Geant4 simulation model of electromagnetic processes in oriented crystals for accelerator physics

Crystalline structure is a unique environment for strong-field QED effects involving both high-energy charged particles and photons. Very strong electric fields of atomic planes or axes reaching up to TeV/cm drastically enhance the processes of particle deflection, radiation and electron-positron pair production. Consequently, a small piece of crystal material could be useful for a wide range of applications in accelerator physics, high-energy frontier physics, nuclear physics, cosmic-ray detectors, dark matter search and radiation therapy.

In particular, a bent crystal is able to deflect charged particle beam. This has already found its application for beam collimation and extraction at such high-energy hadron machines as U70, SPS, RHIC, Tevatron and LHC U70 ; RHIC ; Tevatron ; Tevatron2 ; SPSUggerhoj ; UA9 ; UA92 ; LHC ; LHCion . Since such kind of deflection is equivalent to motion in a magnetic field exceeding 100 T, a bent crystal can be also used for particle spin rotation at an ultrashort distance. Therefore, one can measure a magnetic dipole moment (MDM) and electric dipole moment (EDM) of exotic fast-decaying baryons and $\tau$-leptons at high energy machines such as the LHC Baryshevsky ; Baryshevsky2 ; MDMold ; LHCMDM ; LHCMDMbagli ; PRD to search for a physics beyond the Standard Model.

Due to oscillatory-like motion in a crystal structure, a charged particle, i.e. $e^{\pm}$ produces hard X-ray and $\gamma$ radiation ChannelingRadiation1 ; ChannelingRadiation2 ; CB0 ; CB . This radiation source can be applied in nuclear physics experiments and for radiation therapy as well. Hard $\gamma$-rays can be also converted into electron-positron pair which is suitable for an intense crystal-based positron source PositronSource ; PositronSource2 for both linear and circular $e^{+}e^{-}$ colliders, such as International Linear Collider (ILC), Future Circular Collider (FCC-ee) as well as for muon colliders. Crystal structure is also capable to enhance the process of pair production by hard gamma-ray which together with intense radiation leads to acceleration of electromagnetic shower development PRL2018 ; NIMA . This allows one to design a compact crystalline electromagnetic calorimeter, which can be applied both in accelerator experiments and at a gamma-ray space telescope PRL2018 ; NIMA ; Calorimeters .

Being almost transparent for charged particles in the direction of crystal planes and axes, a crystal can be used as a media for ultra-high gradient plasma acceleration Tajima ; E336 .

The design of each of these applications requires complicated simulations of entire experimental setup, usually involving High Performance Computing (HPC) on a supercomputer. One of the most suitable simulation packages for this purpose is the Geant4 simulation toolkit Geant4 ; Geant42 . It contains a very wide range of simulation models of particles interaction with matter and is applied in high energy, nuclear and accelerator physics, medical and space science. However, most of physics in crystals as well as of the applications mentioned above are still missing in Geant4.

In this paper, we present the new Geant4 simulation model of electromagnetic processes in oriented crystals. We validate it with the experimental data as well as we discuss its perspectives in all the applications mentioned above.

The electromagnetic processes in oriented crystals were observed experimentally in a wide energy range from MeV up to TeV and found a lot of applications. Some of these applications, both already in use and hypothetical are discussed below.

Geant4 is a toolkit used worldwide for the simulation of the passage of particles through matter, including also a rich collection of electromagnetic (EM) effects of scattering and radiation. However, the channeling model existing as a Geant4 process G4Channeling G4Channeling covers only the applications of crystal-based extraction/collimation and MDM, EDM measurement LHCMDMbagli and only for hadrons. It includes neither channeling physics for electrons and positrons nor radiation and pair production in oriented crystals, which is necessary for all the remaining applications mentioned above and for $e^{+}e^{-}$ crystal-based extraction as well.

The main challenge of channeling simulations is the dependence of the value of interplanar/interaxial electric field and multiple scattering cross-section as well on the particle position in a crystal. Moreover, the radiation and pair production processes become completely different in comparison with amorphous matter. All of this requires the modification/replacement of the standard Geant4 physics processes by new ones when a particle propagates in an oriented crystal.

The currently existing realization of G4Channeling makes use of G4biasing in order to change the cross-section of standard Geant4 processes, including single Coulomb scattering, at each Geant4 simulation step. Unfortunately, this functionality does not allow one to modify so-called continuous discrete processes, including multiple scattering, ionization losses, bremsstrahlung and pair production. Though in G4Channeling this limitation it has a small influence on high-energy hadrons under the channeling conditions, it is crucial for $e^{\pm}$.

In order to switch off the standard Geant4 electromagnetic processes entirely inside the crystal volume, a Geant4 Fast Simulation Interface FastSimInterface was chosen as a baseline for the new Geant4 Channeling model ChannelingFastSimModel. The Fast Simulation Interface provides a G4VFastSimulationModel abstract base model that can be inherited by a model developed. This model is active only in a certain G4Region defined in the geometry, only for certain particles declared in the IsApplicable function and only for certain conditions set up in the ModelTrigger function, while all the standard Geant4 physics is suspended at the same Geant4 simulation step. This fully replaces the standard Geant4 physics by the physics implemented in the ChannelingFastSimModel inside a chosen volume and for chosen conditions, but uses the standard physics when these conditions are not fulfilled. Therefore, ChannelingFastSimModel is independent of the Geant4 Physics List and, consequently, is simply transferable to the Geant4 examples already existing.

