Crystal structure and phase transition studies in perovskite-type oxides using powder-diffraction techniques and symmetry-mode analysis: SrLnMRuO6 (Ln = La, Pr, Nd; M = Zn, Co, Mg, Ni, Fe) and ALn2CuTi2O9 (A = Ca, Ba; Ln = La, Pr, Nd, Sm)

Chapters 5 to 10 are mapped below to their materials, figures, tables and the papers they became (ingested 2026-09-28).

status published · visibility public · id doc:thesis-03

Edurne Iturbe Zabalo, 2012 · supervisor Josu M. Igartua · Universidad del País Vasco / Euskal Herriko Unibertsitatea · language en

Files: sources/theses/thesis-03.pdf, sources/theses/thesis-03.md

Chapters

Figures

• Figure 1.1: The schematic representation of the A B O_{3} ideal perovskite structure, where the green, grey and red spheres represent, A, B and oxygen cations, respectively.
• Figure 1.10: Schematic diagram s showing the spin arrangement of (a) paramagnet, (b) ferromagnet, (c) antiferromagnet, (d) ferrimagnet, (e) canted antiferromagnet (where the magnetic components in different directions are represented by dashed arrows), and (f) spin-glass.
• Figure 1.11: Schematic representation of the spin arrangements in (a) A-type, (b) C-type, (c) E-type, and (d) G-type antiferromagnetic ordering.
• Figure 1.12: Examples of magnetic frustration on triangular lattice. (a) and (b) show no frustration, while, (c) and (d) show frustration associated to the disorder of the interaction (c) as well as to the geometry of the lattice (d).
• Figure 1.2: The schematic representation of the A A^{\prime} B B^{\prime} \mathrm{O}_{6} ideal double perovskite structure; where B cation and B \mathrm{O}_{6} octahedra are represented in purple, B^{\prime}-cation and B^{\prime} \mathrm{O}_{6} octahedra in green, A / A^{\prime}-cations in blue, and oxygens in red.
• Figure 1.3: Cation ordering schemes in perovskites. From left to the right: 0D (rock salt), 1D (columnar ordering), and 2D (layered ordering) are shown for A-site ordering in A A^{\prime} B_{2} \mathrm{O}_{6} perovskites (top) and for B-site ordering in A_{2} B B^{\prime} \mathrm{O}_{6} perovskites (bottom).
• Figure 1.4: 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 (red rectangle) and anti-phase boundary defects (blue rectangle); (c) complete disorder. B \mathrm{O}_{6} octahedra: green; B^{\prime} \mathrm{O}_{6} octahedra: purple; A / A^{\prime}-site cations: blue spheres; oxygen: red spheres.
• Figure 1.5: Simulated X -ray diffraction patterns for ordered \left(A A^{\prime}\right) B B^{\prime} \mathrm{O}_{6} cubic structure (space group F m \overline{3} m)(black) and disordered \left(A A^{\prime}\right)\left(B B^{\prime}\right) \mathrm{O}_{6} cubic structure (space group P m \overline{3} m)(blue). (Simulation characteristics: \lambda=1.5406 \AA a_{\mathrm{p}}=3.8873 \AA, 2 a_{\mathrm{p}}.)
• Figure 1.6: The coordination-polyhedron of A-site cation of the A B \mathrm{O}_{3} perovskite; (a) ideal 12-fold coordination (cubo-octahedron), (b) example of 12-fold coordination of a distorted perovskite (yellow bonds larger than the rest), and (c) more suitable 8 -fold coordination for (b) example.
• Figure 1.7: Simulated X -ray diffraction patterns for disordered \left(A A^{\prime}\right)\left(B B^{\prime}\right) \mathrm{O}_{6} cubic structure (black) and the most common distorted symmetries: R \overline{3} c (red), I4/mcm (green) and Pbnm (purple). In the inset is shown the splitting of the peaks. (Simulation characteristics: \lambda=1.5406 \AA (XRPD), a_{\mathrm{p}}=3.8873 \AA, P m \overline{3} m \left(a=a_{\mathrm{p}}\right), R \overline{3} c\left(\mathrm{a} \approx \sqrt{2} a_{\mathrm{p}}, \mathrm{c} \approx 2 \sqrt{6} a_{\mathrm{p}}\right), I 4 / m c m\left(a=\mathrm{b} \approx \sqrt{2} a_{\mathrm{p}}, \mathrm{c} \approx 2 a_{\mathrm{p}}\right), and Pbnm ( \mathrm{a} \approx \mathrm{b} \approx \sqrt{2} a_{\mathrm{p}}, \mathrm{c} \approx 2 a_{\mathrm{p}}.)
• Figure 1.8: Simulated X-ray diffraction patterns for ordered \left(A A^{\prime}\right) B B^{\prime} \mathrm{O}_{6} cubic structure (black) and the most common distorted symmetries: R \overline{3} (red), I 4 / m (green), I 2 / m (blue) and P 2_{1} / n (purple). In the inset is shown the splitting of the peaks. (Simulation characteristics: \lambda=1.5406 \AA (XRPD), a_{\mathrm{p}}=3.8873 \stackrel{\circ}{\mathrm{A}}, F m \overline{3} m\left(a=2 a_{\mathrm{p}}\right), R \overline{3}\left(\mathrm{a} \approx \sqrt{2} a_{\mathrm{p}}, \mathrm{c} \approx 2 \sqrt{6} a_{\mathrm{p}}\right), I 4 / m\left(a=\mathrm{b} \approx \sqrt{2} a_{\mathrm{p}}, \mathrm{c} \approx 2 a_{\mathrm{p}}\right), I 2 / m and P 2_{1} / n\left(\mathrm{a} \approx \mathrm{b} \approx \sqrt{2} a_{\mathrm{p}}\right., \mathrm{c} \approx 2 a_{\mathrm{p}}, \beta \approx 90^{\circ}).)
• Figure 1.9: Cooperative Jahn-Teller distortion of \mathrm{LaMnO}_{3} [58] material.
• Figure 10.1: X-ray powder diffraction patterns (\lambda=1.5406 \AA) for \mathrm{CaLn}_{2} \mathrm{CuTi}_{2} \mathrm{O}_{9}(L n=\mathrm{Pr}, \mathrm{Nd}, \mathrm{Sm}) at room temperature. The groups of peaks are indexed in the orthorhombic system (Pbnm). Inset shows the (022)-(202) peak splitting. The black rectangle in the L n=S m case indicates an unidentified impurity.
• Figure 10.10: Variation of the reduced cell parameters in the orthorhombic phase (space group Pbnm) with temperature for \mathrm{CaLn}_{2} \mathrm{CuTi}_{2} \mathrm{O}_{9}(L n=\mathrm{Pr}, N d, S m) from laboratory X-ray diffraction patterns. Dashed lines between points are guide to the eyes.
• Figure 10.11: X-ray powder diffraction patterns for (a) \mathrm{CaPr}_{2} \mathrm{CuTi}_{2} \mathrm{O}_{9}, (b) \mathrm{CaNd}_{2} \mathrm{CuTi}_{2} \mathrm{O}_{9}, and (c) \mathrm{CaSm}_{2} \mathrm{CuTi}_{2} \mathrm{O}_{9} at RT, (black) before, and (red) after annealing at 1500 K. The irreversibility of the samples is clearly observed.
• Figure 10.12: Amplitudes of \mathrm{GM}_{4}^{+}(fill circles), \mathrm{X}_{3}^{+}(stars) and \mathrm{X}_{5}^{+}irreps (empty circles) conforming the RT phase of the \mathrm{SrLaMRuO}_{6}\left(\mathrm{M}=\mathrm{Zn}^{a}, \mathrm{Mg}^{a}\right) (light-blue), \mathrm{SrLaMRuO}_{6}\left(\mathrm{M}=\mathrm{Cu}^{b}\right. [116], \mathrm{Ni}^{b} [116], \mathrm{Co}^{* c} [111], \mathrm{Co}^{a} [115]), \mathrm{SrNdMRuO}_{6}\left(\mathrm{M}=\mathrm{Zn}^{a}, \mathrm{Co}^{a}, \mathrm{Mg}^{a}, \mathrm{Ni}^{a}\right) (dark-blue) and \mathrm{SrPrMRuO}_{6}\left(\mathrm{M}=\mathrm{Zn}^{a}, \mathrm{Co}^{a}, \mathrm{Mg}^{a}, \mathrm{Ni}^{a}\right) (purple) compounds, and \Psi and \Phi tilt angles (red symbols), as obtained from the NPD { }^{a}, synchrotron radiation { }^{b} or \mathrm{XRPD}^{c} data refinement, in function of calculated tolerance factors. Lines are guide to the eyes.
