We develop an analytical model for ultracold atom-ion collisions using themultichannel quantum-defect formalism. The model is based on the analyticalsolutions of the r^-4 long-range potential and on the application of a frametransformation between asymptotic and molecular bases. This approach allows thedescription of the atom-ion interaction in the ultracold domain in terms ofthree parameters only: the singlet and triplet scattering lengths, assumed tobe independent of the relative motion angular momentum, and the lead dispersioncoefficient of the asymptotic potential. We also introduce corrections to thescattering lengths that improve the accuracy of our quantum-defect model forhigher order partial waves, a particularly important result for an accuratedescription of shape and Feshbach resonances at finite temperature. The theoryis applied to the system composed of a 40Ca+ ion and a Na atom, and compared tonumerical coupled-channel calculations carried out using ab initio potentials.For this particular system, we investigate the spectrum of bound states, therate of charge-transfer processes, and the collision rates in the presence ofmagnetic Feshbach resonances at zero and finite temperature.
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