The entire replacement of the physics requires the introduction of the new physical processes into ChannelingFastSimModel depending on the application. All of the applications require the implementation of a basic channeling model including an electric field function, a multiple and single Coulomb scattering model in an oriented crystal and a geometry of crystal planes/axes either bent or straight inside a crystal volume. All of these features were successfully implemented into ChannelingFastSimModel using the algorithms of the CRYSTALRAD simulation code CRYSTALRAD as a baseline. In addition, a model of Coulomb scattering in an oriented crystal well validated experimentally CSS has been also introduced. This model takes into account the division of the Coulomb scattering process onto a coherent part describing motion in the field of the crystal lattice and an incoherent part, i.e. scattering on single atoms. This model of scattering is a fundamental difference between ChannelingFastSimModel and G4Channeling and is essential for correct simulations of channeling $\sim$ GeV $e^{\pm}$ and potentially improves the accuracy of the simulation of high energy channeling hadrons as well. All differences are summarized in Table 1.

The experimental validation of ChannelingFastSimModel was carried out using the experimental data published in PRL2014 on 855 MeV $e^{-}$ deflection by 30.5 $\mu$m thick silicon (111) (non-equidistant) bent crystal planes with a bending angle of 905 $\mu$rad for both planar channeling and VR cases. The crystal was aligned at 0 and 450 $\mu rad$ far from channeling orientation, respectively. The model particle-to-plane angular range is set up in ModelTrigger as $\pm 100\theta_{L}$ to ensure the activation of ChannelingFastSimModel for all important angles. The comparison of the simulations and experimental data is shown in Figure 8 manifesting a very good agreement. The peak on the right corresponds to the beam fraction deflected under the channeling conditions while the left one includes volume reflection and multiple scattering processes. It is important to underline that the radiation energy losses, being the main process of energy losses of $e^{-}$ at this energy and higher, are almost negligible in this case since the crystal is very short. Therefore, ChannelingFastSimModel is completely adequate at these conditions.

The examples of $e^{-}$ trajectories in co-rotating reference system (with coordinate $z$ directed along and $x$ orthogonally to bent crystal planes) generated using ChannelingFastSimModel for the case of channeling orientation are illustrated in Figure 9. One can observe that only few particles stay in channeling by the end of the crystal while most of them are kicked out by multiple scattering.

The estimated low energy limit of the model applicability is $\sim$100 MeV to ensure the quantum effects become negligible Quantum1 ; Quantum2 . The high energy limit is defined by the level of radiation energy losses which increases with the energy and depends on the crystal thickness and geometry. In many channeling experiments carried out for channeling of $e^{\pm}$ in a Si bent crystal in the energy range 0.1-20 GeV PRL2014 ; EPJC2017 ; SLAC2014 ; PRL2017 this model is applicable.

However, different applications mentioned above may require other simulation models and/or extensions of ChannelingFastSimModel, namely a radiation model of energy losses by charged particles and of photon production, a pair production model (by photons), deposited energy scoring, an ionization model and an hadron scattering model for channeling hadrons etc. The list of the requirements depending on the application is the following:

1. 1.

   Crystal-based extraction/collimation: the basic channeling model, a radiation model for $e^{\pm}$ in the case when the radiation losses become important, an ionization model and an hadron scattering model for hadrons except the case when a crystal is short enough to neglect these processes, a complicated geometry model to describe the case when many particles enter/exit the crystal through its lateral surface (usually when the beam emittance is low enough) miscut . This application also requires the simulation of accelerator dynamics using either existing tracking codes or by introducing a new model into Geant4.

2. 2.

   MDM and EDM measurement: the basic channeling model, an ionization model and an hadron scattering model for exotic baryons, a spin rotation model using BMT-equations Quantum1 .

3. 3.

   Crystalline radiation source: the basic channeling model, a radiation model and a complicated geometry model for CU to simulate periodically bent crystal planes.

4. 4.

   Hybrid crystal-based positron source and compact crystalline e.m. calorimeter: the basic channeling model, a radiation model, a pair production model and deposited energy scoring.

5. 5.

   Plasma wakefield acceleration: the basic channeling model and the particle-in-cell (PIC) method PIC to simulate a complicated collective motion of the drive beam producing plasma waves and the accelerated beam as well.

Electromagnetic processes in oriented crystals can be used in a wide range of applications, namely crystal-based extraction/collimation of charged particles of either sign at modern synchrotrons, MDM and EDM measurement to test the physics beyond the Standard Model, crystalline radiation source and crystalline undulator for nuclear physics and radiation therapy, hybrid crystal-based positron source for future $e^{\pm}$ and muon collider projects, compact crystalline calorimeter for HEP experiments including dark photon search and for $\gamma$-ray space telescopes and plasma wakefield acceleration with extremely high acceleration gradients for a muon collider and, potentially, for many industrial applications, including medical physics.

Geant4 simulation toolkit is a tool that can allow one to design each of these applications and to carry out the simulations of an entire experimental setup. However, the existing Geant4 channeling model partially covers only crystal-based extraction/collimation and MDM and EDM measurement and only for hadrons, being non-applicable for $e^{\pm}$.

In this paper we have introduced a ChannelingFastSimModel Geant4 simulation model that is capable to provide a basic functionality for all the applications mentioned above. It uses Geant4 Fast Simulation Interface to suspend standard Geant4 processes and includes charged particle motion in an electric field in an oriented crystal, a multiple and a single Coulomb scattering mode and a geometry of crystal planes/axes either bent or straight inside of a crystal volume. Fast Simulation Interface makes this model completely independent of the Geant4 Physics List.

ChannelingFastSimModel has been validated with the experimental data for 855 MeV $e^{-}$ deflection in an ultrashort bent silicon crystal at Mainz Mikrotron MAMI showing a good agreement with the experimental data. It is applicable for $e^{\pm}$ motion in a crystal short enough, when the radiation energy losses can be neglected. This covers many experiments carried out in the energy range 0.1-20 GeV. This model will become a baseline for the development of simulation tools for the design of different applications of oriented crystals.