Analyzing structure factor phases in pure and doped single crystals by synchrotron X-ray Renninger scanning
X-ray multiple diffraction has been applied to study the substitutional incorporation of Mg2+ ions into NSH crystals (nickel sulfate hexahydrate, NiSO4·6H2O). Intensity profiles provide information on invariant phases, while angular positions of the multiple diffractions allow accurate determination of lattice parameters. By increasing the atomic disordering only of O2− sites in model structures of doped NSH, the sense and magnitude of induced phase shifts match those necessary to justify the observed changes in the intensity profiles. Causes of disordering and lattice parameter variation are discussed. Although the amount of extra oxygen disordering is relatively large with respect to the small difference in the ionic radii of the metallic ions, this disordering is beyond the resolution power achievable by analyzing diffracted intensities of isolated reflections, such as in standard crystallographic techniques.
Structural analysis of engineered materials is one of the most important issues in materials science. Understanding the relationship between atomic structure and physical properties allows more efficient engineering of new materials. Doping of single crystals is often employed to alter their physical properties, widening their use in many technological applications. Small structural changes induced by doping can be hard to detect and, therefore, characterization methods susceptible to such small changes are of great importance. Diffraction methods (X-ray, neutron and electron) are the number one choice for solving crystal structures whose spatial periodicity defines uniform systems that, in equilibrium state, are characterized by the elemental and structural composition of their unit cells. However, small local changes around dopant ions are diluted in the average unit cell, minimizing the sensitivity of diffraction probes to the doping.
X-rays have intermediate values of atomic scattering cross section, with respect to neutrons and electrons, which allows rescattering of diffracted waves to take place within perfect domains of microscopic size. For neutrons, rescattering would require macroscopic domains, while rescattering processes for electrons are too strong even on atomic scales. Coherent rescattering of diffracted waves, widely known as dynamical diffraction, is the physical basis for accessing information on structure factor phases, i.e. on phases of unit cell scattered waves. Measuring isolated reflections in crystals only provides the amplitudes of the scattered waves, since only intensities are obtained. On the other hand, by promoting interference of multiple diffracted waves – when exciting two or more Bragg reflections – differences between structure factor phases can be measured. In X-ray crystallography these phase differences are known as invariant phases (Hauptman & Karle, 1953).
For decades it has been known that multi-beam diffraction, or multiple diffraction (MD), contains structural phase information and provides a physical tool for invariant phase measurements (Hart & Lang, 1961; Colella, 1974; Post, 1977; Shen, 1986; Weckert & Hümmer, 1997). However, actual applications solving unknown features of crystalline structures are very recent (Shen et al., 2006; Morelhão et al., 2011). In this work, we further exploit phase measurements to achieve a knowledge of the internal stresses in doped crystals that would be hindered in standard diffraction methods susceptible only to structure factor amplitudes, such as powder diffraction. This approach is applied to NSH crystals (nickel sulfate hexahydrate, NiSO4·6H2O; Fig. 1) doped with Mg2+, where substitutional incorporation, , is the most probable process of doping (Su et al., 2008). Experimental and theoretical MD results show remarkable differences between pure and doped samples, implying major shifts of the invariant phases. By increasing the atomic disordering of only one chemical species, oxygen O2− in water molecules, the sense and magnitude of induced phase shifts in model structures match those necessary to explain the experimental data. The amount of induced disordering in oxygen sites is very small and beyond the resolution power achievable by analyzing diffracted intensities of individual reflections.
In standard crystallographic methods, the diffracted intensities of single reflections are directly proportional to the square modulus of the structure factors; K depends on the particular experimental setup, embracing correction factors due to diffraction geometry, polarization and absorption.