• Figure 10.13: Tolerance factor and temperature dependence phase diagram of \mathrm{SrLaMRuO}_{6} (\mathrm{M}=\mathrm{Zn}(\mathrm{PW}), \mathrm{Cu}[116], \mathrm{Mg}(\mathrm{PW}), \mathrm{Ni}[116]), \quad \mathrm{SrNdMRuO}_{6} \quad(\mathrm{M}=\mathrm{Zn}, \mathrm{Co}, \mathrm{Mg}, \mathrm{Ni}) \quad[120] \quad and \quad \mathrm{SrPrMRuO}_{6} (\mathrm{M}=\mathrm{Co}, \mathrm{Mg}, \mathrm{Ni}) [125]. PW stands for present work. Two green tonality are used in P 2_{1} / n phase to distinguish the different transition sequences. Two-phase region separating P 2_{1} / n from R \overline{3} is shown in striped-yellow area. Grilled area highlights two phase transition overlapped area. Vertical dotted line is indicative of the same theoretical tolerance factor for \mathrm{SrLaZnRuO}_{6} and \mathrm{SrLaCoIrO}_{6} [127].
• Figure 10.14: Cation ordering schemes for A-site 2D (layered) ordering in A A^{\prime} B B^{\prime} \mathrm{O}_{6} perovskites, B and B^{\prime} completely (a) disordered and (b) ordered arrangement.
• Figure 10.2: X -ray powder diffraction patterns (\lambda=1.5406 \AA) for \mathrm{BaLn}_{2} \mathrm{CuTi}_{2} \mathrm{O}_{9}(L n=\mathrm{La}, \mathrm{Pr}, \mathrm{Nd}) at room temperature. The groups of peaks are indexed in the tetragonal system (I4/mcm). Inset shows Rietveld refinement results using I4/mcm and P m \overline{3} m models for \mathrm{BaNd}_{2} \mathrm{CuTi}_{2} \mathrm{O}_{9}). The black rectangles indicate, as in previous figure, the presence of an impurity.
• Figure 10.3: Graph of maximal subgroups connecting the space group P m \overline{3} m and its subgroups (left) I4/mcm (ITA No.140) and (right) Pbnm (ITA No. 62) via the (a-b,a+b,2c;0,0,0) and (a+b,a-b,-2c;1/2,1/2,0) transformation matrix, respectively. The active distortion modes are highlighted in orange. Green, blue, and red spheres indicate A / L n, \mathrm{Cu} / \mathrm{Ti}, and O atoms, respectively and the archetype phase structure in Pbnm bases is shown with grey octahedra (red sphere as oxygens), and light green spheres indicating A / \operatorname{Ln} atoms.
• Figure 10.4: Observed ( \circ), calculated ( -), and difference of neutron powder diffraction ( \lambda=1.494 \AA) Rietveld profiles for \mathrm{CaNd}_{2} \mathrm{CuTi}_{2} \mathrm{O}_{9} at room temperature using structural model with Pbnm space group. In the lower part the Bragg-positions of the phase and steel “impurity” are given.
• Figure 10.5: Reduced cell parameters as a function of the lanthanide cation ionic radius for \mathrm{CaLn}_{2} \mathrm{CuTi}_{2} \mathrm{O}_{9}(L n=\mathrm{Pr}, \mathrm{Nd}, \mathrm{Sm})( black ), \mathrm{CaLn}_{2} \mathrm{ZnTi}_{2} \mathrm{O}_{9}(L n=\mathrm{La}, \mathrm{Pr}, \mathrm{Nd}, \mathrm{Eu}) (red) [133] orthorhombic phases ( P b n m), and \mathrm{BaLn}_{2} \mathrm{CuTi}_{2} \mathrm{O}_{9}(\mathrm{Ln}=\mathrm{La}, \mathrm{Pr}, \mathrm{Nd}) (blue) tetragonal phase ( 4 / \mathrm{mcm}) at room temperature. The dashed lines are a guide for the eyes.
• Figure 10.6: The coordination polyhedron for \mathrm{Ca} / \mathrm{Ln} cations occupying the A-site of the A B \mathrm{O}_{3} perovskite; (a) ideal 12-fold coordination (cubo-octahedron), (b) 12-fold coordination of the orthorhombic perovskite (yellow bond larger than the rest), and (c) 8 -fold coordination of the orthorhombic perovskite (yellow bonds eliminated to identified the four-fold antiprisma). In (b), black \mathrm{Ca} / \mathrm{Ln}-O bonds indicate the first coordination sphere, whereas the yellow bonds the second. (a) and (b) show the same view, the former in cubic basis and the later in orthorhombic basis. Green, blue and red atoms are representative for \mathrm{Ca} / \mathrm{Ln}, \mathrm{Cu} / \mathrm{Ti}, and O atoms, respectively.
• Figure 10.7: Variation of (a) mean A-\mathrm{O} bond lengths ( \AA) for first coordination sphere ( <A-\mathrm{O}>(1)) and second coordination sphere ( <A-\mathrm{O}> (2)); (b) A-B interatomic minimum distances and A - O maximum distances ( \AA); (c) first coordination sphere volume ( \AA^{3}) ( A \mathrm{O}_{8}); (d) distortion irrep amplitudes for Pm \overline{3} m \rightarrow Pbnm symmetry breaking; and (e) \mathrm{Cu} / \mathrm{TiO}_{6} octahedra \Phi_{[110] \mathrm{p}} and \Psi_{[001] \mathrm{p}} tilt angles ( { }^{\circ}), for \mathrm{CaLn}_{2} \mathrm{CuTi}_{2} \mathrm{O}_{9} (black circle) and \mathrm{CaLn}_{2} \mathrm{ZnTi}_{2} \mathrm{O}_{9} [133] (red square) compounds at room temperature.
• Figure 10.8: Extracts from the laboratory X-ray diffraction ( \lambda_{\mathrm{K} \alpha 1}=1.5406 \AA and \lambda_{\mathrm{K} \alpha 2}=1.5444 \AA) pattern projections for (a) \mathrm{CaPr}_{2} \mathrm{CuTi}_{2} \mathrm{O}_{9}, (b) \mathrm{CaNd}_{2} \mathrm{CuTi}_{2} \mathrm{O}_{9} and (c) \mathrm{CaSm}_{2} \mathrm{CuTi}_{2} \mathrm{O}_{9} from 800 to 1460 K, showing the evolution of the (111) primitive reflection and (221) cubic reflection as a function of temperature.
• Figure 10.9: Portion of the laboratory X -ray powder diffraction ( \lambda_{\mathrm{K} \alpha 1}=1.5406 \AA and \lambda_{\mathrm{K} \alpha 2}=1.5444 \AA) profiles for \mathrm{CaLn}_{2} \mathrm{CuTi}_{2} \mathrm{O}_{9} showing the evolution in temperature of the (002)-(202) reflections in Pbnm, at 770 \mathrm{K}, 1150 \mathrm{K} and 1450 K. The bars in the lower part of the graphics correspond to the Bragg reflections’ positions; red (lower) bars indicate impurity phase (diffractometer platinum filament).
• Figure 2.1: Schematic representation of the synthesis treatment for double perovskite compounds.
• Figure 3.1: Diffraction of a beam from a set of lattice planes, separated by a distance d.
• Figure 3.2: Cones of diffracted radiation from the incident-beam for a powder sample.
