Outline Superfluid Liquid Helium Supersolid Crystalline Helium Superfluidity and Supersolidity in 4He Author: Lars Bonnes Supervisor: Lode Pollet Proseminar Theoretische Physik: Phase Transitions SS 07 18.06.2007 4 Superfluidity and Supersolidity in He Proseminar Theoretische Physik: Phase Transitions SS 07 Outline Superfluid Liquid Helium Supersolid Crystalline Helium Superfluid Liquid Helium Motivation Two-fluid Model Condensation and Phase Transition Vortices Summary Supersolid Crystalline Helium Kim-Chan Experiment Dependence on the Quality of the Crystal Communicating Vessels Theory Conclusion End 4 Superfluidity and Supersolidity in He Proseminar Theoretische Physik: Phase Transitions SS 07 Outline Superfluid Liquid Helium Supersolid Crystalline Helium Motivation General Information on Helium: 4 Superfluidity and Supersolidity in He Proseminar Theoretische Physik: Phase Transitions SS 07 Outline Superfluid Liquid Helium Supersolid Crystalline Helium Motivation General Information on Helium: ◮ Stable isotopes: 3He and 4He 4 Superfluidity and Supersolidity in He Proseminar Theoretische Physik: Phase Transitions SS 07 Outline Superfluid Liquid Helium Supersolid Crystalline Helium Motivation General Information on Helium: ◮ Stable isotopes: 3He and 4He ◮ In nature: almost only 4He (99.999864%) 4 Superfluidity and Supersolidity in He Proseminar Theoretische Physik: Phase Transitions SS 07 Outline Superfluid Liquid Helium Supersolid Crystalline Helium Motivation General Information on Helium: ◮ Stable isotopes: 3He and 4He ◮ In nature: almost only 4He (99.999864%) ◮ Does not solidify at T = 0 for P < 26 bar 4 Superfluidity and Supersolidity in He Proseminar Theoretische Physik: Phase Transitions SS 07 Outline Superfluid Liquid Helium Supersolid Crystalline Helium Motivation 4 Superfluidity and Supersolidity in He Proseminar Theoretische Physik: Phase Transitions SS 07 Outline Superfluid Liquid Helium Supersolid Crystalline Helium Two-fluid Model Two-Fluid model by Tisza and Landau ◮ Phenomenological model 4 Superfluidity and Supersolidity in He Proseminar Theoretische Physik: Phase Transitions SS 07 Outline Superfluid Liquid Helium Supersolid Crystalline Helium Two-fluid Model Two-Fluid model by Tisza and Landau ◮ Phenomenological model ◮ Explains the frictionless flow of Helium 4 Superfluidity and Supersolidity in He Proseminar Theoretische Physik: Phase Transitions SS 07 Outline Superfluid Liquid Helium Supersolid Crystalline Helium Two-fluid Model Two-Fluid model by Tisza and Landau ◮ Phenomenological model ◮ Explains the frictionless flow of Helium ◮ Gives quantitative predictions in the regime close to T = 0 4 Superfluidity and Supersolidity in He Proseminar Theoretische Physik: Phase Transitions SS 07 Outline Superfluid Liquid Helium Supersolid Crystalline Helium Two-fluid Model T=0 Assume: ◮ Liquid Helium with mass density ρ flowing through capillary with v ◮ T = 0 4 Superfluidity and Supersolidity in He Proseminar Theoretische Physik: Phase Transitions SS 07 Outline Superfluid Liquid Helium Supersolid Crystalline Helium Two-fluid Model T=0 Assume: ◮ Liquid Helium with mass density ρ flowing through capillary with v ◮ T = 0 ◮ He ρ 2 Energy density in the laboratory frame (LF): Elf = 2 v 4 Superfluidity and Supersolidity in He Proseminar Theoretische Physik: Phase Transitions SS 07 Outline Superfluid Liquid Helium Supersolid Crystalline Helium Two-fluid Model T=0 Friction will generate excitations (i.e. phonons). Assume furtherly: 4 Superfluidity and Supersolidity in He Proseminar Theoretische Physik: Phase Transitions SS 07 Outline Superfluid Liquid Helium Supersolid Crystalline Helium Two-fluid Model T=0 Friction will generate excitations (i.e. phonons). Assume furtherly: ◮ One elementary excitation with momentum ~p and energy density ǫ(~p) ◮ He Energy in the rest frame (RFHe ) of the Helium is Erf = ǫ(~p) 4 Superfluidity and Supersolidity in He Proseminar Theoretische Physik: Phase Transitions SS 07 Outline Superfluid Liquid Helium Supersolid Crystalline Helium Two-fluid Model T=0 Friction will generate excitations (i.e. phonons). Assume furtherly: ◮ One elementary excitation with momentum ~p and energy density ǫ(~p) ◮ He Energy in the rest frame (RFHe ) of the Helium is Erf = ǫ(~p) ◮ Apply Galilean transformation to LFHe : He ρ 2 Elf = ǫ(~p)+ ~p~v + 2 v 4 Superfluidity and Supersolidity in He Proseminar Theoretische Physik: Phase Transitions SS 07 Outline Superfluid Liquid Helium Supersolid Crystalline Helium Two-fluid Model Limiting Velocity ρv 2 E He = ǫ(~p)+ ~p~v + lf 2 ∆E | {z } Friction Dissipation of energy from the Helium. ←→ This requires: ∆E < 0 ǫ(~p) v > p 4 Superfluidity and Supersolidity in He Proseminar Theoretische Physik: Phase Transitions SS 07 Outline Superfluid Liquid Helium Supersolid Crystalline Helium Two-fluid Model Limiting Velocity ǫ(~p) v > p Necessary condition for the creation of an elementary excitation! ǫ(~p) v < no friction p ⇐⇒ 4 Superfluidity and Supersolidity in He Proseminar Theoretische Physik: Phase Transitions SS 07 Outline Superfluid Liquid Helium Supersolid Crystalline Helium Two-fluid Model Drop assumption T =0 What happens to the thermal excitations? 