The experimental techniques have evolved to a stage where various examples of nanostructures with nontrivial shapes have been synthesized, turning the dynamics of a constrained particle and the link with geometry into a realistic and important topic of research. Some decades ago, a formalism to deduce a meaningful Hamiltonian for the confinement was devised, showing that a geometry-induced potential (GIP) acts upon the dynamics. In this work we study the problem of prescribed GIP for curves and surfaces in Euclidean space R 3 , i.e., how to find a curved region with a potential given a priori. The problem for curves is easily solved by integrating Frenet equations, while the problem for surfaces involves a non-linear 2nd order PDE. Here, we explore the GIP for surfaces invariant by a 1-parameter group of isometries of R 3 , which turns the PDE into an ODE and leads to cylindrical, revolution, and helicoidal surfaces. Helicoidal surfaces are particularly important, since they are natural candidates to establish a link between chirality and the GIP. Finally, for the family of helicoidal minimal surfaces, we prove the existence of geometryinduced bound and localized states and the possibility of controlling the change in the distribution of the probability density when the surface is subjected to an extra charge.
Density functional theory calculations in different cluster models, the largest one K 38 (C 60 ) 23 with 1,418 atoms, combined with the resonating valence-bond theory show that superconductivity in K 3 C 60 involves interaction of C 60 with K þ , K 0 , and K À in octahedral interstices as well as immobilization of negative charges on positive tetrahedral K atoms. We found that the orbitals involved in the highest occupied molecular orbital-lowest unoccupied molecular orbital gap are mainly from C 60 and octahedral potassium atoms. We suggest that a K 3 C 60 superconductor improvement can be achieved through potassium atom vacancy in octahedral sites.
[1] We describe a mechanism of enhanced terrestrial gamma ray flash production seeding via muon decay in the presence of high electric fields associated with lightning. Our model predicts 10 7 relativistic seed electrons per millisecond at about 15 km altitude with mean energy of 35 MeV and an avalanche multiplication factor of about 10 10
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