Chapter IV
Cryogenic Techniques: Generation and Measurement of Low Temperatures Chapter IV: Cryogenic Techniques
Contents:
2013)
- IV.1 Generation of Low Temperatures 2004 2004 ( IV.1.1 Introduction
IV.1.2 Expansion Machine Institut - IV.1.3 Regenerative Machine
IV.1.4 Joule-Thomson Cooling Meißner - IV.1.5 Summary
IV.1.6 Evaporation Cooling Walther
© © IV.1.7 Dilution Cooling
IV.1.8 Pomeranchuk Cooling A. Marx , Marx, A.
IV.1.9 Adiabatic Demagnetization and
Gross IV.2 Thermometry R. IV.2.1 Introduction IV.2.2 Primary Thermometers IV.2.3 Secondary Thermometers
Chapt. IV - 2 Chapter IV: Cryogenic Techniques
Literature:
1. Tieftemperaturphysik
2013)
- Enss, Hunklinger 2004 2004
( Springer (2000)
Institut - 2. Matter and Methods at Low Temperatures
F. Pobell Meißner - Springer, 2nd edition (1996)
Walther
© © 3. Experimental Low-Temperature Physics
Anthony Kent A. Marx , Marx, A.
American Institute of Physics (1993) and
Gross 4. Cryogenic Systems R. Randall F. Barron Oxford University Press, Oxford (1985)
Chapt. IV - 3 IV.1 Generation of Low Temperatures IV.1.1 Introduction
9 10 center of hottest stars 108 center of the sun, nuclear energies 7
10
6
2013)
- 10
5 2004 2004
( 10 electronic energies, chemical bonding 4
10 Institut - surface of sun, highest boiling temperatures 103
organic life
Meißner 2
- background 10 liquid air
temperature 1 Walther 10 liquid 4He © © in universe 0 universe 10 (2.73 K)
-1 electronic
magnetism A. Marx , Marx, A.
10
and -2
temperature (K) temperature 10
Gross -3 3
superfluid He - R. 10 lowest temperature -4 10 superconductivity accessible in solids -5
10 nuclear
(few µK) -6 magnetism 10 lowest temperatures of condensed matter 10-7
Chapt. IV - 4 IV.1 Generation of Low Temperatures
IV.1.1 Introduction
2013)
- low temperature record 2004 2004
( for nuclear spin system: Institut - • experimental setup
Meißner according to Tauno Knuuttila (2000) -
Walther • lowest temperature: about 100 pK © © by demagnetization of Rhodium nuclei
A. Marx , Marx, A. („temperature of nuclear spins“)
and
PhD Thesis,
Gross Helsinki University of Technology R. (Espoo, Finland)
• problem: spin temperature cannot be transferred to lattice of solid
Chapt. IV - 5 IV.1 Generation of Low Temperatures IV.1.1 Introduction Generation of low temperatures by using cryo-liquids:
19th century: liquefaction of various gases by pressure
except for “permanent gases” (O2, H2, He)
2013)
-
1877: liquefaction of O2 by thermal expansion 2004 2004 ( (L. Cailletet, C.R. Acad. Sci. Paris 85, 1213 (1877); R. Pictet, C.R. Acad. Sci. Paris 85, 1214 (1877))
1884: liquefaction of H2 (precooling with liquid O2) Institut - (K. Olszewski, Ann. Phys. u. Chem. 31, 58 (1887)) 1898: significant amounts of lH for physical experiments Sir James Dewar, Meißner 2 - (1842-1923) (J. Dewar, Proc. R. Inst. Gt. Br. 15, 815 (1898))
Walther 1908: liquefaction of last “permanent gas” He by Kamerlingh Onnes © © (H. Kammerlingh Onnes, Leiden Commun. 105, Proc. Roy. Acad. Sci. Amsterdam 11, 168 (1908))
1922: Kammerlingh Onnes reaches T < 1K
A. Marx , Marx, A.
(H. Kammerlingh Onnes, Leiden Commun. 159, Trans. Faraday Soc. 18 (1922))
and
1926: adiabatic demagnetization of electron spins in Gross
R. paramagnetic salts by Debye and independently (P. Debye, Ann. Phys. 81, 1154 (1926) 1927: by Giauque (W.F. Giauque, J. Am. Chem. Soc. 49, 1864 (1927) Heike Kammerlingh Onnes (1853 – 1926) since 1950th: 3He available Nobelpreis für Physik: 1913 3 He cryostat Peter J. Debye 3 4 He- He dilution refrigerator 1884 - 1966 Chapt. IV - 6 Low Temperature Technology in Germany
1861 study at Polytechnikum Zurich, teachers: Rudolf Clausius, Gustav Zeuner und Franz Reuleaux
1868 offer of chair at the
Polytechnische Schule München (now TUM)
2013)
-
1873 development of cooling machine allowing 2004 2004
( the temperature stabilization in beer Institut
- brewing
21. 6. 1879 foundation of „Gesellschaft für Linde’s Meißner - Eismaschinen AG“ together with two
Walther beer brewers and three other co-founders © ©
1892 - 1910 re-establishment of professorship A. Marx , Marx, A.
12.5.1903 and
patent application: Gross
R. „Lindesches Gegenstrom- verfahren“ liquefaction of oxygen (-182°C = 90 K) Carl Paul Gottfried von Linde * 11. Juni 1842 in Berndorf, Oberfranken † 16. November 1934 in Munich Chapt. IV - 7 IV.1 Generation of Low Temperatures
IV.1.1 Introduction
2013)
-
2004 2004
(
temperatures
Institut
-
low
Meißner
-
Walther © ©
A. Marx , Marx, A. paramagnetic refrigeration
temperatures
and
low
-
Gross
R. ultra
nuclear demagnetization
Year Chapt. IV - 9 IV.1 Generation of Low Temperatures
IV.1.1 Introduction
2013)
- temperature refrigeration technique available typical record 2004 2004
( range since
Tmin Tmin Institut - Kelvin universe 2.73 K
4He evaporation 1908 1.3 K 0.7 K
Meißner -
3He evaporation 1950 0.3 K 0.25 K Walther © © Millikelvin 3He-4He dilution 1965 10 mK 2 mK
Pomeranchuk cooling 1965 3 mK 2 mK
A. Marx , Marx, A.
and
electron spin demagnetization 1934 3 mK 1 mK Gross
R. Microkelvin nuclear spin demagnetization 1956 50 µK 100 pK
Chapt. IV - 10 IV.1 Generation of Low Temperatures IV.1.1 Introduction
cooling techniques:
• expansion of an ideal gas
2013)
- • expansion machine 2004 2004
( • regenerative machine Institut - work against outside world
• expansion of a real gas
Meißner -
• Joule Thomson cooler Walther
© © work against internal interactions
• evaporation of a real gas:
A. Marx , Marx, A.
and work against internal interactions
Gross • dilution cooling (3He/4He) R. work against internal interactions • adiabatic demagnetization (electronic/nuclear moments) work against magnetic ordering
Chapt. IV - 11 IV.1 Generation of Low Temperatures
IV.1.1 Introduction
2013)
- Liquefaction of gases three useful methods: 2004 2004
( Institut
- 1. direct liquefaction by isothermal compression
Meißner 2. letting the gas perform work against external forces at the expense of -
Walther its internal energy © ©
cooling and eventual liquefaction
A. Marx , Marx, A.
and
3. making the gas perform work against its own internal forces by Joule- Gross
R. Kelvin or Joule-Thomson expansion cooling and eventual liquefaction
Chapt. IV - 12 IV.1 Generation of Low Temperatures IV.1.1 Introduction
direct liquefaction of gases by isothermal compression
starting temperature must be smaller than critical temperature Tc
2013)
p -
melting curve 2004 2004
( solid Institut
- liquid Meißner - boiling curve
pc critical point Walther
© © triple point gas A. Marx , Marx, A.
sublimation curve and T
Tc
Gross R.
ammonia (NH3) 406
critical O2 154.5
temperatures Tc N2 126 in K of selected H2 33.2 liquid cryogens 4He 5.2 3He 3.32 Chapt. IV - 13 IV.1 Generation of Low Temperatures IV.1.1 Introduction
p solid
liquid
2013)
- T , p T , p Cryogenic Liquids m m b b Tc , pc 1 at
2004 2004 ( gas
Ttr , ptr
Institut -
Tc T
Meißner -
Walther @ 1 bar
© ©
A. Marx , Marx, A.
and
Gross R.
Chapt. IV - 14 IV.1 Generation of Low Temperatures IV.1.1 Introduction direct liquefaction of gases by expansion (Joule-Thomson-Effect)
starting temperature must be smaller than inversion temperature
2013)
-
cryogen boiling liquefaction latent heat inversion 2004 2004
( point [K] [kJ/I] temp. [K] Institut
- oxygen 90.2 1877: Cailletet and Pictet 240 762 Meißner - nitrogen 77.3 1883: Wroblewski and 160 625
Walther Olszewski © ©
hydrogen 20.4 1898: Dewar 30 203 A. Marx , Marx, A. 4
and Helium 4.2 1908: Onnes 2.6 43.2
Gross 3
R. Helium 3.2 0.5 -
• liquid oxygen and hydrogen have potential hazards • liquid nitrogen and 4He are the most widely used cryogens • liquid 3He is very expensive Chapt. IV - 15 IV.1 Generation of Low Temperatures IV.1.1 Introduction
liquefaction of gases by performance of external work
2013)
-
2004 2004
(
Institut
-
Meißner -
gas molecules are reflected at the moving piston-surface:
Walther © © incoming: laboratory system: 푣푀
A. Marx , Marx, A. piston system: 푣푀 − 푣퐾
and outgoing: piston system: − 푣푀 − 푣퐾 ′
Gross laboratory system: − 푣푀 − 푣퐾 + 푣퐾 = 2푣퐾 − 푣푀 = 푣푀 R.
′ i.e.: 푣푀 = 푣푀 − 2푣퐾 molecule is slower, i.e. colder
average momentum transfer per time to piston = force, force · distance = work external work at the expense of internal energy cooling Chapt. IV - 16 IV.1 Generation of Low Temperatures IV.1.1 Introduction
• Carnot process: - counterclockwise: heat pump (conversion of mechanical work into heat)
- clockwise: heat engine (conversion of heat into mechanical work)
2013)
- • pV diagram: 2004 2004 ( expansion cooling: adiabats p 휅 dQ = 0 (adiabatic) (푝푉 = 푐표푛푠푡, 푑푄 = 0 Q12 Institut 1 - 푐 휅 = 푝 > 1)
퐶푉 Meißner - heat exchange: isotherms W 41 2 T = const (푝푉 = 푐표푛푠푡, 푑푇 = 0) 1
Walther (isothermic) © © work per cycle: 4 W23
A. Marx , Marx, A. 푊 = ∮ 푝푑푉 = 퐚퐫퐞퐚
and dQ = 0 3 T2 = const Q34 Gross • efficiency: R. V W T thermodynamic definition of temperature Q T warm • Carnot process: technologically difficult to realize better: gas circulation, compressor and expansion machine are spatially separated Chapt. IV - 17 IV.1 Generation of Low Temperatures IV.1.2 Expansion Machine
• medium: He gas
Brayton method
e.g. liquefaction of air:
2013)
- - condensation on cold head 2004 2004 ( - distillation in separation columns
Institut N (77.4 K) cooling - 2
Ar (87.3 K) inert gas Meißner
- O2 (90.2 K) welding Walther
© © (should not cause significant resistance
• temperature reduction: for flowing gas e.g. A. Marx , Marx, A.
concentric tubes)
and
Gross R.
