The α-Na0.4R0.6F2.2 crystals (R = Ho–Lu, Y) have been studied by X-ray diffraction analysis at 293 and 85 K. A unified cluster model of nanostructured crystals with a fluorite-type structure, based on the polymorphism of KR3F10 (R = Er, Yb), was used to model their defect structure. The α-Na0.4R0.6F2.2 matrix component contains Na+ and R3+ ions in a ratio of 1 : 1. Part of the matrix anions are shifted from the 8c to 32f site (sp. gr. Fm3̅m ). Excess R3+ cations form, jointly with Na+, octa-cubic clusters with cores in the form of cuboctahedra F12, consisting of interstitial anions at the 48i site. The α-Na0.4R0.6F2.2 cluster component is formed by octa-cubic clusters of type i. The electron diffraction study showed that the clusters are shaped as plates about 5 nm thick with superstructural ordering. Their structural model based on the K0.265Gd0.735F2.47 structure was proposed. Experimental confirmation of the affiliation of α-Na_0.5-xR_0.5 + xF_2 + 2x to nanostructured crystals was obtained for the first time by electron diffraction. When temperature decreases from 293 to 85 K, the type of the cluster component of the defect α-Na0.4R0.6F2.2 structure with R = Ho–Lu, Y does not change. At 293 K, the boundary of the change in the defect structure type in the α-Na_0.5-xR_0.5 + xF_2 + 2x series is located between R = Dy (Z = 66) and Ho (Z = 67). With a decrease in temperature from 293 to 85 K the position of the boundary does not change.

For the first time, the crystal α-Na0.35Dy0.65F2.30 was studied using X-ray diffraction at 293 K and 85 K and electron diffraction at 293 K. A unified cluster model of the defective structure of nanostructured crystals with a fluorite-type structure, based on the polymorphism of ordered phases KR3F10 (R = Er, Yb), was expanded with a matrix part model based on the structure of the KYF4 compound. The unified cluster model was applied to construct the defective structure of α-Na0.35Dy0.65F2.30. It was found that the matrix part of the crystal contains Na+ и Dy3+ cations in a 1:1 ratio. Some of the anions in the matrix are displaced to the 32f positions (space group Fm3m). The excess Dy3+ forms octahedral-cubic clusters with Na+ [Na14–nDynF64+n] with cores in the form of distorted and regular cuboctahedrons {F12}. These are composed of interstitial anions in two 32f positions and one 48i position. The cluster component of the α-Na0.35Dy0.65F2.30 crystal contains octahedral-cubic clusters of f-, f–i- and i-types. Electron diffraction showed that α-Na0.35Dy0.65F2.30 is a nanostructured crystal. Its cluster component is in the form of plate-like inclusions about 5 nm thick with superstructural ordering and individual octahedral-cubic clusters. A model of their structure was proposed. Lowering the temperature to 85 K increases the number of interstitial F(32f)1 anions in the matrix component of the crystal.

The α-Na0.35Dy0.65F2.30 crystal has been investigated for the first time by X-ray diffraction at 293 and 85 K and by electron diffraction at 293 K. The unified cluster model of the defect structure of fluorite-type nanostructured crystals, based on the polymorphism of ordered KR3F10 phases (R = Er, Yb), is extended by the matrix part model based on the KYF4 structure. The unified cluster model is applied to construct the defect structure of α-Na0.35Dy0.65F2.30. The matrix part of the crystal is found to contain Na+ and Dy3+ cations in a ratio of 1 : 1. Some of matrix anions are displaced to 32f sites (sp. gr. Fm 3̅ m). Excess Dy3+ cations and Na+ cations form octa-cubic clusters [ Na_14-nDy_nF_64 + n ] with cores in the form of distorted and regular F12 cuboctahedra. They are composed by interstitial anions at two 32f sites and one 48i site. The cluster component of the α-Na0.35Dy0.65F2.30 crystal contains octa-cubic clusters of the f, f–i, and i types. The electron diffraction study revealed that α-Na0.35Dy0.65F2.30 is a nanostructured crystal. Its cluster component includes platelike precipitates 5 nm thick with superstructural ordering and unit octa-cubic clusters. A model is proposed for their structure. A decrease in the temperature to 85 K increases the number of interstitial F(32f)1 anions in the matrix component of the crystal.

