# Quantum Rainbows in Positron Transmission through Carbon Nanotubes

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## Abstract

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## 1. Introduction

## 2. Theory

#### 2.1. Interaction Potential

#### 2.2. Theory of Rainbow Channeling

#### 2.3. Model of Quantum Rainbow Channeling

## 3. Results

#### 3.1. Interpretation of the Classical Rainbow Effect

#### 3.2. Semi-Classical Interpretation of Quantum Rainbow Effect

#### 3.3. Morphological Interpretation of Quantum Rainbow Effect

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

SWCNT | Single Wall Carbon Nanotubes |

## References

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**Figure 1.**(

**a**) Section of the graphene sheet. Small arrows labeled ${\mathit{a}}_{\mathbf{1}}$ and ${\mathit{a}}_{\mathbf{2}}$ represent primitive vectors of the graphene lattice. Large arrows show chiral vectors ${\mathit{C}}_{\mathit{h}}$ of zig-zag, armchair, and generic chiral single wall carbon nanotubes (SWCNT). Views in direction of axis in the case of: (

**b**) zig-zag, (

**c**) armchair, and (

**d**) chiral SWCNT.

**Figure 2.**(

**a**) Schematic representation of the ion Scattering by the atomic string. The interatomic distance of the string is d. The maximal ion incident angle ${\Theta}_{c}$ and its minimal approach distance ${a}_{\mathrm{sc}}$ are indicated (

**b**) Schematic representation of the ion channeling process. The deflection angles $({\theta}_{x},{\theta}_{y})$ at the exit of the SWCNT are smaller than the critical angle.

**Figure 3.**(

**a**) Family of 1-GeV proton trajectories in the $x0z$ plane of the SWCNT (11,9). (

**b**) Corresponding family of trajectories in the angular space.

**Figure 4.**(

**a**) Angular transmission function ${\Theta}_{x}(b)$ of 1-GeV protons transmitted through 10 $\mathsf{\mu}$m long SWCNT (11,9). (

**b**) The vertical slice through the corresponding angular distribution. All rainbow points are numbered, equivalent rainbow points are designated by the same number.

**Figure 5.**The probability density of the wave-packet of impact parameter $\mathit{b}=(0.624,0)$ nm, in the logarithm scale, at the exit of 200-nm long chiral SWCNT (14,4) in (

**a**) spatial, and (

**b**) angular representation, respectively. The thick dashed line represents the SWCNT boundary.

**Figure 6.**(

**a**) Rainbow diagram of 1-MeV protons transmitted through 200-nm long SWCNT (14,4). Arrows show positions of the classical rainbow points. Motion of the wave packets having impact parameters ${\mathit{b}}_{\mathit{a}}$ = (0,0), ${\mathit{b}}_{\mathit{b}}$ = (0.31, 0), and ${\mathit{b}}_{\mathit{c}}$ = (0.52, 0) nm, respectively, in the: (

**b**) spatial, and (

**c**) angular representations. Initial wave packets are shown by the dashed lines while final wave functions are shown by the solid lines. Intervals containing trajectories giving dominant contribution to wave packets are denoted by the thick red, green, and blue lines, respectively.

**Figure 7.**The spatial (

**a**) and the angular (

**b**) transmission function of the 1-MeV positrons transmitted through 200 nm long SWCNT (14, 4) in the vicinity of the spatial rainbow point ${2}_{s}$. (

**c**) The classical, the semiclassical, and the exact normed probability density shown by the blue, the red and black line respectively.

**Figure 8.**(

**a**) The spatial distribution of the 1-MeV positron beam transmitted through SWCNT (14, 4). (

**b**) The corresponding angular distribution. Positions of the classical rainbow lines are shown by the arrows.

**Figure 9.**Angular distributions along ${\theta}_{x}$ axis for 1-MeV positrons transmitted through 200-nm long SWCNT: (

**a**) (7, 3); (

**b**) (8, 5); (

**c**) (9, 7); (

**d**) (14, 4); (

**e**) (16, 5); and (

**f**) (17, 7). Arrows show positions of maxima in the corresponding classical angular distributions. Symmetrical maxima are designated by the same number.

**Figure 10.**(

**a**) Trajectories of the 1MeV positrons (magenta lines) in the $xOz$ plane and associated envelope lines (blue hue lines). (

**b**) Corresponding Hamilton’s principal function at the z = 150 nm.

**Figure 11.**(

**a**) Vertical slice through spatial probability density of 1-MeV positron beam in the exit plane of SWCNT (11, 9). (

**b**) Corresponding slices through individual wave packets. Magenta lines separate wave packets belonging to the rainbow subensemble. The thick black line shows the corresponding inverse spatial transmission function.

**Figure 12.**(

**a**) The family of wave packet probability densities; (

**b**) The corresponding family of wave packet phase functions. The red line shows the classical Hamilton’s principal functions. The blue line shows the envelope function of the quantum phase function family. Inset show enlarged part of the phase function family in the vicinity of the classical singular point $+{1}^{s}$. The envelope function of the family is shown by the blue line. Members of the rainbow subensemble are designated by magenta lines, remaining wave packets are shown by the gray lines.

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**MDPI and ACS Style**

Ćosić, M.; Petrović, S.; Nešković, N.
Quantum Rainbows in Positron Transmission through Carbon Nanotubes. *Atoms* **2019**, *7*, 16.
https://doi.org/10.3390/atoms7010016

**AMA Style**

Ćosić M, Petrović S, Nešković N.
Quantum Rainbows in Positron Transmission through Carbon Nanotubes. *Atoms*. 2019; 7(1):16.
https://doi.org/10.3390/atoms7010016

**Chicago/Turabian Style**

Ćosić, Marko, Srđan Petrović, and Nebojša Nešković.
2019. "Quantum Rainbows in Positron Transmission through Carbon Nanotubes" *Atoms* 7, no. 1: 16.
https://doi.org/10.3390/atoms7010016