Resonances

Chemical shielding

The chemical shielding is due to a shielding caused by electrons (chemical environment), that modify slightly the interaction between the nuclei and the magnetic field B0, thus, the precession frequency of the nuclei. Here it is worth noting that the chemical shift scale: part per million (ppm) is a frequency scale like in all other spectroscopies but normalized with respect to the magnetic field of the spectrometer to unify the values obtained around the world at different magnetic fields.

Quadrupolar interaction

The quadrupolar interaction only concerns the nuclei with a spin quantum number greater than half. It is due to the interaction between the electric quadrupolar moment of the nucleus and the electric field gradient (EFG) surrounding the nucleus. Here, it is worth mentioning that the electric field gradient is nothing than the gradient of gradient of the electric potential.

Dipole-dipole interaction

The dipole-dipole coupling is the interaction between two nuclei possessing magnetic moments in the magnetic field. It can be either homonuclear or heteronuclear, direct or indirect. The direct dipolar coupling (D) is achieved through space and is intimately related to the distance between the nuclei. The indirect coupling (J) is achieved through the electron shells forming the chemical bonds. The D coupling is much more intense than the J coupling.

Coherence pathway

This concept was introduced by Bodenhausen in 1984 and helped finding, understanding and enhancing many NMR pulse sequences. Briefly, a coherence or coherence order ‘p’ designing a transition in progress between a pair of eigenstates is defined as a fundamental quantum number.

Then each transition is associated with two coherence orders of opposite sign. For a system of K spins ½, p extends from – K to +K. For example, transverse magnetization of a spin ½ is conceived as a coherent superposition of two eigenstates, it corresponds to a particular class of coherence (single quantum, SQ) associated with a change in quantum number p = ± 1.

Within this concept, a R.F. irradiation induces a transfer between coherences while a free precession does not change coherence orders. The route of a particular component of coherence during an NMR experiment is referred to as a coherence transfer pathway.

The sequence of coherence levels that a signal passes through between the first pulse and the time it reaches the receiver is the coherence pathway. Phase cycling is used to accumulate signals of interest, manipulate spin systems and suppress undesired signals and artefacts. Several methods were developed for phase cycling.

 Multi-dimensional NMR

The Fourier transformation of the time scale gives a frequency scale. Then, adding a time variation during the experiment we can transform it into another frequency domain, thus a second dimension. These approaches introduced by Ernst in 1966 allows the detection of the evolution of the system in multiple quantum orders that are undetectable in one-dimension NMR and permits, lately, the recoupling of small interactions that are lost / averaged by the magic angle spinning when seeking a high resolution in solid-state. To obtain pure absorption line shapes, several methods were introduced, e.g. States and time-proportional phase incrementation (TPPI) or a combination of both to mimic the quadrature detection of a one-dimension FID and create the real and imaginary parts to obtain pure absorption line shapes. Every two-dimensional NMR experiment consist of four main elements: preparation, evolution, mixing and detection. 

Single pulse

This is the basic NMR experiment. It consists of N loops or scans of (pulse, detect, wait). The pulse generally consists of a quarter of a nutation cycle (90°) converting the z magnetization (p = 0) into a xy coherence (p = -1) that can be detected as a free induction decay (FID) of the magnetization in time. The signal acquired after N scans is summed and Fourier transformed: converted into a spectrum (in the frequency domain).  The repetition of N scans has as objective the decreasing of the signal to noise ratio.

Echos

Hahn echo

This sequence was developed by Hahn, it consists of a 90° pulse followed by a 180° pulse and was used originally to focus or suppress some coherences in the signals based on their relaxation times, it is also useful to remove broad background signals and to estimate T2, the transversal relaxation time, determining the mobility of nuclei.

Solid echo

When quadrupolar interaction is involved, Hahn echo is not very efficient. The main cause of this is the strength of quadrupolar interaction altering the nutation frequency of the nuclei. Solid echo or quadrupolar echo consists of two consecutives 90° pulses with a specific coherence pathway.

Cross polarization

Cross polarization (CP) is used to ensure a transfer of magnetization from sensitive nuclei (high gamma or high natural abundance) to low gamma or low natural abundant nuclei. It can be seen as thermodynamic transfer from hot to cold reservoirs. The enhancement factor corresponds to the ratio of the gyromagnetic ratios. Since abundant spins are strongly dipolar coupled, subject to large fluctuating magnetic fields resulting from motion, a quickest relaxation occurs when CP is used compared to a simple detection of less coupled nuclei. Polarization is transferred during a spin locking period (the contact time). Cross polarization requires a dipolar coupling between the concerned nuclei and the setting of Hartmann-Hahn conditions: equal nutation frequencies on both channels induce a precession of both nuclei at the same rate in the rotating frame, allowing the transfer of polarization. Ramped and shaped spin locks are often used to enhance the transfer. When the CP is combined with MAS, the optimal Hartmann-Hahn conditions are modulated by the spinning frequency and become narrower. The kinetics of transfer known as ‘dynamics of CP’ tell about the proximity between the concerned nuclei and their mobility.

