Spin Tensors as Infinitesimal Changes of Rotations edit page

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

and perturb it by a rotation about the axis (123) and angle delta=0.01 degree. Since multiplication of rotations is not commutative we have to distinguish between left and right perturbations

We may now ask for the first order Taylor coefficients of the perturbation 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 skew symmetric matrices when multiplied from the left or from the right with the inverse of the reference rotation

A skew 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 perturbation

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 well only if the perturbation has small rotational angle. For large perturbations the matrix logarithm log 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 rotational axis multiplied with the rotational angle.

More directly this disorientation vector may be computed from two rotations using the options SO3TangentSpace.rightVector and SO3TangentSpace.leftVector

## 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 perturbation to a reference rotation rot_ref

## Disorientations under the presence of crystal symmetry

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

Computing the right tangential vector gives us the disorientation vector in crystal coordinates

computing the left tangential vector gives us the disorientation vector in specimen coordinates