Each electron is tracked step by step. One electron track is composed by multiple steps. The electron relativistic velocity β, the corresponding Lorentz factor γ, and its kinetic energy KE are computed at the beginning of each step and at the minimum energy (Figure 2, “Initial and minimum charged particle properties”). The minimum energy is set by the user-specific material optical properties (Material.xml). Using this information, each rectilinear step length (Δx) is calculated from the tabulated particle range (Figure 2 , “Estimation of electron step length”).

By default, Geant4 uses Δx as the electron step length. However, users can limit the maximum % change of the charged particle velocity β per step ² (Δβ) (Figure 2, “Step length update during simulation”). If β represents the electron velocity at the beginning of an electron step, its velocity at the end of the step β _f is estimated using Equation 1. β f = β (   1 −   Δ β ) (1)

Setting a limit to the electron velocity change per step Δβ limits the variation of the kinetic energy Δ K E by controling the Lorentz factor variation Δγ and consequently the electron step length Δl (Equations 2–4). A small Δβ value (e.g., 0.04%, 10%) leads to a small variation of Δγ  (Equation 3), a small variation of the kinetic energy Δ KE per step, and a small step length Δ l (Equation 4; Figures 1B,C). In contrast, when using large Δβ values (e.g., 80%, 90%, or default Geant4 settings), the electron velocity can vary much more at each step, and, consequently, the electron travels in longer steps (Figure 1A). Δ K E = m Δ γ (2) where: Δ γ = 1 1 − β 2 − 1 1 − β 2 ( 1 − Δ β ) 2 (3) Δ l = m Δ γ   d E / d x (4) m is the electron mass (0.511 MeV), and dE/dx is the energy loss per step estimated from the electron total continuous energy loss tables. If this new step length Δl is smaller than the physical particle range Δx, Geant4 stores Δl as the particle step length. If not, Δx is used as the step length instead (Figure 2, “Step length update during simulation”).

Once the charged particle step length is estimated, the number of emitted Cerenkov photons is calculated, and their physical characteristics are computed. The emission time and position of individual Cerenkov photons are calculated from quantities known at the beginning of each electron step.

The theoretical number of Cerenkov photons emitted by a single charged particle is calculated according to the theory of Cerenkov radiation of Frank and Tamm [5]. The number of photons dN/mm generated per unitary track length by an energetic charged particle with a relativistic velocity β in 1 mm of a material with an index of refraction n varying as a function of the Cerenkov energy ε is given by Equation 5: d N   m m = 2 π α z 2   h c ∫ ε min ε max ( 1 − 1 β 2 n 2 ( ε ) ) d ε (5)

Here, α is the fine-structure constant (1/137), z is the charge of the particle (1 for electron), h is the Planck constant, and ε is the energy of the Cerenkov photon. Important to note that dN/mm depends on the Cerenkov energy spectrum [ ε m i n − ε m a x ] defined by the user in the materials. Cerenkov photons emission is predominantly observed in the electromagnetic spectrum visible region [24]. Consequently, all the optical properties defined by the users should be restricted to the necessary optical energy range or the material physical Cerenkov energy range (cut-off) to avoid non-physical results, as this is not restricted by Geant4 [25].

The theoretical number of Cerenkov photons dN/dx emitted in a specific step length dx is calculated using Equation 6 and is invariant when dividing a step into smaller steps [26]. d N d x = S t e p L e n g t h ∗ d N m m (6)

Since a variation of the electron velocity Δβ during a step affects its step length (Equation 4), it will also affect the number of Cerenkov photons generated in each step. The final number of Cerenkov photons emitted per step is a random variable following a discrete Poisson distribution of mean dN/dx in Geant4. The Cerenkov photons are produced evenly along the rectilinear track of the energetic electron. Their momentum direction depends on their relativistic velocity and the refractive index of the medium through the relation: cos ⁡ θ = 1 β n ( ε ) (7)

Cerenkov photons, therefore, have a specific geometric signature since they are emitted along a cone with an angle θ with respect to the charged particle direction.

To investigate the emission, spatial distribution, and timing distribution of electrons and Cerenkov photons as a function of the electron velocity change per step, we performed optical Monte Carlo simulation in a BGO crystal with the toolkit GATE v9.0 (Geant4 10.06.03). BGO scintillation properties were defined as an isotropic emission following a single exponential decay with a time constant of 300 ns, a broad emission spectrum with a peak at 480 nm, and an index of refraction between 2.36 and 2.07, for energies between 320 and 800 nm. The absorption length was modeled in the same range [25]. The fast decay component of BGO (10% of the emission) and the rise time were not included. More details can be found in our previous work [17].

Simulations were conducted using a 511 keV monoenergetic source located at 10 mm from the center of a 3 × 3 × 20 mm³ BGO crystal, perpendicular to the 3 × 3 mm² face. The photodetector was placed between the source emission and the crystal (front-arrangement). It was considered ideal, with a photon detection efficiency (PDE), which includes geometrical and quantum efficiency, of 1. A total of 1,000 gammas were emitted per simulation; only photoelectric interactions with the crystal were considered (photoelectric attenuation coefficient of ∼0.056 cm³/g at 511 keV [27]. The optical photons produced by scintillation and Cerenkov emission from energetic electrons were simultaneously generated and tracked using the Livermore Model [28]. The transport of optical photons was conducted using the LUT Davis model [29–33] with polished surfaces LUTs wrapped in Teflon, except the face in contact with the photodetector, which was modeled coupled to optical grease (index of refraction 1.5) for all simulations.

