Study of crystal structures and phase transitions in perovskite-type oxides using powder-diffraction techniques and symmetry-mode analysis · Perovskite Atlas

Na₀.₅K₀.₅NbO₃, antimonates A₂MSbO₆, tellurates Sr₂MTeO₆ and their Mg, Mn and Fe substitutions, La₂CoMnO₆, and the layered NaLnMWO₆ and NaLaMTeO₆ families. Chapter map is completed in Phase 1.

status published · visibility public · id doc:thesis-04

Brahim Orayech, 2015 · supervisor Josu M. Igartua · Universidad del País Vasco / Euskal Herriko Unibertsitatea · language en

Files: sources/theses/thesis-04.pdf, sources/theses/thesis-04.md

Chapters

chapter	title	materials	published as
8	Sr2MSbO6 (M = Ln, Y): mode-crystallography approach of the structural and high-temperature phase-transition studies	Sr₂NdSbO₆, Sr₂EuSbO₆, Sr₂GdSbO₆, Sr₂DySbO₆, Sr₂HoSbO₆, Sr₂YSbO₆, Sr₂ErSbO₆, Sr₂TmSbO₆, Sr₂YbSbO₆	Effect of the M3+ cation size on the structural and high temperature phase transitions in Sr2 MSbO6 (M = Ln, Y) double perovskites
9	Sr2Co1−xMgxTeO6 (x = 0.1, 0.2, 0.5): structural phase transitions, magnetic and spectroscopic properties	Sr₂CoTeO₆, Sr₂Co_0.9Mg_0.1TeO₆, Sr₂Co_0.8Mg_0.2TeO₆, Sr₂Co_0.5Mg_0.5TeO₆	Structural phase transitions and magnetic and spectroscopic properties of the double perovskites Sr_{2}Co_{1−x}Mg_{x}TeO_{6}(x = 0.1, 0.2 and 0.5)
10	Sr2Ni1−xMgxTeO6 (x = 0.1, 0.2, 0.3, 0.5): structural phase transitions, magnetic and spectroscopic properties	Sr₂NiTeO₆, Sr₂Ni_0.9Mg_0.1TeO₆, Sr₂Ni_0.8Mg_0.2TeO₆, Sr₂Ni_0.7Mg_0.3TeO₆, Sr₂Ni_0.5Mg_0.5TeO₆	The effect of partial substitution of Ni by Mg on the structural, magnetic and spectroscopic properties of the double perovskite Sr_{2}NiTeO_{6}
11	M3+ size effect in the Ca2MSbO6 (M = Ln, Y) and their mode-amplitude changes (Part III, in preparation)	Ca₂NdSbO₆ under review, Ca₂EuSbO₆ under review, Ca₂GdSbO₆ under review, Ca₂DySbO₆ under review, Ca₂HoSbO₆ under review, Ca₂YSbO₆ under review, Ca₂ErSbO₆ under review, Ca₂TmSbO₆ under review, Ca₂YbSbO₆ under review

Figures

Figure 1.1: The schematic representation of the cubic simple perovskite structure A B O_{3}. The A and B cations are shown as yellow and blue spheres. The corners of the octahedra denote the oxygen atoms.
Figure 1.2: Cation ordering schemes in perovskites. From left to the right: rock salt, layered and columnar ordering are shown for B-site ordering in A_{2} B B^{\prime} \mathrm{O}_{6} perovskites.
Figure 1.3: Types of ordering in perovskites with two different B-site cations. (a) Ideal rock-salt ordering; (b) partial rock-salt ordering with local disorder, called antisite (back rectangle) and anti-phase boundary defects (red rectangle); (c) complete disorder. B \mathrm{O}_{6} octahedra: blue; B^{\prime} \mathrm{O}_{6} octahedra: grey; A / A^{\prime}-site cations: green spheres; oxygen: red spheres.
Figure 10.1: Experimental (symbols) and calculated (line) X-ray profiles and neutron profiles for the Rietveld refinement of the \mathrm{Sr}_{2} \mathrm{Ni}_{1-x} \mathrm{Mg}_{x} \mathrm{TeO}_{6} series at room temperature using a structural model with I 2 / m space group for ( \mathrm{x}=0.0,0.1,0.2 and 0.5). The bars in the lower part of the graphics represent the Bragg peak positions.
Figure 10.10: Two dimensional section showing the obtained amplitudes of the irreps (Blue) \mathrm{GM}_{4}^{+}, (Red) \mathrm{X}_{3}^{+}and the two components of the tri-dimensional \mathrm{X}_{5}^{+}: A10 in x-axis and A11 in y, for \mathrm{Ca}_{2} \mathrm{MSbO}_{6}. The results from \mathrm{Sr}_{2} \mathrm{MSbO}_{6} family, studied in Chapter 8, are shown as well, for comparison.
Figure 10.11: Evolution, as a function of the tolerance factor increasing of the unit-cell parameters (a), the unit-cell volume (b) and the monoclinic angle ( \beta) (b) at room temperature of the synthesized materials \mathrm{Ca}_{2} \mathrm{MSbO}_{6}(\mathrm{M}=\mathrm{Nd}, \mathrm{Eu}, \mathrm{Gd}, \mathrm{Dy}, \mathrm{Ho}, \mathrm{Er}, \mathrm{Tm}, \mathrm{Yb} and Y
Figure 10.12: Evolution of the occupancy x (which is the fraction of the Calcium in the B-site occupying by the \mathrm{M}^{3+} element in the formula \left.\left[\mathrm{Ca}_{1+x} \mathrm{M}_{1-x}\right]\left[\mathrm{M}_{x} \mathrm{Ca}_{1-x}\right] \mathrm{SbO}_{6}\right) with the ionic radii of the \mathrm{M}^{3+} in the antimony and the ruthenium families \underline{188}
Figure 10.13: Group-subgroup tree connecting the paraelectric P 4 / n m m phase to the ferroelectric P 2_{1} phase
Figure 10.14: Images in the real space and in the reciprocal space, showing clearly the Nano-Chessboard structure.
