There are some Euler Angles (Bung) (phi1, PHI, Phi2) in BCC structure. How can I find the closet common plane or direction (i.e. (001), (011), (111), and (112)) with its deviation for each one. For example; (0ku) ~ 20°.
Euler angles (e.g. of Bunge) describe an orientation. Primarily they do not have anything to do with the structure, only with the lattice. The Euler angles define the orientation matrix G which is a rotation matrix. The rows give the Cartesian coordinates for the unit vectors of the rotated crystal coordinates described by the reference coordinate system (for orthogonal crystal systems they are parallel to the basis vectors of the unit cell, for all other systems they are only partially parallel to lattice vectors and need to be transformed) . In contrast, the column of the orientation matrix describe the basis vectors of the reference coordinate system (commonly called X,Y or Z ). If you want know the Cartesian description of a certain crystal vector [uvw] you only need to multiply [uvw]*G . This is the reason why the rows of G define [100] which is [g11,.g12,g13] But you can multiply any vector. You certainly noticed, that these are no (hkl) which are lattice planes or normal vectors in reciprocal lattice. But this in not a big deal since correctly the multiplication of [uvw] and G only works for cubic phases. Before you can do this multiplication you have to describe the lattice vector as Cartesian vector which is defined by the so-called crystal matrix A. For cubic phases A=a*1 (unit matrix 1 and lattice parameter a) which simplifies everything extremely. The actually necessary [x,y,z] which you have to multiply with G are given by [x,y,z]=[uvw]*AT. If you are interested in the Cartesian description (x,y,z) of a lattice plane normal (hkl) you need to multiply (hkl)*A-1 (reciprocal matrix of A). In other words: in order to get an arbitrary Cartesian description of (hkl) or [uvw] you need to transform (hkl) or [uvw] first in Cartesian coordinates (x,y,z) or [x,y,z]. These numbers you have to multiply then with G. For cubic you don't have to do this since also A-1=a-1*1 (a-1...reciprocal lattice parameter, which is for cubic simply 1/a). However, for a general application you should not use this simplification since then the danger exist that you do this also with non-cubic phases and then you will get wrong directions.
Your question sounds a bit strange, however, I guess, you want to know the closest indexing of the reference vectors? This is de-facto the same, you only need to multiply from the other side or use GT. But then you should not use (hkl) as description since they reserved for lattice planes only and not for reference vectors or linear combinations of them. However, I am still not sure, whether you know what do you really want... You are mixing too man things, like orientation, lattice plane, lattice structure....and this is a single question. I would recommend you to search for some free available texts (e.g. Ben Brittens tutorial paper in Materials Characterization, 2016, 117, 113 - 126 , or numerous very good lectures of A. Rollett. Or simply read one of the books about EBSD or better crystallography.
Hi
the relationship between Euler angle and miller indices has been summarized by H.Bunge (attached file)
regards
Bunge, H-J. Texture analysis in materials science: mathematical methods. Elsevier, 2013.
Dear Dr. Nolze,
I am grateful for your comments.
Let´s ask like this, how can I find (i) the closest and lowest indexed plane (ii) its tilt angle between them.
Please find the attached index, where found the tilt angle between specific orientation with the closest plane.
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