Feshbach resonances in cold collisions of potassium atoms

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 $F=1,\hspace{0.5em}\hspace{0.5em}{\mu }_F=-1$ 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.