Euler Angle

Grain orientation is also commonly described by three Euler angles. The three Euler angles describe the orientations relationship between the sample reference frame (Ks) and the crystal reference frame (Kc). If the three Euler angles are (0, 0, 0), the three axes of its crystal reference frame are parallel to respectively to those of the sample reference frame. The ( j1, F, j2) defines an orientation that is obtained by a rotation starting from (0,0,0). The operation of three consecutive rotations is as follows: The crystal reference frame is rotated first by j1 degrees counter clockwise around Zc, by F degrees counter clockwise Xc (in its new orientation) and then by j2 degrees around Zc also counter clockwise. (Note: The three axes of the crystal reference frame are not in general parallel to the three crystallographic axes a, b and c of the crystal unit cell. In this case, the interpretation of Euler angles depends on how one choose the crystal reference frame. For our conventions of crystal reference frames, please check the definition of the crystal reference frame. )

g{j1 + 2p, F + 2p, j2 + 2p} = g{ j1, F, j2}

g{j1 + p, 2p - F, j2 + p} = g{ j1, F, j2}

where g stands for orientation. The Euler angle space can be reduced because of the crystal symmetry. For example, in case of cubic symmetry, the range of three Euler angles is 0 - 2p, 0 - p/2, 0 - p/2 for j1, F and j2 respectively. Sample symmetries can also reduce the Euler angle space. For the orthrombic sample symmetry and for the cubic crystal system all three Euler angles are in the ranges of 0 - p/2.