Determination of the ²⁰⁷Pb Chemical Shift Tensor in Crocoite, PbCrO₄, Using Single-Crystal NMR Spectroscopy

Abstract

The full chemical shift tensor of ²⁰⁷Pb (spin $I=1/2$) in the natural mineral crocoite, PbCrO₄, has been determined using single-crystal NMR spectroscopy at room temperature. The eigenvalues of the tensor in its principal axes system are ${\delta }_{11}=-2720\pm 2$ ppm, ${\delta }_{22}=-2319\pm 4$ ppm, and ${\delta }_{33}=-1830\pm 5$ ppm, resulting in an isotropic chemical shift of ${\delta }_{\mathrm{iso}}=-2289\pm 2$ ppm. Additionally, these values were verified using a Herzfeld–Berger analysis of a polycrystalline sample under magic-angle spinning (MAS) conditions.

1. Introduction

Nuclear magnetic resonance (NMR) spectroscopy has become well established as a highly informative technique for structural characterisation of solids [1]. For nuclides with spin $I=1/2$, such as ²⁰⁷Pb, the NMR parameter that may be connected to structural features is mainly the chemical shift, which reflects the electronic shielding of the atomic nuclei [2]. The isotropic part of this chemical shift, ${\delta }_{\mathrm{iso}}$, has been related to structural features such as bond angles [3,4,5], the mean distance of all neighbouring atoms in the coordination sphere [5,6,7,8], as well as to the shortest distance to the closest neighbour [9,10,11]. The simple number ${\delta }_{\mathrm{iso}}$ is, however, the weighted sum of the diagonal elements ${\delta }_{ii}$ of the second-rank chemical shift (CS) tensor $\delta$, which encapsulates the three-dimensional distribution of the electrons around the observed nucleus:

$${\delta }_{\mathrm{iso}}={\displaystyle \frac{1}{3}}\left({\delta }_{11}+{\delta }_{22}+{\delta }_{33}\right)$$

(1)

Therefore, determination of the full CS tensor gives a more complete picture of the electron density surrounding the observed nuclide than is available from only the isotropic shift. For solids, the value of ${\delta }_{\mathrm{iso}}$ is usually derived from magic-angle spinning (MAS) spectra of polycrystalline (’powder’) samples [12]. By measuring powder samples not under spinning, but under static conditions, more detailed information about the chemical shift tensor can be obtained since the individual tensor eigenvalues ${\delta }_{ii}$ may be available from the singularities of the static spectrum [13]. From MAS spectra, the relative intensities of the rotational sideband pattern can be analysed to extract the ${\delta }_{ii}$’s [14] with even higher accuracy [15]. However, the eigenvalues ${\delta }_{ii}$ (plus their corresponding eigenvectors) are available with unrivalled precision from single-crystal NMR experiments, where the orientation dependence of the resonances is directly recorded [16,17,18,19].

Solid-state NMR spectroscopy of ²⁰⁷Pb is of interest in a variety of contexts. Many inorganic Pb(II) compounds exhibit a comparatively strong dependence of the chemical shift on the temperature [20,21,22,23], such that lead nitrate is used routinely for temperature calibrations of MAS probes [20]. In the context of ab initio calculations of electronic structure, ²⁰⁷Pb is classified as a heavy nucleus, so that relativistic effects have to be taken into account [23,24,25,26]. For such calculations, the results depend heavily on the details of the used method, for example the employed wave function, the used relativistic correction, and the selected cluster size. The differences in chemical shift output for the different models are sometimes only a few ppm [24]. Accordingly, there is demand for high-accuracy experimental data to compare these ab initio computation results against. In the current work, the database on ²⁰⁷Pb chemical shift tensor values is expanded using a single-crystal study of the natural mineral crocoite, PbCrO₄. Although PbCrO₄ has been characterised using ²⁰⁷Pb-NMR before [27,28,29,30], only polycrystalline samples under static conditions were investigated. Here, we report the eigenvalues ${\delta }_{ii}$ and eigenvectors of the ²⁰⁷Pb CS tensor of PbCrO₄ to a high degree of accuracy as derived from single-crystal experiments. The ${\delta }_{ii}$ values are corroborated by an additional Herzfeld–Berger analysis [14] of a powder sample under MAS conditions.

2. Results and Discussion

2.1. Single-Crystal ²⁰⁷Pb-NMR of Crocoite

The ‘classical’ strategy of determining NMR interaction tensors from single-crystal samples requires knowledge of the (initial) orientation of the crystal on the goniometer setup [16,17,18], with this information derived from X-ray diffraction (XRD) or optical methods. In many cases, it is however possible to derive orientation information solely from the NMR data [19,31,32]. This strategy has also been applied here, as explained in the following.