We investigated the effect of the maximum electron velocity change per step Δβ with the following values: from 0.01 to 0.05% with 0.01% increments, 0.1, 0.5, 1, 5%, and from 10 to 90% in 10% increments. We also performed simulations using the default Geant4 settings, which do not limit the electron velocity change per step and compute the step length from the range table (see 2.1.1).

Two additional investigations followed the described simulations and studies. First, we carried out simulations in BGO using a 1,000 keV monoenergetic source. Second, we used TlBr as material and 511 keV gammas. The same material characteristics as in our previous works and a 440–800 nm range to define the optical properties were used [18,34].

In all simulations, the characteristics of all emitted electrons and optical photons were stored in text files. The electron kinetic energy at the beginning of each step, the step length, and the number of electron steps per track were saved. We included a tag to identify the nature of each optical photon emitted by the electrons using custom Geant4 source code modifications. This allowed the scintillation and Cerenkov photons that originated from the same electron to be discriminated. We stored the number of generated Cerenkov photons for each electron step. The momentum of the primary electron and that of the i ^th emitted Cerenkov photon (i varying from 1 to 11) were also saved.

The spatial and timing information of the detected optical photons was saved in ROOT files [35].

The text and root output files were analyzed with MATLAB^® to reconstruct the complete history of each electron, identifying the parent gamma event and separating the scintillation and Cerenkov photons emitted at each step.

First, we investigated the effect of limiting the electron velocity change per step Δβ on the electron kinetic energy at the beginning of each step, step length, and number of steps in each track.

Using the step lengths of each electron track, we calculated the electron mean step length and mean track length, as the average of the mean step length per track over all tracks and the average of the summed step length per track over all tracks, respectively. We studied the obtained mean step length and track length as a function of the electron velocity change per step Δβ. We qualitatively compared the simulated mean step lengths with the mean free path of electrons in BGO.

Second, we studied the mean number of Cerenkov photons emitted per 511 keV interaction as a function of the electron mean step length, and consequently to the user-defined Δβ. Considering one 511 keV event, the number of Cerenkov emitted resulted from summing the number of Cerenkov photons per electron step over all steps. The mean number of emitted Cerenkov photons per 511 keV was extracted from the distribution for all events, using a Gaussian fit.

Then, we showed the relationship between the electron mean step length and the primary electron and the ith emitted Cerenkov photon directionality for all photoelectric interactions. Their momentum at the emission was projected on the x-z plane (Figure 1 as reference), and the correspondent angle with respect to the gamma emission was extracted.

Lastly, we investigated how the spatial and timing distributions of the detected Cerenkov photons within the 20 mm BGO are affected by different mean step lengths of the electron.

We studied the electron kinetic energy at the beginning of each step, the step length (all steps in all tracks) and the number of electron steps per track for three representative values of Δβ (Figure 3).

Without limiting Δβ (default Geant4 setting), each primary emitted electron had a kinetic energy of 0.420 MeV at the beginning of its track (Figures 3A,I). This corresponds to the kinetic energy of a recoil electron generated by the photoelectric interaction of a 0.511 MeV gamma with the 0.091 MeV K-edge of bismuth in BGO [27]. A minority of electrons interacted with the second and third energy orbits of bismuth (L-M shells, 0.001–0.016 MeV) and had an initial kinetic energy of ∼0.5 MeV. The majority of these 0.42 and 0.5 MeV electrons traveled ∼0.348 mm and 0.435–0.455 mm during this first step of their track (Figures 3A,II). On average, they performed only two steps per track (Figures 3A,III). They lost most of their energy during the first “long” step, and consequently, the following steps are only a few microns long. This is evidenced by the low peaks of the kinetic energy spectrum and step length distribution (Figure 3A (I-II)).

The same results are obtained with Δβ equal to or greater than 30% (not shown). This indicates that the corresponding step length Δl estimated from Equation 4 is generally larger than the physical step length Δx estimated from the electron range table, as discussed in Section 2.1.1 and summarized in Figure 2. Geant4 therefore always equals the electron step length to Δx for Δβ greater than 30%. Results obtained without limiting Δβ and with Δβ greater than 30% are therefore considered together in the rest of this paper.

When limiting Δβ to 10% and further to 0.04%, the kinetic energy at the beginning of each electron track was always given by the photoelectric interaction of the 511 keV gammas (Figure 3(B-C) (I); note the different y-axis scales in Figure 3I). However, Δβ set a limit on how much the electron velocity could vary per step. This led the electrons to scatter a greater number of times, traveling multiple shorter steps, and losing less energy at each step (Figure 3(B-C)).

Using all step lengths of each electron track, we calculated the electron mean step length and mean track length.

