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We perform a thorough theoretical study of the electron properties of a generic CuO$$_2$$ 2 plane in the framework of Shubin–Kondo–Zener s-d exchange interaction that simultaneously describes the correlation between Tc and the Cu4s energy. To this end, we apply the Pokrovsky theory (J Exp Theor Phys 13:447–450, 1961) for anisotropic gap BCS superconductors. It takes into account the thermodynamic fluctuations of the electric field in the dielectric direction perpendicular to the conducting layers. We microscopically derive a multiplicatively separable kernel able to describe the scattering rate in the momentum space, as well as the superconducting gap anisotropy within the BCS theory. These findings may be traced back to the fact that both the Fermi liquid and the BCS reductions lead to one and the same reduced Hamiltonian involving a separable interaction, such that a strong electron scattering corresponds to a strong superconducting gap and vice versa. Moreover, the superconducting gap and the scattering rate vanish simultaneously along the diagonals of the Brillouin zone. We would like to stress that our theoretical study reproduces the phenomenological analysis of other authors aiming at describing Angle Resolved Photoemission Spectroscopy measurements. Within standard approximations one and the same s-d exchange Hamiltonian describes gap anisotropy of the superconducting phase and the anisotropy of scattering rate of charge carriers in the normal phase.

Abstract. Static distributions of temperature and wind velocity at the transition region are calculated within the framework of magnetohydrodynamics (MHD) of completely ionized hydrogen plasma. The numerical solution of the derived equations gives the width of the transition layer between the chromosphere and the corona as a self-induced opacity of high-frequency Alfvén waves (AW). The domain wall is direct consequence of the self-consistent MHD treatment of AW propagation. The low-frequency MHD waves coming from the Sun are strongly reflected by the narrow transition layer, while the high-frequency waves are absorbed -that is why we predict considerable spectral density of the AW in the photosphere. The numerical method allows consideration of incoming AW with arbitrary spectral density. The idea that Alfvén waves might heat the solar corona belongs to Alfvén, we simply solved the corresponding MHD equations. The comparison of the solution to the experiment is crucial for revealing the heating mechanism. Alfvén model for corona heatingThe discovery of the lines of the multiply ionized iron in the solar corona spectrum [1] posed an important problem for the fundamental physics -what is the mechanism of the heating of the solar corona and why the temperature of the corona is 100 times larger then the temperature of the photosphere.The first idea by Alfvén [2] was that Alfvén waves (AW) [3] are the mechanism for heating the corona. AW are generated by the turbulence in the convection zone and propagate along the magnetic field lines. Absorption is proportional to ω 2 and the heating comes from highfrequency AW. Alfvén's idea for the viscous heating of plasma by absorption of AW was analyzed in the theoretical work by Heyvaerts [4]. In support of this idea is the work by Chitta [5] (Figures 8, 9 therein). The authors came to the conclusion that the spectral density of AW satisfies a power law with an index of 1.59. This gives a strong hint that this scaling can be extrapolated in the nearest spectral range for times less than 1 s and frequencies in the Hz range. Furthermore in the work by Tomczyk [6] it is stated that there exist very few direct measurements of the strength and orientation of coronal magnetic fields, meaning that the mechanisms responsible for heating the corona, driving the solar wind, and initiating coronal mass ejections remain poorly understood. After the launch of Hinode, however, the well-forgotten spatially and temporally ubiquitous waves in the solar corona [7] came again into the limelight and gave strong support for the idea of Alfvén. A clear presence of outward and inward propagating waves in the corona was noted. k − ω diagnostics revealed coronal wave power spectrum with an exponent of ≈ − 3 2 (cf. Fig. 2 of [6]). The observational data for the temperature profile of the solar corona show that the transition layer is extremely thin compared to the radius of the Sun [8]. The width of the transition layer λ may be evaluated using the logarithmic derivative of the temperature, λ = max( ...

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