Cryogenic Techniques: Generation and Measurement of Low … · 2013. 7. 18. · (H. Kammerlingh...
Transcript of Cryogenic Techniques: Generation and Measurement of Low … · 2013. 7. 18. · (H. Kammerlingh...
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Chapter IV
Cryogenic Techniques: Generation and Measurement of
Low Temperatures
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Chapter IV: Cryogenic Techniques
Contents: IV.1 Generation of Low Temperatures IV.1.1 Introduction IV.1.2 Expansion Machine IV.1.3 Regenerative Machine IV.1.4 Joule-Thomson Cooling IV.1.5 Summary IV.1.6 Evaporation Cooling IV.1.7 Dilution Cooling IV.1.8 Pomeranchuk Cooling IV.1.9 Adiabatic Demagnetization
IV.2 Thermometry IV.2.1 Introduction IV.2.2 Primary Thermometers IV.2.3 Secondary Thermometers
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Chapt. IV - 3
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Literature: 1. Tieftemperaturphysik
Enss, Hunklinger Springer (2000)
2. Matter and Methods at Low Temperatures F. Pobell Springer, 2nd edition (1996)
3. Experimental Low-Temperature Physics Anthony Kent American Institute of Physics (1993)
4. Cryogenic Systems Randall F. Barron Oxford University Press, Oxford (1985)
Chapter IV: Cryogenic Techniques
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Chapt. IV - 4
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IV.1 Generation of Low Temperatures IV.1.1 Introduction
background temperature in universe
(2.73 K)
lowest temperature accessible in solids
(few µK)
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
103
104
105
106
107
108
109
tem
per
atu
re (
K)
center of hottest stars
center of the sun, nuclear energies
electronic energies, chemical bonding
surface of sun, highest boiling temperatures
organic life
liquid air
liquid 4He universe
superfluid 3He
lowest temperatures of condensed matter
elec
tro
nic
m
ag
net
ism
nu
clea
r-
ma
gn
etis
m
sup
erco
nd
uct
ivit
y
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• experimental setup according to Tauno Knuuttila (2000)
• lowest temperature: about 100 pK
by demagnetization of Rhodium nuclei („temperature of nuclear spins“)
PhD Thesis,
Helsinki University of Technology (Espoo, Finland)
• problem: spin temperature cannot be transferred to lattice of solid
low temperature record for nuclear spin system:
IV.1 Generation of Low Temperatures IV.1.1 Introduction
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Chapt. IV - 6
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Generation of low temperatures by using cryo-liquids:
19th century: liquefaction of various gases by pressure except for “permanent gases” (O2, H2, He)
1877: liquefaction of O2 by thermal expansion (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) (K. Olszewski, Ann. Phys. u. Chem. 31, 58 (1887)) 1898: significant amounts of lH2 for physical experiments (J. Dewar, Proc. R. Inst. Gt. Br. 15, 815 (1898)) 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 (H. Kammerlingh Onnes, Leiden Commun. 159, Trans. Faraday Soc. 18 (1922)) 1926: adiabatic demagnetization of electron spins in 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) since 1950th: 3He available 3He cryostat 3He-4He dilution refrigerator
Heike Kammerlingh Onnes
(1853 – 1926)
Nobelpreis für Physik: 1913
Sir James Dewar, (1842-1923)
Peter J. Debye 1884 - 1966
IV.1 Generation of Low Temperatures IV.1.1 Introduction
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Carl Paul Gottfried von Linde * 11. Juni 1842 in Berndorf, Oberfranken
† 16. November 1934 in Munich
Low Temperature Technology in Germany
1868 offer of chair at the Polytechnische Schule München (now TUM)
1873 development of cooling machine allowing the temperature stabilization in beer brewing
21. 6. 1879 foundation of „Gesellschaft für Linde’s Eismaschinen AG“ together with two beer brewers and three other co-founders
1892 - 1910 re-establishment of professorship
12.5.1903 patent application: „Lindesches Gegenstrom- verfahren“ liquefaction of oxygen (-182°C = 90 K)
1861 study at Polytechnikum Zurich, teachers: Rudolf Clausius, Gustav Zeuner und Franz Reuleaux
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Chapt. IV - 9
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Year
low
te
mp
era
ture
s
ult
ra-l
ow
te
mp
era
ture
s
paramagnetic refrigeration
