Ordered orthorhombic phases of titanium monoxide

Symmetry analysis is carried out for the ordered phases of cubic monoxide TiO_y with relative oxygen contents y<1 and y>1. It is established that a partially ordered orthorhombic phase (space group Immm)—a derivative of the orthorhombic M₃X₂□ superstructure (at y<1.0) or the inverse superstructure M₂■X₃ (at y>1)—may arise in TiO_y. The distribution of Ti and O atoms, oxygen vacancies □, and titanium vacancies ■ in unit cells of the orthorhombic ordered phases is determined. The phases are formed through the order-disorder transition channel along two rays of a non-Lifshitz star {k ₄}, and the ordering proceeds as a first-order phase transition. The distribution functions of Ti atoms over the sites of metallic and O atoms over the sites of nonmetallic sublattices are calculated for the orthorhombic superstructures of cubic titanium monoxide TiO_y.     

Sequence of disorder-order and order-order transitions accompanying the formation of M<Subscript>2</Subscript>X superstructures

The symmetry analysis of the trigonal, cubic, and tetragonal M₂X superstructures formed in strongly nonstoichiometric compounds MX _y with the B1 structure has been performed. Disorder-order MX _y → M₂X transition channels have been revealed. It has been shown that three physically allowable sequences of transitions associated with the formation of ordered M₂X phases and order-order transitions between them are possible in nonstoichiometric carbides and nitrides MX _y of the transition metals of group IV with a decrease in the temperature.     

Ordering of MX y nonstoichiometric compounds with the formation of M 2 X superstructures

A symmetry analysis of trigonal, cubic, and tetragonal superstructures of the M ₂ X type that are formed in strongly nonstoichiometric compounds MX _y with the B1 structure has been performed. The channels of disorder-order transitions MX _y → M ₂ X have been determined. It has been demonstrated that, in nonstoichiometric carbides and nitrides MX _y of Group IV transition metals, a decrease in the temperature can lead to three physically allowable sequences of transformations associated with the formation of M ₂ X ordered phases.     

Determination of the probability of existence of pair interactions in the formation of M2tX2t?1Superstructures in MXy nonstoichiometric compounds

An analytical method has been proposed for calculating the probabilities P _i ^(2)(s) of existence of X-X, X−□, and □−□ pair interactions in the nonmetal sublattice of M _2tX_2t−1 superstructures formed in strongly nonstoichiometric compounds MX _y (MX _y □_1−y) and M ₂X _y (MX_y/2□_1−y/2) with a high content of structural vacancies □. The main characteristics necessary for the quantitative determination of the probabilities P_i(2)(s) as functions of the composition, degree of long-range order, symmetry, and structure type have been determined for all the known superstructures M _2tX_2t−1.     

Distribution of Vacancies in a Hybrid M<Subscript>(5–11/18)</Subscript>X<Subscript>(5–11/18)</Subscript> Superstructure of a High-Temperature Ordered β-TiO Phase

A new structural model of the high-temperature ordered β-TiO phase of titanium monoxide with vacancies in metallic and nonmetallic sublattices has been proposed. The model suggests that the vacancies are simultaneously distributed over sites of two partially disordered superstructures: monoclinic M₅X_5(mon) (space group C2/m (A2/m)) and cubic M₅X_5(cub) (space group Pm\(\overline 3 \)m). The hybrid superstructure belongs to space group P1m1 of the monoclinic system. The proposed notation M_(5–11/18)X_(5–11/18) of the hybrid superstructure takes into account an expansion of the vacancy sublattice due to the superposition of initial \({M_5}{X_{{5^ - }}}\) type superstructures. It is shown that the arrangements of the superstructure reflections in the diffraction spectra of the M_(5–11/18)X_(5–11/18) and M₅X_5(cub) superstructures are identical.     

