Superfluidity in Hydrogen-Deuterium Mixed Clusters

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1 Journal of Low Temperature Physics - QFS2009 manuscript No. (will be inserted by the editor) Superfluidity in Hydrogen-Deuterium Mixed Clusters Soomin Shim Yongkyung Kwon Received: date / Accepted: date Abstract Path-integral Monte Carlo calculations have been performed on para-h 2 clusters mixed with several ortho-d 2 molecules, (D 2 ) ND (H 2 ) NH with 1 N D 5 and 10 N H 20, to study their superfluid behavior at low temperatures. It is found that heavier D 2 molecules are located near the center of the mixed cluster, and radial density distributions of D 2 and H 2 tend to separate from each other as the number of D 2 molecules increases. We have also found significant suppression of superfluidity at specific cluster sizes when compared to those of the neighboring sizes. This can be understood in terms of the magic number stability previously reported in hydrogen clusters. Finally we present the local superfluid density distributions in the mixed clusters, which shows uniform local superfluidity of H 2 except near D 2 molecules. Keywords hydrogen, deuterium, cluster, superfluidity, path-integral Monte Carlo PACS Ei Mr z 1 Introduction Both experimental and theoretical attempts to find superfluidity in systems other than helium isotopes have been focused on para-h 2, simply H 2 from now on, because of its light mass and the existence of a compound J = 0 boson ground state. Owing to strong interparticle interaction, however, bulk H 2 gets solidified before the condensation occurs, One approach to overcome this problem is sizing down the clusters. Path-integral Monte Carlo (PIMC) calculation of Sindzingre et al. [1] predicted that hydrogen clusters with less than 20 molecules would show superfluidity below 2 K. The first experimental evidence for hydrogen superfluidity was provided by infrared spectroscopic measurements on a linear oxygen carbonyl sulfide molecule surrounded by 14 to 16 H 2 molecules inside large helium droplets [2], whose results were confirmed theoretically by a subsequent PIMC calculation [3]. Division of Quantum Phases and Devices, School of Physics, Konkuk University, Seoul ,Korea Tel.: Fax: ykwon@konkuk.ac.kr

2 2 Since recent Raman spectroscopic measurement [4] on pure (H 2 ) N clusters revealed enhanced stabilities at specific cluster sizes, several theoretical investigations of using quantum Monte Carlo methods have been made on magic number behavior of hydrogen clusters. These studies show that hydrogen clusters, whether pure [5 8] or doped with a single impurity molecule [9, 10], are energetically stable and hydrogen superfluidity is significantly suppressed when the total number of particles is equal to specific values, compared to the clusters of the neighboring sizes. In the mean time two independent PIMC calculations reported different results about the local distribution of superfluidity in pure H 2 clusters. Khairallah et al. claimed that hydrogen superfluidity is most dominant at the cluster surface and minimal in the inner shell region [8]. However, Mezzacapo and Boninsegni showed through their own PIMC calculation that superfluidity is not confined at the surface of the hydrogen cluster and rather displayed uniformly [11]. In this paper we present our PIMC results on H 2 clusters doped with several ortho-d 2 (or D 2 ) molecules, (D 2 ) ND (H 2 ) NH. Mezzacapo and Boninsegni studied these mixed clusters as well, and found that hydrogen clusters of more than 22 molecules became solidlike and experienced drastic suppression of superfluidity by the replacement of one or two molecules with heavier D 2 [12]. The calculation of Mezzacapo and Boninsegni was done with the PIMC method based on the recently developed continuous-space worm algorithm [13], and focused on structural changes in the mixed clusters including more than twenty molecules. On the other hand, we employ a conventional PIMC method whose details are well summarized in Ref. [14], for a systematic study of the changes in the low-temperature properties of these clusters with the numbers of H 2 and D 2 molecules varied between 10 N H 20 and 1 N D 5. We also compute local distributions of hydrogen superfluidity in the mixed clusters by using the local superfluid density estimator based on the microscopic two-fluid model [15], which provides an answer to a controversy about the local distribution of H 2 superfluidity mentioned above. 2 Path-integral Monte Carlo In this study we treat H 2 and D 2 molecules as spherical particles interacting with each other through pairwise interactions. The spherical part of the empirical potential proposed by Buck et al. [16] is used for all pair potentials between different hydrogen isotopes. Therefore the only difference between H 2 and D 2 is that the latter has twice as heavy a mass as the former and has smaller quantum fluctuation effects. In the path-integral representation, the thermal density matrix at a low temperature T is factored into L high-temperature density matrices with time step τ = (Lk B T ) 1. For the high-temperature density matrix we employ the pair-product form of the exact two-body density matrices [14] with τ 1 /k B = 40 K, whose accuracy for the H 2 -H 2 interaction was established in our previous PIMC study of OCS(H 2 ) N [3]. Within the Feynman path-integral formalism, the global superfluid fraction defined by the ratio of quantum and classical moment of inertia, can be expressed in terms of the projected area of the exchange-coupled paths onto a plane normal to a principal axis [14]. This estimator has non-negligible values only when the sizes of the exchange-coupled paths of Bose particles are comparable to the system size. The global estimator for superfluidity can be decomposed to give an estimator for the local distribution of superfluidity [17, 15]. One of us recently proposed the following local superfluid density estimator [15]: ρ s (r) α = 4mk BT A α A α (r) h 2 r 2, (1)

