Zeros of the Macdonald function of complex order

Ferreira, E.; Sesma, J.

doi:10.1016/j.cam.2006.11.014

Cited by 20 publications

(17 citation statements)

References 34 publications

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“…The asymptotic formulas for the zeros x n of the Macdonald function K iμ (x) (also known as the modified Bessel function of the second kind) of pure imaginary order are known in the literature [24] for two limiting cases, μ 1…”

Section: Appendixmentioning

confidence: 99%

Spin-orbit-driven electron pairing in two dimensions

Gindikin

Sablikov

2018

Phys. Rev. B

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We show that the spin-orbit interaction (SOI) arising due to the in-plane electric field of the Coulomb repulsion between electrons in a two-dimensional quantum well produces an attractive component in the pair interaction Hamiltonian that depends on the spins and momenta of electrons. If the Rashba SOI constant of the material is high enough the attractive component overcomes the Coulomb repulsion and the centrifugal barrier, which leads to the formation of the two-electron bound states. There are two distinct types of two-electron bound states. The relative bound states are formed by the electrons orbiting around their common barycenter. They have the triplet spin structure and are independent of the center-of-mass momentum. In contrast, the convective bound states are formed because of the center-of-mass motion, which couples the electrons with opposite spins. The binding energy in the meV range is attainable for realistic conditions.

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Section: Appendixmentioning

confidence: 99%

Spin-orbit-driven electron pairing in two dimensions

Gindikin

Sablikov

2018

Phys. Rev. B

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“…In this weak field regime the problem is amenable to asymptotic analysis. It has been shown by Cochran () and Ferreira & Sesma (, ) that in this limit, the zeros of

K_{i v_{normaln}} (k)

are given by

v_{n} \sim k + s_{normaln} 2^{- 1} k^{1 / 3} + \dots,

where s n = a n 2 2/3 , a n is the modulus of the n th real negative zero of the Airy function Ai (Abramowitz & Stegun ) and omitted terms are of the form k b with b < 1/3. This result can be used to find the asymptotic‐in‐k form of γ by inverting equation to form a bi‐quadratic

γ^{4} + 2 (σ^{2} + \frac{k^{2}}{1 + v_{normaln}^{2}}) γ^{2} + σ^{2} (σ^{2} - 2 \frac{k^{2}}{1 + v_{normaln}^{2}}) = 0,

into which we substitute equation .…”

Section: Vertical Field θ=π/2mentioning

confidence: 99%

“…Asymptotic expressions for γ and k can be found using the results of Cochran () and Ferreira & Sesma () for the (complex) roots of equation ,

v_{normaln} \sim π / (prefixln 2 - γ_{Euler} - prefixln p), p \in C, | p | ≪ 1,

v_{normaln} \sim p + a_{normaln} 2^{- \frac{1}{3}} p^{\frac{1}{3}} + \dots, p \in C, | p | ≫ 1,

where

γ_{Euler} ≃ 0.58

is the Euler constant.…”

Section: Vertical Field θ=π/2mentioning

confidence: 99%

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Quasi-global galactic magnetorotational instability with Braginskii viscosity

Rosin

Mestel²

2012

Monthly Notices of the Royal Astronomical Society

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We present a global‐in‐radius linear analysis of the axisymmetric magnetorotational instability (MRI) in a collisional magnetized plasma with Braginskii viscosity. For a galactic angular velocity profile Ω we obtain analytic solutions for three magnetic field orientations: purely azimuthal, purely vertical and slightly pitched (almost azimuthal). In the first two cases the Braginskii viscosity damps otherwise neutrally stable modes, and reduces the growth rate of the MRI, respectively. In the final case the Braginskii viscosity makes the MRI up to 22 times faster than its inviscid counterpart, even for asymptotically small pitch angles. We investigate the transition between the Lorentz‐force‐dominated and the Braginskii viscosity‐dominated regimes in terms of a parameter ΩνnormalB/B2 where νB is the viscous coefficient and B the Alfvén speed. In the limits where the parameter is respectively small and large, we recover the inviscid MRI and the magnetoviscous instability. We obtain asymptotic expressions for the approach to these limits, and find that the Braginskii viscosity can magnify the effects of azimuthal hoop tension (the growth rate becomes complex) by over an order of magnitude. We discuss the relevance of our results to the local approximation, galaxies and other magnetized astrophysical plasmas. Our results should prove useful for benchmarking codes in global geometries.

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“…In the limit of small β, the roots z n of the Macdonald function K iβ (z) have the following asymptotic behaviour [33]: where γ ≈ 0.577 is Euler's constant. The largest root that we are interested in (n = 1) is then given by…”

Section: Radial Equationmentioning

confidence: 99%

Effect of dipole polarizability on positron binding by strongly polar molecules

Gribakin

Swann

2015

J. Phys. B: At. Mol. Opt. Phys.

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Abstract.A model for positron binding to polar molecules is considered by combining the dipole potential outside the molecule with a strongly repulsive core of a given radius. Using existing experimental data on binding energies leads to unphysically small core radii for all of the molecules studied. This suggests that electron-positron correlations neglected in the simple model play a large role in determining the binding energy. We account for these by including the polarization potential via perturbation theory and non-perturbatively. The perturbative model makes reliable predictions of binding energies for a range of polar organic molecules and hydrogen cyanide. The model also agrees with the linear dependence of the binding energies on the polarizability inferred from the experimental data [Danielson et al 2009 J. Phys. B: At. Mol. Opt. Phys. 42 235203]. The effective core radii, however, remain unphysically small for most molecules. Treating molecular polarization nonperturbatively leads to physically meaningful core radii for all of the molecules studied and enables even more accurate predictions of binding energies to be made for nearly all of the molecules considered.

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Zeros of the Macdonald function of complex order

Cited by 20 publications

References 34 publications

Spin-orbit-driven electron pairing in two dimensions

Spin-orbit-driven electron pairing in two dimensions

Quasi-global galactic magnetorotational instability with Braginskii viscosity

Effect of dipole polarizability on positron binding by strongly polar molecules

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