There is a clear linear relationship between Δβ and the mean step length per track (Figure 4, in log-log scale). For Δβ between 0.01 and 30%, the mean step length increased from 0.051 to 183 μm and plateaued for larger Δβ values.

We compared these values with the electron mean free path calculated with the relativistic equation of electron interactions with bulk materials [3,36]. The mean free path of a ∼400 keV electron in a solid bulk material with Z equal to 73 mainly composed of bismuth or germanium (akin to BGO), was estimated to be ∼0.170 μm. The calculation was validated with measurement up to 200 keV. From 511 keV to 1 MeV, the authors estimated the reported values to be within 0.2% of the correct values.

Based on this assumption, a reasonable electron mean free path in BGO at 511 keV should fall between 0.100 and 0.250 μm (Figure 4), which is obtained with an electron velocity change per step Δβ between 0.02 and 0.05%. The electron velocity change per step has no effect on the electron mean track length (Supplementary Figure S1).

Figure 6 shows the angular distribution of the primary electron, the first and fifth emitted Cerenkov photons at their emission, for each photoelectric interaction and for three mean step lengths values, which correspond to the three representative values of Δβ showed in Figure 3. Each angular distribution was normalized to its maximum to highlight preferred directions of emission. The directionality is also illustrated in Figure 1.

Figure 6A shows the angular distribution of the primary electron momentum at emission. As expected, following a gamma interaction, the energetic electrons are emitted in the forward direction, independently of the length of their steps.

Since the emitted Cerenkov photons’ momentum varies with respect to that of their parent electron (Eq. 7), the momentum distribution of the first emitted Cerenkov photon favors forward directions (Figure 6B).

Shorter electron steps make the track more tortuous and weaken the electron directionality with respect to the incident gamma direction. This effect is directly visible from the momentum distribution of the fifth Cerenkov photon emitted (Figure 6C). When the mean electron step was 183 μm, the electrons traveled only a few steps (Figure 3) and the fifth Cerenkov photon was emitted within the first electron step, thus maintaining the forward direction information. When the mean electron step decreased to 52 μm, the first electron step was still long enough to include the fifth emitted Cerenkov photon and maintain the forward direction. In contrast, when the electron mean step length was in the order of 0.200 μm, the electron rapidly lost directionality, causing the Cerenkov photons to lose their directionality.

We investigated subsequently emitted Cerenkov photons up to the 11^th photon. Figure 7 (x-axis in logarithmic scale) shows the normalized distribution of the x-z photon momentum at 180° with respect to the gamma emission (backward) for different electron mean step lengths.

For a given electron mean step length, the fraction of events with a x-z momentum component at 180° was larger for Cerenkov photons emitted later, meaning that the forward directionality decreased (also illustrated in Figure 1). This effect was more significant for small electron mean step lengths (< 10 μm). It became weaker for larger mean step lengths (e.g., >100 μm, Figure 7) since the first eleven Cerenkov photons were all emitted forward during the first electron step.

The x-z momentum distribution in the 0.100–0.250 μm range (electron mean free path in BGO at 511 keV) was almost constant for all Cerenkov photons (green region highlighted in Figure 7). Therefore, all electron mean step length values in this range led to a similar Cerenkov emission distribution during the simulation. Note that the 0.100–0.250 μm polar momentum distributions for the first and fifth Cerenkov photons correspond to the yellow distributions in Figure 6.

These results directly affect the Cerenkov photon transport within the material, as shown in the next subsection.

When using 1,000 keV gamma interaction, high energetic recoil electrons (∼910 keV) are emitted by the photoelectric interaction with the K-edge of bismuth in BGO. By increasing the gamma energy, the electrons kinetic energy increased, and so did their relativistic velocity β, leading to longer step lengths.

These energetic electrons traveled in average 360 μm when the electron velocity change per step Δβ was not limited (dots in Figure 9A), ∼177 μm more than their mean path at 511 keV (black dots in Figure 9A, already shown in Figure 4). Between 0.01 and 30%, the mean step length increased from a minimum of 0.123 μm to a maximum of 360 μm, respectively. Considering previous electron mean free path calculations [3,36], at ∼1,000 keV electrons travel around 200–300 μm in a solid bulk material composed by bismuth or germanium. Therefore, accurate results could be obtained using a Δβ of 0.02% or 0.03%, which corresponds to ∼55 emitted Cerenkov photons (Figure 9B).

In TlBr, the mean step length increased from 0.045 to 195 μm between 0.01 and 30% (blue dots in Figure 9A). The mean step length trend and values obtained with TlBr were similar to BGO (black dots in Figure 4 and Figure figure9A). This is probably due to their similar bulk characteristics (density and effective atomic number Z_eff) [12]. For this reason, the Δβ values that best approximate the path of the electrons in TlBr at 511 keV are expected to be in the 0.02–0.05%, as with BGO at 511 keV, with electrons traveling 0.100–0.250 μm.

TlBr produced 11 Cerenkov photons per 511 keV interaction when the electron mean step length was smaller than 100 μm. As expected, the effect of Δβ on the number of emitted Cerenkov photons is material-specific (different optical properties) and should be investigated for each case by users.