RLumCarlo: Simulating Cold Light using Monte Carlo Methods
Sebastian Kreutzer
Department of Geography & Earth Sciences, Aberystwyth UniversityJohannes Friedrich
Chair of Geomorphology, University of BayreuthVasilis Pagonis
Physics Department, McDaniel CollegeChristian Laag
Université de Paris, Institut de Physique du Globe de Paris, CNRSEna Rajovic
Chair of Geomorphology, University of BayreuthChristoph Schmidt
Institute of Earth Surface Dynamics, University of Lausanne2021-06-07
Abstract
The luminescence phenomena of insulators and semiconductors (e.g., natural minerals such as quartz) have various application domains. For instance, Earth Sciences and archaeology exploit luminescence as a dating method. Herein, we present the R package RLumCarlo implementing sets of luminescence models to be simulated with Monte Carlo (MC) methods. MC methods make a powerful ally to all kinds of simulation attempts involving stochastic processes. Luminescence production is such a stochastic process in the form of charge (electron-hole pairs) interaction within insulators and semiconductors. To simulate luminescence-signal curves, we distribute single and independent MC processes to virtual MC clusters. RLumCarlo comes with a modularized design and consistent user interface: (1) C++ functions represent the modeling core and implement models for specific stimulations modes. (2) R functions give access to combinations of models and stimulation modes, start the simulation and render terminal and graphical feedback. The combination of MC clusters supports the simulation of complex luminescence phenomena.Introduction
Light is perhaps the most basic everyday experience. Light emission that is not caused by the heating of a substance is called luminescence or ‘cold light’. Various fields exploit this phenomenon. For instance, Earth Sciences and archaeology determine the timing of past events (e.g., last sunlight exposure or heating) with a technique called luminescence dating. Since 2012, the luminescence-dating (or more general trapped-charge dating) community has gradually adapted R as a universal tool to analyze, model, and visualize their data. Relevant related CRAN packages are: BayLum (Bayesian modeling: Philippe, Guérin, and Kreutzer 2019; Christophe et al. 2018), Luminescence (luminescence- data analysis, Kreutzer et al. 2012, 2020), numOSL (luminescence-data analysis, Peng et al. 2013; Peng and Li 2018), RLumModel (luminescence-data modeling, Friedrich, Kreutzer, and Schmidt 2016, 2020), RLumShiny (graphical interface to functions for plotting and calculation in the framework of luminescence-data analysis, Burow et al. 2016; Burow, Wolpert, and Kreutzer 2019), and tgcd (curve deconvolution, Peng, Dong, and Han 2016; Peng 2020).
The luminescence production process is a stochastic process involving discrete random state transitions of subatomic particles. In the case of luminescence, this translates to electrons (and holes) moving to different energy levels, e.g., in the crystal lattice of the natural mineral quartz. Such processes are ideal for Monte Carlo (MC) simulations, and their application has a long and propelling history in physics (cf. Landau and Binder 2015). Figure\(~\)1 summarizes the purpose of RLumCarlo developed to simulate luminescence signals in semiconductors and insulators (e.g., quartz) using MC methods. To that end, RLumCarlo employs simple (energy-band) models that describe the physical processes in, e.g., the quartz crystal, to simulate xy-curves (luminescence curves). The modeling output expresses the evolution of the light production (luminescence process) over time.
Our contribution, and so RLumCarlo, sits on precedent work by Vasilis Pagonis and Chen (2015), Vasilis Pagonis et al. (2014), Vasilis Pagonis et al. (2019), and Vasilis Pagonis et al. (2020). The included collection of energy-band models for different stimulation modes adapted to MC methods are valuable for, e.g., studying the impact of model parameters on the signal-related stochastic uncertainties or statistic effects in tiny, dosimetric systems. Technically, our approach is closely related to the simulation of birth-and-death processes (for a review on birth-and-death process cf. Novozhilov, Karev, and Koonin 2006). Each simulation run describes a Markov process. However, in our case, we allow only a reduction of an initial number of particles (i.e., only death processes).
Herein, we will not derive the full theoretical background of the models, but we will focus on the technical aspects of the package design and the integration of the MC methods. Such a presentation was beyond the scope of previous articles (e.g., Vasilis Pagonis et al. 2019, 2020), but it is likely of interest to a broader community.
