_{2}He ground triplet potential energy surface and quantum scattering calculations

^{1,a)}, Alexandra Viel

^{1,b)}and Jean-Michel Launay

^{1,c)}

### Abstract

We have developed a three-dimensional potential energy surface for the lowest triplet state of the Rb_{2}He complex. A global analytic fit is provided as in the supplementary material [see supplementary material at http://dx.doi.org/10.1063/1.4709433E-JCPSA6-136-034218 for the corresponding Fortran code]. This surface is used to perform quantum scattering calculations of ^{4}He and ^{3}He colliding with ^{87}Rb_{2} in the partial wave *J* = 0 at low and ultralow energies. For the heavier helium isotope, the computed vibrational relaxation probabilities show a broad and strong shape resonance for a collisional energy of 0.15 K and a narrow Feshbach resonance at about 17 K for all initial Rb_{2}vibrational states studied. The broad resonance corresponds to an efficient relaxation mechanism that does not occur when ^{3}He is the colliding partner. The Feshbach resonance observed at higher collisional energy is robust with respect to the isotopic substitution. However, its effect on the vibrational relaxation mechanism is faint for both isotopes.

This work is supported by the Région Bretagne via the project CREATE “4023-HELIUM” and by the “DYNHELIUM” ANR project. Pavel Soldán is warmfully thanked for fruitful discussions.

I. INTRODUCTION

II. FULL-DIMENSIONAL INTERACTION POTENTIAL

A. Electronic structure calculations

B. Analytical fit of the potential energy surface

C. Potential energy surface

III. QUANTUM SCATTERING CALCULATIONS

IV. CONCLUSION

### Key Topics

- Helium-3
- 15.0
- Helium-4
- 15.0
- Potential energy surfaces
- 14.0
- Isotopes
- 10.0
- Rubidium
- 10.0

## Figures

Cuts of the potential energy surface at *r* = 6.117 Å for γ = 0° (black), 30° (green), 60° (blue), and γ = 90° (red) in Kelvin as a function of *R*. The full potential surface is presented in full lines while the restriction to the two-body expansion is shown in dashed lines. The zero of energy corresponds to the separated atoms limit.

Cuts of the potential energy surface at *r* = 6.117 Å for γ = 0° (black), 30° (green), 60° (blue), and γ = 90° (red) in Kelvin as a function of *R*. The full potential surface is presented in full lines while the restriction to the two-body expansion is shown in dashed lines. The zero of energy corresponds to the separated atoms limit.

Contour plots of potential energy surfaces averaged over the *v* = 0 (left panels) and the *v* = 4 (right panels) vibrational states of Rb_{2}. The helium atom is localized by its Cartesian coordinates assuming that Rb_{2} lies along the *x* axis. The contour lines are with respect to the He + Rb_{2} dissociation limit. The contour line spacing is 0.5 K for the upper panel and 0.3 K for the lower panel which focuses on the potential well region.

Contour plots of potential energy surfaces averaged over the *v* = 0 (left panels) and the *v* = 4 (right panels) vibrational states of Rb_{2}. The helium atom is localized by its Cartesian coordinates assuming that Rb_{2} lies along the *x* axis. The contour lines are with respect to the He + Rb_{2} dissociation limit. The contour line spacing is 0.5 K for the upper panel and 0.3 K for the lower panel which focuses on the potential well region.

Vibrational relaxation probability for ^{4}He as a function of the collision energy in Kelvin for initial rovibrational states (*v* _{ i }, *j* = 0) with *v* _{ i } = 1, 2, 3, and 4.

Vibrational relaxation probability for ^{4}He as a function of the collision energy in Kelvin for initial rovibrational states (*v* _{ i }, *j* = 0) with *v* _{ i } = 1, 2, 3, and 4.

Vibrational relaxation probability for ^{4}He as a function of collision energy in Kelvin for initial rovibrational states (*v* _{ i }, *j* _{ i } = 0) with *v* _{ i } = 1, 2, 3, and 4. This figure presents two blow-ups of the data presented in Fig. 3 in the energy ranges of the two resonance features. The openings of the (*v* _{ i } + 1, *j* = 0) channels are materialized by the vertical black lines in the right panel.

Vibrational relaxation probability for ^{4}He as a function of collision energy in Kelvin for initial rovibrational states (*v* _{ i }, *j* _{ i } = 0) with *v* _{ i } = 1, 2, 3, and 4. This figure presents two blow-ups of the data presented in Fig. 3 in the energy ranges of the two resonance features. The openings of the (*v* _{ i } + 1, *j* = 0) channels are materialized by the vertical black lines in the right panel.

Vibrational relaxation probability for *v* _{ i } = 1 as a function of collision energy in K. The black full line corresponds to ^{4}He colliding with Rb_{2} using the full potential. The red dashed line presents the effect of the isotopic substitution of ^{4}He by ^{3}He on the full potential. The blue pointed line is obtained when ^{4}He collides with Rb_{2} on the two-body potential surface.

Vibrational relaxation probability for *v* _{ i } = 1 as a function of collision energy in K. The black full line corresponds to ^{4}He colliding with Rb_{2} using the full potential. The red dashed line presents the effect of the isotopic substitution of ^{4}He by ^{3}He on the full potential. The blue pointed line is obtained when ^{4}He collides with Rb_{2} on the two-body potential surface.

Elastic and inelastic *J* = 0 rate coefficients for ^{4}He + Rb_{2} collisions with the initial rovibrational state *v* _{ i } = 0, …, 4, *j* = 0 of Rb_{2} as a function of the collisional energy in Kelvin.

Elastic and inelastic *J* = 0 rate coefficients for ^{4}He + Rb_{2} collisions with the initial rovibrational state *v* _{ i } = 0, …, 4, *j* = 0 of Rb_{2} as a function of the collisional energy in Kelvin.

Rotational distribution of the (*v* _{ i } = 1, *j* _{ i } = 0) → (*v* _{ f } = 0, *j* _{ f }) transition for three collision energies, (a) 10^{−6} K, (b) 0.15 K, and (c) 2 K for both ^{4}He (black circles) and ^{3}He (red triangle).

Rotational distribution of the (*v* _{ i } = 1, *j* _{ i } = 0) → (*v* _{ f } = 0, *j* _{ f }) transition for three collision energies, (a) 10^{−6} K, (b) 0.15 K, and (c) 2 K for both ^{4}He (black circles) and ^{3}He (red triangle).

## Tables

Long-range coefficients taken from Ref. 32 for RbHe and from Refs. 33 and 34 for Rb_{2}.

Long-range coefficients taken from Ref. 32 for RbHe and from Refs. 33 and 34 for Rb_{2}.

Scattering length as defined in Eq. (6) for ^{4}He and ^{3}He colliding with Rb_{2} in various vibrational states (*v* _{ i }, *j* _{ i } = 0). Values obtained when the three-body term of the potential is removed are also presented (2B).

Scattering length as defined in Eq. (6) for ^{4}He and ^{3}He colliding with Rb_{2} in various vibrational states (*v* _{ i }, *j* _{ i } = 0). Values obtained when the three-body term of the potential is removed are also presented (2B).

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