Crocoite, PbCrO₄, crystallises in the monazite type structure, space group $P{2}_1/n$ (no. 14). In Figure 1, a view of the monoclinic unit cell according to the XRD analysis of Effenberger and Pertlik is shown. The lead atoms (shown in grey in Figure 1) occupy Wyckoff position $4e$ and form two groups of two atoms, which are connected by screw axes parallel to the crystallographic b-axis and by glide planes. Thus, there are two types of magnetically inequivalent ²⁰⁷Pb in the unit cell, and hence, two resonances are expected to show up in the ²⁰⁷Pb-NMR spectra (see Reference [19] for a more detailed explanation of magnetic equivalence). As can be seen from the example spectra in Figure 2, the experiments match these expectations, except for coincidental overlap of resonances.

The chemical shift tensor $\delta$ of lead atom Pb(2) may be generated from the tensor of Pb(1) by action of a screw axis parallel to the b-axis. In the context of NMR spectroscopy, the translational component of a screw axis action is not relevant, therefore the symmetry may be expressed by rotation only. In the $CRY$ frame, that is the (orthogonalised) coordinate system of the unit cell, the relation of the two ${\delta }^{CRY}$ tensors can therefore be expressed by the following:

$$\begin{array}{c}\hfill {\delta }_{Pb\left(2\right)}^{CRY}=\left(\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 1\end{array}\right){\delta }_{Pb\left(1\right)}^{CRY}\left(\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 1\end{array}\right)\end{array}$$

(2)

By convention, only the symmetric part of a chemical shift tensor is considered (since the antisymmetric part has no effect on the resonance position) [35,36], and therefore the tensors for the ²⁰⁷Pb chemical shift in crocoite may be written as follows:

$$\begin{array}{c}\hfill {\delta }_{Pb\left(1\right)}^{CRY}=\left(\begin{array}{ccc}P& Q& R\\ Q& S& T\\ R& T& U\end{array}\right){\delta }_{Pb\left(2\right)}^{CRY}=\left(\begin{array}{ccc}P& -Q& R\\ -Q& S& -T\\ R& -T& U\end{array}\right)\end{array}$$

(3)

The six independent tensor components $P,Q,R,S,T,U$ can be determined by systematically tracing the orientation dependence of the resonance positions. To achieve that, the single crystal is fixed to a goniometer axis (as shown on the right of Figure 1), which allows for controlled step-wise rotation. In most cases—as in ours—this rotation axis is oriented perpendicular to the external magnetic field lines (alternative goniometer set-ups have been suggested [18] but require slightly different data processing). The rotation pattern resulting from recording the resonance positions over a rotation angle range of $\phi$ = 0–${180}^{\circ }$ is shown in Figure 2, with some representative spectra plotted on the right-hand side. These spectra also demonstrate the excellent resolution available from single crystals, as compared to ²⁰⁷Pb powder spectra, which usually are so broad that the finite excitation bandwidth of the radio-frequency pulses becomes a problem [13]. The main reason that the resonance lines in Figure 2 are significantly broader than those seen in solution-state NMR is dipolar interaction between the spins in the crystal lattice. Because of its distance dependence, the dipolar interaction may provide additional structural information, but its effects need to show up strong enough to be clearly resolved, which is rarely the case [37]. In the crocoite structure [34], the shortest distance between ²⁰⁷Pb and ²⁰⁷Pb is 4.283 Å, and that between ²⁰⁷Pb and ⁵³Cr (with 9.5% natural abundance and low resonance frequency) is 3.334 Å. The resulting direct couplings are hence very weak and only cause some unspecific broadening of the resonance lines, which will not be evaluated in detail here.