nuclear demagnetization
IV.1 Generation of Low Temperatures IV.1.1 Introduction
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temperature range
refrigeration technique available since
typical
Tmin
record
Tmin
Kelvin universe 4He evaporation 3He evaporation
1908
1950
1.3 K
0.3 K
2.73 K
0.7 K
0.25 K
Millikelvin 3He-4He dilution
Pomeranchuk cooling
electron spin demagnetization
1965
1965
1934
10 mK
3 mK
3 mK
2 mK
2 mK
1 mK
Microkelvin nuclear spin demagnetization 1956 50 µK 100 pK
IV.1 Generation of Low Temperatures IV.1.1 Introduction
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cooling techniques:
• expansion of an ideal gas
• expansion machine
• regenerative machine
work against outside world
• expansion of a real gas
• Joule Thomson cooler
work against internal interactions
• evaporation of a real gas:
work against internal interactions
• dilution cooling (3He/4He)
work against internal interactions
• adiabatic demagnetization (electronic/nuclear moments)
work against magnetic ordering
IV.1 Generation of Low Temperatures IV.1.1 Introduction
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Liquefaction of gases three useful methods:
1. direct liquefaction by isothermal compression
2. letting the gas perform work against external forces at the expense of
its internal energy
cooling and eventual liquefaction
3. making the gas perform work against its own internal forces by Joule-
Kelvin or Joule-Thomson expansion
cooling and eventual liquefaction
IV.1 Generation of Low Temperatures IV.1.1 Introduction
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direct liquefaction of gases by isothermal compression starting temperature must be smaller than critical temperature Tc
ammonia (NH3) 406
O2 154.5
N2 126
H2 33.2 4He 5.2 3He 3.32
critical temperatures Tc in K of selected liquid cryogens
melting curve
sublimation curve
critical point
triple point
solid
liquid
gas
T
p
Tc
boiling curve pc
IV.1 Generation of Low Temperatures IV.1.1 Introduction
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@ 1 bar
Cryogenic Liquids
Ttr , ptr
solid liquid
gas
T
p
Tc
1 at Tc , pc
Tm , pm Tb , pb
IV.1 Generation of Low Temperatures IV.1.1 Introduction
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cryogen boiling point [K]
liquefaction latent heat [kJ/I]
inversion temp. [K]
oxygen 90.2 1877: Cailletet and Pictet 240 762
nitrogen 77.3 1883: Wroblewski and Olszewski
160 625
hydrogen 20.4 1898: Dewar 30 203
4Helium 4.2 1908: Onnes 2.6 43.2
3Helium 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
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
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gas molecules are reflected at the moving piston-surface:
incoming: laboratory system: 𝑣𝑀 piston system: 𝑣𝑀 − 𝑣𝐾 outgoing: piston system: − 𝑣𝑀 − 𝑣𝐾
laboratory system: − 𝑣𝑀 − 𝑣𝐾 + 𝑣𝐾 = 2𝑣𝐾 − 𝑣𝑀 = 𝑣𝑀′
i.e.: 𝑣𝑀′ = 𝑣𝑀 − 2𝑣𝐾 molecule is slower, i.e. colder
liquefaction of gases by performance of external work
average momentum transfer per time to piston = force, force · distance = work
external work at the expense of internal energy cooling
IV.1 Generation of Low Temperatures IV.1.1 Introduction
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• efficiency: • Carnot process: technologically difficult to realize better: gas circulation, compressor and expansion machine are spatially separated
• Carnot process: - counterclockwise: heat pump (conversion of mechanical work into heat) - clockwise: heat engine (conversion of heat into mechanical work)
• pV diagram: expansion cooling: adiabats (𝑝𝑉𝜅 = 𝑐𝑜𝑛𝑠𝑡, 𝑑𝑄 = 0
𝜅 =𝑐𝑝
𝐶𝑉> 1)
heat exchange: isotherms (𝑝𝑉 = 𝑐𝑜𝑛𝑠𝑡, 𝑑𝑇 = 0)
work per cycle:
𝑊 = ∮ 𝑝𝑑𝑉 = 𝐚𝐫𝐞𝐚
warmT
T
Q
W
thermodynamic definition of temperature
IV.1 Generation of Low Temperatures IV.1.1 Introduction
V
p
Q12
Q34
dQ = 0 (adiabatic)
T1 = const (isothermic)
T2 = const dQ = 0
W23
W41
1
2
4
3
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IV.1 Generation of Low Temperatures IV.1.2 Expansion Machine
• medium: He gas
Brayton method
e.g. liquefaction of air:
- condensation on cold head
- distillation in separation columns
N2 (77.4 K) cooling
Ar (87.3 K) inert gas
O2 (90.2 K) welding
• temperature reduction:
𝜅 = 𝐶𝑝/𝐶𝑉 (= 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
(should not cause
significant resistance
for flowing gas e.g.
