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Physical Review A

Mass dependence of scattering lengths for hydrogen atoms

Michael J. Jamieson, Bernard Zygelman

Published Aug 1, 2001• 35 citations
Publisher

Abstract

We demonstrate that changing the reduced mass in scattering calculations is equivalent to changing the potential. We explore how small changes in the mass alter the scattering length of hydrogen atoms when they interact via the $X{}^{1}{\ensuremath{\Sigma}}_{g}^{+}$ and $b{}^{3}{\ensuremath{\Sigma}}_{u}^{+}$ molecular states. We show how the variation of scattering length with mass depends on the asymptotic potential.

DOI
10.1103/physreva.64.032703

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We demonstrate that changing the reduced mass in scattering calculations is equivalent to changing the potential. We explore how small changes in the mass alter the scattering length of hydrogen atoms when they interact via the X1\ensuremathΣg+X{}^{1}{\ensuremath{\Sigma}}_{g}^{+}X1\ensuremathΣg+​ and b3\ensuremathΣu+b{}^{3}{\ensuremath{\Sigma}}_{u}^{+}b3\ensuremathΣu+​ molecular states. We show how the variation of scattering length with mass depends on the asymptotic potential.
The possibility that the non-adiabatic correction to the scattering length of a pair of hydrogen atoms interacting via the ground state molecular potential, X1Σ+g , of H2 could be made by replacing the mass of each nucleus by a different effective mass is explored. The Born-Oppenheimer potential with adiabatic, relativistic and radiative corrections is used in calculations of the scattering lengths and the mass-dependent shifts of the rotationless vibrational levels with fixed and varying effective masses. The shifts are compared with established values and it is demonstrated that the semi-classical formula for the scattering length accounts well for the effect of changing the mass. A perturbing potential that is equivalent to a change in mass is derived and it is compared to a published local non-adiabatic potential.
Scattering lengths for the collisions of ground and metastable state hydrogen atoms interacting via the B′1∑u +, EF 1∑g +, e3∑u + and h3∑g + molecular potentials are evaluated from published ab initio Born-Oppenheimer potentials with adiabatic corrections, matched to multipole expansions at large interatomic separations. It is demonstrated that the scattering lengths are very sensitive to the connections between the ab initio potentials and their asymptotic forms and the need is shown for accurate interatomic potential calculations at greater separations than are currently available. The density-dependent shift and broadening of the 1S–2S transition frequency in atomic hydrogen caused by two-body elastic collisions are presented.
The scattering lengths for hydrogen and deuterium atoms interacting via the potentials of the X 1 Σ g + , b 3 Σ u + and e 3 Σ u + molecular states are calculated. An estimate of the uncertainty in the X 1 Σ g + scattering length arising from the coupling to the 1 Σ u + states is made. The recommended scattering lengths are −0.67 and 148 bohr for the b 3 Σ u + and e 3 Σ u + interaction potentials, respectively, and 11.1 bohr for the X 1 Σ g + interaction potential. It is suggested that deuterium atoms would be cooled efficiently by collisions with metastable hydrogen atoms.
We derive a simple expression for the change in the s-wave scattering length in terms of zero-energy wavefunctions, we generalize it to obtain an expression for the change in the p-wave scattering volume and we use both expressions to derive the first order differential equations of variable phase theory that are satisfied by the closely related accumulated scattering length and volume. We provide numerical demonstrations for the example of a pair of hydrogen atoms interacting via the X1Σ+g molecular state.
We examine the effect of theoretically varying the collision-system reduced mass in collisions of He with vibrationally excited molecular hydrogen and observe zero-energy resonances for select atomic ``hydrogen'' masses less than 1 u or a ``helium'' mass of 1.951.951.95 u. Complex scattering lengths, state-to-state vibrational quenching cross sections, and a low-energy elastic scattering resonance are all studied as a function of collision-system reduced mass. Experimental observations of these phenomena in the cold and ultracold regimes for collisions of 3He^{3}\mathrm{He}3He and 4He^{4}\mathrm{He}4He with H2{\mathrm{H}}_{2}H2​, HD, HT, and DT should be feasible in the near future.