Channeling of relativistic particles
Coherent effects of ultra-relativistic particles in crystals allow the manipulation of particle trajectories thanks to the strong electric field generated between crystal planes and axes [Tsy76].
When the motion of a charged particle is aligned (or at a small angle) with a string (or plane), a coherent scattering with the atoms of the string (or plane) can occur. In the low-angle approximation we can replace the potentials of the single atoms with an averaged continuous potential. The atomic string (plane) in the continuum approximation gently steers a particle away from the atoms, therefore suppressing the encounters with small impact parameters listed above. The channeling phenomenon is due to the fact that the fields of the atomic axes and planes form the potential wells, where the particle may be trapped. Particles can be trapped between planes or axes, under planar or axial channeling, respectively.
The continuous approximation by Lindhard [Lin65] was developed to describe channeling and its related phenomena. Coherent effects are primary phenomena, i.e., they govern path of particles. Four basic assumptions can be introduced for particles under orientational effects. First, angles of scattering may be assumed to be small. Indeed, scattering at large angles imply complete lost of the original direction. Secondly, because particle move at small angle with respect to an aligned pattern of atoms and collisions with atoms in a crystal demand proximity, correlations between collisions occur. Third, since coherent length \(l\) of scattering process (\(l=2E/q^{2}\), where \(E\) is the particle energy and \(q\) the transferred momentum) is larger than lattice constant, classical picture can be adopted. Fourth, idealized case of a perfect lattice may be used as a first approximation.
By following such assumptions, the continuous approximation can be inferred. Under such approximation, the potential of a plane of atoms \(U(c)\) can be averaged along direction parallel to plane directions. Angle \(\theta\) has to be greater than scattering angle \(\phi\) with a single atom:
The transverse motion of a particle incident at small angle with respect to one of the crystal axes or planes is governed by the continuous potential of the crystal lattice. A charged particle moving in a crystal is in planar channeling condition if it has a transverse momentum that is not sufficient to exceed the barrier to a neighboring channel, in this case the particle can not escape from the channel.
In the limit of high particle momenta the motion of particles in the channeling case (a series of correlated collisions) may be considered in the framework of classical mechanics, even though the single process of scattering is a quantum event . The classical approximation works better at high energy for two reasons: the first is that the wave lengths of incoming particles are sufficiently small to prevent the formation of interference patterns of waves; secondly classical mechanics is applicable thanks to the large number of energetic levels accessible in the interplanar potential (in analogy with the quantum harmonic oscillator). The second condition is always fulfilled for heavy particles, such as ions and protons, but for light particles (electrons, positrons) the classical approach starts to work in the \(10-100 MeV\) range. For motion in the potential \(U(x)\) the longitudinal component of the momentum is conserved for a relativistic particle, implying the conservation of the transverse energy [BCK96]:
The equation which describes the particle motion in the potential well is therefore:
The particle remains trapped within the channel if its transverse energy \(E_{T}\) is less than the potential-well depth \(U_{0}\):
where \(U_{0}\) is the maximum value of the potential barrier at the distance \(d_p/2\) from the center of the potential well, where the plane is located.
Intensity of incoherent interactions for particles under coherent effects strongly depends on local nuclei and electronic density. Thereby, the intensity of interaction in amorphous media has to be weighted with respect to the nuclear and electronic density averaged transverse to the crystal planes or axes [KO73]. Root-mean-square of transverse energy variation in crystal turns into a function of particle position, e.g. it is valid to treat intensity of interactions under planar condition
where \(\left<\frac{dp_{x}^2}{dz}\right>\) is the root-mean-square of transverse energy variation in crystal, \(n(x)\) is the atomic density along the crystal plane, \(n_{am}\) is the average crystal atomic density.
Information on the implementation details can be found in literature [BAB+14][EBaVGuidi13]
Bibliography
- BAB+14
E. Bagli, M. Asai, D. Brandt, A. Dotti, V. Guidi, and D. H. Wright. A model for the interaction of high-energy particles in straight and bent crystals implemented in geant4. The European Physical Journal C, 74(8):2996, 2014. URL: http://dx.doi.org/10.1140/epjc/s10052-014-2996-y, doi:10.1140/epjc/s10052-014-2996-y.
- BCK96
V. M. Biryukov, Y. A Chesnekov, and V. I. Kotov. Crystal Channeling and Its Applications at High-Energy Accelerators. Springer, 1996.
- KO73
M. Kitagawa and Y. H. Ohtsuki. Modified dechanneling theory and diffusion coefficients. Physical Review B, 8(7):3117–3123, oct 1973. URL: https://doi.org/10.1103/PhysRevB.8.3117, doi:10.1103/physrevb.8.3117.
- Lin65
J. Lindhard. Influence of crystal lattice on motion of energetic charged particles. Danske Vid. Selsk. Mat. Fys. Medd., 34:14, 1965.
- Tsy76
E.N. Tsyganov. Some aspects of the mechanism of a charge particle penetration through a monocrystal. Technical Report, Fermilab, 1976. Preprint TM-682.
- EBaVGuidi13
E. Bagli and V. Guidi. Dynecharm++: a toolkit to simulate coherent interactions of high-energy charged particles in complex structures. Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 309(0):124 – 129, 2013. URL: http://www.sciencedirect.com/science/article/pii/S0168583X1300308X, doi:http://dx.doi.org/10.1016/j.nimb.2013.01.073.