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Homonuclear Vanadium-51 Dipolar Couplings in Inorganic Solids Obtained via Hahn Spin Echo Decay NMR Spectroscopy
Solid State Nuclear Magnetic Resonance 19, 73–86 (2001) doi:10.1006/snmr.2001.0023, available online at http://www.idealibrary.com on
Homonuclear Vanadium-51 Dipolar Couplings in Inorganic Solids Obtained via Hahn Spin Echo Decay NMR Spectroscopy Becky Gee Department of Chemistry and Biochemistry, Long Island University–Brooklyn Campus, Brooklyn, New York 11201 Received November 30, 2000; revised February 26, 2001; published online May 3, 2001 Dipolar dephasing of the magnetization following a Hahn spin echo pulse sequence potentially provides a quantitative means for determining the dipolar second moment in solids. In this work, the possibility of employing Hahn spin echo decay spectroscopy to obtain quantitative 51 V–51 V dipolar second moments is explored. Theoretical spin echo response curves are compared to experimental ones for a collection of crystalline vanadium-containing compounds. This work suggests that 51 V dipolar second moments can be obtained by selectively exciting the central m = 1/2 → −1/2 by a Hahn echo sequence for vanadate compounds with line broadening no greater than approximately 220 ppm. For vanadates with greater broadening of the central transition due to chemical shift, second-order quadrupolar, and dipolar interactions, off-resonance effects lead to an oscillatory time dependence of the spin echo. Experimentally determined second moments of the normalized echo decay intensities lie within 10–33% of the calculated values if the second moments are extrapolated to zero evolution time due to the time scale dependence of spin exchange among neighboring vanadium nuclei. Alternatively, the second moments can be obtained to within 10–25% of the calculated values if the broadening of the central transition due to chemical shift and second-order quadrupolar effects can be estimated. © 2001 Academic Press Key Words: 51 V NMR; spin echo decay; dipolar coupling.
INTRODUCTION Dipolar techniques in solid-state NMR have proven highly useful for providing structural information in solids. In particular, techniques well suited to determining distances between I = 1/2 homonuclei have been developed and applied to biological and inorganic solids. Homodipolar techniques performed under magic angle spinning (MAS) conditions such as dipolar restoration at the magic angle (DRAMA) [1], controlled simple excitation for dephasing rotational amplitudes (SEDRA) [2,3], or RF-driven dipolar recovery (RFDR) [4], rotational resonance (R2) [5], and combined rotation with nutation (CROWN) [6] have been used to determine distances between homonuclear I = 1/2 spin pairs in labeled biological solids. Executed on static samples, the Hahn spin echo pulse sequence, 90◦ –t1 –180◦ , has often been used to determine homonuclear dipolar second moments for I = 1/2 nuclei such as 31 P–31 P in disordered inorganic solids [7–10]. 73 0926-2040/01 $35.00
Copyright © 2001 by Academic Press All rights of reproduction in any form reserved.
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Certainly the ability to acquire homonuclear dipole–dipole coupling information from noninteger quadrupolar nuclei would be a powerful method for elucidating the short-range structure of disordered solids. In this regard, there have been a number of recent efforts. Multiple-quantum excitation under MAS conditions has been used to selectively detect nearest neighboring 23 Na spin pairs in Na3 ZrO3 [11]. The second moment of the Hahn spin echo decay response of noninteger quadrupolar nuclei has been derived by a density matrix treatment [12]. Experimental homonuclear dipole-dipole second moments among 23 Na (I = 3/2) nuclei in crystalline and amorphous solids have been obtained via Hahn spin echo decay spectroscopy [13–15]. There certainly exist many noninteger quadrupolar nuclei where the measurement of their homonuclear dipole–dipole couplings in disordered solids can provide a better understanding of local atomic structure. One nucleus in particular is vanadium51 (I = 7/2), which has been the subject of several recent solid-state NMR studies. Quadrupole and chemical shift tensor elements have been determined by fitting 51 V MAS and static wide-line spectra [16–20], while qualitative interpretations of 51 V powder lineshapes and MAS data have provided descriptions of the local bonding environments in supported and bulk oxide catalysts [21–24]. To date, there are no reported attempts to obtain 51 V–51 V homonuclear dipole– dipole couplings in solids. This information would certainly be helpful in further investigations of vanadium structural environments in disordered solids. In this work, the possibility of employing Hahn spin echo decay spectroscopy to obtain quantitative 51 V–51 V dipolar second moments is explored. Theory and Methodology Measurement of the spin echo intensity as a function of the evolution period, 2t1 , in a π/2–t1 –π–acquire pulse sequence potentially provides a means for quantitatively determining dipolar second moments. Experimentally, for short evolution periods and multispin couplings, the normalized echo intensity, I/I0 , can generally be described by a Gaussian function, M I = exp − 2E (2t1 )2 I0 2
(1)
where M2E is the second moment of the spin echo decay arising from homonuclear dipole–dipole couplings and 2t1 is the evolution period of the spin echo pulse sequence. Theoretically, the second moment of a train of echoes is defined by M2E
d 2 tr ρ(2t1 )I+ } = 2t1 =0 d(2t1 )2
(2)
where the density operator, ρ(2t1 ), evolves under the influence of the RF pulses and the appropriate internal interaction Hamiltonians at the relevant periods during
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the pulse sequence. Considering the case of pure homodipolar coupling among neighboring spins, the spin echo response for the case (1)
(2)
||Hz || ||HQ || > ||HRF || > ||Hd || ||HQ ||
(3)
which corresponds to selective excitation of the central transition (m = 1/2 → −1/2), has been calculated by Haase and Oldfield for non-integer quadrupolar nuclei [12]. It has been shown that the homonuclear dipolar interactions among spins with magnetic quantum numbers m = ±1/2 govern the Hahn spin echo decay of the central transition, and for “like” spins the second moment of this decay is given by M2E = EL
µo 4π
2
γ 4 2
3 (1 − 3 cos2 θij ) 2 2 rij3 i>j
(4)
