Simulation of ionization charge carrier cascade time and density for a new radiation detection method based on modulation of optical properties

Abstract

Background:

In time-of-flight PET, image quality and accuracy can be enhanced by improving the annihilation photon pair coincidence time resolution, which is the variation in the arrival time difference between the two annihilation photons emitted from each positron decay in the patient. Recent studies suggest direct detection of ionization tracks and their resulting modulation of optical properties, instead of scintillation, can improve the CTR significantly, potentially down to less than 10 picoseconds CTR. However, the arrival times of the 511 keV photons are not predictable, leading to challenges in the spatiotemporal localization characterization of the induced charge carriers in the detector crystal.

Purpose:

To establish an optimized experimental setup for measuring ionization induced modulation of optical properties, it is critical to develop a versatile simulation algorithm that can handle multiple detector material properties and time-resolved charge carrier dynamics.

Methods:

We expanded our previous algorithm and simulated ionization tracks, cascade time and induced charge carrier density over time in different materials. For designing a proof-of-concept experiment, we simulated ultrafast electrons and free-electron x-ray photons for timing characterization along with alpha and beta particles for higher spatial localization.

Results:

With 3 MeV ultrafast electrons, by reducing detector crystal thickness, we can effectively reduce the ionization cascade time to 0.79 ps and deposited energy to 198.5 keV, which is on the order of the desired 511 keV energy. Alpha source simulations produced a cascade time of 2.45 ps and charge carrier density of 6.39 × 10²⁰cm⁻³. Compared to the previous results obtained from 511 keV photon-induced ionization track simulations, the cascade time displayed similar characteristics, while the charge density was found to be higher. These findings suggest that alpha sources have the potential to generate a stronger ionization-induced signal using the modulation of optical properties as the detection mechanism.

Conclusions:

This work provides a guideline to understand, design and optimize an experimental platform that is highly sensitive and temporally precise enough to detect single 511 keV photon interactions with a goal to advance CTR for ToF-PET.

1. Introduction

Current state-of-the-art time-of-flight positron emission tomography (TOF-PET) systems achieve a coincidence time resolution (CTR) of ~200 ps. Such scintillation-based detection systems use the visible light photons which are created at the final stage of a multi-stage process ^1–3. Once the annihilation photon is stopped in the detection crystal, the induced charge carriers generated migrate to impurity sites to release their energy via visible photons (radiative recombination)⁴. This process is stochastic as it requires the ionized charge carriers to diffuse across the crystal, creating a significant temporal variation in photon emission, and limiting the CTR to the order of 100 picoseconds ³ for 3×3×20 mm³ crystal elements. If instead it was possible to directly detect the transient ionization charge created ^1,5, without waiting for the downstream carrier recombination, one might be able to achieve an order of magnitude better CTR. Considering optical effects, it turns out that the transient charge carriers that result from ionizing interactions produce transient modulation of optical properties on ultrafast time scales (i.e. femtosecond) ⁶. Previous work showed that ionizing photons can generate measurable modulation of optical properties such as refractive index and absorption⁷. This novel approach utilizing modulation of optical properties for ionizing radiation detection shows promise in achieving a less than 10 picoseconds (ps) coincidence time resolution (CTR), which, if successful, would have a significant impact on the field of PET ^1,8,9.

However, thus far we have observed radiation-induced modulation of optical properties by integrating over many 511 keV photons interactions ⁷; but a useful PET detector requires the detection of a single 511 keV photon interaction. To detect a single 511 keV photon interactions, however, two significant challenges must be addressed: temporal response and sensitivity.

A single 511 keV photon interaction typically generates total of ~10⁵ charge carriers along the ionization track for the case of CdTe, therefore requiring high sensitivity to detect ~100 femtojoules deposited in a 50 × 50 × 50μm³ volume ⁶. For the timing, we experimentally confirmed that ionizing radiation induces ultrafast modulation of optical properties ⁶, and we pioneered a technique we refer to as interferometric spectral encoding that can interrogate ionization-induced modulation of optical properties on an ultrafast time scale in a single-shot measurement. To address the two challenges, the experimental setup needs to be built by carefully considering multiple variables of materials, probe wavelength, and optics settings, and therefore having a simulation algorithm that can handle multiple parameters and physics is important.

Here, we expand the previous simulation study and develop a time-resolved simulation framework to precisely understand the cascade time and evolution of charge carrier density over time. In our previous simulation work ¹⁰, we obtained cascade time and charge carrier in CdTe by employing a charge carrier transport model (Lumerical) and incorporating the modulation of optical properties using the Drude model¹¹. These simulations were performed under equilibrium conditions, once the ionization cascade had finished. In this paper, we develop a charge density calculation algorithm that considers each inelastic collision interaction point in a given detector material, incorporating the cascade time during the ionization process. We therefore create a flexible framework that can simulate charge carrier densities in a non-equilibrium state, and screen for materials that can generate high charge carrier densities for strong modulation of optical properties.

