Channeling Fast Simulation Model
Channeling Fast Simulation Model [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 Model (G4BaierKatkov) in order to simulate the production of secondary photons by e+/e-.
Both models are based on the CRYSTALRAD simulation code [STB19, ST15].
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:
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 are carried out by solving the trajectory equations
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. The electric fields are calculated according to the Doyle-Terner approximation [DPW95] and are contained in the data files, each for certain crystal material, the type of the lattice (planes or strings) and their crystallographic direction.
Coulomb scattering
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:
where the coherent part \(\frac{d \sigma}{d \Omega}_{coh}\) is simulated by calculation of the trajectory as described above, while the incoherent fraction
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. 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.
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 suitable for the case of hard radiation, when a charged particle (ultrarelativistic e+/e-) moving along a complicated trajectory within a radiation formation length loses a significant part of its energy to radiate 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 can be written as:
where \(\overline{N}_{21}\) is the radiation polarization matrix, \(\alpha\) is the fine structure constant, \(k = (\omega, \textbf{k})\) 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.
- BKS98
V.N. Baier, V.M. Katkov, and V.M. Strakhovenko. Electromagnetic Processes At High Energies In Oriented Single Crystals. World Scientific Publishing Company, 1998. ISBN 9789814502542.
- DPW95
S.L. Dudarev, L.-M. Peng, and M.J. Whelan. On the doyle-turner representation of the optical potential for rheed calculations. Surface Science, 330(1):86–100, 1995. URL: https://www.sciencedirect.com/science/article/pii/0039602895004645, doi:https://doi.org/10.1016/0039-6028(95)00464-5.
- GBT12
Vincenzo Guidi, Laura Bandiera, and Victor Tikhomirov. Radiation generated by single and multiple volume reflection of ultrarelativistic electrons and positrons in bent crystals. Phys. Rev. A, 86:042903, Oct 2012. URL: https://link.aps.org/doi/10.1103/PhysRevA.86.042903, doi:10.1103/PhysRevA.86.042903.
- MSB+20(1,2)
A. Mazzolari, A. Sytov, L. Bandiera, G. Germogli, M. Romagnoni, E. Bagli, V. Guidi, V. V. Tikhomirov, D. De Salvador, S. Carturan, C. Durigello, G. Maggioni, M. Campostrini, A. Berra, V. Mascagna, M. Prest, E. Vallazza, W. Lauth, P. Klag, and M. Tamisari. Broad angular anisotropy of multiple scattering in a si crystal. The European Physical Journal C, 80(1):63, Jan 2020. URL: https://doi.org/10.1140/epjc/s10052-019-7586-6, doi:10.1140/epjc/s10052-019-7586-6.
- STB19(1,2)
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.
- ST15
A.I. Sytov and V.V. Tikhomirov. Crystal simulation code and modeling of coherent effects in a bent crystal at the lhc. Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 355:383–386, July 2015. URL: http://dx.doi.org/10.1016/j.nimb.2015.02.042, doi:10.1016/j.nimb.2015.02.042.
- 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
Mikhail L. Ter-Mikaelian. High-energy electromagnetic processes in condensed media. New York (N.Y.) : Wiley-Interscience, 1972. ISBN 0471851906.