Ambiguities in Assigning Single‐Crystal NMR Data to Individual Atoms in the Crystal Structure: A Case Study of Hambergite, Be2BO3OH, by 9Be and 11B NMR Spectroscopy, XRD Measurements and DFT Calculations
Jennifer Steinadler, Georg Krach, Kristian Witthaut, Tobias Stürzer, Rupert Hochleitner, Wolfgang Schnick, Thomas Bräuniger
TL;DR
This paper uses NMR and DFT calculations to resolve ambiguities in assigning atomic resonances in the crystal structure of hambergite.
Contribution
A new approach combining NMR, XRD, and DFT calculations is presented to resolve resonance ambiguities in crystal structures.
Findings
NMR interaction tensors for 9Be and 11B in hambergite were determined using single-crystal NMR.
DFT calculations and XRD confirmed the assignment of NMR resonances to specific atomic positions.
Quadrupolar coupling constants and isotropic chemical shifts were precisely measured for 9Be and 11B.
Abstract
The NMR interaction tensors of 9Be and 11B of hambergite, Be BO OH, were derived from single‐crystal NMR experiments. In the orthorhombic crystal structure of hambergite (which we redetermined by single‐crystal XRD, confirming the results of previous studies), both beryllium and boron atoms occupy Wyckoff position 8c, with atoms pairwise related by inversion symmetry. This leads to four magnetically independent 9Be and 11B atoms per site, which are observable in the NMR spectra. Unequivocal assignment of these resonances to atomic positions in the unit cell is generally impossible, as an analysis of the symmetry relations shows. For the hambergite system, this assignment ambiguity could be resolved with the help of DFT calculations using the VASP code, with the resulting eigenvectors compared with the experimental ones. Examination of 9Be–1H dipolar coupling effects, which could be…
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FIGURE 5
FIGURE 6
FIGURE 7
FIGURE 8
FIGURE 9
FIGURE 10
FIGURE 11
FIGURE 12| General atomic position | Symmetry operation | NMR‐relevant rotations |
|---|---|---|
| (1) | (1) 1 | |
| (2) | (2) | (180°) |
| (3) | (3) | (180°) |
| (4) | (4) | (180°) |
| (5) | (5) | |
| (6) | (6) | (180°) |
| (7) | (7) | (180°) |
| (8) | (8) | (180°) |
| Space group | Unit cell | Alternative unit cell by rotation | ||
|---|---|---|---|---|
| setting | (180°) | (180°) | (180°) | |
| ( ) | ( ) | ( ) | ||
| ( ) | ( ) | ( ) | ||
| (a) Assignment | (b) Tensor for | ||||
|---|---|---|---|---|---|
| (180°) | (180°) | (180°) | fixed | ||
| violet | |||||
| teal | |||||
| khaki | |||||
| orange | |||||
| (180°) | (180°) | (180°) | ||
| (90°) | (90°), (180°) | (90°), (180°) | (90°), (180°) | |
| Single‐crystal NMR | DFT (VASP) | ||
|---|---|---|---|
| (MHz) |
(V/Å
| ||
| (MHz) |
(V/Å
| ||
| (MHz) |
(V/Å
| 30.765 ( MHz) | |
| 0.038 | |||
| NMR: | DFT: | ||
|---|---|---|---|
| 11 | 74.6°, 307.0° | 75.0°, 307.7° | 0.8° |
| 22 | 164.2°, 321.1° | 164.6°, 320.8° | 0.4° |
| 33 | 86.3°, 38.0° | 86.7°, 38.6° | 0.7° |
| Single‐crystal NMR | DFT (VASP) | ||
|---|---|---|---|
| Be[1] | |||
| (kHz) |
(V/Å
| ||
| (kHz) |
(V/Å
| ||
| (kHz) |
(V/Å
| 2.458 ( kHz) | |
| 0.943 | |||
| Be[2] | |||
| (kHz) |
(V/Å
| 0.205 | |
| (kHz) |
(V/Å
| 0.972 | |
| (kHz) |
(V/Å
| −1.178 ( kHz) | |
| 0.651 | |||
| NMR: | DFT: | ||
|---|---|---|---|
| Be[1] | |||
| 11 | 23.1°, 79.8° | 18.3°, 77.3° | 4.9° |
| 22 | 89.8°, 170.2° | 89.2°, 169.7° | 0.8° |
| 33 | 66.9°, 260.3° | 71.6°, 259.9° | 4.7° |
| Be[2] | |||
| 11 | 113.4°, 332.5° | 109.3°, 337.4° | 6.1° |
| 22 | 155.0°, 130.8° | 157.8°, 126.7° | 3.2° |
| 33 | 81.8°, 58.9° | 79.5°, 63.7° | 5.3° |
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Taxonomy
TopicsAdvanced NMR Techniques and Applications · Crystallography and Radiation Phenomena · Atomic and Subatomic Physics Research
Introduction
1
Hambergite, Be BO OH, is a relatively rare mineral, which appears in beryllium‐bearing granite pegmatites as an accessory phase. Among other things, it has been utilised for tracing the boron isotope composition in igneous rock complexes [1]. In this study from 2020, also the ^11^B NMR spectrum of Be BO OH was reported, from a polycrystalline sample under magic‐angle spinning (MAS) conditions. In a polycrystalline (powder) sample, all possible orientations are statistically occupied by the small crystallites, leading to a broad NMR line shape which reflects the anisotropic nature of the contributing NMR interactions. The purpose of the MAS technique is to average (or at least scale down) these anisotropies [2, 3]. The situation is different if the NMR sample is a single crystal: here, the resonance lines are usually well resolved, but their position in the spectrum is now strongly dependent on the actual orientation of the crystal to the magnetic field lines. By varying this orientation systematically, full NMR interaction tensors may be determined from single‐crystal measurements [4, 5], generally to higher precision than is available from either static [6] or spinning [7] powder samples. In contrast to single‐crystal experiments, spatial information, that is, orientation of tensor eigenvectors in the crystal lattice, is not available from powder samples, because all orientations are measured simultaneously. For hambergite, single‐crystal NMR has been applied extensively to study the properties of the one‐dimensional zig‐zag chain of protons in the crystal structure, by recording and analysing the ^1^H spectra [8, 9, 10]. Yet, to the best of our knowledge, single crystals of Be BO OH have not been subjected to either ^9^Be or ^11^B NMR, which is the purpose of the current work.
