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Single-crystal 207Pb-NMR of wulfenite, PbMoO4, aided by simultaneous measurement of phosgenite, Pb2Cl2CO3 Otto E.O. Zeman a, Jennifer Steinadler a, Rupert Hochleitner b, Thomas Br€ auniger a, * a b

Department of Chemistry, University of Munich (LMU), Butenandtstr. 5–13, 81377, Munich, Germany Mineralogical State Collection Munich (SNSB), Theresienstr. 4, 80333, Munich, Germany

A B S T R A C T

The effort for determining NMR interaction tensors from orientation-dependent spectra of single crystals may be greatly reduced by exploiting symmetry relations between atoms of the observed nuclide in the unit cell, as is well documented in the literature. In this work, we determined both the full chemical shift (CS) tensor of 207 Pb and the unknown orientation of the rotation axis for the natural mineral phosgenite, Pb2Cl2CO3, from a single rotation pattern, i.e. spectra of crystal orientations from 0 to 180 . In the tetragonal crystal structure of phosgenite, four symmetry-related, but magnetically inequivalent 207Pb are generated by the WYCKOFF multiplicity. The mineral wulfenite, PbMoO4, also crystallises in a tetragonal space group, but the site multiplicity for 207Pb generates only one magnetically inequivalent atom, thus not supplying sufﬁcient experimental data to determine CS tensor and axis orientation from an arbitrary number of rotation patterns. One solution to this problem is to simultaneously acquire data of a known compound with high symmetry and WYCKOFF multiplicity (here: phosgenite), which supplies additional constraints making the solution of the target compound (here: wulfenite) possible. The 207Pb CS tensors thus determined are characterised by the following eigenvalues in PAS PAS PAS PAS PAS ppm: δPAS 11 ¼ ð 2553 1Þ, δ22 ¼ ð 1929 1Þ, δ33 ¼ ð1301 1Þ for phosgenite, and δ11 ¼ ð 2074 1Þ, δ22 ¼ ð 2074 1Þ, δ33 ¼ ð1898 1Þ for wulfenite.

1. Introduction The orientation dependencies of the various interactions governing the response of nuclear spins in a magnetic ﬁeld are customarily expressed in terms of second-rank tensors [1–3]. Both conceptually and in terms of measurement accuracy, magnetic resonance of single crystals is the method of choice to determine these tensors. However, the classic approach is to derive the full tensor from three complete rotations around mutually perpendicular axes [4,5], which has the reputation of being very labour-intensive, to the point that it has been described as ‘tedious' [6]. Yet, by making use of internal crystal symmetries, the necessary amount of data acquisition and evaluation may be greatly reduced [7–11]. Another prerequisite for quantitative evaluation of single-crystal spectra is the knowledge of the orientation of the axis around which the crystal is step-wise rotated, relative to a coordinate system, e.g. the crystal frame. This orientation can be established by either optical or X-ray diffraction (XRD) techniques, but not for systems showing irregular morphology or having high X-ray absorption coefﬁcients. In such cases, it may be possible to extract the precise crystal orientation from just the NMR data, using the internal symmetries mentioned above. This ‘NMR only’ approach has recently been utilised to determine the chemical shift (CS) tensor of 207Pb in a number of natural minerals [12–14], where analysis by XRD with standard laboratory equipment was precluded

because of the high lead content. Obviously, when aiming to exploit symmetry elements in the crystal structure for the purpose of single-crystal NMR experiments, such symmetries have to be present. This is however not the only requirement: these elements also have to generate sufﬁciently high WYCKOFF multiplicities in the unit cell. As a case in point, both natural minerals considered in this work, i.e. phosgenite, Pb2Cl2CO3, and wulfenite, PbMoO4, crystallise in tetragonal space groups (P4=mbm and I41 =a), and thus possess comparable symmetry. From the perspective of 207Pb-NMR, it makes however a substantial difference that in phosgenite, the lead atoms occupy WYCKOFF position 8k [15], which generates four magnetically inequivalent 207Pb in the unit cell, whereas in wulfenite, there exists only one magnetically inequivalent 207Pb on WYCKOFF position 4a [16], see also Fig. 1. As will be discussed in detail below, the presence of four symmetry-related 207Pb resonances in the NMR spectrum of a phosgenite single crystal allows the determination of both the full CS tensor and the orientation of the rotation axis from only one rotation pattern. As will be detailed below, this is in stark contrast to wulfenite, where the single 207 Pb resonance leaves one with a system of equations that is underdetermined, no matter how many rotation patterns we would record. One solution to this quandary, which we demonstrate in the present work, is to measure the compound with low symmetry and/or WYCKOFF multiplicity (here: wulfenite) simultaneously with a compound with high

* Corresponding author. E-mail address: [email protected] (T. Br€auniger). https://doi.org/10.1016/j.ssnmr.2019.101620 Received 2 July 2019; Received in revised form 12 September 2019; Accepted 12 September 2019 Available online 17 September 2019 0926-2040/© 2019 Elsevier Inc. All rights reserved.

