Extending asymmetric convergence and Cauchy condition using ideals

In this paper we use the notion of ideals to extend the convergence and Cauchy conditions in asymmetric metric spaces. The asymmetry (or rather, absence of symmetry) of these spaces makes the whole treatment different from the metric case and we use a genuinely asymmetric condition called (AMA) to prove many results and show that certain classic results fail in the asymmetric context if the assumption is dropped.     

Information entropic measures of a quantum harmonic oscillator in symmetric and asymmetric confinement within an impenetrable box

Information‐based uncertainty measures like Shannon entropy, Onicescu energy and Fisher information (in position and momentum space) are employed to understand the effect of symmetric and asymmetric confinement in a quantum harmonic oscillator. Also, the transformation of the Hamiltonian into a dimensionless form gives an idea of the composite effect of force constant and confinement length (x_c). In the symmetric case, a wide range of x_c has been taken up, whereas asymmetric confinement is dealt with by shifting the minimum of the potential from the origin keeping box length and boundary fixed. Eigenvalues and eigenvectors for these systems are obtained quite accurately via an imaginary‐time propagation scheme. For asymmetric confinement, a variation‐induced exact diagonalization procedure is also introduced, which produces very high quality results. One finds that, in symmetric confinement, after a certain characteristic x_c, all these properties converge to respective values of a free harmonic oscillator. In the asymmetric situation, excited‐state energies always pass through a maximum. For this potential, the classical turning point decreases, whereas well depth increases with the strength of asymmetry. A study of these uncertainty measures reveals that localization increases with an increase of the asymmetry parameter.

     

Synthesis and surface modified hard magnetic properties in Co0.5Pt0.5 nanocrystallites from a rheological liquid precursor

Small crystallites of a metastable phase Co_0.5Pt_0.5 are precipitated by heating a rheological liquid precursor of cobalt–hydrazine complex and platinum chloride H₂PtCl₆·xH₂O in polymer molecules of poly(vinylpyrrolidone) (PVP) in ethylene glycol. The hydrazine co-reduces nascent atoms from the Co²⁺ and Pt⁴⁺ that recombine and grow as Co_0.5Pt_0.5. The PVP molecules cap a growing Co_0.5Pt_0.5 as it achieves a critical size so that it stops growing further in given conditions. X-ray diffraction pattern of a recovered powder reveals a crystalline Co_0.5Pt_0.5 phase (average crystallite size D∼8 nm) of a well-known Fm3m-fcc crystal structure with the lattice parameter a=0.3916 nm (density ρ=14.09 g/cm³). A more ordered L1₀ phase (ρ=15.91 g/cm³) transforms (D≥25 nm) upon annealing the powder at temperature lesser than 700 °C (in vacuum). At room temperature, the virgin crystallites bear only a small saturation magnetization M_s=5.54 emu/g (D=8 nm) of a soft magnet and it hardly grows on bigger sizes (D≤31 nm) in a canted ferromagnetic structure. A rectangular hysteresis loop is markedly expanded on an optimally annealed L1₀ phase at 800 °C for 60 min, showing a surface modified coercivity H_c=7.781 kOe with remnant ratio M_r/M_s=0.5564, and M_s=39.75 emu/g. Crystallites self-assembled in an acicular shape tailor large H_c from ideal single domains and high magnetocrystalline anisotropy of a hard magnet L1₀ phase.