I. THEORY OF CRYSTAL GROWTH Introductory Paper BY N. F. MOTT Received 7th March, 1949 The theory of crystal growth can, it seems to me, conveniently be divided into three parts. These are : (a) The theory of the rate of growth of a surface in contact with a vapour or solution with a given degree of supersaturation. Or, in the case of a crystal growing from the melt, the theory of the rate of growth for a given degree of supercooling. This will include a discussion of the rates of growth of different crystal faces, and the effect on growth rates of impurities which may be adsorbed on the surface, and of imperfections in the crystals them- selves. The solution of the problems under this heading depends, of course, on a knowledge of interatomic forces. (b) The use of results obtained under the heading (a) to determine crystal forms in as far as they depend in the case of growth from solution, or diffusion of the ions or atoms to be deposited, or in the case of growth from the melt on conduction through the material of the heat liberated.Much of the theory of dendrite formation is included in this category. I t forms a part of classical rather than atomic physics, depending as it does on the equations of diffusion and heat flow. (c) Discussions of the crystal form of the deposit. This will include such problems as the formation during growth of screw or edge dislocations in the crystal; a solution of these problems is very important for the theory of mechanical strength. Then there is the question of the possible pseudo- morphic forms of crystalline films grown on a substrate of different composi- tion ; a contribution to this subject is made by van der Menve in a paper to be presented to this conference.And, finally, there is the question of the state of strain and possible cracking of the surface layer treated by MoliGre, Rathje and Stranski. ( a ) Atomic Theory of Growth. The elements of a theory of crystal growth have been laid down by Volmer, Stranski, Becker and Doring, and new contributions made by Frank, Burton and Cabrera (for references, see the contribution of F. C. Frank to this Discussion). This theory applies explicitly to growth from the vapour; but can probably be applied in principle to growth from solution. The problem of growth from the melt remains an open question. The elements of the theory of growth are as follows: consider a flat crystalline surface of low indices (say, (roo) for a simple cubic or (111) for a close-packed structure) in contact with a vapour.Suppose this surface is partly covered by another layer. Then if the pressure of the vapour is raised by a small amount Ap above the equilibrium vapour pressure, theory indicates that the layer will grow, with a speed proportional to A$, until it covers the surface. But in order to start a new layer, a two-dimensional nucleus must be formed, and, like other nucleation phenomena, the rate of nucleation varies with Ap as e-A/*p, where A is a constant at given temperature. It follows that when A@ is below some critical value the rate is negligibly small. It seems likely that the growth rate depends in general on the rate of nucleation, at any rate for surfaces of low indices; for surfaces of high indices, having a step-like formation anyhow, nucleation is much easier.11I 2 THEORY OF CRYSTAL GROWTH But such surfaces of high indices will, of course, by growing quickly tend to disappear, leaving a crystal surrounded by planes of low index only. It should be emphasized that a flat surface in contact with vapour will have a number of atoms adsorbed on it. Two-dimensional nucleation can occur whether or not these are mobile over the surface ; it is not at present quite certain whether their mobility affects the rate of nucleation. Among the papers presented to this Discussion, Becker gives a valuable account of the relation of his theory to Mayer's theory of condensation.Burton and Cabrera, in a paper to be published elsewhere, have made some refinements to the present theory by calculating the shape of the two- dimensional nucleus when it has reached the size beyond which it will normally spread. This puts the theory on a firmer footing, and does not alter the numerical values very much. Frank points out that the theory suggests a growth rate which is negligibly small unless the supersaturation of the vapour is of the order 1-5, and that this is contrary to experiment, in particular to the results of Volmer and Schultze on the growth of iodine crystals; the degree of supersaturation required is of the order 1-01. He suggests that the presence of dislocations is essential for growth at these concentra- tions, and that the growth rate depends essentially on the density of dis- locations in the material. Theory has at present made little contribution to our knowledge of habit modification.It does, however, follow that, if dislocations are essential for crystal growth, very small concentrations of impurity, which could be adsorbed preferentially at the " ledge '' where the dislocation meets the surface, could profoundly affect growth rates and thus lead to habit modifi- cat ion. ( b ) Phenomena Depending on Heat Flow and Diffusion. It is believed that dendrite formation in the solidification of liquid metals is due to the fact that a thin needle, growing into a supercooled solution, will need to get rid of less heat by conduction than a thicker one and so will grow faster.In the same way, in the formation of crystals from solution, a thin needle will grow more quickly than a thick one into supersaturated solution. Probably the clue to the step formation observed by Bunn will be found along these lines. (c) Physical State of the Crystal as a Consequence of the Mechanism of Growth. Frank, in his paper, gives some reasons for believing that, at finite growth rates, dislocations will be formed in the crystal. They are in no sense present in thermodynamic equilibrium and ideally a long enough anneal would get rid of them ; but, in practice, there appear always to remain a certain number. Stranski and his colleagues reopen the very interesting question of the state of strain of the surface layer. The origin of the " Griffith cracks," responsible for the low stress for fracture of brittle materials, has never been explained, and it is possible that this work will provide a clue.In a later section of the Discussion, van der Menve discusses the crystal structure of thin films deposited on a substrate of differing crystal structure. He shows that the question, whether or not the deposit has a pseudomorphic form, depends on whether the first monolayer conforms to the structure of the substrate or not; and that this in turn depends on the degree of misfit . Equilibrium Crystal Forms The study of the shape of a crystal in equilibrium with a vapour forms an interesting field rather apart from the theory of crystal growth. Burton and Cabrera have found that the equilibrium forrn of the two-dimensional crystalline nucleus on a flat substrate is a rounded polygon, if only oneN. F. MOTT I3 atomic or molecular unit is involved. For ionic forces, on the other hand, it appears that the two-dimensional nucleus may have sharp corners. In the case of three dimensions Stranski has shown that the corners of a crystal are rounded off through the presence of afinite number of planes of higher index, and so are not truly rounded. The microstructure of the surface in equilibrium with vapour or solution is also of interest. As already stated, a flat surface will always contain some adsorbed atoms, and there will always be some vacant lattice points. Burton and Cabrera have made an investigation of the concentration of " Frenkel terraces " on a surface in equilibrium. For faces of low index, there will be practically none for a perfect crystal; any which exist depend on the presence of dislocations. A crystal temperature exists, however, at which they form, but this will in general be above the melting point. H. H . Wills Physical Laboratories, Royal Fort, Bristol 8.