Direction cosines coordinate transformation

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... 

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. 

The t2g density is zero where two of the direction cosines x, y, and z are zero, that is, in the planes of the coordinate axes, but peaks along the eight cube diagonals. The density functions are Fourier-transform invariant, as discussed in chapter 3, and expressed by the equation... 

Direction cosines, which can be used to define the direction of a vector in an orthogonal coordinate system, play an essential role in accomplishing coordinate transformations. As illustrated in Fig. A. 1, there is a vector V oriented in a (z,r,9) coordinate system. Because our concern here is only the direction of the vector, the physical dimensions are sufficiently small so that the curvature in the 9 coordinate is not seen (i.e., the coordinate system... 

In addition to representing a vector in a rotated coordinate system, the matrix of direction cosines can also be used to transform a tensor (e.g., the stress tensor) into a rotated coordinate system as... 

Referring to Fig. A.2, assume that the principal coordinates align with z, r, and O. The unit vectors (direction cosines) just determined correspond with the row of the transformation matrix N. Thus, if the principal stress tensor is... 

In the above, ijk denote the Cartesian coordinates of a molecule, IJK those of a crystal (a unit cell), 7Vt is the number of molecules in a unit volume occupying each particular inequivalent site in the unit cell,/ is the number of inequivalent positions of a molecule in a unit cell, and Ng is the number of equivalent positions in a unit cell. The directional cosines are used to transform each of the molecular 0 components to those of the new coordinate system (bUK) and the contributions are summed. 

The components may be expressed in either a space-fixed axis system (p) ora molecule-fixed system (q). The early literature used cartesian coordinate systems, but for the past fifty years spherical tensors have become increasingly common. They have many advantages, chief of which is that they make maximum use of molecular symmetry. As we shall see, the rotational eigenfunctions are essentially spherical harmonics we will also find that transformations between space- and molecule-fixed axes systems, which arise when external fields are involved, are very much simpler using rotation matrices rather than direction cosines involving cartesian components. 

Let us now examine the formulas of transformation necessary to deal with cases where the coordinate system for the equations of flow is not identical with the coordinate system for the stress components. At the point P let x, g and z be in the directions of the principal axes of distortion as defined by Eqn 3-7 and let a, fa, and c be the partial derivatives of flow along these axes. Let x, g and z be a set of orthogonal axes whose orientation is given by the matrix of direction cosines shown below ... 

For a linear molecule, the position of the symmetry axis (the molecule-fixed. z-axis) in space is specified by only two Euler angles, / and 7, which are respectively identical to the spherical polar coordinates 6 and (see Fig. 2.4). The third Euler angle, a, which specifies the orientation of the molecule-fixed x- and y-axes, is unaffected by molecular rotation but appears explicitly as an O- dependent phase factor in the rotational basis functions [Eq. (2.3.41)]. Cartesian coordinates in space- and molecule-fixed systems are related by the geometrical transformation represented by the 3x3 direction cosine matrix (Wilson et al., 1980, p. 286). The direction cosine matrix a given by Hougen (1970, p. 18) is obtained by setting a = 7t/2 (notation of Wilson et al, 1980 6 fi,4)=, x = oi 7t/2). The direction cosine matrix is expressed in terms of sines and cosines of 9 and 4>. Matrix elements (J M O la JMQ), evaluated in the JMQ) basis, of the direction cosines, are expressed in terms of the J, M, and quantum numbers. The direction cosine matrix elements of Hougen (1970, p. 31), Townes and Schawlow (1955, p. 96), and Table 2.1 assume the basis set definition derived from Eq. (2.3.40) and the phase choice a = 7t/2 ... 

Stresses transform from one coordinate-axis system to another according to well-defined transformation laws that utilize direction cosines of the angles of rotation between the final and initial coordinate-axis systems. Matrixes that obey such transformation laws are referred to as tensors (McClintock and Argon 1966). There are three sets of stress relations that are scalar and invariant in coordinate-axis transformations. The first such stress invariant of particular interest is the mean normal stress o- , defined as. 

As with stresses, transformations of strain elements from one coordinate-axis system into another obtained by rotations of axes obey the same transformation laws as those of stresses, utilizing the same direction cosines and making the symmetrical strain matrix also a tensor (McClintock and Argon 1966). 

The use of redundant coordinates requires extensive modification of the lattice dynamical procedure. It is, however, often worth the additional complication to use redundancies if this facilitates the formulation of symmetry coordinates. When the Wigner projection operator (Wigner, 1931) is used to build such symmetry coordinates, it is necessary to first understand the results of the application of all symmetry operations of the applicable group to the displacement coordinates chosen. This is indeed relatively straightforward for the direction cosine displacement coordinates and therein lies their principal value. These coordinates transform like axial vectors in contrast to cartesian coordinates, which transform like polar vectors. 

We use (4.24a) and (4.24b) and the direction cosine displacement coordinates of Walmsley and Pople (1964) to describe the application to a linear molecule as given by Brith, Ron, and Schnepp (1969). Equivalent treatments have been given by Cahill (1968) and by Richardson and Nixon (1968). The components A,(// ) of the unit vector A(// ) are the instantaneous direction cosine of the molecular axis. Part of the components of the matrix of the transformation from molecular (primed) to crystal (unprimed coordinates (4.24b) can be identified with the Aj, i.e.. 

The transformation T" connects the symmetry coordinates with the direction cosine librational displacement coordinates 2.i(lk) as before. We are now in a position to calculate the polarizability derivative tensor components if the symmetry coordinates S or the normal coordinates Q are known in terms of the librational displacement coordinates. 

The relations between the coordinates X, y, z and X, F, Z of a point in space are given by Table I-l. Since the transformation is orthogonal, the table may be read either across or down. Read across it gives the coordinates x, y, z in terms of X, Y, Z read down it gives X, F, Z in terms of x, y, z. The entries in the table are, of course, the cosines of the angles between the various pairs of axes, i.e., the direction cosines gy or (Chap. 3). 

Two sets of angles, (9j, j) and (0, O ), referring to the identical vector rp can be related to each other by the coordinate transformation between O-M1M2M3 and O-X1X2X3 as equation (3), where is a linear transformation operator, and is the direction cosine between the Uj, and x,- axes and the function of (/>, 6 and rj. 

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