Magnetic Irreversible Solution of the Ginzburg-Landau Equations

We present the exact magnetic irreversible solution of the Ginzburg-Landau equations for a cylinder of infinite length (whose Ginzburg-Landau parameter $\kappa$ is unity and whose radius $R$ is three coherence lengths $\xi$ in an axial magnetic field $H_0$ for all values of $H_0$. Solutions for other values of $\kappa$ (0.3 to 3) and $\frac{R}{\xi }$ (2 to 12) are also discussed. We have determined, as a function of $H_0$ and as a function of position, the order parameter, the vector potential, the internal magnetic field, and the current density; and also as a function of $H_0$, the total number of superconducting electrons per unit length of the cylinder and the magnetization per unit volume. This solution is magnetically irreversible and hysteretic because of persistent currents which flow in the sample perpendicular to the applied magnetic field. The magnetization is reversible only over intervals of $H_0$ over which the number of fluxoids is conserved; otherwise it is irreversible. This solution does not depend on defects and is the counterpart to Abrikosov's magnetic reversible mixed-state solution. It is dominant in thin specimens.