A Classical View of the Euler Angles and the Euler Kinematic Equations in NMR

✍ Scribed by David J. Siminovitch

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 213 KB
• Volume: 117
• Category: Article
• ISSN: 1064-1858
• DOI: 10.1006/jmra.1995.0767

✦ Synopsis

The link between spin kinematics and rigid-body kinematics Euler-Rodrigues (åER) parameters. Well before the work made evident in the recently proposed rotation-operator approach of Darboux (2), and even before that of Hamilton (13), is considered here using a classical picture of spin precession. This who is typically credited with the discovery of a composition link is discussed using both the rotation angle/axis { F, n ˆ} and rule for consecutive noncommuting rotations, it is, in fact, the Euler angle {a, b, g} parametrizations of rotations. These Rodrigues (12) who laid the foundation for the quaternion parametrizations are first compared by using the rotation-operator calculus presently used by NMR spectroscopists to calculate approach to derive the kinematic relations for the Euler-Rothe response to multipulse sequences (6, 7,(14)(15)(16)(17)(18). drigues (ER) parameters {cos F/2, n ˆsin F/2} via the rotation Rodrigues' (12) novel use of the ER parameters in his angle/axis { F, n ˆ} parametrization of the rotation operator. Then, study of rigid-body kinematics enabled him to give the first from a classical point of view, a comparison of the rotation angle/ axis { F, n ˆ} and the Euler angle { a, b, g} parametrizations of the purely geometric solution for the resultant of two or more rotation implicit in the Bloch equations is used: (i) to rederive component rotations. If the ER parameters are used to form the same kinematic relations obtained via the rotation-operator the elements of quaternions, discovered by Hamilton three approach for both the ER parameters and for the Euler angles years later, then Rodrigues' composition rule leads directly {a, b, g} (Euler's kinematic equations), and (ii) to solve Euler's to Hamilton's multiplication rule for quaternions. The enorkinematic equations for the time-dependent Euler angles directly mous advantages of the ER parametrization have led to their without using quadratures, in the case of a time-independent effecuse not only in NMR (6, 7), but also in describing Berry's tive field.