#### Abstract

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 *D* and chemical shield *σ* coupling tensors under MAS conditions.

#### 1. Introduction

Developed conceptually in the 1950s [1, 2] Magic Angle Spinning (MAS) has become a routine technique in solid state NMR spectroscopy [3]. This unique concept, together with the exceptional progress made in the field of NMR probe engineering, has made possible the acquisition of high resolution spectra for rigid phase samples. Commonly used, modern probes now allow for spinning the sample under magic angle conditions, reaching rotation frequencies up to dozens of kilohertz [4]. MAS eliminates the broadening of NMR signals by orientation-dependent spin interactions, yielding significant improvements in resolution and signal-to-noise ratio. The excellent resolution observed in the case of liquid samples stems from the fact that during NMR signal acquisition each molecule undergoes extremely fast and random tumbling. Accordingly, both dipolar and quadrupolar interactions are effectively averaged out and only an isotropic chemical shielding is manifested in the spectrum, together with a subtle J coupling effect. In contrast, solid state spectra usually exhibit peaks which are substantially broadened by dipolar coupling and chemical shift anisotropy (and quadrupolar coupling in the case of quadrupolar nuclei). NMR spectra of powdered solids are thus difficult to analyze due to signal overlap. It can be assumed that the nuclear positions within a crystal lattice are roughly time independent and therefore the crystal orientation, with respect to the external magnetic field, defines the effective Larmor frequency for each individual nucleus within the lattice. For single crystal measurements the manipulation of the sample holder orientation itself can lead to a change of the NMR peak positions [5, 6]. On the contrary, powdered samples, consisting of a large ensemble of randomly oriented tiny crystals, reveal broad NMR spectra representing superposition of lines at all possible frequencies (orientations) weighed by their probability. Both the abovementioned results are due to the anisotropic nature of spin interactions. The work presented below (i) introduces the concept of truncated field which represents the effective local magnetic field generated by dipolar nucleus under the influence of Zeeman field, (ii) illustrates the tensorial character of dipolar coupling and chemical shift anisotropy (CSA), and (iii) explains the influence of MAS on spin systems.

#### 2. Discussion

##### 2.1. Dipolar Interactions–Truncated Field Concept

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 1(b). An analog geometry of field surfaces can be observed for an ideal and superconductive coil depicted in Figure 1(a). It is known that in this particular case the magnetic moment depends on current, number of coil turns *N*, and the coil diameter.

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Each nucleus with nonzero spin, placed within external magnetic field obeys the space quantization of angular momentum and its quantum state can be described using discrete Zeeman eigenstates. A nuclear magnetic moment interacting with experiences torque and, accordingly, Larmor precession takes place. It appears that the quantum state of the nucleus can be represented as a linear combination of only two so-called basis states and weighted by complex number coefficients and . Spatial quantization rules imply that in the pure state, spin precess over a double cone surface with a specified apex angle exactly equals the tetrahedral angle (details presented in Supplementary 1). In order to formally describe this mechanism it is possible to analyze one arbitrarily chosen surface (Equation 1 – discussed in Supplementary 1) of local magnetic field and subsequently subject it to rotation transformations.

where and are azimuthal and polar angles, respectively in spherical coordinate system representation. According to spatial quantization rule it is necessary to tilt the surface following the inclination of nuclear magnetic moment vector in . This can be done using rotation matrix with specifically chosen magic angle (half of the tetrahedral angle). The operation is presented in Figure 2(a) and described mathematically below.

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Subsequently, it is necessary to introduce spin precession which is equivalent to the local potential rotation around -axis. In order to do so it is possible to operate another rotation matrix as described below and presented in Figure 2(b).

where is precession angle which can be also called a spin phase, and is Larmor angular frequency. It is worth looking closely at the effective local potential when several different phases are merged together as illustrated in Figure 3. There is something very intruiging about local field precession, namely there is a phase independent, constant magnetic field space resembling the symmetry of 3d spin orbital [7–9]. In the high external magnetic field approximation conditions, this effective, local nuclear magnetic field can be described by the second degree, zero order Laplace’s spherical harmonic and it is called truncated field. Knowing the geometry of the truncated field it is subsequently relatively easy to explain an angular dependence of internuclear interactions and elucidate the idea of MAS technique.

Describing a certain type of spin interactions it is possible to use a general tensorial formula (6) linking the spin and the source of the local field experienced by the spin, taking into consideration an orientational dependence of interaction via a second-rank coupling tensor. In the case of dipolar coupling the source of would be the neighbouring spin and the coupling tensor has the following form [10] (detailed discussion presented in Supplementary 1),

where is the Planck constant, and is the direct dipolar coupling constant which is inversely proportional to the (internuclear vector) and depends on the type of interacting nuclei. It is worth looking closely at the structure of the above mentioned matrix and characterizing its main properties in detail. It is clear that the principal axis of *D* is aligned along *z* direction when is parallel to and that the tensor is symmetrical . The matrix determinant equals zero when is tilted at the magic angle with respect to the principal axis (when ,,- components are equal). This implies that, no matter how close the two adjacent spins and are, they do not “sense” one another as long as the magic angle alignment of is fulfilled.

