# Magic Angle Spinning and Truncated Field Concept in NMR

Authors: J. Jenczyk
Journal: Concepts in Magnetic Resonance Part A
 (2019)

---

## 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 dipolarD and chemical shield σ coupling tensors under MAS conditions.


---

## Body

## 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 I=1/  2 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 Figure1(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 μc depends on current, number of coil turns N, and the coil diameter.Figure 1
(a) Magnetic field geometry produced by a coil, (b) local magnetic field produced by an isolated nucleus having nonzero spin.
(a)
(b)Each nucleus with nonzero spin, placed within external magnetic fieldB0 obeys the space quantization of angular momentum and its quantum state can be described using discrete Zeeman eigenstates. A nuclear magnetic moment interacting with B0 experiences torque and, accordingly, Larmor precession takes place. It appears that the quantum state ψ of the I=1/  2 nucleus can be represented as a linear combination of only two so-called basis states ↑ and ↓ weighted by complex number coefficients c↑ and c↓. 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 S (Equation 1 – discussed in Supplementary 1) of local magnetic field and subsequently subject it to rotation transformations.(1)S=cosφa−acosθsinφa−acosθsinθ.whereφ and θ are azimuthal and polar angles, respectively in spherical coordinate system representation. According to spatial quantization rule it is necessary to tilt the surface S following the inclination of nuclear magnetic moment vector in B0. This can be done using rotation matrix Rxθ with specifically chosen magic angle θm≈54.74°(half of the tetrahedral angle). The operation is presented in Figure 2(a) and described mathematically below.(2)RxθmS=1000cosθm−sinθm0sinθmcosθmcosφa−acosθsinφa−acosθsinθ,(3)RxθmS=cosφa−acosθcosθmsinφa−acosθ−sinθmsinθsinθmsinφa−acosθ+cosθmsinθ.Figure 2
(a)RxθmS and (b) RzφRRxθmS operations.
(a)
(b)Subsequently, it is necessary to introduce spin precession which is equivalent to the local potentialRxθmS rotation around z-axis. In order to do so it is possible to operate another rotation matrix RzφR as described below and presented in Figure 2(b).(4)RzφRRxθmS=cosφR−sinφR0sinφRcosφR0001cosφa−acosθcosθmsinφa−acosθ−sinθmsinθsinθmsinφa−acosθ+cosθmsinθ.(5)RzφRRxθmS=cosω0tcosφa−acosθ−sinω0tcosθmsinφa−acosθ−sinθmsinθsinω0tcosφa−acosθ+cosω0tcosθmsinφa−acosθ−sinθmsinθsinθmsinφa−acosθ+cosθmsinθ.whereφR=ω0t is precession angle which can be also called a spin phase, and ω0 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 Y20φ,θ 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.Figure 3
The truncated field represents phase independent, constant magnetic field space which can be described by Laplace’s spherical harmonicY20φ,θ.Describing a certain type of spin interactions it is possible to use a general tensorial formula (6) linking the spin Ij and the source of the local field Bloc 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 Bloc would be the neighbouring spin Ik and the coupling tensor D has the following form [10] (detailed discussion presented in Supplementary 1),(6)Hdd=hdjkIjDIk.(7)D=rjk2−3x2/rjk2−3xy/rjk2−3xz/rjk2−3xy/rjk2rjk2−3y2/rjk2−3yz/rjk2−3xz/rjk2−3yz/rjk2rjk2−3z2/rjk2,whereh is the Planck constant, and djk is the direct dipolar coupling constant which is inversely proportional to the rjk3 (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 rjk is parallel to z and that the D tensor is symmetrical Dαβ=Dβα. The matrix determinant equals zero when rjk is tilted at the magic angle θ≈54.73° with respect to the principal axis (when x,y,z- components are equal). This implies that, no matter how close the two adjacent spins Ij and Ik are, they do not “sense” one another as long as the magic angle alignment of rjk 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 momentsμj and μk distanced one from another by vector rjk 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 theBo) 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 μ 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).Figure 4
Pair of spins (having the same phase) distanced by vectorrjk presented for two different phases (a) φR=φ0 and (b) φR=φ0+π, respectively. (c) Represents merging of two phases, and (d) illustrates truncated field symmetries for both spins residing in high external magnetic field Bo.
(a)
(b)
(c)
(d)As shown in Figure5(a), the strongest interaction between a pair of spins occurs when the internuclear vector rjk is parallel to the external magnetic field B0. 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 B0. This is illustrated in Figure 5(b). Let us imagine that spin Ij is positioned exactly at the rotor’s main axis (marked in green), while the second spin Ik is slightly off the rotor’s axis. When the sample is spun with a certain angular frequency ωR, the internuclear vector rjk circles a cone (marked in blue) and the θ angle becomes a function of time. Regardless of this, the orientation of the effective local field (produced by the spin Ik) with respect to the B0 remains stable. This feature is also illustrated in animation 1 available in Supplementary 1.Figure 5
