Abstract
In this paper we briefly review the basic steps that allow the calculation of the scattering length in the collision of two alkali-metal atoms in a well defined magnetic polarization state, and in the presence of a static magnetic field. Calculations are actually done for the low-field seeking state $F=1, {\ensuremath{\mu}}_{F}=\ensuremath{-}1$ of bosonic potassium atoms. The electrostatic potentials obtained through Rydberg-Klein-Rees data are connected to a dispersive, long range tail in which the dominant dipole-dipole ${C}_{6}$ term may take different values within a specified range. We show the occurrence of Feshbach resonances in the ultra cold collision of two identical atoms, belonging either to the bosonic species ${}^{39}\mathrm{K}$ or ${}^{41}\mathrm{K}.$ Our results demonstrate that there is a range of ${C}_{6}$ values for which the collision of two ${}^{39}\mathrm{K}$ atoms displays a single resonance, while for other values of ${C}_{6}$ no resonance occurs. On the other hand, Feshbach resonances are present in the collision of two ${}^{41}\mathrm{K}$ atoms for almost all values of the dispersion coefficient ${C}_{6}$ in that range. We also show the origin of the different types of Feshbach resonances that occur in the cold collision of two ${}^{41}\mathrm{K}$ atoms. The detection of such resonances can help establish the actual value of the dispersive coefficient.
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