• Figure 3.3: The irregular function of neutron scattering length against atomic number. The markers joined by dashed line indicate the difference of scattering length among isotopes.
• 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 \operatorname{Sr} L n M R u \mathrm{O}_{6}(L n=\mathrm{La}, \mathrm{Nd}, \mathrm{Pr}; M=\mathrm{Zn}, \mathrm{Co}, \mathrm{Mg}, \mathrm{Ni}) phases. For each space group, the irreducible representation of the distortion mode yielding this symmetry is indicated if existing. For the parent phase (red) and isotropy groups, the corresponding simulated patterns are plotted in the P 2_{1} / n non conventional monoclinic bases. The combination of distortion modes responsible for the F m \overline{3} m \rightarrow P 2_{1} / n symmetry-breaking are plotted for P 2_{1} / n (orange).
• Figure 5.1: Experimental (symbols) and calculated (line) X-ray powder-diffraction profile for the Rietveld refinement of \mathrm{SrNdCoRuO}_{6} at room temperature, using a structural model with the space group P 2_{1} / n. The bars in the lower part of the graphics represent the Bragg peaks positions that correspond to the SrNdCoRuO 6 phase: first row corresponds to the P 2_{1} / n, second row to P 4_{2} / n, third row to P 4 / m n c, fourth row to P 4_{2} / n n m and fifth row to R \overline{3} space group. The black rectangle section is shown bigger in Figure 5.2. Inset shows in detail the appearance of the primitive Bragg peak reflections with h+k+l= 2 n+1. * very small amount of unidentified impurity.
• Figure 5.10: The thermal evolution of the (422), (620) and (642) cubic reflections as determined from XRPD measurements for \mathrm{SrNdZnRuO}_{6} are shown in left, middle and right panels, respectively. Each panel represent a 2 \theta section of the whole pattern at a fixed temperature: a) 340 \mathrm{K}, \mathrm{ b}) 880 \mathrm{K} and c) 1250 K.
• Figure 5.11: Results of the XRPD data refinements at 880 K in \mathrm{SrNdZnRuO}_{6}, using five different models with P 2_{1} / n, P 4_{2} / n, I 2 / m, I 4 / m and R \overline{3} space groups. The failure of I 2 / m, I 4 / m and R \overline{3} symmetries is clear, since small peaks are observed in the pattern which are not indexed by the models. The tetragonal P 4_{2} / n symmetry is suggested for the intermediate phase.
• Figure 5.12: Temperature evolution of the cell parameters and cell-volume of \mathrm{SrNdMRuO}_{6} (M=\mathrm{Zn}, \mathrm{Co}, \mathrm{Mg}, \mathrm{Ni}). The monoclinic a and b cell parameters have been scaled to be comparable with the tetragonal and cubic cell parameters. The transition temperatures ( \mathrm{T}_{1}: P 2_{1} / n P 4_{2} / n; \mathrm{T}_{2}: P 4_{2} / n F m \overline{3} m) for \mathrm{SrNd} \mathrm{MRuO}_{6}(\mathrm{M}=\mathrm{Zn}, \mathrm{Co}, \mathrm{Mg}, \mathrm{Ni}) are the following: \mathrm{T}_{1}=740,660,560,440 \mathrm{K}, and \mathrm{T}_{2}=1060,1000,940, 820 K, respectively.
• Figure 5.13: Graph of maximal subgroups relating the space groups of the parent (highlighted in blue) and distorted P 2_{1} / n and P 4_{2} / n phase (highlighted in red) of \mathrm{SrNdMRuO}_{6} compounds. For each subgroup any irrep yielding this symmetry is indicated and the two primary active irreps evidenced by the Rietveld refinement are highlighted in green. In the scheme we are using the standard settings of the space groups as the BCS does for generating these graphs. P 2_{1} / c is the standard setting of P 2_{1} / n.
• Figure 5.14: Variation of (a) \mathrm{GM}_{4}^{+}distortion irrep amplitude (\AA); (b) \mathrm{X}_{3}^{+}distortion irrep amplitude (\AA); (c) \mathrm{X}_{5}^{+}distortion irrep amplitude ( \AA), and (d) isotropic temperature factors ( \AA^{2}) as function of the temperature ( K) for \mathrm{SrNdCoRuO}_{6}. Label color and symbol indicate three different refinements: black full circle, isotropic temperature factors for \mathrm{O} 1, \mathrm{O} 2 and O 3 independent; blue empty circle, isotropic temperature factors of O1 and O2 refined with the same code and O3 independent; and red star, all isotropic temperature factors refined with the same code. In (d) half-full black and blue symbols, and dotted lines indicate the average value of the isotropic temperature factors for each refinement.
• Figure 5.15: Geometrical representations of the distortion modes of the symmetry breaking from the cubic prototype phase into the tetragonal P 4_{2} / n phase: \mathrm{GM}_{3}^{+}(1), \mathrm{GM}_{3}^{+}(1), \mathrm{GM}_{4}^{+}(1), \mathrm{X}_{2}^{+}(1), \mathrm{X}_{3}^{+}(1), \mathrm{X}_{4}^{+}(2) and \mathrm{X}_{5}^{+}(3). The numbers in parenthesis indicate the number of modes transforming according to the corresponding irrep. The principal modes are highlighted in grey.
• Figure 5.16: and Table 5.6 exhibit the temperature evolution of the amplitude of \mathrm{GM}_{4}^{+}, \mathrm{X}_{3}^{+}and \mathrm{X}_{5}^{+}modes, the rest are negligible, in the monoclinic and in the tetragonal phases. As mentioned, the physical interpretation of the modes taking part in the symmetry breaking is the tilting of the octahedra. In Figure 5.16 we have also shown the octahedral tilt angles (tilt systems: \mathrm{a}^{+} \mathrm{b}^{-} \mathrm{b}^{-}\left(P 2_{1} / n\right) and \mathrm{a}^{+} \mathrm{a}^{+} \mathrm{c}^{-}\left(P 4_{2} / n\right)). The matching of the mode amplitudes and tilt angles is perfect (Figure 5.8). \mathrm{GM}_{4}^{+}and \mathrm{X}_{3}^{+}modes (rotation and inclination
• Figure 5.2: Results of the XRPD data refinements, in the 36-44^{\circ} 2 \theta interval, in \mathrm{SrNdCoRuO}_{6}, using five different models with P 2_{1} / n, P 4_{2} / n, P 4 / m n c, P 4_{2} / n n m and R \overline{3} space groups. P 2_{1} / n and P 4_{2} / n models are able to take into account all the peaks of the pattern, whereas the other three models are not able (see the boxes).
• Figure 5.3: Results of the XRPD a) and b) and SXRPD c) and d) data refinements, in the 19-20^{\circ} and 5.15.3^{\circ} 2 \theta interval respectively, in \mathrm{SrNdCoRuO}{ }_{6}, using two different models with P 2_{1} / n and P 4_{2} / n space groups. The tetragonal space group has only one reflection in this 2 \theta interval, and is unable to take into account the splitting observed in SXRPD; on contrary, the monoclinic space group has three reflections and takes perfectly into account the splitting.
• Figure 5.4: Experimental (symbols) and calculated (line) neutron powder-diffraction profiles for the Rietveld refinement of \mathrm{SrNd} M \mathrm{NuO}_{6}(\mathrm{M}=\mathrm{Zn}, \mathrm{Co}, \mathrm{Mg}, \mathrm{Ni}) at room temperature, using a structural model with the space group P 2_{1} / n. The bars in the lower part of the graphics represent the Bragg peaks positions. In the case \mathrm{SrNdNiRuO}_{6}, the lower set of bars correspond to the impurity \mathrm{NiO}(4.84(7) \%).