This nonintuitive conclusion may be explained relying on the truncated field symmetry in a following way; imagine that there are two nuclear magnetic moments and distanced one from another by vector which is oriented under magic angle with respect to the z axis. We can assume that both spins are on the lower energy state (“parallel” to the) and they have the same phase. Of course in general both spins can have completely different and random phases. (this is done for the sake of presentation only). Furthermore, let us assume that vectors precess over the cone with specified tetrahedral apex angle. Accordingly, we can define local magnetic field structure at certain specific and arbitrarily chosen moment. Figures 4(a) and 4(b) present two different stages differing in phase by 180 degree. In the case of high external magnetic field approximation, it is possible to define certain effective local magnetic fields, averaging the structure over all possible phases. In order to estimate the symmetry of these truncated magnetic fields two Figure 4(a) and 4(b) were merged giving the pattern illustrated in Figure 4(c). Eventually, well known 3d orbital symmetry pattern emerges which is shown in Figure 4(d).

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As shown in Figure 5(a), the strongest interaction between a pair of spins occurs when the internuclear vector is parallel to the external magnetic field . It is also evident that when both spins lay onto the precession cone surface, the dipolar coupling vanishes. It is also worth pointing out that MAS does not affect the orientation of the local effective nuclear magnetic field with respect to . This is illustrated in Figure 5(b). Let us imagine that spin is positioned exactly at the rotor’s main axis (marked in green), while the second spin is slightly off the rotor’s axis. When the sample is spun with a certain angular frequency , the internuclear vector circles a cone (marked in blue) and the angle becomes a function of time. Regardless of this, the orientation of the effective local field (produced by the spin ) with respect to the remains stable. This feature is also illustrated in *animation 1* available in Supplementary 1.

Knowing the truncated field symmetry it is easy to understand and explain experimental data observed by Pake [11]. In Figure 6(a) two interacting dipolar nuclei (placed in strong field) are illustrated together with an expected NMR spectra recorded for different internuclear vector orientations. Characteristic Pake doublet (also presented in the spectra) is a consequence of spin pair dipolar interaction in the case of powdered sample where orientations are randomly distributed. Red signals in Figure 6(a) represent positions of NMR peaks for the same spin pair measured in the case of single crystall sample. Assumed here local nuclear magnetic field symmetry adequately explains angular dependence of dipolar interaction and predicts nicely vanishing of dipolar splitting at magic angle conditions. It is not surprising because any other effective local magnetic field symmetry would not fit to dipolar Hamiltonian structure which involves 2^{nd} Legendre polynomial determining angular dependence of dipolar coupling strength ( form in spherical coordinates is presented in Supplementary 1).

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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 field. Figure 7(a) shows the first specific crystall lattice orientation. It is important to note that due to the crystal symmetry, each closest, neighbouring nucleus (, , , ) of carbon is located onto the magic angle cone surface. In other words, all internuclear vectors are aligned under magic angle conditions with respect to . According to the dipolar Hamiltonian structure, this particular nucleus experiences solely field (taking into consideration only interactions with closest neigbours) and somehow it does not sense any local fields produced by its closest neighbours. This remarkable feature can be nicely explained using truncated field symmetry as it is presented in Figure 7(b). The net local field component along the internuclear vector equals zero. Exactly the same holds for other spin pairs presented.

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Figure 8(a) shows the second specific crystal lattice orientation. In this particular case none of the nuclear pairs fulfill magic angle conditions. Ilustration 8 8(b) reveals that all the adjacent nuclei are located off-magic angle cone. Consequently, nonzero dipolar coupling is expected.

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There are two inequivalent groups of spin pairs: (a) – and (b) – , – , – . Taking into consideration the dipolar Hamiltonian structure and the truncated field symmetry, it is obvious that vector orientation leads to the strongest possible dipolar coupling. It is presented in Figure 9. Three other pairs – , – , – are equivalent in terms of internuclear vector orientation with respect to the external field . All three vectors , , are tilted with respect to with angle . According to 2^{nd} Legendre polynomial relation the dipolar coupling in this particular case should be scaled with prefactor (Figure 9). In order to verify if the truncated symmetry fulfills this condition it is possible to (a) magnify the truncated field produced by µ_{l} until it reaches position and (b) normalize the truncated field to get (and automatically ). This procedure is illustrated on the right hand side in Figure 9. In order to calculate the angular dependent dipolar coupling prefactor we have to find the distance *x*. For normalized truncated field *x* is equal exactly 0.65 which is in agreement with the dipolar Hamiltonian structure.

