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In order to thoroughly comprehend and adequtely interpret NMR data, it is necessary to perceive the complex structure of spin Hamiltonian. Although NMR principles have been extensively discussed in a number of distinguished introductory publications, it still remains difficult to find illustrative graphical models revealing the tensorial nature of spin interaction. Exposure of the structure standing behind mathematical formulas can clarify intangible concepts and provide a coherent image of basic phenomena. This approach is essential when it comes to hard to manage, time-dependent processes such as Magic Angle Spinning (MAS), where the anisotropic character of the spin system interactions couple with experimentally introduced time evolution processes. The presented work concerns fundamental aspects of solid state NMR namely: the uniqueness of the tetrahedral angle and evolution of both dipolar

Developed conceptually in the 1950s [

According to classic electromagnetism an isolated and noninteracting nucleus with nonzero spin should produce local magnetic field having toroidal geometry (magnetic field lines representation in 3D) as presented in Figure

(a) Magnetic field geometry produced by a coil, (b) local magnetic field produced by an isolated nucleus having nonzero spin.

Each nucleus with nonzero spin, placed within external magnetic field

where

(a)

Subsequently, it is necessary to introduce spin precession which is equivalent to the local potential

where

The truncated field represents phase independent, constant magnetic field space which can be described by Laplace’s spherical harmonic

Describing a certain type of spin interactions it is possible to use a general tensorial formula (

where

This nonintuitive conclusion may be explained relying on the truncated field symmetry in a following way; imagine that there are two nuclear magnetic moments

Pair of spins (having the same phase) distanced by vector

As shown in Figure

(a) Graphical representation of angular dependence of dipolar coupling strength described by the 2^{nd} Legendre polynomial. Coloured in red and in orange are effective local magnetic fields produced by

Knowing the truncated field symmetry it is easy to understand and explain experimental data observed by Pake [^{nd} Legendre polynomial determining angular dependence of dipolar coupling strength (

(a) Pake doublet (powdered crystall) and dipolar splitting (single crystall) observed for interacting pair of spins with different

In order to justify and clarify the truncated field concept, it is worth performing the following thought experiment: imagine a diamond monocrystal enriched with ^{13}C isotope and precisely positioned within NMR probe (equipped with goniometer). The idea is that the experimentator can manually orient the monocrystal with respect to the

(a) Diamond crystal orientation with respect to

Figure

(a) Diamond crystal orientation with respect to

There are two inequivalent groups of spin pairs: (a) ^{nd} Legendre polynomial relation the dipolar coupling in this particular case should be scaled with _{l} until it reaches

The truncated field symmetry correctly explains the angular dependence of the dipolar coupling strength.

Schematically presented evolution of a spin pair under MAS conditions for two different internuclear vector orientations

In order to understand the idea of the MAS technique and the process of dipolar coupling averaging, it is worth imagining the internuclear vector evolution under MAS conditions. In Figure _{j} and µ_{k} distanced by _{jk} vector will slide onto the cone surface with apex angle equal to 90°. This precession-like movement will lead to periodic _{k} produces exactly the same truncated magnetic field and its orientation with respect to the external magnetic field is time independent as presented in

(a and c)

(a and c) Illustrate evolution of internuclear vectors (for the case presented in Figures

Another interaction which can be effectively averaged out by MAS is chemical shift anisotropy (CSA). This interaction usually manifests on NMR spectra recorded for powdered crystal samples, which gives rise to broadened (usually asymmetrical) signals. CSA is a direct consequence of the electronic environment of the nucleus. In contrast to the dipolar coupling, the source of CSA is not an adjacent nucleus but the electronic environment and its interaction with external magnetic field. Following the general tensorial formula, the CSA Hamiltonian can be described using Equation (

where

where the principal components of the tensor are denoted as

(a) Schematically illustrated shielding variation for three different monocrystal orientations with respect to the external magnetic field

In order to understand the generation of rotational sidebands (which are a consequence of periodic modulation of NMR signal due to MAS) it is worth imagining the time evolution of the CS tensor under MAS conditions. In Figure

The time evolution of CSA tensors (upper symmetric and lower asymmetric) under MAS conditions. Animation available in Supplementary Materials

Interestingly, shielding evolution _{R} and γ and can be described in the following way

where

where _{2}.

_{2} are presented together with their Fourier transforms representing expected NMR signals.

Basic concepts regarding spin interactions under MAS conditions can be demanding to comprehend due to an experimentally introduced time evolution of the spin system and its anisotropic properties. In order to thoroughly understand the concept of a MAS experiment it is vital to first describe the structure of coupling tensors and their subsequent orientational evolution during NMR experiments. The presented work illustrates the fundamental aspects of dipolar interactions and chemical shift anisotropy, and provides transparent graphical examples and animations explaining the anisotropic character of both.

The data used to support the findings of this study are included within the article.

The authors declare no conflicts of interest.

The research was supported by the H2020-INFRAIA-2016-2017 under research grant “EUSMI – European infrastructure for spectroscopy, scattering and imaging of soft matter”, contract number GA731019, funded under H2020-EU.1.4.1.2.–RIA.

Supplementary Materials include (i) animations revealing CS tensor evolution under MAS conditions, (ii) dipolar Hamiltonian in spherical polar coordinates, and (iii) CS tensor transformations.

^{11}B NMR in single-crystalline YB

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