Both beryllium and boron atoms occupy sites with a Wyckoff multiplicity of 8 in the orthorhombic unit cell of hambergite, with the eight symmetry‐generated atoms being pairwise related by inversion. Therefore, four magnetically inequivalent ^9^Be and ^11^B exist per site and are observable in the corresponding NMR spectra. This makes hambergite a good model system to explore the general dilemma of how to assign individual NMR signals to specific atoms in a set of symmetry‐related atomic positions. While this problem has been identified and briefly discussed before [11, 12], it does not seem to have been systematically examined. In principle, for n atoms with spin I, which are symmetry‐related and magnetically inequivalent, n×2I different NMR signals can be assigned to a particular atom, with all of these assignments leading to mathematically valid data fit solutions. The consequence is that the orientation of tensor eigenvectors cannot be unequivocally assigned to a specific atom within the Wyckoff set. This is a fundamental conundrum, but for the hambergite system studied in this work, it could be resolved by comparing experimental results to the outcome of ab initio calculations and by analysing dipolar coupling effects of ^9^Be–^1^H, which happen to be present in some of the spectra, and provide additional spatial information.
Crystal Structure of Hambergite, Be
BO OH
2
Structure Determination by Diffraction Experiments
2.1
The crystal structure of hambergite has been investigated by X‐ray diffraction experiments, with the first study dating back to 1931 [13, 14, 15] and recently, also by neutron diffraction [16]. Every study found the mineral to crystallise in the orthorhombic and centrosymmetric space group no. 61, and the crystallographic data of all studies were reported in the standard setting of the unit cell, namely, Pbca, following the recommendations outlined in the ‘International Tables for Crystallography’ [17]. Since the details of the hambergite crystal structure are paramount to the interpretation of our NMR results, we decided to redetermine it by XRD from a small single crystal of our sample, which originates from Sahatany Valley/Madagascar (Mineralogical State Collection Munich). Lattice parameters and atomic coordinates of our study resemble the previously reported results [13, 14, 15, 16] very closely and will be used as the reference structure throughout this paper. Further details on the structure determination (Table S1) and a full list of atomic coordinates (Table S2) are given in the Supporting Information. A schematic view of the unit cell of hambergite is provided by Figure 1b. The atoms relevant for ^11^B and ^9^Be NMR spectroscopy both reside on Wyckoff positions 8c, with only a single boron site, B[1], but two beryllium sites, Be[1] and Be[2], present in the crystal structure. To facilitate discussion of the NMR results, the symmetry relations generating the eight atoms of position 8c in the unit cell are listed in full in Table 1. As for the immediate coordination of the nuclei of interest, ^11^B is located in the centres of trigonal‐planar BO units, whereas ^9^Be is tetrahedrally surrounded by oxygen, as depicted in Figure 1c.
(a) Single crystal of hambergite, Be BO OH, from Sahatany Valley/Madagascar; (b) Unit cell of hambergite in the standard setting Pbca, space group no. 61. The beryllium and boron atoms occupy Wyckoff positions 8c and are connected by inversion centres and glide planes (shown are b in 14,b,c and 34,b,c as well as c in a,14,c and a,34,c); (c) Tetrahedral coordination environments of Be[1] and Be[2], which are bridged by an OH group, and the trigonal‐planar BO unit (representation generated with the vesta program [18]).
TABLE 1: General atomic positions of the atoms on Wyckoff position 8c in the space group Pbca, which apply for both 11B and 9Be, and the symmetry operations relating them in the crystal lattice. Also given are the reduced variants relevant for the NMR interactions. (For the first two columns, the notation according to the ITC‐A [17] is retained using x,y,z instead of a,b,c for the fractional coordinates, to differentiate them from the description of the glide planes with a,b,c.)
The last column of Table 1 shows the reduction of the crystal symmetry elements to those relevant for NMR spectroscopy, that is, without the translational and/or inversion components. Thus, screw axes reduce to simple rotations, with the NMR identity operation denoted by e instead of 1.
Crystal Structure: Ambiguities in Space Group Settings
2.2
For interpretation of our NMR results, it is necessary to discuss the matter of space group setting, which is well known in crystallographic research [17], but usually of little consequence. For characterising the NMR interaction tensors in a periodic crystal structure, however, it turns out to be an important aspect, as has been noted before [11, 12]. The basic principle of changing the setting of a space group is illustrated in Figure 2. Here, a fixed arrangement of atoms in space is assumed, with an associated reference frame XYZ. Into this structure, unit cell no. 1 with the axes abc (with the respective lengths l1,l2 and l3) is inscribed; see upper left of Figure 2. Let this unit cell definition correspond to the standard setting of the space group, Pbca. Two elementary transformations, a 90° rotation about Z, designated as Z(90°) here, and a 180° rotation about Y, that is, Y(180°), generate the new setting of unit cell no. 2, where the axes a′b′c′ now point in different directions.
Two different unit cell settings in an orthorhombic crystal structure (left) and the symmetry operations transforming setting 1 into 2 (right); see text for details.
This new coordinate system a′b′c′ actually belongs to space group Pcab, with the alternative axes setting (b,a,−c). The new setting is fully equivalent to the original setting Pbca, in the sense that it allows to describe the atomic structure, and the action of the symmetry elements contained in it, with the same accuracy. Overall, there are six of such equivalent unit cell settings, as listed in Table 2. This will have consequences for the evaluation of the NMR data, as described in the following. The third column of Table 2 lists additional valid variants of the unit cell settings, generated by 180° rotations about the indicated axis. These rotations do not change the relative positions of the symmetry elements in the unit cell and are hence of no crystallographic interest, but since the definition of atomic positions is affected by them, they become important when assigning NMR tensors to specific atoms, as described below.
TABLE 2: Possible settings of space group no. 61. Listed are the standard setting Pbca together with the five equivalent settings and the corresponding configuration of the unit cell axes (v1,v2,v3). Also given are alternative unit cell definitions, generated by a 180° rotation about the specified axis vi, which do not have an influence on the relative positions of symmetry elements, but only on the definition of distinct atomic positions in the unit cell.
