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The whole needle-crystal solution for three-dimensional (3D) dendritic growth is described analytically. Its construction involves the existing 3D selection theory for the tip of the dendrite (M. Ben Amar and E. A. Brener Phys. Rev. Lett. 71, 589 (1993) plus matching of the tail to this tip. This is exhibited here. Both intermediate and final asymptotics of the tail shape are given. This shape, which deviates strongly from an Ivantsov paraboloid, is in qualitative agreement with experiental observations.
Efim A. Brener (Mon,) studied this question.