To fit the experimental data points shown in Figure 2, the dependence of the resonance frequency on the relative orientation of CS tensor and magnetic field in the crystal frame $CRY$ can be expressed in compact fashion by a vector–tensor–vector notation [38,39]:

$$\begin{array}{cc}{\displaystyle \frac{{\nu }^{Pb\left(1\right)}\left(\phi \right)}{{\nu }_0}}\hfill & ={\overrightarrow{b}}_0^T\left(\phi \right)·{\delta }_{Pb\left(1\right)}^{CRY}·{\overrightarrow{b}}_0\left(\phi \right)\hfill \\ \hfill & =P{b}_x^2+2Q{b}_x{b}_y+2R{b}_x{b}_z+S{b}_y^2+2T{b}_y{b}_z+U{b}_z^2\hfill \end{array}$$

(4)

Obviously in the $CRY$ frame, the magnetic field vector ${\overrightarrow{b}}_0$ may adopt any orientation, as opposed to the laboratory frame, where it points along z by definition. For our goniometer axis $\overrightarrow{g}$, the possible orientations of ${\overrightarrow{b}}_0$ are confined to a plane perpendicular to $\overrightarrow{g}$. If $\overrightarrow{g}$ is known from pre-orienting the crystal by optical or XRD means, Equation (4) may be used for the data fit right away. However, as already mentioned above, the orientation of the goniometer axis itself can be made a variable of the data fit and thus also be determined from fitting the rotation pattern(s). This means expressing ${\overrightarrow{b}}_0$ in terms of the components of $\overrightarrow{g}$, with the relevant equations given in Appendix A. Evidently, including the goniometer axis orientation into the data fit further increases the number of free parameters that need to be extracted from the experimental data. The balance of experimental parameters $p_{\exp }$ available from r rotation patterns versus the number of independent unknown tensor components $t_{\mathrm{idp}}$ may be written as follows [19]:

$$p_{\exp }=r\sum _n(2m_n+1)\ge (n·t_{\mathrm{idp}}+3r)$$

(5)

Here n is the number of Wyckoff positions of the observed nuclide, with a respective magnetic multiplicity of $m_n$. For ²⁰⁷Pb-NMR of crocoite, $n=1$, $m_n=2$, and $t_{\mathrm{idp}}=6$ for the chemical shift tensor, so that $p_{\exp }=5r\ge (6+3r)$, which is fulfilled for $r\ge 3$. Consequently, in addition to the rotation pattern depicted in Figure 2, two more rotation patterns with distinctly different goniometer axes orientations were acquired from our crocoite crystal, as shown in Figure 3.

Simultaneously fitting all three rotation patterns gave the following results for the components of ${\delta }^{CRY}$ (with the goniometer orientations calculated by the fit routine listed in the respective figure captions):

$$\begin{array}{cccc}\hfill P=(-1830\pm 5)\mathrm{ppm}& Q=(0.3\pm 0.3)\mathrm{ppm}\hfill & \hfill & R=(-9\pm 17)\mathrm{ppm}\hfill \\ \hfill S=(-2321\pm 4)\mathrm{ppm}& T=(-31.6\pm 0.6)\mathrm{ppm}\hfill & \hfill & U=(-2717\pm 2)\mathrm{ppm}\hfill \end{array}$$

(6)

Diagonalising the chemical shift tensor gave the eigenvalues listed in

Table 1. ²⁰⁷Pb ($I=1/2$) chemical shift tensor eigenvalues ${\delta }_{ii}$, and the resulting isotropic shift values ${\delta }_{\mathrm{iso}}$ of crocoite, PbCrO₄, at room temperature. The eigenvalues are ordered according to the Haeberlen convention, $\left|{\delta }_{33}-{\delta }_{\mathrm{iso}}\right|\ge \left|{\delta }_{11}-{\delta }_{\mathrm{iso}}\right|\ge \left|{\delta }_{22}-{\delta }_{\mathrm{iso}}\right|$. All chemical shift values are reported relative to Pb(CH₃)₄, realised mostly via indirect referencing using other chemicals such as aqueous Pb(NO₃)₂ solutions or TMS [28].

Sample and Method	${\mathit{\delta }}_{11}$	${\mathit{\delta }}_{22}$	${\mathit{\delta }}_{33}$	${\mathit{\delta }}_{\mathbf{iso}}$	Ref.
	(in ppm)
powder, static	$-2719\pm 21$	$-2322\pm 26$	$-1835\pm 19$	$-2292\pm 11$	[27]
powder, static	$-2653\pm 4$	$-2261\pm 2$	$-1795\pm 8$	$-2236\pm 3$	[28]
powder, static	$-2717\pm 6$	$-2312\pm 6$	$-1832\pm 4$	$-2287\pm 3$	[29]
powder, static	$-2711$	$-2313$	$-1827$	$-2284$	[30]
powder, MAS	$-2750\pm 20$	$-2296\pm 17$	$-1810\pm 26$	$-2291\pm 1$	this work
single crystal, static	$-2720\pm 2$	$-2319\pm 4$	$-1830\pm 5$	$-2289\pm 2$	this work