Figure 10.15: Theoretical models describing the Nano-Chessboards structure.
Figure 10.16: Raman spectra of the monoclinic \mathrm{Sr}_{2} \mathrm{MSbO}_{6}(\mathrm{M}=\mathrm{La}, \mathrm{Nd}, \mathrm{Sm} and Eu) measured at room temperature. Experimental data are in open circles, while the fitted curves are in colored lines.
Figure 10.2: (a): Variation, as function of the tolerance factor, of the distortion-amplitude transforming according to the \mathrm{GM}_{4}^{+}irrep and the cell parameters (black) and the unit-cell volume (blue) at room temperature. (b-e): Thermal evolution of the cell parameters refined in the P 2_{1} / n, I 2 / m and I 4 / m space groups for the \mathrm{Sr}_{2} \mathrm{Ni}_{1-x} \mathrm{Mg}_{x} \mathrm{TeO}_{6} series ( \mathrm{x}=0.0,0.1, and 0.2)
Figure 10.3: Temperature dependence of the mode-amplitudes refined in the P 2_{1} / n, I 2 / m and I 4 / m space groups for the \mathrm{Sr}_{2} \mathrm{Ni}_{1-x} \mathrm{Mg}_{x} \mathrm{TeO}_{6} series ( \mathrm{x}=0.3 and 0.5)
Figure 10.4: Thermal evolution of different splitted reflections for the \mathrm{Sr}_{2} \mathrm{NiTeO}_{6} compound.
Figure 10.5: Thermal evolution of different splitted reflections located in the interval ( 75^{\circ}<2 \theta<^{\circ} 77) for the \mathrm{Sr}_{2} \mathrm{Co}_{1-x} \mathrm{Mg}_{x} \mathrm{TeO}_{6} compounds with ( x=0.1,0.2,0.3 and 0.5).
Figure 10.6: Tolerance factor and temperature dependence phase diagram of \mathrm{Sr}_{2} \mathrm{MSbO}_{6} and \mathrm{Sr}_{2} \mathrm{MTeO}_{6}. The backgroupd color are indicating the phase-transition sequence’s domain. An horizontal line is set at 300 K as guide to figure-out the RT space group for each material.
Figure 10.7: (a) Evolution of the laboratory X-ray powder diffraction patterns of \mathrm{Ca}_{2} \mathrm{MSbO}_{6}(\mathrm{M}=\mathrm{Nd}, \mathrm{Eu}, Gd, Dy, Ho, Er, Tm, Yb and Y) at RT. (b) Details from the X-ray powder diffraction patterns of the latter compounds in the selected 2 \theta intervals; 18^{\circ}-26^{\circ}, 29^{\circ}-33^{\circ} and 54^{\circ}-57^{\circ}, to show the continuous evolution of the patterns with the ionic radius of the \mathrm{M}^{3+} cation.
Figure 10.8: Experimental (symbols) and calculated (line) X-ray powder diffraction profiles for the Rietveld refinement of \mathrm{Ca}_{2} \mathrm{EuSbO}_{6} at room temperature using a structural model with P 2_{1} / n space group. Insets shows in detail the presence of the primitive Bragg peaks corresponding to the P 2_{1} / n symmetry.
Figure 10.9: Variation of the \mathrm{GM}_{4}^{+}, \mathrm{X}_{3}^{+}and \mathrm{X}_{5}^{+}distortion irreps in the monoclinic symmetry P 2_{1} / n consolidating the RT phases of \mathrm{Ca}_{2} \mathrm{MSbO}_{6} \mathrm{M}=\mathrm{Ln}(\mathrm{Nd}, \mathrm{Eu}, \mathrm{Gd}, \mathrm{Dy}, \mathrm{Ho}, \mathrm{Er}, \mathrm{Tm}, \mathrm{Yb} and Y materials, as function of the tolerance factor. The red balls stand for the \mathrm{Ca}_{2} \mathrm{YSbO}_{6} amplitudes, since the \mathrm{Yttrium}, \mathrm{Y}, does not belong to the Lanthanides. The dashed lines are guide of the eye to follow the evolution of each irrep.
Figure 3.1: Geometrical illustration of the Braggs’ law.
Figure 3.2: The schematic of the powder diffraction cones produced by a polycrystalline sample while its interaction with an incident-beam.
Figure 4.1: Graph of maximal subgroups relating the F m \overline{3} m and the P 2_{1} / c space groups of room temperature \mathrm{Sr}_{2} M \mathrm{SbO}_{6}(M=L n, Y, In and Sc) phases. For each space group, the irreducible representation of the distortion mode yielding this symmetry is indicated if existing.
Figure 4.2: Graph of maximal subgroups relating the P m \overline{3} m and the A m m 2 space groups of room temperature \mathrm{Na}_{0.5} \mathrm{K}_{0.5} \mathrm{O}_{3} phase. For each space group, the irreducible representation of the distortion mode yielding this symmetry is indicated if existing.
