Single-Crystal NMR Spectroscopy of Spin I = 3: ¹⁰B-NMR of Hambergite, Be₂BO₃OH

Abstract

A monocrystal of the natural mineral hambergite, Be₂BO₃OH, is studied by ¹⁰B-NMR spectroscopy. The nuclide ¹⁰B possesses spin $I=3$, and thus for a single boron site in a periodic solid, the ¹⁰B spectrum is composed of three doublets. For the four inversion-related pairs of boron sites existing in the crystal structure of hambergite, orientation-dependent ¹⁰B spectra are recorded and both the chemical shift and electrical field gradient (EFG) tensor are extracted. The resulting numerical values are in very good agreement with a previous ¹¹B-NMR study of the same sample. Comparing the line widths of the NMR resonances of ¹¹B against ¹⁰B, the concept of inherently better resolution being available from ¹⁰B due to the scaling down of dipolar interactions is confirmed.

1. Introduction

The first solid-state NMR measurements of the nuclide ¹⁰B with spin $I=3$ were reported by Bray and co-workers in the 1970s [1,2,3,4]. However, the preferred isotope for NMR of boron is ¹¹B with spin $I=3/2$, which possesses much higher natural abundance (80.1%, with the remaining 19.9% being ¹⁰B) and a resonance frequency almost triple that of the ¹⁰B isotope. For this reason, the use of ¹¹B-NMR as an analytical tool for solids, also involving P.J. Bray, was pioneered earlier [5,6,7]. Although the spin properties of ¹⁰B look unfavourable at first sight, an argument can be made that quadrupolar interaction parameters can be extracted from spectra of this isotope to higher precision than from ¹¹B [1]. First, dipolar broadening present in many systems adversely affects the line shapes, which are reduced because of the smaller gyromagnetic ratio of ¹⁰B. Second, the inner transitions of spin $I=3$ ($m=0\leftrightarrow \pm 1$) span a much wider spectral range than the central transition of spin $I=3/2$ ($m=+1/2\leftrightarrow -1/2$). Most ¹¹B-NMR studies have restricted themselves to evaluating this transition [8], as its intensity is much stronger than those of the satellite transitions. This advantage in resolution and precision becomes especially significant when investigating systems with a distribution of NMR interaction parameters, such as glasses; in fact, the interest in early ¹⁰B-NMR was mostly motivated by structural studies of borate glasses, and this seems to remain the main motivation in more recent publications [9,10,11], with some more exotic applications also reported [12,13]. In all the work on ¹⁰B published so far, theoretical descriptions and fitting algorithms have been focused on what are in effect polycrystalline samples, which deliver spectra with very broad lines. Invariably, these broad lines lead to loss of spectral resolution, and while this can be partly remedied with advanced methods such as overtone spectroscopy [14], improved resolution may always be obtained from monocrystalline samples [15,16], with the restriction that single crystals of sufficient size must be available.

In this work, the natural mineral hambergite is used as a model system to conduct a ¹⁰B-NMR study of a single crystal. The comparatively rare mineral hambergite, Be₂BO₃OH, appears in beryllium-bearing granite pegmatites as an accessory phase, and forms monocrystals of sufficient size. Single crystals of hambergite have been studied before extensively by ¹H-NMR in order to analyse the properties of the one-dimensional proton chains in the crystal structure [17,18,19]. Characterisation of Be₂BO₃OH by single-crystal ⁹Be- and ¹¹B-NMR, on the other hand, has been performed only very recently by our group [20]. The application of ¹⁰B-NMR to hambergite presented here was carried out on the same crystal specimen with identical orientation of the crystal on the goniometer axis (within experimental error; see below for details). This enables direct comparison between ¹⁰B- and ¹¹B-NMR applied to boron atoms in identical crystallographic environment, allowing us to again test and verify the claims regarding superior accuracy of ¹⁰B-NMR [1] while taking advantage of the superior resolution and defined orientation dependence of single-crystal spectra. Before discussing these results, we briefly review the principles of NMR spectroscopy of spin $I=3$ in the solid state.