, and the following eigenvectors (expressed in polar coordinates $\theta$, $\varphi$ in the $CRY$ frame):

$$\begin{array}{cc}\hfill {\overrightarrow{d}}_{11}& ={(4.6\pm 0.3)}^{\circ },{(82.3\pm 13.3)}^{\circ }\hfill \\ \hfill {\overrightarrow{d}}_{22}& ={(85.5\pm 0.1)}^{\circ },{(270.1\pm 0.2)}^{\circ }\hfill \\ \hfill {\overrightarrow{d}}_{33}& ={(90.6\pm 1.1)}^{\circ },{(0.1\pm 0.1)}^{\circ }\hfill \end{array}$$

(7)

2.2. ²⁰⁷Pb-NMR of Crocoite Under Magic-Angle Spinning (MAS)

For verification of the results obtained from the single-crystal experiments, a polycrystalline (powder) sample of PbCrO₄ was also measured under MAS conditions and subjected to a Herzfeld–Berger analysis [14]. To avoid destruction of the single crystal, the powder sample was synthesised by a simple precipitation reaction, as described in Section 4. The ²⁰⁷Pb-NMR spectrum at 10 kHz MAS frequency is depicted in Figure 4. It can be seen that a well-defined spinning sideband pattern is present, reflecting the strong chemical shift anisotropy of ²⁰⁷Pb²⁺ in PbCrO₄ caused by the electron lone pair (see below for a more detailed discussion).

The sideband intensities were extracted from the MAS spectrum by line shape deconvolution (red line in Figure 4), and then numerically evaluated using the program HBA 1.7.5 made available by K. Eichele [40]. Before comparing the tensor eigenvalues derived from this analysis to our single-crystal results, however, they need to be corrected for temperature effects caused by the friction heating of the MAS rotor. The ²⁰⁷Pb chemical shift is comparatively strongly dependent on temperature for many periodic solids [20,21,22,23], and this is also true for PbCrO₄, with deviations of ${\delta }_{\mathrm{iso}}$ already noticeable when spinning at 10 kHz. This effect can be corrected for by acquiring a number of spectra at different MAS frequencies and then extrapolating the temperature dependence of the chemical shift to zero spinning by plotting positions against the squared MAS frequency [28]. For PbCrO₄, the corresponding plot is shown in Figure 4. The temperature-corrected tensor eigenvalues from the Herzfeld–Berger analysis are listed in Table 1, and it can be seen that within the error margins, they are in good agreement with the single-crystal NMR values.

2.3. Anisotropy of the ²⁰⁷Pb Chemical Shift Tensor in Pb(II) Compounds

One characteristic of Pb(II) compounds is the presence of a lone electron pair at the lead atom. This obviously results in considerable anisotropy of the electron density surrounding ²⁰⁷Pb, which is directly mirrored by solid-state NMR [41,42]: The stronger the anisotropy, the broader the line shape of the static polycrystalline spectrum, or in other words, the larger the difference between the largest and the smallest eigenvalues of the chemical shift tensor. Hence, the span $\Omega ={\delta }_{\max }-{\delta }_{\min }$ [14] is a useful measure of lone pair presence at the ²⁰⁷Pb site.

Table 2. Anisotropies of ²⁰⁷Pb chemical shift tensors in inorganic Pb(II) compounds, gauged by the span $\Omega ={\delta }_{\max }-{\delta }_{\min }$, as determined from single-crystal NMR experiments at room temperature. The entries are ordered with increasing isotropic shift ${\delta }_{\mathrm{iso}}$, with all values reported relative to Pb(CH₃)₄, realised mostly via indirect referencing.

Compound (Mineral)	Wyckoff Position	$\mathit{\Omega }$	${\mathit{\delta }}_{\mathbf{iso}}$	Ref.
		(in ppm)
PbSO4	$4c$	577	$-3615$	[9]
(anglesite)
Pb(NO3)2	$4a$	53	$-3567$ a	[43]
			$-3492$ b	[28]
Pb5(PO4)3Cl	$4f$	180	$-2810$	[44]
(pyromorphite)	$6h$	1149	$-2170$
PbCO3	$4c$	756	$-2626$	[45]
(cerussite)
PbCrO4	$4e$	890	$-2290$	this work
(crocoite)
PbMoO4	$4a$	176	$-2015$	[46]
(wulfenite)
Pb2Cl2CO3	$8k$	1252	$-1928$	[46]
(phosgenite)
Pb5(VO4)3Cl	$4f$	1200	$-1620$	[31]
(vanadinite)	$6h$	1800	$-1729$