(a) Graphical representation of angular dependence of dipolar coupling strength described by the 2nd Legendre polynomial. Coloured in red and in orange are effective local magnetic fields produced by Ij and Ik spins respectively, (b) certain arbitrarily chosen pair of spins under magic angle spinning conditions, and (c) frame of reference for two interacting spins.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 Bo field) are illustrated together with an expected NMR spectra recorded for different internuclear vector rjk orientations. Characteristic Pake doublet (also presented in the spectra) is a consequence of spin pair dipolar interaction in the case of powdered sample where rjk 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 2nd Legendre polynomial determining angular dependence of dipolar coupling strength (Hdd form in spherical coordinates is presented in Supplementary 1).Figure 6
(a) Pake doublet (powdered crystall) and dipolar splitting (single crystall) observed for interacting pair of spins with differentrjk orientations with respect to B0. Red vectors in the spheres represent a set of equivalent rjk orientations. (b) effect of MAS on Pake doublet.
(a)
(b)In order to justify and clarify the truncated field concept, it is worth performing the following thought experiment: imagine a diamond monocrystal enriched with13C 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 B0 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 (μj, μk, μl, μm) of μi carbon is located onto the magic angle cone surface. In other words, all internuclear vectors are aligned under magic angle conditions with respect to B0. According to the dipolar Hamiltonian structure, this particular μi nucleus experiences solely B0 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 μl local field component along the internuclear vector ril equals zero. Exactly the same holds for other spin pairs presented.Figure 7
(a) Diamond crystal orientation with respect toB0. (b) Closest neighbours μj, μk, μl, and μm located onto the magic angle cone surface.
(a)
(b)Figure8(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.Figure 8
(a) Diamond crystal orientation with respect toB0. (b) Closest neighbours μj, μk, μl, and μm located off the magic angle cone surface.
(a)
(b)There are two inequivalent groups of spin pairs: (a)μi – μm and (b) μi – μj, μi – μk, μi – μl. Taking into consideration the dipolar Hamiltonian structure and the truncated field symmetry, it is obvious that rim vector orientation leads to the strongest possible dipolar coupling. It is presented in Figure 9. Three other pairs μi – μj, μi – μk, μi – μl are equivalent in terms of internuclear vector orientation with respect to the external field B0. All three vectors rij, rik, ril are tilted with respect to B0 with angle θ∼70.5°. According to 2nd Legendre polynomial relation the dipolar coupling in this particular case should be scaled with Dil∼0.65 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 µl until it reaches μi position and (b) normalize the truncated field to get p=2 (and automatically q=1). This procedure is illustrated on the right hand side in Figure 9. In order to calculate the angular dependent dipolar coupling prefactor Dil 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.Figure 9
The truncated field symmetry correctly explains the angular dependence of the dipolar coupling strength.Figure 10
Schematically presented evolution of a spin pair under MAS conditions for two different internuclear vector orientationsrjk (marked in red) with respect to the rotor’s main axis (marked in green).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 Figure10  two examples of randomly oriented rjk (with respect to the main rotor axis) are shown. To illustrate the periodical character of θ angle evolution, let us imagine pair of spins µj and µk distanced by rjk=1 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 rjk vector will slide onto the cone surface with apex angle equal to 90°. This precession-like movement will lead to periodic θt variation around the magic angle θm as illustrated in Figure 11(a). Taking into consideration the dipolar Hamiltonian structure it is easy to calculate the dipolar coupling factor D which is a function of time and θ. The Dθ,t 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 Y20φ,θ symmetry, and (b) it shows that the precessing internuclear vector rjk samples this nuclear magnetic field symmetry. Along one rotation period the Dθ,t function reveals two maxima and two minima. Both minima are indicated in the Figure 11(a) and correspondingly two characteristic scaling factors D∼0.91 and D∼1.91 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 μj. 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 D for each stage of rjk vector evolution during MAS and prove that the product Drjk envelopes the Y20φ,θ 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 rjk=1 vector which is oriented 67° with respect to the rotation axis. It is also worth mentioning that the second nucleus µk 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.Figure 11
(a and c)θt and Dθ,t evolution during MAS, (b and d) two extreme internuclear vector (red) orientations with respect to external magnetic field B0. Truncated field symmetry of the local nuclear magnetic field generated by μj.
(a)
(b)
(c)
(d)Figure 12
(a and c) Illustrate evolution of internuclear vectors (for the case presented in Figures11(a) and 11(c)) which slide onto different precession cones, (b and d) show that the same internuclear vectors scaled by D precisely envelope the structure of the truncated field.
(a)
(b)
(c)
(d)