• Figure 5.5: Variation of (a) \mathrm{GM}_{4}^{+}distortion irrep amplitude (Å); (b) \mathrm{X}_{3}^{+}distortion irrep amplitude (Å); (c) \mathrm{X}_{5}^{+}distortion irrep amplitude ( \AA), and (d) isotropic temperature factors ( \AA^{2}), as function of experimentally calculated tolerance factor for \mathrm{SrNd} M \mathrm{RuO}_{6}(M=\mathrm{Zn}, \mathrm{Co}, \mathrm{Mg}, \mathrm{Ni}) at room temperature. Label color and symbol indicate three different refinements: black full circle, isotropic temperature factors for O1, O 2 and O 3 independent; blue empty circle, isotropic temperature factors of O 1 and O 2 refined with the same code and O3 independent; and red star, all isotropic temperature factors refined with the same code. In (d) half-full black and blue symbols, and dotted lines indicate the average value of the isotropic temperature factors for each refinement.
• Figure 5.6: Relative orientation of the unit cell of the prototype cubic simple perovskite ( P m \overline{3} m)(blue), of the prototype cubic double perovskite (F m \overline{3} m) (dashed line), and the distorted monoclinic P 2_{1} / n of the double perovskite (solid line).
• Figure 5.7: Geometrical representations of the distortion modes of the symmetry breaking from the cubic prototype phase into the monoclinic P 2_{1} / n phase: \mathrm{GM}_{3}^{+}(1), \mathrm{GM}_{3}^{+}(1), \mathrm{GM}_{4}^{+}(1), \mathrm{GM}_{5}^{+}(4), \mathrm{X}_{2}^{+}(1), \mathrm{X}_{3}^{+}(1) and \mathrm{X}_{5}^{+}(3). The numbers in parenthesis indicate the number of modes transforming according to the corresponding irrep. The principal modes are highlighted in grey.
• Figure 5.8: Projection along the cubic (simple) perovskite [001]_{p} and [110]_{p} directions, of a section of the unit cell of the double perovskite with A A^{\prime} B B^{\prime} \mathrm{O}_{6} general formula in the P 2_{1} / n and P 4_{2} / n space groups. There is a direct relation between the octahedra tilt and symmetry-adapted mode.
• Figure 5.9: presents the thermal evolution, from XRPD, of the scattering intensity for two 2 \theta intervals, corresponding to 2 \theta=24^{\circ}-26^{\circ}(111;-111) monoclinic primitive reflections and to 2 \theta=92^{\circ}-96^{\circ} (see below, (642) cubic Fm \overline{3} m reflection).
• Figure 6.1: (1)Observed (symbols), calculated (line) and difference (bottom line) XRPD ( \lambda=1.5406 \AA) profile for \mathrm{SrPrMgRuO}_{6} at room temperature. The bars in the lower part of the graphic represent the Bragg-reflections’ positions for three different models: first row, P 2_{1} / n; second row, I 2 / m and third row, P 4_{2} / n. The rectangular section is enlarged in the inset, showing in detail the presence of supercell reflections. The arrows in panel b) indicate the absence of those reflections in the I 2 / m space group model. (2) Observed (symbols), calculated (line) and difference (bottom line) SXRPD ( \lambda=0.8256 \AA) profile for \mathrm{SrPrMgRuO}_{6} at room temperature using a structural model with the P 2_{1} / n space group. The bars indicate the Bragg-reflections’ positions for P 2_{1} / n and P 4_{2} / n models. The inset shows the monoclinic splitting, the tetragonal symmetry does not take into account the observed splitting, ((a) and (c) panels). * indicates very small quantity of an unidentified impurity.
• Figure 6.10: Features of the A-site cation coordination polyhedra ( A \mathrm{O}_{8}) for \operatorname{Sr} L n M R u \mathrm{O}_{6}(M=\mathrm{Zn}, \mathrm{Mg}, \mathrm{Ni}) perovskites. Evolution of (a) the first coordination-sphere’s mean A-O bond-lengths, (b) longest first coordination-sphere ( A \mathrm{O}_{8}) bond-length and shortest second coordination-sphere bond-length, (c) second coordination-sphere’s mean A-O bond-lengths, and (d) longest A-O and shortest A-B bond-lengths, as a function of temperature. Black-rectangle, red-triangle, and purple-rhombus stand for zinc, magnesium and nickel compounds, respectively. In (b) bond-lengths for the trigonal phase for Ni- and Mgcompounds are not indicated. Structural phase transition temperatures are indicate by dotted red and purple lines for magnesium and nickel compounds, respectively.
• Figure 6.11: Schematic of the group-subgroup chains, going from the prototype cubic symmetry (Fm \overline{3} m) to the room temperature space group ( P 2_{1} / n) and to the intermediate-phase space group ( R \overline{3}). The labels for the possible irreducible representations (irreps) capable of yielding those symmetry are indicated. In the symmetry-breaking from the cubic to the monoclinic space group, the irreps marked in green mean the two necessary irreps for the symmetry to be broken, two active modes. From the cubic to the trigonal space group, the irrep marked in red means the primary mode in that case.
• Figure 6.12: Temperature dependence of the amplitudes of the modes conforming the P 2_{1} / n and R \overline{3} structure of \operatorname{SrPr} M R u \mathrm{O}_{6}(M=\mathrm{Zn}( top ), M g( middle ), N i( bottom )), and \Psi and \Phi tilt-angles, as obtained from the NPD data refinement. The amplitudes of the irreps in the high symmetry phase have been scaled (multiplied by 3 / 2) to see the evolution, since AMPLIMODES uses mode normalization within the corresponding primitive unit cell.
• Figure 6.2: Experimental (symbols) and calculated (line) neutron powder-diffraction profiles (D2B \lambda=1.594 \AA) for the Rietveld refinement of \mathrm{SrPr} \mathrm{MRuO}_{6}(\mathrm{M}=\mathrm{Zn}, \mathrm{Co}, \mathrm{Mg}, \mathrm{Ni}) at room temperature, using a structural model with the space group P 2_{1} / n. The bars in the lower part of the graphics correspond to the Bragg-reflections’ positions. In the case \mathrm{SrPrNiRuO}_{6}, the lower set of bars correspond to the impurity \mathrm{NiO}(3.75(6) \%).
• Figure 6.3: Graphical representation of (a) and (c) ideal, and (b) and (d) distorted \mathrm{Sr} / \mathrm{PrO}_{12} cubooctahedral configuration (highlighted in blue) for \mathrm{SrPrZnRuO}_{6} in the P 2_{1} / n space group at room temperature. Projections along the cubic (simple) perovskite axes, (a) and (b) along [001]_{\mathrm{p}}, and (c) and (d) along [010]_{\text {p }}. Yellowish sphere, light and dark blue spheres, and red sphere are A / A^{\prime}-site cation, B and B^{\prime}-site cation, and oxygen cation, respectively. B-\mathrm{O} and B^{\prime}-\mathrm{O} bonds are indicated in orange. In (a) white lines represent A / A^{\prime}-O bonds, and in (b) eight shortest A / A^{\prime}-O bonds are in white and four largest bonds in red. Grey circles highlight the B and B^{\prime}-site cations inside the second coordination-sphere.
• Figure 6.4: Features of the A-site cation coordination polyhedra ( A \mathrm{O}_{8}) for P 2_{1} / n \operatorname{SrLnMRuO} 6 perovskites [115, 116, 120]. Evolution of (a) the first coordination-sphere’s mean A-O bond-lengths, and (b) second coordination-sphere’s mean A-O bond-lengths as a function of the lanthanide ionic radii ( \AA). Black filled rectangle stands for zinc; blue filled circle, cobalt; green star, copper; red triangle, magnesium; and, finally, purple rhombus, nickel.
• Figure 6.5: A-B interatomic minimum distances and A-O maximum distances (indicating the limit of the second coordination-sphere) for (a) lanthanum [115, 116], (b) praseodymium, and (c) neodymium compounds [120], as a function of the B-site cation radii ( \AA); showing which materials have B-site cations inside the second coordination-sphere. Mg compounds are shown in red, to indicate the particular behavior attributed to cation type.