To define proof-of-principle experiments in both timing and sensitivity for single 511 keV annihilation photon interaction detection, we simulated a range of ionizing radiation sources and the associated induced transient charge carrier dynamics. For timing, we simulated both synchronous and asynchronous (randomly emitting) ionizing radiation sources. The Ultrafast Electron Diffraction (UED) source ^12–14 provides synchronous high energy electron sources with sub-picosecond time resolution. X-ray free-electron lasers¹⁵ can generate photons with variable energies. Next, to relax the sensitivity requirement, we simulate asynchronous ionization sources such as alpha and beta particles that have shorter ranges and higher spatial focus of charge carrier generation. The findings will quantitatively guide the design of an appropriate experimental setup that addresses time resolution and sensitivity requirements to detect the single 511 keV annihilation photon interactions via optical detection methods.

2. Materials and methods

2.1. Materials

2.1.1. Synchronized ultrafast x-ray photon and electron ionizing radiation sources

In PET, the 511 keV photons are generated when a positron emitted from a proton rich nucleus annihilates with a nearby electron, and thus the exact time of this event cannot be predicted a priori. As a surrogate gold standard for understanding potential timing and sensitivity that can be achieved from ionization events produced by annihilation photon interactions in a crystal, one possible approach is to utilize synchronized sources. However, a synchronous photon source at 511 keV is unavailable.

There are two ultrafast synchronized ionizing radiation sources available to us, x-ray free electron lasers and ultrafast electron pulses. X-ray free electron laser (XFEL) sources such as Linac Coherent Light Source (LCLS) at Stanford linear accelerator (SLAC) National Accelerator Laboratory can generate up to 20 keV x-ray photons. The x-ray pulses at XFEL exhibit excellent timing characteristics in the femtosecond range,⁶ making them a valuable testbed for evaluating the signal strengths of optical property modulations, which depend on the material properties of the detector crystal. However, in general, the x-ray photons have significantly lower, sub-100 keV energy, and the ionization cascade time needs to be characterized. With the higher energy x-ray laser capabilities under development, such as LCLS-II at 50 keV, our simulation work will provide preparation for experiments at such hard x-ray regimes.

Another synchronized source we have employed utilizes a high-energy ultrafast electron pulse source available at ultrafast electron diffraction (UED) facilities ^12,16,17. However, the energy levels of UED cannot be easily adjusted to the 511 keV range as the more common electron beam energy levels are in the MeV range, exceeding the energy of the annihilation photon.

To fully utilize the ultrafast timing capabilities of the synchronized sources and their implication for asynchronous 511 keV photon ionization-induced modulation of optical properties, we first need to precisely understand the ionization process in different energy ranges and quantify the relative signal strengths of optical property modulations that result. Therefore, we simulate photon and beta particle sources to compare different energy regimes and densities of ionization. This is critical to establishing the relevance of experimental studies at different energies to the 511 keV events we are ultimately interested in.

2.1.2. Asynchronous radiation sources

In addition to synchronous sources, we also simulate asynchronous radiation sources with shorter ranged particles. We modeled Am-241 as a source of alpha particles of 5.5 MeV, as well as the beta endpoint energy of Tl-204, 763 keV, as a monoenergetic beta particle source. We compare simulation results of ionization charge carrier cascade time and density between the above sources and simulated 511 keV photon sources, Ge68 and Na22.

2.1.3. Detector material selection for understanding the optical property modulation strength

Given the relatively small energy deposition and resulting modulation of optical properties from a single 511 keV photon, it is crucial to choose a detector material that demonstrates robust changes in optical properties when exposed to ionizing radiation. In our study, we examine multiple material properties and conduct simulations on a wide range of materials to determine the most suitable crystal for an optical property modulation-based PET detector. The strength of optical property modulation is heavily influenced by the properties of the detector material, including atomic, semiconductor, and optical characteristics. Through simulation, we gain a comprehensive understanding of the contributions of each property. Our goal is to select the optimal materials based on three main considerations: radiation-matter interaction, semiconductor properties, and optical properties.

For radiation detection, one of the most important features is the stopping power, which is essentially the probability of interaction between 511 keV photon and matter. The stopping power is governed by 1) effective density of the material, 2) effective atomic number Z, and 3) material thickness. Semiconductor properties are important as they dictate the number of free carriers generated and how long they live after the generation and before recombination. Per each energy deposition via inelastic scattering (${\text{E}}_{\text{deposited}}$), the electron-hole pair generation energy, ${\text{E}}_{\text{e-h}}$ which is always greater than the bandgap energy, tells us how many electron-hole pairs are generated (${\text{n}}_{\text{carrier}}$), as $n_{carrier}=\frac{E_{deposited}}{E_{e-h}}$. Lower bandgap energy tends to generate more electron-hole pairs, leading to larger modulation, but will have higher dark noise due to thermo-ionic emission. Lastly, for modulation of optical properties, we are seeking materials that can yield the highest modulation for a given number of free carriers produced. The magnitude of changes in optical properties such as refractive index or absorption coefficients from the transient generation of free carriers is a crucial factor.

We first select a few promising materials with good properties, then compare each material with the simulation. Primarily, we selected Cadmium Telluride (CdTe) and Ytterbium Aluminum Garnet (YAG) as the materials to be studied in depth for UED simulations. CdTe is a semiconductor with a bandgap energy of 1.43 eV and good stopping power for high energy photons due to relatively high density and high Z. In recent studies, CdTe has demonstrated strong modulation of optical properties⁷. YAG has shown potential as a time calibration material ^6,18 due to its non-birefringent property. Also, YAG is an insulator, which requires a different simulation approach than semiconductors, similar to that used in inorganic scintillators.

