Hi, researchers,
Recently, I want to use softwares to simulate S/TEM images, but encounter some problems when building the crystal structure.This software considers the beam direction is always along the 3rd (z) direction of the cell, defaultly. Thus, if I want to simulate other certain directions of the cell, such as 110 or 111, I should rotate the crystal to this zone axis and obtain the new-XYZ coordinates of atoms to build the crystal structure files.
For some special directions, such as 100 or 010, I know the transformation matrix
0 0 1
1 0 0
0 1 0
for 010 direction.
But, as to those more general directions,such as 110, 103, or 310, it is beyond my knowledge I think.
I know some software can realize this rotation, such as Vesta or MS, but what is the matrix is the key point. I find a tool (Fig.1) and can get the angles of the rotation, but still do not know how to write the matrix for the rotation of general directions.
I hope to get some help here and will appreciate it if anyone can give me some hints.
I attach the XYZ file.
Best regards
This is a basic linear algebra question. I guess what you're looking for is a general rotation matrix. I'm not sure if it is what you meant, but if I understood it correctly, you want to rotate your crystal (or your coordinate system, that is just a matter of convention) so that let's say the 310 direction becomes the z direction in the new representation (coordinate system). If you want the matrix representation of this rotation you should figure two things first: the rotation axis and the angle. Then there is a formula for the matrix representation of an arbitrary rotation around an given axis. It is there on wikipedia ( https://en.wikipedia.org/wiki/Rotation_matrix#cite_note-5 in subsection Rotation matrix from axis and angle ), I added it as a picture.
The rotation axis should be the normalized cross product of the original z axis (i.e. 001) and the new z axis in the old coordinate system (in our case that would be the 310 vector). The rotation angle is the angle between these two vectors, which you could get from their dot product.
Check this yourself as well, because this might rotate in the wrong direction, but I don't think so.
I hope this helps.
Additional questions:
1- how to ensure the cyrtall still orthorhombic？I try to transform it by such matix but the symmetry has been also changed
cos120° -sin120° 0 sin120° cos120° 0 0 0 1
[along z axis, rotate 120° counterclockwise ]
2- MS can not put in matrix with non-integer elements, any other softwares can do this？
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