We structured our contribution as follows. The introduction continues with a brief paragraph on luminescence and the term ‘cold light’. After that, we detail the rationales for our contribution by recalling conventional modeling approaches in the field. Readers familiar with these topics may safely skip this part. The subsequent section outlines the concept and the implementation of RLumCarlo, including code examples. The remainder addresses (A) the implementation of a virtual dosimetric system to simulate weak spatial correlation of dosimetric cluster groups. (B) We outline how RLumCarlo can simulate more complex models compared to other solutions, with respect to its strengths and limitations. An outlook outlining the potential to implement more interactions between models will close our manuscript.
‘Cold light’ in a nutshell
Light emissions of semiconductors or insulators after exposure to ionizing radiation is a luminescence phenomenon now and then paraphrased as ‘cold light’. The term luminescence relates to light production not purely caused by the heating of a substance, a condition called ‘incandescence’ or black body radiation, but a phenomenon expressing the inherent capacity of a material (dosimeter) to emit light (energy) in the ultraviolet to infrared wavelength range (e.g., Newton Harvey 1957; Mahesh, Weng, and Furetta 1989). Heat-related luminescence phenomena of solids have been explored systematically in physics since the 1930s (Urbach 1930) to characterize materials and understand charge transfers in dosimeters (e.g., McKeever 1983; Mahesh, Weng, and Furetta 1989). The amount of luminescence, in the context of this manuscript, correlates to the energy absorbed by a dosimeter during ionizing irradiation. The closest analogy to a dosimeter is a battery that can be charged by, e.g., \(\gamma\)-radiation and discharges while emitting light. Natural minerals such as quartz or feldspar are dosimeters. Defects and impurities in their crystal lattice can trap charges (electrons or holes) in metastable states between valence and conduction band. The time an electron spends in such a state can vary from a fraction of a second to millions of years, depending on the crystal lattice configuration and the environmental conditions. The amount of (potential) energy held by an electron in such a center is approximately the energy difference between the valence band and the energy level of the center. A transition of the electron to a lower energy state may lead to a photon emission of the form \(E_{photon} = \hbar\omega_{nm} = E_{n} - E_{m}\)1. Energy input (‘stimulation’) can move the electron out of the defect and eventually it recombines with a hole trapped at another defect (‘recombination centre’). Types of stimulation methods relevant for our contribution are heat (thermally stimulated luminescence, TL), visible light (optically stimulated luminescence, OSL), and infrared light (infrared stimulated luminescence, IRSL).
Luminescence phenomena have versatile use in the fields of personal, medical, and accidental dosimetry (e.g., Yukihara and McKeever 2011). As aforementioned, in Earth Sciences and archaeology, the luminescence of natural minerals gained considerable attractiveness as a dating method (luminescence dating). First attempts exploiting luminescence signals as a chronological tool reach back to the 1950s (Daniels, Boyd, and Saunders 1953; Houtermans and Stauffer 1957; Grögler, Houtermans, and Stauffer 1958). Nevertheless, it needed a few decades more before the method took off and became today one of the most frequently used dating methods on sediments for the last 250,000 years and beyond (e.g., Aitken 1985, 1998; Bateman 2019).
Towards Monte Carlo simulations
To explain luminescence production, Johnson (1939) and Randall and Wilkins (1945) introduced the first basic energy-band models. Today, most of the commonly accepted luminescence models use series of more or less complicated systems of differential equations (for an overview, see Reuven Chen and McKeever 1997; Bøtter-Jensen, McKeever, and Wintle 2003; R. Chen, Lawless, and Pagonis 2011) employing energy-band models. Those models provide a proper phenomenological match with measured data for various experimental designs by simulating electronic transitions. ‘Conventional’ energy-band models available to simulate luminescence production are developed as a set of nonlinear differential equations. This brings some limitations:
The models become complex easily and cannot be solved analytically.
If numerical methods are used, some equations are numerically unstable, which may lead to wrong simulation results.
A convenient assumption in many of such models is a great abundance of spatially uniformly distributed traps and recombination centers. However, this is not always the most prudent assumption. A spatial correlation and cluster formation of centers may exist for various reasons (cf. Mandowski and Świaltek 1992; R. Chen, Lawless, and Pagonis 2011; Horowitz, Eliyahu, and Oster 2017).