Nonstoichiometric nanostructured crystals (NSC) alpha-M0.5-xR0.5+xF2+2x (M = Na and K; R = Sc, Y, and La-Lu) and M1-xRxF2+x (M = Ca, Sr, Ba, Cd, and Pb) with a fluorite-type (F) structure are widely used in quantum electronics, optics, solid-state ionics (superionic conductors), and bioimaging. However, the temperature dependence of their defect crystal structure, which is fundamental for understanding the numerous properties of F-NSC, remains unexplored. In this study, temperature changes in the F-NSC defect structure in T-x (temperature-composition) systems were simulated. Data on the KR3F10 (R = Gd-Lu and Y) polymorphism were used for modeling. A fragment representing an octa-cubic cluster (OCC) [M14-nRnF64+n] with a hybrid-type (i-f-) anionic core was isolated in the structure of low-temperature beta-KR3F10 (R = Er and Yb). It includes all types of interstitial anions involved in the formation of i- and f-type clusters in the unified cluster model (UCM) of the F-NSC defect structure. There is a common tendency for hybrid (i + f) and f-type OCCs to form at low T and i-type OCCs at high T. The UCM was applied to verify T changes in the Ca0.94Y0.06F2.06 defect structure within the 293 divided by 1423 K T range.

Negative thermal expansion (NTE) in trifluorides of rare earth elements (REEs – R) is considered. The electronic structure of R divides RF3 into two homological series: 1) short ScF3, YF3, LaF3 of d-elements and 2) long LaF3 and 14 fluorides of 4f-elements (lanthanides - Ln) from CeF3 to LuF3. The type of NTE is different for different series. The Periodic Table of RF3 (RF3 are in the locations of elements R in the Periodic Table of Elements) sets the periodicity of NTE types for the first time. In the short series only ScF3 (ReO3 type) has NTE-I in a wide temperature range. Eight LnF3 (out of 15 RF3) have polymorphic transition (PolTr). The only PolTr β- (β-YF3 type) → t- (LaF3 type) of three possible (β- → t-; β- →α-YF3; t- → α-YF3) gives denser high-temperature t-modification. It is evidence of a giant NTE-II. The unit cell parameters of the model crystal 61(58Ce0.564Gd0.5)F3 («pseudo 61PmF3») were measured in situ by X-ray diffraction to calculate Vform of PmF3. The unit cell parameters of β- ​«pseudo PmF3» were obtained. PmF3 belongs to the structural subgroup (SSG) B of LnF3. PolTrs in these four LnF3 are accompanied by a giant NTE-II up to ∼4.7 %. These LnF3 form a new group of single component materials with non-adjustable NTE-II parameters. Multicomponent materials with adjustable NTE-II parameters arise in LnF3-Ln’F3 systems.

Crystal structures of high-temperature t - (LaF 3 -type) and low-temperature β - (β-YF 3 -type) modifications of 61 ( 58 Ce 0.5 64 Gd 0.5 )F 3 – “ pseudo 61 PmF 3 ” near the temperature of a polymorphic transformation .

