2.1. Crystals

Single crystals of SBN (x = 0.32-0.82) were grown by the Czochralski technique using resistance heating. The detailed description of growth procedures can be found elsewhere (Ulex et al., 2004). The crystals were grown along the [001] direction; they exhibit a slight yellow colouration and show the form of prisms with 16 natural facets. The crystal diameter is between 5 and 7 mm. For the single-crystal X-ray measurements, as-grown (unpoled) single crystals were ground in an agate mortar followed by grinding in a ball mill. These operations produced single-crystalline spherical particles with diameters between 50 and 100m. The samples for the powder X-ray diffractometry were also produced by grinding grown crystals.

2.2. X-ray diffraction measurements

X-ray powder diffraction measurements were performed on crystal powder samples consisting of 40 mg SBN and 12 mg F-phlogopite as standard, using Cu K radiation. The single-crystal X-ray diffraction measurements were carried out on a four-circle kappa diffractometer (Xcalibur from Oxford Diffraction) equipped with a Sapphire2 CCD area detector. Graphite-monochromatized Mo K radiation was used. The sample-to-detector distance was 44.4 mm. With one and four scans, 93.2% of reciprocal space was covered which corresponds to 100% completeness under Laue group 4/m mm at a resolution of 0.8 Å. Data reduction and numerical absorption correction were carried out with the program CrysAlisRED (Oxford Diffraction, 2002). The crystal structure was refined with the program SHELXL97 (Sheldrick, 1997).

All measurements (except those for the temperature dependence of the unit-cell parameters) were carried out at room temperature.

3.1. Unit-cell parameters

Powder X-ray measurements were performed throughout the compositional range x(Sr) from 0.32 to 0.82 with composition steps of ca 0.03. From these measurements, the unit-cell parameters a and c could be derived with an accuracy of approximately 3×10^-5.

The a parameter lies between 1248.5 and 1241.1 pm and the c parameter varies from 397.4 to 389.9 pm. Both a and c unit-cell parameters decrease with increasing Sr concentration. The dependence of the unit-cell parameters on the Sr concentration is shown in Fig. 2. The unit-cell parameters for the composition x = 0.61 are in good agreement with previously reported data (Chernaya et al., 1997). For comparison, the unit-cell parameters resulting from the single-crystal measurements (see below) are also plotted.

Figure 2 Unit-cell parameters of strontium barium niobate SrxBa1-xNb2O6 as a function of the strontium content x derived from powder (circles and squares) and single-crystal (filled triangles) X-ray measurements. Lines denote fits to the powder data.

The composition dependence of the unit-cell parameters can be described using second-order polynomial fits to the powder data

plotted as lines in Fig. 2. A quadratic term is included in both (1) and (2) to account for the slight deviations from Vegard's rule visible in the data. From Vegard's rule, a strictly linear dependence would be expected.

The overall variation of the unit-cell volume is approximately 3%. This can be fully ascribed to the variation of the effective average radii r_eff of the cations occupying the A1 and A2 sites (r_eff = 142-134 pm for r_Ba = 147 pm and r_Sr = 131 pm (Shannon & Prewitt, 1969, 1970), if one takes into account that approximately 15% of the unit-cell volume belongs to the A1 and A2 sites.

In Fig. 3 the measured unit-cell values are compared with the data for other ferroelectrics with the TTB structure. The Sr-rich compositions of SBN show rather small unit-cell parameters. This could be one of the reasons for the high values of the spontaneous polarization P_s and in turn the electrooptic coefficient r_ij in SBN.

Figure 3 SBN unit-cell parameters compared with other ferroelectrics from the TTB family (after Neurgaonkar & Cory, 1986).

3.2. Density

The density of the crystals was measured by hydrostatic technique on SBN samples with a typical volume of some cubic centimeters using distilled water as the immersion liquid. Thus, a good accuracy was achieved. The results are plotted in Fig. 4 together with the calculated X-ray density. For the calculation, the composition-dependent unit-cell parameters as given in (1) and (2) are used. A second-order polynomial fit to the X-ray density yields

Again the quadratic term is included to account for deviations from Vegard's rule.

Figure 4 Density of SBN as a function of the Sr content. Squares denote measured data (Archimedes principle); the line shows the X-ray density calculated using the unit-cell parameters.

3.3. Site-occupancy factors

Single-crystal X-ray diffraction measurements were performed on Sr_xBa_1-xNb₂O₆ crystals with x equal to 0.342, 0.477, 0.613 and 0.822. For all these compositions the structure refinement led to the tetragonal structure type with the space group P4bm.