• Figure 6.6: Amplitudes of \mathrm{GM}_{4}^{+}, \mathrm{X}_{3}^{+}and \mathrm{X}_{5}^{+}irreps (black balls) conforming the RT phase of \mathrm{SrPr} \mathrm{MRuO}_{6} (\mathrm{M}=\mathrm{Zn}, \mathrm{Co}, \mathrm{Mg}, \mathrm{Ni}) compounds, and \Psi( star ) and \Phi( square ) tilt-angles, as function of the observed tolerance factor. M \mathrm{O}_{6}, \mathrm{RuO}_{6} and average tilt-angles are indicated in blue, green and red color, respectively.
• Figure 6.7: shows the temperature dependence of the laboratory X-ray diffraction profile projections around the ( 111;-111) monoclinic primitive reflections ( 2 \theta=24^{\circ}-26^{\circ}) and (642) cubic reflection \left(2 \theta=92^{\circ}-96^{\circ}\right). The reflections shift to lower 2 \theta positions due to thermal expansion. At low temperatures the monoclinic primitive reflection is clearly seen and disappears at higher temperatures, indicating a phase transition to a non-primitive space group. All compounds undergo a temperature induced structural phase transition taking place at \approx 1225,1050,950 and 1025 \mathrm{K}, \mathrm{SrPrMRuO}_{6}(\mathrm{M}=\mathrm{Zn}, \mathrm{Co}, \mathrm{Mg}, \mathrm{Ni}) respectively. The XRPD experiment was not enough good to determine the new phases’ symmetry.
• Figure 6.8: Results of the NPD (D2B \lambda=1.594 \AA) data refinement, for \mathrm{SrPrMgRuO}_{6} at 1123 K, using three different models with the R \overline{3}, I 4 / m and I 2 / m space group. The bars indicate the Bragg reflections. We are showing only the 87.5-90^{\circ} 2 \theta interval.
• Figure 6.9: Temperature evolution of the cell parameters of \operatorname{SrPr} M \mathrm{MuO}_{6}(M=\mathrm{Zn}, \mathrm{Mg}, \mathrm{Ni}) as obtained from the NPD data (HRPT \lambda=1.494 \AA(\mathrm{Zn}, \mathrm{Ni}), \mathrm{D} 2 \mathrm{ B} \lambda=1.594 \AA(\mathrm{Mg})): cell parameters for the monoclinic and trigonal phases. For comparison the parameters have been scaled as indicated by the legends.
• Figure 7.1: Experimental (circles) and calculated (line) neutron diffraction profiles for the Rietveld refinements of \mathrm{SrLaZnRuO}_{6} and \mathrm{SrLaMgRuO}_{6} at room temperature using structural models with P 2_{1} / n space group. Markers below correspond to Bragg reflections. Insets show NPD R-point reflection refinement using monoclinic and orthorhombic models; and XRPD data.
• Figure 7.10: Experimentally calculated tolerance factor and temperature dependence phase diagram of \mathrm{SrLaMRuO}_{6}(\mathrm{M}=\mathrm{Zn}[\mathrm{PW}], \mathrm{Cu}[116], \mathrm{Mg}[\mathrm{PW}], \mathrm{Ni}[116]), \mathrm{SrNdMRuO}_{6}(\mathrm{M}=\mathrm{Zn}, \mathrm{Co}, \mathrm{Mg}, \mathrm{Ni})[120] and \mathrm{SrPrMRuO}_{6}(\mathrm{M}=\mathrm{Co}, \mathrm{Mg}, \mathrm{Ni}) [125]. PW stands for present work. Two green tonality are used in P 2_{1} / n phase to distinguish the different transition sequences. Two-phase region separating P 2_{1} / n from R \overline{3} is shown in striped-yellow area. Grilled area highlights two phase transition overlapped area. Vertical dotted line is indicative of the same theoretical tolerance factor for \mathrm{SrLaZnRuO}_{6} and \mathrm{SrLaCoIrO}_{6} [127].
• Figure 7.2: Amplitudes of \mathrm{GM}_{4}^{+}(filled circles), \mathrm{X}_{3}^{+}(stars) and \mathrm{X}_{5}^{+}irreps (empty circles) in the RT structures of the \mathrm{SrLaMRuO}_{6}\left(\mathrm{M}=\mathrm{Zn}^{a}, \mathrm{Mg}^{a}\right) (PW) (light-blue), \mathrm{SrLaMRuO}_{6} \left(\mathrm{M}=\mathrm{Cu}^{b}[116], \mathrm{Ni}^{b}[116], \mathrm{Co}^{* a}[111], \mathrm{Co}^{c}[115]\right), \quad \mathrm{SrNdMRuO}_{6} \quad\left(\mathrm{M}=\mathrm{Zn}^{a}, \mathrm{Co}^{a}, \mathrm{Mg}^{a}, \mathrm{Ni}^{a}\right) \quad (dark-blue) [120] and \mathrm{SrPrMRuO}_{6}\left(\mathrm{M}=\mathrm{Zn}^{a}, \mathrm{Co}^{a}, \mathrm{Mg}^{a}, \mathrm{Ni}^{a}\right) (purple) [125] compounds, and \Psi and \Phi tilt-angles (red symbols), as obtained from the NPD { }^{a}, synchrotron radiation { }^{b} or XRPD { }^{c} data refinement, in function of calculated tolerance factors ( \mathrm{Sr} / \mathrm{Ln}-\mathrm{O} distance in nine coordination is chosen since it is the highest coordination number available for all tabulated rare-earths). Lines in the figure are guide to the eyes. PW stands for present work.
• Figure 7.3: Temperature evolution of 54-55.5^{\circ} (primitive peak, (-311;311;-131;131)), 87.8-90 { }^{\circ} (cubic (444) reflection) and 95-99^{\circ} (cubic (642) reflection) 2 \theta ranges for \mathrm{SrLaZnRuO}_{6}. The calculated and the difference of the diffraction profiles are obtained after NPD data Rietveld refinement of the structure at each temperature. The structural models used in the refinements have the following symmetries: (a) P 2_{1} / n for 670 K, (b) coexistence of P 2_{1} / n and R \overline{3} phases for 870 K, and R \overline{3} for (c) 1170 K. Markers indicate Bragg reflections. In (b) blue continued line corresponds to P 2_{1} / n phase and red dotted line to R \overline{3} phase.
• Figure 7.4: Refinements results for \mathrm{SrLaZnRuO}_{6}, in the 86-90 { }^{\circ} 2 \theta interval at 920 and 1220 K temperatures, using different models with R \overline{3} and I 4 / m space groups, and R \overline{3} and F m \overline{3} m, respectively. The bars in lower part indicate the Bragg reflections with Miller indices.
• Figure 7.5: Refinements results for \mathrm{SrLaZnRuO}_{6}, in the 22-25^{\circ} (cubic (200) reflection), 81-84^{\circ} (cubic (533) and (622) reflections) and 99-102 { }^{\circ} (cubic (577;731) reflections) 2 \theta interval at 970 K, using different models with R \overline{3} and I 2 / m space groups.
• Figure 7.6: Temperature evolution of the concentration of the P 2_{1} / n (blue) and R \overline{3} (red) phases in the co-existence range for \mathrm{SrLaZnRuO}_{6}. Inset show NPD data Rietveld refinement in combination of two symmetries: blue continued line corresponds to P 2_{1} / n phase and red dotted line to R \overline{3} phase.
• Figure 7.7: Variation of the cell parameters for \mathrm{SrLaZnRuO}_{6} and \mathrm{SrLaMgRuO}_{6} with the temperature. Dashed lines indicate the coexistence temperature range between P 2_{1} / n and R \overline{3} phases.
• Figure 7.8: Temperature evolution of the \mathrm{GM}_{4}^{+}, \mathrm{X}_{3}^{+}and \mathrm{X}_{5}^{+}mode amplitudes, as obtained from refinement of the NPD data for \mathrm{SrLaZnRuO}_{6} (blue, opened symbols) and \mathrm{SrLaMgRuO}_{6} (red, closed symbols). Dashed lines indicate the temperature range of coexistence between the P 2_{1} / n and R \overline{3} phases.