2. NMR Spectroscopy of Spin $\mathit{I}=\mathbf{3}$ in the Solid State

Following the notation used in [16], the resonance frequency of a nuclear spin in a solid-state NMR experiment at high magnetic field may be expressed as a sum of the various interactions contributing to the energy levels of a spin system, with ${\nu }_0$ being the Larmor frequency and ${\nu }_{\mathrm{CS}}$, ${\nu }_{\mathrm{DD}}$, and ${\nu }_J$ being the contributions of the chemical shift, the direct dipolar and the indirect coupling, respectively, at some general orientation $\Omega$:

$$\nu (\Omega )={\nu }_0+{\nu }_{\mathrm{CS}}(\Omega )+{\nu }_{\chi }(\Omega )+{\nu }_{\mathrm{DD}}(\Omega )\left[+{\nu }_J(\Omega )\right]$$

(1)

Regarding the dipolar interaction described by ${\nu }_{\mathrm{DD}}$, it splits the respective resonances of an isolated spin-1/2 pair I and S into a doublet with frequencies ${\nu }^{+}$ and ${\nu }^{-}$. These are placed symmetrically around the unaffected resonance position, with the magnitude of the splitting scaled by the distance $r_{IS}$ between the two spins and with $\beta$ being the angle between the vector connecting the two spins and the external magnetic field lines:

$${\nu }_{\mathrm{DD}}^{\pm }(\Omega )=\pm {\displaystyle \frac{{\mu }_0{\gamma }_I{\gamma }_S\hslash }{8{\pi }^2{r}_{IS}^3}}·{\displaystyle \frac{3{cos}^2\beta -1}{2}}$$

(2)

For nuclei with spin $I>1/2$, the dipolar interaction creates more complex multiplets, with each spin contributing $2I+1$ energy levels. Also, in most samples the spins do not occur as isolated pairs but as part of a large coupled network, which for polycrystalline samples leads to severe broadening of the resonance lines. This is one aspect where ¹⁰B-NMR should have an advantage over ¹¹B-NMR, as the gyromagnetic ratio of the former is only about one-third that of that of the latter [21]:

$${\displaystyle \frac{\gamma {(}^{10}\mathrm{B})}{\gamma {(}^{11}\mathrm{B})}}={\displaystyle \frac{2.8747·{10}^7\mathrm{rad}{\mathrm{s}}^{-1}{\mathrm{T}}^{-1}}{8.5847·{10}^7\mathrm{rad}{\mathrm{s}}^{-1}{\mathrm{T}}^{-1}}}=0.3349$$

(3)

Consequently, the dipolar couplings experienced by ¹⁰B and the concomitant line broadening are scaled down by the same factor.

Another important interaction for ¹⁰B-NMR of solids is the quadrupolar interaction, described by the frequency ${\nu }_{\chi }$, which is usually treated as a perturbation to the main Zeeman interaction energy [22]:

$${\nu }_{\chi }(\Omega )={\nu }_{\chi }^{\left(1\right)}+{\nu }_{\chi }^{\left(2\right)}+{\nu }_{\chi }^{\left(3\right)}+\dots$$

(4)

The product of the nuclear property $eQ$ (with Q being the quadrupole moment of the nucleus, $Q{(}^{10}\mathrm{B})=84.59$ mb [23]) and the largest eigenvalue $V_{33}$ of the electrical field gradient (EFG) tensor $\mathbf{V}$ is called the quadrupolar coupling constant $\chi$:

$$\chi =C_{\mathrm{Q}}={\displaystyle \frac{eQ}{h}}{V}_{33}={\displaystyle \frac{eQeq}{h}}$$

(5)

The ‘quadrupolar frequency’ ${\nu }_{\mathrm{Q}}$ [22] is another useful measure of the interaction strength; in contrast to $\chi$, it scales with the magnitude I of the observed spin:

$${\nu }_{\mathrm{Q}}={\displaystyle \frac{3e^{2}qQ}{2I(2I-1)h}}={\displaystyle \frac{3\chi }{2I(2I-1)}}$$

(6)

For ¹⁰B with $I=3$, this results in

$${\nu }_{\mathrm{Q}}{(}^{10}\mathrm{B})={\displaystyle \frac{3\chi }{6(6-1)}}={\displaystyle \frac{\chi }{10}}$$

(7)

We also find it useful to assign a parameter k to each transition in the spin system, such that [16]:

$$k=m\pm {\displaystyle \frac{1}{2}}\mathrm{for}|m\rangle \to |m\pm 1\rangle$$

(8)

