Determination of the scattering lengths of39Kfrom1uphotoassociation line shapes

We report threshold scattering properties for ${}^{39}\mathrm{K}\mathrm{}(f=1\mathrm{})\mathrm{}$ atoms obtained from analysis of experimental photoassociation spectra of the purely long-range ${}^{39}{\mathrm{K}}_2\hspace{0.5em}{1}_u$ state. The existence of hyperfine structure in the excited ${\mathrm{K}}_2\hspace{0.5em}{1}_u$ 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 ${}^{39}\mathrm{K}$ scattering lengths are between $-3.2$ and $+0.8\mathrm{}\mathrm{nm}(-{60a}_0$ and ${+15a}_0,$ where ${1a}_0=1\mathrm{}\mathrm{a}.\mathrm{u}.\mathrm{}=0.052\mathrm{}9177\mathrm{nm})$ for the $a{}^3{\Sigma }_u^{+}$ molecular state, between +4.8 and +12 nm ${(+90a}_0$ and ${+230a}_0)$ for the $X^1{\Sigma }_g^{+}$ state, and between $-7.9$ and $+1.1\mathrm{}\mathrm{nm}\hspace{0.5em}(-{150a}_0$ and ${+20a}_0)$ for collisions of two $f=1,m=-1$ atoms.