Axis Distribution Function edit page

The misorientation axis distribution is the counterpart of the angle distribution - it is what remains of a misorientation distribution function after integrating out the misorientation angle. It is a function on the sphere and is represented in MTEX by an S2Fun.

The axis distribution of a uniform texture

As for the angle, the reference case of completely random orientations does not give a uniform distribution. The reason is that the maximum possible misorientation angle depends on the direction of the axis - an axis for which the fundamental region extends further contributes more misorientations. The command calcAxisDistribution computes this reference distribution

cs = crystalSymmetry('432');

adf = calcAxisDistribution(cs)
adf = S2FunHarmonicSym (432)
  bandwidth: 256
  antipodal: true

Being an S2Fun it can be plotted, evaluated and integrated like any other spherical function. Restricted to the fundamental sector of the crystal symmetry it looks as follows

plot(adf,'upper')
mtexColorbar

The same plot is produced directly by plotAxisDistribution 

plotAxisDistribution(cs)

The variation is substantial - over the fundamental sector the density ranges between

[min(adf) max(adf)]
ans =
    0.5974    1.5652

i.e. by a factor of more than two. Any interpretation of a measured axis distribution has to be made against this reference, not against a constant.

The axis distribution of measured misorientations

Let us again consider the Magnesium data set

plottingConvention.default('y↑→x');
mtexdata twins silent

grains = calcGrains(ebsd('indexed'),'threshold',5*degree);
grains = smoothBoundary(grains,5);

mori = grains.boundary('Magnesium','Magnesium').misorientation
mori = misorientation (Magnesium → Magnesium)
  size: 2803 x 1
  antipodal: true

The misorientation axes in crystal coordinates are returned by axis 

mori.axis
ans = Miller (Magnesium)
 size: 2803 x 1

and their distribution is displayed by plotAxisDistribution 

plotAxisDistribution(mori,'contourf','halfwidth',5*degree)
mtexColorbar

The single sharp maximum is the twinning axis. Compare this against the uniform reference, which is essentially flat over the same sector

plotAxisDistribution(mori.CS,mori.SS)
mtexColorbar

Crystal versus specimen coordinates

All of the above lives in crystal coordinates - the axis is expressed relative to the crystal lattice and is therefore subject to crystal symmetry. A misorientation axis can equally well be given in specimen coordinates, and this is a different question with a different answer. Passing the two orientations separately rather than the misorientation gives the specimen version

ori1 = ebsd(grains.boundary('Magnesium','Magnesium').ebsdId(:,1)).orientations;
ori2 = ebsd(grains.boundary('Magnesium','Magnesium').ebsdId(:,2)).orientations;

plotAxisDistribution(ori1,ori2,'contourf','halfwidth',5*degree)
mtexColorbar

This is now a plot over the full sphere in specimen coordinates, with only the specimen symmetry imposed. The chapter Tilt and Twist Boundaries uses exactly this distinction to classify boundaries.

The axis distribution of an ODF

Finally, the axis distribution can also be computed for a misorientation distribution function rather than for a list of misorientations. This gives the uncorrelated reference for the given texture

odf = calcDensity(grains('Magnesium').meanOrientation,'halfwidth',10*degree);
mdf = calcMDF(odf);

adfMDF = calcAxisDistribution(mdf)
Warning: The convolution of two SO3FunRBFs could be done fast by pure RBF
method. For big center sizes this yields an SO3FunRBF with lots of
centers, which is not manageable anymore. If you still want to generate an
SO3FunRBF use 'noFourier'. 
 
adfMDF = S2FunHarmonicSym (Magnesium)
  bandwidth: 64
plot(adfMDF,'upper')
mtexColorbar

It differs markedly from the boundary distribution above, which is precisely the statement that the twin boundaries are not a consequence of the texture.