where the prefactor, EL , is a spin quantum number dependent term defined in reference [12]. The spin echo train can be regarded as being primarily governed by the dipolar interactions among the population of spins in the m = ±1/2 states. The contribution to M2E due to flip-flops between the m = ±1/2 and m = ±3/2 states is also included in the prefactor EL . Taking the powder average of Eq. (4) and considering only those spins in the |m| = 1/2 state, the theoretical second moment for spin I = 7/2 nuclei is then given by µ 2
M2E = 47531
o
4π
γ 4 2
i>j
1 rij
6
(5)
Equation (5) describes the homonuclear dipole–dipole couplings among isochromatic nuclei, i.e., both the diagonal and the off-diagonal parts of the dipolar Hamiltonian are relevant. If the resonance frequencies of the nuclei are sufficiently different due to the presence of chemical shift inequivalence and/or second-order quadrupolar interactions, the spin exchange term in the dipolar Hamiltonian may be suppressed for certain, if not all, crystalline orientations in a powdered sample. This results in a theoretical second moment that is reduced by 1/81 of that calculated by Eq. (5). Thus, for a disordered solid or polycrystalline material, suppression of the zero quantum (“flip-flop”) spin transitions for spatially close 51 V nuclei in the m = ±1/2 spin states may impose an important complication. Another limitation may arise if the 51 V nuclei experience dipolar coupling to nonresonant S nuclei which are strongly coupled among themselves. The S spins may influence the spin echo decay in two ways. First, spin exchange among observe (I) nuclei may be suppressed due to modification of the local field by the presence of neighboring S nuclei, resulting in a decelerated echo decay of the I nuclei. Second, spin exchange among strongly coupled S nuclei may decrease the phase coherence in the observe spin system, leading to an acceleration of the echo intensity. Accordingly, it is then inappropriate to consider only the homonuclear dipolar couplings. Calculation of the Hahn echo decay under these circumstances is complicated due
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to the relative influence of each factor. Thus, a common feature of the materials selected for this work is the absence or weakness of dipolar interactions between 51 V and other magnetic nuclei. All compounds selected posses large quadrupole splittings, therefore allowing for selective excitation of the central m = 1/2 → −1/2 transition. However, exceedingly large broadening of the central transition due to chemical shift and/or second-order quadrupolar anisotropies may pose difficulties. For large anisotropies, nonuniform excitation of the central transition will lead to an oscillatory dependence of the echo intensity [25]. Our choice of model compounds was limited to orthovanadates, with typical full widths at half height of 20–140 ppm and pyrovanadates with |δ33 − δ11 | typically ranging from 170 to 220 ppm. Attempts to obtain meaningful echo data from compounds with much greater broadening of the central transition were hampered by off-resonance effects. EXPERIMENTAL Sample Preparation and Specifications The zinc pyrovanadate was prepared by dissolving the appropriate molar ratios of NH4 VO3 (Aldrich, 99.99%) and Zn(NO3 )2 ·6H2 O (Aldrich, 98%) in deionized water. The aqueous solution was thoroughly mixed, followed by evaporation at approximately 110◦ C. The resulting material was heated in air to 630◦ C at a rate of 20◦ C/min. The material was held at 630◦ C for 5 h, cooled at approximately 60◦ C/min to 30◦ C, allowed to furnace cool to ambient temperature, and removed from the furnace. The purity of the resulting compound was confirmed by x-ray powder diffraction (Scintag XI powder diffractometer). The following crystalline compounds were commercially obtained with the following specifications: bismuth orthovanadate (Alfa Aesar, 99.9%), barium orthovanadate (Alfa Aesar, 99.9%), magnesium orthovanadate (Great Western Inorganics, 99%), and potassium orthovanadate (Alfa Aesar, 99%). Samples of magnesium pyrovanadate, strontium pyrovanadate, and lanthanum orthovanadate were provided by the laboratory of Professor Hellmut Eckert (Chemistry, Westfälische Wilhelms-Universit¨at). Solid State NMR Hahn spin echo and one-dimensional nutation 51 V NMR experiments were performed on a Varian Unity Plus 300 solids spectrometer operating at approximately 78.865 MHz with a wide-line probe from Varian. Samples were packed in quartz ampoules under ambient conditions. To minimize resonance offset effects, the spectrometer transmitter frequency was carefully adjusted. For each sample, selective excitation of the central m = 1/2 → −1/2 transition was achieved by adjusting the RF power level and performing a one-dimensional nutation NMR experiment with typical pulse lengths of 1–30 µs incremented at 1–2 µs. A small fraction of the crystallites may be oriented in the static magnetic field such that their satellite transitions lie within the excitation bandwidth of the pulse.
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V DIPOLAR COUPLINGS OBTAINED VIA HAHN SPIN ECHO SPECTROSCOPY 2
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However, typical quadrupolar coupling constants (Cq = e hqQ ) for the vanadates examined in this work range from 0.75 to 5.2 MHz, with quadrupole frequen3Cq ), which therefore range from approximately 54 to 371 kHz. cies (νq = 2I(2I−1) For vanadates with approximate quadrupolar frequencies of 54 kHz, selective 90◦ pulse lengths used in the spin echo experiments were about 22 µs, leading to an excitation bandwidth of roughly 11 kHz. Selective 90◦ pulse lengths of approximately 6 µs (excitation bandwidth of about 42 kHz) were used for those vanadates with approximate quadrupole frequencies of 371 kHz. The excitation bandwidths were therefore 5–10 smaller than those of the satellite spectrum, thus minimizing contributions to the echo decay due to excitation of spins in the m > |1/2| states. Typical acquisition parameters for the spin-echo experiments were dwell time 0.5–1 µs, selective 90◦ pulses of 5.5–22 µs, recycle delays of 10–60 s, and 16–64 scans. A 16-step phase cycle was used to remove weak feed-through signals arising from imperfect 180◦ pulses [26]. Crystallographic data from the literature (Table 1) were used to determine the theoretical 51 V–51 V dipolar second moment of each crystallographically distinct vanadium site within a 10-Å radius. The average second moment is reported for those compounds with crystallographically inequivalent vanadium sites. These include Sr2 V2 O7 with four sites and a maximum difference of ±4% between the M2E value for a site and the reported average and Ba3 (VO4 ) with three crystallographically distinct sites and a difference of < 1% between the M2E value for a site and the reported average.