Additionally, we simulated PbBiGa, CdS, and CdSe with alpha, beta, and photon sources. PbBiGa is a photonic crystal that has high density and stopping power, with lower bandgap which can generate more charge carriers per deposited energy, resulting in strong modulation of optical properties. We added Cd-based materials such as CdS and CdSe as they have relatively good stopping power with good carrier mobility which can lead to strong modulations of optical properties. Finally, for comparison of energy deposition among alpha, beta, and 511 keV photons, we simulated a variety of other materials such as GaP, GaAs, UO₂, BSO, and BGO, with different densities and bandgap energies. For example, GaP and GaAs are low-density semiconductors whereas UO₂ has high density and Z. BGO is a commonly used scintillator and BSO is a highly optoelectric crystal ¹⁹.

The thickness of crystals were either finite, where only a portion of the particles energy is deposited, or infinite, to ensure the full ionization trajectory was absorbed. For simulation using electrons with energy relevant to UED, the finite thickness samples were selected to limit the energy deposited from the ionization trajectory and the resulting cascade time in order to approximate that expected for a 511 keV annihilation photon interaction.

2.2. Methods

We introduce the simulation process and software setup to obtain transient charge carrier density and cascade times. Time-resolved simulations were crucial to determine the timing characteristics of charge carrier generation. We expand the previous algorithm that calculated the free carrier effects arising from the primary electrons resulting from 511keV photon interactions ¹⁰ to simulate synchronous electron and x-ray pulses from UED and LCLS, respectively, as well as asynchronous beta, alpha, and 511 keV photon sources. The entire procedure for simulating the time-resolve charge carrier and modulation of optical properties arising from changes in the complex refractive index can be summarized in four steps: 1) obtaining electron trajectories from Monte Carlo simulation, 2) calculating absorbed energy, primary electron velocity, and cascade time, 3) simulating time-resolved charge carrier density, and 4) calculating optical property modulations. Transmission geometry for detection across the detection crystal was selected to accommodate the interferometric spectral encoding technique. First, the electron trajectory is simulated, then absorbed energy and electron velocity were calculated to obtain cascade time. Next, time-resolved charge carrier densities and resulting modulations of optical properties were calculated.

2.2.1. Simulation of ionization trajectories

The source particle ionization trajectories were simulated using pyPENELOPE for beta sources or GEANT4 for alpha particles. PyPENELOPE is a Monte Carlo simulation (PENELOPE²⁰) package²¹. The input parameters include electron energy, number of electrons, sample density, and geometry. We vary the thickness of the sample to control the absorbed energy and cascade time. The output of pyPENELOPE provides information on the spatial distribution of the deposited energy per injected electron, per each inelastic scatter interaction, simulated via the Monte Carlo method. We varied the electron energy across 350 keV, 750 keV, and 3 MeV.

We used GEANT4²² for simulating alpha trajectories as they are not supported in the pyPENELOPE. GEANT4 is a simulation toolkit that studies the passage of particles through matter ²². As in pyPENELOPE, particles are injected normally onto one base of a cylindrical volume, whose cross-section is large enough to allow no particle escaping sideways. In GEANT4, the cylinder is divided into thin slices internally for the sake of discretizing energy deposition distribution.

2.2.2. Calculation of cascade time

From trajectories generated by pyPENELOPE and GEANT4, we extract the velocity and absorbed energy of the incident electron at each collision step, or interaction, of the trajectory. From special relativity considerations, the electron velocity (v) can be related to the total energy (${\text{E}}_{total}$), rest energy (${\text{E}}_{\text{0}}$), and speed of light (c) as:

$$v=\frac{\sqrt{E_{total}^2-E_0^2}}{E_{total}}c,$$ (1)

which, combined with the distance between events, allows timing information to be extracted.

2.2.3. Time-resolved calculation of induced charge carrier density

With the time and energy information, we obtained a time-resolved charge density profile, which we directly calculate in MATLAB from trajectory data. For the charge density, we calculate the volume with the assumption that the charge carriers diffuse once they are created in the ionization trajectory. Ranges for 50–200 eV electrons can be estimated to be ~1 nm²³, so for those interactions with small energy deposition, charges are created within 1 nm. We track the charge trajectories with cutoff energy of 50 eV (for 350 keV and 750 keV), and 200 eV (for 3 MeV). Once the charges are created from the inelastic collisions, the volume occupied by the charge carriers increases through a diffusion process. Assuming the charge volume at each inelastic collision is spherical, the charge density ${\rho }_{carrier}$ is obtained via

$${\rho }_{carrier}\left(t\right)=\frac{\left(E_{abs}\right)/\left(E_{e-h}\right)}{\frac{4}{3}\pi r(t{)}^3},$$ (2)

where $E_{abs}$ is the absorbed energy, $E_{e-h}$ is the electron-hole pair creation energy, $r(t)$ is distance the charge carrier traveled.