Figure 5.1: a shows the general aspect of the powder sample. Two different types of particles can be observed: large particles of about 2-8 \mu m and the small fragments. The SEM shows the typical KNN morphology consisting of faceted grains with a bimodal grain size distribution. Abnormal grain growth tends to occur as a result of the high temperatures treatments. Such bimodal size distribution of grains was also observed in Li and Sb substituted KNN ceramics 104,105 The small fragments could also have been produced during grinding. The in-set presents an enlargement of a quite well facetted
Figure 5.10: Thermal evolution of the lattice parameters of \mathrm{Na}_{0.5} \mathrm{K}_{0.5} \mathrm{NbO}_{3}, as obtained from XRPD and NPD measurements. Solid circles up 300 K, were obtained in the low-temperature NPD measurement and solid circles, from 300 to 875 K in the high-temperature XRPD measurement. The solid red points correspond to the long acquisition data at 2,10,175 and 300 K temperatures. From 2 to 875 K the compound undergoes tree phase transitions with the following sequence: R 3 c \rightarrow A m m 2 \rightarrow P 4 m m \rightarrow P m \overline{3} m.
Figure 5.11: (a) Relative atomic displacements with respect to the cubic structure, in the P4mm phase: those define polar distortion mode. (b) The four basis symmetry-modes present in a polar distortion along z of the perovskite structure. The 12 -dimensional subspace of all possible polar distortions is spanned by that set of basis modes and their analogous along x and y. it is worth noting that the first three modes involve one atom per unit cell, the mode \epsilon_{4}^{z} involves two atoms per cell and, hence, the normalization condition implies that the atomic displacements in thi tetra-mode basis are smaller by a \frac{1}{\sqrt{2}} factor.
Figure 5.12: (a) Results from the conventional symmetry adapted modes’ amplitude refinements. See the text for details. (b) The structures obtained after having passed to AMPLIMODES, after applying the origin shift proposed by the program. The anomalous evolution of the \mathrm{Na} / \mathrm{K} amplitude, in the whole temperature range, is even more evident.
Figure 5.13: Results of the parameterized refinement. For further discussion, see the text. The dimmed symbols show the results obtained for the first parameterized refinement process without freeing the atomic displacements parameters. Also dimmed in the 0-100 \mathrm{K} interval, the amplitude values in the R 3 m model are shown. With gray stars, we show the octahedral-tilt amplitude values ( \mathrm{R} 4^{+}irrep) from the R 3 c model.
Figure 5.14: Evolution with temperature of the tetra-vector, but split in two parts: amplitudes of the modes \mathrm{Na} / \mathrm{K}(x) versus those of \mathrm{Nb}(\mathrm{y}); amplitude values of the modes acting on the O atoms, \mathrm{O} 1(x) versus O 2 (y). We are plotting two two-dimensional vectors in the x y plane: as black arrows, the evolution in all the distorted phases of the couple of amplitudes; at the vertices of the arrows, in green, the corresponding errors. The four points (1-4), represent values for the amplitudes obtained at the indicated temperatures, taken from \frac{97}{97} passed to AMPLIMODES. Data from single crystal XRD. We have also plot some results from \frac{[6]}{} pertaining to structures of related compounds, all in the tetragonal phase. The matching of the internal direction is very good.
Figure 5.15: Calculated SP (see the text for further details). Different colours represent the points of the SP resulting from the data of different sources, i.e. NPD (black) and XRPD (blue). Dimmed symbols show the polarization values obtained in the R 3 m model, without the octahedral tilt.
Figure 5.2: Experimental (symbols) and calculated (line) a) neutron and b) X-ray profiles for the Rietveld refinement of \mathrm{Na}_{0.5} \mathrm{K}_{0.5} \mathrm{NbO}_{3} at room-temperature using a structural model with Amm2 space group. Insets show in detail the presence of the 011 and 100 Bragg reflections corresponding to the orthorhombic symmetry. The (*)-symbol marked reflection corresponds to the signal of the Vanadium sample container used during the measurement. The bars in the lower part of the graphics represent the Bragg peak positions. In (a), the upper set of bars (in red) correspond to the first wavelength \lambda_{1}=2.52 \AA, and the lower set of bars (in blue) correspond to the second wavelength and \lambda_{2}=1.26 \AA.
Figure 5.3: To the left, chain of maximal sub-groups of P m \overline{3} m space group down-to the experimentally observed room-temperature A m m 2 space group. The calculations were completed in the standard setting of the room-temperature space group phase: A m m 2. To the right, scheme of the structural polarization vectors, projected on the x z plane, of the distortion-modes transforming according to the \mathrm{GM}_{5}^{-}and \mathrm{GM}_{4}^{-} irreps, present in the A m m 2 structure. The arrows represent the displacements of the atoms in the modes. The two distortion-modes are identified as necessary in the cubic-to-orthorhombic symmetry breaking.
Figure 5.4: a) Bright field TEM micrograph of a typical particle in the sample. b), d) and e) SAD patterns acquired along the [011], [2-11] and [110] zone axes, respectively. c) and f) electron microdiffraction acquired along the [101] and [100] zone axes, respectively. The reflections indicated in the patterns were identified by simulations done with the JEMS software, using the neutron data presented in the previous sections.
Figure 5.5: Thermal evolution at low-temperatures from NPD measurements (to the left) of the 101 rhombohedral reflection, and at high-temperatures from XRPD measurements (to the right) of the 011 and 100 orthorhombic reflections. The scattered intensity is represented with shades of grey: black corresponds to high intensity, and white, to low intensity. In the left graph, as temperature increases, the R3c singlet splits into a doublet in the orthorhombic A m m 2 space group: 011 and 100. In right graph, as temperature increases, the two orthorhombic reflections maintain in the tetragonal symmetry, but with their intensities interchanged. After further increasing of temperature, the tetragonal splitting disappears and the double reflection transforms to a single one in the cubic parent space group: 100.
Figure 5.6: Results of the refinements, in the 75^{\circ}-82^{\circ} 2 \theta and 98^{\circ}-106^{\circ} 2 \theta intervals, in \mathrm{Na}_{0.5} \mathrm{K}_{0.5} \mathrm{NbO}_{3}, at 2 K, using two different models with the monoclinic P m and the rhombohedral R 3 c space groups. The rhombohedral space group takes into account perfectly the observed splitting, unlike the monoclinic Pm.
