Crystal Rainbow Research Articles

Crystal rainbow channeling potential for 〈1 0 0〉 and 〈1 1 1〉 cubic crystallographic crystals

In this work we show a way how the crystal rainbow channeling effect can be applied for obtaining the interaction potential between 2 MeV protons and the cubic crystallographic crystals, in the cases of axial channeling of protons along (001) and (111) crystal directions. The procedure for construction of this potential has been based on our previous work in which the rainbow morphology was crucial for their determination. However, important improvements have been made. As a result, the two axial channeling directions can be treated using the same physical arguments leading to more consistent values of the fitting parameters in the ion-atom interaction potential.

Superfocusing and zero-degree focusing in planar channeling of protons in a thin silicon crystal

This is a theoretical study of the superfocusing and zero-degree focusing effects appearing in channeling of protons of energy of 2 MeV in the {1 1 0} channel of a thin silicon crystal. We prove that the ultimate origin of these effects is the crystal rainbow effect, which has been discovered and explored in ion transmission through axial crystal channels, nanotubes and graphene. Moreover, the effect is the origin of the shapes of the whole spatial and angular distributions of channeled protons. The incident proton velocity vector is taken to be parallel to the channel midplane. We use the Molière's approximation of the Thomas-Fermi proton-atom interaction potential and the continuum approximation. The effect of thermal vibrations of the crystal's atoms is included. We solve numerically the proton equation of motion and analyze the spatial and angular proton transmission functions. The extrema of these functions are the spatial and angular rainbow points, respectively. When the whole proton beam is taken into account, a spatial rainbow pattern, composed of the lines emerging from the superfocusing points, and an angular rainbow pattern, consisting of the lines emerging from the origin and the zero-degree focusing points, appear along the channel. We consider the region of crystal thickness comprising the first, second and third superfocusing and zero-degree focusing points. When the effect of thermal vibrations is neglected, each of these rainbow lines look like the bifurcation set of the cusp catastrophe, and when the effect is included, the line resembles the bifurcation set of the butterfly catastrophe. We demonstrate that the spatial and angular distributions of transmitted protons are fully determined by the spatial and angular rainbow lines, respectively. The superfocusing and zero-degree focusing effects weaken with the increase of the crystal thickness. This is a consequence of the anharmonicity of the continuum proton-crystal interaction potential, which makes the proton beam propagation through the channel incoherent. We also explore the influence of the effect of proton collisions with the crystal's electrons on the beam dynamics in the channel. This effect additionally contributes to the incoherence of the beam motion along the channel. Our opinion is that the peaks of the spatial and angular distributions of ions transmitted through the planar channels lying off the origin registered before were in fact the rainbow peaks.

Superfocusing of channeled protons and crystal rainbows

This study is devoted to the effect of superfocusing of protons having the energy of 2 MeV in a 〈1 0 0〉 channel of a Si crystal. The analysis is performed by the theory of crystal rainbows. We analyze the superfocusing effect in the first rainbow cycle. The evolution of the spatial distribution of channeled protons is examined by the numerical solution of the proton equations of motion in the transverse position plane. We demonstrate that the superfocusing effect is a reduced crystal rainbow effect, in which the rainbow line comes to a point.

Crystal rainbows

This review is devoted to ion transmission through axial channels of thin crystals. In this process the rainbows occur. The effect is called the crystal rainbow effect. We shall describe its origin and present the experiments in which it has been observed. We shall explain also how the crystal rainbows can be classified using catastrophe theory. This classification has resulted in a universal, simple and accurate approximation to the continuum potential in the channels. Besides, the periodicity of the angular distributions of transmitted ions with the reduced crystal thickness will be considered. It will be introduced via the effect of zero-degree focusing of channeled ions. In addition, we shall mention the doughnut effect in ion channeling, which has proven to be the rainbow effect with tilted crystals. All these considerations will demonstrate clearly the usefulness of the theory of crystal rainbows, which is the proper theory of ion channeling in thin crystals.

Doughnuts with a 〈110〉 very thin Si crystal

This study is devoted to the angular distributions and transmission patterns of 10 MeV protons channeled in a tilted 〈110〉 very thin Si crystal of thickness 100 nm. The tilt angle of the crystal is varied from zero up to the critical angle for channeling. The angular distributions and transmission patterns were obtained using a computer simulation code based on the continuum approximation. They are dominated by the doughnut effect for larger values of the tilt angle. In order to explain the angular distributions and transmission patterns obtained, we also generated the corresponding crystal rainbow patterns. The analysis showed that the evolution of the angular distribution and transmission pattern with the tilt angle could be explained by the evolution of the rainbow pattern. Also, we calculated the transmission pattern of 3.2 MeV protons channeled in a tilted 〈110〉 thin Si crystal of thickness 190 nm and compared it with the corresponding measured transmission pattern.

