the material is in equilibrium at temperature T , thenFermi–Dirac statistics says that the fraction of trap states N1 that are filled with electrons is

n1

N1= f (E1) = 1

1 + exp(E1 − Ef )/kT≈ exp[−(E1 − Ef )/kT ], (2.10)

where E1 is the energy of the trap and Ef is the Fermi level. Finding the population offree electrons, nc, in the conduction band is slightly more complicated because the freeelectrons have a range of energies. However, after taking into account the density of statesin the conduction band and integrating over energy, one finds

nc

Nc

= f (Ec) = 1

1 + exp(Ec − Ef )/kT≈ exp[−(Ec − Ef )/kT ], (2.11)

14 Thermally and Optically Stimulated Luminescence

Conduction band

Fermi level

Valence band

nc

nυ

n1, N1

εc

εf

ευ

ε1

Energy

Figure 2.1 The energy levels of interest for the thermal equilibrium discussion are shown. Thismaterial may have one or several traps or centers. For our discussion, though, we only need torefer to one

where Ec is the energy of the edge of the conduction band and

Nc = (2πmkT )3/2

h3 , (2.12)

where h is Planck’s constant. Now consider the ratio

nc

n1≈ Nc

N1exp[−(Ec − E1)/kT ]. (2.13)

Note that, in equilibrium, the ratio of the free electron population nc to the trapped electronpopulation n1, depends only on material properties and temperature. Also, since the energydifference Ec − E1 occurs quite frequently in luminescence studies, it is convenient to definea symbol for it

E1 = Ec − E1. (2.14)

E1 is the binding energy of the electron trap; it is the amount of energy that would berequired to raise an electron in the trap up into the conduction band. We can then write

nc

n1= Nc

N1exp(−E1/kT ). (2.15)

2.5 Detailed Balance

Let us consider an electron trap N1 which can capture electrons from the conduction bandwith rate AN1nc or the reverse process can occur via thermal excitation; the electron trap canlose electrons back to the conduction band with rate γ . The charge conservation equation