Defining Crystal Shapes with Smorf

Crystal Shapes shows how a shape is built from face normals and their distances from the origin. Finding those distances for a real mineral is fiddly work by hand, and it is much easier done with a drawing tool that redraws the crystal as the numbers change. This page walks through it on the olivine shape published by Welsch et al. (2013, J. Pet.).

The Drawing Tool

The crystal drawing tool of the Smorf website is a free alternative to the commercial packages, made available by Mark Holtkamp.

Crystal Parameters

Select the point group and enter the cell parameters under celldata - the ones from your own EBSD file, if you have one. For the interpretation of the face distance choose Crystallographic (Kristall2000), which is the convention MTEX follows; the other choices scale the distances differently and the numbers would not carry over.

The same symmetry in MTEX:

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

cs = crystalSymmetry (⊙c→a)
 
  mineral : Forsterite
  symmetry: mmm       
  elements: 8         
  a, b, c : 4.8, 10, 6

The Face Normals

Building a complicated shape face by face is tedious, so start from the views along \(\vec a\), \(\vec b\) and \(\vec c\) and enter every face visible in the published model, each at distance 1.

In MTEX the same set of faces is a list of <Miller.Miller.html Miller> indices.

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

N = Miller (Forsterite)
 size: 1 × 6
  h k l
  0 1 0
  0 0 1
  0 2 1
  1 1 0
  1 0 1
  1 2 0

Adjusting the Distances

Now change the distance of one face at a time - steps of 0.05 work well. A larger distance moves the face away from the origin, reducing its influence on the shape until it may stop cutting the crystal altogether. Fix the largest faces first and keep the overall aspect ratio while moving the others. The drawing does not update by itself; press Draw crystal after each change, and compare against the published crystal until they match.

Then note the indices and their distances.

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

crystalShape takes normals whose length encodes the distance, so the two lists are combined by dividing.

% this defines the crystal shape in MTEX
cS = crystalShape( N ./ dist)

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

% plot the crystal shape
plot(cS,'colored')

The habit of the published crystal is reproduced. cS.faceArea says how much of the surface each face got: the two \((010)\) faces are the largest individual ones at 0.139, which is what makes the crystal tabular, while \((001)\) came out as a small cap of 0.02 and the four faces each of \(\{021\}\) and \(\{110\}\) carry most of the total area.

max(cS.faceArea)

%
% The Smorf mineral database holds many more morphologies, and each of them
% transfers to MTEX in exactly these two lines.

ans =
    0.1390

Next

What the shapes are used for - orientations on a map, twinning, slip systems - is Crystal Shapes.