concentric tubes)
• efficiency:
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• heat pumps: heating and refrigerating machines
- heat pump: heat is generated by mechanical work
- efficiency:
W
Q TT h
11
work performed
at heat generated
- ideal efficiency for reversible Carnot process:
11
21
1
TT
Th
C
C (increases with decreasing temperature difference T1 – T2)
- refrigerating machine: removing heat (generating cold) by mechanical work
01work performed
at heat removed
21
22
TT
Th
TTk CC
V
p Q12
Q34
dQ = 0
T1 = const
T2 = const dQ = 0
W23
W41 1
2
4
3
IV.1 Generation of Low Temperatures IV.1.1 Introduction
(decreases with increasing temperature difference T1 – T2)
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Wikimedia Commons
IV.1 Generation of Low Temperatures IV.1.1 Introduction
Schematic diagram of a heat pump's vapor-compression refrigeration cycle: 1) condenser, 2) expansion valve, 3) evaporator, 4) compressor.
e.g. air conditioning e.g. heating of swimming pool
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realizations of expansion machines:
• piston-cylinder machine
similar to automobile engine
crankshaft, camshaft, valve
brake on turbine axis
controls rotational speed,
annihilates performed work,
use of gas bearings
IV.1 Generation of Low Temperatures IV.1.2 Expansion Machine
• cooling turbine commercially relevant
higher efficiency for larger throughput
principle:
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turbine cooler
(Sulzer machine)
turbine wheel
and nozzle ring
IV.1 Generation of Low Temperatures IV.1.2 Expansion Machine
(Source: Linde Cryogenics Ltd.)
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conclusions:
• expansion machines are technologically simple
• multi-stage arrangements for lower temperatures
almost down to 4.2 K
• but:
efficiency only acceptable for cooling turbines
• no direct liquefaction of gas (mechanical problems)
liquefaction by Joule-Thomson stage
• for small-scale facilities:
regenerative machines better suited
IV.1 Generation of Low Temperatures IV.1.2 Expansion Machine
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IV.1 Generation of Low Temperatures IV.1.3 Regenerative Machines
• regenerator replaces heat exchanger
column with staple of fine
metal meshes (Cu, Pb)
low flow resistance
high heat capacity
low longitudinal heat conductivity
cold gained in step 2 →3 has to be stored and provided in step 4 → 1
• alternating gas flow:
cold gas upward
cooling of meshes
warm gas downward
cooling of gas
• used in Stirling process
V
p Q12
Q34
V = const
T1 = const
T2 = const V = const
Q23
Q41 1
2
4
3
Stirling process (heat engine):
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• Stirling machine: (heat engine)
- periodic expansion and compression of gas along two isotherms and two isochors
- 1 2: isothermal expansion, Q12 is added - 2 3: isochoric cooling, Q23 is removed - 3 4: isothermal compression, Q34 is removed - 4 1: isochoric warming, Q41 is added
- for isochoric steps there is no mechanical work 𝑸𝟐𝟑 = −𝑸𝟒𝟏 = 𝑪𝑽𝚫𝑻
- goal: intermediate storage of Q23 in regenerator to be able to add it again in step 4 1 use of two pistons with phase shift V
p Q12
Q34
V = const
T1 = const
T2 = const V = const
Q23
Q41 1
2
4
3
IV.1 Generation of Low Temperatures IV.1.3 Regenerative Machines
3 – 4 4 – 1 1 – 2 2 – 3
T2
T1
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• (beta) Stirling machine:
IV.1 Generation of Low Temperatures IV.1.3 Regenerative Machines
Power piston (dark grey) has
compressed the gas, the displacer piston (light grey) has moved so that most of the gas is
adjacent to the hot heat exchanger
The heated gas increases in
pressure and pushes the power
piston to the farthest limit of the
power stroke.
The displacer piston now moves,
shunting the gas to the cold end of the
cylinder.