With the above definitions, we can now proceed to specify the individual frequencies ${\nu }_{\chi }^{\left(i\right)}$ in Equation (4). With the general orientation $\Omega$ defined by the azimuthal angle $\alpha$ and polar angle $\beta$ of the magnetic field vector in the principal axes system of the quadrupole coupling tensor $\mathbf{Q}$ (see below), the first-order contribution of transition k for a static sample is given by

$${\nu }_{\chi }^{\left(1\right)}\left(k\right)=k{\displaystyle \frac{{\nu }_{\mathrm{Q}}}{2}}\left(3{cos}^2\beta -1+{\eta }_{\mathrm{Q}}cos2\alpha {sin}^2\beta \right),$$

(9)

where ${\eta }_{\mathrm{Q}}$ is the quadrupolar asymmetry parameter calculated from the components of the diagonalised $\mathbf{Q}$-tensor, which are ordered according to $|Q_{33}| \ge |Q_{22}| \ge |Q_{11}|$:

$${\eta }_{\mathrm{Q}}={\displaystyle \frac{Q_{11}-Q_{22}}{Q_{33}}}$$

(10)

In contrast to NMR spectra of nuclei with half-integer spins, where a central transition with $k=0$ exists, all transitions of spin $I=3$ are affected by the first-order contribution. The maximal resonance frequencies are observed for $\beta =0$, such that

$${\nu }_{\chi }^{(1,\max )}\left(k\right)=k·{\nu }_{\mathrm{Q}}=k·{\displaystyle \frac{\chi }{10}}$$

(11)

The properties of the ¹⁰B resonances in a single-crystal spectrum at maximal displacement are listed in

Table 1. NMR transition properties for spin $I=3$ under quadrupolar interaction to the first order (Equation (9)), with the maximal frequencies ${\nu }_{\chi }^{(1,\max )}$ defined by Equation (11).

Transition:			Relative
$\mathit{m}\to \mathit{m}+\mathbf{1}$	$\mathit{k}\left(\mathit{m}\right)$	${\mathit{\nu }}_{\mathit{\chi }}^{(\mathbf{1},\max )}\left(\mathit{k}\right)$	Intensity
$-3\to -2$	$-{\displaystyle \frac{5}{2}}$	$-{\displaystyle \frac{5}{20}}\chi$	3
$-2\to -1$	$-{\displaystyle \frac{3}{2}}$	$-{\displaystyle \frac{3}{20}}\chi$	5
$-1\to 0$	$-{\displaystyle \frac{1}{2}}$	$-{\displaystyle \frac{1}{20}}\chi$	6
$0\to +1$	$+{\displaystyle \frac{1}{2}}$	$+{\displaystyle \frac{1}{20}}\chi$	6
$+1\to +2$	$+{\displaystyle \frac{3}{2}}$	$+{\displaystyle \frac{3}{20}}\chi$	5
$+2\to +3$	$+{\displaystyle \frac{5}{2}}$	$+{\displaystyle \frac{5}{20}}\chi$	3

.

While the NMR spectrum of a nucleus with spin $I=1$ at a single crystallographic site in a monocrystal consists of a doublet, the corresponding spectrum of ¹⁰B with $I=3$ may be viewed as being composed of outer, middle, and inner doublets with relative intensities 3:5:6. If the quadrupolar interaction is sufficiently small to be fully described by first-order contribution only, then all resonances in the symmetric spectrum are spaced evenly, with the spacing for the maximal displacement being ${\nu }_{\mathrm{Q}}=\chi /10$, as shown in the schematic spectrum in Figure 1b. When considering the quadrupole interaction to the first order only, the relevant terms of the resonance frequency in Equation (1) may also be expressed in a compact tensor notation (with $\overrightarrow{b}$ being the unit vector along the magnetic field lines and ${\overrightarrow{b}}^t$ its transpose), which for spin $I=3$ can be written as shown in Equation (12) below.

$$\nu (\Omega )={\nu }_0+{\nu }_{\mathrm{CS}}(\Omega )+{\nu }_{\chi }^{\left(1\right)}(\Omega )={\nu }_0+{\overrightarrow{b}}^t·\mathit{\delta }·\overrightarrow{b}+{\displaystyle \frac{k}{10}}{\overrightarrow{b}}^t·\mathbf{Q}·\overrightarrow{b}$$

(12)

