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.

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