The structure of large 3He-4He mixed drops around a dopant molecule

We have investigated how helium atoms are distributed within a mixed $^3$He$_{N_3}$-$^4$He$_{N_4}$ large drop with $N_3 \gg N_4$. For drops doped with a SF$_6$ molecule or a Xe atom, we have found that the number of $^3$He atoms within the volume containing the first two solvation shells increases when $N_4$ decreases in a way such that these dopants may be in a superfluid environment for $N_4 \geq$ 60, which gradually disappears as $N_4$ decreases. The result is in qualitative agreement with recent experimental data.

We have investigated how helium atoms are distributed within a mixed 3 HeN 3 -4 HeN 4 large drop with N3 ≫ N4. For drops doped with a SF6 molecule or a Xe atom, we have found that the number of 3 He atoms within the volume containing the first two solvation shells increases when N4 decreases in a way such that these dopants may be in a superfluid environment for N4 ≥ 60, which gradually disappears as N4 decreases. The result is in qualitative agreement with recent experimental data. In a recent experiment, Grebenev et al [1] have carried out the equivalent of the Andronikashvili experiment [2] in a microscopic system, namelly a mixed 3 He- 4 He drop consisting of about 10 4 atoms doped with an oxigen carbon sulfide (OCS) molecule. By analyzing the infrared spectrum of OCS, Grebenev et al (see also Ref. [3]) conclude that the molecule freely rotates when a number of 4 He atoms large enough coats the impurity, preventing the 3 He atoms, which are in the normal phase at a temperature of the order of 150 mK [4,5], from getting too close to the OCS molecule. That number is of the orden of 60, in excelent agreement with path integral [6] and variational [7] Monte Carlo calculations. It is remarkable that the presence of the impurity, which causes the 4 He density to rise up to several times the saturation value, is not destroying its superfluid character, and that in spite of the high densities reached, the first solvation shell remains liquid [8]. An indication of this fluidlike behavior is that the peak density in the first solvation shell continues to increase as the second shell grows [9].
Even if the intrepretation of the microscopic Andronikashvili experiment is on a firm basis, a remaining major question is how 3 He is distributed around the 4 Heplus-impurity complex, and in general, how liquid 3 He is dissolved into 4 He droplets at very low temperatures. These are the questions we want to address in this work.
At zero temperature, it is known that the maximum solubility of 3 He in the bulk of 4 He is ∼ 6.6% [10]. For liquid 4 He systems having a free surface, it is also known that a large amount of 3 He is accumulated on the free surface occupying Andreev states [11,12] before it starts being dissolved into the bulk. In the case of drops made of up to several thousands of atoms, the surface region constitutes a sizeable part of the system [13], and the surface has a large capacity of storing 3 He atoms before they get inside the drop [14]. Due to the wide free surface of both isotopes [15,16] and to the low surface tension of the 3 He-4 He liquid interface [17], one expects that this region plays a prominent role when it constitutes a large part of the system or, as in the present case, when it is close to the foreing atom or molecule.
The structure and energetics of mixed, doped or not, helium droplets has been addressed using a finite-range density functional [14]. That work was carried out before the experiments reported in Ref. [1], and the emphasis was put on improving the density functional to better describe the thermodynamical properties of the liquid mixture, and to study rather small mixed droplets with N 4 ≫ N 3 . Our main goal here is to apply the density functional method to droplets whose characteristics are closer to those of the experiments, with the restriction of spherical symmetry for the He-impurity potential for the sake of simplicity. We have considered Xe and SF 6 as dopants, using for the later a spherically averaged interaction potential. The Xe-He potential is weaker than the SF 6 -He one. In this respect, our results for that atomic impurity should better represent the experimental ones for OCS even if this linear molecule produces deformations in the helium drop that we have not considered here. The density functional and the treatment of the impurity are thoroughly described in Ref. [14].
The large number of 3 He atoms in the droplets we are describing (N 3 > 1000) allows us to employ an Extended Thomas-Fermi method to describe the fermionic component of the mixture. We have used for the 3 He kinetic energy density the expression given in Ref. [18], which contains up to second order density gradient corrections to the standard ∼ ρ 5/3 3 (r) expression, where ρ 3 and ρ 4 will denote the particle density of each isotope. We have checked that this density functional reproduces accurately the Hartree-Fock results [14] obtained for the largest drops there studied (see also Refs. [5,18]). Figure 1 displays the situation in which a 4 He 728 drop, whose size is large enough to clearly distinguish in it a surface and a bulk region, is coated with an increasing number of 3 He atoms, and the limiting situation of the same drop immersed into liquid 3 He. The evolution with N 3 of the 3 He concentration inside the 4 He drop, defined as x 3 ≡ ρ 3 /(ρ 4 + ρ 3 )| bulk , is shown in the insert. Several interesting features of this figure are worth to comment. First of all, a fairly large amount of 3 He is needed before it is appreciably dissolved in the bulk: for N 3 = 1000, ρ 3 near the origin is ∼ 1.4 × 10 −8Å −3 . The solubility is appreciably reduced by finite size effects. Indeed, one can see from the insert that the limiting solubility into the N 4 = 728 drop is ∼ 2.5%, as compared to the 6.6 % value in 1 the liquid mixture. It is also worth to see that for large N 3 droplets, the bulk solubility is slightly higher than the limiting solubility, indicating that finite size effects still appear in rather large drops. Another manifestation of a finite size effect is that the average 3 He density is above the saturation value even for the larger drops, showing that the existence of the outer 3 He surface still causes a visible density compression.