Calculated and Experimental
Compound K3 VO4 BiVO4 Mg3 (VO4 )2 Ba3 (VO4 )2 LaVO4 Zn2 V2 O7 Sr2 V2 O7 Mg2 V2 O7 a
TABLE 1 V–51 V M2E Values with Literature References for Crystallographic Data 51
M2E (106 rad2 /s2 ) calculated 45 216 264 116 155 240 262 167
M2E (106 rad2 /s2 ) experimental by extrapolation to 2t1 = 0
M2E (exp) M2E (calc)
5.0 ± 0.3 5.2 ± 0.3a 20.8 ± 0.2 35.1 ± 0.9 13.7 ± 0.7 22.2 ± 0.3 28.6 ± 0.9 28.2 ± 1.2 21.2 ± 0.5
1.11 1.14 0.96 1.33 1.18 1.43 1.19 1.08 1.27
Ref. 27 28 29 30 31 32 33 34
Results are given for two independently measured data sets. Note. The experimental values are determined by linear extrapolation of the experimental second moment as a function of evolution period.
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RESULTS Shown in Figs. 1 and 2 are, for each compound, the experimental normalized spin echo intensities as a function of evolution time 2t1 compared to spin echo decay curves computed from the crystal structures using Eqs. (1) and (5). The spin echo decay behavior of all compounds studied is fairly similar. At long evolution periods the spin echo intensities clearly exhibit strong deviations from Gaussian behavior. However, at relatively short evolution times there is fairly good agreement between the calculated and experimental data (typically 0 < 2t1 ≤ 014 ms). Table 1 and Fig. 5 present the experimental and calculated second moments obtained by extrapolation of the short evolution time data, as discussed in the following section and presented in Fig. 3 and 4. Experimental second moments obtained by merely
FIG. 1. Normalized experimental spin echo intensities (circles) as a function of evolution time, 2t1 , for four vanadate model compounds. Spin echo decays calculated according to Eqs. (1) and (5) and crystallographic data are shown as solid lines.
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FIG. 2. Normalized experimental spin echo intensities (circles and squares) as a function of evolution time, 2t1 , for four vanadate model compounds. Two independently measured data sets are presented for potassium orthovanadate. The errors for data points represented as open squares are approximately the same as those for the data shown as open circles. Spin echo decays calculated according to Eqs. (1) and (5) and crystallographic data are shown as solid lines.
restricting the data analysis to short evolution periods are compared to calculated ones in Table 2 and Fig. 6. As will be discussed, both methods of data analysis give reasonably good experimental results in comparison to theory. DISCUSSION Similar to previous spin echo studies of 23 Na-containing crystalline compounds [14], there is a noticeable deceleration of the experimental 51 V decay curves compared to the calculated ones at long evolution periods. Fairly good agreement between experimental and calculated data is achieved at short evolution periods
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(approximately 0 < 2t1 ≤ 014 ms) with the experimental data fitting a Gaussianshaped decay well. It is possible that deviations at longer evolution periods result from partial suppression of coupling of the transverse components of neighboring nuclear spins via the B-term of the dipolar Hamiltonian since their difference in resonance frequencies exceeds the strength of the dipolar coupling. This situation is certainly possible for 51 V in powdered diamagnetic solids where a distribution of chemical shifts, second-order quadrupolar shifts, and/or noncoincident chemical shift tensor orientations leads to magnetic inequivalences. The spin echo decay rate also depends on the length of the evolution time 2t1 in the 90◦ –t1 –180◦ –t1 pulse sequence. Spin exchange among nonisochromatic nuclei contributes to the dephasing of the spin echo as long as the condition 2t1 < ("ω)−1
(6)
is met, where "ω(Hz) is the difference in resonance frequencies between the interacting spins. On this timescale, the transverse components of the 51 V spins are not decoupled. Accordingly, Eq. (5) is expected to be applicable in the limit of zero evolution time. In principle, this “zero evolution time limit” can be approached by extrapolation of the experimental M2E values as a function of 2t1 to the limit of zero evolution time. Shown in Figs. 3 and 4 are the experimental M2E values for each crystalline compound obtained by a least-squares fit to all experimental data points from 2t1 max to the shortest 2t1 experimental normalized intensity (i.e., each M2E value was obtained from a least-squares fit of all normalized echo data ranging from 0 ≤ 2t1 ≤ 2t1 -max). A phenomenological linear extrapolation to zero evolution time is also shown in Figs. 3 and 4 for each vanadate. Each compound shows a consistent trend of a linearly increasing M2E as a function of decreasing evolution period. The range of evolution periods examined includes those which give a good fit to the phenomenological linear function. The M2E values obtained by extrapolation are also presented in Table 1 and Fig. 5 with the calculated values. In seven of the eight compounds studied the experimental M2E lies within approximately 10–33% of the calculated one, with experimental M2E values for four of these seven within less than ±20% of the calculated ones. Only lanthanum orthovanadate shows a deviation larger than 33%, with an error of about +43%. Accordingly, this method of extrapolation to zero evolution time provides a reasonable method for analysis of the data. However, by examination of Fig. 5, in all but one of the eight compounds studied (BiVO4 ), extrapolation of the data to zero evolution time overestimates the second moment. As an alternative approach, analysis of the envelope of the echo decay may merely be restricted to short evolution times. This approach produces quite good agreement between experiment and theory with ≤10% error for six of the eight compounds and errors of 18 and 26% for BiVO4 and Sr2 V2 O7 , respectively (Fig. 6 and Table 2). Although this method of data analysis provides quite reasonable agreement between theory and experiment, the range of evolution times analyzed differs for each compound (i.e., each M2E is obtained by analysis of all spin echo data points at times 0 ≤ 2t1 ≤ 2t1 -max. However, 2t1 -max. differs among the compounds studied).