The cooled gas is now compressed by
the flywheel momentum. This takes less energy, since when it is
cooled its pressure dropped.
http://en.wikipedia.org/wiki/File:Beta_Stirling_frame_12.pnghttp://en.wikipedia.org/wiki/File:Beta_Stirling_frame_16.pnghttp://en.wikipedia.org/wiki/File:Beta_Stirling_frame_4.pnghttp://en.wikipedia.org/wiki/File:Beta_Stirling_frame_8.pnghttp://en.wikipedia.org/wiki/File:Stirling_Animation.gifhttp://upload.wikimedia.org/wikipedia/commons/c/c7/Beta_Stirling_frame_12.pnghttp://upload.wikimedia.org/wikipedia/commons/f/f1/Beta_Stirling_frame_16.pnghttp://upload.wikimedia.org/wikipedia/commons/1/1b/Beta_Stirling_frame_8.pnghttp://upload.wikimedia.org/wikipedia/commons/4/4e/Stirling_Animation.gif
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• (alpha) Stirling machine:
IV.1 Generation of Low Temperatures IV.1.3 Regenerative Machines
http://upload.wikimedia.org/wikipedia/commons/0/06/Alpha_Stirling_frame_16.pnghttp://upload.wikimedia.org/wikipedia/commons/e/ea/Alpha_Stirling_frame_4.png
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conclusions (Stirling machine):
•advantages:
high efficiency
well-suited for small systems, especially small coolers
cryocooler
not realizable with turbines
•disadvantages:
mechanically complicated
piston (compressor) at low temperature
•more simple:
Gifford-McMahon machine
but: lower efficiency
IV.1 Generation of Low Temperatures IV.1.3 Regenerative Machines
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Gifford-McMahon machine: uses compressor with switching valve instead of piston
1. warm compression: 2 1 2. isochoric cooling: 1 4 3. expansion in cylinder: 4 3 4. isochoric regeneration: 3 2 5. warm compression: 2 1
IV.1 Generation of Low Temperatures IV.1.3 Regenerative Machines
pressure wave from valve
hot
cold
regenerator
switching valve
cycle:
V
p Q12
Q34
V = const
T1 = const
T2 = const V = const
Q23
Q41 1
2
4
3
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Gifford-McMahon cycle:
IV.1 Generation of Low Temperatures IV.1.3 Regenerative Machines
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3) • the pulse tube refrigerator (PTR) or pulse tube cryocooler is based on operation
principle of Stirling cooler • PTR is made without moving parts in the low temperature part (in contrast with
other cryocoolers, e.g. Stirling cryocooler and Gifford-McMahon cooler) • compact design possible suitable for a wide variety of applications • minimum temperature about 2.5 K (with 4He) and 1.3 K (with 3He)
Pulse – Tube – Refrigerator
IV.1 Generation of Low Temperatures IV.1.3 Regenerative Machines
applications: • industrial applications such as semiconductor fabrication (e.g. cryopumps) • cooling of infrared sensors • cooling of astronomical detectors (e.g. Atacama Cosmology Telescope or the QUBIC
experiment (an interferometer for cosmology studies) • 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
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IV.1 Generation of Low Temperatures IV.1.3 Regenerative Machines
Pulse – Tube – Refrigerator Stirling Cooler
regenerator regenerator
2-1
1-4
4-3
displacer piston
work piston
second (displacer) piston is replaced by pulse tube (gas piston)
3-2
motion of gas volume element equivalent to motion of „displacer piston“
90° phase shift between motion of „displacer piston“ and „work piston“ realized by buffer volume
90° phase shift required for finite heat transport
buffer volume
acoustic impedance
coldest spot between regenerator and pulse tube
regenerator
regenerator
regenerator
regenerator
regenerator
regenerator
Q34
Q12
work piston
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Qc
QH
Qc
QH
Qc: removed heat, QH: generated heat
Pulse – Tube – Refrigerator Stirling Cooler
IV.1 Generation of Low Temperatures IV.1.3 Regenerative Machines
almost sinusoidal motion, phase difference of 90°
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principle
Pulse – Tube – Refrigerator (realizations)
commercially available
pulse tube refrigerator