Rainbows with a tilted 〈111〉 Si very thin crystal

The angular distributions of protons channeled in a tilted 〈111〉 Si very thin crystal together with the corresponding rainbow patterns are considered. The proton energy is 10 MeV while the crystal thickness is 121.1 nm, corresponding to the reduced crystal thickness of 0.15. The tilt angle of the crystal is varied from zero up to the critical angle for channeling. The angular distributions were obtained using the numerical solution of the proton equations of motion and the computer simulation method. The rainbow lines were determined by applying the theory of crystal rainbows. The analysis has shown that the evolution of the angular distribution with the tilt angle can be fully explained by the evolution of the crystal rainbow. We have discovered that the crystal rainbow effect turns into the well-know doughnut effect in ion channeling for the larger values of the tilt angle. The interior of the doughnut is the analogue of the Alexander's dark band, which occurs in between the primary and secondary meteorological rainbows.

Zero-degree focusing of channeled ions and crystal rainbows

We analyze here theoretically the connection between the effect of zero-degree focusing of channeled ions and the crystal rainbow effect. The calculations were performed for Ne 10+ ions and a 〈100〉 Si thin crystal. The ion energy was 60 MeV and the crystal thickness was varied between 630 and 750 atomic layers. These conditions correspond to the neighborhood of the first maximum of the zero-degree yield of the channeled ions. The obtained results show that the effect of zero-degree focusing is influenced strongly by the crystal rainbow effect.

Determination of adatom sort and adsorption sites on single crystal surfaces by ions channeling through thin films

By computer simulation it has been shown that both transmission channeling of low energy medium mass ions through thin metal single crystal films and rainbow effect caused by their interaction with light adatoms on the back side of the films may be used to determine the sort and adsorption sites of adatoms on the single crystal surface and to control the initial stages of adsorption. An experimental test of this technique would permit a realistic evaluation of its capabilities.

Energy loss of ions channeled in a thin crystal

The energy loss distributions of Ne10+ ions channeled in a (100) Si thin crystal are considered. The thickness of the crystal is 500 nm (921 atomic layers), while the ion energy is varied from 1 to 35 MeV per nucleon. The energy loss distributions were obtained by the computer simulation method using the numerical solution of the equations of motion of the ion. We used Lindhard's continuum interaction potential of the ion and the crystal, and the Bethe formula for the stopping power of the electron gas of the crystal. The results of the calculations are explained via the effect of zero-degree focusing of the channeled ions and the crystal rainbow effect.

Angular distributions of ions transmitted through a very thin crystal

The angular distributions of C 6+ ions transmitted through a 〈100〉 Si very thin crystal are considered. The ion energy is 25 MeV while the thickness of the crystal is varied from 54 to 540 atomic layers. The angular distributions of the transmitted ions were obtained using the numerical solution of the equations of motion of the ion with the Lindhard's expression for the continuum interaction potential of the ion and the crystal, and the computer simulation method. The results of the calculations are explained via the crystal rainbow effect.

Crystal rainbows in the cases of square very thin crystals with one atomic string per primitive cell

The transmission of H + ions through the 〈100〉 very thin crystals of 28 elements with the body-centered, face-centered or diamond-type cubic structures starting with C and finishing with Th is studied; the ion energy is 10 MeV, and the crystal thicknesses are 100 nm. These are the cases of square very thin crystals with one atomic string per primitive cell. In this study we use the Lindhard's continuum potential in the crystal as the “exact” one, and model the crystal rainbow effect by the 4X 9 catastrophe. The analysis is focused on the dependences of the parameters of the model continuum potential in the crystal on the position of the element in the periodic table of elements.

The X 9 catastrophe as the organizing center of crystal rainbows

The crystal rainbow effect is modeled by the catastrophe theory. It is shown that the 4X 9 catastrophe can be treated as the organizing center of the crystal rainbows in the cases of square, rectangular, centered rectangular and hexagonal very thin crystals with one atomic string per primitive cell. This results in a universal, simple and accurate approximation to the continuum potential in the crystal.