Spin Tensors as Ininitesimal Changes of Rotations edit page

Spin tensors are skew symmetric tensors that can be used to describe small rotational changes. Lets consider an arbitrary reference rotation

and pertube it by a rotation about the axis (123) and angle delta. Since multiplication of rotations is not communatativ we have to distinguish between left and right pertubations

We may now ask for the first order Taylor coefficients of the pertubation as delta goes to zero which we find by the formula

$T = \lim_{\delta \to 0} \frac{\tilde R - R}{\delta}$

Both matrices T_right and T_left are elements of the tangential space attached to the reference rotation rot_ref. Those matrices are characterized by the fact that they becomes scew symmetric matrices when multiplied from the left or from the right with the inverse of the reference rotation

A scew symmetric 3x3 matrix S is essentially determined by its entries $$S_{21}$$, $$S_{31}$$ and $$S_{32}$$. Writing these values as a vector $$(S_32,-S_{31},S_{21})$$ we obtain for the matrices S_right_R and S_left_L exactly the rotational axis of our pertubation

For the other two matrices those vectors are related to the rotational axis by the reference rotation rot_ref

The Functions Exp and Log

The above definition of the spin tensor works only well if the pertupation rotation has small rotational angle. For large pertubations the matrix logarithm provides the correct way to translate rotational changes into skew symmetric matrices

Again the entries $$S_{21}$$, $$S_{31}$$ and $$S_{32}$$ exactly coincide with the rotional axis multiplied with the rotational angle

More directly this disorientation vector may be computed from two rotations by the command log

The other way round

Given a skew symmetric matrix S or a disorientation vector v we may use the command exp to apply this rotational pertubation to a reference rotation rot_ref

Disorientations under the presence of crystal symmetry

Under the presence of crystal symmetry the order whether a rotational pertupation is applied from the left or from the right. Lets perform the above calculations step by step in the presence of trigonal crystal symmetry

and compute the scew symmetric pertubation matrices