Superposition of M<Subscript>5</Subscript>X<Subscript>5</Subscript> superstructures and X-ray diffraction in TiO<Subscript>1.0</Subscript> titanium monoxide

The interpretation of diffraction spectra of ordered high-temperature phases of solid solutions and strongly nonstoichiometric compounds is discussed. It has been shown that variations of the intensities of superstructure reflections, which cannot be explained within simple ordering models, can be due to the superposition of superstructures with different symmetries in the matrix of the basis crystal structure. Using an example of atom–vacancy ordering in TiO_1.0 titanium monoxide, a model of the order–order transition state formed by the superposition of low-temperature monoclinic (space group A2/m (C2/m)) and high-temperature cubic (space group Pm3?m) M₅X₅ superstructures has been proposed. It has been shown that the transition state is thermodynamically equilibrium and should be implemented instead of the M₅X₅ cubic superstructure. The transition state model can be considered as an M_(5–i)X_(5–i) superstructure (i = 1, 14/18, 11/18) with the monoclinic symmetry (space group P1m1).     

Diffraction spectra of order–order transition structural states in titanium monoxide

A new class of M_(n?i)X_(n?i) superstructures (n is an integer constant and i is a rational variable) that are formed after incomplete equilibrium second-order order–order phase transitions in solid solutions and in compounds with atom–vacancy ordering is proposed. New superstructures are superpositions of partially disordered M _n ?X _n superstructures of various symmetries in the matrix of the basic crystal structure. The model parameters affecting the positions and intensities of superstructure reflections on diffraction patterns have been studied by example of the high-temperature ordered phase β-TiO of titanium monoxide. It has been shown that the diffraction spectra of the low-symmetric M_(5-i)X_(5-i) superstructure (space group P1m1) and the high-symmetric M₅X₅ superstructure (space group Pm3?m) at certain parameters are identical in the positions of superstructure reflections.     

IR spectroscopy of rare-earth aluminum borates RAl3(BO3)4 (R = Y,Pr-Yb)

Rare-earth aluminum borates RAl₃(BO₃)₄ (R = Y, Nd-Yb) obtained by spontaneous high-temperature flux crystallization form two polytype modifications described by the space groups R32 (D ₃ ⁷ ) and C2/c (C _2h ⁶ ). They differ in the symmetry versions in mutual arrangement of layers. These borates have been investigated by mid- and far-IR spectroscopy in combination with factor-group analysis of the vibrations of BO ₃ ^3? ions; translational motions of Al³⁺, R³⁺ (Y, Nd-Yb), and BO ₃ ^3? ions; and BO ₃ ^3? rotations. Rare-earth aluminum borates are assigned to different space groups according to their IR spectra. Borates with large rare-earth Nd and Pr cations are crystallized into the space group C2/c (C _2h ⁶ ), while borates with smaller (Y, Sm-Yb) ions are crystallized into the space group R32 (D ₃ ⁷ ). The rhombohedral structure of the latter compounds includes monoclinically ordered domains, as is evidenced by the presence of monoclinic-phase bands in the IR spectrum. NdAl₃(BO₃)₄ and SmAl₃(BO₃)₄ can form both monoclinic and rhombohedral polytypes. The IR spectrum of the monoclinic SmAl₃(BO₃)₄ phase contains bands due to the rhombohedral phase, while the IR spectrum of NdAl₃(BO₃)₄ (which is described by the space group R32) contains bands due to the monoclinic polytype. The presence of domains with another arrangement of layers in the polytype structure is a characteristic sign that corresponds to the order-disorder theory; this theory explains the structure of polytypes.     

Diffraction of electrons in the CubicTi5O5 superstructure of titanium monoxide

Annealed titanium monoxide TiO_1.087 has been studied by the electron diffraction method. A cubic model of the Ti₅O₅ superstructure (Ti₅O₅ (Ti₉₀?₁₈O₉₀??₁₈)) of nonstoichiometric titanium monoxide Ti _x O _z has been proposed on the basis of experimental data and representations about the disorder-order transition channel. It has been shown that reflections observed on the electron diffraction pattern are identified in the space group $Pm\bar 3m$ . The period of the unit cell of the cubic Ti₅O₅ superstructure is larger than that for the B1 basic disordered structure of Ti _x O _z monoxide by a factor of 3. The disorder-order transition channel Ti _x O _z (space group $Fm\bar 3m$ )-Ti₅O₅ (space group $Pm\bar 3m$ ) includes 75 superstructure vectors of seven stars {k ₁₀}, {k ₇}, {k ₆₍₁₎}, {k ₆₍₂₎}, {k ₄₍₁₎}, {k ₄₍₂₎}, and {k ₁}. The distribution functions of Ti and O atoms over the sites of the cubic Ti₅O₅ (space group $Pm\bar 3m$ ) superstructure have been calculated.     