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Fig. 1. (a) Single crystal of phosgenite, Pb2Cl2CO3, from Monteponi, Sardinia/Italy (mineralogical state collection inventory no. 52634). (b) Unit cell of phosgenite, space group P4=mbm (No. 127), according to Ref. [15]. The lead atoms (grey) at Wyckoff position 8k form two sets of four atoms related by a four-fold rotation axis parallel to the c axis, with the two sets related by an inversion center at the center of the unit cell. All lead atoms are coordinated by ﬁve nearest chlorine atoms (green) and four oxygen atoms (red). (c) Single crystal of wulfenite, PbMoO4, from Bleiberg, Carinthia/Austria (inventory no. 52435). (d) Unit cell of wulfenite, space group I41 =a (No. 88), according to Ref. [16]. The lead atoms (grey) at Wyckoff position 4a, situated on four-fold screw axes parallel to the c axis, are coordinated by eight oxygen atoms (red). a Drawings generated with the VESTA program [17].

0

symmetry and WYCKOFF multiplicity (here: phosgenite), around two different rotation axes. The straightforward analysis of the phosgenite data supplies us with the relative orientation of the two axes. The knowledge of this orientation difference can be carried over to the wulfenite system, supplying an additional ﬁt constraint, which turns out to be sufﬁcient for a successful determination of the 207Pb CS tensor in the wulfenite structure.

B δCRY Pbð1Þ ¼ @ Q 0

For evaluating the single-crystal NMR of phosgenite, which crystallises in the tetragonal space group P4=mbm, we largely follow the procedure previously described for tetragonal γ-LiAlO2 [18,19]. In the unit cell of phosgenite, Pb2Cl2CO3, the lead atoms at WYCKOFF position 8k form two sets of four atoms, which are related by a four-fold rotation axis parallel to the c axis. These two sets are connected to each other by an inversion center at the center of the unit cell, as may be also seen from Fig. 1. Since NMR is invariant to translational elements and inversions, one can only trace four 207Pb atoms in the spectra, which are related by successive 90o rotations about the crystallographic c axis: 90o

90o

x;xþ12;z

xþ12;x;z

x;xþ12;z

Q P

R

1

0

C R A

R

S

Q R

B δCRY Pbð2Þ ¼ @ Q

1

0

C R A S

P R

P

Q R

1

P

C R A

R

R

P

Q R

S 1

B δCRY Pbð4Þ ¼ @ Q R

P

C RA

R

S

(2)

To determine these NMR interaction tensors quantitatively, a single crystal of phosgenite (approximate size 4 3 2 mm) was ﬁxed on a wooden rod and installed in a goniometer probe equipped with a 5 mm coil. The goniometer mechanics of this probe allows a deﬁned change of the rotation angle φ around the goniometer axis ! g , which as in most NMR goniometer designs is oriented perpendicular to the magnetic ﬁeld ! B 0 , to keep the number of harmonic components in the rotation data at the necessary minimum of two (see below, Eq. (5)). According to Eq. (2), we aim to determine the chemical shift tensor in the CRY frame, which is the most useful frame to evaluate electron density in the unit cell. In this ! frame, B 0 can adopt arbitrary orientations, which in the following will ! be described by the normal vector b 0 ¼ ð bx by bz Þ. When aiming to derive magnetic resonance frequencies from an interaction tensor Tin a ! coordinate system different from the LAB frame (where B 0 deﬁnes the z! !T axis), it can be convenient to use the shorthand notation b 0 T b 0 [5, 7]. Thus, the orientation dependence of the resonance position observed for Pb(1), scaled by the Larmor frequency ν0 , is given by:

Pb NMR of phosgenite: single crystal

90o

P

R

207

Pbð1Þ ↔ Pbð2Þ ↔ Pbð3Þ ↔ Pbð4Þ

R

B δCRY Pbð3Þ ¼ @ Q

2. Results and discussion 2.1.

P

(1)

xþ12;x;z

.

Their respective chemical shift tensors in the tetragonal crystal frame (CRY) are therefore described by only four independent tensor components P, Q, R, and S:

!T

!

