207Pb NMR, Powder Diffraction, and Monte Carlo Studies of PbxM1-x

The solid solutions [Pb,Sr](NO3)2 and [Pb,Ba](NO3)2 show a multiplicity of 207Pb NMR signals, arising from shifts of the 207Pb resonances by less pola...
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5892 207Pb

J. Phys. Chem. B 2001, 105, 5892-5899

NMR, Powder Diffraction, and Monte Carlo Studies of PbxM1-x(NO3)2 Solid Solutions Young-Sik Kye, Bruno Herreros, and Gerard S. Harbison* Department of Chemistry, 508 Hamilton Hall, UniVersity of Nebraska at Lincoln, Lincoln, Nebraska 68588-0304 ReceiVed: April 12, 2001

The solid solutions [Pb,Sr](NO3)2 and [Pb,Ba](NO3)2 show a multiplicity of 207Pb NMR signals, arising from shifts of the 207Pb resonances by less polarizable neighbors. The shifts have a unique anisotropic signature that allows tentative assignment of some of the signals. Monte Carlo simulation of the spectral intensities shows that the ions are slightly clustered in the crystal, which disagrees with a conventional but specious analysis of the dependence of the unit cell volume on composition. The extent of the clustering is attributable to the differences between the van der Waals interaction energies of the ions.

Introduction Substitutional solid solutions are of considerable importance in solid-state inorganic chemistry, and they occupy a central place in materials science. Among obvious industrially important examples are many metallic alloys, doped semiconductors, ceramics, and mixed ferroelectrics. However, despite their profound importance, much of the fundamental chemistry of solid solutions is unknown, and in particular, the distribution of the components of such solutions is often uncharacterized. The seminal work of substitutional solid solutions is that of Vegard,1,2 who, based on pioneering powder X-ray diffraction studies, first posited the ideal solid solution, whose lattice constant is linearly proportional to the molar composition (we now know it is more accurately the unit cell volume that depends on composition). In thermodynamic terms, an ideal binary solid solution, according to Vegard, is one where the volume of mixing of the two components is zero:

∆mV ) V h 12 - X1V h 1 - X2V h2 ) 0

(1)

where X1, X2 are the mole fractions of the two components in the solid solution, V h 1, V h 2 their respective molar volumes, and V h 12 the molar volume of the resulting solution. Vegard2 identified the solid solutions formed between the isomorphous nitrates of lead, barium and strontium as an example of ideality, at least within the accuracy of his measurements. Since then, the anhydrous nitrate of calcium3 and the high temperature form of cadmium nitrate4 have been added to this group. All are cubic systems with the space group Pa3, in which the single chemically distinct cation lies at an inversion center, and the nitrate groups at sites of trigonal symmetry. Considered alone, the cations lie in a face-centered cubic lattice, interspersed by the anions. Postulated structures involving a lower-symmetry space group lacking an inversion center were ruled out by 14N NMR.5 Other anomalies remain unexplained, however. For example, crystals in this system often show anomalous birefringence, a phenomenon which is forbidden by the space group symmetry, and Gopalan and Kahr6 postulated that differential incorporation of cations at the 111 and 100 faces of the crystal might lower * Corresponding author. Telephone: (402) 472-9346. Fax: (402) 4729402. E-mail: [email protected]