Deterministic models do not consider stochastic uncertainties and simulated curves are ‘noise free’. These limits subsequent analyses for materials where such uncertainties would matter due to the low, finite, number of charge carriers, and in these scenarios, simulation results are used as reference data to test statistical models used for luminescence data analysis in general.
Modeling code for simulating luminescence production was often written with the tools at hand, e.g., Mathematica (e.g., Vasilis Pagonis, Kitis, and Furetta 2006), which has led to a fragmentation of incompatible solutions. In 2016, Friedrich, Kreutzer, and Schmidt (2016) introduced RLumModel (Friedrich, Kreutzer, and Schmidt 2020), pooling available kinetic (non-MC) models available for the luminescence production in quartz. A tantamount suite of R code was presented simultaneously by Peng and Pagonis (2016). We will compare results from RLumModel and RLumCarlo at the end of this manuscript.
MC simulations offer an alternative and are indispensable if the simulation of defect clusters in combination with the analysis of stochastic uncertainties is desired. Usually, the underlying models are very simple, but can be combined to describe complex systems. Important early work simulating TL using MC methods goes back to Mandowski and Świaltek (1992) and Kulkarni (1994). Mandowski and Świaltek (1992) tried to overcome the prerequisite of a large number of sample carriers, and Kulkarni (1994) investigated MC methods to overcome very long calculation times encountered for numerical calculations in particular scenarios. Kulkarni (1994) (p. 103) also reported a “statistical fluctuation” (noise like scatter) caused by the MC simulations but considered this more as a disadvantage. Later, Vasilis Pagonis et al. (2020) explicitly exploited this as a feature, similar to birth-and-death processes and their related random uncertainties, to investigate specifically the stochastic uncertainties.
Before we start to detail RLumCarlo, a preceding note of caution: Any attempt to answer the question of whether a particular model may better explain the one or the other effect measured in luminescence studies would open Pandora’s box (e.g., Horowitz, Eliyahu, and Oster 2017). Consequently, we will not engage in such a discussion. What we have implemented so far in RLumCarlo can be modified and exchanged. However, the underlying design concept remains applicable.
The concept of RLumCarlo
RLumCarlo implements energy-band models in a modular approach. Each model can simulate only an isolated effect (e.g., a single curve, see below), but the package design allows various combinations, e.g., in the form of clusters. Hence, RLumCarlo can evolve beyond a specific mathematical model through a combination of simple models.
To that end, RLumCarlo differs fundamentally from RLumModel, where the collected models allow the simulation of complex phenomena and even entire measurement sequences (Friedrich, Kreutzer, and Schmidt 2016) but are self-contained by design. In other words, simulations cannot evolve beyond a specific mathematical model selected by the user. In RLumCarlo, the implemented energy-band models can simulate only isolated effects (e.g., a single curve, see below), but the package design allows a combination in the form of clusters. Throughout the text, we will use the word ‘clusters’ to (1) ascribe virtual units used in the MC simulation to run independent random processes (henceforth MC clusters) and (2) to define groups of defects (defect clusters) e.g., defined by their spatial distance. Only the latter carry a physical meaning.
Implemented energy-band models
To date, RLumCarlo ships three simple energy-band models (Figure\(~\) 2) to simulate luminescence production using (A) delocalized transitions, (B) localized transitions, and (C) excited state tunneling transitions. The models are distinguished by the allowed routes of electrons involved in the luminescence process from one energy state to another. Only the first model (Figure\(~\) 2A) involves the conduction band, while the models in Figures\(~\)2B and C limit the allowed electron pathways to energy levels below the conduction band.
While the parameters differ from model to model and depend on the stimulation mode (heat or light, continuous or ramped), key entities remain alike across the models, such as the trap depth (the energy difference of the electron state from the conduction band) \(E\) (eV), the attempt to escape frequency of an electron from the trap (short: frequency factor) \(s\) (s\(^{-1}\)), the temperature \(T\) (K), and the trapped concentration of electrons \(n\) (cm\(^{-3}\)) in the trap. \(N\) (models A and B) is the total number of available electrons in cm\(^{-3}\) and \(\rho'\) is the dimensionless density of recombination centers (model C, Huntley 2006). The symbols \(A_n\), \(A_m\), and \(A\) (model A), \(B\) and \(A\) (model B), and \(B\), \(A\), and \(r'\) (model C) plotted next to the arrows in Figure\(~\) 2 parametrize, simply put, the rates of the electronic transitions.