In this notation, $\mathit{\delta }$ is the chemical shift tensor, the scaled trace of which is the isotropic chemical shift ${\delta }_{\mathrm{iso}}$. The quadrupole coupling tensor $\mathbf{Q}$, on the other hand, is traceless, because it is directly related to the EFG tensor $\mathbf{V}$ by $\mathbf{Q}=(eQ/h)\mathbf{V}$:

$$\begin{array}{ccc}{\delta }_{\mathrm{iso}}={\displaystyle \frac{1}{3}}\sum _i{\delta }_{ii}& & \sum _i{Q}_{ii}=0\end{array}$$

(13)

Both interaction tensors are symmetric, the chemical shift tensor $\mathit{\delta }$ by convention [24] and $\mathbf{Q}$ intrinsically because of the mixed partial derivatives in the definition of the EFG tensor [22]. In an arbitrary coordinate system designated as $xyz$, these two tensors have the following general form:

$${\mathit{\delta }}^{xyz}=\left(\begin{array}{ccc}{\delta }_{xx}& {\delta }_{xy}& {\delta }_{xz}\\ {\delta }_{xy}& {\delta }_{yy}& {\delta }_{yz}\\ {\delta }_{xz}& {\delta }_{yz}& {\delta }_{zz}\end{array}\right){\mathbf{Q}}^{xyz}=\left(\begin{array}{ccc}{Q}_{xx}& Q_{xy}& Q_{xz}\\ Q_{xy}& Q_{yy}& Q_{yz}\\ Q_{xz}& Q_{yz}& Q_{zz}\end{array}\right)$$

(14)

In single-crystal NMR experiments, the dependence of the resonance frequencies on the orientation $\Omega$ of the crystal relative to the magnetic field is systematically traced. This is usually done by mounting the crystal on a goniometer mechanics, rotating it stepwise by an angle ${\phi }_i$, and recording a spectrum for each orientation. In our experimental setup, the goniometer axis is oriented perpendicular to the external magnetic field, which is the most common design, with other designs existing [15]. The rotation of the crystal (and the crystal-frame fixed interaction tensors) about the perpendicular axis makes the individual resonances follow harmonic functions of the following type [25]:

$$\nu \left({\phi }_i\right)=A+Bcos2{\phi }_i+Csin2{\phi }_i\left(+Gcos4{\phi }_i+Hsin4{\phi }_i\right)$$

(15)

The higher harmonics evolving with $4\phi$ only show up when quadrupolar coupling effects are so strong that the first-order corrections ${\nu }_{\chi }^{\left(1\right)}$ do not suffice to describe the observed spectrum, and second-order contributions ${\nu }_{\chi }^{\left(2\right)}$ need to be included as well; see Equation (4). For ¹⁰B resonances of a static sample, these second-order contributions may be written as

$${\nu }_{\chi }^{\left(2\right)}\left(k^2\right)=-{\displaystyle \frac{{\chi }^2}{600{\nu }_0}}\left[\left({\displaystyle \frac{45}{4}}-3k^2\right)g(\alpha ,\beta ,{\eta }_{\mathrm{Q}})-6k^{2}f(\alpha ,\beta ,{\eta }_{\mathrm{Q}})\right],$$

(16)

where the functions g and f have terms depending on ${cos}^4\beta$ and ${cos}^2\beta$, with the dependencies on $\alpha$ and ${\eta }_{\mathrm{Q}}$ encapsulated in the coefficients $A^{\left(2\right)},B^{\left(2\right)},\dots$, which are given in full in Appendix A.1.

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill g(\alpha ,\beta ,{\eta }_{\mathrm{Q}})& =A^{\left(2\right)}{cos}^4\beta +B^{\left(2\right)}{cos}^2\beta +C^{\left(2\right)}\hfill \\ \hfill f(\alpha ,\beta ,{\eta }_{\mathrm{Q}})& =D^{\left(2\right)}{cos}^4\beta +E^{\left(2\right)}{cos}^2\beta +F^{\left(2\right)}\hfill \end{array}\end{array}$$

(17)

Within the framework of equations given above, we can now proceed to analyse the ¹⁰B-NMR spectra of a single crystal of hambergite.

