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Abstract: A generalized relativistic harmonic oscillator for spin 1/2 particles is studied. The eigenfunctions and eigenenergies are obtained for the ring-shaped non-spherical harmonic oscillator by solving Dirac equation with equal mixture of vector and scalar potentials in opposite signs, for which pseudospin symmetry is exact. Several particular cases such as the ring-shaped harmonic oscillator, non-spherical harmonic oscillator, and spherical harmonic oscillator are also discussed. [Copyright &y& Elsevier]

Ti 2 AlNb-based alloy foils are promising for aerospace applications. In this work, we investigate the effects of different hot rolling processes on the microstructure and mechanical properties of hot rolled sheets, as well as on the deformation mechanism, microstructure evolution, and mechanical properties of subsequent cold rolling during the formation of Ti 2 AlNb-based alloy foils. Due to the finer uniform grain size, lower dislocation density, and higher compatible deformation between the phases, the elongation of the sheet after hot rolling in the α 2 + O + B2 phase field is 6.8%, which is 2.5 times higher than that of after hot rolling in the α 2 + B2 phase field. The difference in microstructure and mechanical properties after the hot rolling affects the quality of the formation, the deformation mechanism, and the mechanical properties during the cold rolling. The deformation in the cold rolling is mainly in the form of shear bands. Due to the difference in microstructure, the macro shear bands are oriented parallel or at 30–60° to the rolling direction. As the degree of cold rolling deformation increases, the macro shear bands become denser and wider. When rolling to 80%, the micro shear bands <200 nm thick are formed in the microstructure. In addition, during the cold rolling, work hardening, dislocation strengthening, and grain refinement lead to a gradual increase in strength, with elongation decreasing first and then increasing. The texture evolution affects the elastic modulus of the alloy, where the increase in {111}〈110〉 texture causes a decrease in the elastic modulus, while much anomalous slip system {110}〈001〉 activates the {111}〈112〉 texture and increases the elastic modulus. This process should be considered for the cold rolling preparation of Ti 2 AlNb-based alloy foil because of the superior properties of the sheet after hot rolling in the α 2 + O + B2 phase field, its better strength and elongation and higher yield when cold rolling. • Ti 2 AlNb alloy foils with excellent mechanical properties were prepared by high efficiency and low cost rolling processes. • The deformation mechanism of Ti 2 AlNb alloy under different cold rolling processes was revealed. • Inheritance and transition models have been developed for hot rolled and subsequent cold rolled textures. • The effect of microstructure evolution on mechanical properties of Ti 2 AlNb alloy foils was studied. [ABSTRACT FROM AUTHOR]