Exponential Growth-Rate Constants
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Dialog Box: Exponential Growth-Rate Constants [ Forms List Dialog] [ Forms List for Growth-Rate Constants Dialog]

If the exponential growth model is specified in the Growth Zone Settings dialog, a rate equation, in terms of central distance units and any time units desired, must be specified for each form. The equation is exponential in form:

rate = a1 + a2 t^b2 + a3 t^b3 + a4 t^b4

where t is time. Most continuous functions can be approximated by an equation of this form with a MacLaurin expansion. The coefficients of the equation must be specified for each form; the forms are listed in sequential order, ignoring the crystal to which they belong in the case of epitaxial intergrowths. As a default, the a1 coefficient is simply taken to be the central distance; that is, the default is a linear rate which gives the same crystal shape as that specified by the original central distances, when total time is 1.0. Other a coefficients are set to zero. The default values for b2 through b4 are one, two and three respectively (b1 is not used).

Appearance and disappearance of forms in non-linear growth. In either the exponential or discontinuous models, forms may appear and disappear during growth. If the growth rate of a form is so large that it does not appear on a crystal after a given interval, its central distance is adjusted so that it lies just outside the current corners and edges. When the current growth rate for a form is sufficiently small relative to the other forms over a given growth interval, that form will appear, regardless of how large its rate may have been in previous growth intervals. Thus the central distance of a form is not the integral of the rate equation, if the form does not appear for any time.