The formation of materials with negative thermal expansion (NTE) based on a phase transition-type mechanism (NTE-II) in 50 T-x (temperature-composition) RF3-R'F3 (R = La-Lu) systems out of 105 possible is predicted. The components of these systems are "mother" RF3 compounds (R = Pm, Sm, Eu, and Gd) with polymorphic transformations (PolTrs), which occur during heating between the main structural types of RF3: β-(β-YF3) → t-(mineral tysonite LaF3). The PolTr is characterized by a density anomaly: the formula volume (Vform) of the low-temperature modification (Vβ-) is higher than that of the high-temperature modification (Vt-) by a giant value (up to 4.7%). In RF3-R'F3 systems, isomorphic substitutions chemically modify RF3 by forming R1-xR'xF3solid solutions (ss) based on both modifications. A two-phase composite (β-ss + t-ss) is a two-component NTE-II material with adjustable parameters. The prospects of using the material are estimated using the parameter of the average volume change (ΔV/Vav). The Vav at a fixed gross composition of a system is determined by the β-ss and t-ss decay (synthesis) curves and the temperature T. The regulation of ΔV/Vav is achieved by changing T within a "window ΔT". The available ΔT values are determined using phase diagrams. A chemical classification (ChCl) translates the search for NTE-II materials from 15 RF3 into an array of 105 RF3-R'F3 systems. Phase diagrams are divided into 10 types of systems (TypeSs), in four of which NTE-II materials are formed. The tables of the systems that comprise these TypeSs are presented. The position of Ttrans of the PolTr on the T scale for a short quasi-system (QS) "from PmF3 to TbF3" determines the interval of the ΔTtrans offset achievable in the RF3-R'F3 systems: from -148 to 1186 ± 10 °C. NTE-II fluoride materials exceed known NTE-II materials by almost three times in this parameter. Equilibrium in RF3-R'F3 systems is established quickly. The number of qualitatively different two-component fluoride materials with the giant NTE-II can be increased by more than ten times compared to RF3 with NTE-II.

Multicomponent fluorides of rare earth elements (REEs—R) are phase transition-type negative thermal expansion (NTE-II) materials. NTE-II occurs in RF3-R′F3 systems formed by “mother” single-component dimorphic RF3 (R = Pm, Sm, Eu, and Gd) with a giant NTE-II. There are two structural types of RF3 polymorphic modifications: low-temperature β-YF3 (β−) and high-temperature LaF3 (t−). The change in a structural type is accompanied by a density anomaly: a volume of one formula unit (Vform) Vβ− >Vt−. The empirical signs of volumetric changes ΔV/V of NTE-II materials were considered. For the GdF3-TbF3 model system, an “operating-temperature window ΔT” and a two-phase composition of NTE-II materials follows from the thermodynamics of chemical systems: the phase rule and the principle of continuity. A necessary and sufficient sign of NTE-II is a combination of polymorphism and the density anomaly. Isomorphism in RF3-R′F3 systems modifies RF3 chemically by forming two-component t− and β− type R1−xR’xF3 solid solutions (ss). Between the two monovariant curves of ss decay, a two-phase area with ΔTtrans > 0 (the “window ΔT”) forms. A two-phase composite (t−ss + β−ss) is an NTE-II material. Its constituent t−ss and β−ss phases have different Vform corresponding to the selected T. According to the lever rule on a conode, Vform is calculated from the t−ss and β−ss compositions, which vary with T along two monovariant curves of ss decay. For the GdF3-TbF3 system, ΔV/V = f(T), ΔV/V = f(ΔT) and the “window ΔT” = f(x) dependencies were calculated.