Figure 5.7: Portion of (a) NPD and (b) XRPD profiles for \mathrm{Na}_{0.5} \mathrm{K}_{0.5} \mathrm{NbO}_{3} showing, at selected temperatures 70,150,350,550 and 750 K, the evolution with temperature of the cubic reflections: left panel 100, centre panel 110 and right panel 111.
Figure 5.8: To the left, chain of maximal sub-groups of P m \overline{3} m parent phase space group down-to the experimentally observed low-temperature R 3 c space group phase. To the right, scheme of the polarization vectors of the distortion-modes transforming according to the \mathrm{R}_{4}^{+}and \mathrm{GM}_{4}^{-}irreps, present in the R 3 c structure. The arrows represent the displacements of the atoms in the modes. The two distortion-modes are identified as necessary in the cubic-to-rhombohedral symmetry breaking.
Figure 5.9: Experimental (symbols) and calculated (line) neutron profiles for the Rietveld refinement of \mathrm{Na}_{0.5} \mathrm{K}_{0.5} \mathrm{NbO}_{3} (a) and (b) at 10 K and (c) 175 K, using a structural model with (a) R 3 m, (b) R 3 c and (c) A m m 2 space groups. The double set of vertical bars in (a) and (b) indicates the Bragg reflections from the main wavelength ( 2.52 \AA) (upper) and from the leak ( 1.26 \AA=\frac{1}{2} 2.52) (lower). The blue and red boxes show the Bragg reflections assigned, incorrectly, to the wavelength leak, if the R 3 m space group model is used. The dots in the low vertical bars, (b), are the distinct Bragg reflections due to the wavelength leak in the R 3 c and R 3 m models.
Figure 6.1: (a) Secondary electron image showing the overall appearance of the powder sample. The inset shows an enlargement of a particle. (b) Back-scattered electron image showing the homogeneous character of the powder sample.
Figure 6.2: (a) Bright field image of electron transparent area of one particle. (b) Selected area electron diffraction pattern acquired along the [001] zone axis. (c) and (d) Electron micro-diffraction pattern acquired along the [021] and [010] zone axis, respectively. The reflections indicated in the patterns were identified by simulations done with the JEMS software, using the neutron data.
Figure 6.3: Amplitudes of \mathrm{GM}_{4}^{+}(spheres), \mathrm{X}_{3}^{+}(stars), the global amplitude-value for the tri-dimensional \mathrm{X}_{5}^{+}irreps (squares) and its components: A10 (circles), A11 (up triangle) and A11 (down triangle), in the room-temperature structures of the \mathrm{Ln}_{2} \mathrm{CoMnO}_{6}\left(\mathrm{Ln}=\mathrm{La}[\mathrm{PW}], \mathrm{La}^{* 115}, \mathrm{La}^{* * 119}, \mathrm{Lu}^{128} \mathrm{Nd}^{* 129}, \mathrm{Nd}^{* * 130}\right), \mathrm{Ln}_{2} \mathrm{NiMnO}_{6}\left(\mathrm{Ln}=\mathrm{La} \sqrt{115}, \mathrm{La}^{* * *} \sqrt{134}, \mathrm{ Tb} \sqrt{118}, \mathrm{Sm}{ }^{\underline{118}}, \mathrm{Sm}^{*} \sqrt{135}, \mathrm{Hc} \xrightarrow{131}\right), \mathrm{Y}_{2} \mathrm{NiMnO}_{6} \sqrt{132} and \mathrm{Bi}_{2} \mathrm{NiMnO}_{6} \sqrt{133} compounds, versus the tolerance factor values, calculated using bond distances. PW stands for present work: [\mathrm{PW}] with all modes freed and \left[\mathrm{PW}^{*}\right] only the effectively acting ones, see text.
Figure 6.4: Experimental (symbols) and calculated (line) (a) X-ray profiles and (b) neutron profiles from the Rietveld refinements of \mathrm{La}_{2} \mathrm{CoMnO}_{6}, using a structural model with P 2_{1} / n space group, (a) and (b) at room temperature, from XRPD data and NPD, respectively. In (b), the (*)-symbol marked reflection corresponds to the signal of the Vanadium sample container used in the experiment. The bars in the lower part of the graphics represent the Bragg reflections’ positions. In (b), the upper set of bars correspond to the first wavelength \lambda_{1}=2.52 \AA, and the lower set of bars correspond to the second wavelength \lambda_{2}=1.26 \AA. Insets show in detail (a): ( 22^{\circ}-27^{\circ}) the presence of the superlattice reflections of the form h+k+l=2 n+1, corresponding to the P 2_{1} / n symmetry, \left(85^{\circ}-89^{\circ}\right) the cubic refections ( 444) clearly splitted in the monoclinic symmetry into (044) (-404) (404), (b): selected region, (011) reflection, of the fitted results using an ordered and a disordered structural models.
Figure 6.5: Two dimensional sections showing the mode-amplitudes of the three dimensional space transforming according to the \mathrm{X}_{5}^{+}irrep. The arrows represent a definite combination of those amplitudes deduced from the structures taken from the literature (as indicated), inputed in AMPLIMODES. Dimmed arrows represent the direction defined by the corresponding polarization vector. The full colored arrows indicate the actual distortion in this subspace (see text). (c) panel shows the final effective direction and the actual distortion in the reduced two-dimensional sub-space spanned by the modes A10 and A11. This is the direction used for the parametrization of the refinements.
