Given the a, b and c vectors of a general crystal structure (not necessarily cubic, can be say, monoclinic), is there a general rotation matrix formula for taking the (001) surface of the bulk to any surface. I ask a specific question here: If I wanted to rotate the bulk monoclinic Ga2O3 so the top surface (001) is now the (-201) face, how would I do that. I am familiar with general rotation matrices about a general axis from linear algebra, but I'm not sure how to apply this information to this system. It would be nice to know a general rotation matrix formula for rotating a crystal from one orientation to another regardless of the structure. I really appreciate any help in advance.
Dear Eric,
Rotation of positive definite symmetric matrices or square matrices is well studied subject, however., for other type of matrices we always need some more preprocessing steps. To your query, you may see https://au.mathworks.com/matlabcentral/answers/115726-find-rotation-matrix-using-fsolve and https://au.mathworks.com/matlabcentral/answers/426334-help-with-a-rotation-matrix
I shall restate your problem so that it is more tractable. If my restatement is not your problem I apologize.
A plane through the origin is defined by the unit vector perpendicular to it. So your problem is to find a rotation matrix that will rotate one unit vector (perpendicular to your 001 surface) to another unit vector (corresponding to your second surface). Let the rotation matrix be R, the unit vector perpendicular to the 001 surface be u, and the unit vector perpendicular to the destination surface be v. You need to calculate u and v (a little tedious, but easy), and then solve for a rotation matrix R where Ru=v. (Note: A rotation matrix is orthonormal.)
Alun Wyn-jones thank you for your answer. Let me ask a follow up: How does one find v when one only knows the surface they are interested in?? In other words, v is going to be a 3x3 matrix that represents the 3 basis vectors of the new crystal structure (the x,y,z coordinates of the new crystal basis). It is easy for u since I already have the matrix of basis vectors for my initial crystal, but I only know that the vector I want to intersect at (001) is supposed to be the (-201) plane. I am not sure how to determine the matrix of the v basis vectors so I can then determine the rotation matrix (I am assuming it is a simple inversion of u done on both side from there to solve for R). thank you very much for your correspondence.
Pick any two vectors in your crystal plane and take their vector product. That's a vector perpendicular to the plane. I'm not sure I'm answering your question though.
Is this method possible for a non-cubic crystal like a monoclinic crystal?? The angle between the a and c vector is not 90 degrees, so the vectors won't all be orthogonal. How do I take this into account when trying to find the other two vectors since I know the c vector will be (-201)?? This feels like one of those moments where I learned the base methods in courses in my undergrad and grad school, but the application is much more nuanced and challenging. I look forward to figuring this out. Thank you so much for your continued correspondence Alun Wyn-jones
It's been a long time since I attended a course on crystallography, but here goes... Let the three vectors defining your unit cell be u, v, w. The two vectors, u and v, say, specify a plane, call it F. If you find a rotation matrix, R, say, that rotates F to some other plane F' of your choice (so that F'=RF), then you have one degree of freedom left where to send w because you can rotate the plane F' within itself. By the way, R is not unique, and there are two essentially different ways you can map a plane by rotation to another plane--one being upside-down relative to the other.
I hope this helps--it's 3-d geometry.
I just wanted to share with everyone on this thread this very nice youtube video on finding a rotation matrix using VESTA (https://www.youtube.com/watch?v=2x5E-vboVxc&t=255s). Just make sure you transpose the final rotation matrix you get before inserting into the transfomation matrix in VESTA.
Best,
Eric
Eric Welch
You can neglect the j component, so--
[0,1] dot [-2,1] = sqrt(5) * 1 * cos(theta),
so theta = 63.465 degrees counter-clockwise.
Your new unit cell is
u' = [-ui * sin(63.465), uj, uk * cos(63.465)]
v' = [-vi * sin(63.465), vj, vk * cos(63.465)]
w' = [-wi * sin(63.465), wj, wk * cos(63.465)]
Easy... I just typed that out in about 5 minutes in the coffee shop. For more information, see Marion and Thornton, "Classical Mechanics," chapter 1 (rotation matrices are like page 4, if I remember correctly). Please correct me, if you find an error...
I school Eric every time he has a question, because he slanders me behind my back at every opportunity, with absolutely zero evidence, when I barely even know him.
Edit: changed i,j,k to smaller suffixes
Apparently, half of my answer was deleted, even though it was correct. Someone just tipped me off…
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