A lanthanide contraction(LC) of 14 lanthanides (Ln) from 58Ce to 71Lu consists of the interaction of Ln nucleus with 4f-electrons. Rare earth elements (REEs—R) include Sc, Y, La, and 14 Ln. They are located in 4–6th periods of the subgroup of group III. The electronic structure divides R into short (d- Sc, Y, La) and long (14 f-elements Ce-Lu) homologous series. The most important chemical consequence of LC is the creation of a new conglomerate of 16 RF3 by mixing fluorides of d- (Y, La) and f-elements. This determines the location of YF3 among LnF3. The location of YF3 depends on the structural (formula volumes—Vform) and thermochemical (temperatures and heats of phase transformations, phase diagrams) properties. The location of YF3 between HoF3 and ErF3 was determined by Vform at a standard pressure (Pst) and temperature (Tst). The location of YF3 according to heats of phase transformations ΔHfus and ΔHtrans is in a dimorphic structural subgroup (SSGr) D (Ln = Er-Lu), but without the exact “pseudo ZY”. According to the temperatures of phase transformations (Ttrans) in LnF3 (Ln = Dy-Lu), YF3 is located in the SSGr D between ErF3 and TmF3. The ErF3-YF3 and YF3-TmF3 phase diagrams show it to be between ErF3 and TmF3. The crystals of five β-LnF3 (Ln = Ho-Lu) and β-YF3 were obtained in identical conditions and their crystal structures were studied. Vform (at Pst and Tst) with “pseudo” atomic number ZY = 67.42 was calculated from the unit cell parameters, which were defined with ±5 × 10−4 Å accuracy. It determines the location of YF3 between HoF3 and ErF3.

A specialized empirical (Spec-zd Emp) system of ionic radii (SIR) for R = Y3+, La3+, Ln3+, and F1− (R rare earth elements (REE)) was derived from the dependence of lanthanide contraction (LC) on the atomic number (Z) of lanthanides (Ln). LC decreased the radius of the cation with increasing Z. The structures of t-RF3 (LaF3-NdF3, “pseudo t-SmF3”) of the LaF3 type, 11 β-LnF3 (Ln = Sm-Lu), and β-YF3 of the β-YF3 type were studied. The empirical basis of the shortest (F-F)min and (R-F)min distances was calculated from the structural data for the RF3 complete series. The dependence of (F-F)min on Z reached saturation at Z = 67 (Ho). The base F1− radius r− = 1.2539(16) Å was calculated as the arithmetic mean of five (F-F)min in LnF3 with Ln = Ho-Lu. For the LnF3 series with Ln contributions up to 75 % wt., the dependence of (Ln-F)min on Z reflected the non-uniformity of the 4f orbital filling. SIR was calculated as the difference in the empirical constants of RF3 (ionic radii of (R,Ln)3+ (r+) and F1− (r−)), the change in which was continuous over the series and did not depend on the type of structure: r+ = (ZR-F)min − ½(F-F)min (Z = 57–71). The changes in LC in the LnF3 series were described by a third-degree polynomial. LC reduced r+ by 24% (percentage relative to less) from 1.1671(16) Å (La3+) to 0.9439(17) Å (Lu3+). In the Spec-zd Emp SIR, r+ were constants that did not require corrections for a coordination number (CN). A comparison of r+ in the Spec-zd Emp SIR with other SIRs was performed.

A universal cluster model applicable to the most common nonstoichiometry in inorganic fluorides with the CaF 2 structure.

The combination of 15 RF3 (R - rare earth elements - REEs without ScF3 and YF3) into one system "from LaF3 to LuF3" we call a full quasi-system (QS). It consists of 14 serially linked particular systems (PartSs): LaF3-CeF3; CeF3PrF3; PrF3-NdF3; NdF3-PmF3; PmF3-SmF3; SmF3-EuF3; EuF3-GdF3; GdF3-TbF3; TbF3-DyF3; DyF3-HoF3; HoF3-ErF3; ErF3-TmF3; TmF3-YbF3; YbF3-LuF3. The atomic numbers (Z) of cations from 57 (La) to 71 (Lu) and the temperature (T) are axies of QS. The conditions of including RF3 in PartSs is the neighborhood of R (Ln - lanthanides) positions in Periodic Table of Chemical Elements and |AZ| = 1. This securities of maximum chemical proximity (ChProx) of RF3's components. The QS obeys the phase rule, principles of continuity and conformity. Full QS contains all signs of chemical RF3's interactions identified in 34 studied LnF3-Ln'F3 systems. These signs are dependent of AZ: perfect and limited isomorphism and two varieties (peritectic and eutectic) of structural true morphotropic transformations (MTs) of phases. Modifications of LnF3 forms the solid solutions (ss) Ln1-xLn'xF3 with average Zav=(1-x)NLn + x(N+1)Ln' (N from 57 to 71). These ss reveal the lanthanide contraction (LC) in the regions |AZav|<1. Full QS is an individual sub-level of chemical classification of LnF3-Ln'F3 systems. It is a tool for studies the chemical interactions LnF3, analyzing fine consequences of LC, creating a special system of Ln3+ ionic radii for LnF3, studies of density anomalies (negative thermal expansion), prediction of phase diagrams unexplored systems RF3-R'F3 and other problems of rare earth trifluorides chemical family.

