First and second order phase transitions and magnetic hysteresis in a superconducting plate

The self-consistent solutions of the nonlinear Ginzburg Landau equations, which describe the behavior of a superconducting plate of thickness 2D in a magnetic field H parallel to its surface (provided that there are no vortices inside the plate), are studied. We distinguish two classes of superconductors according to the behavior of their magnetization M(H) in an increasing field. The magnetization can vanish either by a first order phase transition (class-I superconductors), or by a second order (class-II). The boundary SI-II, which separates two regions (I and II) on the plane of variables (D, kappa), is found. The boundary zeta(D, kappa) of the region, where the hysteresis in a decreasing field is possible (for superconductors of both classes), is also calculated. The metastable d-states, which are responsible for the hysteresis in class-II superconductors, are described. The region of parameters (D, kappa) for class-I superconductors is found, where the supercooled normal metal (before passing to a superconducting Meissner state) goes over into a metastable precursor state (p-). In the limit kappa --> 1/root2 and D >> lambda (lambda is the London penetration depth) the self-consistent p- solution coincides with the analytic solution, found from the degenerate Bogomolnyi equations. The critical fields H-1, H-2, H-p, H-r for class-I and class-II superconducting plates are also found.