The conditions of the simulations are defined through these parameters, with \(n\) being the crucial number. Once the electrons have all recombined, the simulation may still continue, but the luminescence signal is zero. As we will detail below, the essential point of the MC simulation, from the physical point of view, is that these concentrations become dimensionless, absolute numbers in a finite system.
Each model supports up to four different stimulation modes (Figure\(~\) 3), i.e., the type of energy input (light or heat) and its modulation (continuous or ramped).
As an example, we will detail the mathematical background and its implementation for delocalized transitions below. For all other models, we may refer to the cited literature as well as the package manual.
Conceptional overview of the implementation
The basic implementation of the MC processes as a software algorithm consists of two nested loops. The outer loop iterates over a time \(0<t\leqslant t_{max}\) with \(t \in \mathbb{R} > 0\). The inner part loops over particles \(0<j\leqslant n\) with \(n \in \mathbb{Z}\). The model tests a random number, newly sampled with replacement in each run, against a threshold \(P\). If the sampled random number is smaller than \(P\), the absolute number of particles is reduced by one. The code below shows the basic algorithm outlined above for the radioactive decay, which we have chosen because it can be found in standard textbooks (e.g., Landau and Binder 2015). Below we used R code for illustrative reasons, while the package implementation is written in C++.
n <- 1000
t <- 1:100
P <- 0.2
remaining <- numeric(length(t))
for (t in t) {
for (j in 1:n) {
if (runif(1) < P && n > 0)
n <- n - 1
}
if (n > 0)
remaining[t] <- n
}For example, the algorithm starts with 1,000 particles. In time instant \(t_1\), the random number was smaller for two particles \(j_6\) and \(j_{576}\). Hence, in time instant \(t_2\), the inner loop iterates only over \(j = \{1,2,...,998\}\) particles. The absolute number of remaining particles for each \(t\) is stored in a vector of length \(t_{max}/\Delta t\). This vector is the observed signal curve (in the case of luminescence, the righthand side graph on the green board in Figure\(~\) 1).
Implementation example for the OTOR model
The implementation for luminescence production in RLumCarlo is very similar. To exemplify the adaptation of the models to be run as an MC simulation, we have selected the one-trap one-recombination center (OTOR) model (based on Halperin and Braner 1960) for TL. Our description below follows Vasilis Pagonis and Chen (2015).
The OTOR model for TL can be expressed with the following set of differential equations:
\[\frac{dn}{dt} = -ns~exp\big(-\frac{E}{k_{B}T}\big) + n_c(N-n)A_{n}\]
\[\frac{dn_{c}}{dt} = -\frac{dn}{dt} - n_{c}mA_{m}\]
\[I_{TL}(t) = -\frac{dm}{dt}= n_{c}mA_{m}.\]
Beyond already mentioned symbols, we used in the equations \(I_{TL}\), the time-dependent intensity, and \(n_c\) (cm\(^{-3}\)), the current concentration of electrons in the conduction band. The concentration of recombination centers is represented by \(m\) (cm\(^{-3}\)), where for reasons of charge neutrality \(m = n + n_{c}\). \(A_{n}\) and \(A_{m}\) (both in cm\(^{3}\,\)s\(^{-1}\)) are the capture coefficients for traps and recombinations centers, respectively. \(k_{B}\) (eV\(\,\)K\(^{-1}\)) is the Boltzmann constant and \(T\) (K), the absolute temperature.
