We present a numerical model of scintillator nonproportionality based on rate equations including carrier and exciton transport in assumed cylindrical electron tracks, solved by finite element method. Framed in volumetric excitation density n, these coupled rate equations describe hot and thermalized carrier transport, bimolecular exciton formation, carrier capture, and nonlinear quenching processes expressed in terms of physical rate constants and transport coefficients that have been measured and/or calculated independently of scintillation measurements in the example of CsI:Tl. The set of equations is solved in the spatial non-uniformities of an electron track and in times spanning the cooling of hot carriers up through thermalized transport and capture. The solution of the rate equations is combined with GEANT4 simulation of electron tracks to model electron energy response from the SLYNCI Compton-coincidence experiment.
The application of advanced theory and modeling techniques has become an essential component to understand material properties and hasten the design and discovery of new ones. This is true for diverse applications. Therefore, current efforts aimed towards finding new scintillator materials are also aligned with this general predictive approach. The need for large scale deployment of efficient radiation detectors requires discovery and development of high-performance, yet low-cost, scintillators. While Tl-doped NaI and CsI are still some of the widely used scintillators, there are promising new developments, for example, Eu-doped SrI2 and Ce-doped LaBr3. The newer candidates have excellent light yield and good energy resolution, but challenges persist in the growth of large single crystals. We will discuss a theoretical basis for anticipating improved proportionality as well as light yield in solid solutions of certain systems, particularly alkali iodides, based on considerations of hot-electron group velocity and thermalization. Solid solutions based on NaI and similar alkali halides are attractive to consider in more detail because the end point compositions are inexpensive and easy to grow. If some of this quality can be preserved while reaping improved light yield and possibly improved proportionality of the mixture, the goal of better performance at the low price of NaI:Tl might be attainable by such a route. Within this context, we will discuss a density functional theory (DFT) based study of two prototype systems: mixed anion NaIxBr1-x and mixed cation NaxK1-xI. Results obtained from these two prototype candidates will lead to further targeted theoretical and experimental search and discovery of new scintillator hosts.
Models of nonproportional response in scintillators have highlighted the importance of parameters such as branching
ratios, carrier thermalization times, diffusion, kinetic order of quenching, associated rate constants, and radius of the
electron track. For example, the fraction ηeh of excitations that are free carriers versus excitons was shown by Payne and
coworkers to have strong correlation with the shape of electron energy response curves from Compton-coincidence
studies. Rate constants for nonlinear quenching are implicit in almost all models of nonproportionality, and some
assumption about track radius must invariably be made if one is to relate linear energy deposition dE/dx to volume-based
excitation density n (eh/cm3) in terms of which the rates are defined. Diffusion, affecting time-dependent track radius
and thus density of excitations, has been implicated as an important factor in nonlinear light yield. Several groups have
recently highlighted diffusion of hot electrons in addition to thermalized carriers and excitons in scintillators. However,
experimental determination of many of these parameters in the insulating crystals used as scintillators has seemed
difficult. Subpicosecond laser techniques including interband z scan light yield, fluence-dependent decay time, and
transient optical absorption are now yielding experimental values for some of the missing rates and ratios needed for
modeling scintillator response. First principles calculations and Monte Carlo simulations can fill in additional
parameters still unavailable from experiment. As a result, quantitative modeling of scintillator electron energy response
from independently determined material parameters is becoming possible on an increasingly firmer data base. This
paper describes recent laser experiments, calculations, and numerical modeling of scintillator response.
The dipole-dipole and free-carrier Auger quenching processes that are generally regarded to be at the root of
nonproportionality depend respectively on the 4th or 6th power of the electron track radius if modeled as cylindrical. In
an initial time interval τ when nonlinear quenching and diffusion compete to reduce the density of excited states, the
track radius expands as (Deffτ)1/2 where Deff is the effective diffusion coefficient for the mixture of excitons and charge
carriers. The range of Deff across semiconductor and scintillator radiation detectors is large, illustrated by 8 decades
between mobilities of self-trapped holes in CsI:Tl and holes in high purity Ge. We present the functional form of
nonlinear quenching predicted by diffusive track dilution and show that the simple model provides a surprisingly good
fit of empirical nonproportionality across a wide range of semiconductor and oxide radiation detectors. We also show
how diffusion drives nonlinear branching between excitons and free carriers in the track when electron and hole
mobilities are unequal, and that this nonlinear branching coupled with linear trapping on defects can produce the "halide
hump" seen in electron yield data for activated halide scintillators. Picosecond time-resolved spectroscopy in alkali
halides, as well as quantitative comparison of recently measured 2nd order quenching rate constants K2 and results of K-dip
spectroscopy, provide experimental benchmarks for consideration of carrier thermalization and the initial track or
cluster radius r0 from which (nearly thermalized) diffusion is assumed to commence. The ratio of initial rate of 2nd order
quenching to that of dilution by diffusion in a cylindrical track is proportional to K2/Deff and does not depend on r0 in
lowest order; however, the absolute rates of both processes decrease with increasing r0.
The effect of high excitation density in promoting nonlinear quenching that is 2nd or 3rd order in electron-hole density is
generally understood to be a root cause of nonproportionality in scintillators. We report and discuss quantitative data on
just how fast these nonlinear channels are in specific cases. Kinetic rate constants for the creation of excitons from
electrons and holes and for their quenching by dipole-dipole transfer have been measured in CsI and NaI. We show in
addition that the strong radial concentration gradient in an electron track gives rise to fast (~ picoseconds) diffusion
phenomena that act both as a competitor in reducing excitation density during the relevant time of nonlinear quenching,
and as a determiner of branching between independent carriers and pairs (excitons), where the branching ratio changes
along the primary electron track. We use the experimentally measured nonlinear quenching rate constants and values of
electron and hole carrier mobilities to carry out quantitative modeling of diffusion, drift, and nonlinear quenching
evaluated spatially and temporally within an electron track which is assumed cylindrical in this version of the model.
Magnitude and inequality of electron and hole mobilities has consequences for quenching and kinetic order that vary
with dE/dx along the path of an electron and therefore affect nonproportionality. It will be demonstrated that in a
material with high mobilities like high-purity germanium, Auger recombination is effectively turned off by diffusive
carrier dilution within < 1 fs in all parts of the track. In alkali halide scintillators like CsI and CsI:Tl, electron
confinement and high-order quenching are accentuated toward the end of a particle track because of hole self-trapping,
while separation of geminate carriers is accentuated toward the beginning of the track, leading to 2nd order radiative
recombination and opening additional opportunities for linear trapping.
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