ODF Characteristics edit page

Let us first begin with some constructed ODFs to be analyzed below

A bimodal ODF:

cs = crystalSymmetry('mmm');
odf1 = unimodalODF(orientation.byEuler(0,0,0,cs)) + ...
  unimodalODF(orientation.byEuler(30*degree,0,0,cs))
odf1 = SO3FunRBF (mmm → y↑→x)
 
  multimodal components
  kernel: de la Vallee Poussin, halfwidth 10°
  center: 2 orientations
 
  Bunge Euler angles in degree
  phi1    Phi   phi2 weight
     0      0      0      1
    30      0      0      1

A fibre ODF:

f001_x = fibre(Miller(0,0,1,cs),xvector)

odf2 = fibreODF(f001_x)
f001_x = fibre (mmm → y↑→x)
 
  h || r: (001) || (1,0,0)
 
odf2 = SO3FunCBF (mmm → y↑→x)
 
  kernel: de la Vallee Poussin, halfwidth 10°
  fibre : (001) || 1,0,0
  weight: 1

An ODF estimated from diffraction data

mtexdata dubna

odf3 = calcODF(pf,'resolution',5*degree,'zero_Range')
pf = PoleFigure (y↑→x)
  crystal symmetry : Quartz (321, X||a*, Y||b, Z||c*)
 
  h = (02-21), r = 72 x 19 points
  h = (10-10), r = 72 x 19 points
  h = (10-11)(01-11), r = 72 x 19 points
  h = (10-12), r = 72 x 19 points
  h = (11-20), r = 72 x 19 points
  h = (11-21), r = 72 x 19 points
  h = (11-22), r = 72 x 19 points
 
odf3 = SO3FunRBF (Quartz → y↑→x)
 
  multimodal components
  kernel: de la Vallee Poussin, halfwidth 5°
  center: 19848 orientations, resolution: 5°
  weight: 1

Modal Orientations

The modal orientation of an ODF is the crystallographic preferred orientation ori_pref of the texture. It is characterized as the maximum of the ODF. In MTEX it is returned as the second output argument of the command max 

[~,ori_pref] = max(odf3)
ori_pref = orientation (Quartz → y↑→x)
 
  Bunge Euler angles in degree
     phi1     Phi    phi2
  133.047 34.5158  207.16

Lets mark this preferred orientation in the pole figures

plotPDF(odf3,pf.allH,'antipodal','superposition',pf.c);
annotate(ori_pref,'marker','s','MarkerFaceColor','black')

Texture Characteristics

Texture characteristics are used for a rough classification of ODFs into sharp and weak ones. The two most common texture characteristics are the entropy and the texture index. The texture index of an ODF \(f\) is defined as:

\[ t = \int_{SO(3)} f({R})^2 dR\]

We may either compute this integral using the command sum directly by

t = mean(odf1.*odf1)
t =
  288.5696

or, more efficiently, by the command norm 

t = norm(odf1)^2
t =
  288.6802

The entropy of an ODF \(f\) is defined as:

\[ H = - \int_{SO(3)} f({R}) \ln f({R}) dR\]

H = entropy(odf2)
H =
   -2.8402

Volume Portions

Volume portions describes the relative volume of crystals having a certain orientation. The relative volume of crystals having a orientation close to a given orientation is computed by the command volume and the relative volume of crystals having a orientation close to a given fibre is computed by the command fibreVolume

The relative volume in percent of crystals with misorientation maximum 30 degree from the preferred orientation ori_pref:

V1 = volume(odf3, ori_pref, 30*degree) * 100
V1 =
   33.8649

The relative volume of crystals with misorientation maximum 20 degree from the preferred fibre in percent:

V2 = volume(odf2,f001_x,20*degree) * 100
V2 =
   95.1633