Crystals of fluorite phases with the CaF2 structure: Ba1-xLaxF2+x (x = 0, 0.05, 0.105, 0.25, 0.35, 0.40) and those of tysonite ones with the LaF3 structure: La1-yBayF3-y (y = 0, 0.05, 0.09) have been studied by electron diffraction and elemental analysis in a transmission electron microscope. Diffraction patterns of Ba1-xLaxF2+x crystals with x = 0.105, 0.25 exhibit diffuse scattering, which indicates the presence of clusters of structural defects. Blurred supercell reflections are seen in diffraction patterns of Ba1-xLaxF2+x crystals with a high lanthanum content (x = 0.35, 0.40) in addition to diffuse scattering. They indicate the existence of nanoscale ordered crystallites as well as clusters. It was found that LaF3 and the tysonite phase La1-yBayF3-y with a low barium content (y = 0.05) is of a low-temperature l-form (space group P (3) over bar c1, Z = 6). A small fraction of tysonite crystallites in La1-yBayF3-y crystals with y = 0.09 is of a high-temperature h-form (space group P6(3)/mmc, Z = 2). An intermediate crystal structure between l- and h-forms was also found. The maximum ionic conductivity sdc of tysonite crystals La1-yBayF3-y at 293 K is 800 times higher than that of fluorite ones Ba1-xLaxF2+x. A correlation was found between logarithm of conductivity and composition: lg sigma(dc)(x) = ax + b for fluorite Ba1-xLaxF2+x and tysonite La1-yBayF3-y crystals. The reason for the additive law of change in ionic conductivity in both types of nonstoichiometric phases in the BaF2-LaF3 system is apparently the similarity of defect structures of fluorite and tysonite phases.

The structural and chemical modelling of PmF3.

The defect structure of Ba0.69La0.31F2.31 single crystals in as-grown state and after annealing at 1173 K for 336 h was studied by X-ray diffraction analysis. Both crystals belong to the CaF2 structure type (sp. gr. Fm3¯m). They have vacancies in the main anion motif and interstitial fluorine anions in Wyckoff positions 48i and 4b. Relaxation (static displacement of some main anions to Wyckoff position 32f) is observed in the annealed crystal. It was established that annealing leads to a change in the type of displacement of the main anions in Wyckoff positions 8c from dynamic to static. Displacement of La3+ cations to Wyckoff position 32f is observed in both crystals. A model of the defect structure of Ba0.69La0.31F2.31 is proposed, according to which interstitial fluorine anions and La3+ cations are aggregated into [Ba14−nLanF64+n] clusters with the cuboctahedral anionic core formed by interstitial fluorine anions in Wyckoff positions 48i. Ba2+ cations are located in the cluster in the centers of the faces, and the La3+ cations are shifted by 0.24 Å from the vertices of the cluster along the three-fold axis towards the center of the cluster. The study establishes the relationship between the defect structure of crystals and their structurally sensitive properties, and to develop approaches to their management.