νPbð1Þ ðφÞ ν0 ¼ b 0 ðφÞ δCRY Pbð1Þ b 0 ðφÞ

¼ P bx bx þ by by þ Q 2bx by þ R 2bx bz 2by bz þ Sðbz bz Þ

(3)

Two auxiliary unit vectors ! u and ! v , which are perpendicular to the

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Fig. 2. (a) 207Pb NMR spectra of a single crystal of phosgenite, Pb2Cl2CO3, rotated counter-clockwise by the indicated angle φ around the rotation axis ! g perpendicular to ! the external magnetic ﬁeld B 0 , starting from an arbitrary point φ ¼ 0. The varying signal intensity is due to the limited excitation length of the applied echo-sequence, with an irradiation frequency offset of 1700 ppm. (b) Full rotation pattern over 180o for the four magnetically inequivalent 207 Pb at WYCKOFF position 8k, acquired by rotating the phosg, genite crystal step-wise by 10o around ! with the drawn lines representing the data ﬁt results (see text for details).

goniometer axis ! g (see also Fig. 7 in the Appendix), can be used to ! describe the step-wise movement of b 0 in the CRY frame by angle φ, with an offset angle φΔ : ! b 0 ðφÞ ¼ ! v sinðφ φΔ Þ þ ! u cosðφ φΔ Þ

0

0

B δ11 0 0 B δPAS ¼ @ 0 δ22 0 A ¼ B B @ 0 0 δ33

δiso

Δδ ð1 þ ηCS Þ 2 0 0

(4)

In our case, the orientation of the goniometer axis ! g itself is however a ﬁt variable, so ! u and ! v need to be expressed in terms of ! g components, as detailed in the Appendix. The offset angle φΔ is also a variable of the data ﬁt, which allows the initial crystal orientation (i.e. resonance positions at φ ¼ 0) to be arbitrary as well. It can be seen from Fig. 2 that the four magnetically inequivalent, but symmetry-linked lead atoms give rise to four resonances in the spectra for almost all recorded crystal orientations. Plotting the rotation pattern, i.e. the resonance positions over a 180o interval, as depicted in Fig. 2b, shows that the resonances follow harmonic functions of the type [4]:

νn ðφÞ = ν0 ¼ An þ Bn cos 2 φ þ Cn sin 2 φ

1

0

1

0

C C C Δδ C δiso ð1 ηCS Þ 0 A 2 0 Δδ þ δiso (6)

In the notation on the right hand side, we have used the deﬁnition of the isotropic chemical shift, δiso ¼ ð1=3Þðδ11 þ δ22 þ δ33 Þ, and two alternative tensor parameters according to the HAEBERLEN convention [20], namely the asymmetry parameter ηCS , and the reduced anisotropy Δδ:

ηCS ¼

δ22 δ11 ; Δδ ¼ δ33 δiso Δδ

(7)

For the above deﬁnitions, the eigenvalues of δ need to be sorted according to: jδ33 δiso j jδ11 δiso j jδ22 δiso j

(5)

(8) 207

Pb CS tensor for phosThe eigenvalues and eigenvectors of the genite are listed in Table 1. The anisotropy parameter Δδ ¼ 627 ppm is comparatively large, despite the high symmetry of the crystal lattice, and hence indicates that the electron lone pair at the lead atoms is signiﬁcantly sp-hybridized [21]. Also, the asymmetry parameter ηCS ¼ 0:995 of the CS reﬂects the low symmetry of the ﬁrst coordination sphere deﬁned by the four surrounding oxygen and ﬁve chlorine atoms.

The three factors An , Bn , and Cn are linear combinations of the CS tensor components and different for each magnetically inequivalent 207 Pb, with n ¼ 1…4. For one physical goniometer axis ! g , the existing four harmonics are linked by one constraint to one another [11], hence nine linear independent parameters may be obtained from one rotation pattern. The chemical shift tensor δCRY we want to calculate according to Eq. (2) possesses only four independent components. This fact provides sufﬁcient free parameters to also include the orientation of the goniometer axis (described by the two angles θ; ϕ in spherical coordinates in ! the CRY frame), and the unknown initial orientation of b 0 (expressed by the offset angle φΔ ), into the data ﬁt. Thus, using Eq. (3) for Pb(1) (and its symmetry-related counterparts for n ¼ 2…4), the rotation pattern in Fig. 2b was subjected to a multiparameter ﬁt, which converged on a global solution for the initial orientation of the goniometer axis ! g in the CRY frame at θg ¼ ð57:5 0:1Þo , ϕg ¼ ð53:2 0:1Þo , and offset angle φΔ ¼ ð42:2 0:1Þo . The components of the chemical shift tensor for 207Pb, extracted from the ﬁt are: P ¼ ð1651:3 0:7Þ ppm, Q ¼ ð277:9 0:7Þ ppm, R ¼ ð206:6 0:5Þ ppm, and S ¼ ð2481 2Þ ppm.1 The lines drawn in Fig. 2b are the pictorial representation of these ﬁt results. Before discussing them further, we note that the comparatively high WYCKOFF multiplicity for 207 Pb in phosgenite makes it a straightforward task to determine both CS tensor and goniometer axis orientation from a single rotation pattern, thus keeping the effort for both data acquisition and evaluation within a manageable scope. The diagonalisation of the CS tensor in the CRY frame transforms it to its PAS frame:

Table 1 Chemical shift tensor of 207Pb in phosgenite, Pb2Cl2CO3. Left: Determined from single-crystal NMR experiments about one general rotation axis at room temperature. The orientation of the corresponding eigenvectors are listed in spherical coordinates (θ, ϕ) in the tetragonal abc crystal frame (CRY) and refer to the atom closest to the origin, i.e. Pb(1). Error values are derived from the ﬁt residuals. Right: Determined from a HERZFELD–BERGER analysis [25] of the rotational side-band pattern at νr ¼ þ22:5 kHz magic-angle spinning (MAS) (i.e., at slightly elevated temperature), with I1 =I0 ¼ 1:0, I2 =I0 ¼ 0:43, and I3 =I0 ¼ 0:14, leading to ρ ¼ 0:04 0:03 and μ ¼ 5:8 0:1. Error values of tensor components are derived from those of ρ and μ.

δPAS 11 δPAS 22 δPAS 33

! d 11 ! d 22 ! d 33 Δδ

ηCS

Single-Crystal NMR

MAS NMR

ð2553 1Þ ppm ð1929 1Þ ppm ð1301 1Þ ppm 13.9

ð2546 10Þ ppma ð1897 20Þ ppm ð1299 10Þ ppma 135.0

90.0

45.0

76.1

315.0

ð627 2Þ ppm 0:995 0:001

ð615 20Þ ppm 1:06 0:02a ð1914 1Þ ppm ð1924 4Þ ppmb

δiso

ð1928 2Þ ppm a

Strict sorting of the δii according to Eq. (8) has been avoided to facilitate comparison, which however leads to ηCS being larger than unity. b Taken from shift extrapolation over squared MAS frequencies, see Fig. 4b.

1 All given uncertainties reﬂect the error derived from the ﬁt residuals and are less than 0.25% for the tensor components.

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Fig. 3. Orientation of the experimentally determined 207Pb chemical shift eigenvectors (blue) in the unit cell of phosgenite, scaled according to the magnitude of the ! associated eigenvalues, such that an absolute value of 750 ppm corresponds to a length of 1 Å. (a) Viewed down along the crystallographic c axis, with d 22 pointing ! along the {110} plane normal. (b) Viewed down along the crystallographic b axis, with the largest eigenvector, d 11 , pointing exactly to the next-nearest oxygen atom.

! Fig. 4. (a) 207Pb magic-angle spinning NMR spectrum of polycrystalline phosgenite, acquired in a magnetic ﬁeld of B 0 ¼ 11:7T at 22.5 kHz spinning speed, with the isotropic band indicated at 1914:5 ppm. (b) 207Pb isotropic chemical shift of phosgenite versus squared MAS frequency, with the linear ﬁt given by the dashed line. An extrapolation to zero spinning frequency results in an isotropic chemical shift of ( 1924 4) ppm.

well known that 207Pb chemical shifts may vary strongly with temperature, so that even the friction heating from fast MAS frequencies can lead to a noticeable change in Refs. δiso [22–24]. Extrapolating the isotropic shift over the squared MAS frequency allows the determination of the MAS isotropic shift at room temperature, as shown in Fig. 4b, giving δMAS ¼ ð 1924 4Þppm, which is in good agreement with iso our single-crystal NMR value. To also extract the three eigenvalues of the CS tensor, we have performed a HERZFELD-BERGER analysis [25] of the MAS spectrum at 22.5 kHz. The results are given in Table 1, showing that the magnitude of the MAS-derived eigenvalues are in very good agreement with the single-crystal values, especially taking into account that discrepancies may arise from the temperature difference between static single-crystal and MAS experiments.