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V DIPOLAR COUPLINGS OBTAINED VIA HAHN SPIN ECHO SPECTROSCOPY
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FIG. 3. Second moment values (•) as a function of the maximum evolution time, 2t1 -max, used to determine M2E . Each M2E value was obtained from a least-squares fit of all normalized echo data ranging from 0 ≤ 2t1 ≤ 2t1 -max. The calculated values ( ) are indicated at 2t1 = 0. Phenomenological linear extrapolations to zero evolution times are shown as solid lines.
This difference may be due to the degree to which spin exchange is restricted in each compound. Figure 7 shows the approximate chemical shift and secondorder quadrupolar broadening of the 51 V m = 1/2 → −1/2 central transition as a function of the maximum evolution time used to determine each M2E shown in Fig. 6. As seen, the range of evolution times examined can be loosely correlated with the broadening of the central transition due to chemical shift and second-order quadrupolar anisotropies and dispersions. The range of suitable evolution times decreases as a function of increasing broadening suggesting that spin exchange among neighboring nuclei is increasingly restricted as there exists a larger population of crystallite orientations such that neighboring nuclei have resonance
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FIG. 4. Second moment values (•) as a function of the maximum evolution time, 2t1 -max, used to determine M2E . Each M2E value was obtained from a least-squares fit of all normalized echo data ranging from 0 ≤ 2t1 ≤ 2t1 -max. The calculated values ( ) are indicated at 2t1 = 0. Phenomenological linear extrapolations to zero evolution times are shown as solid lines. For K3 VO4 , the best linear fit for the data represented by open squares is indicated by the dashed line, while the fit for data plotted as open circles is indicated by the solid line.
frequencies that exceed the dipole–dipole coupling strength. It is also noted that if the maximum 2t1 value is varied by about ±20–40 µs, the resulting experimental second moment values are still within approximately 10–25% of the calculated values. The suitable ranges for data analysis are also shown in Fig 7. This method of analysis suggests the possibility of establishing a calibration plot to determine the range of suitable evolution times that should be considered in the data analysis. With the exception of K3 VO4 , second moments for those compounds with chemical shift and quadrupolar broadening of approximately <110 ppm, such as Mg2 V2 O7 , can be obtained by restricting analysis of the spin echo data to
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V DIPOLAR COUPLINGS OBTAINED VIA HAHN SPIN ECHO SPECTROSCOPY
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FIG. 5. Comparison of calculated and experimental 51 V–51 V dipolar M2E values determined by linear extrapolation of M2E values to zero evolution time.
FIG. 6. Comparison of calculated and experimental 51 V–51 V dipolar M2E values determined by restricting the data analysis to short evolution periods as indicated in Fig. 7 and Table 2.
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Experimental
Compound K3 VO4 BiVO4 Mg3 (VO4 )2 Ba3 (VO4 )2 LaVO4 Zn2 V2 O7 Sr2 V2 O7 Mg2 V2 O7 a
51
TABLE 2 V–51 V M2E Values Obtained by Restricting the Data Analysis to Short Evolution Periods as Indicated M2E (106 rad2 /s2 ) experimental by analysis of short 2t1 data
M2E (exp) M2E (calc)
Evolution period analyzed 0 < 2ti ≤ x(ms)
4.0 ± 0.7a 4.4 ± 0.6 17.8 ± 0.9 25.6 ± 1.9 11.1 ± 1.0 15.2 ± 1.4 22.6 ± 0.5 19.5 ± 1.3 16.1 ± 1.4
0.89 0.98 0.82 0.97 0.96 0.98 0.94 0.74 0.96
x = 028 028 014 012 018 014 010 010 014
Results are given for two independently measured data sets.
FIG. 7. Estimated line broadening of the 51 V central m = 1/2 → −1/2 transition due to chemical shift and second-order quadrupolar effects as a function of the maximum evolution time used to determine M2E as in Fig. 6. For symmetric resonances the broadening is given by the full width at half maximum, while for anisotropic resonances the broadening was estimated as |δ|| − δ⊥ |. Circles indicate the maximum time of the evolution period used to determine the dipolar M2E values given in Fig. 7. The range of maximum evolution times that give experimental M2E values within 10–25% of the calculated values is also plotted.
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evolution times between 0 < 2t1 ≤ 180 µs. For those compounds with broadening >110 ppm, such as Sr2 V2 O7 , M2E may be obtained by limiting the data analysis to 0 < 2t1 ≤ 120µs. CONCLUSIONS This work suggests that using a Hahn spin echo pulse sequence 51 V dipolar second moments can be obtained by exciting the central 51 V transition for compounds with anisotropies of approximately ≤220 ppm. Those with greater line broadening exhibit an oscillatory time dependence of the echo intensity due to off-resonance effects. Accuracies of approximately 10–33% can be achieved if the second moments are linearly extrapolated to zero evolution time. Alternatively, second moments of the echo intensity decay can be obtained within 10–25% of the calculated ones if broadening of the central transition due to chemical shift and second order quadrupolar interactions can be estimated. Although the method of extrapolation tends to overestimate the second moment of spin echo decay intensity, this approach does not require knowledge of the chemical shift or second quadrupolar broadening and is, thus, likely more suitable and provides fairly reasonable results. ACKNOWLEDGMENTS B.G. thanks Professor Ruth Stark and Dr. Hsin Wang (Chemistry, College of Staten Island of the City University of New York) for use of the solid-state NMR instrumentation. B.G. also thanks the laboratory of Professor Hellmut Eckert (Chemistry, Westf¨alische Wilhelms-Universit¨at) for providing the magnesium pyrovanadate, lanthanum orthovanadate, and strontium pyrovanadate samples. Ms. Claudia (n´ee Gheorghe) Liberman (Chemistry, Long Island University) is appreciated for her preparation and characterization of the zinc pyrovanadate and her search of the crystallographic literature. This work was partially supported by the Research Released Time Committee and the Trustees of Long Island University and by the American Association of University Women Educational Foundation American Summer/Short-term Research Publications Grant. Acknowledgement is made to the Donors of The Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research (ACS-PRF 34967-GB5).
