Active and passive rotations with Euler angles in NMR

Abstract Euler angles are commonly used for orientating a coordinate frame relatively to another or for describing the rotation of rigid body about a fixed point. Unfortunately, there is no standard notation for Euler angles and rotation axes in composite rotation matrices. Two types (passive and active) of rotation are possible; each of them can be performed either in rotated or in fixed frames. We suggest a notation harmonization for composite rotation matrices. To this end, we first gather the conditions and conventions available in the literature. Second, the matrices of elementary rotation about basis vectors are identified because they participate in the construction of composite rotation matrices. They allow us to determine the type of composite rotation matrix. Third, we introduce the Euler angles during the description of two composite rotations of bases. This permits us to anticipate the type of the resulting composite rotation matrices. It results the central convention that suggests us to order Euler angles and rotation axes in the definition of composite rotation matrix in the same way as the three elementary rotation matrices. This convention is not always applied in the literature. Fourth, we confirm our anticipation using composite rotation of vector. Fifth, this central convention permits us to define the multiplication order of two composite rotation matrices straightforward. Finally, we apply the matrices of composite active rotation about fixed axes to X‐pulse, spin‐lock pulse, and magnetization inversion pulses. © 2012 Wiley Periodicals, Inc. Concepts Magn Reson Part A 40A: 215–252, 2012.