By assuming quasi-static equilibrium conditions (R. Chen, Lawless, and Pagonis 2011) \[\Big|\frac{dn_{c}}{dt}\Big| \ll \Big|\frac{dn}{dt}\Big|,\Big|\frac{dm}{dt}\Big|~~;~~ n_{c} \ll n, ~~ n \simeq m,\]
the resulting TL intensity becomes the general one trap equation, GOT:
\[I_{TL}(t) = -\frac{dn}{dt} = s~exp(-\frac{E}{k_{B}T})~\frac{A_{m}n^{2}}{(N-n)A_{n} + nA_{m}}.\]
\[T = T_{0} + \beta \times t,\]
with \(T\) (K) and \(T_{0}\) (K) being temperatures, \(\beta\) (K\(\,\)s\(^{-1}\)) the (heating) rate, and \(t\) (s) the simulation time. \(p(t) = s~exp(-\frac{E}{k_{B}T})\) is the rate of thermal excitation, and \(R = \frac{A_{n}}{A_{m}}\) is the dimensionless retrapping ratio. The translation into a finite system with a discrete distribution of charge carriers (cf. Mandowski and Swiatek 1991, 1994), can be expressed through
\[\chi n,\chi N \rightarrow{} \bar{n}, \bar{N},\]
and the differential equation becomes a difference equation:
\[I_{TL}(t) = -\frac{1}{\beta}\frac{\Delta \bar{n}}{\Delta t} = p(t) \frac{\bar{n}^2}{\bar{N}R + \bar{n}(1-R)}.\]
\(\chi\) (cm\(^3\)) is a constant, \(\bar{n},\bar{N} \in \mathbb{Z}\), and \(\Delta t = 1\,\)s is an appropriate time interval. \(R\) is the dimensionless re-trapping ratio in the finite system. To simulate the luminescence process, the related Markov process renders similar to the theory of birth-and-death processes (e.g., Novozhilov, Karev, and Koonin 2006), where the population (here of electrons) decreases over continuous time with the probability to observe a transition within \(\Delta t\) being \(P = \mu_{\bar{n}} \Delta t\) (here \(\mu_{\bar{n}}\) is the “death-rate” in s\(^{-1}\)) until the population is depleted. The so-called ‘brute force’ approach (e.g., Landau and Binder 2015) tests the population of electrons (\(\bar{n}\)) per integer time step sequentially by comparing it against a random number sampled with replacement from a continuous distribution \(r \sim \mathcal{U}(0,1)\) against the conditional probability \(P\) for an electron to get evicted from the trap. In our case, \(P\) is calculated as follows:
\[p(t) \times \delta t \times \frac{n}{\bar{N}R + \bar{n}(1-R)}.\]
The factor \(\delta t\) allows
values of \(\Delta t \neq 1\) while
ensuring that \(P \ll 1\). \(p(t)\) depends on the stimulation mode and
the chosen model. For TL (functions named
run_MC_TL_<model>()) and isothermal TL (functions
named run_MC_ISO_<model>()) applying the localized or
delocalized model \(p(t)\) becomes:
\[p(t) = s~exp(-\frac{E}{k_{B}T}),\]
and for TL, from tunneling transitions it reads:
\[p(t) = s~exp(-\frac{E}{k_{B}T})~exp(-\rho'^{-1/3}r'),\]
with \(\rho'\) being the dimensionless concentration of recombination centers, and \(r'\) being the dimensionless tunneling radius (Huntley 2006). The basic structure in RLumCarlo is, however, identical, except for the models based on excited-state tunneling. Here, an additional outer loop iterates over the dimensionless tunneling radius \(0 \leqslant r' \leqslant 2\) (Huntley 2006).
The package design
In Figure\(~\) 4, we outline the basic layout of RLumCarlo.
Two design decisions stand out: (1) Each stimulation mode/energy-band
model combination has its own exported R function commencing with the
prefix run_MC. (2) These functions interface one C++
function, each via Rcpp (Eddelbuettel et al. 2020) (for an overview, see
the vignette of RLumCarlo).
The R functions provide a convenient user interface, and the C++
functions constitute the workhorse, as shown in the modeling core
(Figure\(~\) 4). While the apparent reason for using C++ was
speed, the implementation could have been programmed more concisely,
i.e., completely in C++ instead of interfacing C++ with R. However, we
wanted to allow code inspection by non-specialists from our field, who
may wish to implement other models alike. We found that the separation
of the user interface (in R) from the modeling core (in C++) aligns best
with our premise of simplicity and flexibility.
As indicated above in the example implementation algorithm, each simulation run (Kulkarni 1994 used the term ‘particle tracking’) starts with \(n > 1\) and ends at \(t_{max}\), while \(I(t) = 0\) for \(n = 0\). In reality, one has to execute several simulation runs separately (henceforth ‘MC clusters’, to be distinguished from defect clusters), either to reduce the statistical fluctuation or to estimate the stochastic error (Kulkarni 1994). RLumCarlo runs the simulations in virtual MC clusters on single or multicore systems using parallel (R Core Team 2020), doParallel (Microsoft Corporation and Weston 2020), and foreach (Microsoft and Weston 2020) supported by helper functions (Figure\(~\) 4), to summarize results and to provide S3-class based graphical output.