From 3D diffusion equation, we have,

$$r\left(t\right)=\sqrt{6*D*t},$$ (3)

where D is the diffusion constant of detector material, and t is the time between the current inelastic collision and the first inelastic collision of the trajectory²⁴.

Then we use Lumerical DEVICE to get the free carrier generation profile with charge transport taken into consideration (Lumerical, 2016). The absorbed energies and charge generation rate were imported into DEVICE as the initial conditions for the charge transport simulation. For 3 MeV electrons, the volume of the entire ionization cloud can reach up to mm³ and exceeds DEVICE computation capabilities, and we divide up the entire volume into smaller volumes, typically, less than (200 μm)³. For lower energies of 350keV and 750keV, the simulation runs with the entire volume. As DEVICE is optimized for semiconductors, YAG needed to be added manually, and the convergence limit was relaxed.

2.2.4. Drude model based calculation of absorption modulation

To illustrate the modulation of optical properties resulting from the 3 MeV asynchronous beta source, we conducted an example calculation by converting the free carrier density to the modulation strength of the absorption coefficient in MATLAB. This calculation assumed a probe laser wavelength of 1.5μm, commonly used in the telecommunication industry, and utilized the Drude model to calculate the change in complex refractive index^11,25,

$$\Delta \alpha =\left(\frac{e^3{\lambda }^3}{4{\pi }^2{c}^3{ϵ}_0}\right)[\frac{\Delta N_e}{{m_{ce}^{\mathrm{*}}}^2{\mu }_e}+\frac{\Delta N_h}{{m_{ch}^{\mathrm{*}}}^2{\mu }_h}],$$ (4)

where $\text{Δ}\alpha$ is the change in absorption coefficient, e is the electronic charge, ${\epsilon }_0$ is the permittivity of free space, $n$ is the refractive index of the unperturbed material, $m_{ce/h}^{\ast }$ is the conductivity effective mass of electrons/holes, ${\mu }_{e/h}$ is the electron/hole mobility, and $N_{e/h}$ is the change in electron/hole carrier density.

2.2.5. Simulation Pipeline

To acquire multiple trajectories, a software pipeline was established to automate the data collection process and develop an umbrella program that serves as an interconnect between the different simulation programs.

As shown in Fig. 1, the overarching script pipeline.py calls all the modalities of pyPENLEOPE, computes the charge generation profile, then sends the information to and runs the DEVICE simulation. The pipeline processes the data in between simulation steps in order to automatically translate it from one program’s output to the next program’s desired input. The flexibility of the pipeline comes from its capability to define multiple properties such as materials and particle energies and run batches of simulations that vary these parameters in ways specified by the user. To accomplish the simulation task, we utilized Sherlock, a high-performance computing (HPC) cluster at Stanford.

Figure 1:

Pipeline Dependency Diagram. Python was selected as the umbrella platform to interface multiple program modalities of pyPENELOPE, Lumerical CHARGE solver, and MATLAB. The user can select cascade.py for timing simulation or batch.py for electron density and optical property modulations simulation.

Due to the compatibility issue of the requirement for a graphical interface for the Lumerical CHARGE simulation, the pipeline was divided into two parts and ran separately. First, the electron trajectories were generated through pyPENELOPE then the charge generation profile through a python script. With the saved charge generation profile, a Lumerical script was run to receive the charge generation profiles and execute CHARGE simulation. A container technology called Singularity²⁶ was used when running the CHARGE simulation, which allows the installation of required libraries that is typically limited on the HPC setting. Once the electron densities were obtained through the DEVICE simulation, a python script was used to simulate the modulation of absorption coefficient based on the Drude model in the transmission geometry, from the electron density. In this work, our focus was primarily on utilizing the simulation pipeline for 3MeV electrons.

3. Results

3.1. Time-resolved simulation for synchronized sources

3.1.1. Simulations for Synchronized Ultrafast Electrons From a UED Source

First, we simulated the ionization trajectories from 100, 3 MeV electrons interacting in CdTe for a crystal with unlimited thickness and for a 200 μm thick portion of that crystal; the results are shown in Fig. 2 (a) and (b). With the electrons incident along Z, its energy is deposited step by step. In the 200μm sample thickness, the spatial dispersion in the x-y plane was reduced because the distance the ionization trajectory spans in z was limited to 200 μm, whereas for the full crystal, the track’s z-dimension was in the mm range.

Figure 2:

Ionization trajectories from 100, 3 MeV electrons in CdTe, for a crystal with unlimited thickness and a 200 μm thick crystal. (a) In the thick CdTe crystal, the ionization tracks span several millimeters in x, y, and z. (b) When the thickness was limited to 200μm, the dispersion in the x, y plane decreased to less than 200μm. Distributions of electron cascade timing for 3 MeV electron in CdTe for the (c) full crystal and (d) 200 μm thickness. In the full crystal, the mean cascade time is 12.3 ps due to the greater electron energy deposited, hence longer ionization tracks. In the 200 μm thick CdTe sample, cascade time decreased to 0.79 ps due to the lower number of collisions/energy deposited and shorter ionization trajectories. (e) An example of a time-resolved charge carrier cascade along a trajectory in YAG for a 350 keV electron. The charge carrier density increases monotonically over the cascade time of 1.7 ps, reaching up to 2.3 × 10¹⁸ cm⁻³ at the end of the trajectory. (f) Distribution of electron density at the end of the trajectory, simulated for 1000, 350 keV electrons traversing unlimited thickness YAG sample.