• efficiency: 휅 = 퐶푝/퐶푉 (= 5/3 for He) expansion from 100 bar to 1 bar
results in T2 = 50 K T2 = 8 K can be reached in a 2 stage cycle Chapt. IV - 18 IV.1 Generation of Low Temperatures IV.1.1 Introduction
• heat pumps: heating and refrigerating machines
p - heat pump:
1 Q12 dQ = 0 2013)
heat is generated by mechanical work - W 41 T = const
2004 2004 1 - efficiency:
( 2
generated heat at T T1 Q1 Institut W - 4 23 h T = const performed work W dQ = 0 Q 3 2 Meißner 34
- V Walther © © - ideal efficiency for reversible Carnot process:
A. Marx , Marx, A. 1 T
1
hC 1 (increases with decreasing temperature difference T1 – T2) and
C T1 T2
Gross R. - refrigerating machine: removing heat (generating cold) by mechanical work
removed heat at T T2 T2 kC hC 1 0 performed work T1 T2
(decreases with increasing temperature difference T1 – T2) Chapt. IV - 19 IV.1 Generation of Low Temperatures IV.1.1 Introduction
e.g. heating of swimming pool e.g. air conditioning
2013)
-
2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
Wikimedia Commons Schematic diagram of a heat pump's vapor-compression refrigeration cycle: 1) condenser, 2) expansion valve, 3) evaporator, 4) compressor.
Chapt. IV - 20 IV.1 Generation of Low Temperatures IV.1.2 Expansion Machine
realizations of expansion machines:
2013) • piston-cylinder machine
-
similar to automobile engine 2004 2004
( crankshaft, camshaft, valve Institut
-
• cooling turbine commercially relevant Meißner - higher efficiency for larger throughput
Walther
© © principle: A. Marx , Marx, A.
brake on turbine axis and
controls rotational speed, Gross
R. annihilates performed work, use of gas bearings
Chapt. IV - 21 IV.1 Generation of Low Temperatures
IV.1.2 Expansion Machine
2013)
- turbine cooler 2004 2004
( (Sulzer machine)
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R. turbine wheel and nozzle ring
(Source: Linde Cryogenics Ltd.) Chapt. IV - 22 IV.1 Generation of Low Temperatures IV.1.2 Expansion Machine
conclusions:
2013)
- • expansion machines are technologically simple 2004 2004
( • multi-stage arrangements for lower temperatures
Institut -
almost down to 4.2 K
Meißner -
• but:
Walther © ©
efficiency only acceptable for cooling turbines
A. Marx , Marx, A.
and
• no direct liquefaction of gas (mechanical problems) Gross R. liquefaction by Joule-Thomson stage • for small-scale facilities: regenerative machines better suited
Chapt. IV - 23 IV.1 Generation of Low Temperatures IV.1.3 Regenerative Machines
• regenerator replaces heat exchanger
column with staple of fine
metal meshes (Cu, Pb)
2013)
-
2004 2004 (
low flow resistance Institut - high heat capacity
Meißner low longitudinal heat conductivity
- Walther
© © • alternating gas flow: cold gas upward
A. Marx , Marx, A. Stirling process (heat engine):
cooling of meshes and p V = const warm gas downward 1 Q12 Gross Q R. cooling of gas 41 2 T1 = const
4 • used in Stirling process Q23 V = const 3 T2 = const Q34 V
cold gained in step 2 →3 has to be stored and provided in step 4 → 1 Chapt. IV - 24 IV.1 Generation of Low Temperatures IV.1.3 Regenerative Machines
• Stirling machine: - periodic expansion and compression of gas along (heat engine) two isotherms and two isochors
p - 1 2: isothermal expansion, Q12 is added V = const - 2 3: isochoric cooling, Q23 is removed 2013) Q
1 12 - - 3 4: isothermal compression, Q34 is removed Q41
2004 2004 - 4 1: isochoric warming, Q is added
( 41 2 T1 = const
- for isochoric steps there is no mechanical work Institut
- 4 푸 = −푸 = 푪 횫푻 Q23 ퟐퟑ ퟒퟏ 푽 - goal: intermediate storage of Q in regenerator
Meißner T = const 23 - V = const 3 2 Q34 to be able to add it again in step 4 1
V use of two pistons with phase shift Walther © © 3 – 4 4 – 1 1 – 2 2 – 3
T1
A. Marx , Marx, A.
and
Gross R.
T2
Chapt. IV - 25 IV.1 Generation of Low Temperatures IV.1.3 Regenerative Machines
• (beta) Stirling machine:
2013)
-
2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Power piston (dark The heated gas The displacer piston The cooled gas is Gross
R. grey) has increases in now moves, now compressed by compressed the pressure and shunting the gas to the flywheel gas, the displacer pushes the power the cold end of the momentum. This piston (light grey) piston to the cylinder. takes less energy, has moved so that farthest limit of the since when it is most of the gas is power stroke. cooled its pressure adjacent to the hot dropped. heat exchanger Chapt. IV - 26 IV.1 Generation of Low Temperatures IV.1.3 Regenerative Machines
• (alpha) Stirling machine:
2013)
-
2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
Chapt. IV - 27 IV.1 Generation of Low Temperatures IV.1.3 Regenerative Machines
conclusions (Stirling machine):
2013)
- •advantages: 2004 2004
( high efficiency Institut
- well-suited for small systems, especially small coolers Meißner - cryocooler
Walther not realizable with turbines © ©
•disadvantages:
A. Marx , Marx, A.
and
mechanically complicated Gross
R. piston (compressor) at low temperature •more simple: Gifford-McMahon machine but: lower efficiency
Chapt. IV - 30 IV.1 Generation of Low Temperatures IV.1.3 Regenerative Machines
Gifford-McMahon machine:
uses compressor with switching valve instead of piston
2013)
- 2004 2004 ( p V = const 1 Q12
Q41
Institut - 2 T1 = const
Meißner 4 - Q23 3 T2 = const
Walther V = const Q34 © © V switching valve
pressure wave from valve
A. Marx , Marx, A.
and cycle: hot
Gross 1. warm compression: 2 1 R. 2. isochoric cooling: 1 4 3. expansion in cylinder: 4 3 regenerator 4. isochoric regeneration: 3 2
5. warm compression: 2 1 cold
Chapt. IV - 31 IV.1 Generation of Low Temperatures IV.1.3 Regenerative Machines
Gifford-McMahon cycle:
2013)
-
2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
Chapt. IV - 32 IV.1 Generation of Low Temperatures IV.1.3 Regenerative Machines
Pulse – Tube – Refrigerator
• the pulse tube refrigerator (PTR) or pulse tube cryocooler is based on operation
2013)
- principle of Stirling cooler
2004 2004 • PTR is made without moving parts in the low temperature part (in contrast with (
other cryocoolers, e.g. Stirling cryocooler and Gifford-McMahon cooler) Institut - • compact design possible suitable for a wide variety of applications
• minimum temperature about 2.5 K (with 4He) and 1.3 K (with 3He)
Meißner
- Walther
© © applications:
• industrial applications such as semiconductor fabrication (e.g. cryopumps) A. Marx , Marx, A.
• cooling of infrared sensors and • cooling of astronomical detectors (e.g. Atacama Cosmology Telescope or the QUBIC
Gross experiment (an interferometer for cosmology studies) R. • precoolers of dilution refrigerators Kurt Uhlig (WMI), “Dry” dilution refrigerator with pulse-tube precooling, Cryogenics 44, (2004), pp. 53–57 • suggested to be used to liquefy oxygen on Mars
Chapt. IV - 33 IV.1 Generation of Low Temperatures IV.1.3 Regenerative Machines
Stirling Cooler Pulse – Tube – Refrigerator
2013)
- regenerator regenerator 2004 2004 ( displacer work work 2-1 piston piston piston
Institut Q - 12 motion of gas volume
regenerator regenerator element equivalent to Meißner - motion of „displacer piston“ buffer
Walther 1-4 volume © © 90° phase shift between
regenerator regenerator motion of „displacer piston“ A. Marx , Marx, A.
and „work piston“ realized and acoustic by buffer volume 4-3 impedance Gross Q34 R. 90° phase shift required for regenerator regenerator finite heat transport 3-2 second (displacer) piston is coldest spot between replaced by pulse tube (gas piston) regenerator and pulse tube
Chapt. IV - 34 IV.1 Generation of Low Temperatures IV.1.3 Regenerative Machines
Stirling Cooler Pulse – Tube – Refrigerator
2013)
-
2004 2004 (
Institut Q
- QH H
Meißner
-
Walther © © Q
Qc c
A. Marx , Marx, A.
and
Gross R.
Qc: removed heat, QH: generated heat
almost sinusoidal motion, phase difference of 90°
Chapt. IV - 36 IV.1 Generation of Low Temperatures IV.1.3 Regenerative Machines
Pulse – Tube – Refrigerator (realizations)
2013)
- commercially
2004 2004 available (
pulse tube Institut - refrigerator
with GM drive
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R. www.cryomech.com 0.5 W @ 4.2 K principle Tmin = 2.3 K (2-stage)
Chapt. IV - 38 IV.1 Generation of Low Temperatures
IV.1.3 Regenerative Machines
2013)
-
2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
pulse tube refrigerator for studies of liquefying oxygen on Mars (580 mm total length)
Chapt. IV - 39 IV.1 Generation of Low Temperatures IV.1.3 Regenerative Machines
conclusion (pulse tube refrigerator):
2013)
- 2004 2004
( •presently very active development Institut
- •no moving parts at low temperatures
long endurance
Meißner -
mobile base stations and satellite applications Walther
© © (e.g. for superconductive microwave filters) A. Marx , Marx, A.
•almost no vibrations
and
Gross •efficiency lower than for displacer R. •only one simpler method: Joule-Thomson cooling
Chapt. IV - 40 IV.1 Generation of Low Temperatures
IV.1.4 Joule-Thomson Cooling
2013)
-
2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R. William Thomson (Lord Kelvin) Born: 26 June 1824, Belfast, Northern Ireland Died: 17 December 1907, Netherhall, Largs Ayrshire, Scotland
Chapt. IV - 41 IV.1 Generation of Low Temperatures IV.1.4 Joule-Thomson Cooling
principle:
• gas performs work against its
2013)
own internal attractive forces
- 2004 2004 ( • working medium/gas (V1) flows
through impedance and
Institut -
expands to V2
Meißner - U U = 0 (adiabatic)
Walther 2 1 © © 1st law of thermodynamics:
A. Marx , Marx, A. U Q W
and
V2 0 Gross
R. p2dV2 p1dV1 p2V2 p1V1 0 V1
U2 p2V2 U1 p1V1 this means: process with constant enthapy: H U pV const.