Principles of Single‐Crystal NMR Spectroscopy of Hambergite
3
Determination of NMR Interaction Tensors
3.1
To understand the NMR spectra of hambergite, the question of how many resonances are expected for a given nuclide needs to be addressed. Regarding ^11^B NMR, there is a single boron site in the unit cell (cf. Table S2), designated B[1], but since there is only one, the label [1] may be dropped. The eight symmetry‐related boron atoms at Wyckoff position 8c are designated according to number of the generating symmetry operation, as listed in Table 1, that is, B(1) and B(2) etc. These eight boron atoms are pairwise related by inversion symmetry: B(1) with B(5), B(2) with B(6) and so forth. Their associated interaction tensors are linked by the same inversion operation and are therefore pairwise magnetically equivalent [19], giving identical ^11^B resonance positions for any crystal orientation. In the context of interpreting the NMR spectra, the symmetry elements connecting the four boron pairs and their respective interaction tensors can be described by 180° rotations around the crystallographic axes abc, as listed in the last column of Table 1, and schematically shown in Figure 3. Thus, in principle, the NMR spectrum of ^11^B (with spin I=3/2) of a hambergite single crystal should show four signal triplets, each composed of a central transition (CT) and two satellite transitions (STs), for each of the magnetically inequivalent boron pairs. The three representative ^11^B spectra displayed on the right hand side of Figure 4 indeed show the expected four pairs of STs resolved, whereas the CTs all overlap for these crystal orientations.
The NMR‐relevant symmetry relations between four pairs of boron atoms on Wyckoff position 8c (see also Table 1) in the orthorhombic unit cell of hambergite (left) and the resulting sign changes in the symmetric NMR interaction tensors (with the redundant symmetric off‐diagonal elements abbreviated by dots) associated with them (right). Association of the tensor Te with boron pair (1,5) is an arbitrary assignment, which will however be kept consistently throughout this paper.
Right: 11B ( I=3/2) spectra of hambergite, Be BO OH, at the indicated rotation angles. The central transitions (CTs) of the four magnetically inequivalent boron pairs in the unit cell overlap, whereas the satellite transitions (STs) are well resolved. Left: Full rotation pattern of the four ST doublets (no CTs) around goniometer axis g→1, which has the orientation θg=(39.72±0.05)°, ϕg=(173.51±0.07)° and an offset angle of φ0=(93.44±0.08)° (see text for details). The lines represent harmonic functions of the type described by Equation (8), with the data belonging to the four magnetically inequivalent boron pairs colour‐coded in violet, teal, khaki and orange.
In solid‐state NMR, the orientation dependence of an observed resonance is customarily expressed using second‐rank tensors. For hambergite, we have determined both the chemical shift tensor δ and the quadrupole coupling tensor Q for the nuclides ^9^Be and ^11^B. In an arbitrary coordinate system, designated xyz, these tensors have the following general form:
The scaled trace of δ is the isotropic chemical shift δiso, as given below. Since the quadrupole coupling tensor Q is directly related to the electrical field gradient (EFG) tensor V by Q=(eQ/h)V (with Q being the nuclear quadrupole moment), Q is traceless [20]:
Whereas the chemical shift tensor δ is symmetric only by convention [21, 22], Q is intrinsically symmetric because of the presence of mixed partial derivatives in the definition of the EFG tensor [20].
In solid‐state NMR spectroscopy, the measured resonance frequency of the observed nuclide depends on the orientation of the relevant interaction tensors relative to the external magnetic field, described by the unit vector b→. For a given orientation Ω, this frequency ν(Ω) may be written as
Here, ν0 is the Larmor frequency, and νχ(1) and νχ(2) are the contributions of the quadrupolar interaction to first and second order, respectively (in the rare cases where νχ(1)≪ν0 does not apply, higher order corrections may have need to be included as well). The effect of the dipolar coupling contribution νDD is detectable only in the ^9^Be spectra for some select orientations of the hambergite crystal and will be discussed in detail below. In the high‐field approximation, the frequency contributions of the chemical shift νCS, and the quadrupolar interaction to first order νχ(1), can be calculated directly from the interaction tensors δ and Q (with b→t being the transposed magnetic field vector):
Here, I is the nuclear spin, and k is a parameter associated with the transitions between spin states with magnetic quantum number m in the following way [5]:
Equation (4) uses the fact that the orientation dependence Ω of the frequency νT, described by a general interaction tensor T, may be computed by a simple vector‐tensor‐vector product [23]:
Here, we have introduced the abbreviations K=Txx and so forth for a more compact representation. For Equation (6) to work, the unit vector b→ and the tensor T need to be defined in the same coordinate system. The orientation dependence Ω of the respective frequency contributions is now encased in the orientation of b→ in the chosen frame (either as Cartesian or spherical coordinates):
To systematically follow the orientation dependence of the resonance frequencies, a single crystal is mounted on a goniometer axis and rotated stepwise about it. The actual crystal of hambergite used in this study is depicted in Figure 1a. In most experimental set‐ups (as in ours), the goniometer axis is orientated perpendicular to the external magnetic field. In any other but the laboratory frame (which is defined by having its z‐axis parallel to the magnetic field lines), this leads to a stepwise motion of b→ in a plane perpendicular to the goniometer axis vector g→ by a defined rotation angle φ. The spectra for each orientation φi differ markedly from each other, as may be seen from the ^11^B spectra of hambergite shown in Figure 4. Tracing the resonance positions over a 180° interval, a rotation pattern is generated (see Figure 4), where the individual resonances νm follow harmonic functions of the type [24]:
The higher harmonics evolving with 4φi only show up when second‐order quadrupolar contributions (i.e., the νχ(2) of Equation 3) are present in the system. To generate a fit function for the experimental data of the rotation pattern, the in‐plane movement of b→ around the goniometer axis g→ may be described using two auxiliary vectors u→ and v→. These vectors are constructed such as to be perpendicular to g→ and to each other. Additionally, an offset angle φ0 is introduced to account for the incidental zero angle of the experimental data:
To define u→ and v→, a reference vector (arbitrary, but non‐parallel to g→) needs to be chosen. When performing the calculations in the normalised orthorhombic crystal frame abc, the crystallographic c axis may, for example, be chosen, that is, c→=(001). Then (with θg being the polar angle of g→ in the crystal frame), the auxiliary vectors are given by
If the frequency contribution νT is due to the interaction described by the tensor T, application of Equation (6) leads to the following expression describing the frequency evolution over the rotation angle:
In principle, the above expression, in combination with Equations (9) and (10), may now be used to fit the experimental data and to extract the tensor components K,L,M,… from this fit. Obviously, the balance of unknown and independent tensor components tidp (six for δ, five for the traceless Q) versus the experimentally available parameters pexp (the A,B,C coefficients in Equation 8) also needs to be considered. The standard procedure to increase pexp is to acquire additional rotation patterns r using different goniometer axis orientations g→i, ideally perpendicular to each other [24]. Since the harmonics acquired for different g→i are linked by one constraint [25], the overall balance is given by [5]:
In addition, for the data fit according to Equation (11) to work, we need to know the exact orientation of the goniometer axis g→ in the chosen frame. This information can be gained from XRD experiments or optical alignment procedures. It is however also possible to derive the orientation from the NMR data directly. This approach makes use of the symmetry‐related, but magnetically inequivalent atoms with their associated tensors in the unit cell. First of all, this reduces the number of necessary rotation patterns around different g→i, which is the so‐called single rotation method [25, 26, 27]. Thus, the harmonic functions plotted in Figure 4, which are caused by the mw=4 magnetically inequivalent boron atoms in the unit cell, may be alternatively viewed as rotation patterns recorded around four different goniometer axes, which are related by the same symmetry operations as the atoms. With r=4, the variable balance of Equation (12) delivers pexp=9, and for either chemical shift ( tidp=6) or quadrupole coupling tensor ( tidp=5), the fit equation system is overdetermined. This allows to also treat the goniometer axis orientation ( θg,ϕg) and the starting point of the rotation pattern ( φ0) as three additional variables of the data fit. With the observed nuclide occupying w crystallographic sites with respective magnetic multiplicities mw, the parameter balance for this strategy (the so‐called minimum rotation method) is given by [5]
The simultaneous extraction of both the interaction tensors Ti and the goniometer vectors g→i from experimental rotation patterns has been successfully applied to several systems [28, 29, 30]. However, there exists a fundamental uncertainty when assigning NMR signals to symmetry‐related atoms in the unit cell, and these uncertainties multiply when also the goniometer axis direction is determined from the NMR data. The hambergite system examined in the current paper is well suited to discuss these problems, as will be done in the following.
Ambiguities in Assigning Tensors to Atoms
3.2
As discussed above, the four magnetically inequivalent boron pairs in the crystal structure of hambergite give rise to four triplets in the single‐crystal NMR spectra, which can be systematically traced, as plotted in Figure 4. To extract the interaction tensors, these four triplets need to be assigned to the boron pairs. If we attribute, say, the harmonics plotted in violet to the boron pair B(1, 5), which is associated to the tensor Te (see Figure 3), then the symmetry‐related tensors Ta,Tb and Tc need to be assigned to the harmonics coloured in teal, khaki and orange, respectively; otherwise, the data fit will not converge. This assignment is represented by the first column in Table 3a.
**TABLE 3: (a) Possible assignments of the symmetry‐generated interaction tensors Ti (associated with the boron atom pairs (n,m) according to Figure 3) to the colour‐coded harmonics plotted in Figure 4. Each column represents a mathematically valid solution with identical goodness of fit. For a fixed orientation of the goniometer axis g→, the data fit will always deliver the same numerical tensor for the harmonics it is assigned to, as shown in column (b). The pattern formed by the interaction tensors Ti in part (a) constitutes the character
The following three columns show that the ‘original’ tensor Te may also be assigned to harmonics of other colour, with the other three tensors still related to Te by rotational symmetry. That is, irrespective of where the ‘origin’ with tensor Te is placed, the data fit will converge, and for a fixed goniometer axis orientation, deliver the same numerical tensor for a given harmonic function, for example, always tensor Tv for the data set plotted in violet; see the first row in Table 3b. In essence, Table 3 demonstrates that the boron pair B(1, 5) may be assigned to each of the four different harmonics in the rotation pattern, always resulting in valid fit solutions with identical goodness‐of‐fit parameters, but delivering different tensors, Tv…To. Because of the fixed symmetry relations between these tensors, the assignment problem affects only the sign pattern of the resulting tensors and not the absolute value of the tensor components K,L,M…, as may be also seen from Figure 3. However, when calculating the eigenvectors for, say, the tensor Te, it is actually not clear to which boron pair they need to be assigned to, leaving four possible options to orient these eigenvectors in the crystal frame of hambergite. This assignment ambiguity is a general and intrinsic problem when matching single‐crystal NMR data to symmetry‐related atoms in the unit cell of the investigated system, even when the goniometer axis orientation has been determined by methods other than NMR. It cannot be resolved without taking additional information into account, such as results of DFT calculations or the orientation dependence of dipolar couplings, which connects the crystal orientation to the external magnetic field. For the hambergite system, we have made use of both these options, as described below.
When attempting to also determine the orientation of the goniometer axis g→ from NMR data, the assignment problems actually become much more severe. With the components of the g→ vector now also being variables of the data fit, the fit algorithm finds equivalent solutions for symmetry‐related tensors, with the associated g→ orientations generated by the same symmetry operation(s), as illustrated by the examples listed in Table 4.
**TABLE 4: Examples of interaction tensors Ti generated by 90° and/or 180° rotations about the indicated crystal frame axes. Also listed are the associated goniometer axis vectors g→i, which have been subjected to the identical rotations. To keep a one‐to‐one connection between Ti and g→i, an assignment change of the harmonics is usually also required. (The full
The Ti listed in the first row of this table are actually identical to those in the first row of Table 3a, that is, the tensors connected by rotations around the crystallographic axes. However, since the goniometer vector g→ is now subjected to the same symmetry operation(s), the numerical fit results would be different from those given in Table 3b. This results in additional permutations of the possible data assignment, as the numerical tensor associated with, for example, the violet harmonic function can not only adopt the numerical form Tv but any of the forms given in the first row of Table 4, with the goniometer orientation changed correspondingly. In fact, there exist many more such possibilities, not just involving the 180° rotations due to crystal symmetry but also 90° rotations, with one such example given in the second row of Table 4. These 90° rotations are connected to the ambiguity inherent in defining the space group setting, as summarised in Table 2. In fact, the consecutive operations c(90°), b(180°) (second row, third column in Table 4) describe the transformation necessary to go from setting Pbca to Pcab, as displayed in Figure 2. When 90° rotations are involved, an accompanying change of assignment of the harmonics to the tensor sequence is usually also required for the fit code to work properly. From considering the ambiguities inherent in assigning (i) the original tensor Te to a harmonic function (Table 3) and (ii) choosing one of the many goniometer axis orientations made possible by different set‐ups of the space group (Table 4), we arrive at 4×24=96 different but mathematically valid solutions. In the following, the methods which have been employed to narrow down these 96 possibilities to a single, credible solution are summarised.