Next, we simulated 1000 electron trajectories to obtain histograms of the cascade time distribution, as shown in Fig. 2 (c) and (d). In the full CdTe crystal, the cascade time exhibits a broad distribution, with a mean of 12.33 ps and standard deviation of 4.02 ps, with overall cascade times ranging in several picoseconds which is greater than the cascade time for a 480 keV photoelectron from a 511 keV photon interaction¹⁰. On the other hand, the 200 μm thick sample gives a much narrower distribution of cascade time than the full crystal case, effectively reducing the cascade time. This reduction in cascade time results from the lower number of inelastic collisions due to the limit in the thickness, hence shorter tracks. As shown in Fig. 2 (d), cascade time becomes less than 1 ps, with a mean of 0.79 ps (std of 0.1 ps) for a 200μm thick sample, which is within a factor of 2 of the 1.15 ps cascade time calculated for 350 keV electrons (see Table 1). This implies that for higher energy electrons (e.g. 3 MeV), if we use a thin detector crystal, owing to incomplete energy deposition, we will obtain on the order of the same number of charge carriers created as that for 511 keV photon interactions, and thus the extracted timing information is relevant to PET. For example, in 1mm thick CdTe, mean absorbed energy of a 3 MeV electron is 1.6 MeV, for 400μm thick CdTe, 500 keV is absorbed, and with 200μm thickness, ~250 keV is absorbed.

Table 1:

Simulated cascade time and charge carrier density for relativistic electrons.

Material	Electron nergy (keV)	Cascade Time (ps)	Total Carrier Density (1/cm3)
CdTe	350 750 3000	1.15 2.79 12.1	8.46 × 1018 2.48 × 1018 1.02 × 1018
YAG	350 750 3000	1.43 3.34 14.4	2.77 × 1018 7.69 × 1017 4.73 × 1017

Then we considered the time-resolved total charge density created by a recoil electron (e.g. from a photoelectric or Compton interaction) along its ionization track. Fig. 2 (e) shows an example of time resolved charge carrier generation and density in YAG, with 350 keV incident electron energy. Over a 1.7 ps time span, the electron density increases and reaches 2.3 × 10¹⁸ cm⁻³ towards the end of the track. We simulated with 1000 electrons to obtain the distribution of electron density at the end of each trajectory as shown in Fig. 2 (f), with 2 × 10¹⁸ cm⁻³ being the most likely charge density.

Compared to YAG, CdTe had higher charge densities, due to its lower bandgap and thus lower average energy to create an electron-hole pair. In both materials, as the electron energy increases, the average charge density decreases due to the longer trajectory, leading to a larger volume that charge carriers spread over.

Results of cascade time with different electron energies simulated in crystals with infinite thickness are summarized in Table 1. As the electron energy increases, the cascade time increases as well. CdTe had a shorter cascade time compared to YAG, due to its higher effective Z and density, hence higher probability of interaction per millimeter of detector material traversed.

As a final step and as an example, we calculated the strength of the modulation of optical properties, using probe beam intensity change as described by the Drude model²⁵. For 3 MeV electrons, we simulated the first 500 electron cascade interactions (inelastic collisions) to match with the experimental condition at UED, where the reduced thickness captures the earlier part for the track, and it corresponded to ~350μm for CdTe and 420μm for YAG. An example of intensity modulation in the defined region, in transmission mode is shown is shown in Fig. 3 (a) and (b).

Figure 3:

An example of optical intensity modulation (in %) generated by a 3 MeV electron in CdTe (a) and YAG (b) to simulate an experimental setting appropriate for UED, where thin samples capture the earlier part of ionization track. For CdTe, the signal is averaged over the first 350μm along the Z direction, and for YAG, 420μm, corresponding to absorbed electron energies of 500 keV and 790 keV, respectively.

Since the electron was injected along the Z direction, we projected the intensity modulation along the Z-axis. For CdTe, averaging the signal over the first 350μm, the absorbed electron energy was 500 keV and for YAG and averaging over the first 420μm, the absorbed energy was 790 keV. In both cases, the maximum intensity modulation reaches above 3%, over a substantial area spanning roughly 120μm by 160μm for CdTe (Fig 3 (a)), and 160μm by 25μm for YAG (Fig 3 (b)), which is well within the detectable range as established by a previously established experimental setup that used a highly-sensitive interferometer or a whispering-gallery-mode resonator²⁷. This suggests that even with 3 MeV electrons and thin samples, the ionization-induced optical property modulation signal is strong enough to be detected via sensitive optical measurement techniques.

3.1.2. Simulation for Synchronized, Ultrafast Photons from Free Electron X-ray Laser System

We simulated cascade times with varying ionizing photon energy from 1–100 keV in the photonic crystal PbBiGa as well as Cd-based, high bandgap semiconductors CdS, CdSe, and CdTe. In Fig. 4, we observe the overall monotonic increase of mean cascade times as the x-ray photon energy increases. However, there is a wide range of cascade time depending on the material. For example, owing to its higher Z and density, PbBiGa showed consistently shorter cascade time compared to other materials across the entire energy range.