Due to the high incompressibility of helium, the bulk density of 4 He decreases when 3 He is dissolved, and the rms radius of the 4 He drop manifests a peculiar N 3 behavior. It decreases when N 3 increases up to a few hundreds due to the initial compression of the outermost 4 He surface, and then steady increases as 4 He is pushed off the center by intruder 3 He atoms. This is a very tiny effect anyway. For example, we have found that the rms radius of the 4 He 728 drop is 15.70Å. It decreases when 3 He is added, reaching a minimum value of 15.64Å for N 3 ∼ 250, and then it steadily increases up to 16.11Å for N 3 = 10000. The rms radius of the 4 He 728 drop immersed into liquid 3 He is 16.14Å.
When a SF 6 molecule is captured by a helium drop, it moves into the bulk producing a drastic rearrangement of the drop density around it [20][21][22][23]. For large 4 He droplets, it is especially noteworthy the appearence of two high density solvation shells with a density-depleted region in between. It is then natural to ask about the possible existence of Andreev-like states arising at the 'inner 4 He surface', and whether a large number of 3 He atoms can be stored there, producing an onion-like structure of alternative 4 He and 3 He shells around the impurity, or even if when N 3 ≫ N 4 , the later can displace the former in the first solvation shell. Figure 2 shows the density profiles of several 4 He 728 + 3 He N3 + SF 6 droplets, giving a positive answer to the first question, and a negative answer to the other two. We have found that indeed, about one 3 He atom is in the inner surface, but that 3 He is mostly coating the 4 He-plus-impurity complex, as in undoped droplets. To check this result we have started the calculations from different initial shapes, some having the onion-like form mentioned before. It has turned out that these always are high energy, metastable configurations, and the mixed droplet eventually evolves towards stable configurations of the kind shown in Fig. 2. The larger zero point motion energy of 3 He makes it energetically more advantageous to fill the first solvation shell with 4 He atoms, and 3 He is expelled to the outer region of the drop.
We are now in position to discuss a physical situation relevant to the microscopic Andronikashvili experiment. We first observe that the first solvation shell [19] can host ∼ 23 4 He atoms in the case of SF 6 as a dopant, and ∼ 15 atoms in the case of Xe [20][21][22]. According to Refs. [6][7][8], these numbers are too small for the 4 He droplet being superfluid. It is thus crucial to know how the second solvation shell is built, especially what is its composition. Too many 3 He atoms in that shell might shrink or even wash out the superfluid environment around the dopant.
The density functional method cannot tell whether a given configuration is superfluid or not, but it can give a quantitative answer to its local composition because it is able to reproduce available microscopic density profiles [20,21,24]. We present examples of such compositions in Figs. 3 and 4. Figure 3 shows the density profiles for 4 He N4 + 3 He 1000 + SF 6 and 4 He N4 + 3 He 1000 + Xe with N 4 = 35, 60 and 100. We have carried out calculations for two different dopants to see what is the influence of the He-impurity potential on the results. It turns out that a weaker attractive potential favors the mixing of both isotopes in the whole allowed volume (the Xe-He and SF 6 -He potentials are plotted in Ref. [22], for instance). However, this is in part a first glance effect, since the number of 3 He atoms in the first solvation shell around Xe is less than one (see Fig. 4). Rather, the relevance of Fig. 3 lies in that it shows how 3 He is filling the second solvation shell as N 4 decreases.
A more quantitative look at this phenomenon is presented in Fig. 4, where we have plotted the number of atoms of each isotope as a function of the radial distance to the center of the drop. Notice that for a given impurity, the number of 4 He atoms in the first solvation shell (extending up to ∼ 5.5Å) is sensibly the same for the three selected N 4 values. It is also worth to look at the ratios N 3 /(N 4 + N 3 ) within the second solvation shell which extends from ∼ 5.5 to ∼ 8.5Å. In the SF 6 case, they are ∼ 8% for N 4 = 100, ∼ 29% for N 4 = 60, and ∼ 65% for N 4 = 35. Considering the content of the two shells, these ratios are ∼ 5%, ∼ 19%, and ∼ 41% which correspond, respectively, to 3, 10, and 22 3 He atoms. The values for Xe are slightly smaller. These numbers make it quite plausible that a SF 6 molecule or a Xe atom in a 4 He N4 + 3 He 1000 drop is in a superfluid environment when N 4 = 100 or 60, whereas it is not when N 4 = 35, as the microscopic Andronikashvili experiment indicates for OCS.
We are indebted to Peter Toennies and Andrej Vilesov for useful discussions. This work has been performed under grants PB95-1249 and PB95-0271-C02-01 from CI-CYT, Spain and Program 1996SGR-00043 from Generalitat of Catalunya.