the symmetry from Pa3 to R3h, thereby making birefringence possible, at least in the mixed crystals. We recently reported7,8 that the 207Pb NMR spectra of [Pb,Sr](NO3)2 are sensitive to the nearest neighbor cation adjacent to the lead probe atoms and may contain up to 13 major lines, arising from 207Pb2+ ions with between zero and twelve Sr2+ ions as nearest neighbors. The magnitude of the shift is around 20 ppm/Sr2+, and the first shifted resonance (i.e., that from 207Pb with 11 leads and 1 strontium neighbor, or Pb2+[Pb11Sr1]) is further split into a doublet, because there are two classes of nearest neighbors; those within a plane normal to the unique 111 axis of the shielding tensor, and those in the layer above or below this plane. This work mirrors similar nearest neighbor effects identified in the solid-state NMR of silicates,9 semiconductors,10 and ionic crystals.11,12 Crundwell et al.13,14 have published similar spectra for both the [Pb,Sr](NO3)2 and [Pb,Ba](NO3)2 solid solutions and advanced several qualitative explanations for the phenomenon. In the present work, we present a quantitative analysis of the chemical shifts, shielding tensors, spectral intensities and energetics of these solid solutions, and discuss their implications for the thermodynamics of ion clustering in such systems. Materials and Methods Barium, lead, and strontium nitrate and calcium nitrate dihydrate were obtained from Aldrich. Saturated solutions of the appropriate compositions were obtained by dissolving suspensions of the mixed powders with slow addition of water. Crystals were then grown by slow evaporation over a period of several weeks. Since the crystals from such solutions are usually different in composition from the solute, and therefore fractionate by crystallization, resulting in crystals of inhomogeneous composition, typically no more than 1 g of mixed crystal was harvested from 500 mL of solution. We estimate that the maximum change in solution composition over the progress of any crystallization was 2%. For 207Pb MAS NMR, pure Pb(NO3)2 was used as a chemical shift reference. Magic angle spinning spectra were obtained at 63.76 MHz (7.1 T) using a simple one pulse sequence. The π/2 pulse length of 207Pb was 4.1 µs, and recovery delays of 8 s were used. Because lead nitrate chemical shifts are highly temperature dependent, and because sample temperature is in

10.1021/jp011387a CCC: $20.00 © 2001 American Chemical Society Published on Web 05/26/2001

PbxM1-x(NO3)2 Solid Solutions

J. Phys. Chem. B, Vol. 105, No. 25, 2001 5893

Figure 1. Molar composition of [Pb,Sr](NO3)2 solid solutions as a function of the composition of the growth solution.

turn dependent on rotor speed, except where otherwise noted, all spectra were obtained at spinning frequencies between 1.9 and 2.1 kHz. Mixed crystal spectra were recorded both with and without a small admixture of pure lead nitrate as an internal standard to account for any differences in temperature between the samples. Peak intensities on all 207Pb MAS NMR spectra were obtained by deconvolution. X-ray powder diffractograms were obtained using CuKR radiation on a Rigaku D-Max/B diffractometer at a temperature of 25 °C, using 99.999% crystalline elemental silicon as an internal standard. Elemental analyses of the crystals were obtained by inductively coupled plasma atomic emission spectrometry (ICP-AES) using a Shimadzu ICP-1000IV spectrometer, and have a reproducibility of (0.8%. Results Composition of Crystals. Unlike materials prepared by annealing at high temperature, solid solutions obtained by crystallization do not necessarily mirror the composition of the solution used for crystal growth, but depend strongly on the relative solubilities of the two components. Figure 1 shows the composition of PbxSr1-x(NO3)2 crystals as a function of the composition of the solution used for crystal growth. As can be seen, the crystals are highly enriched in lead compared with the solution. To obtain crystals that contain 50% lead on a molar basis, crystals are grown from a solution containing approximately 10% lead. MAS Spectra of Solid Solutions. Figure 2 shows 207Pb MAS spectra of lead/strontium nitrate solid solutions. Figure 2a is a typical magic-angle spinning spectrum of the pure Pb(NO3)2. The single resonance is flanked by very weak rotational sidebands. Figure 2b shows the 207Pb spectrum of a polycrystalline sample grown from a solution containing 40% by mole Pb(NO3)2 and 60% Sr(NO3)2. ICP-AES analysis of crystals grown from solutions of this composition shows them to be Pb0.954Sr0.046 (NO3)2. This spectrum shows a weak doublet lower in frequency than the signal from the pure Pb(NO3)2 by -15.6 and -20.0 ppm. The spectrum in Figure 2c, which is for Pb0.761Sr0.239 (NO3)2, shows a much stronger doublet, and a series of other lines. At higher concentrations of strontium, still more lines appear. The spectrum in Figure 2d is for Pb0.441Sr0.559 (NO3)2; that in Figure 2e is Pb0.199Sr0.801 (NO3)2. The shift due to a single strontium substitution, averaged over the two components of the doublet, is -17.8 ppm. The shifts are approximately additive at low strontium incorporations, but tend to increase as the number of strontiums in the second coordination sphere increases. For samples of approximately equimolar composition, the positions of all 13 lines can be measured; these data were fit to a quadratic equation, and from the fit, the magnitude of the shift was determined to be -18.4 ppm per additional strontium ion at zero strontium concentration,

207Pb MAS spectra of [Pb,Sr](NO ) solid solutions. 3 2 (a) Pure Pb(NO3)2; (b) Pb0.954Sr0.046(NO3)2; (c) Pb0.761Sr0.239(NO3)2; (d)Pb0.441Sr0.559(NO3)2; (e) Pb0.199Sr0.801(NO3)2. Rotational sidebands are marked with asterisks.