### 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)(8)HCSA=γhIjσPASBo,whereσPAS is the chemical shift tensor which can be described as(9)σPAS=σxxPAS000σyyPAS000σzzPAS,where the principal components of the tensor are denoted asσxxPAS, σyyPAS, and σyyPAS. 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 z-component of σ in the laboratory frame determined by the external magnetic field B0 direction. In other words, it depends on PAS orientation with respect to B0. 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 σzz (marked in red and parallel to B0) 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 σiso=σxxPAS+σyyPAS+σzzPAS/3. 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.Figure 13
(a) Schematically illustrated shielding variation for three different monocrystal orientations with respect to the external magnetic fieldB0. (b) Effect of MAS on powdered NMR spectrum.
(a)
(b)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 Figure14 the dynamics of two σ (symmetric and asymmetric, both with arbitrary initial orientations) are illustrated for the spinning experiment. As presented, the principal axis of the tensor σzzPAS circles a cone around the rotation axis (marked in red) while the σzz component varies periodically with the angular frequency ωR. This process is depicted in animation 2 available in Supplementary 1.Figure 14
The time evolution of CSA tensors (upper symmetric and lower asymmetric) under MAS conditions. Animation available in Supplementary Materials1.Interestingly, shielding evolutionσzzt 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 σzzt 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 γ 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 ωR and γ and can be described in the following way(10)ω=−ω0σiso+A1cosωRt+α+B1sinωRt+α+A2cos2ωRt+2α+B2sin2ωRt+2α,where(11)A1=223sinγ·cosγcos2βσxxPAS−σzzPAS+sin2βσyyPAS−σzzPAS,B1=223sinβ·cosβ·sinγ·σxxPAS−σzzPAS,A2=13cos2γ·cos2β−sin2βσxxPAS−σzzPAS+cos2γ·sin2β−cos2  βσyyPAS−σzzPAS,B2=−23sinβ·cosβ·cosγ·σxxPAS−σyyPAS,whereα, β, γ are Euler angles describing the orientation of PAS with respect to the rotor axis frame and σiso 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 T2.Figure 15
σ
z
z
t time dependencies presented for two different PAS orientations (various γ) with respect to rotor axis frame. (a) Symmetric CSA tensor, (b) asymmetric CSA tensor. On the right hand side simulated FIDs with specified spin-spin relaxation time T2 are presented together with their Fourier transforms representing expected NMR signals.
(a)
(b)