Figure 4:

Simulation of cascade time and variance in various materials for incident ionizing radiation comprising 1–100 keV energy x-ray photons as a function of the recoil electron energy resulting from the interaction (Compton or photoelectric). (a) As the photon energy increases, the cascade time increases linearly in all materials. Different material properties such as density and Z number contribute to the resulting cascade time. For example, in Cadmium based semiconductors, a higher Z and density exhibits a lower cascade time due to the higher probability of interaction per millimeter of detector material traversed. (b) The variance in cascade time follows a similar trend as the mean cascade time, as expected for a Poisson distribution.

Within the Cd-based semiconductors, the cascade time corresponded with the effective Z values and density of the material, and thus the cascade time decreased in the order of CdS, CdSe, and CdTe. A similar monotonic increase in the variance of cascade time is observed. Therefore, we can conclude that time measurements with lower incoming photon energies can be extrapolated to higher photon energies, and, again be relevant for estimating timing for 511 keV photon interactions. Furthermore, careful selection of material will be beneficial for optimizing the cascade time at various energy points. For the spectral encoding technique⁶, where we need to design spectral bandwidth and temporal chirping, the exact cascade timing will be especially crucial.

3.2. Time-resolved simulation for asynchronous alpha source

An example of simulated trajectories is shown in Fig. 5 (a) for 100 alpha particles in UO₂, injected at the same location, with volume segmented in 1μm layers. As a result of multiple Coulomb scatter off the nucleus, which increases in probability as the alpha particle energy degrades, the trajectories tend to curl around towards the end²⁸ which indicates more energy is deposited per distance as the alpha particle energy decreases.

Figure 5:

(a) Trajectories of 100 alpha particles highlighted in yellow simulated in UO₂ (black). The detection crystal was segmented into 1μm. The trajectories are composed of inelastic excitation and ionization collisions with atomic electrons, and elastic Coulomb scatter collisions off the nucleus. (b) Average cascade time and (c) charge density distribution of a 5.5 MeV alpha particle track, calculated from 1000 trajectories. The average cascade time is 2.45 ps with standard deviation of 0.19 ps. A mean charge density of 3.71 × 10²¹cm⁻³ was observed, with standard deviation of 6.39 × 10²⁰cm⁻³. Energy deposition of (d) 5.5 MeV alpha particles, (e) 750 keV electrons, and (f) 511keV photons.

We then calculate and aggregate the distribution of cascade time and charge density through a similar process as presented in section 3.1.1. Fig. 5(b) shows the cascade time distribution of 1000 alpha particle trajectories. The average cascade time is 2.45 ps and the standard deviation is 0.19 ps. This is close to the average cascade time of 2.29 ps for 480 keV beta particles in CdTe that we obtained in a previous paper¹⁰. Although the 5.5 MeV alpha particles are a magnitude higher in energy, the cascade time is similar to 480 keV beta particles because alpha particles deposit energy faster owing to their larger charge and mass, therefore creating a shorter trajectory. Fig. 5(c) shows the distribution of charge density across 1000 trajectories. A mean charge density of 3.71 × 10²¹cm⁻³ is observed, with standard deviation of 6.39 × 10²⁰cm⁻³.

3.3. Energy deposition for different particle types

To compare how trajectories vary per different particle types, we look at the energy deposition for alpha, beta, and annihilation photon sources in different materials. The energy deposition of 5.5 MeV alpha particles from Am-241 is shown in Fig. 5 (d). Compared to electron and photon trajectories in (e) and (f), the range of alpha particle cascade is much shorter, less than 25 μm for all the materials, with the shortest range of 15 μm. The alpha energy deposition has a characteristic profile such that in all the materials, a sharp peak, known as the Bragg peak, is seen at the end of the trajectory as the primary particle loses energy and the probability of interaction per distance traversed increases. The electron energy deposition profile from 750 keV electrons, which is close to the endpoint energy from a Tl-204 source, exhibit much smoother curves, with a local maximum at the beginning of the trajectory, as seen in Fig. 5 (e). The overall track length is much longer than alpha energy deposition, up to several hundreds of micrometers for lower Z and density materials such as GaP and GaAs. Finally, we examined energy deposition for 511 keV photons, and found that the energy deposition profile is different for high stopping power materials versus lower stopping power materials. In GaP and GaAs, the deposition profile is significantly longer with lower deposited energy per depth, while in UO₂, BSO, and BGO most of the energy was deposited within 20mm thickness.

3.4. Charge carrier densities induced by surrogate ionizing radiation particles

In Figure 6, we present the ratios of the obtained charge carrier densities of the surrogate charged particles to the 480 keV electron ejected from a 511 keV photoelectric interaction in CdTe, which yielded a charge carrier density of 1.20 × 10¹⁹/cm³. The energy range for the electrons was from 100 keV to 4 MeV, which led to carrier densities ranging from 9.72 × 10¹⁷/cm³ to 2.29 × 10¹⁹/cm³. The alpha source with an energy of 5.5 MeV resulted in a charge carrier density of 7.31 × 10²¹/cm³. The ratios obtained from the electrons ranged from 0.08 to 1.85, when compared to the 480 keV photoelectron, while the alpha particle-induced charge carrier density resulted in a significantly higher ratio of 309.