The site-occupancy factors (s.o.f.) for the cation sites A1 and A2, derived from the measurement, are shown in Fig. 5. Site-occupancy factors are defined as

where M_i is the number of atoms per formula unit on site i, N the number of formula units per unit cell and Z the number of symmetry-equivalent sites in the unit cell. For SBN, N = 5, Z = 8, and the sum of the M_i for the A1 and A2 sites is 1. The 12-fold coordinated A1 sites are occupied by Sr cations only, the site-occupancy factor s.o.f.(Sr-A1) depends only slightly on the composition and varies between 0.16 and 0.18 corresponding to a relative occupancy between 66 and 72%. The 15-fold coordinated A2 sites are occupied by both Sr and Ba cations with an approximately linear composition dependence of the respective s.o.f. values. The total relative occupancy for the A2 sites is ca 90%.

Figure 5 Site-occupancy factors for A1 and A2 sites in SBN. Dots indicate measured data: - this work; - Jamieson et al. (1968); - Chernaya et al. (1997); - Woike et al. (2003). The lines represent the results of a model calculation (see text).

The site-occupancy factors and their compositional dependence can be described by a simple statistical model. Assuming that Ba occupies only A2 sites and Sr randomly occupies A1 and the remaining A2 sites with different preferences yields the respective s.o.f. values as a function of the strontium fraction x

K is a factor denoting the preference of Sr for the A2 site compared with A1, with K = 1 for equal preferences. The results of the model calculation for K = 1.1, i.e. a slight preference for A2, are sketched as gray lines in Fig. 5. A satisfactory agreement with the experimental data was found.

3.4. Nb-O bond lengths

The Nb-O bond lengths for all the compositions studied lie between 180 and 214 pm. The mean bond length which characterizes the NbO₆ octahedron size decreases monotonically with increasing Sr concentration and reaches 195.8 pm for the x(Sr) = 0.82, which is only slightly larger than the value of 195.2 pm calculated for the `ideal' oxygen (ion radius 138 pm) octahedron. The maximal difference between Nb-O bond lengths describes the asymmetry of the Nb position within the octahedron and also becomes smaller for the Sr-richer crystals. The measured unit-cell parameters, site-occupancy factors and Nb-O distances are shown in Table 1. All experimental data are given in Table 2.¹

Table 1 Results of the X-ray diffraction measurements on SrxBa1 - xNb2O6 single crystals of different strontium fraction x Crystal   SBN-34 SBN-48 SBN-61 SBN-82 Strontium fraction x 0.342 0.477 0.613 0.822 Cell parameters a (pm) 1248.40 (4) 1248.52 (6) 1245.9 (1) 1241.1 (5)   c (pm) 397.42 (3) 395.66 (5) 393.6 (1) 389.9 (2) Site occupancy Sr-A1 0.1659 (3) 0.1777 (3) 0.1796 (3) 0.1758 (5)   Ba-A2 0.4150 (4) 0.3300 (4) 0.2438 (4) 0.0942 (6)   Sr-A2 0.0444 (4) 0.1170 (4) 0.2019 (4) 0.3550 (6) Nb-O distance Nb1 (pm) 197.10 197.01 196.66 195.78 Nb-O Nb1 (pm) 30.1 28.3 33.4 16.1 Nb-O distance Nb2 (pm) 198.16 197.83 197.19 196.57 Nb-O Nb2 (pm) 30.2 28.2 23.9 7.7