is the structure factor of reflection hkl, where xj, yj and zj are the fractional coordinates, fj is the atomic scattering factor, Cj is the occupation factor and is the Debye–Waller factor of the jth atomic species in the average unit cell. Although Mj is a function of temperature, it also accounts for atomic disordering effects on high-angle reflections since is the mean-squared displacement (m.s.d.) of the jth atoms from their average positions over all unit cells in the diffracting crystal volume. When simulating the diffraction by model structures, scattering factors
Differently from standard methods, simultaneous excitement of two or more Bragg reflections can give rise to interference between wavefields with noncoplanar diffraction vectors. Within the framework of the second-order approximation (Shen et al., 2000; Morelhão & Kycia, 2002), when two reflections, for instance reflections A and B, are excited by the incident X-rays, each diffracted wave inside the crystal is a sum of two waves, and . The primary wave, , is produced by a single reflection (reflection A), while the other wave, , also called the detour wave, comes from a double-bounce reflection formed by reflection B plus a coupling reflection C, whose indices are given by A − B (Fig. 2). By keeping one wave excited and changing the angular condition of the other, characteristic interference profiles are obtained. The most common is the interference profile obtained by an azimuthal scan, where the intensity
is a function of the crystal φ rotation around the diffraction vector of reflection A. Since the primary wave is kept excited during the rotation, its wavefield is taken as constant in this approach. χ is the angle between the oscillation directions of the waves, which can be tuned by a linearly polarized X-ray beam (Morelhão & Kycia, 2002). The total phase difference between the waves is the sum of two phase angles: the invariant phase , where is the structure factor phase of reflection X (= A, B or C); and , the dynamical phase shift of the detour wave, which is a function of the crystal rotation (Weckert & Hümmer, 1997; Shen et al., 2000). In other words, is well known from X-ray diffraction theory and Ψ is the desired invariant phase to be determined from experiments.
In model structures of pure and doped crystals, invariant phases and the relative strength of the detour waves are predicted by calculating structure factor triplets
which are invariant quantities regarding the displacement of the origin since the sum of diffraction vectors is null, , when reflections A and B undergo MD (Hart & Lang, 1961; Post, 1977). Henceforth, we will refer to each MD case by the indices of the reflections B + C whose sum provides the indices of reflection A, i.e. hA = hB + hC, kA = kB + kC and l A = l B + l C, or, occasionally, only by the indices of reflection B.
When the MD intensity profile is dominated by the second-order term [equation (3)], it typically has asymmetric character for Ψ values around 0 or 180°, and symmetric character for values equal to 90°. By shifting Ψ across 90°, the asymmetric aspects of the profiles are changed, for example, from constructive/destructive to destructive/constructive or vice versa. This effect occurs even in crystals with poor crystalline quality, and it has been exploited for analyzing magnetic materials (Shen et al., 2006), semiconductors and doped crystals (Morelhão et al., 2011). Furthermore, long azimuthal scans, recording several MD cases with good angular precision, usually called X-ray Renninger scanning (XRenS) (Renninger, 1937), are also useful for accurate determination of lattice parameters in bulk crystals (Chang & Post, 1975; Avanci et al., 1998) and of tiny residual stresses in semiconductor devices (Morelhão et al., 2005).
is the Dirac delta function and is the reciprocal vector modulus in terms of the d spacing of Bragg planes. In spite of geometric factors, convolution of H(Q) with line broadening functions would provide essentially X-ray diffractograms, such as those obtained from powder samples.
Reference values for pure NSH crystals (Rousseau et al., 2000) were calculated with ions S1.8+ and O0.95- in the tetrahedral units, and O2− in the octahedral units (see Appendix A for atomic scattering factors of these ions). An isotropic r.m.s. displacement of 0.1 Å, i.e. Uj = 0.01 Å2, was assigned to all atoms in the unit cell. The first structure model for the NSH:Mg sample was obtained by changing only the occupation factor of metallic ions, CMg = 0.12 and CNi = 0.88 for Mg2+ and Ni2+, respectively, i.e. 12% of substitutional doping. All other parameters are kept equal to NSH. This provides a histogram that is nearly identical to the reference one (Fig. 3b, top line). In a second model, the disordering of O2− ions was increased from UO2- = 0.01 Å2 to UO2- = 0.0625 Å2 along the c axis only, causing small non-systematic intensity reduction mostly in high-angle reflections with Å−1 (° for X-rays of 8 keV) (Fig. 3b, bottom curve). These small variations in the values emphasize how difficult it would be to detect, with good reliability, changes of fractional atomic coordinates (internal stresses) in neighboring atoms of the dopant ions by standard crystallographic methods.
When comparing structure factor phases, we found that the 008 reflection has a large susceptibility to disordering of octahedral O atoms (O2− ions). Although weak, this reflection is perfectly measurable in single crystals, which makes it suitable to be used as the primary reflection in XRenS. Considering all possible MDs for the 008 primary reflection, the invariant phases of MD cases such as and are the most susceptible ones for checking oxygen disordering. Fig. 4 shows the invariant phases of these MD cases as a function of the O2− disordering. In the NSH:Mg structure with Å2, the intensity profiles of both MDs should have opposite asymmetries with respect to those in the reference structure.