• Figure 7.9: Temperature dependence of the rhombohedral angle \alpha for \mathrm{SrLaZnRuO}_{6}. The solid line is a linear fit to the experimental points data (follows \alpha=60.86(1)-5.6(1) \mathrm{E}^{-4} \mathrm{ T} equation) and the dashed line an extrapolation to \alpha=60^{\circ}, T_{C} \approx 1535 \mathrm{K}.
• Figure 8.1: Thermal variation of the NPD patterns acquired with \lambda=2.52 \AA in the temperature range 10<T<135 \mathrm{K} of a) \mathrm{SrLaCoRuO}_{6}(T=\approx 70 \mathrm{K}), b) \mathrm{SrNdCoRuO}_{6}(T=\approx 80 \mathrm{K}), c) \mathrm{SrPrCoRuO}_{6}(T=\approx 85 \mathrm{K}), d) \mathrm{SrNdNiRuO}_{6}(T=\approx 80 \mathrm{K}) and e) \mathrm{SrPrNiRuO}_{6}(T=\approx 85 \mathrm{K}).
• Figure 8.2: Spin directions in a fitted magnetic models for \mathrm{SrLaCoRuO}_{6}. The moments lie in the (010) plane. \mathrm{Co} / \mathrm{Ru} atoms are shown as light/dark blue balls; green balls represent \mathrm{Sr} / \mathrm{La} atoms. Oxygen atoms have been omitted for clarity. The atoms are shown in the nuclear cell; magnetic propagation vector (1 / 2,0,1 / 2).
• Figure 8.3: Observed (circles), calculated (full line), and difference NPD (D1B, \lambda=2.52 \AA) Rietveld profiles for \mathrm{SrLaCoRuO}_{6}, \mathrm{SrNdCoRuO}_{6}, \mathrm{SrPrCoRuO}_{6}, \mathrm{SrNdNiRuO}_{6} and \mathrm{SrPrNiRuO}_{6} at 4 K. Reflection markers correspond to the positions of Bragg reflections of crystal structure (up) and magnetic structure (bottom). The magnetic reflections are marked with the arrow.
• Figure 8.4: Possible magnetic model of the final refined \left(3 \Gamma 1_{2 a} \oplus 3 \Gamma 2_{2 b} \oplus 3 \Gamma 1_{4 e}\right) magnetic structure for \mathrm{SrPrCoRuO}_{6} at 4 \mathrm{K}. \mu_{\mathrm{y}(\mathrm{Co}, \mathrm{Ru})}=0 was set in order to guarantee the convergence of the model ( \mu_{\text {TOT,Pr }}=1.201 \mu_{\mathrm{B}}, \mu_{\mathrm{TOT}, \mathrm{Co}}=0.326 \mu_{\mathrm{B}}, \mu_{\mathrm{TOT}, \mathrm{Ru}}=2.810 \mu_{\mathrm{B}}). Sr / \operatorname{Ln} green circles, blue vectors; Co, light blue circles, brown vectors; and Ru dark blue circles, green vectors. The atoms are shown in the nuclear cell; magnetic propagation vector (1/2,0,1/2).
• Figure 9.1: Experimental (symbols) and calculated (line) X-ray powder-diffraction profiles for the Rietveld refinement of \mathrm{SrLnFeRuO}_{6}(\mathrm{Ln}=\mathrm{La}, \mathrm{Nd}, \mathrm{Pr}) at room temperature, using a disordered orthorhombic (Pbnm) structural model. The upper (lower) markers indicate the position of the structural \mathrm{SrLnFeRuO}_{6} (impurity: \mathrm{La}_{3} \mathrm{RuO}_{7}, \mathrm{Nd}_{3} \mathrm{RuO}_{7} and \mathrm{Pr}_{3} \mathrm{RuO}_{7}) Bragg reflections. The inset shows the absence of the superstructure reflection, indicated with an arrow. * indicates the impurity.
• Figure 9.10: Thermal evolution of the (101) reflection of (a) \mathrm{SrLaFeRuO}_{6}, (b) \mathrm{SrNdFeRuO}_{6} and (c) SrPrFeRuO \mathrm{O}_{6} collected with \lambda=2.52 \AA between RT and 573 K, in D1B neutron powder diffractometer.
• Figure 9.11: Low angle region of the neutron powder diffraction patterns collected at low temperature, including the observed intensity (open circles), calculated fit to the data (continuous line) and the difference curve. Tick marks correspond to Bragg peak location, (up) \lambda=2.52 \AA and (bottom) \lambda / 2=1.26 \AA. The inset show the data refinement using the crystal and magnetic model. The upper (lower) marks correspond to crystal (magnetic) structure using \lambda=2.52 \AA and \lambda / 2=1.26 \AA.^{*} \mathrm{La}_{3} \mathrm{RuO}_{7}, \mathrm{Nd}_{3} \mathrm{RuO}_{7} and \mathrm{Pr}_{3} \mathrm{RuO}_{7} impurity.
• Figure 9.12: Magnetic structure models, a) \mathrm{SrLaFeRuO}_{6}, 3 \Gamma_{5} solution; b) \mathrm{SrNdFeRuO}_{6}, 3 \Gamma_{3} solution; and c) \mathrm{SrPrFeRuO}_{6}, 3 \Gamma_{3} \oplus 1 \Gamma_{3} solution: \mathrm{Sr} / \mathrm{La}, yellow balls; \mathrm{Sr} / \mathrm{Nd}, green balls; \mathrm{Sr} / \mathrm{Pr}, blue balls, blue arrows; \mathrm{Fe} / \mathrm{Ru}, purple balls, black arrows; and O, red balls.
• Figure 9.13: Temperature dependance of \mathrm{SrLaFeRuO}_{6} 4 a site \mathrm{Fe} / \mathrm{Ru} magnetic momentum for 3 \Gamma_{5} solution.
• Figure 9.2: Experimental (symbols) and calculated (line) neutron powder-diffraction profiles for the Rietveld refinement of \mathrm{SrLnFeRuO}_{6}(\mathrm{Ln}=\mathrm{La}, \mathrm{Nd}, \mathrm{Pr}) at room temperature, using a disordered orthorhombic (Pbnm) structural model. The upper (middle) markers indicate the position of the structural (magnetic) Bragg reflections and the lower markers the impurity ( \mathrm{Nd}_{3} \mathrm{RuO}_{7} and \mathrm{Pr}_{3} \mathrm{RuO}_{7}) Bragg reflections. \mathrm{La}_{3} \mathrm{RuO}_{7} impurity was excluded in order to guarantee the convergence of the refinement.
• Figure 9.3: Graph of maximal subgroups relating the space group of the archetype (high lighted in blue) and distorted Pbnm and R \overline{3} phase (high lighted in pink) of \mathrm{SrLnFeRuO}_{6}. For each subgroup any irrep yielding this symmetry is indicated.
• Figure 9.4: Schematic representation of the five irrep distortions allowed in the Pbnm configuration, corresponding to \mathrm{M}_{2}^{+}, \mathrm{M}_{3}^{+}, \mathrm{R}_{4}^{+}, \mathrm{R}_{5}^{+}and \mathrm{X}_{5}^{+}. Fe/Ru disordered octahedra in purple, the A / A^{\prime}-site Sr/Ln cations in green and, oxygen atoms, in red. As a reference, we are also plotting the simple perovskite cubic cell (red) and the double perovskite orthorhombic cell (black). The polarization vectors are shown by small black arrows.
• Figure 9.5: Amplitudes of \mathrm{R}_{4}^{+}, \mathrm{M}_{3}^{+}and \mathrm{X}_{5}^{+}irreps conforming the RT phase of the \mathrm{SrLaFeRuO}_{6}, \mathrm{SrPrFeRuO}_{6} and \mathrm{SrNdFeRuO}_{6} compounds, and \Psi and \Phi tilt angles, as obtained from the NPD data refinement ( 3 mode refined), in function of the tolerance factors. The tolerance factors are calculated with Sr/Ln-O distance in nine coordination. \Psi and \Phi represent rotation of octahedra about the pseudo-cubic [110]_{\mathrm{p}} and [001]_{\mathrm{p}} axes, respectively.
