Beyond Euler angles: exploiting the angle-axis parametrization in a multipole expansion of the rotation operator.
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).
Siemens
Mark
M
Department of Physics, The University of Lethbridge, Lethbridge, Alta., Canada T1K 3M4.
Hancock
Jason
J
Siminovitch
David
D
D000465:Algorithms; D003198:Computer Simulation; D009682:Magnetic Resonance Spectroscopy / Q000379:methods*; D008956:Models, Chemical; D008958:Models, Molecular; D008968:Molecular Conformation; D013113:Spin Labels
D013113:Spin Labels