Resolving the Assignment Ambiguities
3.3
The following methods have been employed to resolve the ambiguities inherent in assigning NMR interaction tensors to atomic positions:
- The orientation of the hambergite single crystal on its goniometer axis was determined by XRD experiments, with the specifics of this procedure described in the Supporting Information. The goniometer axes orientations established by this method deviated from the respective goniometer axes calculated from NMR data by only 1.6° for ^11^B and 1.7° for ^9^Be, lending strong credibility to the chosen solution.
- The electron density in the unit cell of hambergite was computed by density functional theory (DFT) using the VASP code, as detailed in the Supporting Information. For the energy‐optimised structure (see Table S3 for the resulting atomic positions), the electric field gradient tensors were calculated and the orientations of their eigenvectors compared with the NMR results. This further validated the chosen solution, since the deviations of the NMR eigenvectors compared with EFG were ≤6° for ^9^Be and <1° for ^11^B.
- Analysis of the orientation dependence of ^9^Be–^1^H dipolar couplings, which are observable in some ^9^Be spectra, is quantitatively consistent with the chosen goniometer axes orientations and tensor assignments, as will be outlined below.
Results and Discussion
4
11B NMR Spectroscopy
4.1
The overwhelming majority of NMR spectra of boron‐containing compounds is recorded by observing the nuclide ^11^B [32, 33, 34, 35], which has a natural abundance of 80.1%, and a gyromagnetic ratio of 8.585×107 rad s^–1^ T^–1^, translating to a Larmor frequency of 32.08 MHz per 100 MHz for ^1^H [36]. Although ^11^B has a spin of I=3/2 and thus a nuclear quadrupole moment, it is comparatively small at Q=40.59 mb [37]. The full ^11^B rotation pattern of our hambergite single crystal, acquired with goniometer axis g→1, is plotted in Figure 4, with the orientation of g→1 given in the figure caption. Shown are the evolution of the four ST doublets belonging to the four magnetically inequivalent boron atoms in the unit cell, whereas the overlapping CTs are left out. For ^11^B NMR of hambergite, the deviations of the ST resonances from ν0 are governed by νCS,νχ(1) and νχ(2); see Equation (3). This complexity can be greatly reduced by taking the differences of the satellite positions (also called splittings), Δν±k=ν+k(1)−ν−k(1), which are free of the influence of the chemical shift and second‐order quadrupole effects. For I=3/2 and k=±1, that is, Δk=2, this leads to (see Equation 4):
These Δν are plotted on the left of Figure 5 and, with the b→ vector rotated stepwise around g→1, can be fitted using expressions derived from Equation (11). To improve numerical quality, this fit was combined with a second rotation pattern, which was recorded previously around an axis g→2, with the data shown in Figure S1. Fitting these two rotation patterns simultaneously gave the following quadrupole coupling tensor for ^11^B in the crystal frame of hambergite:
Left: Plot of the ST splittings Δν of the single‐crystal 11B spectra shown in Figure 4, in identical colour coding. The lines represent the global data fit of the quadrupole coupling tensor, with Qe assigned to the harmonic function plotted in violet. The numerical results are given in Equation (15). Right: Orientation dependence of the ST centres (Equation 16), with the contribution of the second‐order quadrupole effect (being in the range of ≤4 kHz) subtracted. The shown fit corresponds to the chemical shift tensor given in Equation (18).
Diagonalising the above quadrupole coupling tensor gives the eigenvalues as listed in Table 5. The quadrupole coupling constant χ=Q33=2.648±0.004 MHz for the trigonally coordinated boron in the hambergite structure (see Figure 1c) falls into the expected range [32]. The corresponding eigenvectors of Qe are listed in Table 6. For comparison, the eigenvalues and eigenvectors of the EFG tensor resulting from our DFT calculations are also shown in Tables 5 and 6.
TABLE 5: Eigenvalues of the Qe tensor for 11B in hambergite (Equation 15), as determined from single‐crystal NMR experiments, and those of the EFG tensor V obtained from DFT calculations. The asymmetry parameter ηT of the respective tensors is defined as ηT=(T11−T22)/T33.
TABLE 6: Eigenvectors of the Qe tensor for 11B in the crystal frame abc of hambergite, as determined from single‐crystal NMR experiments, and those of the EFG tensor V obtained from DFT calculations. NMR and DFT vectors are almost colinear (with the deviation angle listed in the last column), strongly supporting the assignment of tensor Qe to the B(1,5) atom pair in the hambergite unit cell. (Note that the vector q→22 has been inverted to align it with the DFT result.)
As discussed in Section 3, the assignment of the eigenvectors obtained from NMR experiments to one of the four magnetically equivalent boron atoms in the unit cell is ambiguous and rests on the comparison to the vectors calculated by DFT. As may be seen from the last column of Table 6, the deviations between NMR and DFT for the individual vectors spanning the principal axes system are below 1°, strongly supporting the assignment of Qe to the B(1,5) atom pair in the unit cell.
The direction of the eigenvectors are visualised in Figure 6a. It can be seen that eigenvector q→33 is almost perfectly perpendicular to the plane defined by the trigonal BO unit. Furthermore, the larger eigenvectors are always pointing along a ‘void’ in the crystal structure, that is, in the direction where the largest electric field gradient would be expected (these voids are visible only in the more extensive depictions of the crystal structure shown in Figure S2). Overall, the vector directions ‘make chemical sense’, lending additional credibility to the assignment made.
Orientation of the Q tensor eigenvectors (blue arrows) in their respective structural motifs in the unit cell of hambergite, Be BO OH, for the following crystallographic sites: (a) B[1], with 1 Å =^ 1 MHz; (b) Be[1] with 1 Å representing 10 kHz for q→11 and 100 kHz for q→22 and q→33; and (c) Be[2] with 1 Å =^ 30 kHz.