The orientation of the CS tensor eigenvectors in the tetragonal unit cell of phosgenite are depicted in Fig. 3. The eigenvector orientations are in agreement with the crystal symmetry, with one eigenvector for one of ! the lesser tensor components ( d 22 ) aligning exactly along the diagonal of the a and b axis (and its symmetry equivalent directions) in the crystal frame, as may be best seen in Fig. 3a. This is a consequence of the symmetry enforced by the mirror plane in the tetragonal unit cell at WYCKOFF position 8k, on which the 207Pb atoms are situated. The remaining eigenvectors are free to orient according to the electronic environment generated chieﬂy by the surrounding oxygen and chlorine atoms. Interestingly, the eigenvector with the largest corresponding ! eigenvalue, i.e. d 11 , points exactly to the nearest oxygen atom, see Fig. 3b. This peculiar alignment has previously been observed similarly for the two nearest oxygen atoms of 207Pb in the natural mineral anglesite [14].

2.2.

2.3.

207

Pb NMR of wulfenite: single crystal

As already outlined in the Introduction, the crystal structure of wulfenite, PbMoO4, does not provide more than one magnetically inequivalent 207Pb. The four lead atoms in the tetragonal unit cell occupy WYCKOFF position 4a [16], and may be viewed as being generated by a series of inversion operations, to which NMR is invariant. All 207Pb atoms in the unit cell are positioned on four-fold screw axes parallel to the crystallographic c axis, which causes the 207Pb chemical shift tensor to be uniaxial, and the off-diagonal elements to be zero, so that the CRY frame is the principal axis system of the tensor, δCRY ¼ δPAS . Therefore, the CS tensor of wulfenite is described by only two independent tensor components T and U, making it the simplest possible non-isotropic tensor:

207

Pb NMR of phosgenite: polycrystalline

Information about the magnitude of the CS tensor, i.e. the eigenvalues and the isotropic chemical shift derived from them, may also be obtained from a polycrystalline sample, using either static or magic-angle spinning (MAS) NMR spectroscopy. As a useful comparison to our single crystal data, 207Pb-NMR spectra of a sample of phosgenite crushed into a powder were acquired at various MAS spinning speeds, ranging from 12.5 kHz to 22.5 kHz, with the latter shown in Fig. 4a. The isotropic shift derived from this spectrum is 1914:5 ppm, differing signiﬁcantly from the value of our single-crystal experiments (δiso ¼ 1928 ppm). It is however 20

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Solid State Nuclear Magnetic Resonance 103 (2019) 17–24

Fig. 5. 207Pb NMR spectra of single crystals of wulfenite, PbMoO4, and phosgenite, Pb2Cl2CO3, glued together on a teﬂon support and rotated counter-clockwise g 2 . The strong signal belongs to wulfenite, while the four equally intense signals are those of the phosgenite single crystal. Full around the axis (a) ! g 1 and (c) ! g 1 and (d) rotation pattern over 180o for 207 Pb at WYCKOFF position 4a in the wulfenite structure (purple), acquired by rotating the crystals step-wise by 10o around (b) ! ! g 2 . The attenuated background shows the full rotation pattern of the simultaneously measured 207Pb in phosgenite.

0

δCRY Pb

¼ δPAS Pb

T ¼@ 0 0

0 T 0

1 0 0A U

wulfenite (approximate size 1.5 5 2 mm) and a single crystal of phosgenite (approximate size 2.5 3 2 mm) were glued an a 5 mm 5 mm teﬂon support with dual-component adhesive. This allowed the reorientation of the effective rotation axis from ! g 1 to ! g 2, without changing the relative orientation of both crystals. 207Pb-NMR spectra resulting from this setup are shown in Fig. 5, where the four equally intense 207Pb signals of phosgenite can be clearly distinguished from the single, narrow wulfenite signal with much higher intensity. Using the known CS tensor of phosgenite listed in Table 1, the only free parameters in the data ﬁt were θ, ϕ and φΔ for both ! g 1 ðpÞ and ! g 2 ðpÞ in the CRY frame of the phosgenite structure. These could precisely be determined from ﬁtting the rotation patterns in Fig. 5 to:

(9)

Accordingly, compared to phosgenite (Eq. (3)), the expression describing the resonance position is signiﬁcantly simpliﬁed2:

!T

!

