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arXiv:physics/0303031 · Physical Review A

Scaling predictions for radii of weakly bound triatomic molecules

Marcelo Takeshi Yamashita, R.S. Marques de Carvalho, Lauro Tomio, T. Frederico

Published Jul 16, 2003• 21 citations
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Abstract

The mean-square radii of the molecules ${}^{4}{\mathrm{He}}_{3},$ ${}^{4}{\mathrm{He}}_{2}{\ensuremath{-}}^{6}\mathrm{Li},$ ${}^{4}{\mathrm{He}}_{2}{\ensuremath{-}}^{7}\mathrm{Li},$ and ${}^{4}{\mathrm{He}}_{2}{\ensuremath{-}}^{23}\mathrm{Na}$ are calculated using a three-body model with contact interactions. They are obtained from a universal scaling function calculated within a renormalized scheme for three particles interacting through pairwise Dirac-$\ensuremath{\delta}$ interaction. The root-mean-square distance between two atoms of mass ${m}_{A}$ in a triatomic molecule are estimated to be of the order of $\mathcal{C}\sqrt{{\ensuremath{\Elzxh}}^{2}/[{m}_{A}{(E}_{3}\ensuremath{-}{E}_{2})]},$ where ${E}_{2}$ is the dimer and ${E}_{3}$ is the trimer binding energies, and $\mathcal{C}$ is a constant (varying from $\ensuremath{\sim}0.6$ to $\ensuremath{\sim}1),$ which depends on the ratio between ${E}_{2}$ and ${E}_{3}.$ Considering previous estimates for the trimer energies, we also predict the sizes of rubidium and sodium trimers in atomic traps.

DOI
10.1103/physreva.68.012506

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The mean-square radii of the molecules 4He3,{}^{4}{\mathrm{He}}_{3},4He3​, 4He2\ensuremath−6Li,{}^{4}{\mathrm{He}}_{2}{\ensuremath{-}}^{6}\mathrm{Li},4He2​\ensuremath−6Li, 4He2\ensuremath−7Li,{}^{4}{\mathrm{He}}_{2}{\ensuremath{-}}^{7}\mathrm{Li},4He2​\ensuremath−7Li, and 4He2\ensuremath−23Na{}^{4}{\mathrm{He}}_{2}{\ensuremath{-}}^{23}\mathrm{Na}4He2​\ensuremath−23Na are calculated using a three-body model with contact interactions. They are obtained from a universal scaling function calculated within a renormalized scheme for three particles interacting through pairwise Dirac-\ensuremathδ\ensuremath{\delta}\ensuremathδ interaction. The root-mean-square distance between two atoms of mass mA{m}_{A}mA​ in a triatomic molecule are estimated to be of the order of C\ensuremath\Elzxh2/[mA(E3\ensuremath−E2)],\mathcal{C}\sqrt{{\ensuremath{\Elzxh}}^{2}/[{m}_{A}{(E}_{3}\ensuremath{-}{E}_{2})]},C\ensuremath\Elzxh2/[mA​(E3​\ensuremath−E2​)]​, where E2{E}_{2}E2​ is the dimer and E3{E}_{3}E3​ is the trimer binding energies, and C\mathcal{C}C is a constant (varying from \ensuremath∼0.6\ensuremath{\sim}0.6\ensuremath∼0.6 to \ensuremath∼1),\ensuremath{\sim}1),\ensuremath∼1), which depends on the ratio between E2{E}_{2}E2​ and E3.{E}_{3}.E3​. Considering previous estimates for the trimer energies, we also predict the sizes of rubidium and sodium trimers in atomic traps.
The ground state of weakly bound dimers and trimers with a radius extending well into the classically forbidden region is explored, with the goal to test the predicted universality of quantum halo states. The focus of the study is molecules consisting of T↓, D↓, ^{3}He, ^{4}He, and alkali atoms, where the interaction between particles is much better known than in the case of nuclei, which are traditional examples of quantum halos. The study of realistic systems is supplemented by model calculations in order to analyze how low-energy properties depend on the interaction potential. The use of variational and diffusion Monte Carlo methods enabled a very precise calculation of both the size and binding energy of the trimers. In the quantum halo regime, and for large values of scaled binding energies, all clusters follow almost the same universal line. As the scaled binding energy decreases, Borromean states separate from tango trimers.
The usefulness of a scale-independent approach to identify Efimov states in three-body systems is shown by comparing such an approach with a realistic calculation in the case of three helium atoms. We show that the scaling limit is realized in practice in this case, and suggest its application to study other similar systems, including the case where two kinds of atoms are mixed. We also consider the observed large scattering length of the87Rb\mathrm{the}{ }^{87}\mathrm{Rb}the87Rb dimer to estimate the critical value of the ground-state energy of the corresponding trimer (> 1.5mK)(>~1.5 \mathrm{mK})(> 1.5mK), in order to allow for one Efimov state above the ground state.
The experimental three-atom recombination coefficients of the atomic states ${}^{23}\mathrm{Na}|{F=1,m}_{F}=\ensuremath{-}1〉,$ ${}^{87}\mathrm{Rb}|{F=1,m}_{F}=\ensuremath{-}1〉,$ and ${}^{85}\mathrm{Rb}|{F=2,m}_{F}=\ensuremath{-}2〉,$ together with the corresponding two-body scattering lengths, allow predictions of the trimer bound-state energies for such systems in a trap. The recombination parameter is given as a function of the weakly bound trimer energies, which are in the interval $1<m(a/\ensuremath{\Elzxh}{)}^{2}{E}_{3}<6.9$ for large positive scattering lengths a.a.a. The contribution of a deep-bound state to our prediction, in the case of ${}^{85}\mathrm{Rb}|{F=2,m}_{F}=\ensuremath{-}2〉,$ for a particular trap, is shown to be relatively small.
We calculate root-mean-square radii for a three-body system confined to two spatial dimensions and consisting of two identical bosons (AAA) and one distinguishable particle (BBB). We use zero-range two-body interactions between each of the pairs, and focus thereby directly on universal properties. We solve the Faddeev equations in momentum space and express the mean-square radii in terms of first-order derivatives of the Fourier transforms of densities. The strengths of the interactions are adjusted for each set of masses to produce equal two-body bound-state energies between different pairs. The mass ratio, A=mB/mA\mathcal{A}={m}_{B}/{m}_{A}A=mB​/mA​, between particles BBB and AAA are varied from 0.01 to 100, providing a number of bound states decreasing from 8 to 2. Energies and mean-square radii of these states are analyzed for small A\mathcal{A}A by use of the Born-Oppenheimer potential between the two heavy AAA particles. For large A\mathcal{A}A the radii of the two bound states are consistent with a slightly asymmetric three-body structure. When A\mathcal{A}A approaches thresholds for binding of the three-body excited states, the corresponding mean-square radii diverge inversely proportional to the deviation of the three-body energy from the two-body thresholds. The structures at these three-body thresholds correspond to bound ABABAB dimers and one loosely bound AAA particle.