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Homonuclear Vanadium-51 Dipolar Couplings in Inorganic Solids Obtained via Hahn Spin Echo Decay NMR Spectroscopy Becky Gee Department of Chemistry and Biochemistry, Long Island University–Brooklyn Campus, Brooklyn, New York 11201 Received November 30, 2000; revised February 26, 2001; published online May 3, 2001 Dipolar dephasing of the magnetization following a Hahn spin echo pulse sequence potentially provides a quantitative means for determining the dipolar second moment in solids. In this work, the possibility of employing Hahn spin echo decay spectroscopy to obtain quantitative 51 V–51 V dipolar second moments is explored. Theoretical spin echo response curves are compared to experimental ones for a collection of crystalline vanadium-containing compounds. This work suggests that 51 V dipolar second moments can be obtained by selectively exciting the central m = 1/2 → −1/2 by a Hahn echo sequence for vanadate compounds with line broadening no greater than approximately 220 ppm. For vanadates with greater broadening of the central transition due to chemical shift, second-order quadrupolar, and dipolar interactions, off-resonance effects lead to an oscillatory time dependence of the spin echo. Experimentally determined second moments of the normalized echo decay intensities lie within 10–33% of the calculated values if the second moments are extrapolated to zero evolution time due to the time scale dependence of spin exchange among neighboring vanadium nuclei. Alternatively, the second moments can be obtained to within 10–25% of the calculated values if the broadening of the central transition due to chemical shift and second-order quadrupolar effects can be estimated. © 2001 Academic Press Key Words: 51 V NMR; spin echo decay; dipolar coupling.
INTRODUCTION Dipolar techniques in solid-state NMR have proven highly useful for providing structural information in solids. In particular, techniques well suited to determining distances between I = 1/2 homonuclei have been developed and applied to biological and inorganic solids. Homodipolar techniques performed under magic angle spinning (MAS) conditions such as dipolar restoration at the magic angle (DRAMA) [1], controlled simple excitation for dephasing rotational amplitudes (SEDRA) [2,3], or RF-driven dipolar recovery (RFDR) [4], rotational resonance (R2) [5], and combined rotation with nutation (CROWN) [6] have been used to determine distances between homonuclear I = 1/2 spin pairs in labeled biological solids. Executed on static samples, the Hahn spin echo pulse sequence, 90◦ –t1 –180◦ , has often been used to determine homonuclear dipolar second moments for I = 1/2 nuclei such as 31 P–31 P in disordered inorganic solids [7–10]. 73 0926-2040/01 $35.00
Copyright © 2001 by Academic Press All rights of reproduction in any form reserved.
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Certainly the ability to acquire homonuclear dipole–dipole coupling information from noninteger quadrupolar nuclei would be a powerful method for elucidating the short-range structure of disordered solids. In this regard, there have been a number of recent efforts. Multiple-quantum excitation under MAS conditions has been used to selectively detect nearest neighboring 23 Na spin pairs in Na3 ZrO3 [11]. The second moment of the Hahn spin echo decay response of noninteger quadrupolar nuclei has been derived by a density matrix treatment [12]. Experimental homonuclear dipole-dipole second moments among 23 Na (I = 3/2) nuclei in crystalline and amorphous solids have been obtained via Hahn spin echo decay spectroscopy [13–15]. There certainly exist many noninteger quadrupolar nuclei where the measurement of their homonuclear dipole–dipole couplings in disordered solids can provide a better understanding of local atomic structure. One nucleus in particular is vanadium51 (I = 7/2), which has been the subject of several recent solid-state NMR studies. Quadrupole and chemical shift tensor elements have been determined by fitting 51 V MAS and static wide-line spectra [16–20], while qualitative interpretations of 51 V powder lineshapes and MAS data have provided descriptions of the local bonding environments in supported and bulk oxide catalysts [21–24]. To date, there are no reported attempts to obtain 51 V–51 V homonuclear dipole– dipole couplings in solids. This information would certainly be helpful in further investigations of vanadium structural environments in disordered solids. In this work, the possibility of employing Hahn spin echo decay spectroscopy to obtain quantitative 51 V–51 V dipolar second moments is explored. Theory and Methodology Measurement of the spin echo intensity as a function of the evolution period, 2t1 , in a π/2–t1 –π–acquire pulse sequence potentially provides a means for quantitatively determining dipolar second moments. Experimentally, for short evolution periods and multispin couplings, the normalized echo intensity, I/I0 , can generally be described by a Gaussian function, M I = exp − 2E (2t1 )2 I0 2
(1)
where M2E is the second moment of the spin echo decay arising from homonuclear dipole–dipole couplings and 2t1 is the evolution period of the spin echo pulse sequence. Theoretically, the second moment of a train of echoes is defined by M2E
d 2 tr ρ(2t1 )I+ } = 2t1 =0 d(2t1 )2
(2)
where the density operator, ρ(2t1 ), evolves under the influence of the RF pulses and the appropriate internal interaction Hamiltonians at the relevant periods during
51
V DIPOLAR COUPLINGS OBTAINED VIA HAHN SPIN ECHO SPECTROSCOPY
75
the pulse sequence. Considering the case of pure homodipolar coupling among neighboring spins, the spin echo response for the case (1)
(2)
||Hz || ||HQ || > ||HRF || > ||Hd || ||HQ ||
(3)
which corresponds to selective excitation of the central transition (m = 1/2 → −1/2), has been calculated by Haase and Oldfield for non-integer quadrupolar nuclei [12]. It has been shown that the homonuclear dipolar interactions among spins with magnetic quantum numbers m = ±1/2 govern the Hahn spin echo decay of the central transition, and for “like” spins the second moment of this decay is given by M2E = EL
µo 4π
2
γ 4 2
3 (1 − 3 cos2 θij ) 2 2 rij3 i>j
(4)
where the prefactor, EL , is a spin quantum number dependent term defined in reference [12]. The spin echo train can be regarded as being primarily governed by the dipolar interactions among the population of spins in the m = ±1/2 states. The contribution to M2E due to flip-flops between the m = ±1/2 and m = ±3/2 states is also included in the prefactor EL . Taking the powder average of Eq. (4) and considering only those spins in the |m| = 1/2 state, the theoretical second moment for spin I = 7/2 nuclei is then given by µ 2