3. Results and Discussion

3.1. Boron NMR of Hambergite

To understand boron NMR of hambergite, an important question is the number of expected resonances in a single-crystal spectrum, which of course is intrinsically linked to the crystal structure of Be₂BO₃OH. There have been several diffraction studies of hambergite over the years [26,27,28,29], including a recent one from our group [20]. All studies agree that the mineral crystallises in the orthorhombic and centrosymmetric space group no. 61, with lattice parameters and atomic coordinates always reported in the standard setting Pbca. Figure 2a shows a view of the unit cell of hambergite, where boron occupies a site with a Wyckoff multiplicity of eight. Since the boron atoms are pairwise related by inversion, four magnetically inequivalent pairs of ¹⁰B respectively ¹¹B atoms exist, with their additional symmetry relations summarised in Figure 2b. These four pairs are observable in the corresponding NMR spectra; however, when attempting to relate these NMR signals to atoms in the unit cell, a number of problems arise. For orthorhombic systems such as hambergite, several choices for the setting of the unit cell exist, all being fully equivalent in terms of interpreting the NMR spectra. Also, the problem of assigning NMR resonances to individual atoms in the crystal structure is of fundamental nature [20,30,31], and can only be resolved by resorting to additional information derived from ab initio calculations of the electron density or other orientation-dependent data such as dipolar couplings. For single-crystal NMR of boron in the hambergite structure, these problems and their solutions have been discussed extensively for ¹¹B-NMR in our previous publication [20], and the interested reader is referred to it. In the current work, we use these solutions and assignments for evaluating the ¹⁰B-NMR spectra without deriving them again.

3.2. Determination of the Quadrupole Coupling Tensor

A crystal specimen of hambergite was mounted on a wooden support axis, as shown in Figure 2c. This is the same crystal and for all practical purposes (see below) the same axis orientation as was used for ¹¹B-NMR in the previous work [20]. Recording a complete ¹⁰B-NMR spectrum usually required the acquisition of several spectral windows, with a representative spectrum shown in Figure 3.

Subsequently, a set of spectra was acquired over the rotation angle range of $\phi =0^{\circ }\dots {180}^{\circ }$, in steps of 10 degrees. Plotting the orientation-dependent resonance frequencies over $\phi$ gives the full ¹⁰B-NMR rotation pattern of hambergite, as displayed on the left of Figure 4.

The recorded ¹⁰B multiplets are not fully symmetric around zero, which is apparent from the multiplet spacings given on top of the spectrum shown in Figure 3. This asymmetry is caused by second-order contributions of the quadrupole interaction. The presence of these contributions makes it impossible to use Equation (12) to extract the quadrupole coupling tensor $\mathbf{Q}$. However, they can be removed by taking the differences between the transition frequencies (Equation (1)) of the $\pm k$ doublets, resulting in the following expression:

$$\Delta \nu \left(\phi \right)=\nu (+k)-\nu (-k)=\Delta {\nu }_{\chi }^{\left(1\right)}(\pm k)={\displaystyle \frac{\Delta k}{10}}{\overrightarrow{b}}^t·\mathbf{Q}·\overrightarrow{b}$$

(18)

These differences $\Delta \nu$, which are also called splittings, are plotted on the right of Figure 4. In principle, for the various transitions k listed in Table 1, fit code expressions can be derived from Equation (18), which can be used to extract the quadrupole coupling tensor $\mathbf{Q}$ from the experimental data. However, in many spectra across the rotation pattern, the resonances belonging to the outer transitions with $k=\pm 5/2$ were inadequately defined because of their comparatively low intensities, cf. Figure 1b. Therefore, these transitions were not included in the data fit. To first order, all transitions k are connected by constant factors; thus, only some redundancy is lost when excluding those with $k=\pm 5/2$.