3 3 where r is the distance from the principle axis x α and Aα (r) a local contribution to the projected area Aα at position r. This estimator was shown to provide a consistent and rigorous analysis for the response of quantum fluids to the rotation of external field [15]. 3 Results Fig. 1 (color online) The global superfluid fractions of (D2 )ND (H2 )NH clusters at T = 1.6 K versus the total number of particles, Ntot = NH + ND. The data for ND = 0 are taken from Ref. [8], and the ND = 1 ones from Ref. [10]. The lines are just guides to eyes. Figure 1 shows the global superfluid fraction of (D2 )ND (H2 )NH at T = 1.6 K as a function of the total number of particles Ntot, with different numbers of D2 included. One can see that the superfluid fractions get noticeably reduced at all cluster sizes as more H2 molecules are replaced with D2 molecules. This suggests that the presence of D2 impurity molecules tend to interrupt exchange coupling among H2 molecules. Detailed discussion of impurity effects on superfluidity of Bose clusters can be found elsewhere [18, 19]. We also find that regardless of the number of D2 molecules, the clusters with Ntot = 13 and 19, which correspond to the magic sizes of pure hydrogen clusters [4,8], feature significantly suppressed superfluid fractions, compared to those of the neighboring sizes. This leads us to a conclusion that the replacement of H2 with D2 molecules in (D2 )ND (H2 )NH does not change the overall structure of the cluster and the magic number stability of pure (H2 )N holds true even in the mixed cluster. It is seen in Fig. 1 that the mixed clusters including as many as four D2 molecules show minimal superfluid fractions for Ntot = 12 to 20. For further confirmation of this we have computed superfluid fractions of bigger clusters with nineteen or twenty H2 molecules doped by one to five D2 molecules, at a lower temperature of T = 0.8 K. When added with four or more D2 molecules (ND 4), both NH = 19 and NH = 20 mixed clusters show totally quenched superfluid fractions. This suggests that impurity effects with four D2 molecules are strong enough to prohibit long exchange coupling among H2 molecules and to suppress their superfluidity completely. Near complete quenching of superfluidity is also found in the

