Dialog Box: Donnay-Harker Morphology [ Input1 Menu]
This option will give a list of forms with central distances as a function of X-ray d-spacing or relative reticular area.
According to the Law of Bravais, later extended by Donnay and Harker, the morphological importance of a form is inversely related to the interplanar spacing, or X-ray d-spacing. Since the d-spacings depend on the presence of non-primitive Bravais lattices, screw axes and glide planes, it is necessary to know the space group as well as the unit-cell parameters.
This relationship is essentially empirical, and the quantitative relationship between d-spacing and growth rate is difficult to establish. Analysis of the equation for growth rate by the surface-nucleation mechanism (E. Dowty, "Crystal Growth and Nucleation Theory", in R.B. Hargraves, Ed. , Physics of Magmatic Processes), suggests a relationship of the form
R = k1 d exp( - k2 d )
where k2 is inversely proportional to the supersaturation (or supercooling in the case of growth from the melt). Of course this is a great simplification, and ignores the influence of different surface energy on different faces.
For small values of k2, corresponding to large supersaturation or rapid growth, the rate of growth may show a maximum as a function of d-spacing. For the mechanism of growth by the spiral dislocation mechanism, there is apparently no inverse dependence of growth rate on d-spacing, rather the principal influence seems to be surface energy (included in k2 in the above equation). Thus the Bravais/Donnay-Harker law seems to apply to slow growth by the surface-nucleation mechanism, and adherence to this law is not expected in all cases, particularly rapid growth in the laboratory.
SHAPE offers the choice of three functions for the dependence of growth rate or central distance on d-spacing; a simple inverse dependence (R = 1 / d), the above exponential relation, and the above relation without the d in the pre-exponent.
The function (R = 1/d) has special consequences, in that it necessarily causes some faces to lie exactly on corners or edges. For example, in the orthorhombic group mmm , the first-ranked forms are {100}, {010}, and {001} - these are the only forms which appear as actual faces. However, the faces of forms {110}, {101} and {011} lie exactly on the edges and those of form {111} lie exactly on the corners. The faces of these forms appear in the listings for each corner, which may cause confusion. Because of the finite precision of the calculation, there may sometimes be extra corners or other anomalies (see Precision, Tolerances, etc. )
Whichever relation is chosen, SHAPE generates a set of hkl form indices consistent with the symmetry. The forms are sorted according to the growth rate or central distance, and normalized to the smallest growth rate. This list of forms will replace any forms currently present.
The space group may be specified by typing in either the H-M (Herman-Maughin or International ) symbol, the Hall symbol, or the number of the space group in the International Tables for X-ray Crystallography . However, because of different possibilities for origin and orientation, the best way to specify the space group is by selection in the list box. This selection is not entered until the selection is double-clicked or the Select button is clicked.
SHAPE requires monoclinic crystals to be in the second setting, i.e. unique or symmetry axis b, rather than c. If you enter a symbol which does not explicitly give the orientation, such as C2/c, the second setting will be assumed. Do not enter a symbol which does not conform to this standard (such as P112/m)
It is important to realize that an empirical list of forms in order of morphological importance for a given crystal usually represents a statistical compilation, not a typical or ideal morphology. To put it in another way, such compilations or observations are usually based on frequency of occurrence of forms, rather than measurements of central distance for any particular case, or averages of such measurements. While such a list may have dozens of entries, actual crystals typically show a small number, perhaps only 1 or 2 for a cubic crystal. Presumably the ideal relative growth rates as predicted theoretically on the basis of crystal structure and geometry alone are modified in each case by environmental factors, such as composition of the growth fluid and overall growth rate. The Bravais-Donnay-Harker list as derived by SHAPE should thus be considered as a ranking of forms in order of probability of occurrence.
Note that space groups in some trigonal, tetragonal and hexagonal point groups may have alternate orientations of the symmetry elements with respect to the crystallographic axes - for example P321 and P312 are different space groups. These differences in orientation are now included in the choices of point group in the Symmetry dialog, and it may be necessary to change the point group or crystal class to agree with the space group.
You may only make a D-H list of forms for a single crystal - if you use this option for an epitaxial intergrowth, all crystals except the main one will be deleted. In order to apply D-H morphology to an epitaxial crystal, you must do that crystal separately and save the results in a file (Save or Saveas in the File menu). Then that file may be read as a second or subsequent crystal in the Epitaxy option (Input1 menu). Of course it is advisable to remove any forms which do not appear in the actual calculated crystal - forms are marked as PRESENT or ABSENT in the Listing of calculation results (File menu).