Table of Contents
Name | Direction | Type | Default | Description |
---|---|---|---|---|
PeaksWorkspace | InOut | PeaksWorkspace | Mandatory | An input workspace. |
a | Input | number | Mandatory | Lattice parameter a |
b | Input | number | Mandatory | Lattice parameter b |
c | Input | number | Mandatory | Lattice parameter c |
alpha | Input | number | Mandatory | Lattice parameter alpha |
beta | Input | number | Mandatory | Lattice parameter beta |
gamma | Input | number | Mandatory | Lattice parameter gamma |
Given a set of peaks (Q in the goniometer frame, HKL values), and given lattice parameters , it will try to find the U matrix, using least squares approach and quaternions 1. Units of length are in in , angles are in degrees.
The algorithm calculates first the B matrix according to Busing and Levi.
Given a set of peaks in the reference frame of the inner axis of the goniometer, , indexed by , we want to find the U matrix that maps peaks in the reciprocal space of the sample to the peaks in the goniometer frame
(1)
For simplicity, we define
(2)
In the real world, such a matrix is not always possible to find. Therefore we just try minimize the difference between the two sets of p
(3)
In equation (3), , so the first two terms on the left side are U independent. Therefore we want to maximize
(4)
We are going to write the scalar product of the vectors in terms of quaternions 2. We define , and the rotation U is described by quaternion
Then equation (4) will be written as
(5)
We define matrices
(6)
and
(7)
Then, we can rewrite equation (5) using matrices 3, 4:
(8)
The problem of finding that maximizes the sum can now be rewritten in terms of eigenvectors of . Let and be the eigenvalues and corresponding eigenvectors of , with . We can write any vector as a linear combination of the eigenvectors of :
(9)
(10)
(11)
where is a unit quaternion, (12)
Then the sum in equation (11) is maximized for
Therefore U is the rotation represented by the quaternion , which is the eigenvector corresponding to the largest eigenvalue of .
Categories: Algorithms | Crystal\UBMatrix