I have an orientation map in terms of three Bunge Euler angles (fi1, fi, fi2) = f(x,y,z) and need to calculate an orientation distribution function (ODF). I want to write my own script for this purpose, without using MTEX. I looked through Bunge "Texture Analysis".
I begin with a calculation of volume for each set of (fi1,fi,fi2)=g with a given step dfi. However, I am confused about how to proceed and calculate an ODF. According to Bunge, ODF(g)dg = deltaV(g)/V, where dg = sin(fi) dfi1 dfi dfi2 / (8*pi2), deltaV(g) is the volume taken by the orientation g, V is the whole specimen volume. I can't understand what ODF(g)dg is from the numerical point of view.
I would be grateful if you could give me any advice.
The original derivation from Bunge was developed for series expansion methods that derive the 3D Fourier expansion coefficients (for describing the actual orientation distribution function ODF) for the case of generalized spherical harmonics (used as library functions depending on the specific crystal symmetry and on sample symmetry), extreacted from incomplete diffraction pole figures (which also first have to be developed into series on the basis of spherical harmonics).
For a set of such pole figures (available as pole density - polar data sets ) also series expansions are developed (using several such pole figures) from which an incomplete set of three-dimensional Fourier coefficients can be calculated.
This set of equations is usually underdetermined so that only a limited set of coefficients can be gained, leading to the so-called incomplete series expansion method, which has to be iterated further by using assumptions such as the non-negativity rule for instance etc.
This changes and becomes actually more simple when you have a discrete set of Euler angles for a certain set of discrete orientations (as you describe in your note).
For this purpose it is usually much easier to use a binning method that directly calculates the odf, correcting for instance for symmetry and Euler space distortion (please note that standard binning needs to be corrected since the classical orientation distribution function which is plotted in Eulerian space comes with very strong distortions and must be corrected in a binning procedure).
For such cases it is therefore more convenient to use quarternions and plot the orientation distribution function better in Rodriguez space (and not in Euler space).
However, if you do prefer to use the Bunge method anyway, for instance for the convenience of retrieving Fourier expansion coefficients , which are very handy (and much more elegant and fast compared to the today commonly used discrete methods) when it comes to property calculations, then you must better choose a different method.
In this case I suggest to superimpose all discrete orientations by individual Eulerian Gauss components and develop them by using spherical harmonics into an ODF.
I attach some of my own programmes as a guideline, but I would rather strongly advise to use existing well tested binning or similar software (which considers all Euler distorions or use quarternions, we also coded some of that in our damask packages but there is some more software around I guess). My software I attach is a bit older, not sure if it runs OK.
Good luck !
PS sorry - these system does not let me upload exe
Based on the sports laws used and the good mathematical subject matter.
Recently, program language is replaced from FORTRAN. Microsoft Windows 10 operating system does not have FORTRAN dynamic link library. However, Intel sell the Visual FORTRAN compiler. Please use the Intel Visual FORTRAN compiler if you want to use your software, although I cannot guarantee whether your software does not run by the Visual FORTRAN compiler. At least it necessary to use the Visual FORTRAN compiler in Microsoft Windows 10 operating system.
If you have knowledge of FORTRAN, it is worthy to be attempted.
Ok, maybe this comes a bit late, nevertheless let me add some points to the discussion. As I understand, the point is to describe a preferred orientation of crystallites in a polycrystal. And measuring/describing it using a set of individual orientations is perfectly valid. Many softwares, e.g., VPSC, work with sets of individual orientations. But there is also the ODF. Which is the probability density for material to contain a volume element with a certain orientation. And as a probability density, it is a statistical entity. It means that to transfer from individual orientations to ODF we need to be sure that such transition is statistically sound. There are certain requirements for the actual number of measured crystallites to be statistically relevant, see, e.g., works by Pospiech et al. 1994 (Article Statistical Analysis of Single Grain Orientation Data Genera...
) and Matthies & Wagner 1996 (Article On a 1/n Law in Texture Related Single Orientation Analysis
); it is also discussed in more detail in a book "Introduction to texture analysis" by Engler and Randle (Paragraph 9.3). Here the tricky part is that normally we have a measured/modeled set of individual orientations, but for statistical "tests" we need a number of crystallites, and for ODF we need not a number, but volume fractions. Once we confirmed that we can do the statistically relevant estimation of the ODF with desired resolution, there are generally two possibilities. One is to split the orientation space to discrete cells, e.g., 5x5x5 deg., similar to direct ODF reconstruction methods as WIMV or ADC. Into every cell at Euler angles (alpha_i,beta_i,gamma_i), we will put all individual orientations with alpha_i-2.5
I use the orientation distribution function with the series-expansion method such as the Bunge method. This was made by myself by Fortran language in consideration of textural change in the sheet depth direction. I do not use the other software. I cannot answer about them. Not EBSD pole figures but X-ray pole figures with a 5 degrees step in alpha and beta angles were used in this analysis.
I calculated the complete ODF by using the Bunge method of 22 even parts with odd parts up to 21 odd terms for ghost correction.
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