• Figure 9.6: Thermal evolution of the (111) primitive peak and (321) cubic reflections of \mathrm{SrPrFeRuO}_{6} obtained from XRPD experiment. The scattered intensity is represented with yellow-red scale, lightyellow being high intensity and dark-red lowest intensity. At 1075 K the intensity of the peaks related to the primitive cell disappear, transforming into a trigonal cell.
• Figure 9.7: Temperature evolution of the cell parameters of \mathrm{SrPrFeRuO}_{6} as obtained from the NPD measurements: cell parameters of the orthorhombic and trigonal phase. The parameters have been scaled to the setting of the double perovskite cubic structure, \mathrm{a}_{c}=2 \mathrm{a} p, in order to compare.
• Figure 9.8: Temperature dependence of the amplitudes of the irreps conforming the P b n m and R \overline{3} c structure of \mathrm{SrPrFeRuO}_{6}, and \Psi and \Phi tilt angles, as obtained from the NPD data refinement. The amplitudes of the irreps in the high symmetry phase have been scaled (multiplied by 3 / 2) to see the evolution, since the program AMPLIMODES uses mode normalization within the corresponding primitive unit cell.
• Figure 9.9: XRPD data at RT (blue line), NPD data at 2 K (red dotted line) and RT (green crossed line) in d space (in Å). All patterns are re-normalized with respect to the peak height of the main (112/200) index peak at 2 \theta \sim 32.6^{\circ}(d \sim 2.7 \AA).
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Tables

• Table 1.1: Example of perovskite type material properties.
• Table 1.2: Selected ionic properties of the rare-earth elements [72].
• Table 10.1: Refined distortion mode amplitudes ( \AA) at room temperature obtained by AMPLIMODES for FullProf [93] and FullProf [85]. The primary mode, if any, is written in bold letters. The column headed as Dim referees to the number of basis modes necessary in general to describe the corresponding irrep distortion; and the direction in the irrep space corresponding to the irrep distortion in the convention of ISOTROPY [99] is indicated in the bottom of the columns.
• Table 10.2: Lattice parameters, Wyckoff sites, atomic fractional coordinates, occupancies, and isotropic temperature factors of the room temperature structures for A \mathrm{Ln}_{2} \mathrm{CuTi}_{2} \mathrm{O}_{9}(A=\mathrm{Ca}, \mathrm{Ba}; \mathrm{Ln}=\mathrm{La}, \mathrm{Pr}, \mathrm{Nd}, \mathrm{Sm}) obtained from XRPD patter refinements.* fixed in 0.5 \AA to guarantee the convergence of the refinement.
• Table 10.3: Bond lengths ( \AA), and selected angles ( { }^{\circ}) for \mathrm{CaLn}_{2} \mathrm{CuTi}_{2} \mathrm{O}_{9}(L n=\mathrm{Pr}, N d, S m) at room temperature calculated from XRPD pattern with Pbnm space group.
• Table 10.4: Bond lengths ( \AA), octahedra tilt angles ( { }^{\circ}), and selected angles ( { }^{\circ}) for \mathrm{BaLn}_{2} \mathrm{CuTi}_{2} \mathrm{O}_{9} ( \mathrm{Ln}=\mathrm{La}, \mathrm{Pr}, \mathrm{Nd}) at room temperature calculated from XRPD pattern with I4/mcm space group.
• Table 10.5: Crystal chemistry for \mathrm{CaLn}_{2} \mathrm{CuTi}_{2} \mathrm{O}_{9} and \mathrm{CaLn}_{2} \mathrm{ZnTi}_{2} \mathrm{O}_{9}[133](\mathrm{Ln}=\mathrm{La}, \mathrm{Pr}, \mathrm{Nd}, \mathrm{Sm}, \mathrm{Eu}) perovskites with orthorhombic space group P b n m at room temperature.
• Table 2.1: Synthesis last treatment temperature (K) and duration (h) for double perovskite materials.
• Table 4.1: Transformation of the P 2_{1} / n monoclinic non-conventional structure to the conventional P 2_{1} / c setting by PSEUDO; the transformation matrix: c,b,-a-c;0, 0, 0.
• Table 4.2: 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 [99], 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 4.3: Reference structures (parent phase) for ordered ( F m \overline{3} m, No.225) and disordered ( P m \overline{3} m, No.221) perovskite structures. The same virtual reference structures have been used for all the compounds.
• Table 5.1: Crystal structure data and refinement results for {\mathrm{SrNd} M R u \mathrm{O}_{6}}(M=\mathrm{Zn}, \mathrm{Co}, \mathrm{Mg}, \mathrm{Ni}) from NPD at RT after the refinement of all modes. \mathrm{Sr} / \mathrm{Nd}(4 \mathrm{e}(x, y, z), o c c=0.5 / 0.5), M\left(2 \mathrm{ b}(0,0,1 / 2), B_{\text {iso }}=0.5 \AA^{2}\right., o c c=1), \operatorname{Ru}\left(2 \mathrm{a}(0,0,0), B_{i s o}=0.5 \AA^{2}, o c c=1\right), \mathrm{O}(4 \mathrm{e}(x, y, z), o c c=1). The B_{i s o} value for M and Ru cations was fixed in 0.5 \AA^{2} in order to guarantee the convergence of the refinement.
• Table 5.2: Octahedra tilt angles ( { }^{\circ}), volume of the octahedra ( \AA^{3}), main bond distances ( \AA) and selected angles ( { }^{\circ}) for \mathrm{SrNdMRuO}_{6}(M=\mathrm{Zn}, \mathrm{Co}, \mathrm{Mg}, \mathrm{Ni}) at room temperature. The values were obtained refining all mode amplitudes.
• Table 5.3: Resume of the irreps taking part in the symmetry breaking from the totally symmetric, cubic, space group (F m \overline{3} m) to the room temperature P 2_{1} / n monoclinic and P 4_{2} / n tetragonal 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.
• Table 5.4: Values after the refinement of the amplitudes (in \AA) of the irreps taking part in the symmetry breaking from F m \overline{3} m space group to the roomtemperature P 2_{1} / n monoclinic space group, refining (a) all modes, (b) 4 and (c) 3 highest amplitudes modes for \mathrm{SrNdMRuO}_{6}(M=\mathrm{Zn}, \mathrm{Co}, \mathrm{Mg}, \mathrm{Ni}) compounds.
• Table 5.5: Crystal structure data and refinement results for \mathrm{SrNdCoRuO}_{6} from NPD at 653 K using P 4_{2} / n model. Cell parameters: a=7.8688(4) \AA; c=7.8728(8) \AA; V=487.47(6) \AA^{3}. Reliability factors (NPD): R_{p}=8.75 \%, R_{\text {wp }}=11.6 \%, R_{\text {exp }}=4.80 \%, R_{\text {Bragg }}=15.8 \%. The B_{\text {iso }} value for Co and Ru cations was fixed in 0.5 \AA^{2} in order to guarantee the convergence of the refinement.
• Table 5.6: Amplitude of the modes (in \AA) of phase P 2_{1} / n and P 4_{2} / n for \mathrm{SrNdCoRuO}_{6}. In column labeled (a) calculated from all refined amplitudes and in (b), with some amplitudes set to zero. RT and HT data set are not measured in the same experimental conditions.
• Table 6.1: Crystal structure data and refinement results for \mathrm{SrPr} \mathrm{MRuO}_{6}(\mathrm{M}=\mathrm{Zn}, \mathrm{Co}, \mathrm{Mg}, \mathrm{Ni}) from NPD at RT, refining all mode amplitudes. \mathrm{Sr} / \operatorname{Pr}(4 \mathrm{e}(x, y, z), o c c=0.5 / 0.5), M(2 \mathrm{ b}(0,0,1 / 2), o c c=1), \mathrm{Ru}(2 \mathrm{a} (0,0,0), o c c=1), \mathrm{O}(4 \mathrm{e}(x, y, z), o c c=1).
