# Structural Disorder and NMR of Quadrupolar Nuclei

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## Abstract

Growing computational power in connection with an increasing accuracy of nuclear magnetic resonance (NMR) measurements requires precise models for fitting NMR spectra. The occurrence of structural disorder in the material causes an inaccuracy of NMR signals fitted with standard methods. This article presents the calculation of distribution of the quadrupolar coupling constant and asymmetry parameter of the electric field gradient (EFG) tensor in solids with local disorder. In this work, the calculation was made directly from the disturbance of the initial electric charge system surrounding the quadrupolar nucleus without using an approach based on EFG tensor invariants.

## 1 Introduction

The precise fitting of powder nuclear magnetic resonance (NMR) spectra of quadrupolar nuclei requires powerful computers. The necessity of integration over all orientation of crystalites gives a spectrum without an analytic representation. In addition to that, the use of the well-known models [1] in connection with a high accuracy of NMR measurements causes a perfect fit not being possible at each attempt [2, 3]. This may be because of many reasons: from some unidentified apparatus errors to an inaccuracy of the physical model. For many compounds, such as glasses with coordination number less or equal to 3 (e.g. boron glasses) [4], nanomaterials [5, 6] and other substances with structural disorder, a similar approach as for powders [1] cannot be used anymore. Random deformations occurring in such materials should be taken into account in high precision measurements.

The basic problem in this situation is to determine a joint distribution of parameters describing quadrupolar interaction. In this work, such a distribution is obtained without using an approach based on electric field gradient (EFG) tensor invariants [7, 8, 9]. This approach can be useful for compounds with structural disorder

## 2 EFG Tensor

*Q*is the nuclear quadrupole moment and \({\varvec{V}}\) is the electric field gradient (EFG) tensor with components in the Cartesian coordinates:

The complete definition of the EFG tensor requires five quantities. Instead of \(V_{ik}\) components, the other five are used: \(V_{zz}^\mathrm{{PAS}}\), asymmetry parameter \(\eta =\left( V_{xx}^\mathrm{{PAS}}-V_{yy}^\mathrm{{PAS}}\right) /V_{zz}^\mathrm{{PAS}}\) and three Euler angles [1, 7] with the convention \(\left| V_{zz}^\mathrm{{PAS}}\right| \geqslant \left| V_{yy}^\mathrm{{PAS}}\right| \geqslant \left| V_{xx}^\mathrm{{PAS}}\right| \). As the splitting of nuclear energy levels does not depend on the Euler angles, the occurrence of powder NMR signals is determined by \(V_{zz}^\mathrm{{PAS}}\) and \(\eta \) only. The superscripts “PAS” can be omitted as long as it will cause no confusion.

## 3 Slightly Deformed Charge System

*N*th charge of a small vector \(\Delta {\varvec{r}}=(\Delta x,\Delta y,\Delta z)\) makes changes of the EFG tensor components (see Fig. 1).

The new charge system is not in PAS and requires re-diagonalization. This fact makes the problem of impact of \(\Delta {\varvec{r}}\) randomization on \(V_{zz}\) and \(\eta \) very complicated, because not only diagonal components of the EFG tensor change but also non-zero off-diagonal elements appear.

*N*th charge makes a small change of the EFG diagonal components:

*N*) will be omitted in \(x_{k}^{(N)}\) and \(r^{(N)}\) for the

*N*th charge. Then, the explicit form of \(A_{ik}\) can be written as

## 4 Distribution \(P_{V}\left( V_{xx},V_{yy}\right) \)

## 5 Distribution \(P\left( V_{zz},\eta \right) \)

*f*has no unambiguous inverse transformation. Therefore, it is necessary to use a general method of densities transformation [13]:

Because direct integration of distribution \(P_{V}\left( V_{xx},V_{yy}\right) \) in the form given in (11) over all six areas is very complicated, let us consider some special cases.

### 5.1 \(V_{zz}^{(0)}=0\)

The case where both \(V_{xx}^{(0)}=0\) and \(V_{yy}^{(0)}=0\) corresponds to a regular structure in which there is no electric gradient in the quadrupolar nucleus. The asymmetry parameter is then undetermined. Because quadrupolar interactions are among the strongest in NMR, this situation is defect sensitive. Moreover, because there are no limitations to dispersion \(\sigma \), the case \(V_{zz}^{(0)}=0\) can be used with isotropic, amorphous solids.

### 5.2 \(V_{zz}^{(0)}\gg 0\)

## 6 NMR Spectra

These spectra were obtained according to the theory presented in [1] for central transition (CT) NMR of quadrupolar nuclei in the magic angle spinning (MAS) regime by taking into account the dispersion (14) calculated in this work.

## 7 Summary

The joint dispersion of electric field gradient tensor components was made directly from the disturbance of the initial electric charge system surrounding the quadrupolar nucleus. In addition to that, the dependence between the structural and the obtained dispersion parameters was calculated, which resulted in the reduction of the number of dispersion parameters. The theory presented in this work can be useful, owing to the analytic form of the obtained distributions, for the modeling of NMR signals in disordered compounds with quadrupolar interactions where parameters are scattered around initial magnitudes. The influence of the obtained theory on NMR signals has been presented in this work with the example central transition MAS NMR spectra for several cases.

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