Figure 6.6: Evolution, at room temperature, of the cell parameters along the analyzed structures in the families: \mathrm{Ln}_{2} \mathrm{CoMnO}_{6} (blue) \left(\mathrm{Ln}=\mathrm{La}[\mathrm{PW}], \mathrm{La}^{* 115}, \mathrm{La}^{* * 119}, \mathrm{Lu}^{128} \mathrm{Nd}^{* 129}, \mathrm{Nd}^{* * 130}\right); and, \mathrm{Ln}_{2} \mathrm{NiMnO}_{6} (black) \left(\mathrm{Ln}=\mathrm{La}^{\underline{115}}, \mathrm{ Tb}^{\underline{118}}, \mathrm{Sm}^{\underline{118}}, \mathrm{Hd}^{\underline{131}}\right), \mathrm{Y}_{2} \mathrm{NiMnO}_{6}{ }^{\underline{132}} and \mathrm{Bi}_{2} \mathrm{NiMnO}_{6} \stackrel{133}{\text {. As in figure6.3, we are using }} the experimental tolerance factor, calculated using the experimentally obtained bond-distances. PW stands for present work.
Figure 6.7: Thermal evolution of selected ranges in 2 \theta, obtained from the XRPD experiment: (a) 24.5-27^{\circ}, corresponding to the (111) primitive peak and (b) 56.5-60^{\circ} presenting a clear region to follow the evolving of the intensity. (c) Represents the 66.5-69^{\circ} 2 \theta interval corresponding to the (440) cubic reflections. In all the figures, the horizontal lines in blue, a guide to the eye, show the transition temperatures. The other dashed lines, in black, are a guide to follow the disappearance of the splittings.
Figure 6.8: Temperature evolution of the amplitudes of the modes breaking the symmetry from the cubic to the P 2_{1} / n and R \overline{3} symmetries, as obtained from the NPD ( 2-300 \mathrm{K})and XRPD (300-1600 K) data refinements. The \mathrm{GM}_{4}^{+}, \mathrm{X}_{3}^{+}, \mathrm{X}_{5}^{+}\left(P 2_{1} / n\right) and \mathrm{GM}_{4}^{+}(R \overline{3}) are the irreps of the symmetry-modes whose presence sets the distortions in the mentioned distorted phases. In the case of the \mathrm{X}_{5}^{+}irrep, in the lowtemperature range, we are showing separately the effectively freed two components of the 3 dimensional sub-space the irrep spans. In the case of the R \overline{3} space group symmetry, we are also showing, dimmed, the two components of the \mathrm{GM}_{5}^{+}irrep: they do not contribute effectively, so the final values of \mathrm{GM}_{4}^{+}for this symmetry correspond to them having been freed alone. The lower part of the graph (blue symbols) shows the relative difference, in the whole temperature range, between the \mathrm{GM}_{4}^{+}values in two refinements: only the effectively acting modes and all the modes freed. The continuos line in red is the fitting of the trigonal \mathrm{GM}_{4}^{+}amplitude values to the equation: A_{G M_{4}^{+}}=A(0)_{G M_{4}^{+}}\left[T_{C}-T\right]^{\alpha}, with the refined values A(0)_{G M_{4}^{+}}= 0.127(8), T_{c}=1523(1) \mathrm{K} and \alpha=0.26(1).
Figure 6.9: Variation with temperature of the lattice parameters for \mathrm{La}_{2} \mathrm{CoMnO}_{6}, as obtained from the NPD (2-300 K)(dimmed black symbols) and XRPD (300-1600 K) data refinements. In the studied temperature range (2-1600 \mathrm{K}), the compound undergoes two phase transitions: first-order one, discontinuous, at \simeq 450 K transforming the cell from a monoclinic to a trigonal; and a second one of the second-order occurring at \simeq 1520 \mathrm{K} from a trigonal to a non-distorted cell, cubic. The phase-transition sequence is the following: P 2_{1} / n \rightarrow R \overline{3} \rightarrow F m \overline{3} m.
Figure 7.1: Energy projections, in the \mathrm{X} 5+ two-dimensional space (A10, A11), for a fixed value of \mathrm{X}_{3}^{+}as indicated and for the set of values of \mathrm{GM}_{4}^{+}, as indicated. The values for the amplitudes, are taken from a regular grid (16 points in each dimension) created to construct the energy, one virtual distorted structure for the given set of amplitude-values in the grid. The energy tetra-surface is constructed after the relaxed structures’ energies, obtained form the simulations, are fitted to the polynomial constructed with the invariant, up to the fourth-order, terms describing the symmetry breaking form the prototype F m \overline{3} m cubic to the P 2_{1} / n monoclinic.
Figure 7.2: (a) Evolution with the tolerance factor of the partial distortion of the \mathrm{Sr}_{2} \mathrm{MM}^{\prime} \mathrm{O}_{6} given by the three components of the \mathrm{X}_{5}^{+}: A10,A11 and A12. The evolution of the global amplitudes of the tri-dimensional irrep \mathrm{X}_{5}^{+}with the tolerance factor, is shown in the panel (b)
Figure 7.3: (a) Representation of the \mathrm{X}_{5}^{+}direction given by the components c 1 et c 2 for the \mathrm{Sr}_{2} \mathrm{MM}^{\prime} \mathrm{O}_{6} and (b) their corresponding distortion given by the amplitudes of A10 and A11.
Figure 8.1: (a) Evolution of the synchrotron X-ray powder diffraction patterns at room temperature of the \mathrm{Sr}_{2} \mathrm{MSbO}_{6}(\mathrm{M}=\mathrm{Ln}, \mathrm{Y}) materials. The first black rectangle section is showing the presence of the primitive Bragg peak reflections with h+k+l=2 n+1, for all the materials. The second black rectangle section, shown bigger the lower graph (b), presents selected regions of diffraction pattern. The regions selected are those that correspond to the cubic reflections (444) and (800), which split in the monoclinic symmetry into a triplet and doublet, respectively.
