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Journal of Physics B Atomic Molecular and Optical Physics

Binding energy and scattering length for diatomic systems

Bo Gao

Published Oct 27, 2004• 46 citations
Publisher

Abstract

We discuss the relationship between scattering length and binding energy for a diatomic system with long-range van der Waals interaction. The well-known relation, 0 = −(ℏ 2 /2μ)(1/ a 0 ) 2 , is generalized to much higher orders for the s wave. Analytic results for the binding energies of p and higher angular momentum states are also presented.

DOI
10.1088/0953-4075/37/21/004

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•Jan 1, 2010•0 citations
We discuss the relationship between scattering length and binding energy for a diatomic system with long-range van der Waals interaction. The well-known relation, 0 = −(ℏ 2 /2μ)(1/ a 0 ) 2 , is generalized to much higher orders for the s wave. Analytic results for the binding energies of p and higher angular momentum states are also presented.
Using an l-insensitive quantum defect theory, we derive a set of universal properties, both spectral and scattering, that are shared by diatomic systems with an asymptotic van der Waals interaction and having a zero-energy bound or a quasibound state of angular momentum lb.{l}_{b}.lb​. The 133Cs{}^{133}\mathrm{Cs}133Cs dimer is studied as an example. Systems that are close to having a bound or a quasibound state at the threshold are also discussed.
At long distances, interactions between neutral ground-state atoms can be described by the van der Waals potential. In the ultracold regime, atom-atom scattering is dominated by sss-waves phase shifts given by an effective range expansion in terms of the scattering length ${\ensuremath{\alpha}}_{0}$ and the effective range r0{r}_{0}r0​. We show that while the scattering length cannot be predicted for these potentials, the effective range is given by the universal low-energy theorem ${r}_{0}=A+B/{\ensuremath{\alpha}}_{0}+C/{\ensuremath{\alpha}}_{0}^{2}$, where AAA, BBB, and CCC depend on the dispersion coefficients Cn{C}_{n}Cn​ and the reduced diatom mass. We confront this formula to about 100 determinations of r0{r}_{0}r0​ and ${\ensuremath{\alpha}}_{0}$ and show why the result is dominated by the leading dispersion coefficient C6{C}_{6}C6​. Universality and scaling extend much beyond naive dimensional analysis estimates.
A theoretical relation between the electron scattering length and the van der Waals approximation to the equation of state is described. The theory provides a simple calculation for the scattering length of neutral atomic scatterers such as rare gases, using two experimental parameters: the hard-core radius (calculated from the van der Waals equation of state, using critical data) and the atomic polarizability. The results for the calculated electron scattering length agree with measured values derived either from the energy shifts of highly excited molecular Rydberg states, perturbed by He, Ne, Ar, Kr, and Xe, or from swarm electron cross-section data for the same gases.
Long-range states of diatomic molecules have average values of internuclear separations $\ensuremath{\langle}R\ensuremath{\rangle}$ at least one order of magnitude larger than the equilibrium value of RRR. For example, the helium dimer 4He2{{}^{4}\mathrm{He}}_{2}4He2​ has a single bound state with $\ensuremath{\langle}R\ensuremath{\rangle}$ of about 50 \AA{}. We show that the properties of these states, such as $\ensuremath{\langle}R\ensuremath{\rangle}$, the dissociation energy, or the sss-wave scattering length, can be related by simple, yet very accurate, formulas if a potential-energy curve is known. By examining a range of ab initio and empirical helium dimer potentials as well as scaling these potentials, we found that the formulas remain accurate, even if very approximate potentials were used. In addition to 4He2{{}^{4}\mathrm{He}}_{2}4He2​, we present results for 9Be2,20Ne2{{}^{9}\mathrm{Be}}_{2},\phantom{\rule{0.28em}{0ex}}{{}^{20}\mathrm{Ne}}_{2}9Be2​,20Ne2​, and KRb.
The use of scattering length of particle-target interaction due to real- valued potential to study the bound states of the particle-target system is well known in nuclear and atomic physics. In view of the current interest in using eta-nucleus scattering length to infer the existence of eta-mesic nucleus, we derive general analytic expressions that relate the binding energy and half-width of an unstable bound state to the complex-valued scattering length due to the same particle-target interaction.