Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Possibility of a /sup 4/He/sub 2/ bound state, effective range theory, and very low energy He--He scattering

Yea-Hwang Uang, W. C. Stwalley

Published May 15, 1982• 0 citations
OpenAlex

Abstract

The best available intermolecular potential for helium by Aziz, Nain, Carley, Taylor, and McConville is shown here for the first time to have a /sup 4/He/sub 2/ bound state. Two numerical analyses, namely, eigenvalue solution and effective range theory, are used to support this conclusion. Unlike usual chemically bound species, the binding energy of this very weakly bound level is found to be only 8.3 x 10/sup -4/ K, which is four orders of magnitude smaller than the potential well depth epsilon = 10.8 K. The scattering length for He+He collisions, determined from effective range theory, is used to calculate the elastic cross section in the very low energy limit. The results (1.878 x 10/sup 5/ A/sup 2/ for /sup 4/He+/sup 4/He and 6.035 x 10/sup 2/ A for /sup 3/He+/sup 3/He) are consistent with measurements at the lowest velocities yet attained. In terms of the estimated uncertainties of the parameters of the potential of Aziz and co-workers, it is shown that it is very likely that a bound state of the /sup 4/He/sub 2/ molecule does in fact exist.

Related papers

The possibility of a 4He2 bound state, effective range theory, and very low energy He–He scattering

Yea-Hwang Uang, William C. Stwalley

The best available intermolecular potential for helium by Aziz, Nain, Carley, Taylor, and McConville is shown here for the first time to have a 4He2 bound state. Two numerical analyses, namely, eigenvalue solution and effective range theory, are used to support this conclusion. Unlike usual chemically bound species, the binding energy of this very weakly bound level is found to be only 8.3×10−4 K, which is four orders of magnitude smaller than the potential well depth ε = 10.8 K. The scattering length for He+He collisions, determined from effective range theory, is used to calculate the elastic cross section in the very low energy limit. The results (1.878×105 Å2 for 4He+4He and 6.035×102 Å for 3He+3He) are consistent with measurements at the lowest velocities yet attained. In terms of the estimated uncertainties of the parameters of the potential of Aziz and co-workers, it is shown that it is very likely that a bound state of the 4He2 molecule does in fact exist.

The Journal of Chemical Physics•May 15, 1982•66 citations

Scattering of cold He4 on He4Li6,7 and He4Na23 molecules

M. A. Shalchi, A. Delfino, T. Frederico +1 more

We predict $s$-wave elastic cross-sections $\ensuremath{\sigma}$ for low-energy atom-molecule collisions with kinetic energies up to 40 mK for the $^{4}\mathrm{He}$ collision with weakly bound diatomic molecules formed by $^{4}\mathrm{He}$ with $^{7}\mathrm{Li},\phantom{\rule{0.28em}{0ex}}^{6}\mathrm{Li}$, and $^{23}\mathrm{Na}$. Our scattering calculations are performed by using diatomic and triatomic molecular binding energies obtained from several available realistic models as input in a renormalized zero-range model as well as a finite-range one-term separable potential in order to quantify the relevance of range corrections to our predictions. Of particular relevance for possible experimental realization, we show the occurrence of a zero in $\ensuremath{\sigma}$ for the collision of cold $^{4}\mathrm{He}$ on a $^{4}\mathrm{He}\phantom{\rule{0.16em}{0ex}}^{23}\mathrm{Na}$ molecule below 20 mK. Also our results for the elastic collision $^{4}\mathrm{He}$ on $^{4}\mathrm{He}\phantom{\rule{0.16em}{0ex}}^{6,7}\mathrm{Li}$ molecules suggest that $\ensuremath{\sigma}$ varies considerably for the realistic models studied. As the chosen molecules are weakly bound and the scattering energies are very low, our results are interpreted on the light of the Efimov physics, which explains the model independence and robustness of our predictions despite some sensitivity on the potential range.

arXiv:1809.09140•Physical Review A•Sep 24, 2018•4 citations

Modern He–He potentials: Another look at binding energy, effective range theory, retardation, and Efimov states

Alec R. Janzen, R. A. Aziz

We compare a number of helium–helium potentials with respect to their predictions of dimer binding energy, scattering length, effective range and Efimov states. We also study the effect of retardation on the ‘‘best’’ potential. All realistic potentials support a weakly bound dimer, while none supports an Efimov state. We agree with other authors that retardation decreases the binding energy by about 10%. Finally, we investigated the effect on the binding energy from the application of retardation over different ranges of separation. The precise effects of retardation at short range in realistic potentials require further study.