νPb ðφÞ ν0 ¼ b 0 ðφÞ δCRY Pb b 0 ðφÞ ¼ 2Tb2x þ Ub2z

(10)

To extract the 207Pb CS tensor for wulfenite from one rotation pattern using Eq. (10), ﬁve linear independent parameters are required: tensor components T and U, and axis parameters θ, ϕ and φΔ . Yet, any rotation pattern of wulfenite will only supply a single harmonic function with the three linear independent parameters A, B, and C according to Eq. (5). The seemingly obvious solution to this problem, i.e. acquisition of an additional rotation pattern with a different rotation axis ! g 2 , does not work here. Although this would supply a new harmonic function with parameters A2 , B2 , and C2 , also three new ﬁt parameters θ2 ; ϕ2 and φΔ2 are added, in effect making acquisition of additional rotation patterns a zerosum game. One possible way out of this quandary is to acquire two rotation patterns of wulfenite, while simultaneously measuring a single crystal of phosgenite. Evaluation of the phosgenite data then provides independent information about the two (randomly chosen) rotation axes, and thereby reduces the amount of ﬁt variables. To this end, a single crystal of

! g 1 ðpÞ : θ1 ¼ ð17:5 0:1Þo ϕ1 ¼ ð36:0 0:1Þo φΔ1 ¼ ð30:4 0:1Þo ! g 2 ðpÞ : θ2 ¼ ð96:9 0:1Þo ϕ2 ¼ ð120:1 0:1Þo φΔ2 ¼ ð153:8 0:1Þo Δ! g ðpÞ : Δθ ¼ ð79:4 0:2Þo Δϕ ¼ ð84:1 0:2Þo ΔφΔ ¼ ð123:4 0:2Þo (11) The two rotation measurements on the two glued crystals are hence linked by the known difference of goniometer axis orientation Δ! g ðpÞ ¼ ! g 1 ðpÞ and offset angles ΔφΔ . While we do not know anything g 2 ðpÞ ! yet about the absolute orientation of the two goniometer axes ! g 1 ðwÞ and ! g 2 ðwÞ in the unit cell of wulfenite, we do know that the difference must be the same as for phosgenite, Δ! g ðwÞ ¼ Δ! g ðpÞ. Therefore, the motion of ! the ﬁeld vector b 0 in the CRYframe of wulfenite of the second rotation pattern (Fig. 5d) can be related back to the ﬁrst rotation pattern (Fig. 5b) by θ2 ¼ θ1 þ Δθ, ϕ2 ¼ ϕ1 þ Δϕ, and φΔ2 ¼ φΔ1 þ ΔφΔ . This reduces the amount of free ﬁt parameters from eight to exactly the ﬁve that are needed to deﬁne the system. Therefore, it is now possible to simultaneously ﬁt the two rotation patterns according to Eq. (10), and to obtain

2 It should be noted that the strong reduction of independent tensor elements by the crystal symmetries in our test system wulfenite is a coincidence – the general strategy of tensor determination outlined here does not depend on it.

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! Fig. 6. (a) 207Pb magic-angle spinning NMR spectrum of polycrystalline wulfenite, acquired in a magnetic ﬁeld of B 0 ¼ 11:7 T at 22.5 kHz spinning frequency, with 207 Pb isotropic chemical shift of wulfenite versus squared MAS frequency, with the linear ﬁt given by the dashed line. the isotropic band indicated at 2006:6 ppm. (b) An extrapolation to zero spinning results in an isotropic chemical shift of ( 2015 2) ppm. (c) 207Pb static powder spectrum of wulfenite, acquired in a magnetic ﬁeld ! of B 0 ¼ 11:7 T. The dashed line was calculated with the SIMPSON package [28], using the chemical shift values listed in Table 2(right).

both δCRY and ! g 1 ðwÞ, ! g 2 ðwÞ. The ﬁt converged on a global solution, Pb giving the CS tensor elements to T ¼ ð2074 1Þ ppm and U ¼ ð1898 1Þ ppm, plus the orientations of the two goniometer axes (the errors reﬂect the ﬁt residuals and the uncertainties in Δ! g ðpÞ determined from the phosgenite rotation patterns):

! d 22 placed in the abplane. The tensor parameters agree reasonably well with those previously reported from static powder spectra of polyPAS PAS crystalline wulfenite: δPAS 11 ¼ δ22 ¼ 2067 ppm, δ33 ¼ 1880 ppm, and Δδ ¼ 123 ppm [24].

! g 1 ðwÞ : θ1 ¼ ð17:3 0:2Þo ! g 2 ðwÞ : θ2 ¼ ð96:7 0:4Þo

2.4.

ϕ1 ¼ ð153:5 0:7Þo ϕ2 ¼ ð237:6 0:9Þo

φΔ1 ¼ ð59:2 0:4Þo φΔ2 ¼ ð2:6 0:6Þo (12)

Table 2 Chemical shift tensor of 207Pb in wulfenite, PbMoO4. Left: Determined from single-crystal NMR experiments about two rotation axes at room temperature. The orientation of the corresponding eigenvectors is listed in spherical coordinates (θ, ϕ) in the tetragonal abc crystal system CRY. Error values are derived from the ﬁt residuals and uncertainties in Δ! g determined from the phosgenite rotation patterns (Eq. (11)). Right: Determined from a static polycrystalline powder sample at room temperature. Error values of tensor components are derived from a SIMPSON [28] simulation.