4 4 Fig. 2 The D 2 (solid lines) and H 2 (dashed lines) radial density distributions of the mixed clusters at T = 1.6 K with respect to the centers of mass of the clusters. The left axes are for D 2 density scale and the right axes for H 2 density. The top panel represents the results for (D 2 ) 2 (H 2 ) 12 cluster and the bottom panel for (D 2 ) 2 (H 2 ) 17. The length and the density are in units of Å and Å 3, respectively. N H = 20 cluster with only three D 2 molecules. However, this is understood to be mostly due to the magic number stability at N tot = 23, because the cluster with one less H 2 molecules, N H = 19 and N D = 3, is found to have a significant superfluid fraction of 0.6. Radial density distributions of D 2 and H 2 molecules in (D 2 ) 2 (H 2 ) NH with respect to the center of mass are shown in Fig. 2. In the top panel of Fig. 2, one can see that the D 2 density distribution in (D 2 ) 2 (H 2 ) 12 has a sharp peak near the cluster center. A small peak located at 3.4 Å from the center is hardly visible in this scale with peak height of Å 3. This tells us that one D 2 molecule is located at the cluster center and the other one is distributed around 3.4 Å from the center. On the other hand all twelve H 2 molecules are placed off the center and their density distribution is peaked around 3.9 Å from the center. We find that these structural features are shared by all smaller clusters with 10 N H 14 and N D = 2. For a larger cluster of (D 2 ) 2 (H 2 ) 17 the D 2 density distribution has a single peak located at 1.7 Å from the center (see the bottom panel of Fig. 2), which says that both D 2 molecules are distributed off the cluster center unlike in smaller clusters. In addition H 2 distribution of the larger cluster is shown to be more structured than that of the smaller cluster. These features characterize the radial density distributions of the mixed clusters with 15 N H 18 and N D = 2. Radial density distributions in the clusters of twenty H 2 molecules doped with one to five D 2 molecules are computed for further analysis of the cluster structure. Figure 3 shows both D 2 (solid lines) and H 2 (dashed lines) radial density distributions of (D 2 ) ND (H 2 ) 20 with N D = 1,4 at T = 0.8 K. The calculations at a higher temperature of T = 1.6 K turn out to produce the density distributions nearly identical to those shown in Fig. 3, which implies that thermal effects on the cluster structure are minor between these two temperatures. As the number of additional D 2 molecules increases, heavier D 2 molecules push away more H 2 molecules from the cluster center and at the end the entire inner shell region is occupied by D 2 molecules. This trend is seen by a clear separation of D 2 and H 2 radial density

5 5 Fig. 3 The D 2 and H 2 density distributions of (D 2 ) ND (H 2 ) 20 with N D = 1 (left panel) and N D = 4 (right panel) with respect to the center of mass. The D 2 density distributions are represented by solid lines and the H 2 density distributions by dashed lines. The length and the density unit are Å and Å 3, respectively. distributions from each other for N D = 4, which is responsible for complete disruption of long exchange coupling among H 2 molecules and the total quenching of H 2 superfluidity discussed above. Fig. 4 The total (solid line) and local superfluid (dashed line) density distributions of H 2 with respect to the center of mass are presented in the left column. The right column shows the local superfluid fraction, defined as the ratio between the total density and the superfluid density. The results for N H = 12 and N D = 2 are shown in the top row and the ones for N H = 17 and N D = 2 in the bottom row. The length and the density unit are Å and Å 3, respectively. Using the local estimator of Eq. (1), we have evaluated the local superfluid density distributions of H 2 in the mixed clusters, whose results for N H = 12,17 with N D = 2 are shown in