Figure 2.

increasing in magnitude to -24.4 ppm as the number of strontium neighbors reaches 12. The total shift of the lead surrounded by 12 strontiums is -262 ppm. Figure 3 shows the 207Pb spectra of [Pb,Ba](NO3)2 solid solutions. Except for the absence of a doublet splitting of the first barium substitution peak, they mirror closely the spectra of [Pb,Sr](NO3)2 solid solutions, albeit with somewhat smaller shifts. These data were fit in the same way used for the [Pb,Sr](NO3)2 samples. From the fit, the magnitude of the shift is -16.3 ppm per additional barium ion at low barium concentration, increasing in magnitude to -22.2 ppm as the number of barium neighbors reaches 12. The total shift of the lead surrounded by 12 bariums is -229 ppm. These shift data are plotted and compared for both sets of solid solutions in Figure 4. Despite repeated attempts, we have been unable under slowgrowth conditions to reproduce the extra peak noted by Crundwell et al.13,14 in [Pb,Ca](NO3)2 samples produced by rapid precipitation. Lead nitrate precipitated by rapidly mixing a saturated lead nitrate solution with an excess of the highly viscous saturated solution of calcium nitrate showed a weak, broad peak centered 12 ppm to lower frequency of the main 207Pb peak (Figure 5a); this may well be a peak due to a single calcium replacement, but the failure to observe peaks due to multiple replacement, and the possibility that other forms of disorder may be present in such microcrystalline systems, makes this assignment at best tentative. Notably, this peak was absent from samples produced by slow crystallization, where only weak and unreproducible signals could be detected on the baseline. Spectra of lead nitrate crystals harvested from highly viscous solutions containing saturated calcium nitrate were ground and examined to signal/noise levels of 1000:1 at high spinning

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207Pb MAS spectra of Pb(NO ) samples crystallized 3 2 from solutions containing a large excess of Ca2+ (a) by slow crystallization and (b) by rapid precipitation. A possible peak arising from Ca2+ incorporation is arrowed.

Figure 5.

207Pb MAS spectra of [Pb,Ba](NO ) solid solutions. 3 2 (a) Pure Pb(NO3)2; (b) Pb0.886Ba0.114(NO3)2; (c) Pb0.752Ba0.248(NO3)2; (d)Pb0.371Ba0.629(NO3)2. Rotational sidebands are marked with asterisks.

Figure 3.

207Pb isotropic chemical shifts of peaks in the spectra of Pb0.441Sr0.559(NO3)2 (filled squares) and Pb0.371Ba0.629(NO3)2 (open circles) relative to unshifted Pb(NO3)2.

Figure 4.

speeds. They showed no peaks of intensity greater than 0.2% of the main peak that could not be attributed to MAS sidebands or other artifacts (Figure 5b). It is possible that our peak and the extra peak noted by the earlier authors is due to incorporation of calcium into a different site under kinetic conditions, or to 207Pb2+ in some other disordered environment. Regardless, it appears that at equilibrium calcium does not incorporate appreciably into lead nitrate crystals even at very high molar ratios of Ca2+. Attempts to obtain shifted NMR signals from solid solutions containing other metal ions were similarly unsuccessful. Pb(NO3)2 crystallized from solutions containing high concentrations of Cd2+ or Hg2+ were indistinguishable from those of pure