Table 2 Experimental details   k2shape sr477 shape137 sbn159k3 Crystal data Chemical formula Ba0.67Nb2O6Sr0.33 Ba0.52Nb2O6Sr0.48 Ba0.39Nb2O6Sr0.61 Ba0.14Nb2O6Sr0.86 Mr 402.26 395.44 388.68 378.29 Cell setting, space group Tetragonal, P4bm Tetragonal, P4bm Tetragonal, P4bm Tetragonal, P4bm Temperature (K) 298 (2) 298 (2) 298 (2) 298 (2) a, c (Å) 12.4840 (4), 3.9742 (3) 12.4844 (5), 3.9572 (3) 12.4575 (7), 3.9382 (3) 12.4179 (9), 3.9074 (5) V (Å3) 619.38 (5) 616.77 (6) 611.17 (7) 602.54 (10) Z 5 5 5 5 Dx (Mg m-3) 5.392 5.323 5.282 5.216 Radiation type Mo K Mo K Mo K Mo K (mm-1) 13.33 13.78 14.31 15.13 Crystal form, colour Sphere, colourless Sphere, colourless Sphere, colourless Sphere, colourless Crystal size (mm) 0.26 × 0.22 × 0.19 0.18 × 0.17 × 0.16 0.20 × 0.19 × 0.13 0.17 × 0.17 × 0.12           Data collection Diffractometer Xcalibur CCD Xcalibur CCD Xcalibur CCD Xcalibur CCD Data collection method and and and and Absorption correction Numerical Numerical Numerical Numerical Tmin 0.097 0.159 0.130 0.099 Tmax 0.177 0.221 0.267 0.169 No. of measured, independent and observed reflections 11 682, 2077, 1852 11 771, 2157, 1635 11 458, 2096, 1740 11 273, 2153, 1365 Criterion for observed reflections I > 2(I) I > 2(I) I> 2(I) I > 2(I) Rint 0.034 0.038 0.039 0.060 max (°) 40.3 41.3 39.5 43.3           Refinement Refinement on F2 F2 F2 F2 R[F2 > 2(F2)], wR(F2), S 0.027, 0.065, 1.12 0.030, 0.055, 1.00 0.033, 0.076, 1.14 0.045, 0.091, 0.96 No. of reflections 2077 2157 2096 2153 No. of parameters 65 65 65 65 Weighting scheme w = 1/[2(Fo2) + (0.0303P)2 + 1.1155P], where P = (Fo2 + 2Fc2)/3 w = 1/[2(Fo2) + (0.021P)2], where P = (Fo2 + 2Fc2)/3 w = 1/[2(Fo2) + (0.0334P)2 + 0.3182P], where P = (Fo2 + 2Fc2)/3 w = 1/[2(Fo2) + (0.0383P)2], where P = (Fo2 + 2Fc2)/3 (/)max 0.009 0.005 0.001 0.010 max, min (e Å-3) 2.18, -2.79 2.12, -2.80 2.54, -3.17 3.63, -3.54 Extinction method SHELXL SHELXL SHELXL SHELXL Extinction coefficient 0.0137 (3) 0.00421 (12) 0.0108 (3) 0.0118 (4) Absolute structure Flack (1983), 807 Friedel pairs Flack (1983), 877 Friedel pairs Flack (1983), 866 Friedel pairs Flack (1983), 891 Friedel pairs Computer programs used: CrysAlisCCD Oxford Diffraction, 2002), SHELXL97 (Sheldrick, 1997), WinGX (Farrugia, 1999).

3.5. Temperature dependence of the unit-cell parameters

Usually the unit-cell parameters of crystals increase monotonically with increasing temperature owing to the amplitudes of the ionic oscillations, increasing with increasing temperature. Thermal expansion measurements of SBN with x = 0.5 (Shorrocks et al., 1982) and x = 0.6 (Bhalla et al., 1987), however, show a negative temperature coefficient for the unit-cell parameter c at low temperatures and a positive one at higher temperatures with a minimum value of c near the phase-transition temperature. In contrast to this, the unit-cell parameter a always strictly increases with increasing temperature. Dilatometric measurements, however, are problematic in ferroelectric crystals owing to domain effects. Recent X-ray measurements (Qadri et al., 2005) for SBN with x = 0.75 confirm this anisotropic behavior - with a negative temperature coefficient for c, a positive one for a - throughout the temperature range investigated. To clarify the behavior, we extended the measurements of Quadri et al. (2005) up to higher temperatures on SBN with x = 0.822. The results are plotted in Fig. 6.

Figure 6 Temperature dependence of the unit-cell parameters a and c of SBN with an Sr content of 0.822 (black dots with error bars). For lower temperatures, the results taken from Quadri et al. (2005) were plotted (squares) and converted to different compositions (0.75 versus 0.822). The gray arrows denote the piezoelectric contributions to the thermal expansion (see text).

Our measurements show that X-ray measurements exhibit a similar behavior as found by thermal expansion measurements for other compositions: the a parameter strictly increases with temperature, whereas c decreases at low temperatures and increases at higher ones. An analog behavior has just been shown by neutron scattering measurements (Schefer et al., 2006).

The explanation for this behavior had already been given by Cross et al. (1980) and Shrout et al. (1981a,b), who derived an additional electrostrictive contribution to the thermal expansion of ferroelectrics from energetic considerations using the Landau-Ginsburg-Devonshire phenomenological theory. These contributions, Q₃₁P² for the unit-cell parameter a and Q₃₃P² for the unit-cell parameter c, are plotted as gray arrows in Fig. 6. For the calculation, the electrostriction constants Q_ik given by Cross et al. (1980) and a typical saturation polarization P = 0.3 C m^-2 were used.