Beyond Euler angles: Exploiting the angle–axis parametrization in a multipole expansion of the rotation operator

Siemens, Mark E.; Hancock, Jason; Siminovitch, D.

doi:10.1016/j.ssnmr.2006.12.001

Cited by 21 publications

(16 citation statements)

References 42 publications

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“…Few things in elementary physics look as easy and are as deadly as three-dimensional rotations -Leonard Euler's original error of selecting an invalid parameterization for SO (3) is so profound and has such far-reaching consequences [98] that the internal policy of the Spinach kernel specifically forbids any interaction specification other than the 3 Â 3 matrix and any orientation specification other than the Wigner matrix. All other conventions are converted at the kernel entry point and all internal functions are written accordingly.…”

Section: Rotationsmentioning

confidence: 99%

See 1 more Smart Citation

Spinach – A software library for simulation of spin dynamics in large spin systems

Hogben

Krzystyniak

Charnock

et al. 2011

Journal of Magnetic Resonance

424

486

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Section: Rotationsmentioning

confidence: 99%

“…This is not a well-defined procedure from the Lie algebraic point of view [98], and it should be avoided if possible. The above mentioned issue of eigenvector matrix not always being equal to DCM also applies.…”

Section: W=dcm2wigner(s);mentioning

confidence: 99%

Spinach – A software library for simulation of spin dynamics in large spin systems

Hogben

Krzystyniak

Charnock

et al. 2011

Journal of Magnetic Resonance

424

486

View full text Add to dashboard Cite

“…Some of them are: direction cosine matrix, 62 Euler angles, 3,24,55,63 angle-axis of rotation 24,55,62-64 denoted by (h, n), the four Cayley-Klein parameters, 63,65 and quaternions. 25,64 Siemens et al 66 discussed in details the benefits of using angle-axis of rotation parametrization over that of Euler angles in NMR. In contrast, we focus on Euler angle parametrization, which is extensively used in the literature other than that about NMR.…”

Section: Rotation Of Space Function In Terms Of Euler Anglesmentioning

confidence: 99%

Wigner active and passive rotation matrices applied toNMRtensor

Man¹

2016

Concepts Magnetic Resonance

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NMR Hamiltonians are double contraction of two spherical rank‐2 tensors, their space parts are represented by a spherical tensor and their spin parts are composed of spherical tensor operators. The comprehension of modern NMR experiments is very often based on the rotation of these tensors. We present the active and passive rotations in a progressive way from position vector to spherical tensor operator via space function, spherical harmonic, and vector operator. The passive rotation of a physical quantity is described by the rotation of coordinate system. Both the left‐ and right‐handed rotation conventions are applied whereas the right‐handed rotation convention is mainly used in the literature. Throughout the article, we explore the equivalence between the active rotation of a physical quantity in one direction and the rotation of the coordinate system in the opposite direction. The article presents redundant mathematical demonstrations between active and passive rotations, but they clarify the meanings of some important expressions not well developed in the literature.

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“…In this article, by focusing on a multipole operator expansion of the rotation operatorDðÈ,nÞ [16][17][18], we show for the first time how Racah algebra may be used to greatly simplify the calculus of rotations in NMR.…”

Section: Introductionmentioning

confidence: 99%

“…With one very notable exception, the elegant use by Llor et al [24] of the multipole operator expansion ofDðÈ,nÞ Equation (6) to analyse coherent isotropic averaging in zero-field NMR, until very recently, neither the rotation operator nor the rotation matrix expansions have seen much use in NMR or NQR applications. A recent review discussion by Siemens et al [18] of NMR applications of these expansions would suggest they have a very promising future. This review, and the original references cited therein, constitute the foundation core of all the results that we derive and discuss in this article.…”

Section: Introductionmentioning

confidence: 99%

Angular momentum coupling in NMR: a new view of rotation matrix products via Racah algebra

Siminovitch¹

2008

Mol. Phys.

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Choosing the rotation angle-rotation axis parameters fÈ,ng instead of the conventional Euler angle parameters {, , } to parametrise rotations, and a spherical tensor operator basis, Happer was the first to propose a novel multipole operator expansion of the rotation operatorDðÈ,nÞ ¼ exp½ÀiÈðn Á JÞ [W. Happer, Ann. Phys. (N.Y.) 48, 579 (1968)]. As Happer pointed out, such an expansion in irreducible spherical tensor operators readily lends itself to the methods of Racah algebra, a feature we exploit to simplify the calculus of rotation matrix products. Working with a Clebsch-Gordan coefficient expansion [W. Happer, ibid. cit.; M.S. Marinov, Yad. Fiz. 5, 943 (1967)] of the corresponding rotation matrices D J MM 0 ðÈ,nÞ hJMj exp½ÀiÈðn Á JÞjJM 0 i, and closure relations for these (2J þ 1)-dimensional irreducible representations of the rotation group, we state, and prove for the first time, simple recoupling coefficient expansions for rotation matrix products. With an eye towards applications in NMR, where multiple rotations in spin or coordinate space often require the efficient evaluation of such rotation matrix products, we focus on expansions defining the matrices representing the result of two or three consecutive rotations. We show that the coefficients of these expansions are defined by 6 À j symbols and bipolar spherical harmonics (two rotations), or by 9 À j symbols and tripolar spherical harmonics (three rotations). We show that these recoupling coefficient expansions, in conjunction with Racah algebra, are the cornerstone of an unusually simple and direct method of (1) proving the Euler-Rodrigues (quaternion) parameter f, ,g fðÈ,nÞ, ,ðÈ,nÞg ¼ fcosðÈ=2Þ,n sinðÈ=2Þg composition rules for two and three consecutive rotations and (2) constructing rotational invariants expressed in Weyl-invariant form. These methods are distinguished by their independence on the usual algebraic approaches, either those of operator algebra, or ordinary algebra.

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Beyond Euler angles: Exploiting the angle–axis parametrization in a multipole expansion of the rotation operator

Cited by 21 publications

References 42 publications

Spinach – A software library for simulation of spin dynamics in large spin systems

Spinach – A software library for simulation of spin dynamics in large spin systems

Wigner active and passive rotation matrices applied toNMRtensor

Angular momentum coupling in NMR: a new view of rotation matrix products via Racah algebra

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