Simple illustrative examples
Simulations start with a call of the respective function, e.g., for
TL using the DELOC model
run_MC_TL_DELOC() or run_MC_TL_LOC() for the
LOC model, respectively (see Figures\(~\)2A-B).
results <- run_MC_TL_DELOC(
s = 3.5e12,
E = 1.45,
clusters = 10,
n_filled = 200,
R = 1,
times = 100:450)The function parameter names follow the terminology used for the
mathematical expression of the models as closely as possible. For
example, E and s are \(E\) (trap depth in eV) and \(s\) (frequency factor in s\(^{-1}\)). R (\(R\) in Figure\(~\) 3) is the
so-called re-trapping ratio, expressing the chance an electron of
not passing into the conduction band, and n_filled
is \(\bar{n}\), the number of electrons
at the beginning of the simulation. The duration of the simulation on
the time domain (which is not the duration of the computation) is set by
times. The parameter clusters sets the number
of MC clusters, i.e., the number of Markov chains. High numbers in
clusters increase the confidence in the simulation output
at the cost of more computation time.
The output can be passed to a dedicated plot function
(plot_RLumCarlo()). The function supports a couple of
standard plot arguments, such as main for the title of the
plot, which is passed down to graphics::plot.default() via
... (type ?dots in the R terminal).
plot_RLumCarlo(
object = results,
legend = TRUE,
main = "(A) Delocalized transition")
The parameter n_filled can be a vector enabling
different starting conditions for each MC cluster. Figure\(~\)5A shows the
graphical output for delocalized transitions along with the simulation
results for TL stimulation using localized (Figure\(~\) 5B) and
tunneling transitions (Figure\(~\) 5C). The output is an object of class
RLumCarlo_Model_Output, which is a list
comprising a multi-dimensional array (one slice per MC
cluster) with the resulting luminescence signal and a
numeric vector for the stimulation time.
Currently we provide S3-generics for summary() and
c(). The first one is also used internally by
plot_RLumCarlo() to melt the array into a
data.frame before plotting. The plot output adapts to the
used stimulation mode provided via an attribute with each output
object.
A straightforward application for this kind of simulation is the study of the impact of physical parameters on the luminescence signal output and the estimation of the stochastic uncertainties, which cannot be achieved with the deterministic approach of differential equations.
We provide more, always up-to-date examples with the package vignette, where we also compiled a table with meaningful physical parameter ranges for each model.
Advanced examples and further considerations
The examples so far presented may not appear very sophisticated, and still, they allow insight that goes beyond a simple educational purpose of simulating luminescence based on phenomenological models. Vasilis Pagonis et al. (2020), who used a preliminary version of RLumCarlo, addressed in detail the stochastic uncertainties of TL and OSL models. These uncertainties come into play in nano-dosimetric materials with a small number of defect clusters where the “finite-size” (Mandowski and Swiatek 1991) of the system starts to matter in terms of a presumed spatial correlation of defect cluster groups. To some extent, this should also be true for systems exposed to high-energy radiation causing defect clusters (e.g., Mandowski and Swiatek 1991; Mandowski and Świaltek 1992). Previously in this paper, we have used the term ‘MC clusters’. For a start, in RLumCarlo, ‘MC clusters’ entail independent and continuous Monte Carlo Markov chains employed to simulate luminescence production, starting with a particular number of electrons in the system. Whether the processes are run in parallel or sequentially has no impact on the outcome, except for computation speed. In other words, ‘MC clusters’ carry no meaning regarding the underlying physics. However, as mentioned above, ‘MC clusters’ from different models (with the same stimulation mode) can be concatenated (see Figure\(~\) 6B-C) to simulate defect clusters (also, dosimetric clusters), to which we can attribute physical meaning.
Spatial correlation
To simulate a three-dimensional (dosimetric) system, we can add meaning to MC clusters by reinterpreting them as dosimetric clusters. From the modeling perspective, nothing changes, but MC clusters gain a connotation of having a physical meaning.