- for ideal gas: p1V1 = p2V2 and hence U1 = U2 resp. T1 = T2 no cooling !! Chapt. IV - 43 IV.1 Generation of Low Temperatures IV.1.4 Joule-Thomson Cooling
real gas: transformation of gas into liquid on decreasing T and (or) short-distance repulsion increasing p due to 4
2013) work against attractive interaction between the
- Epot(r) 3
molecules 2
2004 2004 ( weak long-range attraction: 1
Energy 0 Institut
- tends to keep molecules closer together, same effect -1
as additional compression of the gas. -2
-3 Meißner
- a 1.5 2.0 2.5 3.0 3.5 4.0 distancer peff p 2
Walther V © © a is a measure of the long-range attraction long-distance
strong short-range repulsion: attraction A. Marx , Marx, A.
molecules are rigid: p and as soon as the molecules “touch” each other. expansion (decrease of pressure): Gross R. low pressure: Veff V b attraction costs work b (≈ 4휋휎3/3): “excluded volume” per particle cooling of gas Van der Waals equation: high pressure: repulsion provides work a p V b RT heating of gas 2 m Vm Chapt. IV - 44 IV.1 Generation of Low Temperatures IV.1.4 Joule-Thomson Cooling
E
interaction potential: pot
2013)
-
r
2004 2004
(
Institut -
repulsive attractive
Meißner -
Walther with U (p,T) © ©
T = const.
A. Marx , Marx, A.
and
p Gross R. attractive repulsive
minimum
Chapt. IV - 45 IV.1 Generation of Low Temperatures IV.1.4 Joule-Thomson Cooling
• more detailed analysis of Joule-Thomson process (isenthalpic expansion):
H H
H T p 0 2013)
- T p p T 2004 2004 ( H H with Cp CpT p
Institut
- T p p T Meißner - 1 H T Joule-Thomson coefficient JT 휇퐽푇 > 0: cooling on expansion
Cp p T p H Walther
© © 휇퐽푇 < 0: heating on expansion A. Marx , Marx, A.
H S
with H TS Vp T V and
p T p T Gross R. S V p T T p
T 1 H 1 V JT T V p H Cp p T Cp T p
Chapt. IV - 46 IV.1 Generation of Low Temperatures IV.1.4 Joule-Thomson Cooling
V R V ideal gas: pV RT JT 0
T p p T
2013)
- 3 5 H 2004 2004 ( H(T,p) U pV NkBT NkBT NkBT const. µJT 0
2 2 p T
Institut -
equipartition theorem ideal gas law @ T = const Meißner
- for monoatomic gas
Walther
© ©
A. Marx , Marx, A.
and
areas of Gross
R. H = const. lines of intersection with H = const.
Chapt. IV - 47 IV.1 Generation of Low Temperatures IV.1.4 Joule-Thomson Cooling
a 5 real gas: p V b RT 2 m H(T,p) U pV NkBT U(p,T) const.
Vm 2
a
2013)
- at low densities: we use approximation p , V b and obtain
V 2
2004 2004 ( a
pV pb RT, ...p Institut
- V T Meißner - V a V V R p 2 R, T p V T p T p a Walther p 2
© © V A. Marx , Marx, A. T 1 H 1 V insert into JT T V and
p H Cp p T Cp T p Gross R. T 1 a JT 2 b p H CP RT
μJT > 0 for T < 2a/bR cooling on expansion 2a inversion temperature: Tinv μJT < 0 for T > 2a/bR heating on expansion bR Chapt. IV - 48 IV.1 Generation of Low Temperatures IV.1.4 Joule-Thomson Cooling
inversion curve
2013)
-
2004 2004
(
Institut -
areas of lines of intersection with Meißner
- H = const. H = const.
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
Chapt. IV - 49 IV.1 Generation of Low Temperatures IV.1.4 Joule-Thomson Cooling maximum Joule-Thomson coefficient (without approx.): inversion 2 temperature (2a RT)(1 b V ) b JT
2013) 2
- Cp 1 2a VRT 1 b V
2004 2004 (
large volume (p >> a/V2, V >> b) :
Institut - 1 a
JT 2 b Meißner - CP RT
Walther inversion curve: points where µ = 0 © © JT
2
A. Marx , Marx, A. vdW gas: (2a/RT)(1-b/V) = b
and
inversion temperature Tinv: Gross
R. 2 2a b T 1 inv bR V
equation of state gives Tinv(p,T) 2a maximum inversion temperature: T inv bR Chapt. IV - 50 IV.1 Generation of Low Temperatures IV.1.4 Joule-Thomson Cooling
experimental data for N2:
2013)
-
2004 2004
( Institut - T repulsion
Meißner JT -
p H Walther © © slope of attraction
isenthalps
A. Marx , Marx, A.
and
Gross R.
Chapt. IV - 51 IV.1 Generation of Low Temperatures IV.1.4 Joule-Thomson Cooling
gas maximum inversion temperature [K]
2013) Helium-3 (23)
-
2004 2004 Helium-4 45 (
Hydrogen 205
Institut -
Neon 250 Meißner - Nitrogen 621
Walther Air 603 © ©
Carbon monoxide 652 A. Marx , Marx, A.
Argon 794
and
Gross Oxygen 761 R. Methane 939 Carbon dioxide 1500 Ammonia 1994
vdW gas can be liquefied only for T < T !!! inv Chapt. IV - 52 IV.1 Generation of Low Temperatures IV.1.4 Joule-Thomson Cooling
closed cycle cooling: “Linde-process”
2013)
-
2004 2004
(
Institut - Carl von Linde
Meißner (1842 – 1934)
-
Walther
© ©
A. Marx , Marx, A.
and
Gross
R. • gas is cooled by JT-expansion until liquid drops out the impedance • patent application by Carl von Linde on May 12, 1903 (liquefaction of oxygen)
(Source: PTB Braunschweig) Chapt. IV - 53 IV.1 Generation of Low Temperatures IV.1.4 Joule-Thomson Cooling
Lindesche
Gasverflüssigungsanlage (1895)
2013)
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2004 2004
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Institut
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© ©
A. Marx , Marx, A.
and
Gross R.
Chapt. IV - 54 IV.1 Generation of Low Temperatures IV.1.4 Joule-Thomson Cooling
Professor Dr. Carl Paul Gottfried von Linde (* 11. Juni 1842 in Berndorf, Oberfranken; † 16. November 1934 in München) war ein deutscher Ingenieur, Erfinder und Gründer eines heute internationalen Konzerns, der Linde AG.
Linde begann 1861 ein Studium am Polytechnikum Zürich, wo Rudolf Clausius, Gustav Zeuner und Franz Reuleaux seine Lehrer waren. 1864 beendete er sein
Studium. Reuleaux vermittelte ihm eine Lehrstelle in der Baumwollfabrik Kottern in Berlin, die er im selben Jahr antrat. Es war aber nur kurze Zeit, bevor er nach
München zog, um als Konstrukteur bei der Lokomotivenfabrik Krauss zu arbeiten. 1866 heiratete er Helen Grimm: aus der 53-jährigen Ehe folgten sechs Kinder.
1868 folgte er einem Ruf der Polytechnischen Schule München, wo er zunächst - mit erst 26 Jahren - außerordentlicher Professor, 1872 dann ordentlicher
2013)
- Professor für Maschinenbau wurde. Am Polytechnikum richtete Linde ein Maschinenlabor ein, an dem unter anderem Rudolf Diesel ausgebildet wurde. 1871
veröffentlichte Linde einen Aufsatz über verbesserte Kältetechnikverfahren. Viele Brauereien interessierten sich dafür, und bald versorgte Linde sie mit den neuen 2004 2004 ( Maschinen, an denen er ständig arbeitete.
Linde schuf wesentliche Grundlagen der modernen Kältetechnik. 1871 konzipierte er eine mit Methylether arbeitende Kältemaschine, die er in der Institut - Maschinenfabrik Augsburg (heute MAN AG) herstellen ließ. Die zweite, 1876 folgende Generation von Kühlmaschinen arbeitete mit Ammoniak. Das Prinzip der Abkühlung von Gas, das vorher mechanische Arbeit geleistet hatte, war beiden gemeinsam. Ein Preisausschreiben für eine Kühlanlage zum Auskristallisieren von
Paraffin war 1873 für den Hochschullehrer der Anreiz zum Bau einer Kühlmaschine, die beim Bierbrauen die Gärung bei konstanter Temperatur zuließ. Brauereien Meißner - in ganz Europa (so Dreher in Triest, die Mainzer Actien-Bierbrauerei, Spaten in München, Heineken in den Niederlanden, Carlsberg in Dänemark) interessierten sich prompt für die neue Kältetechnik.
Am 21. Juni 1879 gab der Erfinder sein Lehramt auf und rief mit zwei Brauern und drei anderen Gründern die "Gesellschaft für Linde’s Eismaschinen AG" ins Leben Walther
© © (heute Linde AG). Nach relativ kurzer Zeit war das Unternehmen in Europa führend auf kältetechnischem Gebiet, auch begünstigt durch einen milden Winter 1883/1884. Es kam deshalb zu einer Knappheit bei Natureis, das zum Kühlen des Gerstensaftes in Bierkellern eingesetzt wurde. Bisherige Vorbehalte der Brauer gegen das Kunsteis schmolzen dahin, Kühlmaschinen waren plötzlich gefragt und Linde lieferte umgehend.
Kühlhäuser für Lebensmittel und mehrere Eiswerke ließ Linde nach und nach sogar selbst bauen. Doch auch auf Eislaufbahnen, in Molkereien und bei der A. Marx , Marx, A.
Verflüssigung von Chlor und Kohlensäure war sein Verfahren gefragt. Die Firma florierte, 1890 zog sich Linde aus dem operativen Geschäft in den Aufsichtsrat and
seiner Aktiengesellschaft zurück. In den Jahren 1892 bis 1910 nahm er seine Professur wieder auf.
Gross Auf der Grundlage der Arbeiten von James Prescott Joule, Sir William Thomson (Lord Kelvin of Largs) und der Einführung des Gegenstromverfahrens konnte Linde
R. 1895 erstmals größere Mengen Luft verflüssigen (Linde-Verfahren). Damit schuf er die Möglichkeiten für physikalische Tieftemperaturuntersuchungen und zur Trennung der Luftbestandteile durch fraktionierte Destillation. 1901 folgte die Errichtung einer Anlage zur Gewinnung von Sauerstoff und (ab 1903) Stickstoff.
Linde war Mitglied wissenschaftlicher und Ingenieurvereinigungen, unter anderem gehörte er dem Kuratorium der Physikalisch-Technischen Reichsanstalt und der Bayerischen Akademie der Wissenschaften an. Er wurde vom bayerischen König Ludwig II in den nicht erblichen Adelsstand erhoben. Linde war 1916 der erste Preisträger des Siemens-Rings.
