We analyze the problem of onset of morphological instability and the increasing supersaturation gradient in the particular case of float-zone traveling solvent growth of LaB6 crystals in excess of La, taking into account the actual liquidus line on the relevant phase diagram of Storms and Mueller [J. Phys. Chem. 82, 51 (1978)]. We show that the action of Marangoni convection decreases the concentration gradient at the solidification interface and inhibits the morphological instability and the increasing supersaturation gradient. We suggest that this effect allows the implementation of float-zone traveling solvent growth of LaB6 crystals with pulling rates of congruent to cm/h without the appearance of defects structures associated with the morphological instability and increasing supersaturation gradient. (C) 1996 American Institute of Physics.