M. Ćosić

Rainbows with Positrons and Carbon Nanotubes

This chapter is devoted to the rainbows occurring in channeling of positrons of incident kinetic energy of E0 = 1 MeV in (11, 9) chiral single-wall carbon nanotubes. As it has been said and explained in Subsect. 2.3.1, the process will be treated using quantum mechanics. In this case, the mass of the projectiles is sufficiently small and their incident kinetic energy sufficiently low that they clearly exhibit their wave properties. As it has been demonstrated in Chap. 1, in such a case, each rainbow is composed of a principal rainbow and one or more supernumerary rainbows. The principal rainbow is the primary, secondary, or a higher order rainbow.

Rainbows in Proton Channeling in Silicon Crystals

In this chapter, some of the high-resolution proton channeling measurements with a very thin (100) Si crystal conducted by the Singapore group [45–48] will be thoroughly analyzed. Those extraordinary measurements have proven to be crucial for verification of the theory of crystal rainbows as the proper theory of ion channeling in crystals. We shall present the results of three experimental and theoretical studies that were induced by those measurements and performed jointly by us and the Singapore group. The first study leads to the very accurate ion-atom interaction potentials [133]. As a continuation, we shall additionally explore the process that is inverse to the transmission process under consideration, and, thus, fully answer the question of its multiplicity, which is directly connected to the essence of the crystal rainbow effect, being the ion focusing along a line reflecting the symmetry of the illuminated crystal. The second study is devoted to the effect of superfocusing of channeled ions, which is the effect of spatial focusing occurring in the middle of each rainbow cycle [134]. The third study contains the proof that the doughnut effect in ion channeling, occurring with tilted crystals, which has been seen and measured many times, is in fact a crystal rainbow effect [130]. Before presenting the results of the second and third studies, we shall describe in detail the superfocusing and doughnut effects, respectively.

Rainbows with Protons and Carbon Nanotubes

Carbon nanotubes were discovered by Iijima in 1991 (Iijima Nature 354:56, 1991). They can be described as sheets of carbon atoms rolled up into cylinders with the atoms lying on the hexagonal lattice sites. The diameters of nanotubes are of the order of a few nanometers, and their lengths can be above a hundred micrometers. Nanotubes can be single-walled and multiwalled ones, depending on the number of cylinders they include. A nanotube is achiral or chiral. If it is achiral, the nanotube includes the atomic strings parallel to its axis. If it is chiral, the nanotube contains the atomic strings that spiral around its axis. Nanotubes have remarkable geometrical and physical properties (Saito R, Dresselhaus G, Dresselhaus MS (2001) Physical Properties of Carbon Nanotubes. Imperial College Press, London). As a result, they have begun to play an important role in the field of nanotechnologies (R.H. Baughman, A.A. Zakhidov, W.A. de Heer, Carbon nanotubes – The route toward applications. Science 297, 787 (2002)).

Group-theoretical approach to Bloch electron in magnetic field problem

In this paper magnetic-translation group theory is extended to include full rotational symmetry of Hamiltonian. Proper generalization of small representation and star of the representation concepts are derived. Irreducible representations of magnetic-translation group and magnetic-space group are presented. Correct form of symmetrized basis function is derived, reflecting symmetry of the magnetic-point group. From viewpoint of group theory reduction of Hamiltonian symmetry group caused by magnetic field and splitting of energy levels is investigated.