To determine the chemical shift tensor, we need to rely on the satellite transitions, since in the ^11^B spectra of our hambergite crystal, all central transitions overlap; see Figure 4. Therefore, the centres of the STs are tracked [34], and the following quantity is evaluated:
To isolate the chemical shift distribution, the second‐order quadrupole contribution νχ(2) needs to be subtracted, but since the full Q tensor is known, this can be accomplished. Detailed description of this procedure may be found in References [38, 39]. The data derived from this subtraction are plotted together with their fit on the right of Figure 5, with the eigenvalues δii of the chemical shift tensor given below. To discuss these results, it is helpful to make use of the following conventions, which order the δii according to their magnitude and introduce the span as a measure of the observable width of the corresponding powder pattern [40]:
Thus, the δ tensor of ^11^B in the hambergite structure is described by the following numbers:
Obviously, the data points in Figure 5 show appreciable scatter, resulting in comparatively large error values on the tensor components, but it needs to be remembered that the overall variation of the chemical shift observed over the entire the rotation pattern is of the order of ≈5 ppm, which has been extracted from resonance lines with a full‐width‐half‐maximum (fwhm) of about 30 ppm. The chemical shift anisotropy of ^11^B in hambergite, as characterised by the span of 4.3±2.3 ppm, is similar to that observed for boron atoms in trigonal coordination in the natural minerals borax (Na B O H O, span 5.4 ppm) and colemanite (CaB O (OH) H O, span 8.6 ppm) [34]. Likewise, the determined isotropic shift value of 18.1±1.0 ppm for hambergite is well within the range expected for trigonally coordinated boron [33, 34, 35], and in very good agreement with the value determined by us from the MAS spectrum of a powder sample, δiso≈18.3 ppm, as discussed below.
9Be NMR Spectroscopy
4.2
In principle, ^9^Be is a nucleus that is well suited for NMR spectroscopy, as it occurs with 100% natural abundance, and has a gyromagnetic ratio of −3.760×107 rad s^–1^T^–1^, which translates to a Larmor frequency of 14.05 MHz per 100 MHz for ^1^H [36]. While ^9^Be, which has spin I=3/2, possesses a nuclear quadrupole moment, it is moderate with Q=52.88 mb [37]. In spite of these favourable properties, relatively few ^9^Be NMR studies can be found in the literature [41], which is probably due to safety concerns over handling of beryllium [42]. Whereas most existing NMR studies on ^9^Be were conducted on polycrystalline samples under MAS conditions [43, 44, 45, 46, 47, 48, 49], a few single‐crystal studies have been published as well, notably on the natural minerals beryl [50] and chrysoberyl [51].
Evaluation of the ^9^Be rotation patterns shown in Figure 7 has been carried out using the same strategy described above for the ^11^B data, with the chief difference being that two crystallographic 8c sites are occupied by beryllium in the hambergite crystal structure (see Table 1), as opposed to a single site for boron.
Right: 9Be ( I=3/2) spectra of hambergite, Be BO OH, at the indicated rotation angles. The central transitions (CTs) of the two sets of four magnetically inequivalent 9Be pairs overlap, with the satellite transitions (STs) largely resolved. Left: Full rotation pattern of the eight ST doublets (no CTs) around goniometer axis g→1′, which has the orientation θg′=(39.29±0.13)°, ϕg′=(172.39±0.08)° and an offset angle of φ0′=(83.94±0.07)°; see text for details.
In consequence, two sets of four magnetically equivalent pairs of ^9^Be are present, so that a maximum of 2×4×2I=24 resonance lines may show up in the single‐crystal spectrum. However, analogous to the ^11^B spectra, the CTs always overlap, leaving a maximum of 16 ST positions for the analysis. Since the hambergite crystal (glued on a support rod) had been removed between the ^11^B and ^9^Be measurements, and could only be approximately aligned visually, the orientation of the goniometer axis g→1′ and the offset angle φ0′ differ slightly from the values used for ^11^B, as comparison of the actual values in the respective figure captions shows.
The quadrupole coupling tensors of ^9^Be were derived from fitting the splittings of the STs, as plotted in Figure 8, resulting in
Plot of the ST splittings Δν of the single‐crystal 9Be spectra shown in Figure 7, with the two sets of four magnetically equivalent beryllium pairs separated. Left: Be[1], with the harmonics plotted in violet assigned to atom pair Be(1, 5) with the tensor Qe, teal to Be(4, 8) with Qa, khaki to Be(3, 7) with Qb and orange to Be(2, 6) with Qc. Right: Be[2], with blue to Be(1, 5) with Qe, red to Be(4, 8) with Qa, black to Be(3, 7) with Qb and green to Be(2, 6) with Qc. The lines represent the data fit of the two Q tensors given in Equations (19) and (20).
Diagonalising the above Qe tensors gives the eigenvalues listed in Table 7. For the two beryllium sites in hambergite, the magnitudes of the observed quadrupole coupling constants, χ[1]=222.6±0.6 kHz and χ[2]=−121.2±0.4 kHz, are well within the expected range for tetrahedrally coordinated ^9^Be [41].
**TABLE 7: Eigenvalues of the Qe tensors for 9Be for the two beryllium sites in hambergite, as derived from single‐crystal NMR experiments, and those of the EFG tensor V obtained from DFT calculations. The sign of χ[2] has been made to conform to the DFT result, with the absolute sign not being available from experimental NMR data. The definition of the asymmetry parameter η is given in the caption of
The corresponding eigenvectors of the two Qe's are listed in Table 8, with the eigenvalues and eigenvectors of the DFT‐calculated EFG tensors also given in Tables 7 and 8 for comparison.
TABLE 8: Eigenvectors of the two Qe tensors for 9Be (Equations 19 and 20) in the crystal frame abc of hambergite, as determined from single‐crystal NMR experiments, and those of the EFG tensor V obtained from DFT calculations. Also listed are the divergence angles between the respective eigenvectors belonging to Qe and V. (For Be[2], the vectors q→11 and q→22 have been inverted to align them with the DFT result.)
Although the deviation between the experimental and calculated eigenvectors for ^9^Be is somewhat larger than those observed for ^11^B, the assignment of the Qe tensors to the first atom position for each beryllium site, that is, Be1 and Be2 is still convincing. Visualising the direction of these vectors in the crystal structure lends further support to this assignment, with the orientation of some eigenvectors following ‘chemical intuition’: q→33 of Be[1] is oriented nearly parallel to one of the Be–O bonds (see Figure 6b), and q→11 of Be[2] lies very close to the plane spanned by the bond vectors of Be to O and O (see Figure 6c), which is, due to the tetrahedral geometry, also the bisecting plane between the two bonding vectors of Be to O and O . Additional evidence for tensor assignment and goniometer axis orientation is provided by the presence of dipolar couplings between ^9^Be and the protons in the crystal structure, as will be explained below.