M2E = 47531
o
4π
γ 4 2
i>j
1 rij
6
(5)
Equation (5) describes the homonuclear dipole–dipole couplings among isochromatic nuclei, i.e., both the diagonal and the off-diagonal parts of the dipolar Hamiltonian are relevant. If the resonance frequencies of the nuclei are sufficiently different due to the presence of chemical shift inequivalence and/or second-order quadrupolar interactions, the spin exchange term in the dipolar Hamiltonian may be suppressed for certain, if not all, crystalline orientations in a powdered sample. This results in a theoretical second moment that is reduced by 1/81 of that calculated by Eq. (5). Thus, for a disordered solid or polycrystalline material, suppression of the zero quantum (“flip-flop”) spin transitions for spatially close 51 V nuclei in the m = ±1/2 spin states may impose an important complication. Another limitation may arise if the 51 V nuclei experience dipolar coupling to nonresonant S nuclei which are strongly coupled among themselves. The S spins may influence the spin echo decay in two ways. First, spin exchange among observe (I) nuclei may be suppressed due to modification of the local field by the presence of neighboring S nuclei, resulting in a decelerated echo decay of the I nuclei. Second, spin exchange among strongly coupled S nuclei may decrease the phase coherence in the observe spin system, leading to an acceleration of the echo intensity. Accordingly, it is then inappropriate to consider only the homonuclear dipolar couplings. Calculation of the Hahn echo decay under these circumstances is complicated due
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BECKY GEE
to the relative influence of each factor. Thus, a common feature of the materials selected for this work is the absence or weakness of dipolar interactions between 51 V and other magnetic nuclei. All compounds selected posses large quadrupole splittings, therefore allowing for selective excitation of the central m = 1/2 → −1/2 transition. However, exceedingly large broadening of the central transition due to chemical shift and/or second-order quadrupolar anisotropies may pose difficulties. For large anisotropies, nonuniform excitation of the central transition will lead to an oscillatory dependence of the echo intensity [25]. Our choice of model compounds was limited to orthovanadates, with typical full widths at half height of 20–140 ppm and pyrovanadates with |δ33 − δ11 | typically ranging from 170 to 220 ppm. Attempts to obtain meaningful echo data from compounds with much greater broadening of the central transition were hampered by off-resonance effects. EXPERIMENTAL Sample Preparation and Specifications The zinc pyrovanadate was prepared by dissolving the appropriate molar ratios of NH4 VO3 (Aldrich, 99.99%) and Zn(NO3 )2 ·6H2 O (Aldrich, 98%) in deionized water. The aqueous solution was thoroughly mixed, followed by evaporation at approximately 110◦ C. The resulting material was heated in air to 630◦ C at a rate of 20◦ C/min. The material was held at 630◦ C for 5 h, cooled at approximately 60◦ C/min to 30◦ C, allowed to furnace cool to ambient temperature, and removed from the furnace. The purity of the resulting compound was confirmed by x-ray powder diffraction (Scintag XI powder diffractometer). The following crystalline compounds were commercially obtained with the following specifications: bismuth orthovanadate (Alfa Aesar, 99.9%), barium orthovanadate (Alfa Aesar, 99.9%), magnesium orthovanadate (Great Western Inorganics, 99%), and potassium orthovanadate (Alfa Aesar, 99%). Samples of magnesium pyrovanadate, strontium pyrovanadate, and lanthanum orthovanadate were provided by the laboratory of Professor Hellmut Eckert (Chemistry, Westfälische Wilhelms-Universit¨at). Solid State NMR Hahn spin echo and one-dimensional nutation 51 V NMR experiments were performed on a Varian Unity Plus 300 solids spectrometer operating at approximately 78.865 MHz with a wide-line probe from Varian. Samples were packed in quartz ampoules under ambient conditions. To minimize resonance offset effects, the spectrometer transmitter frequency was carefully adjusted. For each sample, selective excitation of the central m = 1/2 → −1/2 transition was achieved by adjusting the RF power level and performing a one-dimensional nutation NMR experiment with typical pulse lengths of 1–30 µs incremented at 1–2 µs. A small fraction of the crystallites may be oriented in the static magnetic field such that their satellite transitions lie within the excitation bandwidth of the pulse.
51
V DIPOLAR COUPLINGS OBTAINED VIA HAHN SPIN ECHO SPECTROSCOPY 2
77
However, typical quadrupolar coupling constants (Cq = e hqQ ) for the vanadates examined in this work range from 0.75 to 5.2 MHz, with quadrupole frequen3Cq ), which therefore range from approximately 54 to 371 kHz. cies (νq = 2I(2I−1) For vanadates with approximate quadrupolar frequencies of 54 kHz, selective 90◦ pulse lengths used in the spin echo experiments were about 22 µs, leading to an excitation bandwidth of roughly 11 kHz. Selective 90◦ pulse lengths of approximately 6 µs (excitation bandwidth of about 42 kHz) were used for those vanadates with approximate quadrupole frequencies of 371 kHz. The excitation bandwidths were therefore 5–10 smaller than those of the satellite spectrum, thus minimizing contributions to the echo decay due to excitation of spins in the m > |1/2| states. Typical acquisition parameters for the spin-echo experiments were dwell time 0.5–1 µs, selective 90◦ pulses of 5.5–22 µs, recycle delays of 10–60 s, and 16–64 scans. A 16-step phase cycle was used to remove weak feed-through signals arising from imperfect 180◦ pulses [26]. Crystallographic data from the literature (Table 1) were used to determine the theoretical 51 V–51 V dipolar second moment of each crystallographically distinct vanadium site within a 10-Å radius. The average second moment is reported for those compounds with crystallographically inequivalent vanadium sites. These include Sr2 V2 O7 with four sites and a maximum difference of ±4% between the M2E value for a site and the reported average and Ba3 (VO4 ) with three crystallographically distinct sites and a difference of < 1% between the M2E value for a site and the reported average.