Figure 8.2: Graph of maximal subgroups relating the space groups of the parent phase (highlighted in blue) and distorted phase (highlighted in green) of \mathrm{Sr}_{2} \mathrm{MSbO}_{6} compounds in the P 2_{1} / n and R \overline{3}. For each subgroup any irrep yielding this symmetry is indicated and the primary active irreps evidenced by the Rietveld refinement are highlighted in orange. In the scheme we are using the standard settings of the space groups as the Bilbao Crystallographic Server does for generating these graphs. The P 2_{1} / c is the standard setting of P 2_{1} / n
Figure 8.3: Experimental and calculated powder diffraction profiles for the Rietveld refinement of \mathrm{Sr}_{2} \mathrm{NdSbO}_{6} (a) and \mathrm{Sr}_{2} \mathrm{YbSbO}_{6} (b) at room temperature, using a structural model with P 2_{1} / n space group. The bars in the lower part of the graphics represent the Bragg peak positions.
Figure 8.4: a): Amplitudes of \mathrm{GM}_{4}^{+}, \mathrm{X}_{3}^{+}and \mathrm{X}_{5}^{+}irreps consolidating the RT phases of \mathrm{Sr}_{2} \mathrm{MSbO}_{6} \mathrm{M}=\mathrm{Ln} (Nd, Eu, Gd, Dy, Ho, Er, Tm, Yb and Y materials in the monoclinic symmetry P 2_{1} / n, as function of the tolerance factor. The red balls stand for the \mathrm{Sr}_{2} \mathrm{YSbO}_{6} amplitudes, since the Yttrium, Y, does not belong to the Lanthanides. b): Two dimensional section showing the obtained amplitudes of the irreps (Blue) \mathrm{GM}_{4}^{+}, (R e d) \mathrm{X}_{3}^{+}and the two components of the tri-dimensional \mathrm{X}_{5}^{+}: A10 in x-axis and A11 in y.
Figure 8.5: Evolution, as a function of the tolerance factor increasing of the unit-cell parameters (a), the unit-cell volume (b) and the monoclinic angle ( \beta) (b) at room temperature of the synthesized materials \mathrm{Sr}_{2} \mathrm{MSbO}_{6}(\mathrm{M}=\mathrm{Nd}, \mathrm{Eu}, \mathrm{Gd}, \mathrm{Dy}, \mathrm{Ho}, \mathrm{Er}, \mathrm{Tm}, \mathrm{Yb} and Y).
Figure 8.6: Thermal evolution of the 22-26^{\circ} 2 \theta interval, corresponding to the monoclinic primitive reflections (-111) and (111) (top) which disappears as temperature increases and the second 2 \theta interval 85.5-90 { }^{\circ} (bottom) corresponding to the (642) cubic reflection splits clearly in the trigonal phase obtained from XRPD experiment. The scattered intensity is projected and represented with shades of gray. Black corresponds to high intensity, and white to low intensity. The horizontal lines are a guide to the eye, and mark the temperatures at which the phase transitions in the \mathrm{Sr}_{2} \mathrm{MSbO}_{6} compounds were observed.
Figure 8.7: Temperature evolution of the amplitudes of some of the symmetry adapted modes in P 2_{1} / n and R \overline{3}, as obtained from the XRPD data refinements for (top) \mathrm{Sr}_{2} \mathrm{NdSbO}_{6} and \mathrm{Sr}_{2} \mathrm{YSbO}_{6} (bottom). \mathrm{GM}_{4}^{+}, \mathrm{X}_{3}^{+}, \mathrm{X}_{5}^{+}and \mathrm{GM}_{4}^{+}, \mathrm{GM}_{5}^{+}are the freed modes in the refinements for the P 2_{1} / n and R \overline{3} respectively. The amplitudes of the irreps in the high symmetry phase, R \overline{3}, have been scaled (multiplied by 3 / 2) to see the evolution, since the program AMPLIMODES uses mode normalization within the corresponding primitive unit cell and the obtained values were fitted with the equation A_{G M_{4}^{-}}=A(0)_{G M_{4}^{-}}\left[T_{c}-T\right]^{\alpha} with A(0)_{G M_{4}^{-}}=0.62(5), T_{c}=1225 \mathrm{K} and \alpha=0.20(8) for \mathrm{Sr}_{2} \mathrm{NdSbO}_{6} and A(0)_{G M_{4}^{-}}=0.46(1), T_{c}=875 \mathrm{K} and \alpha=0.22(4) for \mathrm{Sr}_{2} \mathrm{YSbO}_{6}.
Figure 8.8: Temperature evolution of the lattice parameters of \mathrm{Sr}_{2} \mathrm{MSbO}_{6}. The cell parameters of the trigonal phase are shown in the hexagonal notation. The monoclinic and hexagonal parameters have been scaled to be comparable with the edge of the centered cubic. The \mathrm{Sr}_{2} \mathrm{MSbO}_{6} materials undergo two phase transitions: a first-order one, discontinuous, transforming the cell from a monoclinic to a trigonal and a second one of the second-order from a triginal to a cubic cell. The phase-transition sequence is the following: P 2_{1} / n \rightarrow R \overline{3} \rightarrow F m \overline{3} m.
Figure 8.9: Phase diagram (phase transition temperatures versus tolerance factors) for the \mathrm{Sr}_{2} \mathrm{MSbO}_{6} ( \mathrm{M}=\mathrm{Ln}, \mathrm{Y}) compounds
Figure 9.1: Splitting of the singlet cubic reflection (444) { }_{\mathrm{c}} into a triplet: (404), (-404) and (044) reflections. The plot shows a fit with three reflections.