To isolate the chemical shift tensor, the centres of the STs were computed according to Equation (16), and the second‐order quadrupole contribution subtracted, with the resulting data points plotted together with their fit in Figure 9.
Plot of the 9Be ST centres (Equation 16), with the contribution of the second‐order quadrupole effect (being in the range of ≤110 Hz) subtracted. The shown fits (left: Be[1], right: Be[2]) represent the chemical shift tensors given in Equation (21), with the isotropic chemical shift values taken from the analysis of the MAS spectra; see text for details.
To ensure consistency with the evaluation of the MAS spectrum, the isotropic chemical shifts used for simulation of the MAS data (see below) were used as fixed parameters for the fits shown in Figure 9. The thus derived chemical shift tensor values for the Be[1] and Be[2] sites in the hambergite structure are:
Similar to the data for ^11^B in Figure 5, the data points for ^9^Be show considerable scatter, due to the fact that chemical shift variation of about 8 ppm had to be derived from spectra consisting of resonances with a linewidth (fwhm) of about 30 ppm, which leads to comparatively large error margins. The isotropic chemical shifts of ^9^Be in hambergite fall within the expected range for beryllium tetrahedrally coordinated by oxygen, as, for example, observed for BeO with −0.6 ppm [45]. Regarding the chemical shift anisotropy of ^9^Be, literature data are extremely scarce. Schurko and co‐workers [46] have studied a number of homoleptic beryllocenes with organic ligands, but due to their highly asymmetric structures, these compounds exhibit unusually large shift spans of 50–60 ppm. A more appropriate comparison is provided by the compound Be(acac) [47], where as in hambergite, ^9^Be is tetrahedrally coordinated by oxygen. This results in isotropic chemical shifts of 0.7 and 0.5 ppm for the two crystallographic sites, and shift spans of 3.4 and 6.6 ppm.
9Be–1H Dipolar Couplings
4.3
For some crystal orientations, the effects of dipolar coupling to protons, which are present in the structure of hambergite as part of OH groups, show up in the ^9^Be spectra. One such spectrum is shown in Figure 10 (black line), where the outer satellite transitions exhibit a doublet splitting caused by heteronuclear dipolar coupling to ^1^H.
9Be spectrum of the hambergite single crystal at a rotation angle of φ≈89.5° acquired by single pulse using 40 scans (black line), and under proton decoupling using 20 scans (pink line), with the recycle delay being 3000 s for both spectra.
For an isolated spin‐1/2 pair I and S, the dipolar interaction splits their respective resonances into a doublet with frequencies ν+ and ν− placed symmetrically around the unaffected resonance position:
For a given spin pair, the strength of the dipolar interaction varies with the distance rIS between the two spins and with the angle β describing the orientation of their connecting vector relative to the external magnetic field. For ^9^Be with I=3/2, the transitions k=−1,0,+1 may be viewed as three fictitious spin‐1/2 transitions [52, 53], with Equation (22) applying to each of them. The satellite transitions positioned at approximately ±65 kHz, which are an accidental overlap of the STs of Be1 and Be1, show the expected doublet, as may be seen from Figure 10. As all central transitions of the four magnetically inequivalent ^9^Be pairs overlap, the doublet splittings are not resolved on the CTs. When ^1^H decoupling is applied, the ST doublets collapse to a single line, as demonstrated by the spectrum plotted in pink in Figure 10. (Interestingly, the resonances at ±50 kHz show the opposite behaviour: Here, a doublet becomes visible under proton decoupling, as weak dipolar broadening is removed and the independent contributions of Be[1](4, 8) and Be[1](2, 6) become resolved.) While the ^9^Be–^1^H pairs in the hambergite structure do not represent perfectly isolated spin pairs, couplings to other, more remote spins are sufficiently small to only affect the line width, leaving a clear doublet structure. For 6 out of 18 recorded crystal orientations, the ^9^Be spectra displayed such well‐resolved dipolar coupling doublets, which could be fitted to extract the coupling magnitude. Many more indications of coupling effects were detectable in the spectra, such as broad lines showing signs of a small doublet separation, or shoulders, but these were not quantitatively evaluated. In contrast, for the goniometer axis orientations used here, no such dipolar coupling effects were observable in the ^11^B spectra.
Since the magnitude of the dipolar coupling is orientation‐dependent with respect to the magnetic field vector (see Equation 22), additional spatial information can be derived from them. This approach has been utilised successfully before by the group of Harbison, who analysed the ^14^N–^1^H dipolar couplings for a single crystal of sulfamic acid [11]. Regarding hambergite, the experimentally observed couplings can be systematically compared with theoretical predictions. For crystal rotation around the axis g→1′, as depicted in Figure 7, the P2(β) values for the shortest beryllium–hydrogen distances of Be[1] and Be[2] are plotted in Figure 11, with the experimentally observable couplings indicated by black crosses. It can be seen that while the experimental values are generally somewhat smaller than the predicted ones, the overall situation is reproduced well: doublets are only observed at rotation angles where the predicted magnitude of the dipolar coupling is large, and their occurrence follows the amplitude of the harmonic functions assigned to the respective beryllium sites. This further validates our goniometer axis choice and tensor assignments.
Theoretical magnitude of the P2(β) polynomial (see Equation 22) of the 9Be–1H dipolar couplings in hambergite for the rotation pattern shown in Figure 7, using the same colour coding as before, with the data for (a) Be[1] and (b) Be[2] plotted separately. The experimentally observable couplings (normalised by bIS=1.75 kHz for Be[1] and 1.56 kHz for Be[2], with the 9Be–1H distances taken from the DFT‐optimised crystal structure) are indicated by black crosses ( ×).
MAS‐NMR of Hambergite
4.4
To corroborate the interaction parameters derived from single‐crystal NMR and DFT calculations, MAS spectra of a powdered sample of hambergite were recorded. Under MAS, both first‐order quadrupole and dipolar interaction, plus the chemical shift anisotropy are (partly) averaged. The MAS central transition spectra of ^11^B and ^9^Be are displayed in Figure 12.