Pb(NO3)2, while addition of Cu2+ to Pb(NO3)2 solutions resulted in blue-colored lead nitrate crystals, but showed no NMR evidence of near neighbor shifts. Slow MAS of Solid Solutions. Figure 6 shows 207Pb magic angle spinning spectra of [Pb0.954Sr0.046](NO3)2, collected at the very low spinning speeds of 768, 838, and 957 Hz. At this composition, only the unshifted Pb[Pb12Sr0] peak and the two components of the Pb[Pb11Sr1] doublet have any appreciable intensity. The centerbands are arrowed; they are in each case flanked by rotational sidebands, which in some cases overlap with centerband resonances. The deconvolved sideband intensities were fit by the method of Herzfeld and Berger,15 using a program that simultaneously fits spectra obtained at several different spinning speeds. Simulated spectra calculated using the Herzfeld-Berger intensities are shown below the experimental spectrum in all three cases; as can be seen, the correspondence between simulation and experiment is excellent. The chemical shielding tensors obtained are given in Table 1. From this analysis, a chemical shift anisotropy (∆σ) of 45 ( 2 ppm and an asymmetry parameter (η) of 0.6 was obtained for the left peak of the doublet, and ∆σ ) 50 ( 2 ppm and η ) 0.7 for the right peak, respectively. Powder Diffraction. Figure 7 shows the X-ray powder diffractograms of several mixed crystal samples. These diffractograms were obtained by crushing the same polycrystalline samples used for NMR. It should be noted immediately that the peaks in mixed crystal diffractograms are as narrow as those of the pure materials, indicating that the samples are macroscopically homogeneous. If the powder were composed of material whose composition varied macroscopically, somewhat different lattice parameters would be obtained from different regions, and an overall broadening would be observed. From these diffractograms, the lattice parameters and thence the unit cell volumes of the solid solutions were obtained as a

PbxM1-x(NO3)2 Solid Solutions

J. Phys. Chem. B, Vol. 105, No. 25, 2001 5895

Figure 7. Powder diffractograms of (a) Pb0.984Sr0.016(NO3)2; (b) Pb0.441Sr0.559(NO3)2; (c) Pb0.199Sr0.801(NO3)2. Peaks from the elemental silicon standard are arrowed.

Figure 6. Slow spinning 207Pb MAS spectra of [Pb0.954Sr0.046](NO3)2, collected at speeds of (a) 768 Hz, (b) 838 Hz, and (c) 957 Hz. The experimental spectrum in each case is shown above a spectrum simulated using centerband and sideband intensities calculated by the method of Herzfeld and Berger.15 Isotropic peaks are arrowed.

TABLE 1: 207Pb Chemical Shift Tensor Data for Pure Lead Nitrate and for the Two Components of Singly Substituted Doublet in Pb0.954Sr0.046(NO3)2 Pb[Pb12Sr0] Pb[Pb11Sr1]a Pb[Pb11Sr1]b

σ11

σ22

σ33

σi

∆σ

η

18.2 14.5 8.4

18.2 -21.4 -14.9

-36.5 -39.9 -53.5

0 -15.6 -20

-54.7 -45.2 -50.3

0 0.6 0.7

function of composition (Figure 8), with the errors given being the standard deviation in the lattice parameter over the average of values obtained from the positions of all of the measured diffraction peaks. In both cases, a slight concave deviation is observed from the ideal dependence. Discussion Monte Carlo Simulation of 207Pb Spectral Intensities in Solid Solutions. In an ideal solid solution, the two atoms or ions are distributed randomly in the crystal lattice. This means that at any site adjacent to a 207Pb ion the probability of finding another lead is given by XPb, the mole fraction of lead in the lattice. Since each site is independent of all others (a condition of randomness), if there are m sites adjacent to each lead, the probability of n lead atoms in these m sites is given by

Figure 8. Unit cell volume of PbxSr1-x(NO3)2 (filled squares) and PbxBa1-x(NO3)2 (open circles) as a function of composition. The lines are the best fits to the Margules equation, as described in the text.

Pn )

m! X n(1 - XPb)m-n n!(m - n)! Pb

(2)

This is, of course, a binomial distribution. In the divalent metal nitrate, each lead is surrounded by twelve nearest neighbors, so m ) 12, and 0 ) n ) m. Since we can obtain XPb by elemental analysis, we can compare the normalized intensities of the 207Pb[PbnSr12-n] NMR lines with those predicted by the model. This is done in Figure 9. It can be seen that the binomial distribution reproduces the experimental data qualitatively but not in detail, suggesting small but significant deviations from ideality.