Calculated and Experimental
Compound K3 VO4 BiVO4 Mg3 (VO4 )2 Ba3 (VO4 )2 LaVO4 Zn2 V2 O7 Sr2 V2 O7 Mg2 V2 O7 a
TABLE 1 V–51 V M2E Values with Literature References for Crystallographic Data 51
M2E (106 rad2 /s2 ) calculated 45 216 264 116 155 240 262 167
M2E (106 rad2 /s2 ) experimental by extrapolation to 2t1 = 0
M2E (exp) M2E (calc)
5.0 ± 0.3 5.2 ± 0.3a 20.8 ± 0.2 35.1 ± 0.9 13.7 ± 0.7 22.2 ± 0.3 28.6 ± 0.9 28.2 ± 1.2 21.2 ± 0.5
1.11 1.14 0.96 1.33 1.18 1.43 1.19 1.08 1.27
Ref. 27 28 29 30 31 32 33 34
Results are given for two independently measured data sets. Note. The experimental values are determined by linear extrapolation of the experimental second moment as a function of evolution period.
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BECKY GEE
RESULTS Shown in Figs. 1 and 2 are, for each compound, the experimental normalized spin echo intensities as a function of evolution time 2t1 compared to spin echo decay curves computed from the crystal structures using Eqs. (1) and (5). The spin echo decay behavior of all compounds studied is fairly similar. At long evolution periods the spin echo intensities clearly exhibit strong deviations from Gaussian behavior. However, at relatively short evolution times there is fairly good agreement between the calculated and experimental data (typically 0 < 2t1 ≤ 014 ms). Table 1 and Fig. 5 present the experimental and calculated second moments obtained by extrapolation of the short evolution time data, as discussed in the following section and presented in Fig. 3 and 4. Experimental second moments obtained by merely
FIG. 1. Normalized experimental spin echo intensities (circles) as a function of evolution time, 2t1 , for four vanadate model compounds. Spin echo decays calculated according to Eqs. (1) and (5) and crystallographic data are shown as solid lines.
51
V DIPOLAR COUPLINGS OBTAINED VIA HAHN SPIN ECHO SPECTROSCOPY
79
FIG. 2. Normalized experimental spin echo intensities (circles and squares) as a function of evolution time, 2t1 , for four vanadate model compounds. Two independently measured data sets are presented for potassium orthovanadate. The errors for data points represented as open squares are approximately the same as those for the data shown as open circles. Spin echo decays calculated according to Eqs. (1) and (5) and crystallographic data are shown as solid lines.
restricting the data analysis to short evolution periods are compared to calculated ones in Table 2 and Fig. 6. As will be discussed, both methods of data analysis give reasonably good experimental results in comparison to theory. DISCUSSION Similar to previous spin echo studies of 23 Na-containing crystalline compounds [14], there is a noticeable deceleration of the experimental 51 V decay curves compared to the calculated ones at long evolution periods. Fairly good agreement between experimental and calculated data is achieved at short evolution periods
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BECKY GEE
(approximately 0 < 2t1 ≤ 014 ms) with the experimental data fitting a Gaussianshaped decay well. It is possible that deviations at longer evolution periods result from partial suppression of coupling of the transverse components of neighboring nuclear spins via the B-term of the dipolar Hamiltonian since their difference in resonance frequencies exceeds the strength of the dipolar coupling. This situation is certainly possible for 51 V in powdered diamagnetic solids where a distribution of chemical shifts, second-order quadrupolar shifts, and/or noncoincident chemical shift tensor orientations leads to magnetic inequivalences. The spin echo decay rate also depends on the length of the evolution time 2t1 in the 90◦ –t1 –180◦ –t1 pulse sequence. Spin exchange among nonisochromatic nuclei contributes to the dephasing of the spin echo as long as the condition 2t1 < ("ω)−1
(6)
is met, where "ω(Hz) is the difference in resonance frequencies between the interacting spins. On this timescale, the transverse components of the 51 V spins are not decoupled. Accordingly, Eq. (5) is expected to be applicable in the limit of zero evolution time. In principle, this “zero evolution time limit” can be approached by extrapolation of the experimental M2E values as a function of 2t1 to the limit of zero evolution time. Shown in Figs. 3 and 4 are the experimental M2E values for each crystalline compound obtained by a least-squares fit to all experimental data points from 2t1 max to the shortest 2t1 experimental normalized intensity (i.e., each M2E value was obtained from a least-squares fit of all normalized echo data ranging from 0 ≤ 2t1 ≤ 2t1 -max). A phenomenological linear extrapolation to zero evolution time is also shown in Figs. 3 and 4 for each vanadate. Each compound shows a consistent trend of a linearly increasing M2E as a function of decreasing evolution period. The range of evolution periods examined includes those which give a good fit to the phenomenological linear function. The M2E values obtained by extrapolation are also presented in Table 1 and Fig. 5 with the calculated values. In seven of the eight compounds studied the experimental M2E lies within approximately 10–33% of the calculated one, with experimental M2E values for four of these seven within less than ±20% of the calculated ones. Only lanthanum orthovanadate shows a deviation larger than 33%, with an error of about +43%. Accordingly, this method of extrapolation to zero evolution time provides a reasonable method for analysis of the data. However, by examination of Fig. 5, in all but one of the eight compounds studied (BiVO4 ), extrapolation of the data to zero evolution time overestimates the second moment. As an alternative approach, analysis of the envelope of the echo decay may merely be restricted to short evolution times. This approach produces quite good agreement between experiment and theory with ≤10% error for six of the eight compounds and errors of 18 and 26% for BiVO4 and Sr2 V2 O7 , respectively (Fig. 6 and Table 2). Although this method of data analysis provides quite reasonable agreement between theory and experiment, the range of evolution times analyzed differs for each compound (i.e., each M2E is obtained by analysis of all spin echo data points at times 0 ≤ 2t1 ≤ 2t1 -max. However, 2t1 -max. differs among the compounds studied).
