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The spinel stoichiometric compounds of the CuCr 2 − x Sb x Se 4 family were obtained for the first time and prepared with the assumption of the valence of Sb ions equal to 3. The magnetization isotherms at 4.2 K and at the magnetic stationary fields up to 14 T as well as the temperature dependencies of magnetic susceptibility were studied in the temperature range of 4.2 ÷ 400 K. These results were compared with analogous measurements performed earlier by the authors for the nonstoichiometric spinel family built of the same elements and obtained under the assumption of the valence of Sb ions equal to 5. From this comparison it follows that these two spinel series reveal significant differences: (1) The interval of possible concentrations of Sb ions in the obtained stoichiometric spinel samples under study is up to 1.0, whereas for the nonstoichiometric spinels is (0.30 ÷ 0.48). (2) The corresponding intervals for the appearance of the spin glass states are: (0.2 ÷ 1.0) and (0.30 ÷ 0.48), whereas the freezing temperatures do not differ significantly. (3) The magnetization processes in the stoichiometric samples depend strongly on the Sb concentration, whereas in the nonstoichiometric samples they run very hardly.

The experimental evidence of metamagnetism, giant negative magnetoresistance and spin glass state is presented for the compounds under study. An attempt is made to explain the jump-like phase transition of the colossal negative magnetoresistance of −4433% at 153.2 K into the colossal positive magnetoresistance of +980% at 154.2 K for the spinel sample with selenium. It was emphasized that in all the phenomena described in this paper the double exchange magnetic interaction plays the main role.

High degree of defectiveness both in the stoichiometric CuCr 2− x Sb x S 4 ( x = 0.3, 0.4, 0.5) and nonstoichiometric Cu 1 + x Cr 1.5 + y Sb 0.5 + z Se 4 + t (where: −0.02 ≤ x ≤ 0.01, 0.03 ≤ y ≤ 0.35, −0.2 ≤ z ≤ −0.02, 0.01 ≤ t ≤ 0.08) spinels doped with Sb has been determined with the aid of the vacancy model. From the analysis carried out of the effective magnetic moments and the exchange integrals, it follows that in the spinels with sulphur mainly the cation subarrays are defected whereas in the spinels with selenium the defectiveness appears mainly in the anion subarrays. It turned out that in the spinels with sulphur the ferromagnetic cluster spin glass states appear accompanied by the large negative magnetoresistance, whereas in the spinels with selenium the corresponding clusters are antiferromagnetic and the giant magnetoresistance has been observed both positive and negative. A jump-like phase transition was here observed changing both the sign and the measured value of the magnetoresistance from −4433% at 153.2 K up to +980% at 154.24 K. Moreover, in the spinels with selenium both semiconductor–metal and metal–semiconductor phase transitions have been observed from the temperature dependence of the resistance.

The X-ray absorption K-edges of sulfur in CuCr2−xSbxS4 (x = 0.3, 0.4, 0.5) have been measured and compared to linearized augmented plain wave (LAPW) numerical calculations. The near edge region is generated by magnetically polarized and hybridized S 2p energy levels and shows small change with increased Sb content, which can be attributed to the movement of Cr 3d spin-down states up on the energy scale. The LAPW result confirms this effect. Moreover, the calculations show that the antimony in those spinels is rather in +3 than in +5 valence state. The calculated magnetic moment on Cr equals to 2.72μB. In EXAFS, there is no visible change between x = 0.3 and 0.5 spectra, which indicates that the local structure does not change significantly.

Powder samples of CuCr 2− x Sb x S 4 were obtained for x =0.3, 0.4 and 0.5. All the samples have spinel structure. The magnetization measurements were carried out with the use of the induction method in the temperature range of 4.2–300 K in stationary fields with inductions up to 14 T. Both the electrical conductivity and the magnetoresistivity were measured in the same range of temperatures with the use of the four-point method. For the magnetoresistivity measurements both stationary fields up to 14 T and pulsed fields of 38 T were applied. A lack of magnetic saturation is observed for all the samples. Moreover, an increase of the antiferromagnetic component of the magnetization is observed with the increase of the Sb concentration. The absolute values of magnetoresistivity as a function of x exhibit at 4.2 K and 14 T a maximum at about x =0.4, assuming zero at x =0.5. For the two concentrations, i.e. for x =0.3 and 0.5, the magnetoresistivity at room temperature changes its sign twice as a function of the magnetic field. A peak clearly seen in the temperature dependence of the relative derivatives of both the magnetization and susceptibility seems to testify to a spin-flip transition at 13 K and 1 T.