Ab 1910 zog sich Linde als Direktor seiner inzwischen ungeheuer erfolgreichen Aktiengesellschaft zurück und reichte sie an seinen Söhne Friedrich und Richard weiter. Die Weltwirtschaftskrise von 1929 versetzte der Linde AG einen starken Schlag; das Unternehmen erholte sich aber, und die Gewinne fingen schon wieder an zu steigen, bevor Linde 1934 im Alter von 92 Jahren starb. Chapt. IV - 55 IV.1 Generation of Low Temperatures
IV.1.4 Joule-Thomson Cooling
2013)
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2004 2004
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Institut
-
Meißner
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Walther
© ©
A. Marx , Marx, A.
and
Gross R.
schematics of a Helium liquefier
Chapt. IV - 56 IV.1 Generation of Low Temperatures
IV.1.5 Summary
2013)
-
2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
Chapt. IV - 57 IV.1 Generation of Low Temperatures
IV.1.5 Summary
2013)
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2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
Chapt. IV - 58 IV.1 Generation of Low Temperatures IV.1.5 Summary
Northrop Grumman's HEC cryocooler
Sumitomo Heavy Industries
2013)
-
2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and specification for cryocooler: Stirling cycle miniature cryocooler:
Gross • 1 Watt of cooling @ 80 K, rejecting heat at 300 K - lightweight cooler, ideal for cooling of R. • 10 year life sensors and other electronics when low • 230 K to 340 K survival temperature power consumption is important • survival of launch vibration (non-operating) - mean time before failure of 24,000 hours • low exported vibration - cooling capacity of 1 W @ 80 K • high efficiency - power consumption of only 55 W. • no maintenance possible oil-free
Chapt. IV - 59 IV.1 Generation of Low Temperatures
IV.1.6 Evaporation Cooling
2013)
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2004 2004
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Institut
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Walther
© ©
A. Marx , Marx, A.
and
Gross R.
Chapt. IV - 60 IV.1 Generation of Low Temperatures IV.1.6 Evaporation Cooling
• everyday experience: sweating, wind direction, cooling of coffee, …
moisten finger, evaporation cooling
2013)
-
2004 2004 (
• microscopically:
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R. • evaporation: work required to overcome binding forces only the fastest molecules will do it high-energy particles are lost liquid cools down
Chapt. IV - 61 IV.1 Generation of Low Temperatures IV.1.6 Evaporation Cooling
• limit of evaporation cooling: kBT becomes too small compared to Hvap
(heat of evaporation)
2013)
- • Hvap should be small to reach large cooling power at low temperatures 2004 2004
( • numbers: about 1 K can be reached with 4He, about 0.3 K with 3He
Institut
- Meißner - • boiling point can be calculated by using the Clausius-Clapeyron equation, if
Walther heat of vaporization and the vapor pressure of the liquid at a certain
© © temperature is known
A. Marx , Marx, A.
and
Gross R.
Chapt. IV - 62 IV.1 Generation of Low Temperatures IV.1.6 Evaporation Cooling
Clausius-Clapeyron equation:
Hvap: molar latent
2013) heat [J/mole]
-
2004 2004 ≈ 90 J/mole (
• approximate expression using pV = RT (ideal gas): for 4He
Institut
-
Meißner
- Walther
© © • integration yields (assuming that Hvap is constant over the considered T range):
A. Marx , Marx, A.
and
Gross
R. • normal boiling temperature: pressure above liquid (boiling point corresponds to the temperature at which the vapor pressure of the liquid equals the surrounding environmental pressure) boiling temperature at p0 Chapt. IV - 63 IV.1 Generation of Low Temperatures
IV.1.6 Evaporation Cooling
2013)
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2004 2004
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Institut
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Meißner
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© ©
A. Marx , Marx, A.
and
Gross R.
Chapt. IV - 64 IV.1 Generation of Low Temperatures
IV.1.6 Evaporation Cooling
2013)
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2004 2004
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Institut
-
Meißner
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Walther
© ©
A. Marx , Marx, A.
and
Gross R.
www.mallister.com/graphics/vapor3.jpg Chapt. IV - 65 IV.1 Generation of Low Temperatures IV.1.6 Evaporation Cooling
liquid 4He (c.f. chapter II)
2013) •
boson - •
2004 2004 liquid down to 0 K (@ 1 atm) (
• superfluid 4Helium at 2.17 K Institut - – Bose condensation: macroscopic number
of atoms in ground state Meißner - – very low viscosity
Walther – very high heat conduction © ©
– strange thermomechanical effects A. Marx , Marx, A.
– creeping on vertical surfaces and
– vortex core with radius 0.8Å @ 0.6K Gross
R. – explained by a two-fluid model • density 125 kg/m3
Chapt. IV - 67 IV.1 Generation of Low Temperatures IV.1.6 Evaporation Cooling
Liquid Helium cryostats:
2013)
- 2004 2004
( • LHe has small latent heat Institut - good thermal insulation by vacuum
Meißner LHe container of poor thermal conductivity, glass or -
stainless steel Walther
© © thermal radiation shield at liquid Nitrogen temperature A. Marx , Marx, A.
to reduce black-body radiation
and
Gross R. bath cryostat - sample is immersed in the LHe gas flow cryostat - sample is located in cold He gas
Chapt. IV - 68 IV.1 Generation of Low Temperatures IV.1.6 Evaporation Cooling
Liquid Helium container
2013)
-
2004 2004
(
Institut
-
Meißner
- Walther © © narrow neck to minimize
A. Marx , Marx, A.
- heating by radiation and
vacuum
- heating by thermal conduction radiation shields
Gross R. typical losses LHe - 1 l of LHe / day
Chapt. IV - 69 IV.1 Generation of Low Temperatures IV.1.6 Evaporation Cooling
He-bath cryostat
2013)
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2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
Chapt. IV - 70 IV.1 Generation of Low Temperatures IV.1.6 Evaporation Cooling
He-gas flow cryostat
2013)
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2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
Chapt. IV - 71 IV.1 Generation of Low Temperatures IV.1.6 Evaporation Cooling
He-gas flow cryostat (II)
2013)
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2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
• no liquid Nitrogen required • radiation shield cooled by cold helium return gas
Chapt. IV - 72 IV.1 Generation of Low Temperatures IV.1.6 Evaporation Cooling
Liquid 4He temperature < 4.2 K Clausius-Clapeyron equation:
• reducing the vapour pressure over
2013)
- bath of 4Helium
Hvap: molar latent heat [J/mole] 2004 2004 ( 4 • temperature down to 1.2 K ≈ NA∙Ebinding: 90 J/mole for He
3 Institut
- at pumping speed of 10 m /h
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
cooling power:
Hvap Q n Hvap p exp gas RT rate of atoms going to gas phase up to 10 mW cooling power @ 1.2K Chapt. IV - 73 IV.1 Generation of Low Temperatures IV.1.6 Evaporation Cooling
Liquid 3Helium (c.f. chapter II):
• fermion
2013)
- • superfluid at 2.5mK
2004 2004 (
– formation of weakly bound fermions: Cooper pairs Institut - • density 59 kg/m3
4 Meißner - • higher vapour pressure than He due to smaller latent heat: 3
Hvap = 40 J/mole cooling power ≈ 80mW @ 1.2K and 10 m /h pumping Walther
© © speed
• 0.3 K by pumping 3He vapour A. Marx , Marx, A.
3 and – some cm
Gross – 0.1mW cooling power @ 0.3K R.
• 3He obtained by nuclear reactions • extremely expensive • 1 liter of 3He gas costs about US $5.000 (2012)
Chapt. IV - 75 IV.1 Generation of Low Temperatures IV.1.6 Evaporation Cooling
3 Liquid He cryostat 3 latent heat of He: Hvap = 40 J/mole 3 as compared to 90 J/mole for 4He
He
larger cooling power 2013)
backflow - 4 3
2004 2004 He He ( Hvap Q n Hvap p exp
pump pump gas RT Institut - 3
Meißner ≈ 80mW @ 1.2 K for He as compared to -
≈ 10mW @ 1.2 K for 4He
Walther © ©
4 4He impedance
A. Marx , Marx, A. He
and bath 3 ≈ 1.2K 4He condensation of backflowing He gas
Gross 4.2 K R. vacuum flow restriction for condensed 3He
3He ≈ 0.3 K minimum temperature: ≈ 300 mK (cooling power ≈ 0.1 mW)
Chapt. IV - 76 IV.1 Generation of Low Temperatures IV.1.7 Dilution Cooling
revision of some facts on 3He/4He mixtures
cf. chapter I.6
2013)
- makes use of miscibility gap of 3He/4He mixtures (compare section I.6)
2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
Chapt. IV - 77 IV.1 Generation of Low Temperatures IV.1.7 Dilution Cooling – revision of I.6
• bonding: A A A B B B
VAA VAB VBB
2013)
- 1 • 푉퐴퐵 > 푉퐴퐴 + 푉퐵퐵 complete miscibility (e.g. water and alcohol)
2004 2004 2 (
1 Institut - • 푉 + 푉 > 푉 phase separation (e.g. water and petrol)
2 퐴퐴 퐵퐵 퐴퐵 Meißner - optimization of binding energy
complete phase separation @ T = 0
Walther © © increasing mixing with increasing T:
critical point
A. Marx , Marx, A.
and
F = U – TS min
Gross R. miscibility gap binding thermal energy motion
minimization of free energy
Chapt. IV - 79 IV.1 Generation of Low Temperatures IV.1.7 Dilution Cooling – revision of I.6
3 3 4 • binding energy of He in He (V33) and He (V34):
3 3 (i) He in He: binding energy is given by the latent heat of evaporation L3: 퐿 휇 2013) 3 3,푐 3 - binding energy for single He atom: 휖3,푐 = − =
푁퐴 푁퐴
2004 2004 ( 휇3,푐 = chemical potential of pure (concentrated)
liquid
Institut -
3 4
(ii) single He atom in liquid He: Meißner - 휇 푥 →0 3 3,푑 3 binding energy for single He atom: 휖3,푑(푥3 → 0) =
푁퐴 Walther
© © 휇3,푑 = chemical potential of dilute phase
• 휖3,푑 0 > 휖3,푐 or vice versa ? A. Marx , Marx, A.
and 3He has smaller mass larger zero point fluctuations occupies larger volume
Gross 3 4 3 4
R. binding of He is larger in He than in He due to larger density of He
|휖3,푑 0 | > |휖3,푐|
1 corresponds to case 푉 > 푉 + 푉 complete miscibility expected, 퐴퐵 2 퐴퐴 퐵퐵 why miscibility only up to 6.5% ?? Chapt. IV - 80 IV.1 Generation of Low Temperatures IV.1.7 Dilution Cooling – revision of I.6
Questions:
• why can‘t we dissolve more than 6.5% of 3He in 4He at T = 0 ?
