Energetic theory of face growth
Hartmann and Bennema’s theory (1980)
Experience teaches us that a crystal can usually be considered as a set of continuous pyramids, each with a shared summit that is known as the initial point. The bases of these pyramids are the crystal faces.
The surface of each base can be written as:
hi : pyramid height
ai: geometrical coefficient characteristic of index i pyramid.
The apparition of a crystal corresponds with the search for the minimum molecular attachment energy. Indeed, on a crystal face, the energy binding molecules to the crystal below is the attachment energy. The total of these attachment energies is consequently:
Furthermore, the crystal volume is:
The aim is to identify a relationship that minimizes E^fix , while keeping
the crystal volume D constant. This problem is typical of those for which we apply the Lagrange multiplier method. This method implies adding the differentials of E^fix and D, having multiplied the latter by the coefficient
X. This total must be zero, which will imply the invalidity of d (E^fix) and
thereby the presence of an extremum for this function.
Accordingly, height hi is proportional to orthogonal growth rate R i of face i. Finally:
Growth rate is proportional to the attachment energy of the face. This relationship was verified experimentally by Hartman and Bennena in 1980 [HAR 80]. However, this theory only concerns growth from vapor at low pressure.