Siemens M, Hancock J, Siminovitch D. Beyond Euler angles: exploiting the angle-axis parametrization in a multipole expansion of the rotation operator.
SOLID STATE NUCLEAR MAGNETIC RESONANCE 2007;
31:35-54. [PMID:
17267183 DOI:
10.1016/j.ssnmr.2006.12.001]
[Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/01/2006] [Revised: 10/10/2006] [Accepted: 12/27/2006] [Indexed: 05/13/2023]
Abstract
Euler angles (alpha,beta,gamma) are cumbersome from a computational point of view, and their link to experimental parameters is oblique. The angle-axis {Phi, n} parametrization, especially in the form of quaternions (or Euler-Rodrigues parameters), has served as the most promising alternative, and they have enjoyed considerable success in rf pulse design and optimization. We focus on the benefits of angle-axis parameters by considering a multipole operator expansion of the rotation operator D(Phi, n), and a Clebsch-Gordan expansion of the rotation matrices D(MM')(J)(Phi, n). Each of the coefficients in the Clebsch-Gordan expansion is proportional to the product of a spherical harmonic of the vector n specifying the axis of rotation, Y(lambdamu)(n), with a fixed function of the rotation angle Phi, a Gegenbauer polynomial C(2J-lambda)(lambda+1)(cosPhi/2). Several application examples demonstrate that this Clebsch-Gordan expansion gives easy and direct access to many of the parameters of experimental interest, including coherence order changes (isolated in the Clebsch-Gordan coefficients), and rotation angle (isolated in the Gegenbauer polynomials).
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