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Physical Review Letters

Stability of Small Mixed Clusters of H4e and H3e Atoms

R. Guardiola, Jesús Navarro

Published Oct 17, 2002• 17 citations
Publisher

Abstract

The existence of small helium clusters containing a variable number of $^{\mathrm{4}}\mathrm{H}\mathrm{e}$ and $^{\mathrm{3}}\mathrm{H}\mathrm{e}$ atoms is studied within a variational Monte Carlo calculation employing the Aziz HFD-B(HE) pair interaction. The clusters ${^{\mathrm{4}}\mathrm{H}\mathrm{e}}_{2}$ support one and two $^{\mathrm{3}}\mathrm{H}\mathrm{e}$ atoms; however, the system with three $^{\mathrm{3}}\mathrm{H}\mathrm{e}$ atoms is metastable, and the next bound system requires at least 18 fermions. All clusters obtained by adding $^{\mathrm{3}}\mathrm{H}\mathrm{e}$ atoms to the trimer ${^{\mathrm{4}}\mathrm{H}\mathrm{e}}_{3}$ and the tetramer ${^{\mathrm{4}}\mathrm{H}\mathrm{e}}_{4}$ are bound, but the clusters ${^{\mathrm{4}}\mathrm{H}\mathrm{e}}_{3}$-${^{\mathrm{3}}\mathrm{H}\mathrm{e}}_{3,4,5}$ and ${^{\mathrm{4}}\mathrm{H}\mathrm{e}}_{3,4}$-${^{\mathrm{3}}\mathrm{H}\mathrm{e}}_{9}$ are metastable. All remaining clusters with three or more bosons and any number of fermions are stable.

