# 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.