! d 11 ! d 22 ! d 33 Δδ

ηCS

Single-Crystal NMR

Polycrystalline-Powder NMR

ð2074 1Þ ppm ð2074 1Þ ppm ð1898 1Þ ppm 90.0 , β a

ð2079 5Þ ppm ð2079 5Þ ppm ð1887 10Þ ppm

3. Conclusions In the present work, we have outlined a strategy to determine NMR interaction tensors from single crystals of compounds with low symmetry and/or WYCKOFF multiplicity. For our test system wulfenite, PbMoO4, knowledge of the rotation axis is required to be able to extract all elements of the chemical shift (CS) tensor of 207Pb from the orientationdependent data. However, resolving crystal orientation by optical means is usually fraught with difﬁculties, and the high X-ray absorption coefﬁcient of wulfenite (caused by the high lead content) prohibits axis orientation by standard XRD methods. This makes it desirable to determine the axis orientation solely from the ﬁt of the NMR data, as demonstrated before for the CS tensor of 207Pb in several lead-bearing minerals [12–14]. The system of equations for such a ﬁt is however underdetermined because of the low WYCKOFF multiplicity of lead in the wulfenite structure. To obtain additional information on the rotation axes orientations, we acquired two full rotation patterns of wulfenite, while simultaneously measuring a single crystal of phosgenite, Pb2Cl2CO3. In the phosgenite structure, four magnetically inequivalent 207Pb are

90.0 , β þ 90:0∘ a 0.0 , 0.0 ð117 3Þ ppm 0

δiso

ð2015 1Þ ppm

Pb NMR of wulfenite: polycrystalline

To compare the isotropic chemical shift calculated from the singlecrystal NMR experiments (δiso ¼ 2015 ppm) to MAS experiments, NMR spectra of a polycrystalline sample of crushed wulfenite were acquired at spinning speeds from 10 kHz to 22.5 kHz. The 207Pb chemical shift of wulfenite is known to exhibit a signiﬁcant change with increasing temperature [27]. Accordingly, the 207Pb MAS NMR spectra at 22.5 kHz spinning speed (Fig. 6a) shows the isotropic band at 2006:6 ppm, while the extrapolation of the squared spinning frequency to zero spinning shown in Fig. 6b results in an isotropic shift of δiso ¼ 2015 ppm, in perfect agreement with the single-crystal NMR values. The absence of a noticeable rotational sideband pattern (due to the small anisotropy of the 207 Pb CS tensor) precludes CS tensor eigenvalue determination using a HERZFELD–BERGER analysis [25]. Instead, the eigenvalues were estimated from a static 207Pb spectra of polycrystalline wulfenite, shown in Fig. 6c. To this end, the recorded spectra was compared to various spectra calculated with the SIMPSON package [28]. The best agreement between the experimental and computed spectra was found using the eigenvalues given in Table 2(right). While these eigenvalues are very similar to those derived from single-crystal NMR, their error margins are much larger, conﬁrming again that NMR of single crystals is the ‘gold standard’ [29] for tensor determination.

It may be noted that the error margins on the orientations of the goniometer axes are much smaller for the phosgenite system than for the wulfenite. We attribute this to the fact that the free ﬁt parameters for phosgenite are heavily overdetermined, whereas for wulfenite, the number of free parameters exactly matches the data supplied by the experiment, as explained above. These small errors further illustrate the suitability of the phosgenite system for determination of goniometer axes orientation. The CS tensor of wulfenite is also listed in Table 2. In contrast to phosgenite, we ﬁnd the anisotropy of δPAS in the wulfenite structure to be small (Δδ ¼ 117 ppm), which indicates that the electron lone pair at the lead atoms has predominantly s-character [21]. The asymmetry parameter ηCS ¼ 0, and the orientation of the eigenvectors are a consequence of ! ! the crystal symmetry, with d 33 aligning exactly along the c axis and d 11 ,

δPAS 11 δPAS 22 δPAS 33

207

ð128 20Þ ppm 0 ð2015 10Þ ppm ð2015 2Þ ppmb

a

Indeterminate in the ab plane because of the cylindrical symmetry of the tensor. b Taken from shift extrapolation over squared MAS frequencies, see Fig. 6b. 22