Figure 6:

Ratio of the charge densities of the surrogate charged particles compared to that from 511 keV photoelectric interactions, represented by 480 keV photoelectron induced charge carrier density. The simulated particles were 100 keV, 200 keV, 350 keV, 480 keV, 1 MeV, 2MeV, 3MeV, 4 MeV electrons (sample thickness of 1mm for 3 and 4 MeV electrons) and 5.5 MeV alpha particles.

4. Discussion

4.1. Simulation settings

The simulation pipeline uses multiple software modules, ranging from electron trajectory simulation to modulation of optical properties, as presented in section 2.2.1. The interplay between each module is important and a script was written to coordinate data movement among the modules. We chose python as the overarching platform that interconnects cascade trajectory simulation, charge carrier generation, electron density calculation, and optical property modulation. For integration with GEANT4, having another language such as C++ as the base language might be desirable, which will enable a simulation pipeline that addresses all three particle types studied, alpha, beta, x-ray, and annihilation photons.

We utilize the developed simulation pipeline in the UED setting, as presented in section 3.1.1. Examples of optical property modulations in the first sections of ionization trajectories were presented in Figures 2 and 3. Due to the large range of trajectories (up to millimeters), full volume simulation is not readily possible, even with using the resources from HPC. We suggest simulating over segmented volumes to reduce the simulation load in the future, as the ionization trajectories are tortuous 3D curves going through a volume.

4.2. Synchronized electron simulation

In section 3.1.1, particle energy of 3 MeV was chosen which represents nominal synchronous electron energies at MeV-UED facilities. As the 3 MeV energy is higher than the annihilation photon energy of 511 keV, we simulated both a crystal with unlimited thickness and with limited thickness to reduce the absorbed energy. The ionization trajectory was shorter for the limited thickness (Fig 2 (a), (b)) and therefore had less cascade time and jitter (Fig 2 (c), (d)). By using the time-resolved simulation algorithm (Fig 2 (e), (f)), we were able to compare the total carrier densities from YAG and CdTe, and furthermore obtaining intensity changes of above 3% (Fig 3) which is in detectable range (e.g. see ²⁷). This indicates that by reducing the sample thickness, it will be possible to obtain findings more relevant to PET. In case there are other energies that can be used, the same simulation pipeline can be used, and also similar trends in the reduction of cascade time with thinner samples are expected. For the detector crystals, we selected CdTe and YAG as examples, but the same methodology can be easily expanded to other materials with atomic elements and density known. For the charge density calculation, we considered only charge diffusion which takes place during the cascade, after each inelastic collision, leading to an increase in the charge carrier density over time. Simulation over longer time scales to describe the decay of the charge carrier density would be of interest to define the carrier cascade lifetime and how the modulations of optical properties will be affected at these timescales.

4.3. X-ray pulse simulation

For the timing calculation for synchronous x-ray photons in section 3.1.2., Fig 4, in the future, for a number of detector crystals over the 1–100keV range, we would like to increase the variety of detector materials to conduct a comprehensive and quantitative comparison of the cascade timing performance and compare with experimental data.

4.4. Asynchronous source simulation

The simulation results obtained from both alpha and beta sources provide an important foundation for our progress towards detecting 511 keV photons. While synchronized sources offer valuable insights into the modulation of optical properties induced by radiation, there are two primary challenges to overcome in order to detect single 511 keV photons for PET.

The first challenge pertains to timing. Positron decays occur spontaneously, and the arrival time of annihilation photons is unpredictable. This necessitates an effective means to continuously monitor the detection crystal throughout a wide temporal window. This is in contrast to the synchronized source scenario, where the monitoring window can be precisely located in time and can be very narrow, covering only a small fraction of a second (typically in picoseconds).

The second challenge lies in the intensity of the signal generated by single 511 keV photons, which is exceedingly small when compared to the substantial number of ionizing particles emitted by synchronized sources.

Attempting to address both challenges at once is exceptionally difficult, and an intermediate step is required, involving the relaxation of one condition at a time. Consequently, an asynchronous ionizing radiation source, such as alpha or beta sources, capable of generating more pronounced modulation in a confined volume, becomes critical for the understanding of and practical formulation of experimental methods to detect 511 keV photons using the resulting modulation of optical properties. More specifically, with a detection setup designed for asynchronous detection (operating without a predefined timing trigger), alpha and beta sources can serve as initial tools to optimize detection sensitivity to a level sufficient for the identification of individual 511 keV photon interactions.

