Supersolidity and Superfluidity of Grain Boundaries

We have looked for dc mass transport through solid 4He in a simple experiment with two communicating vessels filled with solid 4He in equilibrium with liquid 4He. Through good quality crystals, we have observed no mass transport, in contradiction with the hypothesis of a Bose–Einstein condensation of vacancies. Through crystals containing grain boundaries, we have found superfluid flow along these grain boundaries. We discuss these results in the context of other experiments on supersolidity. PACS 67.80-s · 67.90+z We present new evidence for the “supersolid” behavior which was recently discovered by Kim and Chan (KC) and confirmed by three other groups [1–5]. KC’s experiment has triggered a considerable interest because it could be interpreted in terms of a Bose–Einstein condensation (BEC) of vacancies present in the crystals. If confirmed, it would be a remarkable coexistence of order in real space (crystalline order) and in momentum space (BEC leading to superfluidity). However this interpretation is controversial [6–12]. Furthermore, Rittner and Reppy [3] have found that annealing crystals destroys supersolidity. This is why we looked for effects of disorder in a different experiment. We have found mass transport along grain boundaries, which S. Sasaki ( ) · R. Ishiguro · F. Caupin · S. Balibar Laboratoire de Physique Statistique de l’Ecole Normale Supérieure, Associé au CNRS et aux Universités Paris 6 et 7, 24 rue Lhomond, 75231 Paris Cedex 05, France e-mail: [email protected] Present address: R. Ishiguro Graduate School of Science, Osaka City University, Osaka 559-8585, Japan H.-J. Maris Department of Physics, Brown University, Providence, RI 02912, USA 666 J Low Temp Phys (2007) 148: 665–670 Fig. 1 a The cell contains a 1 cm diameter test tube and is partially filled with solid helium in equilibrium with liquid helium above. The solid inside the tube is higher by h than outside. Schematic grain boundaries (width w and thickness e) connecting the inner and the outer liquid are also shown. b A crystal grown from normal liquid 4He at 1.8 K. c A crystal (# 3) with cusps at the liquid–solid interface indicating the existence of grain boundaries are superfluid as proposed by Burovski et al. [6]. This mechanism is very different from the BEC of vacancies invoked by many authors [7, 13–15]. The principle of our experiment is simple: we have placed a glass tube (inner diameter D = 1 cm) in the cell of our optical cryostat which we filled with solid 4He in equilibrium with superfluid 4He above. The top of the tube is closed and we could bring the liquid–solid interface higher inside the tube than outside (Fig. 1a). Then we watched the possible relaxation of the height difference between the inside and the outside of the tube. Since the liquid density ρL is smaller than the solid density ρC , any level change inside the tube required mass transport through the solid. According to KC, a superfluid density (ρs = 0.01 ρC) flows at a critical velocity vc = 10 μm/s below 50 mK. These numbers correspond to natural 4He containing about 0.1 ppm of 3He, which is what we also used. We thus expected the height difference to relax at a velocity V = ρsvc/(ρC −ρL)= 1 μm/s. As explained below, this is not what we have found. In order to grow the solid inside the tube, we had to apply a substantial pressure difference from the outside. For this we had the inside of the tube at 1.3 K (consequently at the melting pressure Pm(1.3 K) = 25.7 bar) and the outside at 1.4 K (and at Pm(1.4 K) = 26.0 bar) for about 10 s. To do it, we used a heater or temporarily increased the flow rate through the fill line. After such a treatment, several cusps often appeared at the liquid–solid interface (Fig. 1c), meaning that grain boundaries (GB) had formed inside the crystal. These cusps result from the mechanical equilibrium between the GB surface tension and the liquid–solid interfacial tension. Once solid helium had entered the tube, we could cool down to 50 mK, lower the outside level 1 cm below the inside level by opening a valve on the fill line and watch the relaxation with a video camera. We studied 13 crystals. For most of them, no cusps could be seen inside the tube, and we found no relaxation of the level within 50 μm over 4 hours, meaning that the interface velocity V was smaller than 3.5 × 10−3 μm/s. This observation rules out simple interpretations of KC’s experiment in terms of 1% of the crystal mass being a 3D superfluid of vacancies with a critical velocity of 10 μm/s as mentioned above. It also puts a constraint on models involving a superfluid layer near the glass wall as proposed by Dash and Wettlaufer [10] and by Khairallah and Ceperley [11]. If this J Low Temp Phys (2007) 148: 665–670 667 layer had a thickness e, a superfluid density ρs and a critical velocity v c , we would find