• Table 6.2: Octahedra volume ( \AA^{3}), selected bond-lengths ( \AA) and angles ( { }^{\circ}) for \mathrm{SrPr} \mathrm{MRuO}_{6} (\mathrm{M}=\mathrm{Zn}, \mathrm{Co}, \mathrm{Mg}, \mathrm{Ni}) at room temperature, refining all mode amplitudes.
• Table 6.3: Crystal chemistry of \mathrm{SrLnMRuO}_{6}(\mathrm{Ln}=\mathrm{La}, \mathrm{Pr}, \mathrm{Nd}; \mathrm{M}=\mathrm{Zn}, \mathrm{Co}, \mathrm{Cu}, \mathrm{Mg}, \mathrm{Ni}) perovskites [115, 116, 120] at room temperature.
• Table 6.4: Irrep amplitude (in \AA) value after the refinement for \operatorname{SrPr} M R u \mathrm{O}_{6}(M=\mathrm{Zn}, \mathrm{Co}, \mathrm{Mg}, \mathrm{Ni}) compounds at room temperature for cubic ( F m \overline{3} m) to monoclinic ( P 2_{1} / n) phase transition; from which \mathrm{GM}_{4}^{+} and X_{3}^{+}irreps are the ones breaking the symmetry.
• Table 6.5: Details of the \mathrm{SrPrMgRuO}_{6} and \mathrm{SrPrNiRuO}_{6} intermediate-phase structures obtained from NPD data refinement, refining all mode amplitudes.
• Table 6.6: Resume of the irreps taking part in the symmetry-breaking from the totally symmetric, cubic, space group ( F m \overline{3} m) to the intermediate phase R \overline{3} trigonal space group. The list is part of the output of the AMPLIMODES for FullProf. The number in parenthesis indicate the number of modes transforming according to the irrep and the primary mode is written in bold letters.
• Table 6.7: Mode amplitude (in \AA) value after the refinement for \mathrm{SrPr} \mathrm{MRuO}_{6}(\mathrm{Mg}, \mathrm{Ni}) compounds at 1173 K and 1200 K, respectively, for cubic ( F m \overline{3} m) to trigonal ( R \overline{3}) phase transition; refining (a) all modes, (b) 3 and (c) 1 highest amplitude modes.
• Table 7.1: A summary of the symmetries and phase transitions reported for compounds with the general formula \mathrm{Sr}^{2+} L n^{3+} M^{2+} \mathrm{Ru}^{5+} \mathrm{O}_{6}^{2-}. t stands for tolerance factors calculated from (a) the ionic radii in [48] and (b) Rietveld refinement interatomic distances (the strontium and rare-earth radius in nine coordination is chosen since it is the highest coordination number available for all tabulated rare-earths); symmetry of the structure at room temperature; transition temperature; intermediate-phase symmetry and high temperature phase symmetry; literature reference. PW stands for present work.
• Table 7.2: Crystal structure data and refinement results for \mathrm{SrLaZnRuO}_{6} and \mathrm{SrLaMgRuO}_{6} at RT obatined from NPD using a P 2_{1} / n structural model.
• Table 7.3: Octahedra volume ( \AA^{3}), main bond lengths ( \AA), selected angles ( { }^{\circ}) and bond-valence sum [86] for \mathrm{SrLa} \mathrm{MRuO}_{6}(\mathrm{M}=\mathrm{Zn}, \mathrm{Mg}) at room temperature.
• Table 7.4: Refined values of the irrep amplitudes (in \AA) taking part in the symmetry breaking from F m \overline{3} m space group to P 2_{1} / n monoclinic (RT) and intermediate phase R \overline{3} space group ( 923 K for \mathrm{SrLaZnRuO}_{6} and 673 K for \mathrm{SrLaMgRuO}_{6}). F m \overline{3} m \rightarrow P 2_{1} / n: (a) all modes, (b) 4 and (c) 3 highest irrep amplitudes. F m \overline{3} m \rightarrow R \overline{3}: a) all modes, (b) \mathrm{GM}_{1}^{+}fixed to zero and (c) only \mathrm{GM}_{4}^{+}.
• Table 7.5: Crystal structure data and refinement results for \mathrm{SrLaZnRuO}_{6} and \mathrm{SrLaMgRuO}_{6} from NPD using R \overline{3} space group, at 1173 K and 773 K, respectively.
• Table 8.1: Basis vectors of \Gamma_{1}, \Gamma_{2}, \Gamma_{3}, and \Gamma_{4} irreducible representations for the sites Co or \operatorname{Ni}(2 b)=(0,0,1 / 2), \operatorname{Ru}(2 a)=(0,0,0) and \operatorname{Pr}(4 e)=(\simeq 0, \simeq 1 / 2, \simeq 3 / 4).
• Table 8.2: Magnetic representations, magnetic moments at 4 K, for Co and Ru atoms, and w_{\mathrm{RP}} values (multipattern refinement) for four magnetic structure models for \mathrm{SrLaCoRuO}_{6}. Graphical representation in Figure 8.2.
• Table 9.1: Structural details of \mathrm{SrLaFeRuO6}^{6}, \mathrm{SrPrFeRuO}_{6} and \mathrm{SrNdFeRuO}_{6} obtained at room temperature from NPD data using Pbnm structural model, refining all mode amplitudes. The B_{\text {iso }} value for Fe/Ru cations was fixed in 0.5 \AA in order to guarantee the convergence of the refinement.
• Table 9.2: Bond lengths ( \AA), octahedra tilt angles ( { }^{\circ}) selected angles ( { }^{\circ}) and bond-valence sum for \mathrm{Sr} L n \mathrm{FeRuO}_{6}(L n=\mathrm{La}, \mathrm{Pr}, \mathrm{Nd}) at room temperature obtained from NPD data using Pbnm structural model and refining all mode amplitudes. \mathrm{Sr} / \mathrm{Ln}-O distance in nine coordination is chosen since it is the highest coordination number available for all tabulated rare-earths.
• Table 9.3: Summary of the irreps involved in the symmetry breaking from the totally symmetric, cubic, space group ( P m \overline{3} m) to the room temperature P b n m orthorhombic space group and R \overline{3} c trigonal space group.The list is part of the output of the AMPLIMODES for FullProf. The number in parenthesis indicate the number of modes transforming according to the irrep.
• Table 9.4: NPD data refinement results for the amplitudes (in \AA) of irreps taking part in the symmetry breaking from P m \overline{3} m space group to the room temperature Pbnm orthorhombic space group, refining (a) all the modes and (b) 3 and (c) 2 highest amplitudes modes for SrLnFeRuO6 ( \mathrm{Ln}=\mathrm{La}, \mathrm{Pr}, \mathrm{Nd}) compounds.
• Table 9.5: Crystal structure data and refinement results for \mathrm{SrPrFeRuO}_{6} from NPD (HRPT, PSI) at 1200 K using R \overline{3} c model.
• Table 9.6: Input information for AMPLIMODES for FullProf: high symmetry phase information (virtual prototype structure), low symmetry structure information (cell parameters obtained by XRPD data refinement) and transformation matrix. \operatorname{Pm} \overline{3} m (ITA No. 221, standard setting), Pbnm (ITA No. 62, nonstandard setting) and R \overline{3} c (ITA No. 167, standard setting).
• Table 9.7: Basis vectors of 4 possible irreducible representations for the sites Fe and \operatorname{Ru}(4 a)=\left(0, \frac{1}{2}, 0\right) and \operatorname{Pr}(4 c)=\left(0.001(1),-0.0255(4), \frac{1}{4}\right).
• Table 9.8: Magnetic moment components (in \mu_{B}) determined for each Pr and \mathrm{Fe} / \mathrm{Ru} site and Rietveld reliability factors for \mathrm{SrLnFeRuO}_{6}(\mathrm{Ln}=\mathrm{La}, \mathrm{Nd}, \mathrm{Pr}).
• Table 9.9: Structural details of \mathrm{SrLaFeRuO}_{6} obtained at different temperatures from NPD (D1B, ILL) data using Pbnm structural model with \mathrm{R}_{4}^{+}, \mathrm{M}_{3}^{+}and \mathrm{X}_{5}^{+}modes refined.