A chemical classification of rare earth (R) trifluorides and some studied RF3-R'F-3 systems with their participation by the difference Delta Z = vertical bar Z(R)-Z(R')vertical bar of atomic numbers R and R' has been developed. The classification has three levels. The first level divides 16 RF3 (without ScF3) in structure and phase transformations into four structural subgroups (SSs): A (LaF3-NdF3); B (PmF3-GdF3); C (TbF3 -HoF3), and D (ErF3-LuF3, YF3) with PmF3 belonging to SS B. The second level forms from 10 types of RF3-R'F-3 systems (TSs), in which chemical interactions of the components increase with increasing Delta Z(max). Jumps of Delta Z(max) between some TSs distinguish the third level - four groups of systems (GSs). The jumps in Delta Z(max) are associated with changes in the topological features of the phase diagrams. GS-4 systems are not well understood.

The anionic nonstoichiometry in inorganic fluorides is at result of substitution of F1– for O2– in the anionic sublattice. All families of fluorides exhibit the initial stage of anionic nonstoichiometry (ISAN), which was previously studied for trifluorides of rare-earth elements (REEs), RF3. Partial substitution of F1– for O2– in RF3 occurs in reactions with H2O vapor upon heating (pyrohydrolysis), exchange reactions of RF3 and R2O3 in melts, hydrothermal solutions, solid phase, and during mechanochemical synthesis. The ISAN is based on the formation of RF3 – 2xOx oxofluorides, the type and structure of which depend on the crystalline RF3 forms. Congruently melting tys-RF3 – 2xOx compounds are formed based on the tysonite forms tys-RF3 (R = La–Gd, the LaF3 type). Berthollide phases ~tys-RF3 – 2xOx, which are isostructural to the above-mentioned phases and melts incongruently above the corresponding RF3 compounds, are formed with R = Tb–Ho. The effect of stabilization of tys-RF3 – 2xOx when moving up the temperature scale (+ΔTfus) changes nonmonotonically along the REE series, exhibiting a maximum of ~100°C in the range of Gd–Tb. There are no F1– → O2– substitutions in the β-RF3 forms (R = Er–Lu, Y) of the β-YF3 type. The α-RF3 – 2xOx phases of the α-YF3 (α-UO3) type melt incongruently and decompose at high temperatures. The ISAN products in RF3 may serve as sources of new congruently melting fluorine–oxygen materials.

X-ray diffraction data on the Sr1 – xLaxF2 + x (x = 0.11, 0.24, 0.32, 0.33) single crystals with different thermal histories (quenched and as grown) and post-growth treatments (annealing at 1300, 900, and 750°С for 36, 96, and 192 h, respectively) are reported. All the crystals belong to the CaF2 structural type (sp. gr. $$Fm\bar {3}m$$ ) and have vacancies in the main anionic motif and interstitial fluorine anions in two 32f sites. The Sr0.89La0.11F2.11 crystal annealed at 900°С contains interstitial anions in the 4b site. It was established that annealing at 900°C of the as grown crystal with x = 0.11 leads to an increase in the number of vacancies in the main anionic motif and to a decrease in the number of relaxed anions F(32f)1. Annealing at 750°C of the as grown crystal with x = 0.33, whose composition is close to congruent, leads to a decrease of ~ 2.5 times in the number of relaxed anions F(32f)1, the predominance of the tetrahedral configuration of the F(32f)3 anions does not change during heat treatment.

The concentration series of nonstoichiometric crystals Ca1–xYxF2+x (x = 0.01–0.14) was obtained from a melt by directional crystallization to refine the composition of the temperature maximum on the melting curves. A precision (±9 × 10−5 Å) determination of lattice parameters of the Ca1–xYxF2+x crystals with the structure of fluorite (sp. gr. Fm-3m) was performed, and a linear equation of their concentration dependence was calculated: a(x) = 5.46385(5) + 0.1999(4) x. The distribution of yttrium along the crystals Ca1–xYxF2+x, the content of which is determined by the precision lattice parameters, is studied. The congruently melting composition x = 0.105(5) of the Ca1–xYxF2+x phase is refined by the method of directional crystallization.