CrystalDirections.m

%% Miller Indices
%
% Miller indices are used to describe directions with respect to the
% crystal reference system.
%
%% Crystal Lattice Directions
%
% Since lattice directions are always subject to a certain crystal
% reference frame, the starting point for any crystal direction is the
% definition of a variable of type <crystalSymmetry.crystalSymmetry.html
% |crystalSymmetry|>.

cs = crystalSymmetry('triclinic',[5.29,9.18,9.42],[90.4,98.9,90.1]*degree,...
  'X||a*','Z||c','mineral','Talc');

%%
% The variable |cs| contains the geometry of the crystal reference frame
% and, in particular, the alignment of the crystallographic $\vec a$, $\vec
% b$, and, $\vec c$ axis.

a = cs.aAxis
b = cs.bAxis
c = cs.cAxis

%%
% A lattice direction $\vec m = u \cdot \vec a + v \cdot \vec b + w \cdot
% \vec c$ is a vector with coordinates $u$, $v$, $w$ with respect to these
% crystallographic axes. Such a direction is commonly denoted by $[uvw]$
% with coordinates $u$, $v$, $w$ called Miller indices. In MTEX a lattice
% direction is represented by a variable of type <Miller.Miller.html
% |Miller|> which is defined by

m = Miller(1,0,1,cs,'uvw')

%%
% for values $u = 1$, $v = 0$, and, $w = 1$. To plot a crystal direction as
% a <SphericalProjections.html spherical projections> do

plot(m,'labeled','grid')

annotate([a,b,c],'label',{'a','b','c'},'backgroundcolor','w','textAboveMarker')

%%
% Note that MTEX by default aligns spherical projections of crystal
% directions such that the b-axis points towards east and the z-axis points
% out of the plane. This behaviour can be changed by the commands
% |plota2east|, |plota2north|, |plota2west|, |plota2south|, |plotb2east|,
% |plotb2north|, |plotb2west|, |plotb2south|, or |plotaStar2East|.
%
%% Crystal Lattice Planes
%
% A crystal lattice plane $(hkl)$ is commonly described by its normal
% vector $\vec n = h \cdot \vec a^* + k \cdot \vec b^* + \ell \cdot \vec
% c^*$ where $\vec a^*$, $\vec b^*$ and $\vec c^*$ describe the reciprocal
% crystal coordinate system. In MTEX a lattice plane is defined by

m = Miller(1,0,1,cs,'hkl')

%%
% By default lattice planes are plotted as normal directions. Using the
% option |'plane'| we may alternatively plot the trace of the lattice plane
% with the sphere.

hold on
% the normal direction
plot(m,'upper','labeled')

% the trace of the corresponding lattice plane
plot(m,'plane','linecolor','r','linewidth',2,'add2all')
hold off

%%
% Note that for non Euclidean crystal frames uvw and hkl notations usually
% lead to different directions.
%
%% Trigonal and Hexagonal Convention
%
% In the case of trigonal and hexagonal crystal symmetry often four digit
% Miller indices $[UVTW]$ and $(HKIL)$ are used, as they make it more easy
% to identify symmetrically equivalent directions. This notation is
% redundant as the first three Miller indeces always sum up to zero, i.e.,
% $U + V + T = 0$ and $H + K + I = 0$. The syntax is

% import trigonal Quartz lattice structure
cs = loadCIF('quartz');

% a four digit lattice direction
m = Miller(2,1,-3,1,cs,'UVTW')

% a four digit plane normal
n = Miller(1,1,-2,3,cs,'hkil')

plot(m,'upper','labeled','backgroundColor','white','grid','on')
hold on
plot(n,'upper','labeled')
hold off

%%
% In order to switch the output format, e.g. from UVTW to uvw do

m.dispStyle = 'uvw';
round(m)

%%
% or from reciprocal to direct coordinates

n.dispStyle = 'UVTW';
round(n)

%%
% Note, that this does not change the vector but only the display of the
% coefficients. Internally, all vectors are stored with respect to the
% cartesian coordinate system.
%