MQMAS

Multiple quantum Magic Angle Spinning (MQMAS) pulse sequence is used to average the second order quadrupolar interaction, an impossible task using MAS. It was introduced by Frydmann in 1995. DAS and DOR being not so practical, Frydmann has noticed that 2nd rank quadrupolar interactions (4th rank space term) can be averaged by detecting multiple quantum transitions. However, they cannot be directly detected, coherence order ≠ -1, they can be seen in the indirect dimension, this is the reason why it is a two-dimensional experiment. Several coherence transfer pathways were proposed e.g. zero quantum filter (ZQF), echo/anti echo and split t1. These methods have the advantage to be widely applied with the MAS technology. STMAS (Satellite transition Magic angle spinning) follows a similar approach and allows obtaining isotropic chemical shifts in the indirect dimension by probing single quantum satellite transitions instead of multiple quantum transitions. In both, MQMAS and STMAS, an additional shearing operation is needed to synchronize the echoes and extract the isotropic chemical shift values precisely in the indirect dimension.

Trigonometry - Euler and Fourier

Many NMR theorems and operators are related to rotation, it is worth mentioning the importance of the golden Euler equation:  formulating simply all rotations we deal with in NMR. The Fourier transformation, is also of high importance, it allows the conversion of time domain signals in frequency domains by decomposing the free induction decay (FID) signal into its corresponding sinus functions.

Legendre polynomials

The polynomials of Legendre often used to show why at specific angular orientations (magic angles), 2nd and 4th rank anisotropic interactions anneal.  We can see that 54.74° is the magic angle for 2nd rank anisotropic interactions while 30.56° and 70.12° are magic angles for 4th rank anisotropic interactions. Magic Angle Spinning (MAS), Dynamic Angle Spinning (DAS) and Double Rotation (DOR) respectively are three technologies that derive from these equations. MAS is the most widely used. The 2nd and 4th Legendre polynomials are the concerned ones.

Frequency vs. phase

For waves, three parameters are important to define, their amplitude, their frequency and their phase. Frequency determines the number of repetition of the wave in time, usually measured as waves/second or Hertz (Hz). The amplitude determines the intensity of the wave. However, phase is not a property of just one signal but instead involves the relationship between two or more signals. Two waves with the same frequency may be shifted in phase if their consecutive maximum amplitudes are not coherent.

Tensors

They are mathematical objects. A point in the space is a scalar with zero orientation, described with 30 elements: 1 element, it is a 0th order tensor. A vector has a unique orientation, described with 31 elements: 3 elements, it is a 1st order tensor. An ellipsoid has two orientations, described with 32 elements: 9 elements, it is a 2nd order tensor. The latter are usually used to describe the NMR interactions in solid-state, faithfully representing their anisotropy.

Second rank tensors, when diagonalized in their ‘principal axis system’ (PAS) are defined with at least 2 parameters: the magnitude (a z component) and the asymmetry showing the 'shape' of the ellipsoid, if the tensor is traceless. However, if the trace (sum of the diagonal components) is different than zero, the trace determines the isotropic part.

Depending on the rank (n) of the tensors used, they can be composed by 3^n elements in cartesian coorinates and (2n+1) elements in spherical coordinates.

It is important not to confuse order with rank in NMR notations. An interaction of nth order is itself represented with tensors developed with increasing ranks depending on the size and the symmetry of the interaction.

The diagonalization of a vector towards its Principal Axis System (PAS) is equivalent to the finding of all its eigenvectors in all dimensions! Determinants tell about the size of a matrix scaling after its transformation. Those with a determinant equal to zero shrink after transformation and can be neglected.

Rotations, frames and Euler angles

Rotations

The location of any point in space may be defined using cartesian coordinates (x, y, z) or spherical coordinates (r, tetha, phi) called radial, azimuthal, polar successively.

Spherical coordinates are often used for mathematical simplification of the equations. The rotation of tensors of a certain rank in spherical coordinates basis is done using the so-called Wigner matrices of similar rank. Here, the magic angle can be easily understood as the angle that mainly anneals the central element of a second rank Wigner matrix, constituted by 25 elements (5 x 5): . This was demonstrated independently by Andrew and Lowe in 1959. The magic angle is annealing the term : ½ (3cos2theta -1), theta = arccos (1/√3) = 54.7°.