As shown in Fig 5(a), the ionization trajectory in UO₂ from asynchronous 5.5 MeV alpha particles were limited to a 5 μm thick volume, exhibiting a significantly shorter ranges when compared to 511 keV annihilation photon case. The cascade time was similar to 511 keV case, on the order of a few picoseconds (Fig 5 (b)); the induced charge density was three orders of magnitude higher (Fig. 5 (c)) than the charge density induced from 511 keV photons (Table 1). One possible mechanism that can contribute to this higher charge density is that the interaction probability of the alpha particles is significantly higher due to their higher mass and charge. We expect higher strength of optical property modulations will be induced from the alpha particle, with similar time scales, which will be informative in establishing an experimental setup to detect single 511 keV photons. When we compared the energy deposition profiles from the three particle types of alpha, beta, and annihilation photons (Fig 5 (e–f)), we confirmed that the alpha and beta particles have much shorter ranges and can be used as surrogate ionization sources for the intermediate experimental setup that probe much less optical volume.

4.5. Consideration for experimental setup

Based on our simulation result and preliminary experimental results from the synchronized x-ray sources ⁶, we layout potential experimental techniques that will utilize these surrogate charged particle radiation sources.

First, to examine the relationship between the surrogate charged particles and the 511 keV source, we compared the charge densities originating from radiation sources with different energies to the 511 keV annihilation photon scenario. As shown in Figure 6, the charge carrier densities arising from the electrons with energies ranging from 100 keV to 4MeV were relatively similar to the 511 keV case (480 keV photelectron ejected) as the ratio was within a factor of 2. On the other hand, the alpha particles gave a much higher (>100-fold) ionization charge density. With this result, we can anticipate stronger modulations of optical properties from alpha particles.

Subsequently, we explored potential experimental configurations and considered how the simulation results could guide us in optimizing sensitivity and timing. We suggest that conducting investigations using synchronized sources will be advantageous for obtaining preliminary and fundamental insights to radiation-induced modulations, as the particle flux is tunable, allowing us to generate a range of modulation signal intensities. Using this surrogate source, it will be possible to obtain timing characteristics of the optical property modulations and screen candidate detector crystals which will yield the strongest modulation signal.

To guide our final goal of 511 keV photon detection, we used surrogate asynchronous radiation sources such as alpha and beta sources to extract quantitative information that will help to optimize the setup (Fig 7)). In the schematic depicted in Fig 7, we propose using the interferometric spectral encoding technique^6,29 with strong matching of the probe laser beam diameter and ionization track dimensions. To increase the detection sensitivity, for example, it will be possible to select a pair of polarizers with higher extinction ratio and configure imaging condition and fiber bundle geometry. Another crucial factor to consider is the detector crystal, as the modulation of optical properties heavily relies on the material properties, including charge carrier generation, as well as modulation parameters like bandgap structure and absorption coefficients. The specific characteristics vary depending on the dominant mechanism, whether it is Drude free carrier absorption ¹¹ or bandgap modification ^30,31. Our simulation framework will be utilized in a feedback-like loop to refine the design of the experimental setup in all the aforementioned aspects. This iterative process will allow us to narrow down the sensitivity and timing requirements more precisely for detection of single 511 keV annihilation photon interactions.

Figure 7:

Schematic of experimental setup utilizing the interferometric spectral encoding technique, with components that can be optimized to improve the sensitivity. Major components include detection crystal, crossed polarizers, and fiber bundle.

4.6. Considering mechanisms behind modulation of optical properties

Finally, it is important to note that the mechanism employed for calculating the modulation of optical properties in Figure 3, based on the Drude model, represents only one of the possible modulation mechanisms. For example, when probing the detector crystal near its bandgap, the charge carriers from ionization can affect the crystal’s band structure and a strong dependency of the modulation strength on the probe wavelength may become dominant. In this regime, theoretical frameworks that consider the modification of band structure such as the band-filling effect or bandgap renormalization may become more relevant ³¹. Regardless, the main focus of our study is on the charge carrier density and cascade time and these quantities should be applicable to all the different mechanisms behind for modulation of optical properties.

5. Conclusions

In this paper, we developed a simulation algorithm, expanding from our previous work ¹⁰ and simulated multiple radiation sources with various detector materials. This work improves our understanding of radiation-matter interaction in ultrafast timescales and allows us to compare similarities and differences in the ionization events leading to surrogate experiments that can provide insights into the optical property modulation strength from 511 keV photon interaction-induced ionization. With a 3 MeV synchronized electron source, in samples with a thickness of 200 μm, cascade time is 3 ps with modulation of intensity change of > 3%, which is comparable to the modulation strength simulated for 511 keV photon-induced ionization cascade events⁶. Shorter range, asynchronous particles of alpha and beta particles were simulated for assessing localized ionization trajectories and charge carrier densities. With 5.5 MeV alpha particles, the average trajectory spanned 20 μm along one direction, and calculated charge density was 3.71 × 10²¹cm⁻³ with 2.45 ps cascade time. For beta particles with 750 keV energy, a longer average trajectory length of 200μm was observed. We observed the longest trajectory lengths reaching up to several centimeters when 511 keV photon source was simulated. We conclude that both alpha and beta particles are capable of producing detectable modulation of optical properties that can be utilized to develop experimental setups applicable to the 511 keV annihilation photon detection.

Data availability

The software suite used in this work can be accessed here: https://gitlab.com/miil-stanford/penelope-pipeline. The simulation datasets utilized in this study can be obtained by submitting a reasonable request to the corresponding author.