Crystal screw dislocation theory

The screw-dislocation theory (sometimes referred to as the BCF theory because of its development by Burton, Cabrera, and Frank) is based on a mechanism of continuous movement in a spiral or screw of a step or ledge on the crystal surface. The theory shows that the dependence of growth rate on supersaturation can vary from a parabolic relationship at low supersaturations to a linear relationship at high supersaturations. In the BCF theory, growth rate is given by ... 

The difference between the observed and theoretical growth rates has been reconciled by the Frank screw-dislocation theory. Actual space lattices of real crystals are far from perfect and crystals have imperfections called dislocations. Planes of particles on the surfaces and Mtliin the crystals are displaced, and several kinds of dislocations are known. One common dislocation is a screw dislocation (Fig- 27.9), where the individual particles are shown as cubical building blocks. The dislocation is in a shear plane perpendicular to the surface of the crystal, and the slipping of the crystal creates a ramp. The edge of the ramp acts like a portion of a two-dimensional nucleus and provides a kink into which particles can easily fit, A complete face never can form, and no nucleation is necessary, As growth... 

A number of theories have been put forth to explain the mechanism of polytype formation (30—36), such as the generation of steps by screw dislocations on single-crystal surfaces that could account for the large number of polytypes formed (30,35,36). The growth of crystals via the vapor phase is beheved to occur by surface nucleation and ledge movement by face specific reactions (37). The soHd-state transformation from one polytype to another is beheved to occur by a layer-displacement mechanism (38) caused by nucleation and expansion of stacking faults in close-packed double layers of Si and C. 

The classical crystal growth theory goes back to Burton, Cabrera and Frank (BCF) (1951). The BCF theory presents a physical picture of the interface (Fig. 6.9c) where at kinks on a surface step - at the outcrop of a screw dislocation-adsorbed crystal constituents are sequentially incorporated into the growing lattice. 

Surface Spiral Step Control. Many crystals grow faster at small supersaturation than allowed by Equation 7. This lead Frank (17) to suggest that steps may also originate from the presence of a screw dislocation, and that this kind of steps is not destroyed by spreading to the crystal edge, but continues infinitely. The rate law according to this theory is parabolic (7). We shall use the following version of the kinetic equation (10)... 

Next, let us compile some quantitative relations which concern the stress field and the energy of dislocations. Using elastic continuum theory and disregarding the dislocation core, the elastic energy, diS, of a screw dislocation per unit length for isotropic crystals is found to be... 

Burton, Cabrera, and Frank [56] and Bennema and Gilmer [57] have developed a theory to predict the crystal growth rate for screw dislocations. The growth rate will depend on the shape of the growth spiral. For an Archimedian spiral, shown in Figure 6.14 [58], the distance between the steps of the spiral yo is... 

Figure 5.17. Image profiles for screw dislocations calculated using the kinematical theory for various vaiues of in a crystal of thickness z = 20/s. The dislocation is located at /3 = 0. (After Hirsch, Howie, and Whelan I960.)...

FIGURE 20.8 Crystal growth proceeds in a spiral fashion, known as BCF theory, in which screw dislocation serves as a continuous source to generate growth sites so that the stepped growth will continue. 

KRI 63] KRIVOGLAZ M.A., RYBOSHAPKA K.P., Theory of X-ray scattering by crystals containing dislocations, screw and edge dislocations randomly , Fiz. Metal. Metalloved., vol. 15, p. 18-31,1963. 

LEV 97] LEVINE L.E., THOMSON R., X-ray scattering by dislocations in crystal. General theory and application to screw dislocations ,4cto Cryst A, vol. 53, p. 590-602,1997. 

A noteworthy aspect of the above solution is that it does not involve either lattice dilatation du/dz or layer undulation V n. Therefore, within the approximations of the linear theory considered here, screw dislocations in smectic A have no self energy (apart from the core), nor do they interact amongst themselves. In this respect they are entirely different from screw dislocations in crystals. 

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