4 Superfluidity and Supersolidity in He Proseminar Theoretische Physik: Phase Transitions SS 07 Outline Superfluid Liquid Helium Supersolid Crystalline Helium Two-fluid Model T =0 6 Assume: ◮ The criterion on creating new excitations remains valid. 4 Superfluidity and Supersolidity in He Proseminar Theoretische Physik: Phase Transitions SS 07 Outline Superfluid Liquid Helium Supersolid Crystalline Helium Two-fluid Model T =0 6 Assume: ◮ The criterion on creating new excitations remains valid. ◮ Thermal excitations form a dilute gas of quasi particles (only valid close to T = 0) 4 Superfluidity and Supersolidity in He Proseminar Theoretische Physik: Phase Transitions SS 07 Outline Superfluid Liquid Helium Supersolid Crystalline Helium Two-fluid Model T =0 6 Assume: ◮ The criterion on creating new excitations remains valid. ◮ Thermal excitations form a dilute gas of quasi particles (only valid close to T = 0) ◮ Gas (“COM”) moves with ~v ′ relative to the liquid 4 Superfluidity and Supersolidity in He Proseminar Theoretische Physik: Phase Transitions SS 07 Outline Superfluid Liquid Helium Supersolid Crystalline Helium Two-fluid Model T =0 6 Assume: ◮ The criterion on creating new excitations remains valid. ◮ Thermal excitations form a dilute gas of quasi particles (only valid close to T = 0) ◮ Gas (“COM”) moves with ~v ′ relative to the liquid ◮ ′ In RFHe they are distributed via n(ǫ(~p) ~p~v ; T ) − 4 Superfluidity and Supersolidity in He Proseminar Theoretische Physik: Phase Transitions SS 07 Outline Superfluid Liquid Helium Supersolid Crystalline Helium Two-fluid Model Momentum Density Calculate momentum density P~ : P~ = dτ~pn(ǫ ~p~v) Z − Expand n(ǫ ~p~v ′) n(ǫ) ~pv~′ dn + .... − ≈ − dǫ Average over spatial dimensions: ~pv~′ 1 ppˆv~′. ≈ 3 1 dn(ǫ) P~ = v~′ dτ p2 Z 3 − dǫ 4 Superfluidity and Supersolidity in He Proseminar Theoretische Physik: Phase Transitions SS 07 Outline Superfluid Liquid Helium Supersolid Crystalline Helium Two-fluid Model T =0 6 1 dn(ǫ) P~ = v~′ dτ p2 Z 3 − dǫ ←→ ′ P~ = v~ ρn 1 2 dn(ǫ) Flow carries mass: ρn = dτ p 3 − dǫ R 4 Superfluidity and Supersolidity in He Proseminar Theoretische Physik: Phase Transitions SS 07 Outline Superfluid Liquid Helium Supersolid Crystalline Helium Two-fluid Model T =0 6 The quasi particles may scatter with the capillary walls. ←→ Energy can dissipate from the system. ←→ The mass current will feel friction. 4 Superfluidity and Supersolidity in He Proseminar Theoretische Physik: Phase Transitions SS 07 Outline Superfluid Liquid Helium Supersolid Crystalline Helium Two-fluid Model Two-fluid model The mass current experiencing friction has a mass density of ρn. Two-fluid model: Write ρ = ρs + ρn. ◮ ρs : Density of superfluid Helium 4 Superfluidity and Supersolidity in He Proseminar Theoretische Physik: Phase Transitions SS 07 Outline Superfluid Liquid Helium Supersolid Crystalline Helium Two-fluid Model Two-fluid model The mass current experiencing friction has a mass density of ρn. Two-fluid model: Write ρ = ρs + ρn. ◮ ρs : Density of superfluid Helium ◮ ρn: Density of normal viscous Helium (gas of elementary excitations) 4 Superfluidity and Supersolidity in He Proseminar Theoretische Physik: Phase Transitions SS 07 Outline Superfluid Liquid Helium Supersolid Crystalline Helium Two-fluid Model Two-fluid model The mass current experiencing friction has a mass density of ρn. Two-fluid model: Write ρ = ρs + ρn. ◮ ρs : Density of superfluid Helium ◮ ρn: Density of normal viscous Helium (gas of elementary excitations) ◮ Both components interpenetrate eachother without interaction. 4 Superfluidity and Supersolidity in He Proseminar Theoretische Physik: Phase Transitions SS 07 Outline Superfluid Liquid Helium Supersolid Crystalline Helium Two-fluid Model Two-fluid model Note: The superfluid velocity is not an observable but the current ~j = ~vρ = ~vnρn + ~vs ρs 4 Superfluidity and Supersolidity in He Proseminar Theoretische Physik: Phase Transitions SS 07 Outline Superfluid Liquid Helium Supersolid Crystalline Helium Two-fluid Model Goal: Calculate ρn Phonons: ǫphonon(p)= cp 2 roton (p−p0) Rotons: ǫ (p)=∆+ 2m′ with ∆ = 8.7 K 4 Superfluidity and Supersolidity in He Proseminar Theoretische Physik: Phase Transitions SS 07 Outline Superfluid Liquid Helium Supersolid Crystalline Helium Two-fluid Model Particle Distribution Bose statistics: n(ǫ; T ) = [eǫ/T 1]−1 − ∆ is large compared to T : approximate nrot (ǫ; T ) with Boltzmann factor: −ǫ/T nrot (ǫ; T )= e 4 Superfluidity and Supersolidity