Magic‐angle spinning NMR of a polycrystalline hambergite sample. (a) 11B spectrum at 12 kHz spinning frequency (black line). The fitted spectrum (green line) was obtained with dmfit [54], delivering values of χ=2.646 MHz, ηQ=0.025 and δiso=18.3 ppm. (b) 9Be spectrum at 10 kHz spinning frequency (black line). The simulated spectrum (green line) was generated with simpson [55], using the quadrupole coupling parameters listed in Table 7, and δiso values of 1.6 and 1.4 ppm for Be[1] and Be[2], respectively; see text for details.
For ^11^B, the sizeable quadrupole coupling leads to second‐order broadening of the resonance to a distinct ‘canonical’ line shape, which can be subjected to an unrestrained fit to obtain the NMR interaction parameters. The resulting numerical values are listed in the caption of Figure 12 and are in good agreement with the values derived from single‐crystal ^11^B NMR, as described above. In contrast, in the ^9^Be spectrum, the small quadrupole coupling leads only to an unspecific broadening of the line, with the contributions of the two beryllium sites also largely overlapping. Therefore, the line shape was simulated using the quadrupole coupling parameters derived from single‐crystal NMR. The isotropic chemical shifts for the two ^9^Be sites were determined by calculating the quadrupole‐induced shift (QIS) according to Samoson [56], which came out to −0.3 and −0.1 ppm for Be[1] and Be[2], respectively. These values were applied as corrections to the experimentally observed line position (1.3 ppm) and used as δiso values for the simulation. While obviously not providing the strong evidence of an independent data fit, this procedure at least demonstrates that the ^9^Be MAS spectrum can be simulated using the interaction parameters extracted from the single‐crystal experiments.
Conclusions
5
The NMR interaction tensors for ^9^Be and ^11^B in hambergite, Be BO OH, have been determined from single‐crystal spectra. In the orthorhombic unit cell of hambergite, both the beryllium and the boron atoms occupy Wyckoff positions 8c, where the eight symmetry‐generated atoms are pairwise related by inversion. This results in four magnetically inequivalent ^9^Be and ^11^B per site and makes hambergite a good model system to examine the problem of assigning a set of NMR signals to a set of symmetry‐related atoms in the unit cell. The points addressed in the current work may be summarised as follows:
- (i)Whenever magnetically inequivalent, but symmetry‐related nuclides are present in the unit cell of a system, the assignment of the NMR resonances to specific atoms in this symmetry‐related set is a fundamental problem, which cannot be solved by NMR and XRD experiments alone. For n atoms related by a symmetry element, n mathematically valid assignments of the harmonics to these atoms are possible; see Table 3a for the example of hambergite with n=4. The permutation of these solutions does not change the eigenvalues of the respective NMR interaction tensors, but it does change (permute) the directions of the eigenvectors in the unit cell. The problem may be aggravated by the existence of several unit cell settings in terms of defining the crystallographic axes, which affects the pattern of the tensor elements; see Table 4 for the situation in hambergite. This ambiguity inherent in assigning NMR signals to XRD structure data is of general nature and can be resolved only by obtaining additional information via other methods. One possibility is the application of ab initio quantum mechanical calculations of sufficient quality, with subsequent comparison of eigenvector orientation to the experiment. Analysis of dipolar coupling patterns (if they are observable in the system, such as the ^9^Be–^1^H couplings in hambergite) may also be helpful, because they provide additional spatial information.
- (ii)Assignment ambiguities become even more complex when attempting to also determine the goniometer axis orientation from NMR data only [28]. For the fitting routine to converge, all that matters is the relative orientation between goniometer axis vector and the eigenvectors of the evaluated interaction tensor. The fitted tensor may be assigned to any one of the n symmetry‐related atoms, respectively, to the harmonic function in the rotation pattern created by that one atom. If the remaining (n−1) harmonics are then sequentially assigned according to the generating symmetry element, the data fit will deliver a mathematically valid result, with the fitted goniometer axis orientation depending on which atom has been chosen as the origin. Thus, in addition to the conundrum described in point (i) above, a number of possible goniometer axis orientations exists, which are all viable solutions with identical goodness of fit. The exact number of available options depends on the symmetry of the crystal system. The choice of axis orientation affects the structure of the resulting tensors, and thus the orientation of the eigenvectors (without changing the eigenvalues), as shown in Tables 4 and S3 for the hambergite system. Except for the ambiguities caused by the unit cell setting, the problem of axis orientation may however be avoided by determining the orientation of the single crystal on the goniometer axis by XRD or optical methods.
Experimental Details
6
Single‐crystal NMR spectra were measured on a Bruker Avance‐III 500 WB spectrometer at LMU Munich. The hambergite crystal was glued to a wooden axis and spanned into a goniometer mechanics (built by NMR Service GmbH, Erfurt, Germany) allowing controlled rotation perpendicular to the external magnetic field within a solenoid coil. The Larmor frequencies of the investigated nuclides were ν0(9Be)=70.277 MHz and ν0(11B)=160.462 MHz. For the beryllium spectra, a recycle delay of 1.25 h was used, with 16–60 scans acquired. Baseline correction and a line‐broadening factor of 500 Hz were applied during spectra processing. Heteronuclear ^1^H dipolar decoupling for the ^9^Be spectra was realised by application of a SW ‐TPPM sequence [57], using a linear sweep profile [58], which scaled the optimal pulse width by factors between 0.68−1.32 over 11 pulse pairs, using a phase angle of ±12.5°. The ^11^B measurements were run using recycle delays between 1.5 and 3 h, and a varying number of scans from 4 to 12, with a line‐broadening factor of 1500 Hz and baseline correction applied to the spectra. The MAS spectra were acquired at rotation frequencies of 10 kHz (^9^Be) and 12 kHz (^11^B), with an echo sequence [59] employed to minimise baseline roll for ^11^B. All NMR spectra were referenced against the secondary reference of the ^1^H resonance of 1% Si(CH ) in CDCl . NMR interaction tensors were extracted from the single‐crystal NMR data by global fits performed with the igor pro software [60]. Tensor diagonalisations were carried out with the online service wolfram | alpha, which is based on the software mathematica [61]. The error estimations on the eigenvalues were carried out according to the method of Nelson [62].
Details concerning XRD experiments (for determination of both crystal structure and orientation of the single crystal used for NMR experiments), and the DFT calculations to derive EFG tensors for comparison are given in the Supporting information.
Conflicts of Interest
The authors declare no conflicts of interest.
Supporting information
hambergite_si.pdf.
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