51
V DIPOLAR COUPLINGS OBTAINED VIA HAHN SPIN ECHO SPECTROSCOPY
81
FIG. 3. Second moment values (•) as a function of the maximum evolution time, 2t1 -max, used to determine M2E . Each M2E value was obtained from a least-squares fit of all normalized echo data ranging from 0 ≤ 2t1 ≤ 2t1 -max. The calculated values ( ) are indicated at 2t1 = 0. Phenomenological linear extrapolations to zero evolution times are shown as solid lines.
This difference may be due to the degree to which spin exchange is restricted in each compound. Figure 7 shows the approximate chemical shift and secondorder quadrupolar broadening of the 51 V m = 1/2 → −1/2 central transition as a function of the maximum evolution time used to determine each M2E shown in Fig. 6. As seen, the range of evolution times examined can be loosely correlated with the broadening of the central transition due to chemical shift and second-order quadrupolar anisotropies and dispersions. The range of suitable evolution times decreases as a function of increasing broadening suggesting that spin exchange among neighboring nuclei is increasingly restricted as there exists a larger population of crystallite orientations such that neighboring nuclei have resonance
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BECKY GEE
FIG. 4. Second moment values (•) as a function of the maximum evolution time, 2t1 -max, used to determine M2E . Each M2E value was obtained from a least-squares fit of all normalized echo data ranging from 0 ≤ 2t1 ≤ 2t1 -max. The calculated values ( ) are indicated at 2t1 = 0. Phenomenological linear extrapolations to zero evolution times are shown as solid lines. For K3 VO4 , the best linear fit for the data represented by open squares is indicated by the dashed line, while the fit for data plotted as open circles is indicated by the solid line.
frequencies that exceed the dipole–dipole coupling strength. It is also noted that if the maximum 2t1 value is varied by about ±20–40 µs, the resulting experimental second moment values are still within approximately 10–25% of the calculated values. The suitable ranges for data analysis are also shown in Fig 7. This method of analysis suggests the possibility of establishing a calibration plot to determine the range of suitable evolution times that should be considered in the data analysis. With the exception of K3 VO4 , second moments for those compounds with chemical shift and quadrupolar broadening of approximately <110 ppm, such as Mg2 V2 O7 , can be obtained by restricting analysis of the spin echo data to
51
V DIPOLAR COUPLINGS OBTAINED VIA HAHN SPIN ECHO SPECTROSCOPY
83
FIG. 5. Comparison of calculated and experimental 51 V–51 V dipolar M2E values determined by linear extrapolation of M2E values to zero evolution time.
FIG. 6. Comparison of calculated and experimental 51 V–51 V dipolar M2E values determined by restricting the data analysis to short evolution periods as indicated in Fig. 7 and Table 2.
84
BECKY GEE
Experimental
Compound K3 VO4 BiVO4 Mg3 (VO4 )2 Ba3 (VO4 )2 LaVO4 Zn2 V2 O7 Sr2 V2 O7 Mg2 V2 O7 a
51
TABLE 2 V–51 V M2E Values Obtained by Restricting the Data Analysis to Short Evolution Periods as Indicated M2E (106 rad2 /s2 ) experimental by analysis of short 2t1 data
M2E (exp) M2E (calc)
Evolution period analyzed 0 < 2ti ≤ x(ms)
4.0 ± 0.7a 4.4 ± 0.6 17.8 ± 0.9 25.6 ± 1.9 11.1 ± 1.0 15.2 ± 1.4 22.6 ± 0.5 19.5 ± 1.3 16.1 ± 1.4
0.89 0.98 0.82 0.97 0.96 0.98 0.94 0.74 0.96
x = 028 028 014 012 018 014 010 010 014
Results are given for two independently measured data sets.
FIG. 7. Estimated line broadening of the 51 V central m = 1/2 → −1/2 transition due to chemical shift and second-order quadrupolar effects as a function of the maximum evolution time used to determine M2E as in Fig. 6. For symmetric resonances the broadening is given by the full width at half maximum, while for anisotropic resonances the broadening was estimated as |δ|| − δ⊥ |. Circles indicate the maximum time of the evolution period used to determine the dipolar M2E values given in Fig. 7. The range of maximum evolution times that give experimental M2E values within 10–25% of the calculated values is also plotted.
51
V DIPOLAR COUPLINGS OBTAINED VIA HAHN SPIN ECHO SPECTROSCOPY
85
evolution times between 0 < 2t1 ≤ 180 µs. For those compounds with broadening >110 ppm, such as Sr2 V2 O7 , M2E may be obtained by limiting the data analysis to 0 < 2t1 ≤ 120µs. CONCLUSIONS This work suggests that using a Hahn spin echo pulse sequence 51 V dipolar second moments can be obtained by exciting the central 51 V transition for compounds with anisotropies of approximately ≤220 ppm. Those with greater line broadening exhibit an oscillatory time dependence of the echo intensity due to off-resonance effects. Accuracies of approximately 10–33% can be achieved if the second moments are linearly extrapolated to zero evolution time. Alternatively, second moments of the echo intensity decay can be obtained within 10–25% of the calculated ones if broadening of the central transition due to chemical shift and second order quadrupolar interactions can be estimated. Although the method of extrapolation tends to overestimate the second moment of spin echo decay intensity, this approach does not require knowledge of the chemical shift or second quadrupolar broadening and is, thus, likely more suitable and provides fairly reasonable results. ACKNOWLEDGMENTS B.G. thanks Professor Ruth Stark and Dr. Hsin Wang (Chemistry, College of Staten Island of the City University of New York) for use of the solid-state NMR instrumentation. B.G. also thanks the laboratory of Professor Hellmut Eckert (Chemistry, Westf¨alische Wilhelms-Universit¨at) for providing the magnesium pyrovanadate, lanthanum orthovanadate, and strontium pyrovanadate samples. Ms. Claudia (n´ee Gheorghe) Liberman (Chemistry, Long Island University) is appreciated for her preparation and characterization of the zinc pyrovanadate and her search of the crystallographic literature. This work was partially supported by the Research Released Time Committee and the Trustees of Long Island University and by the American Association of University Women Educational Foundation American Summer/Short-term Research Publications Grant. Acknowledgement is made to the Donors of The Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research (ACS-PRF 34967-GB5).
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