Calculation of spontaneous radiation during channeling of relativistic positrons in non-chiral nanotubes using a quadratic approximation

The work investigates the conditions for the possibility of using the quadratic approximation U(ρ) = αρ2 for the interaction potentials of channeled positrons with the inner walls of non-chiral carbon nanotubes of types (n, 0) and (n, n). In particular, (8, 0), (10, 0), (12, 0) and (8, 8), (10, 10), (12, 12) nanotubes were selected. In this case, when calculating the single-particle potential of the carbon atom, only the contribution of valence electrons was taken into account. As a result of this approximation, the parameters α were determined for all the nanotubes studied. Using wave functions and the corresponding quantum levels of transverse energy obtained by solving the Schrödinger equation, the probabilities of occupation of these levels were calculated for positron beams with zero angular dispersion moving along the axes of nanotubes. Based on this information, values of the longitudinal energy of positrons for which the quadratic approximation is applicable were determined for all the studied nanotubes. Spectral distributions of spontaneous radiation were calculated in the dipole approximation for non-dispersive relativistic positron beams, both within the framework of quantum-mechanical and classical approaches.


Introduction
There have been relatively few studies dedicated to the investigation of channeling of positively charged particles (protons, positrons) in carbon nanotubes (CNTs).In particular, interesting results in the case of channeling of relativistic positrons were obtained, for example, in works [1][2][3][4], which have stimulated the present research.
This study focuses on the search for and description of optimal conditions for application the quadratic approximation  (ρ) = αρ 2  (1.1) during the channeling relativistic positrons in non-chiral nanotubes of types (, 0) and (, ).Additionally, when calculating the single-particle potential of a carbon atom at distances where expression (1.1) is valid, only the contribution from valence electrons is taken into account.
The study also includes an analysis of spontaneous radiation spectra using quantum mechanical and classical approaches for non-dispersive positron beams moving along the axes of nanotubes.
The values of parameters α for all investigated non-chiral CNTs are given in table 1.
α, eV•A −2 1.14 ± 0.14 0.40 ± 0.08 0.12 ± 0.03 0.06 ± 0.02 0.012 ± 0.006 0.003 ± 0.001 As seen from table 1, the values of the parameter α are specified with certain uncertainties, the consideration of which justifies the possibility of using the quadratic approximation.It is also clear that non-chiral CNTs of the (, 0) types are more preferable from the perspective of using formula (1.1) at lower longitudinal energies of particles, as further elaborated below.With the increasing CNT radii, the depths of the central regions of potential wells, where quadratic approximation is applicable, sharply decrease (see figures 1a, b).Additionally, spectral distributions shift towards low frequencies due to the proportionality of the parameters α, as will be demonstrated later.

JINST 19 C05031
3 Analysis of the channeled motion of positron beams in non-chiral carbon nanotubes using quantum mechanical and classical approaches

Calculation of populations of quantum states of transverse motion and matrix elements of dipole transitions between these states
By solving the Schrödinger equation with potential (1.1), the wave functions and corresponding energy levels of transverse energy are obtained [8] and have the following form: Here 1  1 (, , ) -degenerate hypergeometric function, ω 0 = √︁ 2α/μγ, λ = μω 0 γ/ℏ, μ -positron mass.In accordance with the theory of sudden perturbations (see [9]), for a positron moving at an angle θ relative to the symmetry axis of the CNT, the probabilities of its capture into quantum states (3.1) of transverse motion correspond to the expression:  Moreover, as seen from figure 2, during the channeling of such an ideal positron beam, states with  ρ = 1, . . ., 21 will be populated with sufficiently large probabilities   ρ .Hence, taking into account the table 1 and formula (3.2) it follows that approximation (1.1) can be used for such Lorentz-factors: -3 -γ ≥ 8.4 • 10 3 for CNTs of type (8, 0), γ ≥ 4.5 • 10 3 for CNTs of type (10, 0) and γ ≥ 9.5 • 10 3 for CNTs of type (12, 0) (it is evident that CNTs of type (10, 0) are optimal from the perspective of using lower longitudinal energies).For non-chiral CNTs of types (8,8), (10,10), (12,12) such minimum values of Lorentz-factors will be even larger, as the depths of potential wells (1.1) are very small, as seen from the values of parameter α in table 1 and from figure 1b.
Radiative dipole transitions will occur to levels (3.2) in states with  ′ ρ <  ρ and  ′ = ±1.In this case, the square of the modulus of the matrix elements of these dipole transitions is expressed as It should be noted that when calculating further spectral distributions, expression (3.4) needs to be multiplied by two.This is necessary to take into account the multiplicity of degeneracy of levels (3.2) in this case.

Classical analysis of the transverse motion of positrons in carbon nanotubes
In the classical study of the transverse motion of a positron (finding ì ρ() = ì   () + ì   ()) we will proceed from the following equation [9]: When analyzing this movement, the conservation of angular momentum law [10] must be satisfied: Excluding the value φ from formulas (3.5) and (3.6), we obtain the equation for the function ρ().

Calculation of spectral distributions within the classical approach
Since, in addition to the inequality εγ/μ 2 ≪ 1, the inequality μγ 2 ≫ ε also holds, the calculation of spectral distributions can be carried out within the framework of classical consideration using the following formula [9]: √ 3ρ min , then the spectral distributions (4.1) will be the same for these two cases (they are shown for the Lorentz-factor γ = 10 4 in figure 3b).
In conclusion, it is worth noting that the spectral distributions for dispersive positron beams do not differ significantly from those shown in figure 3a, which is due to the insufficiently high probabilities of populating states with  ≠ 0 (in this case, the functions   ρ , || have substantial values only for  ρ = 0, . . ., 21).