2013)
- two effects: 2004 2004 (
(i) 3He forms degenerate Fermi liquid
Institut - 푇퐹 increases with 푥3
Meißner for 푥 > 6.5%, the Fermi energy exceeds the gain in binding energy - 3
Walther (ii) 휖 푥 > 휖 (푥 = 0) © © 3,푑 3 3,푑 3
effective attraction between two ³He atoms (magnetic and volume effect)
A. Marx , Marx, A.
and
Gross • why don‘t we have a complete phase separation into 3He and 4He at T = 0 ? R.
results in finite disorder, violation of 3. law of thermodynamic ? no: we have degenerate Fermi gas, ordering in k-space
Chapt. IV - 81 IV.1 Generation of Low Temperatures IV.1.7 Dilution Cooling – revision of I.6
energy 0.065
gaseous ³He 0
2013)
- x3
2004 2004
(
Institut
- Meißner - pure liquid ³He −휖3,푐
Walther 푬퐩퐨퐭 풙ퟑ = ퟏ + 풌퐁푻퐅 풙ퟑ = ퟏ 푬 풙 + 풌 푻 풙
© © 퐩퐨퐭 ퟑ 퐁 퐅 ퟑ A. Marx , Marx, A.
³He diluted in 4He −휖3,푑(0)
and
Gross 푬 풙 R. 퐩퐨퐭 ퟑ
for 푥3 > 6.5%: 푬퐩퐨퐭 풙ퟑ + 풌퐁푻퐅 풙ퟑ > 흐ퟑ,풄 = 푬퐩퐨퐭 풙ퟑ = ퟏ + 풌푩푻퐅 풙ퟑ = ퟏ
separation of pure ³He Chapt. IV - 82 IV.1 Generation of Low Temperatures IV.1.7 Dilution Cooling – revision of I.6
• operation principle: concentrated remove ³He atoms from the dilute phase below Tk = 0.87 K (lighter)
transport of ³He atoms across phase boundary
to maintain equilibrium concentration
2013) ³He
- corresponds to evaporation of ³He from concentrated phase 2004 2004 ( cools as the latent heat of evaporation is removed diluted,6.5%
Institut (heavier) - • for Fermi liquid: 2/3 ³He
Meißner C < C (x = 0.065) (퐶푉 ∝ 푇/푇퐹, 푇퐹 ∝ 푛 ) - concentrated diluted 3
Walther • with we therefore obtain: © ©
A. Marx , Marx, A. Uconcentrated (T) < Udiluted (T) U
and Udiluted • on transition across phase boundary: Gross Uconcentrated R. F ≈ U = const
increase of U results in decrease of temperature ! T ³He/4He dilution refrigerator Chapt. IV - 83 IV.1 Generation of Low Temperatures IV.1.7 Dilution Cooling – revision of I.6
• assumption: one mole of ³He crosses boundary
• removed heat depends on enthalpy difference between concentrated and dilute phase
2013)
- 2004 2004 (
Institut • cooling power -
Meißner
-
• since there is no volume change Walther
© ©
A. Marx , Marx, A.
and
Gross • with and (standard expressions for Fermi liquid) R.
we obtain the entropy
Chapt. IV - 84 IV.1 Generation of Low Temperatures IV.1.7 Dilution Cooling – revision of I.6
plausibility consideration thermally
kBT excited 3He atoms
2013)
- Fermi 3 3 large He density
2004 2004 He, concentrated ( sphere large Fermi sphere
high TF
Institut - k T
Meißner B -
3 small 3He density Walther He, diluted Fermi © © small Fermi sphere sphere
low TF A. Marx , Marx, A.
fraction of thermally and excited 3He atoms
Gross increases ( T/TF) R.
entropy increases going from concentrated to diluted phase
removed heat: dQ = TdS
Chapt. IV - 85 IV.1 Generation of Low Temperatures IV.1.7 Dilution Cooling
2 • large cooling power Q 84 n 3 T requires large throuphput 3 pumping of He generates
from ≈ 1.2 K 2013)
osmotic pressure - 3He condenser 3 3He flows from mixing chamber
2004 2004 He ( pump to still
3 Institut
- only possible if He atoms cross
flow impedance phase boundary Meißner
- cooling Walther
© © heater to allow for > 90% 3He still effective pumping of 3He
(≈ 0.7 K) < 1% 3He A. Marx , Marx, A.
heat exchangers
and
Gross dilute concentrated R. phase phase
mixing 100% 3He concentrated phase minimum temperature: chamber 6.5% 3He dilute phase ≈ 1.5 mK (≈ 0.01 K)
Chapt. IV - 86 IV.1 Generation of Low Temperatures IV.1.7 Dilution Cooling
• example: desired cooling power: 10-5 W
still temperature: 0.7 K 2013)
mixing chamber temperature: 10 mK
- 2004 2004 ( • what is the required pumping speed ??
Institut - 105 2 n3 0.0012 mole / s Q 84 n3 T Meißner 2 2 - 84(10 )
Walther we assume that 3He is an ideal gas (R = 8.31 J / mole K) © ©
A. Marx , Marx, A. V n RT / p 푅 = 8.31 J/mole K
3
and
3
Gross • numbers: vapor pressure of He at 0.7 K: 0.0828 mbar = 8.28 Pa R. @ 300 K we obtain:
V 0.00128.31300/8.28 0.363 m3 /s 360 l/s
large 3He pump is required
Chapt. IV - 87 IV.1 Generation of Low Temperatures
IV.1.7 Dilution Cooling
2013)
-
2004 2004
( Institut
- still
Meißner
- Walther © © heat
exchangers
A. Marx , Marx, A.
and
Gross R.
mixing chamber
Chapt. IV - 88 IV.1 Generation of Low Temperatures
IV.1.7 Dilution Cooling
2013)
-
2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
JANIS Model JDR-500 Dilution Refrigerator Chapt. IV - 89 IV.1 Generation of Low Temperatures IV.1.8 Pomeranchuk Cooling
104 3He shows minimum in melting phase diagram of (compare I.5.2) curve at T = 0.32 K 4 3 fcc
He and He
2013)
- can be used for cooling of ³He 2004 2004
( Pomeranchuk effect
103
Institut -
3He
Meißner -
Walther 4He © ©
(bar) hcp hcp
p p 2 A. Marx , Marx, A.
10 and
bcc
Gross R. bcc
Isaak Jakowlewitsch Pomeranschuk liquid liquid (Исаак Яковлевич Померанчук; 101 born: 20. Mai 1913 at Warschau; 0.1 1 10 died: 14. Juli 1966 at Moscow) Russian physicist. T (K) Chapt. IV - 90 IV.1 Generation of Low Temperatures IV.1.8 Pomeranchuk Cooling
• Clausius-Clapeyron equation:
• always: Vsol < Vliq
2013)
- disorder larger in
2004 2004 • for T < 0.32 K:
( solid than in liquid Institut - • explanation: phase !!
solid phase: atoms are ordered,
Meißner - spins are disordered and determine entropy: Ssolid = R ln2
Walther at low T: antiparallel ordering of spins, S decreases towards zero © ©
liquid phase: atoms are spatially disordered, but ordering in k-space (Fermi liquid) A. Marx , Marx, A.
entropy of Fermi liquid: and S
R ln2
Gross R. solid • cooling power: liquid 2 T 320 mK Q n3T(Ssolid - Sliquid) TRln2- 2 TF 0 lnT Chapt. IV - 91 IV.1 Generation of Low Temperatures IV.1.8 Pomeranchuk Cooling
3.4 stamp stainless steel 3.3
bellow solid
liquid 3.2
2013)
-
3.1 liquid
2004 2004 (
(MPa) 3.0 p p
Institut solid pressure cell -
2.9
Meißner T
- min • precooling to T < T 2.8 min -3 -2 -1 0 10 10 10 10
Walther © © • adiabatic compression 0.8 ln 2 solid
solidification and cooling A. Marx , Marx, A.
0.6 and
• lowest T: ≈ 1.5 mK Gross
R. limitation due to antiparallel 0.4
spin ordering in solid 3He R / S 0.2 liquid
0.0 10-3 10-2 10-1 100 T (K) Chapt. IV - 92 IV.1 Generation of Low Temperatures IV.1.9 Adiabatic demagnetization
magnetic refrigeration: based on the magnetocaloric effect
2013)
-
magnetocaloric effect: 2004 2004
( Institut
- - magneto-thermodynamic phenomenon
Meißner - - reversible change in temperature is
caused by exposing a material to a Walther
© © changing magnetic field
A. Marx , Marx, A.
- also known as and
adiabatic demagnetization
Gross R.
Chapt. IV - 93 IV.1 Generation of Low Temperatures IV.1.9 Adiabatic demagnetization
Adiabatic magnetization: thermodynamic cycle: - substance is placed in an insulated environment - increasing external magnetic field (+H) causes magnetic
ordering reduction of magnetic entropy and heat capacity
- overall energy is not lost and therefore total entropy is not
2013)
- reduced (T + ΔTad).
2004 2004 ( Isomagnetic enthalpic transfer:
- added heat is removed by coupling to heat sink (-Q) Institut - - magnetic field is held constant
- after heat removal, magnetocaloric material and the coolant Meißner - are separated (H = 0).
Walther Adiabatic demagnetization: © © - the substance is decoupled from heat sink no heat
exchange with environment entropy stays constant A. Marx , Marx, A.
- magnetic field is decreased, the thermal energy causes the and
magnetic moments to overcome the field, and thus the sample
cools energy (and entropy) transfers from thermal entropy Gross
R. to magnetic entropy (disorder of the magnetic dipoles).
Isomagnetic entropic transfer: - the magnetic field is held constant to prevent the material from heating up - the material is brought in thermal contact with the environment being refrigerated cooling effect - magnetic materials heats up (+Q) Chapt. IV - 94 IV.1 Generation of Low Temperatures IV.1.9 Adiabatic demagnetization
• generation of temperatures below 1 mK
• dQ = TdS = dU – dW = dU + pdV – B dM ≈ dU – B dM (for solid: pdV small)
2013)
-
• adiabatic cooling: dQ = 0 dU = B dM
2004 2004 (
– step1: magnetize sample with field B, work must be done on sample and Institut - heat is released to thermal bath at Ti
– step2: thermally isolated sample and remove magnetic field B, sample
Meißner - uses internal energy to demagnetize and temperature falls to Tf < Ti
Walther © ©
퐵푖 퐵푓 퐵푓
A. Marx , Marx, A. • 푆 = 푆 푇 = 푇 ⋅ adiabatic 푓 푖
푇푖 푇푓 퐵푖
and
• experiment:
Gross R. use of copper: Bi = 3 T, Bf = 0.3 mT, Ti = 10 mK Tf ≈ 1μK
Chapt. IV - 95 IV.1 Generation of Low Temperatures IV.1.9 Adiabatic demagnetization
more detailed discussion:
• which amount of heat Qspin can be absorbed by the spin systems ?
