Role of the Cluster Structure of Amorphous Fe<Subscript>67</Subscript>Cr<Subscript>18</Subscript>B<Subscript>15</Subscript> Alloy in Magnetism and in the Changing of Electron Scattering Mechanisms under the Influence of Ion (Ar<Superscript>+</Superscript>) Irradiation

The contribution of clusters of different sizes to magnetism and the switching of electron scattering mechanisms in amorphous Fe₆₇Cr₁₈B₁₅ alloy during ion Ar⁺ irradiation is studied. The cluster magnetism is found to be related to the presence of clusters of the following two types: large α-(Fe, Cr) clusters of size D = 150–250 Å and small (D = 40–80 Å) clusters in a random intercluster medium. The generation of small ferromagnetic and antiferromagnetic clusters during ion irradiation leads to the formation of cluster glass, which affects the electrical properties of the alloy and causes a magnetic frustration. The temperature dependence of the barrier height is shown to characterize the magnetic state of the alloy in low fields. On the whole, the temperature dependence of the order parameter is a universal characteristic of the system. The temperature dependence of resistivity of initial alloys in the temperature range 98–300 K (ρ(T) ∝ T²) is determined by electron scattering by quantum defects, and the transition into a ferromagnetic state is revealed when the derivative ?ρ/?T ∝ T is analyzed. The increase in resistivity and the relation ρ ∝ T^1/2 in strongly inhomogeneous samples after irradiation at a dose Φ = 1.5 × 10¹⁸ ions/cm² are caused by weak localization effects, and the transition to a ferromagnetic state becomes obvious when the derivative ?ρ/?T ∝ T^–1/2 is considered. Irradiation by fluence Φ = 3 × 10¹⁸ ions/cm2 induces a giant (twofold) increase in the alloy density, restores the ferromagnetism of large clusters, decreases the resistivity by 37%, and restores the relation ρ(T) ∝ T², which results from the overlapping of the irradiation-induced small clusters when their concentration increases and from an increase in the alloy density. The overlapping of clusters lowers the barrier height and decreases the sensitivity of the alloy to an applied field. The relation ρ(T) ∝ T² is valid for the entire temperature range T = 2–300 K because of the partial screening of the magnetic moments of large clusters by a medium having the properties of cluster glass.     

On Sums of Projections

In the paper, for all n, we describe the set _n of all real numbers admitting a collection of projections P ₁,...,P _n on a Hilbert space H such that _k=1 ⁿ P _k= I (I is the identity operator on H) and study the problem to find all collections of this kind for a given _n .     

Special features of the structure and properties of La<Subscript>0.6</Subscript>Sr<Subscript>0.2</Subscript>Mn<Subscript>1.2−<Emphasis Type="Italic">y</Emphasis></Subscript>Cr<Subscript>y</Subscript>O<Subscript>3</Subscript> magnetoresisitive ceramics

Atomic order and magnetoresistive parameters are studied in La_0.6Sr_0.2Mn_1.2?yCr_yO₃ ceramics, in which Cr substituted for the above stoichiometric Mn content serves to enhance cluster formation in a solid solution. It is shown that in a range of 0.05<y<0.1, La_0.6Sr_0.2Mn_1.2?yCr_yO₃ experiences a phase transition from a partially disordered solid solution with a rhombohedral lattice \((R\bar 3c)\), in the form of planes possessing a long-range order and clusters of mesoscopic order, to a chemically microstratified solid solution, in the form of planes of an orthorhombic (Pnma) crystal lattice having a matrix structure and mesoscopic formations (clusters) coherently combined with a matrix of plane fragments peculiar to oxides of γ-MnO₂ type or, at y>0.1, of Cr₂O₃ type. It is shown that the structural phase transition is accompanied by maximum changes of the magnetoresistive parameters in the same range of Cr concentrations, i.e., 0.05<y<0.10.     

Nonhomogeneous evolutionary equations with impulse effects

For evolutionary equations with impulse effects basic theorems of perturbations theory are proved. An example from the theory of partial differential equations with impulse effects is given.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 1, pp. 93–100, January, 1992.     

Self-organization of atomic order and electronic structure in LaSrMnO films

A linear relationship between the critical temperatures T_max and T_min in the temperature dependences of the resistance of La_0.6Sr_0.2Mn_1.2O₃ single-crystal films that have a mesoscopic irregularity (metallic clusters in an insulating matrix) is found. A correlation between the atomic order and electronic structure of the films is studied by taking X-ray diffraction patterns and optical absorption spectra. It is shown that a rise in T_max and a simultaneous decrease in T_min cause correlated local changes in cluster areas of the structure. Namely, the volume occupied by a family of Mn-O planes with large interplanar spacings (d=2.04–2.08 Å) shrinks, while the volume occupied by a family of closer spaced (d=1.90–1.99 Å) planes grows. In the electronic subsystem, the density of states at ?ω=1.5 and 2.4 eV, which are due to Mn³⁺ and Mn⁴⁺ ions, increases, and the contribution from Mn²⁺ states at ?ω=0.9 eV decreases. As the charge states associated with Mn³⁺ and Mn⁴⁺ ions become dominant, the Mn-O binding energy grows. As a result, the contribution of the structural states with smaller d increases, thereby raising the density of states in the electronic subsystem at energies between 0.5 and 2.7 eV. The effect of self-organization in the multicomponent LaSrMnO system shows up in the transition from the heavily distorted rhombohedral to the less distorted orthorhombic structure.     

On Identities in Algebras Q n,λ Generated by Idempotents

We investigate the presence of polynomial identities in the algebras Q _n, generated by n idempotents with the sum e ( and e is the identity of an algebra). We prove that Q _4,2 is an algebra with the standard polynomial identity F ₄, whereas the algebras Q _4,, 2, and Q _n,, n 5, do not have polynomial identities.     

Weakly nonlinear boundary-value problems for operator equations with pulse influence

We consider the problem of finding conditions of solvability and algorithms for construction of solutions of weakly nonlinear boundary-value problems for operator equations (with the Noetherian linear part) with pulse influence at fixed times. The method of investigation is based on passing by methods of the Lyapunov—Schmidt type from a pulse boundary-value problem to an equivalent operator system that can be solved by iteration procedures based on the fixed-point principle. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 272–288, February, 1997.