To ensure that differences in crystalline perfection of pure and doped samples will not compromise experimental data analysis, MD intensity profiles were simulated by X-ray dynamical diffraction theory (Weckert & Hümmer, 1997) in (001) crystal slabs of different thicknesses. The primary extinction distances of the involved reflections are all above 4 µm, and in (001) slabs the shorter penetration depth under MD conditions is 3 µm for reflection 316 (Freitas, 2011). Hence, we chose 1 µm as the smallest thickness (thin slab) for dynamical simulation near the kinematical diffraction regime. Fig. 5 shows that inversion of asymmetries is expected to occur when increasing UO2-, independently of slab thickness. The other MD cases, , and , show similar results. As discussed elsewhere (Morelhão et al., 2011), by changing the slab thickness, the contribution of higher-order terms in the intensity profile [equation (3)] should also change. Therefore, MDs whose asymmetrical aspects are not affected by slab thickness can provide experimental evidence of shifting in Ψ values, regardless of some reduction in the lattice coherence length of doped samples.
Single crystals were grown by the slow evaporation method under conditions of controlled temperature and pH (304 K and pH 4.0), starting with supersaturated solutions of NiSO4·6H2O in distilled water to obtain NSH crystals, and adding MgSO4·5H2O to the solution to obtain NSH:Mg crystals. Final samples have rectangular a ×b ×c shapes with dimensions greater than 3 mm along all axes. Regarding physical properties, thermogravimetric analysis has found an increase of about 20% in the dehydration temperature of NSH:Mg crystals: 10% of weight loss occurring at 359 K, against 345 K in pure crystals.
X-ray data collection was carried out at the Brazilian Synchrotron Light Laboratory (LNLS), in a bending-magnet diffraction station (XRD1) with X-rays of eV, effective divergences of 18 (vertical) × 24 (horizontal) arcseconds, and a beam size at the sample position of 0.5 ×0.5 mm. Goniometry for azimuthal scans with adjustable polarization was provided by a three-axis goniometer with an angular resolution (minimum step size) of 0.0002° in the , θ and φ rotation axes. In XRenS, the crystal φ rotation sense is clockwise when the primary diffraction vector (of reflection 008) points towards the observer. was chosen as the azimuth in which the  direction lies on the diffractometer's incidence plane, pointing towards the X-ray source. All scans were performed in σ polarization.
Fig. 6 shows portions of the 008 XRenS in pure and doped samples. As can be seen, there are differences in the MD positions, implying a small variation of lattice parameters. For tetragonal lattices, every combination of two independent MDs provides one set of a and c parameters since b = a and = 90°. By combining position values of all MDs identified in the XRenS of Fig. 6, the obtained average lattice parameters (and standard deviations) are a = 6.782 (6) Å and c = 18.283 (2) Å for the NSH sample; and a = 6.783 (5) Å and c = 18.35 (2) Å for the NSH:Mg sample. Within the experimental resolution, only the variation in the c parameter, of 0.4 (1)%, could be detected. An increase in the c parameter has been observed by standard single-crystal diffractometry in small pieces of the samples, as well as in powder samples (Melo, 2012), but it was much smaller, of about 0.19 (2)%. Strain in the c parameter can also be seen when comparing rocking curves of the 008 reflection (Fig. 7). In the NSH sample, the observed width is purely instrumental since the intrinsic width is very small (′′; inset of Fig. 7). After deconvoluting instrumental broadening, the NSH:Mg sample has a width of 12′′ [ = (21.62 − 17.92)1/2], which is mostly due to strain along the c axis.
High-resolution azimuthal scans of those MDs predicted theoretically (§3) as susceptible to oxygen disordering are presented in Fig. 8. Besides small drifts in their φ positions due to the variation in the c parameter – which are not large enough for switching their positions or for bringing other cases into the scanned range – inversion of asymmetry in the doped sample with respect to the reference one is evident for both MDs. To enhance visual perception of such inversion of asymmetry in the experimental profiles, a line profile function described elsewhere (Morelhão et al., 2011) has been used, as also shown in Fig. 8. Moreover, each profile stands for two coincident three-beam cases, i.e. a four-beam case (Chang, 1984; Weckert & Hümmer, 1997). For instance, 312/316 should be read as MDs and occurring at the same φ position. Although each one of these three-beam cases may have differences in strength, they carry the same invariant phase and asymmetric aspect. Hence, the observed inversion of asymmetries implies invariant phase shifts with sense and magnitude close to those obtained by dynamical simulations (e.g. Fig. 5).