DOI
10.1103/physrevlett.89.193401

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The existence of small helium clusters containing a variable number of 4He^{\mathrm{4}}\mathrm{H}\mathrm{e}4He and 3He^{\mathrm{3}}\mathrm{H}\mathrm{e}3He atoms is studied within a variational Monte Carlo calculation employing the Aziz HFD-B(HE) pair interaction. The clusters 4He2{^{\mathrm{4}}\mathrm{H}\mathrm{e}}_{2}4He2​ support one and two 3He^{\mathrm{3}}\mathrm{H}\mathrm{e}3He atoms; however, the system with three 3He^{\mathrm{3}}\mathrm{H}\mathrm{e}3He atoms is metastable, and the next bound system requires at least 18 fermions. All clusters obtained by adding 3He^{\mathrm{3}}\mathrm{H}\mathrm{e}3He atoms to the trimer 4He3{^{\mathrm{4}}\mathrm{H}\mathrm{e}}_{3}4He3​ and the tetramer 4He4{^{\mathrm{4}}\mathrm{H}\mathrm{e}}_{4}4He4​ are bound, but the clusters 4He3{^{\mathrm{4}}\mathrm{H}\mathrm{e}}_{3}4He3​-3He3,4,5{^{\mathrm{3}}\mathrm{H}\mathrm{e}}_{3,4,5}3He3,4,5​ and 4He3,4{^{\mathrm{4}}\mathrm{H}\mathrm{e}}_{3,4}4He3,4​-3He9{^{\mathrm{3}}\mathrm{H}\mathrm{e}}_{9}3He9​ are metastable. All remaining clusters with three or more bosons and any number of fermions are stable.
A stability chart of mixed 4He{}^{4}\mathrm{He}4He and 3He{}^{3}\mathrm{He}3He clusters has been obtained by means of the diffusion Monte Carlo method, using both the Aziz HFD-B and the Tang-Toennies-Yiu atom-atom interaction. The investigated clusters contain up to eight 4He{}^{4}\mathrm{He}4He atoms and up to 20 3He{}^{3}\mathrm{He}3He atoms. One single 4He{}^{4}\mathrm{He}4He binds 20 3He{}^{3}\mathrm{He}3He atoms, and two 4He{}^{4}\mathrm{He}4He bind 1, 2, 8, and more than 14 3He{}^{3}\mathrm{He}3He atoms. All clusters with three or more 4He{}^{4}\mathrm{He}4He atoms are bound, although the combinations 4He33He9,10,11{}^{4}{\mathrm{He}}_{3}^{3}{\mathrm{He}}_{9,10,11}4He33​He9,10,11​ and 4He43He9{}^{4}{\mathrm{He}}_{4}^{3}{\mathrm{He}}_{9}4He43​He9​ are metastable. Clusters with 2, 8, and 20 3He{}^{3}\mathrm{He}3He atoms are particularly stable and define magic 3He{}^{3}\mathrm{He}3He numbers.
We have studied the stability of mixed 3He/4He^{3}\mathrm{H}\mathrm{e}/^{4}\mathrm{H}\mathrm{e}3He/4He clusters in L=0L=0L=0 states by the diffusion Monte Carlo method, employing the Tang-Toennies-Yiu He-He potential. The clusters 3He4HeN^{3}\mathrm{H}\mathrm{e}^{4}\mathrm{H}\mathrm{e}_{N}3He4HeN​ and 3He24HeN^{3}\mathrm{H}\mathrm{e}_{2}^{4}\mathrm{H}\mathrm{e}_{N}3He24​HeN​ are stable for N>1N>1N>1. The lighter atoms tend to move to the surface of the cluster. The minimum number of 4He^{4}\mathrm{H}\mathrm{e}4He atoms able to bind three 3He^{3}\mathrm{H}\mathrm{e}3He atoms in a L=0L=0L=0 state is nine. Two of three fermionic helium atoms stay on the surface, while the third one penetrates into the cluster.
We report results for the ground-state energy and structural properties of small (4)He-T↓ clusters consisting of up to four T↓ and eight (4)He atoms. These results have been obtained using very well-known (4)He-(4)He and T↓- T↓ interaction potentials and several models for the (4)He- T↓ interatomic potential. All the calculations have been performed with variational and diffusion Monte Carlo methods. It takes at least three atoms to form a mixed bound state. In particular, for small clusters the binding energies are significantly affected by the precise form of the (4)He- T↓ interatomic potential but the stability limits remain unchanged. The only exception is the (4)He(2)T↓ trimer whose stability in the case of the weakest (4)He- T↓ interaction potential is uncertain while it seems stable for other potentials. The mixed trimer (4)He(T↓)(2), a candidate for the Borromean state, is not bound. All other studied clusters are stable. Some of the weakest bound clusters can be classified as quantum halo as a consequence of having high probability of being in a classically forbidden region.
Small helium (4He) clusters containing the lighter isotope He3 are studied by means of quantum Monte Carlo methods. Accurate ground state energies and structural properties are obtained using accurate trial wave functions and the Tang–Tonnies–Yiu (TTY) helium–helium pair potential. The dimer He4–3He is not bound; as well as the trimer He4He23. The smallest cluster containing He3 is He24He3 with a nonrigid structure having a marked linear contribution. Interestingly, this weakly bound system, with an energy one order of magnitude less than the He34 trimer, is able to bind another He3 atom, forming the tetramer He24He23, which shows the odd feature of having five out of six unbound pairs. In general, the substitution of a single He4 atom in a pure cluster with a He3 atom leads to an energetic destabilization, as the pair He4–3He is not bound. The isotopic impurity is found to perturb only weakly the distributions of the remaining He4 atoms, which retain the high floppiness already found in the pure clusters. As the number of atoms increases the isotopic impurity has the marked tendency to stay on the surface of the cluster. This behavior is consistent with the formation of the so-called “Andreev states” of a single He3 in liquid He4 helium and droplets, where the impurity tends to form single-particle states on the surface of the pure He4.
We report accurate ground state energies and structural properties for small clusters of 4He computed with the diffusion quantum Monte Carlo (DMC) method combined with high quality trial wave functions and using the recent analytical pair potential of Tang, Toennies, and Yiu [Phys. Rev. Lett. 74, 1546 (1995)]. Calculations based on the older HFD-B(He) potential are reported for comparison. The clusters are found to be extremely floppy and to be characterized by very diffuse wave functions. The DMC results for 4He2 and 4He3 are in excellent agreement with other calculations using conventional methods. 4He3 is found to have a noticeable contribution from nearly linear geometries. The internal structure of the clusters is described by a three particle correlation function which reveals a significantly non-spherical internal cluster structure. The energies for all cluster sizes are found to be slightly higher than those obtained with the HFD-B(He) pair potential. Exploratory calculations on the helium trimer indicate that the effects of three body interactions do not exceed the uncertainty margin provided by different state of the art pair potentials.