Rotations in NMR: Part II. Applications of the Euler-Rodrigues parameters

✍ Scribed by Siminovitch, David J.

• Publisher: John Wiley and Sons
• Year: 1997
• Tongue: English
• Weight: 249 KB
• Volume: 9
• Category: Article
• ISSN: 1043-7347
• DOI: 10.1002/(sici)1099-0534(1997)9:4<211::aid-cmr2>3.0.co;2-1

✦ Synopsis

A quantum-mechanical context is used to discuss the advantages offered ( ) by a Euler ᎐ Rodrigues ER parameterization of the rotation implicit in the propagators of 1 1

spin-and spin-1 systems. To appreciate these advantages, we begin with a spin-2 2 { } { } propagator, using either a rotation angle ᎐ axis ⌽, n or a Euler angle ␣ , ␤, ␥ parameteri-ẑation of the rotation operator. The kinematic relations for the ER parameters and the Euler angles are then derived via the rotation operator approach, and used to compare the two parameterizations. As an illustration of the utility of the ER parameterization, the role of 1 time symmetry in composite pulse design for spin-and spin-1 systems is elucidated. A 2 1 { ( ) ( )} phase-alternating composite pulse pair I = , / 2 I = 1 is considered, and well-estab-2 1 lished analogies are shown between the respective spin-and spin-1 propagators. Simple 2 relationships among the phase and amplitude factors of the pulse pair are easily established using the respective ER parameters.