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In quest of 4 He supersolid a work with J. Peter Toennies (MPI-DSO Göttingen), Franco Dalfovo (Uni Trento), Robert Grisenti & Manuel Käsz (Uni Frankfurt), Pablo Nieto (Automoma Madrid) History of a conjecture: BEC in a quantum solid ? Vacancy diffusivity and solid 4 He Poisson ratio The Geyser effect in solid 4 He vacuum expansion Bernoulli flow of a nominal 4 He solid Suppression of flow anomalies by 1% 3 He 4 He vacuum expansion from low -T sources Firenze 2005 - 1

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History of a conjecture: BEC in a quantum solid? 1969 Andreev $ Lifshitz 1970 Chester Leggett 1977 Greywall 2004 Kim & Chan 2004 Ceperley & Bernu Firenze 2005 - 2

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Kim & Chan 2004 measurements of non-classical rotational inertia Firenze 2005 - 3

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no trend ? Kim & Chan Firenze 2005 - 4

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Galli & Reatto 2001 (a) no ground state vacancies but only thermal vacancies (b-d) ground state + thermal vacancies (for different vacancy formation energies) what about injected (non-equilibrium) vacancies? Firenze 2005 - 5

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Vacuum expansion of solid 4 He Firenze 2005 - 6

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continuity Bernoulli Firenze 2005 - 7

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4 He phase diagram Firenze 2005 - 8

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The Geyser effect Firenze 2005 - 9

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Period vs. T at constant pressure 40.7 bar 35.0 bar 32.0 bar Firenze 2005 - 10

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Period versus P 0 at constant temperature Bernoulli Firenze 2005 - 11

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P s/l information on dynamical processes inside solid 4 He P information on Poisson ratio of solid 4 He Firenze 2005 - 12

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Poisson ratio of solid 4 He Firenze 2005 - 13

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Plastic flow motion of dislocation motion of vacancies dominant in solid He (high diffusivity!) Polturak et al experiment (PRL 1998) vacancy injection at s/l interface + sweeping by pressure gradient Firenze 2005 - 14

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Vacancy drift solid 4 He p-type SC Firenze 2005 - 15

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V a = V* - V a V a = 35.15 Å 3 (atomic volume) V* 0.45V a (vacancy isobaric formation volume) A0A0 A s/l L Virtual volume to be filled by vacancies in the time L/u 0 u0u0 The vacancy mechanism Firenze 2005 - 16

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accumulation of vacancies up to a critical concentration X c drift + diffusion diffusion Pressure distance from s/l interface 0L COLLAPSE! Geyser mechanism vacancy bleaching & resetting of initial conditions

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Data on vacancy diffusivity and concentration in 4 He Firenze 2005 - 17

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Transport theory Generation function surface generation velocity Firenze 2005 - 18

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Solution for L Excess vacancies Current at the s/l interface (x = 0) due to excess vacancies = surface depletion layer thickness Firenze 2005 - 19

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- the shape of the current depends on 2 parameters (, ) - the time scale implies another parameter ( v ) - the ratio of the oscillation amplitude to the constant background is measured by X 0 V a u v /u 0 and is of the order of a few percent (as seen in experiment) fitting reduced form: Firenze 2005 - 20

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Theory vs. experiment D v = 1.3·10 -5 cm 2 /s v = 5.4·10 10 s/g u v = 2.0·10 -3 cm/s u s = 2u v s = 60 s v = 13 s * = 10.7 s 0 = 82 s P 0 = 31 bar T 0 = 1.74 K best fit with = 4 = 1.214 Firenze 2005 - 21

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better fits are obtained with finite L (one more parameter) large means fast recombination Firenze 2005 - 22

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Period 0 vs. diffusivity finite L approximate solution by Greens function method X c = critical concentration Firenze 2005 - 23

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Firenze 2005 - 24

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Anomalies below the point! Firenze 2005 - 25

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a sharp transition in the flow regime at 1.58 K ! Firenze 2005 - 26

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Effects of 3 He on the anomalies from R. Richardson et al Firenze 2005 - 27

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3 He-vacancy binding energy Firenze 2005 - 28

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normal behaviour induced by less than 1% 3 He ! Firenze 2005 - 29

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CONCLUSIONS 1.The geyser effect indicates (via Bernoullis law) an oscillation of the s/l (quasi-)equilibrium pressure at a given T: vacancy concentration appears to be the only system variable which can give such effect. 2. Below the temperature flow anomalies are observed: (a) The most dramatic one is the occurrence of a Bernoulli flow corresponding to pressures > Pm, at which 4 He should be solid. (b) Below 1.58 K a sharp drop of the geyser period signals a dramatic change in the flow properties of solid 4 He. These anomalies, suggesting superflow conditions, are attributed to injected excess vacancies, and agree with Galli and Reatto predictions for a vacancy-induced (Andreev-Lifshitz) supersolid phase. 3.A 3 He concentration of 0.1% is shown to suppress the flow anomalies, suggesting a quantum nature of the superflow. Firenze 2005 - 30

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2 I = flow (current), assumed approximately constant over a period A 0 = tube section A = average flow cross section in the s/l interface region (A is slightly < A 0 ) g 0 = conductivity far away from the s/l interface due to the equilibrium concentration of vacancies X 0 : g 0 = X 0 v where v is the vacancy mobility g = conductivity near the s/l interface: g = X v where X is the actual vacancy concentration near the s/l interface. Immediately after the collapse (brown and red lines in the figure) X << X 0 and g << g 0 whereas just before the collapse (green line) X >> X 0 and g >> g 0. When X = X 0 (purple line) the gradient is the same between 0 and L 0. The corresponding gradients are inversely proportional (see figure)! 1 Pressure gradients: 3 Length L of the gradient near the s/l interface (solve the above system for P L and L): where the term in parenthesis is constant. For A A 0 it appears that L grows with g/g 0 = X/X 0 as qualitatively shown in the figure. Thus the sensor during the period measures a pressure varying from P 0 to P s/l

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