Three-dimensional systems, Euler angles

In a case (a) basis set, the electron spin angular momentum is quantised along the linear axis, the quantum number E labelling the allowed components along this axis. Because we have chosen this axis of quantisation, the wave function is an implicit function of the three Euler angles and so is affected by the space-fixed inversion operator E. An electron spin wave function which is quantised in an arbitrary space-fixed axis system,. V. Ms), is not affected by E, however. This is because E operates on functions of coordinates in ordinary three-dimensional space, not on functions in spin space. The analogous operator to E in spin space is the time reversal operator. 

Recall from Section 2.2.1 that the coordinate axes of any individual grain, or crystal, can be transformed to the sample reference coordinate system through a series of three Euler angle rotations. With the ODF, tme three dimensional representations of intensity... 

A different set of coordinates for three atoms in 3-dimensional space has been introduced by Shipsey (1969, 1972). This set is intended to treat in an optimum manner reaction paths with small radius of curvature and seems useful for systems with a bent intermediate configuration. Euler angles are introduced to relate space-fixed and body-fixed frames, and parabolic coordinates are defined in this last frame. The coordinates are illustrated for the reaction 1 eO + 14N 1 sO - 1 sO + 14N 160, and it is shown that the orientation coordinate for relative motion becomes the bending normal coordinate of N02 in the interaction region. 

Figure 5.1 The laboratory (LAB) frame, space-fixed (SF) frame, and body-fixed (BF) frame for the A+BC system in Jacobi coordinates. All are right handed axis systems. The SF frame originates on the center of mass (CM) of the A-fBC system, and is parallel with the LAB frame, which originates on the experimental apparatus. The BF frame also originates on the center of mass of the A+BC system, but rotates in space with the system so that the BF z-axis lies on the Jacobi scattering coordinate R, and the BF x-axis lies in the plane of the three paxticles. The transformation from the SF frame to the BF frame is a three dimensional rotation symbolized by 7, which may be specified in terms of the Euler angles (, 0, ). [The two versions of A+BC in this Figure are identical in every way. The axis systems, however, axe different in the two pictures.] The details and conventions axe discussed in the main text.

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