We report threshold scattering properties for
39K(f=1) atoms obtained from analysis of experimental photoassociation spectra of the purely long-range
39K21u state. The existence of hyperfine structure in the excited
K21u state and a knowledge of the Hamiltonian structure for two colliding ground state alkali atoms enables us to obtain information on the sign and magnitudes of certain scattering lengths for ground state collisions. Analysis of the experimental photoassociation line shapes shows that the
39K scattering lengths are between $\ensuremath{-}3.2$ and $+0.8\mathrm{nm}(\ensuremath{-}{60a}_{0}$ and
+15a0, where
1a0=1a.u.=0.0529177nm) for the $a{}^{3}{\ensuremath{\Sigma}}_{u}^{+}$ molecular state, between +4.8 and +12 nm
(+90a0 and
+230a0) for the ${X}^{1}{\ensuremath{\Sigma}}_{g}^{+}$ state, and between $\ensuremath{-}7.9$ and $+1.1\mathrm{nm} (\ensuremath{-}{150a}_{0}$ and
+20a0) for collisions of two $f=1,m=\ensuremath{-}1$ atoms.
We demonstrate a number of two-colour spectroscopy techniques where the first step is the photoassociation of laser-cooled and trapped sodium atoms to form bound states of . High-resolution spectra of are obtained. Spectra of the `purely long-range' state demonstrates the use of two-colour spectroscopy to open up an ionization channel for detecting the occurrence of photoassociation. The utility of photoassociation as a technique for producing cold molecular samples in a well defined rovibrational state is demonstrated with spectra showing and (Condon internal diffraction) transitions after the photoassociation. Starting from states in the and potentials, these transitions include upward transitions to autoionizing potentials dissociating near 3P + 3P and downward transitions back to the 3S + 3S ground-state potentials. Spectroscopic constants for six levels of a doubly excited state of are given.