Precision measurement of the last bound states in H${}_2$ and determination of the H + H scattering length

Abstract

The binding energies of the five bound rotational levels $J=0-4$ in the highest vibrational level $v=14$ in the X${}^1{\Sigma }_g^{+}$ ground electronic state of H${}_2$ were measured in a three-step ultraviolet-laser experiment. Two-photon UV-photolysis of H${}_2$S produced population in these high-lying bound states, that were subsequently interrogated at high precision via Doppler-free spectroscopy of the F${}^1{\Sigma }_g^{+}$ - X${}^1{\Sigma }_g^{+}$ system. A third UV-laser was used for detection through auto-ionizing resonances. The experimentally determined binding energies were found to be in excellent agreement with calculations based on non-adiabatic perturbation theory, also including relativistic and quantum electrodynamical contributions. The $s$-wave scattering length of the H + H system is derived from the binding energy of the last bound $J=0$ level via a direct semi-empirical approach, yielding a value of $a_s$ = 0.2724(5) $a_0$, in good agreement with a result from a previously followed theoretical approach. The subtle effect of the $m{\alpha }^4$ relativity contribution to $a_s$ was found to be significant. In a similar manner a value for the $p$-wave scattering volume is determined via the $J=1$ binding energy yielding $a_p$ = -134.0000(6) $a_0^3$. The binding energy of the last bound state in H${}_2$, the ($v=14$, $J=4$) level, is determined at 0.023(4) cm${}^{-1}$, in good agreement with calculation. The effect of the hyperfine substructure caused by the two hydrogen atoms at large internuclear separation, giving rise to three distinct dissociation limits, is discussed.

11 pages, 9 figures