Frames

Different frames are usually used when dealing with NMR tensors, we can cite, the laboratory frame (L), the rotor frame (R), the molecular frame (M) and the interaction frame (L) or principal axis system (PAS). It is possible to switch from one frame to another using adequate rotation matrices. While conventional matrices are used in the cartesian coordinate basis, Wigner matrices are the ones used in the spherical coordinates’ basis. Chains of matrices may be applied.

Euler angles

A set of 3 angles Ω {alpha, beta, gamma } (0 < alpha < 2 pi, 0 < beta <pi, 0 < gamma < 2 pi ), that describe the relative orientations of two tensors. By rotating a tensor by alpha, beta, gamma  with respect to another one, their axis systems match together. Different conventions are used to define the rotations. The ZYZ convention consists of three consecutive positive rotations about the three Z, Y and finally Z axis. Euler angles may be determined experimentally by precise modelling of spectra, however, it is a delicate procedure.

Spherical harmonics

They faithfully represent the NMR interactions, s (0) for the isotropic part, p (1) and d (2) for the anisotropies. Here it worth mentioning that the chemical shift anisotropy (CSA) for example, is a ‘d’ interaction, thus, the usefulness of the MAS at b = 54.7°. Spherical harmonics also may be written using cartesian or spherical coordinates.

Here, it is worth introducing spherical tensors (A) designing the space terms and the spherical tensor operators (T) designing the spin terms, they can be faithfully presented using spherical harmonics and are often used to formulate the Hamiltonians corresponding to NMR interactions according to Levitt.

Note again, to rotate a spherical tensor (A) or a spherical tensor operator (T) of a certain rank j, from a frame to another oriented relatively to a set of Euler angles, we use Wigner matrices with similar rank. The rank of the Wigner matrix used should be the same as the rank of the tensor we want to rotate. Levitt has assigned different ranks for space and spin terms and called them ‘rotational signatures. In the original paper, a field rank is also defined.

 It is worth noting that all Hamiltonians that give real physical observables are scalar, their global rank is 0. They are invariant with respect to rotation, thus independent of the selected frame and independent of Earth rotation, this is a consequence of angular momentum conservation principle of Noether.

In a spherical coordinate basis, we can manipulate either space terms [A] which are spherical tensors (ST) or spin terms [T] which are spherical tensor operators (STO). As stated earlier, both ST and STO can be seen as spherical harmonics in analogy with electronic orbitals (sharp, principal, diffuse, and fundamental: s, p, d, f). These manipulations are not a trivial task because of the narrowness of the optimal conditions and the number of parameters to optimize but a wealth of information of high interest may be extracted.

NMR powder spectra

In solid state, when dealing with powders packed in rotors, the orientation of the crystallites is random with respect to B0, this gives an anisotropy that is reflected by the shape of the spectrum. The spectral shape depends on the spin quantum number and on the governing interaction but also on the probability of orientation of the crystallites and their corresponding tensors in the space.

For a spin ½ governed by a symmetric Chemical shift Anisotropy, for tetha = 0° (CSA tensor parallel to B0), there is only one possible orientation of the tensor giving a low intensity. For tetha= 90° (tensor perpendicular to B0), there are many orientations giving a higher intensity. When the asymmetry parameter is different than zero (x ≠ y), the spectrum takes intermediate shapes.

When rotating at the magic angle (54.7°), we observe an envelope of spinning side bands separated by the spinning frequency. Only, the isotropic peak does not change its position whatever the spinning speed, a way to identify it is to record the spectra at different spinning rates.

For a quadrupolar nucleus (I = n/2), (2 I) transitions constitute the powder spectrum. For half integer spins, a spectrum consists of a combination of one intense central transition (CT) and (2I – 1) low intensity satellite transitions spread over the entire spectral width. The central transition of this particular kind of nuclei is broadened by an intense quadrupolar interaction that cannot be truncated with a first order approximation, thus cannot be averaged by the usual (MAS) and exhibits complex shape. Dynamics Angle Spinning (DAS), Double Rotation (DOR), Multiple Quantum Magic Angle Spinning (MQMAS) and Satellite Transitions Magic Angle Spinning (STMAS) are used to obtain high resolution NMR spectra. For integer spins e.g. 14N, 2H etc. (I = 1), a ‘Pake doublet’ shape is usually observed, it corresponds to two transitions between three eigenstates, mainly perturbed by the quadrupolar interaction.