5896 J. Phys. Chem. B, Vol. 105, No. 25, 2001

Kye et al. relative probabilities

p 1-p

) exp(-∆E/RT)

(4)

where p is the probability of switching. This simplifies to

p)

1 1 + exp(∆E/RT)

(5)

In our model, we are only concerned with energy differences, we can therefore arbitrarily define the interaction energy between like atoms to be zero (EA-A ) EB-B ) 0) and assign the interaction energy between unlike atoms to some value f. Each atom can have 0 to 12 unlike neighbors leading to atomic energies (Ei) from 0 to 12f in our arbitrary units. The lattice energy is calculated by half the sum of the energy of each atom EL ) 1/2(∑i)1NEi). Switching two unlike atoms creates an energy difference in the lattice ∆EL. The switch affects the energy of the two atoms involved in the switch and the energy of their 12 respective neighbors. A negative value of f indicates an energy configuration that favors clustering, whereas positive values of f cause anticlustering. The probability of switching two atoms becomes

p)

Figure 9. Comparison of experimental intensities (black) with those calculated using a random ion distribution (hatched) and the best fit for a Monte Carlo simulation (white) with an unlike pair energy of 120 J/mol (see text) for three samples: (a) Pb0.199Sr0.801(NO3)2; (b) Pb0.441Sr0.559(NO3)2; and (c) Pb0.886Ba0.114(NO3)2.

The most straightforward way to deal with these deviations is to assign probabilities of unlike nearest neighbor pairs different from those of like pairs and from the mole fractions. Such different probabilities can arise, even if the crystal growth occurs near equilibrium conditions, if the pair interaction energy of an unlike pair is different from that of a like pair. A closed form mathematical expression similar to that in eq 2 could probably be obtained under these assumptions, but it is easier to calculate the intensities of the NMR lines by Monte Carlo methods. For these calculations we employed the Monte Carlo/ Metropolis algorithm, a computational method that models in an intuitively obvious way the approach of a system to equilibrium. The Monte Carlo method used in this work is a threedimensional Metropolis model.16 We begin with a simulated lattice N unit cells on a side, with periodic boundary conditions, and populate the lattice randomly with lead and strontium atoms with probabilities of XPb and (1 - XPb), respectively. Counting the numbers of nearest neighbors at this stage of this calculation reproduces the binomial result within an error determined by the sampling statistics. We also define an unlike pair energy ∆E, corresponding to the energy difference between an unlike nearest neighbor pair and a like pair.

∆E )

(21)(2E

Pb-Sr

- EPb-Pb - ESr-Sr)

(3)

We now randomly select atoms in the lattice, two at a time, and either choose to switch them, or not to switch them, with

1 1 + exp(∆EL/RT)

(6)

A lattice consisting of 8 × 8 × 8 unit cells (2048 atoms), 10% Sr2+ atoms in a matrix of atoms of type A and an unlike pair energy f ) RT reached effective equilibrium after 2000 effective switches, showing the complete clustering of the lattice. The influence of the lattice size has been studied, and it appears that lattice dimensions over 5 × 5 × 5 unit cells (500 atoms) gives similar averaged values, but standard deviations decrease with an increasing number of unit cells. After systematically examining the effect of lattice size and equilibration time on computed results, all calculations employed 8 × 8 × 8 unit cells (2048 atoms) with periodic boundary conditions and a time of prior equilibration of 10 000 switches. The number of near neighbors was then counted over 30 000 further switches and used to determine the mean number of neighbors and standard deviations. The mole fractions of lead atoms used in the simulations were obtained by elemental analysis. The value of f is determined from the best fit of the experimental data and should remain constant for all simulations since the pair energies are assumed not to depend on the composition of the crystals. The best fit to all of the lead/ strontium and lead/barium spectra were obtained with a value of f ) 0.05RT corresponding to a difference in energy EA-B (EA-A + EB-B)/2 ≈ 120 ( 10.8 J/mol. This result indicates a slight clustering of the lattice. Monte Carlo simulations obtained with this energy are compared in Figure 9 with experimental data, and with intensities expected for a random distribution, for three of the solid solution samples. Crystal Inhomogeneities and the Question of Equilibrium. It is well nigh impossible to establish conclusively that crystals grown from solution, an inherently nonequilibrium process, are at or are even arbitrarily close to thermodynamic equilibrium. In particular, two distinct mechanisms have been identified by which these solid solutions in particular may deviate from equilibrium. The first is the fractionation that occurs upon crystallization because the crystals have a different composition than the solution. This fractionation can be minimized by the simple expedient of collecting a small amount of material from a large volume of solution and by slow crystallization, which