## 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 Figure1(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 μc depends on current, number of coil turns N, and the coil diameter.Figure 1
(a) Magnetic field geometry produced by a coil, (b) local magnetic field produced by an isolated nucleus having nonzero spin.
(a)
(b)Each nucleus with nonzero spin, placed within external magnetic fieldB0 obeys the space quantization of angular momentum and its quantum state can be described using discrete Zeeman eigenstates. A nuclear magnetic moment interacting with B0 experiences torque and, accordingly, Larmor precession takes place. It appears that the quantum state ψ of the I=1/  2 nucleus can be represented as a linear combination of only two so-called basis states ↑ and ↓ weighted by complex number coefficients c↑ and c↓. 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 S (Equation 1 – discussed in Supplementary 1) of local magnetic field and subsequently subject it to rotation transformations.(1)S=cosφa−acosθsinφa−acosθsinθ.whereφ and θ are azimuthal and polar angles, respectively in spherical coordinate system representation. According to spatial quantization rule it is necessary to tilt the surface S following the inclination of nuclear magnetic moment vector in B0. This can be done using rotation matrix Rxθ with specifically chosen magic angle θm≈54.74°(half of the tetrahedral angle). The operation is presented in Figure 2(a) and described mathematically below.(2)RxθmS=1000cosθm−sinθm0sinθmcosθmcosφa−acosθsinφa−acosθsinθ,(3)RxθmS=cosφa−acosθcosθmsinφa−acosθ−sinθmsinθsinθmsinφa−acosθ+cosθmsinθ.Figure 2
(a)RxθmS and (b) RzφRRxθmS operations.
(a)
(b)Subsequently, it is necessary to introduce spin precession which is equivalent to the local potentialRxθmS rotation around z-axis. In order to do so it is possible to operate another rotation matrix RzφR as described below and presented in Figure 2(b).(4)RzφRRxθmS=cosφR−sinφR0sinφRcosφR0001cosφa−acosθcosθmsinφa−acosθ−sinθmsinθsinθmsinφa−acosθ+cosθmsinθ.(5)RzφRRxθmS=cosω0tcosφa−acosθ−sinω0tcosθmsinφa−acosθ−sinθmsinθsinω0tcosφa−acosθ+cosω0tcosθmsinφa−acosθ−sinθmsinθsinθmsinφa−acosθ+cosθmsinθ.whereφR=ω0t is precession angle which can be also called a spin phase, and ω0 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 Y20φ,θ 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.Figure 3
The truncated field represents phase independent, constant magnetic field space which can be described by Laplace’s spherical harmonicY20φ,θ.Describing a certain type of spin interactions it is possible to use a general tensorial formula (6) linking the spin Ij and the source of the local field Bloc 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 Bloc would be the neighbouring spin Ik and the coupling tensor D has the following form [10] (detailed discussion presented in Supplementary 1),(6)Hdd=hdjkIjDIk.(7)D=rjk2−3x2/rjk2−3xy/rjk2−3xz/rjk2−3xy/rjk2rjk2−3y2/rjk2−3yz/rjk2−3xz/rjk2−3yz/rjk2rjk2−3z2/rjk2,whereh is the Planck constant, and djk is the direct dipolar coupling constant which is inversely proportional to the rjk3 (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 rjk is parallel to z and that the D tensor is symmetrical Dαβ=Dβα. The matrix determinant equals zero when rjk is tilted at the magic angle θ≈54.73° with respect to the principal axis (when x,y,z- components are equal). This implies that, no matter how close the two adjacent spins Ij and Ik are, they do not “sense” one another as long as the magic angle alignment of rjk 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 momentsμj and μk distanced one from another by vector rjk 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 theBo) 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 μ 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).Figure 4
Pair of spins (having the same phase) distanced by vectorrjk presented for two different phases (a) φR=φ0 and (b) φR=φ0+π, respectively. (c) Represents merging of two phases, and (d) illustrates truncated field symmetries for both spins residing in high external magnetic field Bo.
(a)
(b)
(c)
(d)As shown in Figure5(a), the strongest interaction between a pair of spins occurs when the internuclear vector rjk is parallel to the external magnetic field B0. 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 B0. This is illustrated in Figure 5(b). Let us imagine that spin Ij is positioned exactly at the rotor’s main axis (marked in green), while the second spin Ik is slightly off the rotor’s axis. When the sample is spun with a certain angular frequency ωR, the internuclear vector rjk circles a cone (marked in blue) and the θ angle becomes a function of time. Regardless of this, the orientation of the effective local field (produced by the spin Ik) with respect to the B0 remains stable. This feature is also illustrated in animation 1 available in Supplementary 1.Figure 5