Calorimetric and optical studies of orthorhombic and cubic CsLiCrO4 crystals

Careful calorimetric and polarization optical studies of a metastable cubic modification of a CsLiCrO₄ crystal (space group ( $F\overline 4 3m$ )) are performed. Anomalies of the heat capacity indicative of structural distortions at 92.4 and 69.1 K were revealed, and the values of the thermodynamic parameters of the phase transitions were determined. The crystal twinning in the intermediate phase was studied. A comparative analysis of the phase transitions occurring in a series of ALiMO₄ crystals (A = NH₄, Cs; M = S, Cr, Mo, W) was performed. First studies of the monotropic structural transformation $F\overline 4 3m$ → Pmcn and the enantiotropic phase transition Pmcn ? P112₁/n were carried out using scanning calorimetry, and the thermodynamic parameters of these transitions were determined.     

Ultraviolet to infrared refractive indices of tetragonal double tungstate and double molybdate laser crystals

Sellmeier parameters of thirteen tetragonal (space group $I\overline{4}$ ) double tungstate and double molybdate laser crystals with M⁺T³⁺(X⁶⁺O₄)₂ composition have been calculated using the room temperature refractive indices determined from the ultraviolet band gap of the crystals to ????2 ??m. All considered crystals are uniaxial but only crystals with Bi in their composition show a significant birefringence (??n>10^?2). The refractive index value increases for the sequence T³⁺=La, Y, Gd, Lu, and Bi independently of the M⁺ and X⁶⁺ cation pair. Implications for the design of laser waveguides and laser pulse dispersion are discussed.     

Eine scheinbare Abschwächung der Lokalitätsbedingung. II

If for a relativistic field theory the expectation values of the commutator (Ω|[A (x),A(y)]|Ω) vanish in space-like direction like exp {? const|(x-y ²|^α/2#x007D; with α>1 for sufficiently many vectors Ω, it follows thatA(x) is a local field. Or more precisely: For a hermitean, scalar, tempered fieldA(x) the locality axiom can be replaced by the following conditions 1. For any natural numbern there exist a) a configurationX(n): $$X_1 ,...,X_{n - 1} X_1^i = \cdot \cdot \cdot = X_{n - 1}^i = 0i = 0,3$$ with \(\left[ {\sum\limits_{i = 1}^{n - 2} {\lambda _i } (X_i^1 - X_{i + 1}^1 )} \right]^2 + \left[ {\sum\limits_{i = 1}^{n - 2} {\lambda _i } (X_i^2 - X_{i + 1}^2 )} \right]^2 > 0\) for all λ _i ≧0i=1,...,n?2, \(\sum\limits_{i = 1}^{n - 2} {\lambda _i > 0} \) , b) neighbourhoods of theX _i 's:U _i (X _i )?R ⁴ i=1,...,n?1 (in the euclidean topology ofR ⁴) and c) a real number α>1 such that for all points (x):x ₁, ...,x _n?1:x _i ∈U _i (X _r ) there are positive constantsC ⁽ⁿ⁾{(x)},h ⁽ⁿ⁾{(x)} with: $$\left| {\left\langle {\left[ {A(x_1 )...A(x_{n - 1} ),A(x_n )} \right]} \right\rangle } \right|< C^{(n)} \left\{ {(x)} \right\}\exp \left\{ { - h^{(n)} \left\{ {(x)} \right\}r^\alpha } \right\}forx_n = \left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ r \\ \end{array} } \right),r > 1.$$ 2. For any natural numbern there exist a) a configurationY(n): $$Y_2 ,Y_3 ,...,Y_n Y_3^i = \cdot \cdot \cdot = Y_n^i = 0i = 0,3$$ with \(\left[ {\sum\limits_{i = 3}^{n - 1} {\mu _i (Y_i^1 - Y_{i{\text{ + 1}}}^{\text{1}} } )} \right]^2 + \left[ {\sum\limits_{i = 3}^{n - 1} {\mu _i (Y_i^2 - Y_{i{\text{ + 1}}}^{\text{2}} } )} \right]^2 > 0\) for all μ _i ≧0,i=3, ...,n?1, \(\sum\limits_{i = 3}^{n - 1} {\mu _i > 0} \) , b) neighbourhoods of theY _i 's:V _i(Y _i )?R ⁴ i=2, ...,n (in the euclidean topology ofR ⁴) and c) a real number β>1 such that for all points (y):y ₂, ...,y _n y _i ∈V _i (Y _i there are positive constantsC _(n){(y)},h _(n){(y)} and a real number γ_(n){(y)∈a closed subset ofR?{0}?{1} with: γ_(n){(y)}\y ₂,y ₃, ...,y _n totally space-like in the order 2, 3, ...,n and $$\left| {\left\langle {\left[ {A(x_1 ),A(x_2 )} \right]A(y_3 )...A(y_n )} \right\rangle } \right|< C_{(n)} \left\{ {(y)} \right\}\exp \left\{ { - h_{(n)} \left\{ {(y)} \right\}r^\beta } \right\}$$ for \(x_1 = \gamma _{(n)} \left\{ {(y)} \right\}r\left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 1 \\ \end{array} } \right),x_2 = y_2 - [1 - \gamma _{(n)} \{ (y)\} ]r\left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 1 \\ \end{array} } \right)\) and for sufficiently large values ofr.     