When replacing Ni2+ with coordination number VI (six water molecules as first neighbors of the metallic ion; Fig. 1), the Mg2+ ion has a 3 pm larger ionic radius (Shannon, 1976), justifying an enlargement of the unit cell that is mainly along the c parameter. Attempts to characterize the doping with single-crystal diffractometry have provided best fitting of relative intensities for substitutional incorporation of 15%, and an average oxygen disordering of UO2- = 0.034 Å2 against 0.032 Å2 in the pure sample.
The substitutional incorporation alone has not been able to promote phase shifts that could explain the remarkable inversion of asymmetries observed in the experimental MD profiles. In model structures with no extra oxygen disordering, substitutional doping induces both invariant phases to shift by = −15°, which is in the opposite sense required for inversion of profile asymmetries. It could only enhance the original asymmetric aspect of the profiles. Disordering of tetrahedral O atoms, O0.95- ions in the SO4 units, has to be as large as 0.12 Å2 to increase the invariant phases in Fig. 4 by no more than 10°, failing to explain the experimental results. On the other hand, an extra r.m.s. displacement of just 0.15 Å on octahedral oxygen sites shifts the invariant phases by the necessary amount to explain the data. Since 008 is the reflection promoting the major phase shift (inset of Fig. 4), there is no information in the data regarding the anisotropic nature of the disordering. We can only say that this extra disordering occurs at least along the c axis.
Within the hypothesis that all changes in the structure are caused by the 3 pm difference in ionic radii, the major axes of an octahedron with Mg2+ inside would increase by 6 pm, which is practically the 7 (2) pm observed variation in the c parameter. Since the a and b parameters remain unchanged, and the O2− extra r.m.s. displacement of 15 pm implies larger displacements than 6 pm, inflated octahedra may be twisted or reorientated by several degrees to accommodate along the c axis the difference in bonding length between oxygen and the metallic ion.
Differently from any other method in X-ray crystallography based on structure refinement of intensity data, the presented method pinpoints a specific feature of the structure and directly proves its existence beyond any reliability parameter or goodness-of-fitting value. In the particular case studied in this work, to inspect enhanced disordering of water molecules in doped crystals, we developed a systematic procedure based on comparing structure factor phases to identify multiple diffraction cases whose intensity profiles could exhibit drastic inversion of asymmetry exclusively because of this feature (enhanced oxygen disordering). By observing such inversion experimentally, definitive proof is given of its occurrence in NSH crystals doped with Mg2+. To cause the observed inversion of asymmetry, the minimum amount of disordering has been estimated as 15 pm of r.m.s. displacement above thermal vibration in pure crystals. Accurate lattice parameter determination via XRenS, i.e. by recording MD azimuthal positions, shows that the unit cell of the doped crystals has a larger c parameter. A hypothesis of local distortion of octahedral units containing dopant ions has been raised to provide a physical explanation for the observed magnitude of both oxygen disordering and lattice parameter variation. We emphasize the practical application of multi-wave diffraction techniques for probing structural features that are very difficult to detect by standard crystallographic techniques.
Atomic scattering factors for non-tabulated ions
The tetrahedral units have an ionic charge of −2e, balanced as 4Ox- and Sy+, where y = 4x-2. In the case of x = 0.5 and y = 0, atomic scattering factors could be estimated from the available values in the literature for S0, O0 and O1− (Prince, 2006), since for . However, to obtain theoretical MD profiles with asymmetries similar to those observed in pure NSH (Figs. 8a and 8b), we used x = 0.95 and y = 1.8. Scattering factors for Sy+ ions were estimated as (Fig. 9), where Å2 is the value that fits , as shown in the inset of Fig. 9. Cromer–Mann coefficients for fS6+, which are not available in the literature, were then obtained by curve fitting as = [1.211915, 6.034215, 1.913354, 0.808553, 0.030582, 1.550274, 0.568617, 9.531798, 4.240228]. For fO2-, the theoretical values given by Tokonami (1965) were better adjusted by curve fitting, providing = [0.45629, 0.562170, 4.998626, 2.565331, 1.415686, 33.476352, 9.042665, 32.917736, 0.432043]. In equation (2), .
This work has been supported by FAPESP (project No. 2012/01367-2), CNPq (project No. 306982/2012-9) and LNLS.
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