Harmonic Series Expansion of an ODF

Harmonic Series Expansion of an ODF edit page

mtexdata dubna

odf3 = calcODF(pf,'resolution',5*degree,'zero_Range')

pf = PoleFigure
  crystal symmetry : Quartz (321, X||a*, Y||b, Z||c*)
  specimen symmetry: 1
 
  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 = ODF (Quartz → xyz)
 
  Radially symmetric portion:
    kernel: de la Vallee Poussin, halfwidth 5°
    center: 19836 orientations, resolution: 5°
    weight: 1

Fourier Coefficients

The Fourier coefficients allow for a complete characterization of the ODF. The are of particular importance for the calculation of mean macroscopic properties e.g. the second order Fourier coefficients characterize thermal expansion, optical refraction index, and electrical conductivity whereas the fourth order Fourier coefficients characterize the elastic properties of the specimen. Moreover, the decay of the Fourier coefficients is directly related to the smoothness of the ODF. The decay of the Fourier coefficients might also hint for the presents of a ghost effect. See Ghost Correction.

transform into an odf given by Fourier coefficients

fodf = FourierODF(odf3,32)

fodf = ODF (Quartz → xyz)
 
  Harmonic portion:
    degree: 32
    weight: 1

The Fourier coefficients of order 2:

reshape(fodf.components{1}.f_hat(11:35),5,5)

ans =
  Columns 1 through 4
   0.0000 - 0.0000i   0.0000 - 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
  -0.0000 + 0.0000i  -0.0000 + 0.0000i  -0.0000 + 0.0000i  -0.0000 - 0.0000i
   0.1329 - 0.3932i   1.3900 - 0.8349i   1.9650 - 0.0000i   1.3900 + 0.8349i
  -0.0000 + 0.0000i  -0.0000 + 0.0000i  -0.0000 - 0.0000i  -0.0000 - 0.0000i
   0.0000 - 0.0000i   0.0000 - 0.0000i   0.0000 - 0.0000i   0.0000 + 0.0000i
  Column 5
   0.0000 + 0.0000i
  -0.0000 - 0.0000i
   0.1329 + 0.3932i
  -0.0000 - 0.0000i
   0.0000 + 0.0000i

The decay of the Fourier coefficients:

close all;
plotFourier(fodf)

ODFs given by Fourier coefficients

In order to define a ODF by it Fourier coefficients the Fourier coefficients C has to be given as a literally ordered, complex valued vector of the form

\[ C = [C_0,C_1^{-1-1},\ldots,C_1^{11},C_2^{-2-2},\ldots,C_L^{LL}] \]

where \(l=0,\ldots,L\) denotes the order of the Fourier coefficients.

cs   = crystalSymmetry('1');    % crystal symmetry
C = [1;reshape(eye(3),[],1);reshape(eye(5),[],1)]; % Fourier coefficients
odf = FourierODF(C,cs)

plot(odf,'sections',6,'silent','sigma')

odf = ODF (1 → xyz)
  antipodal:         true
 
  Harmonic portion:
    degree: 2
    weight: 1

plotPDF(odf,[Miller(1,0,0,cs),Miller(1,1,0,cs)],'antipodal')