Reconstructing a Crystal Habit with Smorf

Crystal Shapes explains how MTEX builds a polyhedron from face normals. This page shows how to estimate the relative face distances from a published drawing with the Smorf crystal drawing tool, then transfer the result to MTEX.

The example reproduces the olivine growth form in Welsch et al. (2013).

Notice the broad (010) and (110) faces, the narrow (001) cap and the bevels formed by the other three crystal forms. These relative face sizes are the target of the reconstruction.

A crystal habit is the characteristic external shape of a crystal and the development of its forms. The model reconstructed here is an idealized habit, not the measured three-dimensional morphology of an EBSD grain. Its fitted distances are geometric parameters, not growth rates or surface energies.

plottingConvention.default('y↑→x');

Match the Symmetry and Unit Cell

Smorf is a free browser-based drawing tool maintained by Mark Holtkamp. Select the point group and enter the unit-cell lengths and angles from the source you want to reproduce. For an imported phase, these values are available from its crystal symmetry.

Choose Crystallographic (Kristall2000) for the face-distance interpretation. This is the convention that transfers to the MTEX construction below; the other choices scale the face normals differently.

The screenshot pairs point group mmm with the three forsterite lattice parameters and highlights the required distance convention. The same crystal symmetry in MTEX is

cs = crystalSymmetry('mmm',[4.756 10.207 5.98], ...
  'mineral','Forsterite');

Enter the Crystal Forms

A crystal form is the set of faces generated from one face by the point group. Enter one representative Miller index for every form visible in views along \(\vec a\), \(\vec b\) and \(\vec c\). Start every form at distance 1 so that differences in the first drawing come only from the lattice metric and symmetry.

The slide sets the published views along \(\vec a\), \(\vec b\) and \(\vec c\) beside the Smorf drawing of the same views, with the interfacial angles to compare against. The equal-distance drawing in the previous screenshot is only a starting block: at distance 1 the (010), (001) and (110) forms do not reach the surface at all. The same six plane normals are a list of Miller indices in MTEX.

N = Miller({0,1,0},{0,0,1},{0,2,1}, ...
  {1,1,0},{1,0,1},{1,2,0},cs);

Tune the Face Distances

Change one distance at a time and redraw the crystal. A larger distance moves that form away from the origin, so the neighbouring forms cut its faces back and they shrink, until the form can disappear from the polyhedron altogether. Steps of 0.05 are small enough for this example.

Match the largest faces and the overall aspect ratio first. Then tune the smaller bevels. Smorf does not redraw automatically, so press Draw crystal after each change.

dist = [0.4, 1.3, 1.4, 1.05, 1.85, 1.35];

The tuned drawing now has the broad (010) and (110) sides, the small (001) cap and the bevels of the other three forms seen in the source figure. Only relative distances matter: multiplying every entry of dist by the same number leaves the normalized shape unchanged.

Transfer the Distances to MTEX

crystalShape describes each limiting plane by a normal \(\mathbf{n}\) and the half-space \(\mathbf{x}\cdot\mathbf{n}\leq 1\). A longer normal therefore moves its plane inward. To preserve the distances entered in Smorf, divide each Miller normal by its corresponding distance:

\[\mathbf{n}_{\mathrm{MTEX}} = \mathbf{n}_{hkl} / d_{\mathrm{Smorf}}.\]

cS = crystalShape(N ./ dist)

cS = crystalShape
 mineral: Forsterite (mmm)
 vertices: 36
 faces: 20

plot(cS,'colored');

The MTEX figure reproduces the published habit. The (010) and (110) faces form the broad prism sides, the (021) faces terminate the ends around the small (001) caps, and the narrow (120) and (101) faces truncate the remaining edges and corners.

Check the Developed Faces

The constructor applies the point group to each input normal. The multiplicity is the number of symmetry-related faces in a form. The table separates the area of one face from the total area of its complete form.

multiplicity = N.multiplicity;
areaPerFace = zeros(size(multiplicity));
formArea = zeros(size(multiplicity));
for k = 1:length(N)
  areas = cS(N(k).symmetrise('unique')).faceArea;
  areaPerFace(k) = mean(areas);
  formArea(k) = sum(areas);
end
formName = ['(010)';'(001)';'(021)';'(110)';'(101)';'(120)'];
formSummary = table(formName,dist(:),multiplicity,areaPerFace,formArea, ...
  'VariableNames',{'Form','Distance','Multiplicity','AreaPerFace','FormArea'})

formSummary =
  6×5 table
    Form     Distance    Multiplicity    AreaPerFace    FormArea
    _____    ________    ____________    ___________    ________
    (010)       0.4           2            0.13897       0.27793
    (001)       1.3           2           0.019766      0.039533
    (021)       1.4           4             0.1384       0.55362
    (110)      1.05           4             0.1346       0.53841
    (101)      1.85           4           0.033199       0.13279
    (120)      1.35           4           0.043238       0.17295

The six input forms generate 20 faces. An individual (010) face is the largest at 0.13897, just above an individual (021) face at 0.13840. The four (021) and four (110) faces have the largest total form areas, 0.55362 and 0.53841. The two (001) caps are the smallest at 0.01977 each.

These areas belong to MTEX's normalized polyhedron and have no physical unit. Use them to compare relative face development within this shape.

Further Reading

Next

Return to Crystal Shapes to rotate, scale and place this idealized habit as an orientation glyph. See Miller Indices for the distinction between plane normals and crystal directions, and Crystal Symmetries for point groups and unit-cell definitions.