A hypothesis is put forward concerning the existence of the general relation between aperiodic crystal structures and the invariants of the group of the automorphisms of the d-dimensional torus. A relation is proved between the one-dimensional and two-dimensional displacive modulations and the rationality of some invariants of the group of automorphisms of both the circle S 1 and the torus T 2 .

The evolution of a probe D-brane in the p-brane background is considered. The anisotropic evolution of the world-volume of the D-brane with a given topology of a world-volume in the form of a direct product of a n-dimensional flat space and (3 − n)-dimensional sphere is formulated. In this case the anisotropy is described with the aid of two parameters (Hubble parameters) The special case of this evolution, namely the isotropic evolution corresponds to equality of these two parameters. In the latter case the masses and charges of the background p-branes are obtained.

The Dirac-Born-Infeld (DBI) action of the 3-dimensional brane for its dynamical embeddings and the gauge fields has been studied. The evolution of both the D3-brane and the ambient space has been obtained. For the special constraint put on the transverse coordinates a family of the contracting and expanding spaces has been found.

The cosmological constant on a D-brane is analyzed. This D-brane is in the background produced by the p-brane solutions. The energy–momentum tensor in this model has been found and the form of the cosmological constant has been derived. This energy–momentum tensor is interpreted as an energy–momentum tensor for a perfect fluid on the D-brane. The energy density and the pressure for this fluid have been derived. As it turned out the pressure is negative but the speed of sound is real.

We consider gravity in the system consisting of the Bogomol'nyi-Prasad-Sommerfield (BPS) $D3$-brane embedded in a flat background geometry, produced by the solutions of supergravity. The effective action for this system is represented by the sum of the Hilbert-Einstein and Dirac-Born-Infeld actions. We derive the Wheeler-DeWitt equation for this system and obtain analytical solutions in some special cases. We also calculate the tunneling probability from the Planckian size of the $D3$-brane to the classical regime.

The paper concerns the application of the fibre bundle approach to the description of the magnetic structures and their symmetry groups. Hence the explicit formulas describing both the variety of magnetic structures and their symmetry groups have been derived. The assumption was made that the bundle sections correspond to magnetizations of the separate crystal planes multiplied by a certain Gaussian factor defined in ℝ3, the last factor making the problem continuous and more physical.

We consider a novel kind of plot for presenting the behaviour of null geodesics in Schwarzschild spacetime. Our diagram depicts various features of the light rays as recorded by different classes of observers. The idea stems from the use of phase portraits for illustrating the null geodesics in Schwarzschild spacetime, above and below the horizon of a black hole, but they reveal unphysical characteristics. The plots we propose are free from such anomalies. Moreover they allow us to discover new under-horizon features.

The behavior of the structure amplitude is analyzed in terms of the harmonics of a modulation function for the case of one-dimensional longitudinal displacive modulation. This modulation function is given by a finite series of N sine functions. It turns out that: (1) for the large values of the amplitudes of the modulation function the structure amplitude tends to zero for the arbitrary values of the order m of the satellite reflections; (2) for certain values of the amplitudes of the modulation function the modulus of the structure amplitude assumes its maximum value and (3) for the latter case the intensity of an arbitrary satellite reflection is a function of the intensities of both the closest main reflection and the intensities of the finite set of the neighbor satellite reflections.

The authors have found the general description of all the eight fundamental magnetic structures and their symmetry groups (including spin glass state - sgs) with the use of the fibre bundle approach [1]. The structure of sgs turns out to consist in the Gaussian-like randomness of the orientations of all the separate atomic magnetic moments in the sample. It means that the magnetization vector M is also subject to Gaussian distribution. Thus the structure of the sgs is represented by the magnetization vector situated along the generatrix of a certain cone whose axis coincides with the direction of an internal spontaneous magnetic field indispensable for stability of the sgs in a sample [2]. The axis of the 3-dimensional Gaussian distribution mentioned above coincides also with the same direction. Every precession angle of M around this field makes the symmetry operation of sgs once the angle between M and this field remains constant. Thus the symmetry group of sgs turns out to be SO(2). The temperature dependence of magnetic susceptibility of sgs in the alloys CuMn [3] corresponds when heating to the gradual degeneration of the Gaussian distribution of the magnetization vector, and when cooling to its gradual recovery with no hysteresis. In both cases an internal spontaneous magnetic field mentioned above plays an essential role. The interrelation of this field, of the Gaussian distribution mentioned above, of the average kinetic energy of magnetic atoms and of the frustration of orientations of the magnetic moments placed on these atoms will be discussed.

The conditions of stability of the spin glass state and its percolation limit in the spinel families have been analyzed on the base of the de Almeida-Thouless theory. The temperature range was determined as a function of the antimony concentration for which the spin glass state is stable.