Figure\(~\)7A illustrates the situation of model combinations transferred into a virtual, three-dimen-sional dosimetric system. Since all defect clusters are distributed evenly over the system, the distance to each neighboring point is identical, and it is a constant rather than a variable. In other words, the spatial distance between neighboring points does not matter and is of no relevance for the simulation but here chosen for illustrative reasons only. Figure\(~\)7B represents a situation that takes one step further. Here, the points are randomly distributed over the system, and points form groups (defect cluster groups). Additionally, RLumCarlo supports the mixing of models for the same stimulation mode as in Figure\(~\) 7A (not shown here). The driving idea of the implementation is the assumption of an individual spatial ordering of defects in a, e.g., quartz crystal to which the luminescence production process might be assigned based on models mentioned above.
Such a system can be created in RLumCarlo
via create_ClusterSystem(). The function distributes points
randomly with their coordinates:
\[x_1,y_1,z_1,...,x_k,y_k,z_k \sim \mathcal{U}(0,1)~\mathtt{|}~k \in \mathbb{Z}.\]
Then, the Euclidean distance between the points is determined with
stats:dist(), which is used by stats:hclust()
to group the defect clusters (\(\Xi\)).
To avoid too many small groups, we then cut the cluster tree using
stats:cutree(), with the outcome shown in Figure\(~\) 7B. The
selection of stats:hclust() and stats:cutree()
for defining the clusters is somewhat arbitrary and might be refined in
the future. Therefore, more research, supported by measurements, is
needed.
Now, any function from RLumCarlo
can be used, and the output of create_ClusterSystem() is
taken as input for the argument clusters. For example:
run_MC_TL_LOC(s = 3.5e12, E = 1.45, n_filled = 1000,
clusters = create_ClusterSystem(100))creates a system with 100 randomly distributed defect clusters. If
the simulation is run in such a mode, the meaning of
n_filled changes. Previously, it defined the number of
electrons in each cluster (\(\bar{n}_{cl_i}\)). However, now the same
parameter defines the total number of electrons in the entire system.
The number of electrons in the \(i\)th
cluster (\(\bar{n}_{cl_i}\)) is then an
integer fraction of electrons available in each cluster group (\(\bar{n}_{\Xi_{i}} = \mathtt{n\_filled}/
N_{\Xi}\), with \(N_{\Xi}\) the
total number of cluster groups). The more clusters are in one group, the
less electrons are available per cluster in the group and vice versa.
While this is a very simple approach, it allows us to simulate basic
spatial correlation. Figure\(~\)7C drafts a better way of mimicking spatial
interaction of clusters, which is, however, not yet part of RLumCarlo.
While it would be, based on the designed system, easy from the
programming perspective, the needed equations to describe to exchange
electrons are yet to be developed.
Comparison to RLumModel
In the remainder, we want to compare simulation results from RLumCarlo, with other types of solutions, such as RLumModel which uses coupled differential equations to simulate luminescence production. RLumModel was selected since it was developed by some of the authors of this contribution. However, in theory, simple scripts using any existing models to simulate luminescence should work as well (as long as the models are comparable).
In contrast to RLumCarlo, RLumModel input values for physical parameters are preset. RLumModel encourages users to write a virtual luminescence signal measurement sequence, which is processed based on a pre-defined model with preset physical parameters.
For the comparison, we have selected a TL curve simulated with the luminescence model for quartz by Bailey (2001).