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Solid State Nuclear Magnetic Resonance 103 (2019) 17–24

generated by the WYCKOFF multiplicity, allowing the determination of both 207Pb CS tensor and axis orientation from a single rotation pattern. Use of the previously determined CS tensor of phosgenite made it feasible to resolve the relative orientation of the two rotation axes for the simultaneous measurements of the two mineral crystals. This orientation difference could be applied as an additional ﬁt constraint for the data ﬁt of wulfenite, making the determination of the full 207Pb CS tensor possible, without using other crystal alignment techniques. The approach presented herein for determination of the full chemical shift tensor in compounds with arbitrary crystal lattice symmetry and WYCKOFF multiplicity is, of course, not limited to 207Pb NMR, but may be applied to any nucleus with spin I ¼ 1=2. It can also easily be adapted to determine the EFG tensor for nuclei with spin I > 1=2, thus expanding the capabilities of single-crystal NMR spectroscopy.

temperature (about 20 ∘ C) with a clip-on goniometer build by NMR Service GmbH (Erfurt, Germany). This goniometer was installed on a wide-bore Bruker static NMR probe equipped with a home-made solenoid coil with 6 mm inner diameter. For the MAS spectra, a polycrystalline sample was prepared by crushing single crystals of phosgenite and wulfenite with an agate mortar, and measured using a 2.5 mm rotor. All spectra were referenced indirectly to 1H in 100% TMS at 0:1240 ppm, which is equivalent to the common Pb(NO3)2-powder referencing at δiso ¼ 3492 ppm. The global ﬁt of the rotation pattern, including the ! orientation of the magnetic ﬁeld vector b 0 in the CRY frame, was performed with the program IGOR PRO 7 from WaveMatrics Inc., which delivers excellent non-linear ﬁtting performance.

4. Experimental details

The authors would like to thank Igor Moudrakovski (MPI-FKF, Stuttgart) for support and advice during the phosgenite single-crystal data acquisition. We are also grateful to Constantin Hoch (LMU Munich) for helpful discussions regarding crystallographic symmetry operations.

Acknowledgements

207

Pb NMR spectra were acquired on a Bruker AVANCE-III 500 Spectrometer at LMU Munich, with Larmor frequency ν0 (207Pb) ¼ 104.63 MHz, using echo acquisition with 2= 4 μs pulse duration to minimize base line roll [30], and a recycle delay of 60 s. The angular dependent single-crystal spectra were acquired at room

! Appendix. b 0 in the CRY frame

Fig. 7. : Coordinate system and relevant vectors for analysis of the phosgenite rotation patterns: a, b, and c are the unit vectors of the tetragonal crystal system (CRY), ! u and ! v are auxiliary unit vectors needed to describe the g is the unit vector along the goniometer axis with its orientation deﬁned by the polar angles θg and ϕg , and ! ! step-wise rotation of the magnetic ﬁeld unit vector b 0 by rotation angle φ according to Eq. (4), with φΔ being the initial offset angle.

! For evaluation of the rotation patterns, the step-wise rotation of the magnetic ﬁeld vector b 0 in a plane perpendicular to the goniometer axis ! g ! needs to be speciﬁed. In spherical coordinates, the orientation of the unit vector g in the tetragonal CRY frame is described by the angles θg and ϕg :

1 0 sinθg cosϕg ! @ g ¼ sinθg sinϕg A cosθg

(13)

u and ! v (see However, θg and ϕg are themselves variables that need to be determined from ﬁtting the data. By deﬁning two auxiliary unit vectors ! ! ! u Fig. 7), an expression for the movement of b 0 in terms of the spherical coordinates of g can be constructed. In previously published work [14,26], ! and ! v were deﬁned starting from a projection of the crystallographic c axis. Due to the fact that the goniometer axis ! g ðwÞ of the ﬁrst rotation pattern of 1

wulfenite was comparatively close to the crystallographic c axis (with θg ¼ 17:3∘ ), this construct did not work well for the ﬁtting routine, which failed to ! converge. Therefore, in the current work, we used b ¼ ð0; 1; 0Þ along the crystallographic b axis as the reference vector instead, leading to the following expressions for the auxiliary vectors: 23

O.E.O. Zeman et al.

! v ¼

1 ! qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð! g b Þ; cos2 θg þ sin2 θg cos2 ϕg

! u ¼

1 ! ! v ! g ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð! g bÞ! g 2 2 2 cos θg þ sin θg cos ϕg

Solid State Nuclear Magnetic Resonance 103 (2019) 17–24

(14)

! 1 ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ b ! g cosθg 2 2 2 cos θg þ sin θg cos ϕg ! The thus deﬁned vectors ! u and ! v are employed in Eq. (4) to express the step-wise rotation of b 0 in the CRY frame.

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