The minimal number of 3He atoms required to form a bound cluster has been
estimated by means of a Diffusion Monte Carlo procedure within the fixed-node
approximation. Several importance sampling wave functions have been employed in
order to consider different shell-model configurations. The resulting upper
bound for the minimal number is 32 atoms.Comment: 2 pages, no figure

Guardiola, R.

Navarro, J.

English

arXiv

The minimal number of 3He atoms required to form a bound cluster has been estimated by means of a Diffusion Monte Carlo procedure within the fixed-node approximation. Several importance sampling wave functions have been employed in order to consider different shell-model configurations. The resulting upper bound for the minimal number is 32 atoms

Guardiola Barcena, Rafael

Navarro Faus, Jesús

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

arXiv:physics/0408127v1  [physics.atm-clus]  30 Aug 2004Minimal mass-size of a stable3He clusterR. Guardiola∗Institute de Recherches Subatomiques, IN2P3-CNRS/Universite´ Louis Pasteur, F-67037 Strasbourg, FranceJ. NavarroIFIC (CSIC-Universidad de Valencia), Apartado Postal 22085, E-46071-Valencia, SpainThe minimal number of 3He atoms required to form a bound cluster has been estimated by meansof a Diffusion Monte Carlo procedure within the fixed-node approximation. Several importancesampling wave functions have been employed in order to consider different shell-model configurations.The resulting upper bound for the minimal number is 32 atoms.PACS numbers: 36.40.-c 61.46.+wHelium clusters are unique systems for studying the properties of finite sized quantum objects. The first systematicmicroscopic calculation of their ground state properties was carried out by Pandharipande and coworkers [1] eighteenyears ago. One interesting finding is that, contrary to the 4He case, a minimum number of 3He atoms is requiredto create a stable cluster, as a consequence of the large zero-point motion and the Pauli effect. The calculations ofRef. [1] were based on an optimized trial wave function, where the fermions are assumed to fill harmonic oscillatorsingle-particle states 1s21p61d102s21f142p6 . . . The outcome was that twenty atoms are not enough to form a boundsystem, but forty atoms are bound, and in between a critical or threshold number exists for having stable clusters.From the experimental point of view, there is only indirect evidence of such a critical number, since the smaller 3Hesystems detected insofar contains thousand of atoms.The critical number was calculated in Ref. [2], by using a non-local finite-range density functional approach. Theusual Kohn-Sham procedure provided the residual interaction required to perform a configuration interaction calcu-lation. The obtained critical number was 29 atoms. Afterwards, we carried out [3, 4] a variational microscopic studybased on the two-body He-He interaction known as HFD-B(He) interaction [5], which reproduces accurately the prop-erties of both 4He and 3He liquid. The trial wave function contained a Jastrow-correlated part and a self-adjustabletranslationally invariant configuration interaction part, including up to three particle-hole excitations. Due to thevariational character of the computational procedure, only an upper bound was obtained, and we concluded that thecritical number is less or equal to 34 atoms.In order to improve this estimate we have carried out several calculations within the fixed-node Diffusion MonteCarlo (DMC) procedure [6, 7], for selected systems near the previously determined critical number. The importance-sampling wave functions have been constructed in a similar way to the variational forms previously used in Refs. [3, 4],but with some modifications. First of all, the confining part of the two-body Jastrow correlation used here has anexponential shape, instead of a gaussian one. Because of the longer range of the exponential shape one may expectto be more appropriate for systems near the binding threshold. Apart from this, the same backflow correlation wasused, but the configuration interaction part has not been included, with the objetive of having a fast difussion MonteCarlo algorithm.The other relevant modification is related to the antisymmetric part of the wave function. The fermionic antisym-metry is considered by means of the product of two Slater determinants, one for each spin orientation. In the presentcalculation we have assumed several shell-model orderings for these determinantal parts, and not simply the orderingrelated to the harmonic-oscillator single-particle potential previously considered. Actually, nothing is known aboutthe ordering of shells. The only indirect information comes from the study of a single 3He atom tied to a core of 