In order to relate the orientation of the magnetic field vector $\overrightarrow{b}$ to the rotation angle $\phi$, the orientation of the goniometer axis $\overrightarrow{g}$ must be known. With sufficient data available from single-crystal NMR experiments, the orientation of $\overrightarrow{g}$ can be fitted from these data as well without resorting to additional crystal alignment techniques [16,33,34,35]. A brief overview of the equations needed for fitting the ¹⁰B-NMR data is given in Appendix A.2. As already mentioned above, for the current study we can take advantage of the previously published characterisation of the same crystal specimen by ¹¹B-NMR [20] and make use of the goniometer axis orientation and tensor assignments derived there. Although the studied crystal remained glued to the goniometer axis rod, the rod itself had been removed from the experimental setup in between the ¹¹B- and ¹⁰B-NMR measurements; thus, a small misalignment in both the direction of $\overrightarrow{g}$ and the zero-angle offset ${\phi }_0$ was expected. In order to allow for correction of these small misadjustments, a ¹¹B rotation pattern was additionally recorded while collecting the ¹⁰B data of Figure 4. Then, the splittings of this ¹¹B rotation pattern (as shown in Appendix A.3) were fitted simultaneously with the ¹⁰B data, with only the goniometer axis direction and ${\phi }_0$ as fit variables, i.e., with the ¹¹B quadrupole coupling tensor elements fixed. The elements of the $\mathbf{Q}$-tensor of ¹⁰B, on the other hand, were treated as free fit variables, resulting in the following tensor for the atom pair B(1,5).

$${\mathbf{Q}}_e^{abc}/\mathrm{kHz}=\left(\begin{array}{ccc}2310\pm 60& 3972\pm 14& 463\pm 12\\ .& 438\pm 12& 288\pm 16\\ .& .& (-2748\pm 72)\end{array}\right)$$

(19)

Transforming the above tensor, which is expressed in the normalised orthorhombic crystal frame $abc$, into its own principal axes system (PAS) leads to a quadrupolar coupling constant of

$$\chi {(}^{10}\mathrm{B})={\displaystyle \frac{e^{2}qQ{(}^{10}\mathrm{B})}{h}}=Q_{33}^{\mathrm{PAS}}=5490\pm 40\mathrm{kHz}$$

(20)

Furthermore, with the elements of the quadrupole coupling tensor related to those of the electric field gradient tensor by $Q_{ij}=(eQ/h)V_{ij}$, the components of the EFG tensor can be calculated and compared to those derived from previous ¹¹B-NMR measurements [20], with the results listed in

Table 2. Eigenvalues and asymmetry parameter of the electrical field gradient (EFG) tensor $\mathbf{V}$ for boron in the crystal structure of hambergite, Be₂BO₃OH, as derived from ¹⁰B- and ¹¹B-NMR measurements of a single crystal. The quadrupole moments used for the conversion from the experimentally determined $\mathbf{Q}$ tensor are $Q{(}^{10}\mathrm{B})=84.59$ mb and $Q{(}^{11}\mathrm{B})=40.59$ mb [23]. For comparison, the values derived from DFT calculations [20] are also listed.

	10B-NMR (This Work)	11B-NMR (Ref. [20])	DFT (VASP) (Ref. [20])
$V_{11}$/(V/Å2)	$-13.05\pm 0.15$	$-12.970\pm 0.029$	$-14.792$
$V_{22}$/(V/Å2)	$-13.79\pm 0.30$	$-14.01\pm 0.08$	$-15.973$
$V_{33}$/(V/Å2)	$26.84\pm 0.20$	$26.98\pm 0.05$	$30.765$
${\eta }_{\mathrm{V}}$	$0.028\pm 0.017$	$0.039\pm 0.005$	$0.038$

.

From comparing the eigenvalues of the EFG tensors, it can be seen that the errors on the EFG tensor components are much larger for ¹⁰B than for ¹¹B. This is chiefly due to the quality of the individual spectra, with ¹⁰B usually having a much worse signal-to-noise ratio (s/n). The usual way to improve the s/n is by acquiring more scans, but this was a limited option for ¹⁰B, with the complete spectrum for one orientation usually consisting of several sub-spectra, each requiring several days of measurement time (see Experimental for further details). Nevertheless, the error ratios of our single-crystal work do not invalidate the claim made by Bray and co-workers [1] about quadrupolar interaction parameters being available from ¹⁰B spectra to higher precision. This argument was made for polycrystalline samples, where the measurement of the larger span of the inner transitions of spin $I=3$ ($m=0\leftrightarrow \pm 1$) was compared to measuring the much smaller broadening of the central transition (CT) of spin $I=3/2$ ($m=+1/2\leftrightarrow -1/2$) under second-order effects. However, the situation is fundamentally different for a single crystal, where the positions of the satellite transitions are evaluated to extract the quadrupole parameters. Since these positions cover a much larger frequency range than the comparatively small changes of the CT, the above argument does not apply.