output <- RLumModel::model_LuminescenceSignals(
sequence = list(IRR = c(20, 10, 1), TL = c(20, 400, 1)),
model = "Bailey2001"
)The results are shown in Figure\(~\) 8 (here already with the simulation results from RLumCarlo plotted on top of it). The output of RLumModel is a three-peak-shaped curve. To simulate the same curve in RLumCarlo, we used the parameters from the model by Bailey (2001) (his Table 1), e.g., for the first peak:
TL110 <- RLumCarlo::run_MC_TL_DELOC(
s = 5e+12, E = 0.97, R = 5e-10, times = seq(20, 400, 2),
N_e = output$`conc. level 1 (TL)`[1,2] / 1e+5)N_e was divided by a constant to reduce the computation
time. The dimensionless parameter R corresponds to \(B\) (s\(^{-1}\)) in Bailey
(2001). The other two peaks were simulated alike (objects
TL230 and TL325) before all three objects were
combined via:
object <- c(TL110, TL230, TL325)and plotted on the top of the curve derived from RLumModel:
RLumCarlo::plot_RLumCarlo(
object = object,
plot_value = "sum",
add = TRUE,
FUN = function(x) {
x * 1/(1 + (1e+7 * exp(-0.61/(8.617e-5 * (object$time + 273)))))}
)The argument plot_value = "sum" was used to plot the
total count sum instead of its average. The additional function injected
via the argument FUN corrects the TL curves for a
phenomenon known as thermal quenching (Wintle
1975). This is a reduction of luminescence production efficiency
at higher temperatures. The chosen quenching parameters follow roughly
data measured for quartz by Friedrich, Kreutzer,
and Schmidt (2018). In summary, the results in Figure\(~\) 8 show that
even complex luminescence models can be simulated through the
combination of clusters, which brings us back to the initial
‘simplicity’ premise of RLumCarlo.
Still, a big ‘but’ remains. Luminescence models such as those proposed
by Bailey (2001) or V. Pagonis et al. (2008) go beyond single curve
simulations. Their purpose is to deliver a general kinetic model for
luminescence production of, e.g., quartz, including the simulation of
trap filling by irradiation and the simulation of the thermal activation
history of the mineral. By contrast, so far all simulations in RLumCarlo
start with a predefined number of electrons in a trap and are not by
default limited to a specific dosimeter. RLumCarlo
can model more complex luminescence phenomena, but not in a
pre-described way out of the box. Instead, RLumCarlo
is more like a patch box with each model representing a socket ready to
be flexibly rewired in many ways to simulate cascades of luminescence
production. Due to the nature of the chosen MC approach, in theory
(adhering to the patch box picture), the number of sockets is not
limited.
Summary
The modeling of luminescence phenomena (cold light) of semiconductors and insulators after having received ionizing radiation is a challenging task. MC methods allow setting up flexible and simple systems to simulate luminescence with a finite number of charge carriers. This enables users to address effects usually observed for nano-dosimetric systems, and it provides insight into the stochastic uncertainty structure. We presented RLumCarlo, which renders, to our best knowledge, the first open-source and ready-to-use compilation of basic MC luminescence models for different stimulation modes (so far CW-OSL, LM-OSL, ISO-TL, and TL). We showed that the output from different models, which are simulated in separate MC chains in virtual clusters, can be combined to either simulate more complex systems or to mimic simple spatial correlations between cluster groups. The way of the implementation does not limit RLumCarlo to a specific dosimeter (e.g., quartz). In this light, RLumCarlo can be used in education, and research to test the impact of model parameters, such as cluster sizes and related stochastic uncertainties. Furthermore, RLumCarlo can help in in formulating research hypotheses and test them with commonly accepted or new models, still to be developed.
Future work will implement more models to run as MC simulation, e.g., for irradiation processes in crystals (including its luminescence output: radiofluorescence) and for an advanced interaction of clusters.
Acknowledgements
We thank two anonymous reviewers for their thorough reviews and constructive suggestions, which helped to improve our manuscript. Furthermore, we are grateful to the CRAN team in general for their tireless efforts in keeping such a great resource alive and in particular for their patience during the initial submission of RLumCarlo. Alex Roy Duncan is thanked for his support on the package development during our stay in Westminster. The development of RLumCarlo benefited from the support of various funding bodies. The initial work by JF, SK and CS was supported by the Deutsche Forschungsgemeinschaft (2015–2018, DFG SCHM 3051/4-1, “Modelling quartz luminescence signal dynamics relevant for dating and dosimetry”). Later financial support was secured through the project “ULTIMO: Unifying Luminescence Models of quartz and feldspar” granted by the Deutscher Akademischer Austauschdienst (DAAD PPP USA 2018, ID: 57387041). SK was supported by the LabEx LaScArBx (ANR - n.\(^{\circ}\)ANR-10-LABX-52) until 2019. From 2020, SK has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 844457 (CREDit). This is IPGP contribution number 4202.
\(\hbar\) (eV\(~\)s): Planck constant divided by \(2\pi\); \(\omega\) (radians per s): frequency; \(E_{n}\) (eV): higher energy state; \(E_{m}\) (eV): lower energy state↩︎