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 10 two examples of randomly oriented (with respect to the main rotor axis) are shown. To illustrate the periodical character of θ angle evolution, let us imagine pair of spins µ_{j} and µ_{k} distanced by vector which is oriented 45° with respect to the rotation axis (green arrow) as depicted in Figure 11(b). It is clear that under spinning conditions the r_{jk} vector will slide onto the cone surface with apex angle equal to 90°. This precession-like movement will lead to periodic variation around the magic angle as illustrated in Figure 11(a). Taking into consideration the dipolar Hamiltonian structure it is easy to calculate the dipolar coupling factor which is a function of time and . The evolution is also presented in Figure 11(a). It is worth examining this function in detail because: (a) its evolution proves that the nucleus produces the effective magnetic field (truncated field) having symmetry, and (b) it shows that the precessing internuclear vector samples this nuclear magnetic field symmetry. Along one rotation period the function reveals two maxima and two minima. Both minima are indicated in the Figure 11(a) and correspondingly two characteristic scaling factors and can be identified. The two maxima exhibit exactly the same scaling which is equal to 1. In order to fully understand this specific evolution let us introduce the truncated field generated by . As depicted in Figure 11(b) this approach enables us to find directly the two characteristic scaling factors corresponding to both minima. Moreover, it is possible to calculate the scaling factor for each stage of vector evolution during MAS and prove that the product envelopes the surface as it is depicted in Figure 12. Both Figures 11(c), 11(d), 12(c), and 12(d) illustrate the same analysis performed for vector which is oriented 67° with respect to the rotation axis. It is also worth mentioning that the second nucleus µ_{k} produces exactly the same truncated magnetic field and its orientation with respect to the external magnetic field is time independent as presented in *animation 1* available via Supplementary 1.

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##### 2.2. Chemical Shift Anisotropy

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 (8)

where is the chemical shift tensor which can be described as

where the principal components of the tensor are denoted as , and . They represent the principal axis system (PAS) of the tensor. It is clear from relations equation 6 that effective shielding is directly proportional to the -component of *σ* in the laboratory frame determined by the external magnetic field direction. In other words, it depends on PAS orientation with respect to . This orientational dependence is illustrated in Figure 13, where there are three different orientations of hypothetical monocrystals comprising of nuclei exhibiting anisotropic electronic environment (represented by a symmetrical tensor). The component (marked in red and parallel to ) varies as the monocrystal is rotated. Accordingly, an expected NMR signal changes its position on the spectrum. As a result, substantially broadened and asymmetrical, static NMR spectra are often recorded for powdered crystalline samples. Figure 13(b) illustrates that MAS technique allows finding the position of NMR peak characterized for isotropic shielding . It is easy to distinguish this particular signal from rotational sidebands (marked with stars) due to the fact that its position does not depend on spinning frequency.

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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 14 the dynamics of two (symmetric and asymmetric, both with arbitrary initial orientations) are illustrated for the spinning experiment. As presented, the principal axis of the tensor circles a cone around the rotation axis (marked in red) while the component varies periodically with the angular frequency . This process is depicted in *animation 2* available in Supplementary 1.

Interestingly, shielding evolution strongly depends on the orientation of PAS with respect to the rotation axis. This inclination can be described by γ angle defined in 15. As indicated, a slightly different initial orientation of the CS tensor leads to significant changes in dependence. This effect is even more pronounced for an asymmetric tensor as presented in 15(b). Evolutions of symmetric and asymmetric CS tensors with different positioning are illustrated in *animation 3* and *animation 4* respectively (available in Supplementary 1). It is possible to derive an analytical expression describing the above mentioned evolution and, consequently, calculate the principal CS tensor components directly from the spinning sidebands pattern [12, 13]. It has been shown [13, 14] that NMR signal position is a complex function (10) which depends on both ω_{R} and γ and can be described in the following way

where

where , , are Euler angles describing the orientation of PAS with respect to the rotor axis frame and represents the isotropic shielding. Detailed derivation of equation 9 and all necessary CS tensor transformations are presented in Supplementary 1. Alternatively, it is also possible to utilize Equation 9 in the case of a static experiment, provided there is a monocrystalline sample available together with an NMR probehead equipped with a goniometer 15. This allows for sampling chemical shielding for various PAS orientations (manifested as a chemical shift change) and accordingly leads to the determination of principal CS tensor components. As shown in Figure 15 the sideband pattern varies (in terms of amplitudes) for different CS tensor orientations. Presented here NMR spectra are Fourier transforms of simulated FID signals with specified spin-spin relaxation time *T*_{2}.

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#### 3. Conclusions

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.

#### Data Availability

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

#### Conflicts of Interest

The authors declare no conflicts of interest.

#### Acknowledgments

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

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