3.3. Determination of the Chemical Shift Tensor

To determine the chemical shift tensor, the movement of the centres of gravity (CoGs) of the $\pm k$ doublets around the Larmor frequency ${\nu }_0$ are evaluated:

$${\nu }_{\mathrm{CoG}}\left(\phi \right)={\displaystyle \frac{\nu (+k)+\nu (-k)}{2}}-{\nu }_0={\nu }_{\mathrm{CS}}\left(\phi \right)+{\nu }_{\chi }^{\left(2\right)}\left(\phi \right)$$

(21)

Here, the individual frequencies $\nu (\pm k)$ are those given by Equation (1). Since the effect of the dipolar interaction as described by ${\nu }_{\mathrm{DD}}$ leads only to an unspecific broadening of the single-crystal resonance lines, it has been omitted from the above equation; the first-order contributions of the quadrupole interaction with opposing signs $\pm k$ cancel each other, leaving only the chemical shift and the second-order contribution. For illustrative purposes, these data points are plotted for two $k=\pm 1/2$ transitions in Figure 5, together with the second-order contributions calculated from Equation (16) using the $\chi$ value of Equation (20).

The application of Equation (16) requires knowledge of the polar coordinates $\beta ,\alpha$ of the magnetic field vector $\overrightarrow{b}$ in the principal axes system (PAS) of the $\mathbf{Q}$-tensor. However, for this data fit the movement of $\overrightarrow{b}$ is calculated in the crystal frame (CRY), as outlined in Appendix A.2. One possible procedure to derive $\beta ,\alpha$ from the orientation of $\overrightarrow{b}$ in the CRY frame is described in Appendix A.4.

After subtracting ${\nu }_{\chi }^{\left(2\right)}$ from the centres of gravity described by Equation (16), the remaining data points are affected by chemical shift only, as plotted for all $k=\pm 1/2,\pm 3/2$ transitions in Figure 6.

Simultaneous fit of these points according to equations similar to those in Appendix A.2 (but without pre-factor) results in the chemical shift tensor of ¹⁰B listed in

Table 3. Eigenvalues of the chemical shift tensor $\mathit{\delta }$ and the corresponding isotropic chemical shift values for boron in the crystal structure of hambergite, Be₂BO₃OH, as derived from ¹⁰B- and ¹¹B-NMR measurements of a single crystal.

	Source	${\mathit{\delta }}_{11}$/ppm	${\mathit{\delta }}_{22}$/ppm	${\mathit{\delta }}_{33}$/ppm	${\mathit{\delta }}_{\mathbf{iso}}$/ppm
10B	this work	$20\pm 4$	$19\pm 10$	$12\pm 8$	$17\pm 8$
11B	Ref. [20]	$19.7\pm 0.9$	$19.1\pm 0.7$	$15.4\pm 1.4$	$18.1\pm 1.0$

. Comparison to the previously determined values for ¹¹B [20], as listed in the same table, shows that the values are identical for both isotopes within error margins, but that the errors themselves are much larger for ¹⁰B. As discussed for the quadrupole tensor above, the errors are chiefly caused by the poor signal-to-noise ratio of the individual ¹⁰B spectra.

3.4. Comparison of ¹⁰B- and ¹¹B-NMR Line Widths

For the comparatively small residual line width of the boron single-crystal resonances, the heteronuclear dipolar interaction as described by Equation (2) has been shown in our previous study to be the dominant cause for broadening of the spectral lines for ¹¹B [20]. Resolved dipolar doublets due to direct ¹H–¹¹B couplings could be observed for some crystal orientations, with the doublets disappearing and line widths generally decreasing under application of proton decoupling [20]. Consequently, because of the lower gyromagnetic ratio of ¹⁰B, those line widths should be even smaller in the ¹⁰B spectra. Figure 7 shows the full width at half maximum (fwhm) in kHz for both the ¹⁰B and ¹¹B resonances for the B(1,5) pair, acquired for the current paper, across the full respective rotation patterns of the single crystal. It can be seen that some variations of the fwhm show up over the rotation angle because of the orientation dependence of the dipolar interaction. However, averaging the fwhm values over the entire rotation pattern leads to an average ratio of ¹⁰B/¹¹B = 0.410, very close to the theoretical value of $0.335$ predicted by Equation (3). Therefore, our single-crystal results fully validate the claim advanced by Bray and co-workers [1] about superior resolution being available from ¹⁰B spectra as compared to ¹¹B whenever the main broadening mechanism is dipolar coupling.