2013)
- 2004 2004 ( Ti Ti Sspin Qspin (B 0) CspindT T dT (cooling capacity)
Institut T T
- f f T B
Meißner - • entropy of spin system with spin quantum number J for gµBB << kBT:
Walther
© © 2 2 2 2 g J(J 1)B B Bint S NkB ln(2J 1) 2 2
A. Marx , Marx, A. 6k T
B
and
• final temperature
Gross R. B2 B2 T T f int f i 2 2 Bi Bint remaining internal field due to finite magnetic interactions (should be as small as possible) Chapt. IV - 96 IV.1 Generation of Low Temperatures IV.1.9 Adiabatic demagnetization
Ti Ti S spin Qspin (B 0) CspindT T dT Tf Tf
T B
NkB ln(2J 1)
2013)
-
2004 2004
( heat sink Institut
- heat
entropy S entropy Bf = 0 Bi = 5T switch Meißner - 1
medium Walther
© © 2
A. Marx , Marx, A. 0 0 T T and f i T
Gross R. 1 switch on magnetic field at constant Ti (coupling to heat sink)
2 switch off magnetic field for thermally isolated sample cooling to Tf
Chapt. IV - 97 IV.1 Generation of Low Temperatures IV.1.9 Adiabatic demagnetization
• paramagnetic salts:
e.g MAS = MnSO4 · (NH4)2SO4 · 6H2O
cooling of electron spins 2013) - material with large entropy S/R but large Bint lowest temperatures T ≈ 100 mK 2004 2004 f
( large cooling capacity Institut -
e.g CMN = 2Ce(NO3)3 · 2Mg(NO3)2 · 24H2O Meißner - cooling of electron spins
material with small entropy S/R but small Bint Walther
© © lowest temperatures Tf ≈ 2 mK
small cooling capacity A. Marx , Marx, A.
and • nuclear demagnetization:
Gross e.g 63Cu (L = 3/2) or 65Cu (L = 3/2) (B ≈ 0.3 mT, T ≈ 10 mK, B ≈ 3 T)
R. int i i cooling of nuclear spins
Tf (Bf = 0) ≈ 1 µK problem: transfer of spin temperature to lattice long spin-lattice relaxation time
141 195 other materials: PrNi5 (L=5/2), Pt (L=1/2) Chapt. IV - 98 IV.1 Generation of Low Temperatures IV.1.9 Adiabatic demagnetization
• spin-lattice coupling cools the lattice
– metals: ≈ 1000 s (hyperfine interaction: t = CKorringa/Te, Korringa relation)
– non-metals: ≈ hours to several days
2013)
- 2004 2004
( • spin-lattice coupling: spin temperature increases and lattice temperature decreases
in thermal equilibrium Institut - Teq Teq
Meißner - celectrondTe cnucleardTn 0
Ti Tf Walther
© © • calculation of equilibrium temperature for copper: (Bint ≈ 0.3 mT, Ti ≈ 10 mK, Bi ≈ 3 T)
A. Marx , Marx, A.
Teq = 1.03 μK and
• lowest reported experimental temperature Gross
R. demagnetization of Pt: Teq = 2 μK (F. Pobell et al., (1996))
Chapt. IV - 99 IV.1 Generation of Low Temperatures IV.1.9 Adiabatic demagnetization
Cu demagnetization stage
(length: 525 mm, diameter: 78 mm)
2013)
-
2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
R. Gloss et al., J. Low Temp. Phys. 73, 101 (1988)
Chapt. IV - 100 IV.1 Generation of Low Temperatures IV.1.9 Adiabatic demagnetization
µK facility of PTB Berlin:
2013) Lattice temperatures measured on the
- 105-mol-copper stage of the Berlin 2004 2004 ( microkelvin facility with Pt-NMR. The
achieved minimal temperature was Institut - 23.3 µK. The red line depicts the
calculated course of temperature for Meißner - the thermodynamically optimized
demagnetization function. Walther
© © heat leak: below 1.5 nW.
A. Marx , Marx, A.
and
Gross R.
Chapt. IV - 101 IV.1 Generation of Low Temperatures IV.1.9 Adiabatic demagnetization
Cryogen-free Two Stage Adiabatic
Demagnetization Refrigerator from Janis
2013)
-
2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross A cryogen-free two stage adiabatic demagnetization refrigerator using a 4 K pulse tube cryocooler. R. Gallium Gadolinium Garnet (GGG) and Ferric Ammonium Alum (FAA) paramagnetic pills were used for the first and second stage of the ADR, with Kevlar string supports for each stage. The FAA stage reaches a base temperature below 50 mK, and remains below 100 mK for more than two days.
Chapt. IV - 102 IV.1 Generation of Low Temperatures IV.1.9 Adiabatic demagnetization
Continuous Adiabatic Demagnetization
Refrigerator (CADR) under development at
2013) NASA´s Goddard Space Flight Center
- 2004 2004 ( - CADR to cool from below 5K to ≈ 35 mK
Institut - advantage: no stored cryogens -
maximizing the lifetime/mass Meißner
- ratio for the instrument
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
Chapt. IV - 104 Contents:
IV.1 Generation of Low Temperatures
IV.1.1 Introduction
2013)
- IV.1.2 Expansion Machine 2004 2004
( IV.1.3 Regenerative Machine
IV.1.4 Joule-Thomson Cooling Institut - IV.1.5 Summary
IV.1.6 Evaporation Cooling Meißner - IV.1.7 Dilution Cooling IV.1.8 Pomeranchuk Cooling
Walther IV.1.9 Adiabatic Demagnetization © ©
IV.2 Thermometry A. Marx , Marx, A. IV.2.1 Introduction
and
IV.2.2 Primary Thermometers Gross
R. IV.2.3 Secondary Thermometers
Chapt. IV - 105 IV.2 Thermometry IV.2.1 Introduction
temperature and temperature scales • temperature of a system in thermodynamic equilibrium:
defined as the relation between the amount of heat δQ incident on the system during
2013)
- an infinitesimal quasi-static transformation, and the variation δS of its entropy during 2004 2004 ( this transformation:
Q Institut - T S
Meißner - Q Walther for reversible Carnot process (dS = 0): 0
© © T A. Marx , Marx, A.
and
• Lord Kelvin (1854): there is an absolute zero of temperature scale Gross
R.
T0 = 0 K = - 273.15°C 1 K = 1°C see http://www.its-90.com
Kelvin scale Celsius scale (1742)
Chapt. IV - 106 IV.2 Thermometry IV.2.1 Introduction
26 June 1824(1824-06-26) Born Belfast, Co. Antrim, Ireland
17 December 1907 (aged 83)[1]
Died Largs, Ayrshire, Scotland [1]
2013)
- 2004 2004 ( Cambridge, England Residence
Glasgow, Scotland Institut
- United Kingdom of Great Britain Nationality and Ireland
Meißner Institutions University of Glasgow - Glasgow University Alma mater
Peterhouse, Cambridge Walther © © A variety of physical phenomena and concepts with
which Thomson is associated are named Kelvin: A. Marx , Marx, A.
• Kelvin material and • Kelvin water dropper Lord Kelvin by Hubert von Herkomer Gross • Kelvin wave http://br.geocities.com/saladefisica3/fotos/kelvin.gif R. • Kelvin-Helmholtz instability • Kelvin-Helmholtz mechanism William Thomson (Lord Kelvin) • Kelvin-Helmholtz luminosity • The SI unit of temperature, kelvin • Kelvin transform in potential theory • Kelvin's circulation theorem • Kelvin-bridge (also known as Thomson-bridge) Chapt. IV - 107 IV.2 Thermometry IV.2.1 Introduction
• SI temperature scale - the SI temperature scale is the Kelvin scale. It defines the triple point of water as the
numerical value of 273.16, i.e., 273.16 K. The unit of temperature in this scale is the
Kelvin (K).
2013)
-
• Celsius scale:
2004 2004 (
the Celsius scale has units of °C (degrees Celsius) with the size of the unit equal to Institut
- 1 Kelvin.
T(°C) = T(K) – 273.15 Meißner - • agreement of bureaus of standards:
Walther
© © ITS-90 temperature scale for T > 0.65 K (Comité International des Poids et Messures 1990)
A. Marx , Marx, A. - the ITS-90 is defined by 17 fixed points and 4 defining instruments. It spans a
and
temperature range from 0.65 K to 10 000 K. For cryogenic purposes the three defining
instruments are helium vapor pressure thermometry, gas thermometry, and platinum Gross
R. resistance thermometry.
PLTS-2000 for lower T (Provisonal Low Temperature Scale, melting curve of 3He ) - the PLTS-2000 is defined by a polynomial, relating the melting pressure of 3He to temperature from the range 0.9 mK to 1 K. The pressure to temperature relationship is based on primary thermometers such as Johnson noise and nuclear orientation. Chapt. IV - 108 IV.2 Thermometry IV.2.1 Introduction
The Water Triple Point
The triple point of water is the most important defining
thermometric fixed point used in the calibration of
2013)
- thermometers to the International Temperature Scale of
1990 (ITS-90). 2004 2004 ( It is the sole realizable defining fixed point common to
Institut the Kelvin Thermodynamic Temperature Scale (KTTS) - and the ITS-90; the assigned value on these scales is
Meißner 273.16 K (0.01°C) -
(374°C, 21800 kPa)
Walther
© ©
A. Marx , Marx, A.
and critical
Gross point R. liquid triple solid point
gaseous(0.01° C, 603 Pa)
273.15 K 273.16 K Chapt. IV - 109 IV.2 Thermometry IV.2.1 Introduction
Defining Fixed Points of the ITS-90
a b Number Temperature Substance State Wr (T90)
T90/K t90/°C
1 3 to 5 -270.15 to -268.15 He V
2013)
-
2 13.8033 -259.3467 e-H2 T 0.001 190 07 2004 2004 ( 3 ~17 ~-256.15 e-H2 (or He) V (or G)
4 ~20.3 -252.85 e-H2 (or He) V (or G) Institut - 5 24.5561 -248.5939 Ne T 0.008 449 74
6 54.3584 -218.7916 O2 T 0.091 718 04
Meißner - 7 83.8058 -189.3442 Ar T 0.215 859 75
8 234.3156 -38.8344 Hg T 0.844 142 11
Walther © © 9 273.16 0.01 H20 T 1.000 000 00
10 302.9146 29.7646 Ga M 1.118 138 89 A. Marx , Marx, A.
11 429.7485 156.5985 In F 1.609 801 85 and 12 505.078 231.928 Sn F 1.892 797 68
Gross 13 692.677 419.527 Zn F 2.568 917 30 R. 14 933.473 660.323 Al F 3.376 008 60 15 1234.93 961.78 Ag F 4.286 420 53 16 1337.33 1064.18 Au F 17 1357.77 1084.62 Cu F
a 3 All substances except He are of natural isotopic composition, e-H2 is hydrogen at the equilibrium concentration of the ortho- and para-molecular forms. b V: vapour pressure point; T: Triple Point (temperature at which the solid, liquid and vapour phases are in equilibrium); G: gas thermometer point; M,F melting point, freezing point (temperature, at a pressure of 101 325 Pa, at which the solid and liquid phases are in equilibrium) see http://www.its-90.com Chapt. IV - 110 IV.2 Thermometry IV.2.1 Introduction
temperature measurement
• definition of temperature via reversible Carnot process is not well suited for
establishing useful measuring methods
2013)
- 2004 2004
( • in practice: use of fixpoints and interpolation polynoms
Institut -
• primary thermometers: Meißner - measured quantity is related directly to temperature
(in a theoretically predictably way) Walther
© © no calibration is required A. Marx , Marx, A.
