Channeling Fast Simulation Model

Channeling Fast Simulation Model (G4ChannelingFastSimModel) [SBC+23] is a model allowing one to simulate the trajectories of charged particles in an oriented crystal in the field of atomic planes (1D model) and atomic strings (2D model). The model also includes incoherent multiple and single Coulomb scattering depending on the particle coordinate in the crystal lattice and taking in account the suppression [MSB+20] of incoherent Coulomb scattering due to presence of the coherent part, i.e. motion in the field of atomic planes/strings. The model also includes ionization energy losses.

Furthermore, the model exploits the Baier-Katkov method [BKS98], [GBT12], [STB19] (G4BaierKatkov) in order to simulate the production of secondary photons by e+/e-.

Both models are based on the CRYSTALRAD simulation code [STB19].

To simulate an electromagnetic shower in an oriented crystal, the model of coherent \(e^+e^-\) pair production by a high energy photon has been developed (G4CoherentPairProduction). This model exploits the Baier-Katkov method modified for the pair production case [BKS98], [TM72] as well as the same algorithms and crystal data as G4ChannelingFastSimModel to simulate the dynamics of e+/e-.

Applicability range

The Channeling Fast Simulation Model is a classical model calculating the classical trajectories. This sets its low energy limit of applicability, being 100 MeV for e+/e- and less than MeV for hadrons. However, additional limitations for hadrons are connected with hadronic physics not introduced yet into the model. Therefore, channeling of hadrons can be used only at the crystal thicknesses much smaller than the corresponding interaction lengths of hadronic processes at a given energy.

Angular range of the model is described by the Lindhard angle being a critical channeling angle:

(296)\[\theta_{L}=\sqrt{2U_{0}/pv}\]

where, \(U_{0}\) is the potential well depth (depending on the material and the lattice type typically a few tens of eV for atomic planes and a few hundred of eV for atomic strings), \(p\) and v particle momentum and velocity, respectively. The channeling model should be used until a particle feels the influence of transverse electric field of the planes/strings, i.e. up to at least 10-20 \(\theta_{L}\).

Both angular and energy limits can be adjusted by the user. Outside these limits a standard Geant4 physics is active.

Trajectory simulation

The simulations of trajectories of charged particles is carried out by solving the trajectory equations (in G4ChannelingFastSimModel)

(297)\[\begin{split}\begin{cases} \frac{d^{2}x}{dz^{2}}+\frac{U'_{x}}{pv}+\frac{1}{R}=0,\\ \frac{d^{2}y}{dz^{2}}+\frac{U'_{y}}{pv}=0,\\ \end{cases}\end{split}\]

where \(x\) and \(y\) are the transverse coordinates, \(z\) being the longitudinal coordinate directed along an atomic plane/string, either straight or bent (only in the horizontal direction), \(R\) the crystal bending radius (\(1/R\) is missing for a straight crystal), \(U'_x\) and \(U'_y\) the electric field functions (derivative of the potential \(U\)) in \(x\) and \(y\) directions, respectively (the second equation is missing in the case of the 1D model).

The system \(x\), \(y\), \(z\) is fixed w.r.t. the coordinate system of the logical volume though they do not necessarily coincide. The functions of transformations of both particle coordinates and angles both forward and inverse have been also introduced (in G4ChannelingFastSimCrystalData). The electric fields are calculated according to the Doyle-Terner approximation [DPW95] and are contained in the data files in G4CHANNELINGDATA, each for certain crystal material, the type of the lattice (planes or strings) and their crystallographic direction. They are uploaded into the model through G4ChannelingFastSimCrystalData into G4ChannelingFastSimInterpolation. There is also an option to use an external datafile which does not belong to G4CHANNELINGDATA.

Coulomb scattering and ionization losses

The Coulomb scattering in an oriented crystal does not coincide with scattering in amorphous materials since it is splited onto an incoherent part of scattering on single atoms and a coherent part of motion in the field of crystal lattice [MSB+20], [TM72]. The cross-section of this scattering is described by:

(298)\[\frac{d \sigma}{d \Omega} = \frac{d \sigma}{d \Omega}_{coh}+ \frac{d \sigma}{d \Omega}_{inc},\]

where the coherent part \(\frac{d \sigma}{d \Omega}_{coh}\) is simulated by calculation of the trajectory as described above, while the incoherent fraction

(299)\[ \frac{d \sigma}{d \Omega}_{inc} = D \frac{d \sigma}{d \Omega}_{Yukawa}(x,y)\]

is calculated using the cross-section based on Coulomb screened (Yukawa) atomic potential \(\frac{d \sigma}{d \Omega}_{Yukawa}(x,y)\). This cross-section depends on the particle position through the atomic density map contained in the data files of the electric fields. \(D\) is the Debay-Waller factor which represents the fundamental difference between incoherent scattering in amorphous and crystalline materials.

The Coulomb scattering on single electrons is simulated using the Rutherford cross-section in G4VChannelingFastSimCrystalData. It requires the maps of electron density and minimum energy of ionization calculated using the same model as the electric fields and contained in the same data files.

