Tensor colinearity

Fig. 1. Force and stress acting on a parallelepipedic element. Sx and Fx denote a surface element perpendicular to the. v-axis and a force colinear to the. v-axis, respectively. For stress tensor elements <7 the subscripts i and j (x,y or z) relate to the orientations of the force and the surface element, respectively.

In addition to g tensor anisotropy, EPR spectra are often strongly affected by hyperfine interactions between the nuclear spin I and the electron spin S. These interactions take the form T A S, where A is the hyperfine coupling tensor. Like the g tensor, the A tensor is a second-order third-rank tensor that expresses orientation dependence, in this case, of the hyperfine coupling. The A and g tensors need not be colinear in other words, A is not necessarily diagonal in the coordinate systems which diagonalize g. 

However, in the general case, D and E will not be colinear. Therefore a tensor is necessary to describe the relationship between D and E ... 

As an illustrative example, focus on the element in the first row and second column, pVjcVy, for the matrix representation of the convective momentum flux tensor. The subscript x on the first velocity component indicates that pv Vy is a force per unit area acting across a simple surface oriented with a unit normal vector in the x direction. The subscript y on the second velocity component reveals that this force acts in the y direction. If we perform this analysis for all nine components in the matrix (8-6) the three entries in the first row represent X, y, and z components, respectively, of the vector force per unit area that is transmitted across the simple surface defined by a constant value of the x coordinate, which means that the unit normal vector to the surface is colinear with the X direction. In vector-tensor notation, this is... 

Similarly, the three entries in the second row of the matrix (8-6) represent x, y, and z components, respectively, of the vector force per unit area that is transmitted across the simple surface defined by a constant value of the y coordinate, which means that the unit normal vector to the surface is colinear with the y direction. This information can be condensed into vector-tensor notation as... 

If we depart from pure classical electrostatics, things become slightly more complicated. The deviation from spherical symmetry of an atomic nucleus can be described by a nuclear electric quadrupole tensor Q. A natural axis system for this tensor is one where one axis is colinear with the nuclear spin axis. By virtue of the effective cylindrical symmetry of a rapidly spinning deformed nucleus, the components of the quadrupole tensor are determined by a single scalar quantity Q and in addition by the components of the nuclear spin vector i. 

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