The Journal of Chemical Physics•Dec 8, 1995•90 citations

On the Quantum Theory of the Third Virial Coefficient

A. Pais, G. E. Uhlenbeck

The quantum theory of the third virial coefficient $C$ is discussed. Three types of intermolecular pair forces must be distinguished. (1) No bound or low-lying two- and/or three-body states exist. The first four terms of the low-temperature expansion of ${C}_{\mathrm{BE}}$ are obtained. They depend on the scattering length, the effective range, and a third length which cannot be inferred from scattering data. The limitations of the applicability of such expansions are discussed, both for ${\mathrm{He}}^{4}$ and ${\mathrm{He}}^{3}$, by means of a comparison of the corresponding expansion for the second virial coefficient $B$ with detailed numerical results known for specific potentials. (2) Existence of a near zero energy level both for the two- and the three-body system. It is shown how in this case the actual potentials may be replaced by suitably matched boundary conditions on the two- and the three-body wave functions near the respective coordinate origins. It is first explained how the method applies to $B$. Then the leading term of $C$ is explicitly determined. (3) Existence of strongly bound two- and three-body states. An approximate expression for $C$ is given by treating the single atoms and the binary and ternary compounds as a system of three ideal gases in chemical equilibrium.

Physical Review•Oct 15, 1959•88 citations

Zero-energy scattering and bound states in the He4 trimer and tetramer

S. Nakaichi-Maeda, T. K. Lim

Using the HFDHE2 He-He potential of Aziz et al. [J. Chem. Phys. 90, 4330 (1979)] with the Faddeev-Yakubovsky theory, we have calculated the $^{4}\mathrm{He}$ atom-triatom scattering length and found it to be - 116 \AA{}. We reduced the $s$-wave two-body $t$ matrix to a separable form through the unitary-pole approximation and did the same for the $s$-wave [3+1] and [2+2] subamplitudes with the energy-dependent-pole approximation in our work. Also, the binding energies of all the states in the trimer and tetramer, and the atom-diatom scattering length, already determined in our previous work with other potentials, were recalculated for the new interaction. We found an excited trimer bound state, apparently not an Efimov state, near the atom-diatom threshold and an atom-diatom scattering length of 195 \AA{}. A feature of our results was the absence of an excited state in the tetramer, but in its place, was a resonancelike state which led to the large atom-triatom scattering length.

Physical Review A•Aug 1, 1983•46 citations

Spectra of helium clusters with up to six atoms using soft-core potentials

M. Gattobigio, Alejandro Kievsky, M. Viviani

In this paper, we investigate small clusters of helium atoms using the hyperspherical harmonic basis. We consider systems with $A=2,3,4,5,6$ atoms with an interparticle potential which does not present a strong repulsion at short distances. We use an attractive Gaussian potential that reproduces the values of the dimer binding energy, the atom-atom scattering length, and the effective range obtained with one of the widely used He-He interactions, the Aziz and Slaman potential, called LM2M2. In systems with more than two atoms, we consider a repulsive three-body force that, by construction, reproduces the trimer binding energy of the LM2M2 potential. With this model, consisting of the sum of a two- and three-body potential, we have calculated the spectrum of clusters formed by four, five, and six helium atoms. We have found that these systems present two bound states, one deep and one shallow, close to the threshold fixed by the energy of the $(A\ensuremath{-}1)$-atom system. Universal relations between the energies of the excited state of the $A$-atom system and the ground-state energy of the $(A\ensuremath{-}1)$-atom system are extracted, as well as the ratio between the ground state of the $A$-atom system and the ground-state energy of the trimer.