Ti5O5 superstructures of cubic titanium monoxide

A cubic model is proposed for the Ti₅O₅ (Ti₅?O₅□ ≡ Ti₉₀?₁₈O₉₀□₁₈) superstructure of nonstoichiometric titanium monoxide Ti _x O _z with double imperfection. The unit cell of the cubic Ti₅O₅ superstructure has the threefold lattice parameter of the unit cell of the basis disordered B1 structure of Ti _x O _z monoxide and belongs to space group $Pm\bar 3m$ . The channel of the disorder-order transition, i.e., Ti _x O _z (space group $Fm\bar 3m$ )-Ti₅O₅ (space group $Pm\bar 3m$ ), includes 75 superstructure vectors of seven stars {k ₁₀}, {k ₇}, {k ₆₍₁₎}, {k ₆₍₂₎}, {k ₄₍₁₎}, {k ₄₍₂₎}, and {k ₁}. The distribution functions of Ti and O atoms over the sites of the cubic Ti₅O₅ superstructure are calculated. A comparison of the X-ray and electron diffraction data obtained for ordered TiO_1.087 monoxide with the theoretical simulation results supports the existence of the cubic Ti₅O₅ superstructure. The cubic (space group $Pm\bar 3m$ ) Ti₅O₅ superstructure is shown to be a high-temperature structure relative to the well-known monoclinic (space group C2/m) superstructure of the same type.     

Atomic-vacancy ordering in the lowest tungsten carbide W2C

Atomic-vacancy ordering in the lowest tungsten carbide W₂C with an L′3-type basic hexagonal structure has been studied by neutron diffraction and X-ray diffraction. In the temperature range 2700–1370 K, the only ordered phase of the lowest tungsten carbide is shown to be the trigonal ?-W₂C phase (space group P $\overline 3 $ 1m). This trigonal ?-W₂C phase is found to form via a disorder-order phase transition channel, which includes three superstructure vectors (k ₁₅ ⁽¹⁾ , k ₁₅ ⁽²⁾ , k ₁₇ ⁽¹⁾ ) of two Lifshitz stars ({k ₁₅}, {k ₁₇}, and to be described by two long-range order parameters (η₁₅, η₁₇). The distribution function of carbon atoms in the trigonal ?-W₂C superstructure is calculated, and the corresponding region of the allowable values of the long-range order parameters η₁₅ and η₁₇ is found. Symmetry analysis of other possible superstructures of the lowest tungsten carbide W₂C is performed, and the physically acceptable sequence of phase transformations in W₂C is determined.     