Figure 9.2: Selected 2 \theta intervals for the four compositions of the \mathrm{Sr}_{2} \mathrm{Co}_{1-x} \mathrm{Mg}_{x} \mathrm{TeO}_{6} series. a) Vanishment of the reflection that violates the I-centering condition on the h k l. b-c) Change of the cubic reflection splitting with the increase of the magnesium content.
Figure 9.3: Experimental (symbols) and calculated (line) X-ray profiles and neutron profiles for the Rietveld refinement of the \mathrm{Sr}_{2} \mathrm{Co}_{1-x} \mathrm{Mg}_{x} \mathrm{TeO}_{6} series at room temperature using a structural model with P 2_{1} / n space group for ( \mathrm{x}=0.0 and 0.1) and I 2 / m for ( \mathrm{x}=0.2 and 0.5). The bars in the lower part of the graphics represent the Bragg peak positions.
Figure 9.4: Variation, as function of the tolerance factor, of the three distortion irrep amplitudes \mathrm{GM}_{4}^{+} (black), \mathrm{X}_{3}^{+}(red) and \mathrm{X}_{5}^{+}(blue), and the cell parameters (black) and the unit cell volume (blue) for the \mathrm{Sr}_{2} \mathrm{Co}_{1-x} \mathrm{Mg}_{x} \mathrm{TeO}_{6} series at room-temperature. The colored backgrounds indicate the regions of the two space groups, P 2_{1} / n and I 2 / m, observed for the four compositions.
Figure 9.5: Disappearance, as temperature increases, of the primitive reflections (311), (-311), (131) and (-131) of the \mathrm{Sr}_{2} \mathrm{Co}_{0.9} \mathrm{Mg}_{0.1} \mathrm{TeO}_{6} compound.
Figure 9.6: Thermal evolution of the (444) cubic reflection splitting located in the interval ( 82^{\circ}<2 \theta<{ }^{\circ} 86) for the \mathrm{Sr}_{2} \mathrm{Co}_{1-x} \mathrm{Mg}_{x} \mathrm{TeO}_{6} compounds with (x=0.1,0.2,0.5). The splitting of the triplet (404), (-404) and (044) is clearly seen at 300 K and as temperature increases, the reflections get closer, up to 870 K where they remain as a single peak.
Figure 9.7: Temperature dependence of (a-c) the amplitudes of the modes related to the P 2_{1} / n, I 2 / m and I 4 / m symmetries and (d-f) their corresponding cell parameters, as obtained from NPD and XRD data refinements, for the \mathrm{Sr}_{2} \mathrm{Co}_{1-x} \mathrm{Mg}_{x} \mathrm{TeO}_{6} compounds with ( x=0.0,0.1,0.2,0.5)
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Tables

Table 10.2: Main bond distances ( \AA), selected angles ( { }^{\circ}), and \Phi\left(\Phi^{\prime}\right) and \Psi\left(\Psi^{\prime}\right) tilt angles, for \mathrm{Sr}_{2} \mathrm{Ni}_{1-x} \mathrm{Mg}_{x} \mathrm{TeO}_{6}(x=0.0,0.1,0.2,0.3 and 0.5) at room temperature. The values were obtained refining the mode breaking the symmetry from the cubic F m \overline{3} m to the monoclinic I 2 / m.
Table 10.4: Crystal structure data and refinement results, at room temperature for \mathrm{Ca}_{2} \mathrm{MSbO}_{6}; \mathrm{M}=\mathrm{Ln}(\mathrm{Nd}, \mathrm{Eu}, \mathrm{Gd}, \mathrm{Dy}, \mathrm{Ho}, \mathrm{Er}, \mathrm{Tm}, \mathrm{Yb}) and Y), from X -ray powder diffraction. Sb atoms occupy the site 2 \mathrm{c}(0,1 / 2,0); \mathrm{M} 2 / \mathrm{Ca} 2 atoms occupy the site 2 \mathrm{ d}(1 / 2,0,0); Sr atoms occupy the site 4 \mathrm{e}(\mathrm{x}, \mathrm{y}, \mathrm{z}) and O atoms occupy the site 4 \mathrm{e}(\mathrm{x}, \mathrm{y}, \mathrm{z}). The B_{\text {iso }} values for M and Ca cations were fixed in 0.5^{2} in order to guarantee the refinement convergence.
Table 10.5: The possible materials and their corresponding phase-transitions sequences.
Table 2.1: Last temperature treatment (\mathrm{K}) and duration (\mathrm{h}) for the \mathrm{A}_{2} \mathrm{MSbO}_{6} compounds.
Table 4.3: Summary of the symmetry mode decomposition, with respect to the ideal cubic perovskite F m \overline{3} m, of the room temperature monoclinic phase ( P 2_{1} / n), as given by AMPLIMODES. The primary mode, if existing, is written in bold letters. For each irrep the number of basis modes associated to the indicated atoms is shown in parenthesis. Atom labels refer to the F m \overline{3} m structure. The column headed Dimension refers to number of basis modes necessary in general to describe the corresponding irrep distortion; whereas the one headed as Direction indicates, in the convention of ISOTROPY [ISOTROPY], the direction in the irrep space corresponding to the irrep distortion. Mode amplitudes are given in the last column and their standard deviations in parenthesis.
Table 4.4: Input information for AMPLIMODES: high-symmetry phase information (virtual prototype structure), low-symmetry structure information and transformation matrix. Amm2 (ITA No.38, standard setting), and P m \overline{3} m (ITA No.221). Structures are given in the Server format, ready to be used.
Table 4.5: lists the number of basis symmetry modes and their irreps, that describe the
Table 4.5-2: lists the number of basis symmetry modes and their irreps, that describe the
Table 5.1: Input information for AMPLIMODES for FullProf: high-symmetry phase information (virtual prototype structure), low-symmetry structure information and transformation matrix. Amm2 (ITA No.38), R 3 c (ITA No.161), P4mm (ITA No.99) and P m \overline{3} m (ITA No.221). Structures are given in the Server format, ready to be used.