arXiv:1106.3853•Physical Review A•Nov 4, 2011•61 citations
The best available intermolecular potential for helium by Aziz, Nain, Carley, Taylor, and McConville is shown here for the first time to have a /sup 4/He/sub 2/ bound state. Two numerical analyses, namely, eigenvalue solution and effective range theory, are used to support this conclusion. Unlike usual chemically bound species, the binding energy of this very weakly bound level is found to be only 8.3 x 10/sup -4/ K, which is four orders of magnitude smaller than the potential well depth epsilon = 10.8 K. The scattering length for He+He collisions, determined from effective range theory, is used to calculate the elastic cross section in the very low energy limit. The results (1.878 x 10/sup 5/ A/sup 2/ for /sup 4/He+/sup 4/He and 6.035 x 10/sup 2/ A for /sup 3/He+/sup 3/He) are consistent with measurements at the lowest velocities yet attained. In terms of the estimated uncertainties of the parameters of the potential of Aziz and co-workers, it is shown that it is very likely that a bound state of the /sup 4/He/sub 2/ molecule does in fact exist.
The best available intermolecular potential for helium by Aziz, Nain, Carley, Taylor, and McConville is shown here for the first time to have a 4He2 bound state. Two numerical analyses, namely, eigenvalue solution and effective range theory, are used to support this conclusion. Unlike usual chemically bound species, the binding energy of this very weakly bound level is found to be only 8.3×10−4 K, which is four orders of magnitude smaller than the potential well depth ε = 10.8 K. The scattering length for He+He collisions, determined from effective range theory, is used to calculate the elastic cross section in the very low energy limit. The results (1.878×105 Å2 for 4He+4He and 6.035×102 Å for 3He+3He) are consistent with measurements at the lowest velocities yet attained. In terms of the estimated uncertainties of the parameters of the potential of Aziz and co-workers, it is shown that it is very likely that a bound state of the 4He2 molecule does in fact exist.
We predict sss-wave elastic cross-sections $\ensuremath{\sigma}$ for low-energy atom-molecule collisions with kinetic energies up to 40 mK for the 4He^{4}\mathrm{He}4He collision with weakly bound diatomic molecules formed by 4He^{4}\mathrm{He}4He with 7Li,6Li^{7}\mathrm{Li},\phantom{\rule{0.28em}{0ex}}^{6}\mathrm{Li}7Li,6Li, and 23Na^{23}\mathrm{Na}23Na. Our scattering calculations are performed by using diatomic and triatomic molecular binding energies obtained from several available realistic models as input in a renormalized zero-range model as well as a finite-range one-term separable potential in order to quantify the relevance of range corrections to our predictions. Of particular relevance for possible experimental realization, we show the occurrence of a zero in $\ensuremath{\sigma}$ for the collision of cold 4He^{4}\mathrm{He}4He on a 4He23Na^{4}\mathrm{He}\phantom{\rule{0.16em}{0ex}}^{23}\mathrm{Na}4He23Na molecule below 20 mK. Also our results for the elastic collision 4He^{4}\mathrm{He}4He on 4He6,7Li^{4}\mathrm{He}\phantom{\rule{0.16em}{0ex}}^{6,7}\mathrm{Li}4He6,7Li molecules suggest that $\ensuremath{\sigma}$ varies considerably for the realistic models studied. As the chosen molecules are weakly bound and the scattering energies are very low, our results are interpreted on the light of the Efimov physics, which explains the model independence and robustness of our predictions despite some sensitivity on the potential range.
We compare a number of helium–helium potentials with respect to their predictions of dimer binding energy, scattering length, effective range and Efimov states. We also study the effect of retardation on the ‘‘best’’ potential. All realistic potentials support a weakly bound dimer, while none supports an Efimov state. We agree with other authors that retardation decreases the binding energy by about 10%. Finally, we investigated the effect on the binding energy from the application of retardation over different ranges of separation. The precise effects of retardation at short range in realistic potentials require further study.
The quantum theory of the third virial coefficient CCC is discussed. Three types of intermolecular pair forces must be distinguished. (1) No bound or low-lying two- and/or three-body states exist. The first four terms of the low-temperature expansion of CBE{C}_{\mathrm{BE}}CBE​ are obtained. They depend on the scattering length, the effective range, and a third length which cannot be inferred from scattering data. The limitations of the applicability of such expansions are discussed, both for He4{\mathrm{He}}^{4}He4 and He3{\mathrm{He}}^{3}He3, by means of a comparison of the corresponding expansion for the second virial coefficient BBB with detailed numerical results known for specific potentials. (2) Existence of a near zero energy level both for the two- and the three-body system. It is shown how in this case the actual potentials may be replaced by suitably matched boundary conditions on the two- and the three-body wave functions near the respective coordinate origins. It is first explained how the method applies to BBB. Then the leading term of CCC is explicitly determined. (3) Existence of strongly bound two- and three-body states. An approximate expression for CCC is given by treating the single atoms and the binary and ternary compounds as a system of three ideal gases in chemical equilibrium.
Using the HFDHE2 He-He potential of Aziz et al. [J. Chem. Phys. 90, 4330 (1979)] with the Faddeev-Yakubovsky theory, we have calculated the 4He^{4}\mathrm{He}4He atom-triatom scattering length and found it to be - 116 \AA{}. We reduced the sss-wave two-body ttt matrix to a separable form through the unitary-pole approximation and did the same for the sss-wave [3+1] and [2+2] subamplitudes with the energy-dependent-pole approximation in our work. Also, the binding energies of all the states in the trimer and tetramer, and the atom-diatom scattering length, already determined in our previous work with other potentials, were recalculated for the new interaction. We found an excited trimer bound state, apparently not an Efimov state, near the atom-diatom threshold and an atom-diatom scattering length of 195 \AA{}. A feature of our results was the absence of an excited state in the tetramer, but in its place, was a resonancelike state which led to the large atom-triatom scattering length.
In this paper, we investigate small clusters of helium atoms using the hyperspherical harmonic basis. We consider systems with A=2,3,4,5,6A=2,3,4,5,6A=2,3,4,5,6 atoms with an interparticle potential which does not present a strong repulsion at short distances. We use an attractive Gaussian potential that reproduces the values of the dimer binding energy, the atom-atom scattering length, and the effective range obtained with one of the widely used He-He interactions, the Aziz and Slaman potential, called LM2M2. In systems with more than two atoms, we consider a repulsive three-body force that, by construction, reproduces the trimer binding energy of the LM2M2 potential. With this model, consisting of the sum of a two- and three-body potential, we have calculated the spectrum of clusters formed by four, five, and six helium atoms. We have found that these systems present two bound states, one deep and one shallow, close to the threshold fixed by the energy of the $(A\ensuremath{-}1)$-atom system. Universal relations between the energies of the excited state of the AAA-atom system and the ground-state energy of the $(A\ensuremath{-}1)$-atom system are extracted, as well as the ratio between the ground state of the AAA-atom system and the ground-state energy of the trimer.