6 6 the left column of Fig. 4, along with the total H 2 density distributions of the corresponding clusters. One can see that the superfluid distribution of the N H = 12 cluster has a significantly higher peak than that of the N H = 17 cluster, which reflects the fact that the latter is a magic cluster. Furthermore it is shown that the superfluid density distributions have similar structures to the corresponding total density distributions in both clusters. This implies that H 2 superfluidity in the mixed cluster is uniformly distributed rather than confined either in the surface or in the inner part of the cluster. In order to quantify this we have computed the local superfluid fraction defined by the ratio between the superfluid density and the total density, f s (R) = ρ s (R)/ρ tot (R). Unlike the claim of Khairallah et al. that the superfluidity is dominant near the cluster surface, we find a rather uniform distribution of H 2 superfluidity in the mixed cluster (see the right column of Fig. 4), which is in agreement with the main conclusion of Ref. [11]. Weakly structured features in the local superfluid fraction f s (R) are due to the suppression of H 2 superfluidity near D 2 molecules as well as large statistical noise in the region where the total density has minimal values. 4 Conclusion The low-temperature properties of the mixed cluster of (D 2 ) ND (H 2 ) NH have been calculated by using the PIMC method. It is found that the global superfluid fraction of H 2 is significantly suppressed when the total number of particles is equal to 13, 19, and 23, compared to those of the clusters of the neighboring sizes. This is due to magic number stability of hydrogen clusters which had been studied extensively for pure (H 2 ) N clusters [4,8]. As more H 2 molecules are replaced with heavier D 2 molecules, the inner part of the cluster gets occupied completely by D 2 molecules, resulting in the total suppression of H 2 superfluidity in the clusters with N D 4. Finally the local superfluid response of the H 2 -D 2 mixed clusters is discussed. We find that H 2 superfluidity is not restricted either to the outer surface or to the inner shell of the cluster, but displayed equivalently in all regions except near D 2 molecules. Acknowledgements This work was supported by Basic Science Research Program (R ) and by WCU program (R ) through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology. References 1. P. Sindzingre, D. M. Ceperley, and M. L. Klein, Phys. Rev. Lett. 67, 1871 (1991). 2. S. Grebenev, B. Sartakov, J. P. Toennies, and A. F. Vilesov, Science 289, 1532 (2000). 3. Y. Kwon and K. B. Whaley, Phys. Rev. Lett. 89, (2002). 4. G. Tejeda, J. M. Fernández, S. Montero, D.Blume, and J. P. Toennies, Phys. Rev. Lett. 92, (2004). 5. R. Guardiola, J. Navarro, Phys. Rev. A 74, (2006). 6. J. E. Cuervo and P. N. Roy, J. Chem. Phys. 125, (2006). 7. F. Mezzacapo and M. Boninsegni, Phys. Rev. Lett. 97, (2006). ; Phys. Rev. A 75, (2007). 8. S. A. Khairallah, M. B. Sevryuk, D. M. Ceperley, and J. P. Toennies, Phys. Rev. Lett. 98, (2007). 9. S. Baroni and S. Moroni, Chem. Phys. Chem. 6, 1884 (2005). 10. J. Choo and Y. Kwon, J. Kor. Phys. Soc. 52, 259 (2008) ; J. Low Temp. Phys. 150, 358 (2008). 11. F. Mezzacapo and M. Boninsegni, Phys. Rev. Lett. 100, (2008). 12. F. Mezzacapo and M. Boninsegni, Phys. Rev. A 76, (R) (2007). 13. M. Boninsegni, N. V. Prokof ev, and B. V. Svistunov, Phys. Rev. E 74, (2006). 14. D. M. Ceperley, Rev. Mod. Phys. 67, 279 (1995). 15. Y. Kwon, F. Paesani, and K. B. Whaley, Phys. Rev. B 74, (2006). 16. U. Buck, F. Huisken, A. Kohlhase, D. Otten, and J. Schaeffer, J. Chem. Phys. 78, 4439 (1983). 17. E. W. Draeger and D. M. Ceperley, Phys. Rev. Lett. 90, (2003). 18. Y. Kwon and K. B. Whaley, Phys. Rev. Lett. 83, 4108 (1999). 19. Y. Kwon, P. Huang, M. V. Patel, D. Blume, and K. B. Whaley, J. Chem. Phys. 113, 6469 (2000).

arxiv:cond-mat/ v1 5 Aug 2002

arxiv:cond-mat/ v1 5 Aug 2002 Superfluidity in a Doped Helium Droplet E. W. Draeger and D. M. Ceperley Department of Physics and National Center for Supercomputing Applications, University of Illinois Urbana-Champaign, 68 Path Integral