• secondary thermometers: and
measured quantity varies with temperature in a reproducible way Gross
R. must be calibrated using a primary thermometer
• requirements for temperature measurement: good thermal contact between thermometer and sample low self-heating fast response to temperature changes Chapt. IV - 111 IV.2 Thermometry IV.2.1 Introduction
typical temperature range of some thermometers
2013)
-
2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
T(K) Chapt. IV - 112 IV.2 Thermometry IV.2.1 Introduction
most common thermometers for 1K < T < 300K
• gas thermometer: p = p(T)
2013) – Helium gas ≈ ideal gas down to 10K:
- 2004 2004 ( • vapour pressure thermometer: Tliquid = f(pvapor)
– pressure of 10 Pa corresponds to 0.4K for 3He Institut -
• thermocouples: Vth = Vth(T) Meißner -
Walther • resistance thermometry: R = R(T)
© © – 1K - 300K – semiconductors (e.g. Ge doped with Arsenic has 100-500 Ω/K @ 4.2K,
self-heating around 10μA) A. Marx , Marx, A.
– p-n junction diode (problem with high bias current self heating) and
Gross • capacitance thermometry: C = C(T) R. – based on temperature change of dielectric properties – virtually no magnetic field-induced errors
• noise thermometer: S = S(T)
– Johnson noise in resistor: SV = 4kBTR – like gas thermometer, but with electrons – with SQUID measurements: 0.1% @ 1K Chapt. IV - 113 IV.2 Thermometry IV.2.1 Introduction
most common thermometers for T < 1K
1mK ≤ T ≤ 1K:
2013)
-
2004 2004 • magnetic suceptibility thermometer ( M C M: magnetization Curie’s law: B: applied magnetic field
Institut 0
- B T C: Curie constant
Meißner - mutual inductance between two coils: m m0 f
Walther Cerium magnesium nitrate (CMN) useful from 1K - 10mK
© © low temperature limit set by magnetic ordering at ≈ 1mK A. Marx , Marx, A.
• resistance thermometers
and
Gross R. T < 1mK:
• Nuclear Magnetic Resonance (NMR) thermometer
temperature dependence of spin relaxation platinum ideal choice for NMR thermometry Chapt. IV - 114 Contents:
IV.1 Generation of Low Temperatures
IV.1.1 Introduction
2013)
- IV.1.2 Expansion Machine 2004 2004
( IV.1.3 Regenerative Machine
IV.1.4 Joule-Thomson Cooling Institut - IV.1.5 Summary
IV.1.6 Evaporation Cooling Meißner - IV.1.7 Dilution Cooling IV.1.8 Pomeranchuk Cooling
Walther IV.1.9 Adiabatic Demagnetization © ©
IV.2 Thermometry A. Marx , Marx, A. IV.2.1 Introduction
and
IV.2.2 Primary Thermometers Gross
R. IV.2.3 Secondary Thermometers
Chapt. IV - 115 IV.2 Thermometry IV.2.2 Primary Thermometers
gas thermometers
• ideal gas would be a perfect thermometer:
2013)
- measure pressure at constant volume
2004 2004 pV nRT
(
Institut -
• for real gases life is more complicated deviations from ideal behavior
Meißner - 2 3
Walther pV nRT b(T )p c(T )T d(T )T
© © A. Marx , Marx, A.
virial coefficients (tabulated ITS-90 values)
and
Gross
R. • systematic errors: dead volumes thermal expansion of cell, elastic deformation of cell adsorption and desorption from walls
mainly used in calibration laboratories ! Chapt. IV - 116 IV.2 Thermometry IV.2.2 Primary Thermometers
vapour pressure thermometry
dp L(T) Hvap(T) • Clausius-Clapeyron equation:
2013) dT (V V )T V T
gas liquid gas -
2004 2004 ( limited Hvap(T)
Institut T range
- • for ideal gas (pV = RT): Rlnp const. 2 dT
T Meißner -
• if Hvap(T) is known determine T of liquid (e.g. He) via measurement of He pressure Walther
© © above liquid A. Marx , Marx, A.
• in practice, a set of secondary vapour pressure scales is used: and
Gross ITS-90:
R. i (lnp B) (in principle no primary T Ai thermometer !!) i C
• with 3He and 4He: ITS defined down to 0.65 K Chapt. IV - 117 IV.2 Thermometry IV.2.2 Primary Thermometers
values of the constants for the helium vapour-pressure, and the temperature range for which each equation,
identified by its set of constants, is valid (see http://www.its-90.com).
3 4 4
2013) He He He
-
0.65K to 3.2K 1.25K to 2.1768K 2.1768K to 5.0K
2004 2004 (
A0 1.053 447 1.392 408 3.146 631
Institut -
A1 0.980 106 0.527 153 1.357 655 Meißner - A2 0.676 380 0.166 756 0.413 923 A 0.372 692 0.050 988 0.091 159
Walther 3 © ©
A4 0.151 656 0.026 514 0.016 349 A. Marx , Marx, A.
A5 -0.002 263 0.001 975 0.001 826
and
A6 0.006 596 -0.017 976 -0.004 325 Gross R. A7 0.088 966 0.005 409 -0.004 973
A8 -0.004 770 0.013 259 0
A9 -0.054 943 0 0 B 7.3 5.6 10.3 C 4.3 2.9 1.9 Chapt. IV - 118 IV.2 Thermometry IV.2.2 Primary Thermometers
vapour pressure thermometry
3He
2013)
- 4He
2004 2004
(
Institut
-
Meißner -
4 Walther
© © He
A. Marx , Marx, A.
and
Gross R.
Chapt. IV - 119 IV.2 Thermometry
IV.2.2 Primary Thermometers
2013)
-
2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
www.mallister.com/graphics/vapor3.jpg Chapt. IV - 120 IV.2 Thermometry IV.2.2 Primary Thermometers
vapour pressure thermometry
2013)
-
2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
www.bm-industries.com
Chapt. IV - 121 IV.2 Thermometry IV.2.2 Primary Thermometers
3He melting curve thermometry
• use of melting curve of 3He to define PLTS-2000 temperature scale down to 0.9 mK
2013)
-
2004 2004 • polynom for melting curve: ( 9
i Institut - p iT
i 3 Meißner -
coefficients given by PLTS-2000 Walther © © also use of 4 fix points (minimum of melting curve, transition temperatures
to A and B phase and afm order of nuclear spins in solid 3He)
A. Marx , Marx, A.
and
Gross R.
Chapt. IV - 122 IV.2 Thermometry IV.2.2 Primary Thermometers 3He melting curve thermometry
melting curve of 3He
2013)
-
2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
Source: R.L. Rusby et al. (2001)
Chapt. IV - 123 IV.2 Thermometry IV.2.2 Primary Thermometers noise thermometry
• Nyquist theorem: S 4k TR
2013) V B
-
2004 2004 ( voltage noise resistance
power spectral density Institut - temperature (V²/Hz)
Meißner - valid only in the low frequency limit f << kBT / h (≈ 20 GHz @ 1K)
Walther © ©
• temperature determined by measurement of SV and R A. Marx , Marx, A.
and
• example: R = 10 kW, T = 1K, band width f = 105 Hz Gross
R. 1/2 -7 (SV f) ≈ 2·10 V sensitive amplifier required SQUID preamplifier
P ≈ 10-18 W
Chapt. IV - 124 IV.2 Thermometry IV.2.2 Primary Thermometers
superconducting fix point thermometers
• based on the precise measurement of the transition temperatures of superconductors 2013)
-
• available from NIST at Boulder
2004 2004
( Institut
- ITS-90 NIST fixpoint device
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
Chapt. IV - 125 Contents:
IV.1 Generation of Low Temperatures
2013) IV.1.1 Introduction
- IV.1.2 Expansion Machine 2004 2004 ( IV.1.3 Regenerative Machine
IV.1.4 Joule-Thomson Cooling Institut - IV.1.5 Summary
IV.1.6 Evaporation Cooling Meißner - IV.1.7 Dilution Cooling
IV.1.8 Pomeranchuk Cooling Walther
© © IV.1.9 Adiabatic Demagnetization
A. Marx , Marx, A.
IV.2 Thermometry and IV.2.1 Introduction
Gross IV.2.2 Primary Thermometers R. IV.2.3 Secondary Thermometers
Chapt. IV - 126 IV.2 Thermometry
IV.2.3 Secondary Thermometers
2013)
-
2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
Chapt. IV - 127 IV.2 Thermometry IV.2.3 Secondary Thermometers
resistance thermometers
• required: well established relation between resistance and temperature,
2013)
- sufficiently large dR/dT
2004 2004 (
• advantage: resistance easy to measure resistance thermometry very popular Institut -
• fact: temperature variation of resistance may have very different physical origin Meißner -
• commonly used: Walther
© ©
Pt resistors (PT-100, PT-1000) A. Marx , Marx, A.
RhFe resistors and
carbon resistors (Speer, Allen-Bradley) Gross
R. carbon glass resistors Ge resistors
RuO2 resistors
Chapt. IV - 128 IV.2 Thermometry IV.2.3 Secondary Thermometers
Platinum resistors
The platinum resistance thermometer (PRT) is very widely used below 500 °C as a thermometric sensor. There is a wide
range of quality of PRT available, from the standard instrument (SPRT) of the ITS-90 to some industrial types (IPRT) that
2013)
- are accurate only to within a few tenths of a kelvin or, perhaps, even a kelvin or more. The major difference of the
industrial type of fabrication from the standard type is not just the purity of platinum, but also the less strain-free 2004 2004 ( mounting of the film or wire which is embedded (partially or totally) in a cement (glass or refractory). Furthermore, in
most cases, the thermometer body is not hermetically sealed.
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
Source: Lake Shore Cryotronics, Inc. Chapt. IV - 129 IV.2 Thermometry IV.2.3 Secondary Thermometers Platinum resistors
• ITS-90 requirement for Pt resistance thermometer (PRT)
temperatures are determined in terms of the ratio of the resistance R (T ) at a
2013) 90
-
temperature T90 and the resistance R(273.16 K) at the triple point of water: 2004 2004 ( W(29.7646 °C) ≥ 1.118 07 W(T90) = R(T90)/R(0.01 °C)
Institut W(-38.8344 °C) ≤ 0.844 235
- Meißner
- • industrial PRT Walther
© © for 0 < T < 100°C
R = R0 · (1 + a · T)
A. Marx , Marx, A. -3
a = 3.85 · 10 / K and
Gross R. allowed errors in °C: classA: dT = ± (0.15 °C + 0,002 · T) class B: dT = ± (0.30 °C + 0,005 · T) 1/3 class B: dT = ± 1/3 · (0.30 °C + 0.005 · T)
Chapt. IV - 130 IV.2 Thermometry IV.2.3 Secondary Thermometers
• for the range 13.8033 K to 273.16 K the following reference function is defined:
2013)
-
2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
Chapt. IV - 131 IV.2 Thermometry IV.2.3 Secondary Thermometers
RhFe resistor thermometer
2013)
- Rhodium with 2004 2004
( 0.5% Fe
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
Source: Lake Shore Cryotronics, Inc. Chapt. IV - 132 IV.2 Thermometry IV.2.3 Secondary Thermometers carbon resistors commercial carbon-composition resistors manufactured by Allen-Bradley were introduced as
temperature sensors by Clement and Quinnell in 1952. The carbon composition resistor is a small
2013) cylinder consisting of graphite with a binder encased in an outer phenolic shell.
