, , If we introduce a more general and consistent notation of then α = α111, β = α110 and γ = α113, whereas α100 = 1. Note that Vn is invariant with respect to reflections and permutations of the axes. We shall further refer to αn as the growth parameter of the n plane. The equation of the n plane at time t is where dn is a dimensionless parameter which controls the initial position of the plane (at t = 0) and l0 is taken to be the initial size of the crystal along the x -axis. In fact, the distance from the origin to the n plane at time t = 0 is equal to . It is now natural to introduce dimensionless normalized Cartesian coordinates rˆ, such that Obviously, division by t is not practical in computer calculations, but using dimensionless coordinates makes sense in a theoretical treatment. Then the equation of the n plane πn becomes which is equivalent to where We see that the set of parameters L, dn and αn completely control the shape of the crystal at all times t , with L being the only parameter with dimensions (L controls the scaling of the time axis and the initial size of the crystal). Computationally, the easiest way to specify a plane is by providing its normal vector n and the point on that plane closest to the origin, which is the intersection of the plane with the straight line going along n. Suppose that this point is Then, from the equation of the plane,
the origin to the new inclined plane. Figure 2 displays several examples of initial seed shapes. “Opacity” controls the transparency of the faces of the polyhedron. 3.2.2. Growth settings “t = ∞” sets the time to infinity and the polyhedron shown on the right-hand side is the ultimate shape. “Time of growth (hrs)” sets the time t in hours at which the polyhedron P (t) is calculated. “ (mm/hr)” sets the value of the growth speed in millimeters per hour. “α ”, “β”, “γ ” are the three main growth parameters defined in the first section. The four sliders “Vicinal cut Vn multiplier” control the multipliers mn for n = (100), (110) , (111) , (113) as described in Subsection Figure 3 displays a example of the evolution of the growth for a particular seed shape and growth parameter settings. Section settings It is possible to view the growth sectors inside the crystal in the form of their traces on the section plane x = c. Note that each section comprises many thin colored polygons, therefore it may appear transparent if their number is too low. Each of the polygons is the section of the polyhedron P (τ) at some point in time τ between 0 and t with its edges colored in agreement with the colors of the faces they were in. Note that there is always equal amount of time between two neighboring polygons. There may be a hole in the middle of the section. It corresponds to the case when the section plane has a normal vector which coincides with one of the normal vectors of the polyhedron and thus the corresponding face of the crystal “hits” the section plane at some positive time , therefore giving the nonzero size of the earliest nonempty section. The “Cut at time t” checkbox enables the section mode. The “Draw gray section face” checkbox controls the visibility of the gray face that is drawn behind the section polygons. Enabling this option is recommended if the section plane looks transparent and you do not want to increase the number of polygons. With this setting on it is recommended to turn up the “Opacity” setting to 1 to avoid clipping. The three sliders “Cut normal ” specify the normal vector of the cutting plane.
Experimentally, the values reported for in microwave plasma CVD of single crystal diamond growth are in the range[11-13, 20] We present in figure 5 the morphology of an isolated penetration twin on the {100} surface for a few values of in the range listed above. We have also analytically determined the stability conditions for a penetration twin growing on a {111} surface in the same manner used above. The conditions for a stable penetration twin on a {111} surface are: , or And for a contact twin on a {111}, the conditions are the same as for the existence of {111} surfaces: We have not yet implemented a specific cdf program to show the morphology of the penetration twin on {111}, but hopefully this will be available by the time of publication. CONCLUSION: A geometric model of growth of a cubic crystal has be implemented in readily available software. New conditions for the stability of penetration twins on {100} and {111} surfaces have been determined. Such modeling has value in designing growth conditions optimized to particular properties. For example, the incorporation of impurities, dopants, and the formation of structural defects depends on the local surface structure. Each of the low index planes in the model has a different stereochemical structure and thus the bulk material grown under a particular low index surface (often referred to as the ‘growth sector’) will have different physical properties reflecting the local defects and impurities in the growth sector. For example, the <111> growth sectors in