(a) Graphical representation of angular dependence of dipolar coupling strength described by the 2nd Legendre polynomial. Coloured in red and in orange are effective local magnetic fields produced by Ij and Ik spins respectively, (b) certain arbitrarily chosen pair of spins under magic angle spinning conditions, and (c) frame of reference for two interacting spins.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 Bo field) are illustrated together with an expected NMR spectra recorded for different internuclear vector rjk orientations. Characteristic Pake doublet (also presented in the spectra) is a consequence of spin pair dipolar interaction in the case of powdered sample where rjk 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 2nd Legendre polynomial determining angular dependence of dipolar coupling strength (Hdd form in spherical coordinates is presented in Supplementary 1).Figure 6
(a) Pake doublet (powdered crystall) and dipolar splitting (single crystall) observed for interacting pair of spins with differentrjk orientations with respect to B0. Red vectors in the spheres represent a set of equivalent rjk orientations. (b) effect of MAS on Pake doublet.
(a)
(b)In order to justify and clarify the truncated field concept, it is worth performing the following thought experiment: imagine a diamond monocrystal enriched with13C 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 B0 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 (μj, μk, μl, μm) of μi carbon is located onto the magic angle cone surface. In other words, all internuclear vectors are aligned under magic angle conditions with respect to B0. According to the dipolar Hamiltonian structure, this particular μi nucleus experiences solely B0 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 μl local field component along the internuclear vector ril equals zero. Exactly the same holds for other spin pairs presented.Figure 7
(a) Diamond crystal orientation with respect toB0. (b) Closest neighbours μj, μk, μl, and μm located onto the magic angle cone surface.
(a)
(b)Figure8(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.Figure 8
(a) Diamond crystal orientation with respect toB0. (b) Closest neighbours μj, μk, μl, and μm located off the magic angle cone surface.
(a)
(b)There are two inequivalent groups of spin pairs: (a)μi – μm and (b) μi – μj, μi – μk, μi – μl. Taking into consideration the dipolar Hamiltonian structure and the truncated field symmetry, it is obvious that rim vector orientation leads to the strongest possible dipolar coupling. It is presented in Figure 9. Three other pairs μi – μj, μi – μk, μi – μl are equivalent in terms of internuclear vector orientation with respect to the external field B0. All three vectors rij, rik, ril are tilted with respect to B0 with angle θ∼70.5°. According to 2nd Legendre polynomial relation the dipolar coupling in this particular case should be scaled with Dil∼0.65 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 µl until it reaches μi position and (b) normalize the truncated field to get p=2 (and automatically q=1). This procedure is illustrated on the right hand side in Figure 9. In order to calculate the angular dependent dipolar coupling prefactor Dil 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.Figure 9
The truncated field symmetry correctly explains the angular dependence of the dipolar coupling strength.Figure 10
Schematically presented evolution of a spin pair under MAS conditions for two different internuclear vector orientationsrjk (marked in red) with respect to the rotor’s main axis (marked in green).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 Figure10  two examples of randomly oriented rjk (with respect to the main rotor axis) are shown. To illustrate the periodical character of θ angle evolution, let us imagine pair of spins µj and µk distanced by rjk=1 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 rjk vector will slide onto the cone surface with apex angle equal to 90°. This precession-like movement will lead to periodic θt variation around the magic angle θm as illustrated in Figure 11(a). Taking into consideration the dipolar Hamiltonian structure it is easy to calculate the dipolar coupling factor D which is a function of time and θ. The Dθ,t 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 Y20φ,θ symmetry, and (b) it shows that the precessing internuclear vector rjk samples this nuclear magnetic field symmetry. Along one rotation period the Dθ,t function reveals two maxima and two minima. Both minima are indicated in the Figure 11(a) and correspondingly two characteristic scaling factors D∼0.91 and D∼1.91 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 μj. 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 D for each stage of rjk vector evolution during MAS and prove that the product Drjk envelopes the Y20φ,θ 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 rjk=1 vector which is oriented 67° with respect to the rotation axis. It is also worth mentioning that the second nucleus µk 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.Figure 11
(a and c)θt and Dθ,t evolution during MAS, (b and d) two extreme internuclear vector (red) orientations with respect to external magnetic field B0. Truncated field symmetry of the local nuclear magnetic field generated by μj.
(a)
(b)
(c)
(d)Figure 12
(a and c) Illustrate evolution of internuclear vectors (for the case presented in Figures11(a) and 11(c)) which slide onto different precession cones, (b and d) show that the same internuclear vectors scaled by D precisely envelope the structure of the truncated field.
(a)
(b)
(c)
(d)


