Cosine Direction

For an isotropic material, all orientations are equally probable and all such products that have an odd number of Tike direction cosines will vanish upon averaging-. This restricts the nonvanishing tensor elements to those such as xVaaa abba - Similarly for the elements Such orientational averaging is crucial in... 

Consider the behavior of various tensors under the transformation of coordinates in Figure A-1 where a rotation about the z-axis is made. That is, the x, y, z coordinates are transformed to the x , y , z coordinates where the z-direction coincides with the z -direction. The direction cosines for this transformation are... 

If the magnitude and direction of a vector are known, its components are the products of the magnitude and the respective direction cosines. In the case of the velocity vector, for example, the components are... 

Figure 6-3. Top Structure of the T6 single crystal unit cell. The a, b, and c crystallographic axes are indicated. Molecule 1 is arbitrarily chosen, whilst the numbering of the other molecules follows the application of the factor group symmetry operations as discussed in the text. Bottom direction cosines between the molecular axes L, M, N and the orthogonal crystal coordinate system a, b, c. The a axis is orthogonal to the b monoclinic axis.

Table 1 The direction cosines expressed as functions of the Eulerian... 

For orientation measurements, this tensor also needs to be expressed in the coordinate system OXYZ, axrz, using the matrix transformation u.xyz = Oaxyz / where O is a matrix whose elements are the direction cosines of the coordinate axes and is its transposed matrix [44]. 

A sinusoidal plot of grf>2 vs.

crystal plane gives another set of Ks that depend on other combinations of the gy, eventually enough data are obtained to determine the six independent values of gy (g is a symmetric matrix so that gy = gy,). The g-matrix is then diagonalized to obtain the principal values and the transformation matrix, elements of which are the direction cosines of the g-matrix principal axes relative to the crystal axes. An analogous treatment of the effective hyperfine coupling constants leads to the principal values of the A2-matrix and the orientation of its principal axes in the crystal coordinate system. 

Corrpute direction cosines li(Q,cp) with Equation 5.3 Corrpute g-value g(gi,li) with Equation 5.5 Corrpute intensity 1(g) with Equation 6.4 Corrpute resonance field Bres(g) with Equation 5.4 DO STEP in dB from Bmln to Bmax... 

The k s are the direction cosines of Bx with respect to the xyz-axes system. Equations 8.14 and 8.15 combine into an expression with a system-dependent prefactor and a part that depends on the specific spin system under consideration,... 

In powder EPR simulators we use the orientation of the static-field vector B with respect to the molecular xyz-axes system as the definition of molecular orientation. The orientation is defined in terms of the polar angles 0, solve Equation 8.17 we have to define the direction cosines k, of B, in terms of the direction cosines li of B. 

A mathematically trivial, but experimentally important case is that of parallel-mode EPR, in which Bt IIB and, therefore, k = / . Substituting the expressions for the direction cosines of Equation 5.3 into Equation 8.17 gives the parallel-mode relative intensity... 

For powder EPR in perpendicular mode Bx lies in a plane perpendicular to B, and Bx samples all possible orientations in this plane. We define a vector v in this plane by means of an angle a between Bx and v. Then, the direction cosines of Bx are... 

As a starting point let us be faithful to the history of the subject and try a simple physical model due to (Johnston and Hecht, 1965) if the inhomogeneous EPR line reflects a distribution in g-values, then the anisotropy in the linewidth should be scalable to the anisotropy in the g-value. In other words, the analytical expression for g-anisotropy in terms of direction cosines, between B and the... 

Note that cos a and cos p are referred to as the direction cosines of the line segment described. To summarize in expanded notation ... 

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