Vegard B. Sørdal Thermodynamics of 4He-3He mixture and application in dilution refrigeration
1. Introduction 2. Crogenic methods Contents of the presentation 3. Properties of Helium 4. Superfluid Helium 4 -Two fluid model -Quantum vortex 5. 4He - 3He mixtures 6. Dilution refrigeration
Importance of low temperatures in mesoscopic physics No fundamental arguments prohibit mesoscopic effects at room temperature Limitations: important mesoscopic quantities like mean free path and coherence length. These quantities depends on T and feature size.
Typical lengthscales for mesoscopic effects T = 300 K L < 10 nm T = 77 K L < 100 nm T = 4.2 K L < 5000 nm
How do we reach low T? Doppler laser cooling Isentropic demagnetization Mechanical compression refrigiration Liquid evaporation refrigiration Dilution refrigiration + Many other novel techniques or variations of the above
Doppler laser cooling ~μk
RW in momentum space due to spontaneous photon emission/absorption Transitions have finite frequency width atoms can scatter from wrong laser Minimal consentration exited atoms colliding unexited atom release kinetic energy and photon falls back to ground state Limited available atoms: - Hyperfine structure (more ways to photon from upper state and not return to ground state) - Laser power required too high for λ < 300 nm
Isentropic demagnetization
Isothermal magentization Low B High B Insulate thermally Isentropic demagentization T 2 = T 1 B loc B
Magnetocaloric effect intrinsic magentic property Depends on magnetic ordering temperature (Néel Temperature) Thermal response highest close to T N High spin entropy before B application is better 100 pk obtained by nuclear DM + dilution refrigeration by Aalto University
Mechanical compression refrigeration pv = T Porous material with high C / V COP = Tl/(T a T l )
Properties of Helium 4He: Boson 3He: Fermion Abundance: 4He - 99.999863% 3He - 0.000137% Liquid transition temperature: 4He 4.23 K 3He 3.19 K Latent vaporization heat: 4He - 0.0829 kj/mol 3He - 0.026 kj/mol ~25% of normal mass in the universe
Zero-point fluctuation energy larger than VDW attraction. - P 30 bar required for crystallization Zero-point energy high due to low mass VDW is low since it is a noble gas and has no dipole moment (4He) Easy to separate two Helium atoms - latent heat of vaporization : 5 cal/g - for water : 500 cal/g Large distance between neighbouring atoms - 13% mass density of water
Joule-Thompson Effect Attraction between atoms Atoms perform work against expansion and cool down Exceptions: Helium, Hydrogen, Neon, etc.
Liquid Helium-3 Gas-liquid transition at 3.18 K Fermions form Cooper pairs and condensate 3He B: BW state 3He A: ABM state
Liquid Helium-4 He I: Normal fluid He II: Superfluid Only ~250 cal to go from He I to He II. However! Large part of original He evaporates in transition
Superfluid Absence of bubbling indicate large heat conductivity. Heat conductivity increase by 10^6 at transition No capillary viscosity. But viscous drag is observed second sound two fluid model Zero entropy: Heat cannot flow from cold to hot, but this seems to be broken in He II. Heating induce net flow to right chamber. Connection only traversable by He II. He II carries no heat, any internal energy no longer thermally available. Heat energy carried in normal component.
Two interpenetrating elements: super/normal ρ = ρ n + ρ s = const Two-fluid Model Normal component acts like an ordinary fluid Superfluid component has the unique properties Elastic and supports sound as oscillating pressure waves Can also transport heat in form of waves, with a characteristic speed: the speed of second sound
Second Sound Normal heat conduction is a diffusion process: Superfluids can conducts heat in wave form ρ = ρ n + ρ s j = v s ρ s + v n ρ n First sound: v s = v n Traditional wave where the whole fluid moves as one unit Second sound: j = 0 and ρ = constant The relative densities oscillate, not the total density! Second sound is an entropy wave in the medium and can be generated by an oscillating heat source
F = (μ m + M m gz) μ m = U ST + PV ρ n v n t = ρ n ρ P + ρ ss T + η 2 v n ρ s v s t = ρ s ρ P ρ ss T
Two-fluid model only gives qualitative description of He II He II is a liquid capable of two types of motion, but we can not claim that they happen in different parts of the fluid True nature of He II can only be described by quantum hydrodynamics Many models: vortex rings, hard-sphere models, Gaussian cluster theories Quantum vortex: topological defect with superfluid circulating around. Carries angular momentum and allow superfluid to rotate
Simple quantum vortex Impurity: air, vacuum, excited particle, etc. Superfluid wavefunction: n is the number density and S is the phase Wavefunction must return to same value after n circulations, thus φ = 2πn Circulation around a vortex is quantized!
Phase separation below 0.8 K depending on 3He consentration. 3He - 4He mixtures 3He has smaller mass than 4He, thus larger ZPE Triple point 3He rich phase is a Fermi fluid with m 3m 4He acts as inert background in this phase Even at T=0, 6.6% is 3He in 4He rich phase
M 3He 3 4 M 4He 4He-3He VDW is stronger than 3He-3He! 3He component 3He 4He + 3He
3He is circulated in system by pump operating at p c Precooled by liquid nitrogen Impedance: capillary tubes designed to keep vapor pressure p v (T) below p c Cooling power comes from moving 3He across the phase boundary in the mixing chamber. Dilution refrigeration
1. 4He bath cools whole system to 4.2 K 2. At the1k pot, 3He condense and the pot absorb heat of condensation (p c p v ) 3. Exchange heat with still to ~0.7K 4. All impedances chosen to keep p local > p v 5. Further cooled by 3He moving from the mixing chamber 6. Enters mixing chamber and crosses phase boundary. Heat needed for dilution is the cooling power of the refrigerator 7. Leaves the mixing chamber from the diluted phase, enters still and evaporate. Circulation is continued.
The still is heated so 3He can evaporate and be extracted by the pump. 1% 3He has lower boiling point than 4He, so the still is fine tuned to keep 4He liquid By reducing 3He concentration in the still (~1%) osmotic pressure drives 3He from mixing chamber to still More 3He can cross the phase boundary, and absorb heat as they do so. The entropy increases as 3He moves from concentrated to dilute phase (entropy of mixing) 6% Cooling comes from the difference in Fermi temperature of the two phases 3He moving from concentrated phase to diluted phase «expand into vacuum»
Cooling power If mixing is reversible we have: S is molar entropy, n is circulation rate. T M = mixing chamber temperature. x is the 3He concentration Fermi gas heat capacity: Both the concentrated and dilute phase can be approximated as Fermi gas S = C V T dt Cooling really based on increase in molar volume of 3He!
References Second Sound in He II, Otis Chodosh, Jeremy Hiatt, Samir Shah, and Ning Yan Mesoscopic Electronics in Solid State Nanostructures, Thomas Heinzel Second Sound in Liquid Helium II, C. T. Lane et. al Refrigeration and thermometry below one Kelvin, D. S Betts Second sound and the superfluid fraction in a Fermi gas with resonant interactions, Leonid A. Sidorenkov The two-fluid theory and second sound in liquid helium, Russell J. Donnelly Introduction to dilution refrigeration, A.T.A.M. de Waele Daniel P. Lathrop's Nonlinear Dynamics Lab (http://complex.umd.edu/)