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J. Phys. Chem. B, Vol. 105, No. 25, 2001 5897

allows the rate of ion transport by diffusion within the solution to exceed the rate of selective removal at the bottom of the flask by crystallization. The second is the differential incorporation of ions into different faces of the crystal, reported by Gopalan and Kahr,6 which would cause crystals growing predominantly along 111 or symmetry-equivalent faces to differ in composition from those growing predominantly along 100 or equivalent. As those authors note, this fascinating hypothesis also results in a reduction of crystal symmetry from cubic to hexagonal. However, it must be emphasized that this phenomenon must be kinetic rather than thermodynamic; there cannot be two equilibrium states of different composition in a homogeneous system. Under our growth conditions, we have been unable to find evidence of the sorts of crystal inhomogeneities reported by Crundwell et al. for PbxBa1-x(NO3)2. In particular, our powder diffractograms (Figure 7) do not show the sort of dramatic evidence of inhomogeneity they report. Only when crystals are obtained from solutions crystallized near to dryness, a process which, as we have noted, inevitably leads to fractionation, do such diffractograms in our hands become broader than those of the pure materials. Volume of Mixing of Lead and Strontium Nitrates. The powder diffraction data fully justify Vegard’s exemplification of this system as a near-ideal solid solution; the plots of lattice volume versus composition for both [Pb,Sr](NO3)2 and [Pb,Ba](NO3)2 deviate from linearity by only slightly more than the error of the measurement. The deviations are nonetheless significant, and were quantified by fitting the curves to a Margules equation.17

V ) X1V h 1 + X2V h 2 + 4X1X2V h 12

(7)

This equation generally fits nearly ideal liquid solutions satisfactorily and has the advantage that it provides a volume of mixing of the two components. At equal mole fractions of the two components, this volume of mixing V h 12 is -0.3 mL/ mol of Pb for the lead-barium system and -0.06 mL/mol of for lead/strontium. Obviously, these are very small volumes, less than 1% of the molar volume. Nonetheless, their sign is surprising. A negative volume of mixing has generally18 been attributed to energetically favorable interactions between the two components of the solid solution, the rationalization being that a favorable interaction is accompanied by net attraction between the components, and a consequent reduction in the cell volume. Of course, this logic is directly contradicted by the Monte Carlo simulations; more importantly, it has no thermodynamic validity. The volume of mixing is related not to ∆mE but to the partial derivative of the free energy of mixing with respect to pressure, and there is no particular reason why a negative δ∆mG/δP should not be accompanied by a positive ∆mE. Energies of Mixing and the van der Waals Force. Our Monte Carlo simulations indicate that the cations tend to be slightly clustered in the lattice with unlike pairs of atoms being favored by approximately 120 J compared to like pairs. These numbers can be contrasted with solid solutions containing covalent bonds, such as the aluminosilicates, for which Dove9 has reported an energy difference E(Al-O-Al) + E(Si-O-Si) - 2 E(Al-O-Si) ∼ 40 kJ/mol. The magnitude of the energy differences involved here are obviously much closer to those of van der Waals interactions than they are to bond energies. The van der Waals interaction between two species can be estimated from the energy of the lowest excited state and the dipole polarizability of the two

TABLE 2: Polarizabilities and Energies of the First Excited State Used in Dispersion Calculations Sr2+ Ba2+ Pb2+

E (cm-1)

R/4π0 (m3 × 10-30)

177698.2 134568.2 95341.6

1.6 2.5 4.9

species involved:

〈V(r)〉 ) -

(

)

R1R2 3 E1E2 2 E1 + E2 16π22r6 0

(8)

where E is the first excitation energy, R is the polarizability of each ion, and r is the distance between ions 1 and 2. The values assumed in this work are given in Table 2 and include dipole polarizabilities reported by Tessman et al.;19 first excitation energies reported by Smith,20 Persson et al.,21 and Hellentin;22 and internuclear distances obtained from the crystal structure. They yield van der Waals energy differences