^a Ambiguities exist because the use of a Pb(NO₃)₂ solution as reference, where the resonance frequency depends markedly on concentration and temperature. ^b Derived from polycrystalline powder [28].

lists the $\Omega$ parameter for a range of inorganic Pb(II) compounds (most of them natural minerals like crocoite), as determined from high-precision NMR experiments on single crystals. From this table, it can be seen that in terms of ²⁰⁷Pb chemical shift parameters, crocoite is fairly average (in the sense of non-extreme) for an inorganic Pb(II) compound, with the values for both the span $\Omega$ and the isotropic chemical shift ${\delta }_{\mathrm{iso}}$ placed in the respective middle ranges of the reported values.

3. Conclusions

The natural mineral crocoite, PbCrO₄, has been the subject of several ²⁰⁷Pb-NMR studies; however, until now, only polycrystalline (powder) samples under static conditions have been measured [27,28,29,30]. In the present work, single-crystal experiments were carried out to derive the chemical shift tensor eigenvalues (Table 1) of PbCrO₄ with unprecedented precision, and the eigenvectors in the crystal lattice (Equation (7)), which have been determined for the first time. Additionally, a Herzfeld–Berger analysis [14] has been applied to a powder sample under MAS conditions to corroborate the findings from the single crystal. The resulting anisotropy of the chemical shift tensor, quantified by a span of $\Omega ={\delta }_{\max }-{\delta }_{\min }=890$ ppm, indicates a medium strength influence of the electron lone pair at the ²⁰⁷Pb nuclei in the crocoite structure.

Finally, we return to the relation of the chemical shift tensor to structural parameters of the compound [1,2,3,4,5,6,7,8], as already discussed in the Introduction. Establishing clear relations of chemical shift parameters, such as the isotropic shift ${\delta }_{\mathrm{iso}}$, to structural features such as bond angles or distances to neighbouring atoms works best when restricting this analysis to one class of compounds, for example correlating ${\delta }_{\mathrm{iso}}$(²⁹Si) to silicon-oxygen bond lengths in silicates [3]. In the context of ²⁰⁷Pb-NMR of lead bearing minerals, a correlation of the isotropic chemical shift to the shortest lead-oxygen distance in the crystal structure has been suggested [9]. The value ${\delta }_{\mathrm{iso}}\approx -2290$ ppm determined here from a single crystal deviates only slightly from previously reported values [27,28,29,30]; indeed, this fits nicely into the suggested fit [9]. When comparing ²⁰⁷Pb-NMR results for closely related minerals, such as the apatites vanadinite, pyromorphite and mimetite, other relationships may be uncovered. In the apatite case, a dependence of ${\delta }_{\mathrm{iso}}$ on the ratio $c/a$ of the unit cell parameters has been proposed [47]. Obviously, in our current work about crocoite, we only have a single chemical shift tensor available, so establishing relations of the type discussed above is not possible.

4. Materials and Methods

All ²⁰⁷Pb NMR spectra were recorded on a Bruker Avance-III 500 spectrometer, with a Larmor frequency of ${\nu }_0$(²⁰⁷Pb) = 104.63 MHz (i.e., $B_0=11.7$ T). To reduce baseline roll, the single-crystal spectra were acquired with a spin-echo sequence [48], with the crystal orientation change realised by a goniometer mechanics built by NMR Service GmbH (Erfurt, Germany). Longitudinal $T_1$ relaxation times were measured on the PbCrO₄ single crystal, using the saturation recovery method, resulting in values between 9 s and 17 s, depending on the orientation of the crystal. Consequently, a recycle delay of 60 s was used for spectra acquisition, with the number of scans between 60 and 120. All spectra were referenced indirectly to ¹H in 100% TMS at $-0.1240$ ppm. According to our measurements, this is equivalent to the commonly used reference of a static Pb(NO₃)₂ powder spectrum at room temperature, with ${\delta }_{\mathrm{iso}}=-3492$ ppm, or $-3474$ ppm at the highest point of the line shape, in good agreement with previously reported values [28]. Rotation patterns were fitted with the program Igor Pro 7 (WaveMetrics Inc., Lake Oswego, OR, USA). A polycrystalline sample of PbCrO₄ was synthesised by adding 30 mL of a 0.73 mmol Pb(NO₃)₂ solution to 15 mL of a 0.73 mmol K₂CrO₄ solution. The resulting yellow precipitation was filtered, washed twice with 10 mL water, and dried for 24 h at 100 °C.