Critical vortex line length near a zigzag of pinning centers

A vortex line passes through as many pinning centers as possible on its way from one extremety of the superconductor to the other at the expense of increasing its self-energy. In the framework of the Ginzburg-Landau theory we study the relative growth in length, with respect to the straight line, of a vortex near a zigzag of defects. The defects are insulating pinning spheres that form a three-dimensional cubic array embedded in the superconductor. We determine the depinning transition beyond which the vortex line no longer follows the critical zigzag path of defects.Comment: 8 pages, 25 figures with low resolution option, 1 table. To be published in Eur. Phys. Jour.

Effects of boundaries in mesoscopic superconductors

A thin superconducting disk, with radius $R=4\xi$ and height $H=\xi$, is studied in the presence of an applied magnetic field parallel to its major axis. We study how the boundaries influence the decay of the order parameter near the edges for three-dimensional vortex states.Comment: To appear in Physica C as a special issue of M2S-HTS

On Exact and Approximate Solutions for Hard Problems: An Alternative Look

We discuss in an informal, general audience style the da Costa-Doria conjecture about the independence of the P = NP hypothesis and try to briefly assess its impact on practical situations in economics. The paper concludes with a discussion of the Coppe-Cosenza procedure, which is an approximate, partly heuristic algorithm for allocation problems.P vs. NP , allocation problem, assignment problem, traveling salesman, exact solution for NP problems, approximate solutions for NP problems, undecidability, incompleteness

Energy dependence of a vortex line length near a zigzag of pinning centers

A vortex line, shaped by a zigzag of pinning centers, is described here through a three-dimensional unit cell containing two pinning centers positioned symmetrically with respect to its center. The unit cell is a cube of side $L=12\xi$, the pinning centers are insulating spheres of radius $R$, taken within the range $0.2\xi$ to $3.0\xi$, $\xi$ being the coherence length. We calculate the free energy density of these systems in the framework of the Ginzburg-Landau theory.Comment: Submitted to Braz. Jour. Phys. (http://www.sbfisica.org.br/bjp) 11 pages, 6 figures, 1 table, LaTex 2

Effect of the boundary condition on the vortex patterns in mesoscopic three-dimensional superconductors - disk and sphere

The vortex state of mesoscopic three-dimensional superconductors is determined using a minimization procedure of the Ginzburg-Landau free energy. We obtain the vortex pattern for a mesoscopic superconducting sphere and find that vortex lines are naturally bent and are closest to each other at the equatorial plane. For a superconducting disk with finite height, and under an applied magnetic field perpendicular to its major surface, we find that our method gives results consistent with previous calculations. The matching fields, the magnetization and $H_{c3}$, are obtained for models that differ according to their boundary properties. A change of the Ginzburg-Landau parameters near the surface can substantially enhance $H_{c3}$ as shown here.Comment: 7 pages, 4 figures (low resolution

Paramagnetic excited vortex states in superconductors

We consider excited vortex states, which are vortex states left inside a superconductor once the external applied magnetic field is switched off and whose energy is lower than of the normal state. We show that this state is paramagnetic and develop here a general method to obtain its Gibbs free energy through conformal mapping. The solution for any number of vortices in any cross section geometry can be read off from the Schwarz - Christoffel mapping. The method is based on the first order equations used by A. Abrikosov to discover vortices.Comment: 14 pages, 7 figure

Vortex patterns in a superconducting-ferromagnetic rod

A superconducting rod with a magnetic moment on top develops vortices obtained here through 3D calculations of the Ginzburg-Landau theory. The inhomogeneity of the applied field brings new properties to the vortex patterns that vary according to the rod thickness. We find that for thin rods (disks) the vortex patterns are similar to those obtained in presence of a homogeneous magnetic field instead because they consist of giant vortex states. For thick rods novel patterns are obtained as vortices are curve lines in space that exit through the lateral surface.Comment: 4 pages, 4 figues, Proceeding of the Sixth International Conference in School Format on Vortex Matter in Nanostructured Superconductors (VORTEX VI

Three-dimensional Ginzburg-Landau simulation of a vortex line displaced by a zigzag of pinning spheres

A vortex line is shaped by a zigzag of pinning centers and we study here how far the stretched vortex line is able to follow this path. The pinning center is described by an insulating sphere of coherence length size such that in its surface the de Gennes boundary condition applies. We calculate the free energy density of this system in the framework of the Ginzburg-Landau theory and study the critical displacement beyond which the vortex line is detached from the pinning center.Comment: Submitted to special issue of Prammna-Journal of Physics devoted to the Vortex State Studie

New Faraday lines through Four Bosons EM

Field physics was founded by Faraday introducing magnetic fields (1831), electric fields (1837) and light as an EM wave (1846), initiating the process where nature is made by matter and fields. Consider that, ordinary space is full of fields. The Faraday view is basis for modern quantum field theory. The concept of fields set up a physicality in development. Physics would like to know how far matter is created by fields. Generate matter from nonlinear fields. Faraday lines of force relating physical entities as electric charge and mass depending on fields. Our purpose is on Faraday lines for nonlinear abelian electromagnetism. Introduce the Four Bosons EM. The phenomenology of a generic charge $\{+,0,-\}$ transmitted by four bosons $\{A_{\mu}, U_{\mu}, V_{\mu}^{\pm}\}$. Nonlinear equations constituted. New Faraday lines were introduced. The potentials fields of physics are developed. Granular and collective fields strengths expressed. Four types of fields charges are derived. They are electric charge, modulated, neutral, Bianchi. This work introduces a systematic procedure of associative physics. Mass and charge are generated due to the four fields interrelationships. Masses are derived without spontaneous symmetry breaking. It is obtained naturally from gauge symmetry, London, and mixing terms. Electric charge is written by fields through the Noether theorem. EM interactions not necessarily coupled with electric charge are proposed. An enlargement of EM energy is derived.Comment: 20 pages, 0 figure