4Heatoms, where calculations indicate an order based on the orbital angular momentum, namely 1s 1p 1d 1f . . ., withprobably the 2s level being interleaved between the n = 1 levels [8, 9]. Therefore, we have considered configurationswith a fixed occupancy 1s2 1p6 1d10, the N = 18 major shell, and several distributions of the remaining particlesbetween the 2s and the 1f shell, which are classified by the value of the total spin S.The actual calculations have been carried out with a time slice τ = 0.00025 K−1, with an average population of1000 walkers and for 10000 time steps, plus 1000 previous steps in order to reach the stability. To obtain an estimateof the variance, the sampling steps were grouped into blocks of 100 moves.The obtained results are presented in Table I. The N = 34 cluster appears clearly bound, as well as one of theconfigurations chosen for N = 33. This last result has the interest of suggesting the 2s shell to be deeper bound than∗On leave of absence from Departamento de F´ısica Ato´mica y Nuclear, Facultad de F´ısica, E-46100-Burjassot, Spain2TABLE I: Ground state energies E (in K) obtained for several clusters near the critical binding number. For a given numberN of atoms, the shell-model configuration and the value of the total spin S are indicated.N Config. S E (K)31 2s21f11 3/2 Unbound31 2s21f11 1/2 Unbound32 2s21f12 0 −0.27 ± 0.0332 2s21f12 1 −0.23 ± 0.0332 2s01f14 0 Unbound33 2s11f14 1/2 Unbound33 2s21f13 1/2 −0.86 ± 0.0434 2s21f14 0 −1.52 ± 0.04the 1f shell.Regarding the N = 32 cluster we had an initial guess in favor of the configuration 1s0 1f14, because it wouldcorrespond to a closed shell. However the actual calculations show that this configuration does not result in a boundstate, preferring instead to complete the 2s shell. This is not surprising once it has been established the shell orderingin the N = 33 case. Note that the total spin of the configuration 2s21f12 can have the two values S = 0 and 1. Takinginto account the statistical errors, one may conclude that these states are bound with the same binding energy. Eitherthese states are degenerate or the difference in energy is smaller than our statistical errors, i.e. of the order of tenmK. Given that the obtained energies are very close to zero, the imaginary time evolution was carried out for thesetwo cases for as much as 40000 time steps.Finally we have considered the N = 31 cluster, with two spin states (S = 1/2 and S = 3/2) of the 2s21f11configuration. It turns out that neither of these states is bound, their energy being close to zero but positive.In conclusion, we have find N = 32 as an upper bound to the minimal mass-size of a stable 3He cluster. Thevarious configurations here considered correspond in practice to probe different nodal surfaces. It is worth stressingthat our results indicate that, within the computational statistical errors, the binding energy is independent of the spincoupling, depending only on the chosen configuration. Finally, it should also be mentioned that the separation energyfor N = 34 (0.66± 0.06 K) is almost the same as that of N = 33 (0.59± 0.05, 0.64± 0.05 K), thus suggesting a single-particle structure of these fermionic clusters, with a residual interaction compatible with zero. The determination ofthe precise critical number should require a calculation beyond the variational fixed-node approximation, and it isnot excluded that the result could depend on the He-He interaction employed in the practical calculation.AcknowledgmentsThis work has been supported by MCyT/FEDER (Spain), grant number BMF2001-0262 and GV (Spain), grantnumber GV2003-002. One of us (RG) acknowledges the IReS (Strasbourg) by his hospitality.[1] V. R. Pandharipande, S. C. Pieper, and R. B. Wiringa, Phys. Rev. B 34, 4571 (1986).[2] M. Barranco, J. Navarro and A. Poves, Phys. Rev. Lett. 78, 4729 (1997).[3] R. Guardiola and J. Navarro, Phys. Rev. Lett. 84, 1144 (2000).[4] R. Guardiola, Phys. Rev. B 62, 3416 (2000).[5] R.A. Aziz, F.R. McCourt and C.C.K. Wong, Mol. Phys. 61, 1487 (1987).[6] P.J. Reynolds, D.M. Ceperley, B.J. Alder and W.A. Lester Jr., J. Chem. Phys. 77, 5593 (1982).[7] J.W. Moskowitz, K.E. Schmidt, M.A. Lee and H.M. Kalos, J. Chem. Phys. 77, 349 (1982).[8] J. Navarro, A. Poves, M. Barranco and M. Pi, Phys. Rev. A 69, 023202 (2004).[9] S. Fantoni, R. Guardiola and J. Navarro, Phys. Rev. A 70, 023206 (2004).

Minimal mass size of a stable He-3 cluster

R. Guardiola

J. Navarro

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Minimal mass size of a stable
                    
                      
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Minimal mass-size of a stable 3He cluster

Minimal mass-size of a stable 3He cluster

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