The energy loss resulting from energy transfer to an electron is also utilized to model fluctuations in ionization losses in G4VChannelingFastSimCrystalData. This is achieved by replacing the energy-transfer-dependent terms of the Bethe formula, while the remaining terms are computed using the standard Bethe formula [Nav24]. The corresponding deposited energy is added to the current Geant4 simulation step.

Baier-Katkov Model

The Baier-Katkov (BK) quasiclassical method [BKS98], [GBT12], [STB19] is a radiation integral made over the classical trajectory such as calculated using the Channeling model, but taking into account quantum recoil. Thereby, this method is siutable for the case of hard radiation, when a charged particle (ultrarelativistic e+/e-) moving along a complicated trajectory within a radiation formation length losses a significant part of its energy for radiation a single photon. The energy range of the photons produced using the BK method is from ~ MeV (can be adjusted by user) up to the charged particle energy.

The BK quasiclassical formula expressing the radiation probability can be written as (in natural unit system):

(300)\[ dP_{rad} = \frac{\alpha }{4 \pi^2 }\frac{d^3 k}{\omega}\int \int dt_1 dt_2 \overline{N}_{21} \exp\left[i k^\prime (x_1-x_2)\right],\]

where \(\overline{N}_{21}\) is the radiation polarization matrix averaged over the initial particle polarization, \(k = (\omega, \textbf{k})\) is the 4-vector of the photon momentum including radiated energy \(\omega\) and 3-momentum \(\textbf{k}\), \(k^\prime = \varepsilon k/\varepsilon^\prime\), where \(\varepsilon\) and \(\varepsilon ^\prime = \varepsilon - \omega\) are the particle energy before and after the photon emission, respectively, \(x_{1,2}\) the particle coordinate 4-vector.

The Baier-Katkov Model (G4BaierKatkov) can potentially be used beyond the channeling model, for instance to simulate the hard radiation in magnetic and electric field in accelerator elements (e.g. magnets) as well as in beam-beam electric fields.

Coherent Pair Production Model

Unlike G4ChannelingFastSimModel, the model of Coherent Pair Production (G4CoherentPairProduction) is realized as G4VDiscreteProcess and is configured via G4CoherentPairProductionPhysics. This model is also based on the Baier-Katkov method [BKS98]. However, unlike the case of radiation, which involves calculating the probability of photon emission along a fixed trajectory of a charged particle, the pair production process requires a different approach. Specifically, it involves randomly generating a \(e^-e^+\) pair and tracking its motion over a short distance. This trajectory is then used within the Baier-Katkov method to calculate the probability of generating of this pair.

The BK quasiclassical formula expressing the \(e^+e^-\) pair production probability can be written as (compare to (300)):

(301)\[ dP_{PP} = \frac{\alpha }{4 \pi^2 }\frac{d^3 p}{\omega}\int \int dt_1 dt_2 \overline{N}_{p} \exp\left[i k^\prime (x_1-x_2)\right],\]

where \(\overline{N}_{p}\) represents the pair production polarization matrix, averaged over the initial photon polarization. The definitions align with those in (300), except that \(p\) represents the momentum of the charged particle whose trajectory is being integrated, and \(\varepsilon\) is the energy of this particle, while \(\varepsilon^\prime = \omega - \varepsilon\) denotes the energy of the other particle in the \(e^+e^-\) pair.

The G4CoherentPairProduction exploits G4ChannelingFastSimCrystalData and requires that G4ChannelingFastSimModel is configured with the same crystal data for the same oriented crystal. The combination of G4ChannelingFastSimModel with G4BaierKatkov activated and G4CoherentPairProduction enables the simulation of an electromagnetic shower in an oriented crystal.

Bibliography

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STB19(1,2,3): A. I. Sytov, V. V. Tikhomirov, and L. Bandiera. Simulation code for modeling of coherent effects of radiation generation in oriented crystals. Phys. Rev. Accel. Beams, 22:064601, Jun 2019. URL: https://link.aps.org/doi/10.1103/PhysRevAccelBeams.22.064601, doi:10.1103/PhysRevAccelBeams.22.064601.
SBC+23: Alexei Sytov, Laura Bandiera, Kihyeon Cho, Giuseppe Antonio Pablo Cirrone, Susanna Guatelli, Viktar Haurylavets, Soonwook Hwang, Vladimir Ivanchenko, Luciano Pandola, Anatoly Rosenfeld, and Victor Tikhomirov. Geant4 simulation model of electromagnetic processes in oriented crystals for accelerator physics. Journal of the Korean Physical Society, 83(2):132–139, Jul 2023. URL: https://doi.org/10.1007/s40042-023-00834-6, doi:10.1007/s40042-023-00834-6.
TM72(1,2): Mikhail L. Ter-Mikaelian. High-energy electromagnetic processes in condensed media. New York (N.Y.) : Wiley-Interscience, 1972. ISBN 0471851906.