The situation becomes more complex when distributions of chemical shift or quadrupolar interaction are involved. Such distributions are present in glasses [1,2,3,4,5], but may also occur in single crystals of natural minerals, for example as domain distributions caused by crystal mosaicity [36]. The concomitant distribution of the NMR interaction tensor orientation leads to increased NMR line widths, which may in turn show orientation dependence [37]. If these line widths are expressed in kHz, as in Figure 7, then ¹⁰B again has an advantage over ¹¹B for a chemical shift distribution because of the lower Larmor frequency. For the quadrupole interaction, the maximal displacement observable to first order in both single or polycrystalline sample is described by Equation (11). For ¹⁰B with spin $I=3$, the largest available value for k is $k_{\max }=\pm 5/2$, while for ¹¹B with spin $I=3/2$ it is $k_{\max }=\pm 1$. With the quadrupolar frequency (Equation (6)) of ¹¹B being ${\nu }_{\mathrm{Q}}{(}^{11}\mathrm{B})=\chi /2$, for the two isotopes this gives

$$\begin{array}{ccc}{\nu }_{\chi }^{(1,\max )}{(}^{10}\mathrm{B})=\pm {\displaystyle \frac{1}{4}}\chi & & {\nu }_{\chi }^{(1,\max )}{(}^{11}\mathrm{B})=\pm {\displaystyle \frac{1}{2}}\chi \end{array}$$

(22)

Because $\chi$ for ¹⁰B is about double that of ¹¹B, the line broadening caused by a distribution of the quadrupolar interaction is very similar for the two isotopes. In practice, the outer transitions of ¹⁰B with $k_{\max }=\pm 5/2$ prove difficult to record (we omitted them in our single crystal analysis here), and the signal-to-noise conditions will be even worse for polycrystalline samples; hence, when basing the data analysis on the inner ¹⁰B transitions with $\left|k\right|<5/2$, the maximal possible displacement (cf. Table 1) is again smaller, and in the presence of distributions should give better spectral resolution compared to ¹¹B.

4. Conclusions

¹⁰B-NMR spectroscopy has been applied to a single crystal of the natural mineral hambergite, Be₂BO₃OH. In a solid-state NMR spectrum, three doublets are expected for ¹⁰B with spin $I=3$, and these were observed for the four inversion-related pairs of boron sites in the crystal structure of hambergite. Evaluation of the full rotation pattern (with the goniometer axis orientation known from a previous study [20]) allowed for determination of both the electrical field gradient (EFG) and the chemical shift tensor. The numerical values are in very good agreement with those derived from ¹¹B-NMR spectroscopy of the same sample [20]. The previously suggested notion [1] about inherently better resolution being available from ¹⁰B in comparison to ¹¹B because of the lower gyromagnetic ratio of the former could be confirmed by our single-crystal data.

5. Materials and Methods

NMR spectra were recorded on an Avance-III 500 WB spectrometer (Bruker, Karlsruhe, Germany) at LMU Munich, with a Larmor frequency of ${\nu }_0{(}^{10}\mathrm{B})=53.72$ MHz. The sample crystal was glued onto a wooden axis which was spanned into a goniometer mechanics (built by NMR Service GmbH, Erfurt, Germany), permitting its rotation perpendicular to the external magnetic field. This setup was used with a wide-bore static NMR probe (Bruker), with a solenoid coil placed around the sample. Spectra were referenced against the secondary reference of the ¹H resonance of 1 % Si(CH₃)₄ in CDCl₃. The program Igor Pro [38] was used to extract the NMR interaction tensors from the NMR data, using global fits across several datasets. The obtained tensors in the CRY frame were diagonalised using the online tool Wolfram|Alpha, which is based on the software Mathematica [39]. Eigenvalue errors were estimated according to the method of Nelson [40]. For each crystal orientation, between three and five ¹⁰B spectra with varying transmitter offset had to be acquired in order to cover the entire spectral range. Using a recycle delay of 600 s and acquiring between 400 (for the stronger inner transitions) and 800 (for weaker outer transitions) scans, recording all resonance frequencies for one particular crystal orientation took between one and three weeks. Acquisition parameters for the ¹¹B rotation pattern were identical to those described in [20].