Abstract The diffusion problem of growth units on a crystal surface which contains a given step pattern is analyzed. An integral equation for the step advance velocity is derived and applied to the case of a growth spiral. The resulting non-linear integral equation for the spiral shape is studied further. An exact relation is obtained between the central radius of curvature of the spiral and the interstep distance at large distances from the centre. A second, approximate, relation is obtained, using the Taylor and asymptotic series of the non linear integral equation, and assuming a smooth change over from the solution in the inner and in the outer spiral regions. Combining these two relations spiral growth rates and slopes of growth hillocks are obtained.