Electronic color centers in SrF2-Na crystals

It was shown that optical bleaching of M _A ⁺ color centers at 80 K in SrF₂-Na crystals causes the core of an M _A ⁺ -center to transform into the V _a ⁺ Me ⁺ V _a ⁺ configuration, in which all three point defects are arranged diagnonally in the cube cell. Reirradiation of an optically bleached crystal by x-rays generates F _D centers in it: V _a ⁺ Me ⁺ V _a ⁺ + e ^? → V _a ⁰ Me ⁺ V _a ⁺ ≡ F _D. The F _D → M _A ⁺ transformation in SrF₂-Na crystals proceeds at T = 135 K, in contrast to the F _A → M _A ⁺ transformations, which take place at T > 200 K.     

Hard photon emission in spectatorB meson decays

We study the emission of a hard photon in the spectator, free quark approximation ofB meson decays. We thus obtain an estimate for the high end of the photon spectrum in the processb→X _u,c+γ and in the processesb→ \(\bar c\) +X _u,c+γ andb→l \(\bar v\) +X _u,c+γ, whereX _q=q+X. We show that the hard photon spectrum will allow to distinguish between decays via spectator and “annihilation” amplitudes, and could be used as well for an independent estimate of |V _ub/V _cb|. Further implications of our calculation are discussed.     

Reichenbachian Common Cause Systems

A partition C_{i i∈ I} of a Boolean algebra $\mathcal{S}$ in a probability measure space $(\mathcal{S},p)$ is called a Reichenbachian common cause system for the correlated pair A,B of events in $\mathcal{S}$ if any two elements in the partition behave like a Reichenbachian common cause and its complement, the cardinality of the index set I is called the size of the common cause system. It is shown that given any correlation in $(\mathcal{S},p)$ , and given any finite size n>2, the probability space $(\mathcal{S},p)$ can be embedded into a larger probability space in such a manner that the larger space contains a Reichenbachian common cause system of size n for the correlation. It also is shown that every totally ordered subset in the partially ordered set of all partitions of $\mathcal{S}$ contains only one Reichenbachian common cause system. Some open problems concerning Reichenbachian common cause systems are formulated.     

The influence of disorder on magnetocaloric effect and the order of transition in ferromagnetic manganite (La1−x Nd x )2/3(Ca1−y Sr y )1/3MnO3

Magnetocaloric effect and order of transition in (La_1?x Nd _x )_2/3(Ca_1?y Sr _y )_1/3MnO₃, prepared by conventional solid-state reaction, have been investigated. Using Banerjee criterion, we demonstrate first-order transition for (J1) and (J2 ) as well as second-order transition for (J3 ), (J4 ), and (J5 ) samples. The ΔS _M ^max is ranging between 9.18 Jkg^?1 K^?1 and 4.87 when Nd and Sr content changes leading to relative cooling power (RCP) varying between 330 and 229.35 J/kg. Both ΔS _M ^max and the RCP are found sensitive to the disorder σ ². The universal behavior obtained from ΔS variation curves confirmed the first-order transition for (J1) and (J2 ) samples and second-order transition for (J3), (J4), and (J5 ) samples obtained by Banerjee criterion. All samples with second-order phase transition exhibit inhomogeneous character estimated from local exponent n.     

Order-disorder phase transformations and specific heat of nonstoichiometric vanadium carbide

AbstractA study has been made of the order-disorder phase transformations in the homogeneity region of the VC_y nonstoichiometric cubic vanadium carbide (0.66<y<0.88). It has been established that an ordered V₆C₅ phase with monoclinic (space group C2/m) or trigonal (P3₁) symmetry, and a cubic (space group P4₃32) ordered V₈C₇ phase can form in the VC_y carbide below 1450 K, depending on the actual composition. The effect of off-stoichiometry and structural vacancy ordering on the specific heat of the VC_y carbide has been investigated. The temperatures and heats of the reversible order-disorder equilibrium transitions have been determined. The ordering in the VC_y carbide is shown to be a first-order phase transition. An equilibrium diagram of the V-C system taking into account ordering in the nonstoichiometric vanadium carbide has been constructed. Fiz. Tverd. Tela (St. Petersburg) 41, 529–536 (March 1999)