EvdW,Pb-Sr - (EvdW,Pb-Pb + EvdW,Sr-Sr)/2 ) 121.5 J/mol for lead-strontium and

EvdW,Pb-Ba - (EvdW,Pb-Pb + EvdW,Ba-Ba)/2 ) 78.1 J/mol for lead-barium. These results are remarkably close to the experimental values, considering the approximate nature of the calculation and the fact that possible dielectric shielding by intervening nitrate ions has been ignored (although at frequencies close to the first excitation energy of these ions, the dielectric constant of nitrate is probably close to unity). This van der Waals energy, resulting from the differing polarizabilities of the two species, is the limiting factor on the ideality of liquid solutions, and it appears in this case also to limit the ideality of nearly ideal solid solutions. Origin of the Shifts. We should note initially the presence of a substantial discrepancy between our results and those of Crundwell et al. with regard to the magnitude and the linearity of the shifts. Those authors report an essentially linear dependence of shift on the number of nearest neighbor ions; in our hands, there is a decided nonlinearity (Figure 4), which is even evident in some of the spectra (examine the spacings between the lines in Figure 2d and 2e). Crundwell et al.14 attribute the shift to a size discrepancy between the ions. We disagree. Pb2+ ions are slightly larger than Sr2+ ions and considerably smaller than Ba2+ ions. However, the effects of substitution of Pb2+ by either Sr2+ or Ba2+ are identical in sign and similar in magnitude, indicating that this effect is not merely a result of the size of the replacing ion. It is true that the magnitude of the size mismatch could be the important factor, although we find it difficult to come up with a physical mechanism for this. One plausible origin for these shifts, first proposed by Buckingham in 1960,23 lies in dispersion effects. Fluctuations in the electric dipole moment of the adjacent ion lead to a fluctuating electric field at the lead site; the time average of this field is zero, but the average of its square is nonzero. Therefore, shielding hyperpolarizability terms in the expression for the chemical shift of the form

σdisp ) B1〈E2〉

(9)

will give rise to a shift. This model was proposed by Jameson and deDios24 as a source of intermolecular chemical shifts in the rare gases. More recent ab initio studies25 have called into

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question whether actual shielding hyperpolarizabilities are in fact sufficiently large to account for the magnitude of the observed shifts in noble gas systems. Nonetheless, the Buckingham model gives excellent empirical fits to these and other data, and it is possible that it may be satisfactorily modified, for example, by allowing the dispersion energy to cause a displacement of the ion in the lattice, and this in turn causing chemical shift changes. In any case, it is probably wise to treat B1 as an adjustable parameter in the equation

(

)

E1E2 R2 3 σdisp ) - B1 2 32 E1 + E2 π 2r6 0

(10)

Using eq 10 and values given in Table 2, we obtain B1/(4π0) values of 1.35 × 10-13 and 1.50 × 10-13 m3J-1 from the PbSr and Pb-Ba systems, respectively; the agreement between the two is satisfactory given the considerable errors inherent in these calculations. Similar shifts seen in the system [Pb,Sr]TiO3 are approximately 70 ppm. In these perovskites, the nearest neighbor distance is 4.0 rather than 5.5 Å; the ratio of the observed shifts in titanates and nitrates is approximately the inverse sixth power of the ratio of intercationic distances. Moreover, we have recently observed that the 133Cs spectra of Cs2SO4/Tl2SO4 differ in the sign of the nearest neighbor shifts, i.e., replacing a cesium with a thallium ion shifts the neighboring cesiums to high frequency. The crucial difference here is that replacing a cesium with a thallium replaces a less polarizable ion with a more polarizable ion, and so the sign of the shift is inverted. Chemical Shielding Anisotropies and Assignments. We have already8 attributed the splitting of the Pb[Pb11Sr1] line into a doublet to orientational inequivalence of the nearest neighbors with respect to the unique 111 axis. The chemical shielding tensors principal values obtained in the slow MAS experiments give us an opportunity to assign tentatively the components of the Pb[Pb11Sr1] doublet. It is notable, first of all, that in contrast with the axial symmetry of the main peak (dictated by the point symmetry of the lattice site) the two doublet components have a large asymmetry parameter, a clear indication that the point symmetry is broken by the introduction of a Sr into a neighboring lattice site. In the lead nitrate crystal structure, there are 12 such sites; six of these neighbors form a hexagon in the plane perpendicular to 111, and six a distorted octahedron, with 3-fold rotational symmetry about the 111 axis. In our earlier work, we assigned the components of the doublet to these two distinct substitutions; Crundwell et al. came to a similar conclusion with respect to the doublet splitting of the Pb[Pb1Ba11] line. However, those authors reported no discernible asymmetry parameter in the first substituted barium site. These tensor changes can be used to produce a tentative assignment of the doublet components. The less shifted doublet component shows a substantial shift to high frequency of a single, in-plane tensor element (σ22), while, in contrast, the righthand peak has substantial shifts of all three tensor elements. Let us start by assuming that the next-nearest-neighbor effect results in an equal high-frequency shift of the two tensor elements orthogonal to the perturbation; in other words, if the perturbation is along the y axis, the shift has the form