Table 5.2: Summary of the irreps taking part in the symmetry breaking from, space group ( P m \overline{3} m) to the orthorhombic A m m 2, the rhombohedral R 3 c, the rhombohedral R 3 m and the tetragonal P 4 m m. The list is part of the output of AMPLIMODES for FullProf. The number in parenthesis indicates the number of modes transforming according to the corresponding irrep. WP stands for Wyckoff positions.
Table 5.3: Values after the refinement of the amplitudes (in Å) of the irreps modes, transforming according to the listed irreps, taking part in the symmetry breaking from P m \overline{3} m space group to the room-temperature Amm 2 orthorhombic space group for the \mathrm{Na}_{0.5} \mathrm{K}_{0.5} \mathrm{NbO}_{3} in the present work (PW), compared to the one published in [NKN2] and to \mathrm{KNbO}_{3} in [106].
Table 5.4: Crystal structure data and refinement results for \mathrm{Na}_{0.5} \mathrm{K}_{0.5} \mathrm{NbO}_{3} determined using neutron diffraction data at 10 K using the R 3 c space group. Cell parameters: a=5.6188(6) \AA, c=13.7873(4) \AA, V= 376.96(1) \AA^{3}.
Table 6.4: Main bond-distances ( \AA) for \mathrm{La}_{2} \mathrm{CoMnO}_{6} obtained at different temperatures from NPD (D1B, ILL) data using the space group P 2_{1} / n (ITA No.14, non-standard setting)
Table 7.1: Atomic displacement parameters of the forty materials of the \mathrm{Sr}_{2} \mathrm{MM}^{\prime} \mathrm{O}_{6}\left(\mathrm{M}^{\prime}=\mathrm{W}, \mathrm{Mo}, \mathrm{Sb}, \mathrm{Ru}\right. and Te) families.
Table 8.1: Preparation conditions of \mathrm{A}_{2} \mathrm{MSbO}_{6} compounds. The data shown are as follows: Ionic radii in { }^{3}; \Delta \mathrm{r}=\left|r_{M^{3+}}-r_{S b^{5+}}\right|; tolerance factor; highest temperature of treatment; the detected impurities; symmetry of the crystal structure at room temperature and literature reference. PW stands for present work.
Table 8.2: Crystal structure data and refinement results, at room temperature for \mathrm{Sr}_{2} \mathrm{MSbO}_{6}; \mathrm{M}=\mathrm{Ln}(\mathrm{Nd}, \mathrm{Eu}, \mathrm{Gd}, \mathrm{Dy}, \mathrm{Ho}, \mathrm{Er}, \mathrm{Tm}, \mathrm{Yb}) and Y), from Synchrotron powder diffraction. The atomic positions and isotropic atomic displacement parameters were refined in the space group P 2_{1} / n. \mathrm{Sb} atoms occupy the site 2 \mathrm{a}(0,0,0); \mathrm{M} atoms occupy the site 2 \mathrm{ b}(0,0,1 / 2); Sr and O atoms occupy the site 4 \mathrm{e}(\mathrm{x}, \mathrm{y}, \mathrm{z}).
Table 9.1: Input information for AMPLIMODES for FullProf: high-symmetry phase information (prototype structure), low-symmetry structure information and transformation matrix for \mathrm{Sr}_{2} \mathrm{Co}_{1-x} \mathrm{Mg}_{x} \mathrm{TeO}_{6}. P 2_{1} / n (ITA No.14, non-standard setting), I2/m (ITA No.12, non-standard setting), I4/m (ITA No.87, standard setting) and F m \overline{3} m (ITA No.225, standard setting). The second part shows a list of the irreps taking part in the symmetry breaking from the cubic space group ( F m \overline{3} m) to the monoclinic P 2_{1} / n and I 2 / m space groups observed experimentally at RT, and to the tetragonal space group I 4 / m, observed as intermediate high-temperature space group. The list is part of the output of AMPLIMODES for FullProf. The number in parenthesis indicate the number of modes transforming according to the irrep. The irreps in bold are the ones representing the order parameters: 2 in the P 2_{1} / n and one in the I 2 / m space groups, respectively (see text)
Table 9.2: Amplitudes of the modes (in \AA) of the irreps taking part in the symmetry breaking from F m \overline{3} m space group to the monoclinic space groups P 2_{1} / n and I 2 / m for \mathrm{Sr}_{2} \mathrm{Co}_{1-x} \mathrm{Mg}_{x} \mathrm{TeO}_{6}(x=0.0,0.1,0.2,0.5) compounds at room-temperature after the refinement of (a) all modes and (b) only the effectively acting mode-amplitudes.
Table 9.3: Crystal structure data and refinement results for \mathrm{Sr}_{2} \mathrm{Co}_{1-x} \mathrm{Mg}_{x} \mathrm{TeO}_{6}(x=0.0,0.1,0.2 and 0.5) from NPD at room-temperature after the refinement of (a) all modes and (b) only the effectively acting mode. \mathrm{Co} / \mathrm{Mg}(2 \mathrm{ b}(0,0,1 / 2)), \mathrm{Te}(2 \mathrm{a}(0,0,0))
Table 9.4: Main bond distances ( \AA) and selected angles ( { }^{\circ}) for \mathrm{Sr}_{2} \mathrm{Co}_{1-x} \mathrm{Mg}_{x} \mathrm{TeO}_{6}(x=0.0,0.1,0.2 and 0.5) at room temperature. The values were obtained refining (a) all modes and (b) only the effectively acting mode.