## 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)(8)HCSA=γhIjσPASBo,whereσPAS is the chemical shift tensor which can be described as(9)σPAS=σxxPAS000σyyPAS000σzzPAS,where the principal components of the tensor are denoted asσxxPAS, σyyPAS, and σyyPAS. 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 z-component of σ in the laboratory frame determined by the external magnetic field B0 direction. In other words, it depends on PAS orientation with respect to B0. 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 σzz (marked in red and parallel to B0) 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 σiso=σxxPAS+σyyPAS+σzzPAS/3. 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.Figure 13
(a) Schematically illustrated shielding variation for three different monocrystal orientations with respect to the external magnetic fieldB0. (b) Effect of MAS on powdered NMR spectrum.
(a)
(b)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 Figure14 the dynamics of two σ (symmetric and asymmetric, both with arbitrary initial orientations) are illustrated for the spinning experiment. As presented, the principal axis of the tensor σzzPAS circles a cone around the rotation axis (marked in red) while the σzz component varies periodically with the angular frequency ωR. This process is depicted in animation 2 available in Supplementary 1.Figure 14
The time evolution of CSA tensors (upper symmetric and lower asymmetric) under MAS conditions. Animation available in Supplementary Materials1.Interestingly, shielding evolutionσzzt 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 σzzt 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 γ 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 ωR and γ and can be described in the following way(10)ω=−ω0σiso+A1cosωRt+α+B1sinωRt+α+A2cos2ωRt+2α+B2sin2ωRt+2α,where(11)A1=223sinγ·cosγcos2βσxxPAS−σzzPAS+sin2βσyyPAS−σzzPAS,B1=223sinβ·cosβ·sinγ·σxxPAS−σzzPAS,A2=13cos2γ·cos2β−sin2βσxxPAS−σzzPAS+cos2γ·sin2β−cos2  βσyyPAS−σzzPAS,B2=−23sinβ·cosβ·cosγ·σxxPAS−σyyPAS,whereα, β, γ are Euler angles describing the orientation of PAS with respect to the rotor axis frame and σiso 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 T2.Figure 15
σ
z
z
t time dependencies presented for two different PAS orientations (various γ) with respect to rotor axis frame. (a) Symmetric CSA tensor, (b) asymmetric CSA tensor. On the right hand side simulated FIDs with specified spin-spin relaxation time T2 are presented together with their Fourier transforms representing expected NMR signals.
(a)
(b)


## 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.



---

*Source: 5895206-2019-12-04.xml*