( )

σ 0 0 3 n σ jn ) 0 0 0 2 0 0 σn

(11)

If the shift arises from an atom in the same layer, the

perturbation will be orthogonal to the unique direction of the unperturbed 207Pb chemical shielding tensor and the overall shifted tensor will be

(

σn + σu,⊥ 0 0 3 σu,⊥ 0 σ j)σ jn + σ ju ) 0 2 σn + σu,⊥ 0 0

)

(12)

leading to a tensor in which the perpendicular and one of the parallel elements is shifted by the same amount, as is seen in the more shifted of the two doublet components. In contrast, if the shift arises from an atom in the next layer, it will be oriented at an angle of 35.3° from the z axis and, in the frame of reference of the unperturbed shielding tensor, will have the form

(

σn 0 0 3 0 1/3σ -x2/3σn n σ jn ) 2 0 -x2/3σn 2/3σn

)

(13)

The off diagonal elements do not commute with the unperturbed tensor, and therefore have minimal influence on the overall shift. The result is a tensor with a large shift in one of the perpendicular elements and a smaller shift in the parallel element. This is reasonably close to the pattern in the less-shifted line. This suggests (although it by no means confirms) that the more shifted doublet component arises from strontium replacement in the same layer perpendicular to 111. It is worth noting that since the occupied valence pz orbital of the lead is oriented along 111, the polarizability of the ion is likely to be highest along this direction, and therefore the largest shifts should arise from substitutions perpendicular to this direction, in agreement with our tentative assignment. Conclusions Resolved shifts of the sort analyzed in this paper are likely observable for most spin 1/2 nuclei in the lower half of the periodic table, and in some instances for quadrupolar nuclei. Where they can be detected, they allow the local environment of probe nuclei in solid solutions to be probed with a high degree of accuracy and detail. Although the magnitudes of the shifts themselves can at present be understood only in a semiempirical fashion, it is hoped that incorporation of relativistic corrections in ab initio methods will enable a more quantitative understanding of the phenomenon. Acknowledgment. This research was supported by the NSF under grant number MCB-9604521, and by the Research Council of the University of Nebraska at Lincoln. We thank Cynthia Jameson for useful discussions and for sending a manuscript prior to publication. References and Notes (1) Vegard, L. Z. Kristallogr. 1921, 5, 17. (2) Vegard, L.; Dale, H. Structurbericht 1928, 67, 148. (3) Ewald, P. P.; Hermann, C. J. Solid State Chem. 1931, 1, 304. (4) Louer, M.; Louer, D. D. J. Phys. Chem. 1976, 17, 231. (5) Santos, R. A.; Tang, P.; Chien, W.-J.; Kwan, S.; Harbison, G. S. J. Solid State Chem. 1990, 94, 2717. (6) Gopalan, P.; Kahr, B. Inorg. Chem. 1993, 107, 563. (7) Kye, Y.-S.; Harbison, G. S. Mater. Res. Soc. Symp. Proc. 1998, 37, 6030. (8) Kye, Y.-S.; Herreros, B.; Harbison, G. S. Geoderma 1999, 547, 339. (9) Dove, M. T. Phys. ReV. B 1997, 80, 353.

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