Cartesian and Spherical Tensors in NMR Hamiltonians

Man, Pascal P.

doi:10.1002/cmr.a.21289

Cited by 13 publications

(38 citation statements)

References 111 publications

(417 reference statements)

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“…We focus on operators and coordinate systems, whereas state vector is hidden. The active transformation may be involved in one or two coordinate systems . Two coordinate systems are involved in passive transformation.…”

Section: Active Passive and Canonical Transformations Of Operatormentioning

confidence: 99%

“…In fact, the active transformation corresponds to our active rotation in one coordinate system, the fixed coordinate system associated with the unprimed operator. The canonical transformation corresponds to our active rotation in two coordinate systems: the fixed coordinate system is associated with the unprimed operator and the body‐attached coordinate system is attached to the operator and rotates with it. Equation means that the primed operator in the body‐attached coordinate system has the same matrix elements as those of the unprimed operator in the fixed coordinate system before the active rotation.…”

Section: Active Passive and Canonical Transformations Of Operatormentioning

confidence: 99%

“…Viewed in its body‐attached coordinate system, the space function remains motionless. We previously shown that Rose used this approach for deducing the active rotation operator involving Euler angles. As Rose did not use figures to support his presentation, Wolf and Bouten understood that Rose dealt with passive rotation of space function.…”

Section: Rotation Of Space Function About a Single Axismentioning

confidence: 99%

“…We manipulated the expression of

R_{A}^{normalrot} false(normalγ, normalβ, normalα false)

defined in Eq. , which depends on angular momentum operators of successively rotated body‐attached coordinate systems, so that the expression depends on angular momentum operators of fixed coordinate system.…”

Section: Rotation Of Space Function In Terms Of Euler Anglesmentioning

confidence: 99%

“…NMR Hamiltonians may be defined with Cartesian or spherical tensors . In the latter case, they are double contractions of two spherical rank‐2 tensors, one representing the space part and the other the spin part.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Active Passive and Canonical Transformations Of Operatormentioning

confidence: 99%

Section: Active Passive and Canonical Transformations Of Operatormentioning

confidence: 99%

Section: Rotation Of Space Function About a Single Axismentioning

confidence: 99%

“…We manipulated the expression of

R_{A}^{normalrot} false(normalγ, normalβ, normalα false)

Section: Rotation Of Space Function In Terms Of Euler Anglesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Wigner active and passive rotation matrices applied toNMRtensor

Man¹

2016

Concepts Magnetic Resonance

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Cartesian and Spherical Tensors in NMR Hamiltonians

Man

2013

Concepts Magnetic Resonance

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NMR Hamiltonians are anisotropic due to their orientation dependence with respect to the strong, static magnetic field. They are best represented by product of two rank‐2 tensors: one is the space‐part tensor T and the other is the spin‐part tensor A. We reformulated the dot product of Cartesian tensors and the dyadic product of spherical tensors in NMR Hamiltonian as the double contraction of these two tensors. As the double contract has two definitions (double inner product and double outer product of two rank‐2 tensors), there are two sets of spherical tensor components in terms of Cartesian tensor components for any rank‐2 tensor, two Cartesian Hamiltonians, and two spherical Hamiltonians. We succeeded in determining the spherical components of tensors A and T that verify the Cartesian Hamiltonian defined by the double inner product of two rank‐2 tensors and the spherical Hamiltonian defined by the double outer product of two rank‐2 tensors. In particular, we established the spherical components of space‐part tensor T in terms of Cartesian tensor components provided by Cook and De Lucia. Throughout the article, Wigner active rotation matrix is used to illustrate the active rotation of spherical vector, spherical harmonics, and spherical tensor as well as the passive rotation via their rotational invariants. © 2014 Wiley Periodicals, Inc. Concepts Magn Reson Part A 42A: 197–244, 2013.

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Investigation of the Li-ion conduction behavior in the Li10GeP2S12 solid electrolyte by two-dimensional T1-spin alignment echo correlation NMR

Paulus

Graf

Harks

et al. 2018

Journal of Magnetic Resonance

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LiGePS (LGPS) is the fastest known Li-ion conductor to date due to the formation of one-dimensional channels with a very high Li mobility. A knowledge-based optimization of such materials for use, for example, as solid electrolyte in all-solid-state batteries requires, however, a more comprehensive understanding of Li ion conduction that considers mobility in all three dimensions, mobility between crystallites and different phases, as well as their distributions within the material. The spin alignment echo (SAE) nuclear magnetic resonance (NMR) technique is suitable to directly probe slow Li ion hops with correlation times down to about 10 s, but distinction between hopping time constants and relaxation processes may be ambiguous. This contribution presents the correlation of the Li spin lattice relaxation (SLR) time constants (T) with the SAE decay time constant τ to distinguish between hopping time constants and signal decay limited by relaxation in the τ distribution. A pulse sequence was employed with two independently varied mixing times. The obtained multidimensional time domain data was processed with an algorithm for discrete Laplace inversion that does not use a non-negativity constraint to deliver 2D SLR-SAE correlation maps. Using the full echo transient, it was also possible to estimate the NMR spectrum of the Li ions responsible for each point in the correlation map. The signal components were assigned to different environments in the LGPS structure.

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Cartesian and Spherical Tensors in NMR Hamiltonians

Cited by 13 publications

References 111 publications

Wigner active and passive rotation matrices applied toNMRtensor

Wigner active and passive rotation matrices applied toNMRtensor

Cartesian and Spherical Tensors in NMR Hamiltonians

Investigation of the Li-ion conduction behavior in the Li10GeP2S12 solid electrolyte by two-dimensional T1-spin alignment echo correlation NMR

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