DOI: https://doi.org/10.30970/jps.01.134
PSEUDOPOTENTIAL WITHIN THE FRAMEWORK OF PHASE FUNCTIONS METHOD. THE STRUCTURE OF MODEL PSEUDOPOTENTIAL OF TRANSITION AND RARE-EARTH METALS
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Journal of Physical Studies 1(1), 134–147 (1996)
DOI: https://doi.org/10.30970/jps.01.134 PSEUDOPOTENTIAL WITHIN THE FRAMEWORK OF PHASE FUNCTIONS METHOD. THE STRUCTURE OF MODEL PSEUDOPOTENTIAL OF TRANSITION AND RARE-EARTH METALS |
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V. V. Fourman, P. M. Yakibchuk
Ivan Franko Lviv State University, Chair of Theoretical Physics
12 Drahomanov Str., Lviv UA-290005, Ukraine
The criteria of constructing model pseudopotentials for the transition and rare-earth metals on the basis of the phase functions formalism and the scattering theory taking into account the effects of hybridization and spin-orbit coupling have been determined. An integral equation for the partial component scattering amplitude is received on the basis of Lippmann-Shvinger equation. Taking into account the analyticity of the scattering amplitude as well as the scattering matrix in the presence of pseudopotential the conditions imposed on its structure have been analysed.
Since the poles of the partial scattering amplitude that are on the positive imaginary semiaxis correspond to the energies of the bound states, with the help of the phase functions method the criterion for finding the parameters of model pseudopotentials has been suggested. A phase equation that contains pseudopotential for the properties of partial scattering amplitude for the bound states was obtained. Asymptotic solutions were found for this equation. The phase equation for a partial scattering amplitude with the partial component of pseudopotential and asymptotics valid for arbitrary model pseudopotentials was suggested. On the basis of this equation a method for finding model pseudopotential parameters has been proposed.
For the real values of ${\bf k}$ the scattering amplitude determines a partial scattering cross-section and for the complex values of ${\bf k}$ it describes stationary and quasi-steady bounded states of the system. Proceeding from on the scattering amplitude behaviour the structure of model pseudopotential for transition metals has been established.
The peculiarities of the scattering matrix and the amplitude for complex energies are exploited for constructing model pseudopotential of transition metals taking into account the effects of hybridization. Making use of the dispersion relation and the optical theorem for the partial scattering amplitude a form of hybridization term in model pseudopotential for transition metals has been proposed. Judging by the analysis of properties of scattering cross-section and partial scattering amplitude during resonance that correspond to quasi-steady bounded state the expression hybridized component model pseudopotential of transition metals is received. Phase equation for electronic eigenfunction of $d$-state is received for finding limits of $d$-band of transition metals and its width.
The scheme that takes into account spin-orbit coupling determines the scattering process with a pseudopotential. The behaviour of the scattering amplitude was analyzed taking into account spin-orbit coupling. The expressions for calculating formfactors model pseudopotential of rare-earth metals, considering spin-orbit coupling in pseudopotential have been received provided that the parity conservation for scattering matrix elements is observed.