Analytical solutions of the kinetic equation for rounded spirals in effectively isotropic systems

A method for finding analytical solutions to the Cabrera and Levine differential equation is proposed for the case of the spiral formation on a screw dislocation when there is no difference between the growth and evaporation of a crystal. The method is based on finding asymptotic solutions for small spiral angles, exact solutions in the interval of dimensionless spiral radii equal and exceeding unity, and asymptotic solutions for large radii. Then, the method of matching the obtained solutions and their derivatives by dimensionless radius is used for two radius values determined from the problem conditions. In the particular case, for angular velocity omega 1 = 0.33, the exact solution is the result of matching the asymptotic solution of the original equation for small spiral angles, the exact solution in the interval of dimensionless spiral radii equal and exceeding unity, and the asymptotic solution for large radii. For large spiral radii, the asymptotic solution is the general one of the second kind Abel equation. Analysis of the obtained solution showed that the spiral pitch is not constant and equal to 19, but monotonically decreases with increasing spiral radius.