-
2004 2004 The carbon resistors used as thermometers are generally characterized by their room temperature (
resistance and their wattage [see Rubin (1980)], and have come largely from the following Institut
- manufacturers:
• Allen-Bradley Meißner - • Airco Speer (usually referred to simply as Speer)
• Ohmite Walther
© © • Matsushita,
• CryoCal
A. Marx , Marx, A.
and
Gross R. A: thermistor, B: 68 Ω Allen-Bradley, C: 220 Ω Speer (grade 1002), D: 51 Ω Speer (grade 1002), E: 10 Ω Speer (grade 1002)
Chapt. IV - 133 IV.2 Thermometry IV.2.3 Secondary Thermometers carbon glass resistors A porous glass is prepared by removing the boron-rich phase from a borosilicate
alkaline glass to leave a material having the appearance of silicate spheres of about 30
nm diameter, randomly distributed and separated by 3 to 4 nm pores. The spaces are
2013)
- then partially filled with high-purity carbon to form amorphous fibres better 2004 2004
( stability than carbon resistors.
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
Source: Lake Shore Cryotronics, Inc. Chapt. IV - 134 IV.2 Thermometry IV.2.3 Secondary Thermometers Cernox resistors
Patents:
#5,363,084, Nov. 1994, “Film Resistors Having Trimmable Electrodes”
2013)
-
#5,367,285, Nov. 1994, “Cernox™”, “Metal Oxy-nitride Resistance Films and 2004 2004 ( Methods of Making the Same,”
Institut
- small magnetic field coefficient: T/T typically smaller than 0.5% @ 19 T
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
Chapt. IV - 135 IV.2 Thermometry IV.2.3 Secondary Thermometers
Cernox resistors
2013)
-
2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
Low temperature thermometry in high magnetic fields VII. Cernox sensors to 32 T, B.L. Brandt et al, Rev. Sci. Instrum., vol 70, No 1, 1999, pp 104-110. Chapt. IV - 136 IV.2 Thermometry IV.2.3 Secondary Thermometers
Germanium resistors
2013)
-
2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
Source: Lake Shore Cryotronics, Inc. Chapt. IV - 137 IV.2 Thermometry IV.2.3 Secondary Thermometers
Germanium resistors
2013)
-
2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
Source: Lake Shore Cryotronics, Inc. Chapt. IV - 138 IV.2 Thermometry IV.2.3 Secondary Thermometers
Thermocouples
Thermocouples are pairs of dissimilar metal wires joined at least at one end, which
2013) generate a net thermoelectric voltage between the the open pair according to the size of
- the temperature difference between the ends, the relative Seebeck coefficient of the wire 2004 2004
( pair and the uniformity of the wire-pair relative Seebeck coefficient.
Institut
- Meißner - • based on Seebeck effect: V S T S = thermopower
Walther
© © • measurement of difference of thermovoltages of two different materials
A. Marx , Marx, A.
and
VA Cu
Gross R.
VB Cu T1 T2 well-known reference temperature
Chapt. IV - 139 IV.2 Thermometry IV.2.3 Secondary Thermometers
Thermocouples
• Chromel-Gold/Iron (0.07%)
2013)
- consists of a Gold (Au)-0.07 at % Iron (Fe) as the negative thermoelement and a 2004 2004 ( Ni-Cr alloy (Chromel) as the positive thermoelement. This thermocouple is more
widely used because of ist relatively high thermoelectric sensitivity (>15 µV/K Institut - above 10 K).
Meißner - • Type E (Chromel (Ni-Cr-alloy) / Constantan (Cu-Ni-alloy) )
Walther has the highest sensitivity among the three standard thermocouple types © © typically used at low temperatures (types E, K, and T). The best choice
A. Marx , Marx, A. for temperatures down to 40 K.
and
• Type K (Chromel (Ni-Cr-alloy) / Alumel (Ni-Al-alloy) ) Gross R. recommended for continuous use in inert atmospheres. Has a sensitivity of 4.1 mV/K at 20 K (about ½ of Type E).
• Type T (Copper / Constantan)
many more !! Chapt. IV - 140
IV.2 Thermometry IV.2.3 Secondary Thermometers Thermocouples
Thermocouple Type Names of Materials Useful Application Range
Platinum30% Rhodium (+) 2500 -3100F 2013)
- B Platinum 6% Rhodium (-) 1370-1700°C 2004 2004 ( W5Re Tungsten 5% Rhenium (+) 3000-4200F
C W26Re Tungsten 26% Rhenium (-) 1650-2315°C
Institut - Chromel (+) 200-1650F
E Constantan (-) 95-900°C
Meißner - Iron (+) 200-1400F
J Constantan (-) 95-760°C Walther © © Chromel (+) 200-2300F
K Alumel (-) 95-1260°C A. Marx , Marx, A.
Nicrosil (+) 1200-2300F and N Nisil (-) 650-1260°C
Gross Platinum 13% Rhodium (+) 1600-2640F R. R Platinum (-) 870-1450C Platinum 10% Rhodium (+) 1800-2640F S Platinum (-) 980-1450°C Copper (+) -330-660F T Constantan (-) -200-350°C
Chapt. IV - 141 IV.2 Thermometry IV.2.3 Secondary Thermometers
Thermocouples
2013)
-
2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
Chapt. IV - 142 IV.2 Thermometry IV.2.3 Secondary Thermometers Diode Thermometers The temperature-indicating parameter is the forward-biased junction voltage,
which decreases approximately linearly with increasing temperature when the current is kept constant, since I exp(eV /k T)
2013) B
- The typical I-V characteristic is such as to make the internal impedance 2004 2004 ( of the device very high (easily greater than 100 kΩ) at small currents; or else - using a larger current -
one encounters unacceptably high power dissipation at low temperatures. Institut - There is a transition region in the conduction mechanism around 20 K that makes fitting a V-T
characteristic over the whole temperature range difficult for GaAs and impossible for Si Meißner
- Walther
© © Si Si
A. Marx , Marx, A.
and
Gross R.
Source: Lake Shore Cryotronics, Inc. Chapt. IV - 143 IV.2 Thermometry IV.2.3 Secondary Thermometers
Capacitive Thermometers
• based on the well defined relation between the dielectric constant and temperature
2013)
- 2004 2004 ( • temperature is determined via a capacitance measurement
Institut
- • advantage: virtually no magnetic field dependence
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
Chapt. IV - 144 IV.2 Thermometry IV.2.3 Secondary Thermometers
Capacitive Thermometers
2013)
-
2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
Source: Lake Shore Cryotronics, Inc. Chapt. IV - 145 IV.2 Thermometry IV.2.3 Secondary Thermometers
magnetic field effects
2013)
-
2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
Chapt. IV - 146 IV.2 Thermometry IV.2.3 Secondary Thermometers
magnetic field effects
2013)
-
2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
Chapt. IV - 147 IV.2 Thermometry
IV.2.3 Secondary Thermometers
2013)
-
2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
Chapt. IV - 148 IV.2 Thermometry
IV.2.3 Secondary Thermometers
2013)
-
2004 2004
(
Institut
-
Meißner
-
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
Chapt. IV - 149 IV.2 Thermometry IV.2.3 Secondary Thermometers
Magnetic Susceptibility Thermometers
C
• Curie-Weiss law:
2013)
- T TC 2004 2004 ( measure as a function of T, calibaration requires determination of C and T
Institut C
- Meißner - • experimental techniques:
Walther
© © inductance bridge: compare inductance of coil containing the magnetic
material to empty reference coil
A. Marx , Marx, A.
and SQUID magnetometer: sample
Gross I R. resolution: Mi Vout -10 T/T ≈ 10 @ 1K first order L Ic F Ic increases ≈ 1/T gradiometer p L reference coil i L
T.C.P. Chui et al., Phys. Rev. Lett. 69, 3005 (1992) Bext Chapt. IV - 150 IV.2 Thermometry IV.2.3 Secondary Thermometers
Magnetic Susceptibility Thermometers
• materials (use of electronic magnetic susceptibility):
2013)
-
2004 2004 paramagnetic salts: e.g. Ce-Mg-nitrate (CMN) (
++ very low ordering temperature: TC ~ 2 mK Institut - ++ large Curie constant
- - long time constant at low T (~ 100 s) Meißner - - - cannot be used in vacuum (instable)
Walther
© © nonmagnetic metals: e.g. PdFe or AuEr
A. Marx , Marx, A.
and
Gross R. - usable down to about 0.3 mK - response time: 1 s @ 10 mK
M. Jutzler, B. Schröder, K. Gloos, F. Pobell, Z. Phys. B64, 115 (1986) Chapt. IV - 151 IV.2 Thermometry IV.2.3 Secondary Thermometers
Magnetic Susceptibility Thermometers
2013)
• materials (use of nuclear susceptibility): -
2004 2004
( use of nonmagnetic metals: e.g. Cu Institut
- - - nuclear moments much smaller sensitive SQUID magnetometer
- - perturbing magnetic impurities very pure materials Meißner - (e.g. 1 ppm Fe in Cu gives similar signal as all Cu nuclear spins)
++ can be used below 1 mK
Walther
© ©
A. Marx , Marx, A.
and
Gross R.
Chapt. IV - 152 IV.2 Thermometry IV.2.3 Secondary Thermometers
Nuclear Magnetic Resonance Thermometers
• are not based on static orientation of nuclear moments as in susceptibility 2013)
thermometers -
2004 2004
( • have selectivity to specific nuclear moments (less sensitive to magnetic impurities) Institut - • usable at temperatures below 1 mK
Bz Meißner -
• materials: predominantly 195Pt Walther
© © • experimental techniques: stationary and pulsed NMR B cos wt
A. Marx , Marx, A. y
and
Gross measures resonance absorption measures decay of induced R. of high frequency signal My(t) after 90° pulse allows to determine induced amplitude dMy(t = 0) or integral of My M0(T) 1/TN at resonance decay curve M0(T) 1/TN requires small By ≈ 1µT to avoid saturation effects
general problem: one measures temperature TN of nuclear spins are they in thermal equilibrium with lattice ?? Chapt. IV - 153 IV.2 Thermometry IV.2.3 Secondary Thermometers
Nuclear Magnetic Resonance Thermometers
2013)
- comparison of the Curie and Korringa 2004 2004 ( temperatures of a Pt sample:
Institut - the inverse nuclear susceptibility 1/N
(arb. units) is plotted against the Meißner - electron temperature -3
Walther Te = (1/t1) 29.9 x 10 K sec.
© ©
A. Marx , Marx, A.
and
Gross R.
A.I. Ahonen et al., J. Low Temp. Phys. 25, 421 (1976)
Chapt. IV - 154