Long-range interaction of the 39K(4s)+39K(4p) asymptote by photoassociative spectroscopy. I. The g− pure long-range state and the long-range potential constants

This paper reports on a comprehensive study of the long-range interaction of the ${}^{39}\mathrm{K}{\text{(4s)+}}^{39}\mathrm{K}\text{(4p)}$ asymptotic system. We present a detailed discussion of the $R$ -dependent angular momentum couplings and correlation between the Hund’s case (a) and case (c) molecular states. Analytical expressions for the 16 adiabatic Hund’s case (c) long-range potential curves are derived including the higher order dispersion forces and the effects of retardation. Experimentally, six Hund’s case (c) long-range molecular states ($0_u^{+},$ $1_g,$ and $0_g^{-}$ dissociating to the ${\text{4\hspace{0.17em}}}^2{S}_{\text{1/2}}{\text{+4\hspace{0.17em}}}^2{P}_{\text{3/2}}$ asymptote and $0_u^{+},$ $1_g,$ and $0_g^{-}$ to the ${\text{4\hspace{0.17em}}}^2{S}_{\text{1/2}}{\text{+4\hspace{0.17em}}}^2{P}_{\text{1/2}}$ limit) are observed with rovibrational resolution by photoassociative spectroscopy of ultracold ${}^{39}\mathrm{K}$ atoms in a high density magneto-optical trap (MOT). Among the six observed long-range states, the upper $0_g^{-}$ “pure long-range” state has negligible short-range chemical exchange contributions and the measured molecular binding energies $\text{(v=0–26)}$ are used to precisely determine the long-range potential constants of the $\text{4s+4p}$ interaction. We determine: $C_3^{\Pi }\text{=8.436(14)\hspace{0.17em}}\mathrm{a.u.},$ $C_3^{\Sigma }\text{=16.872(28)\hspace{0.17em}}\mathrm{a.u.},$ $C_6^{\Pi }\text{=6272(94)\hspace{0.17em}}\mathrm{a.u.},$ and $C_6^{\Sigma }\text{=9365(141)\hspace{0.17em}}\mathrm{a.u.}.$ Molecular constants for the three special pure long-range states, the $0_g^{-}$ and $1_u$ (dissociating to the ${\text{4\hspace{0.17em}}}^2{P}_{\text{3/2}}$ limit and with potential minimum) and the $1_u$ (dissociating to the ${\text{4\hspace{0.17em}}}^2{P}_{\text{1/2}}$ and with potential maximum), are reported. The internal consistency of the theoretical model used in this work is confirmed by the excellent agreement between the long-range potential curve of the $1_g$ state obtained in present work (from the $0_g^{-}$ state) and the long-range portion of the RKR potential curve of the ${\text{1\hspace{0.17em}}}^1{\Pi }_g$ state previously determined by conventional molecular spectroscopy. The radiative lifetime of the K $\text{4p}$ state derived from the dipole–dipole interaction constant $C_3^{\Pi }$ is also in excellent agreement with a recent fast-beam measurement.