with GM drive
0.5 W @ 4.2 K Tmin = 2.3 K (2-stage)
www.cryomech.com
IV.1 Generation of Low Temperatures IV.1.3 Regenerative Machines
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pulse tube refrigerator for studies of liquefying oxygen on
Mars (580 mm total length)
IV.1 Generation of Low Temperatures IV.1.3 Regenerative Machines
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conclusion (pulse tube refrigerator):
•presently very active development
•no moving parts at low temperatures
long endurance
mobile base stations and satellite applications
(e.g. for superconductive microwave filters)
•almost no vibrations
•efficiency lower than for displacer
•only one simpler method:
Joule-Thomson cooling
IV.1 Generation of Low Temperatures IV.1.3 Regenerative Machines
-
Chapt. IV - 41
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IV.1 Generation of Low Temperatures IV.1.4 Joule-Thomson Cooling
William Thomson (Lord Kelvin) Born: 26 June 1824, Belfast, Northern Ireland Died: 17 December 1907, Netherhall, Largs Ayrshire, Scotland
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Chapt. IV - 43
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principle:
• gas performs work against its
own internal attractive forces
• working medium/gas (V1) flows
through impedance and
expands to V2
1122
0
0
1122
2
1
VpVpdVpdVpV
V
WQU
12 UU
111222 VpUVpU
IV.1 Generation of Low Temperatures IV.1.4 Joule-Thomson Cooling
1st law of thermodynamics:
= 0 (adiabatic)
this means: process with constant enthapy: .constpVUH
- for ideal gas: p1V1 = p2V2 and hence U1 = U2 resp. T1 = T2 no cooling !!
-
Chapt. IV - 44
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weak long-range attraction: tends to keep molecules closer together, same effect as additional compression of the gas. a is a measure of the long-range attraction strong short-range repulsion: molecules are rigid: p as soon as the molecules “touch” each other. b (≈ 4𝜋𝜎3/3): “excluded volume” per particle Van der Waals equation:
IV.1 Generation of Low Temperatures IV.1.4 Joule-Thomson Cooling
Epot(r)
short-distance repulsion
long-distance attraction
-3
-2
-1
0
1
2
3
4
1.5 2.0 2.5 3.0 3.5 4.0
distance
Energ
y
r
real gas: transformation of gas into liquid on decreasing T and (or) increasing p due to work against attractive interaction between the molecules
2V
appeff
bVVeff expansion (decrease of pressure): low pressure: attraction costs work cooling of gas high pressure: repulsion provides work heating of gas RTbV
V
ap m
m
2
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IV.1 Generation of Low Temperatures IV.1.4 Joule-Thomson Cooling
interaction potential: Epot
r
repulsive attractive
p
repulsive attractive
U (p,T)
T = const.
minimum
with
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IV.1 Generation of Low Temperatures IV.1.4 Joule-Thomson Cooling
0
p
p
HT
T
HH
Tp
• more detailed analysis of Joule-Thomson process (isenthalpic expansion):
pp
HTCC
T
H
T
pp
p
JT
HTpp
T
p
H
C
1
Vp
ST
p
HpVSTH
TT
pTT
V
p
S
V
T
VT
Cp
H
Cp
T
ppTpH
JT
11
Joule-Thomson coefficient
with
with
𝜇𝐽𝑇 > 0: cooling on expansion
𝜇𝐽𝑇 < 0: heating on expansion
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ideal gas: 0
JT
p T
V
p
R
T
VRTpV
IV.1 Generation of Low Temperatures IV.1.4 Joule-Thomson Cooling
0 .2
5
2
3),(
T
JTBBBp
HµconstTNkTNkTNkpVUpTH
equipartition theorem for monoatomic gas
ideal gas law
areas of H = const.
lines of intersection with H = const.
@ T = const
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IV.1 Generation of Low Temperatures IV.1.4 Joule-Thomson Cooling
real gas: RTbVV
ap m
m
2
at low densities: we use approximation and obtain bVV
ap ,
2
pT
RTpbV
apV ...,
2
2,
V
ap
R
T
VR
T
V
V
a
T
Vp
ppp
b
RT
a
Cp
T
PH
JT 21
μJT > 0 for T < 2a/bR cooling on expansion
μJT < 0 for T > 2a/bR heating on expansion inversion temperature:
bR
aTinv
2
.),(2
5),( constTpUTNkpVUpTH B
V
T
VT
Cp
H
Cp
T
ppTpH
JT
11
insert into
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IV.1 Generation of Low Temperatures IV.1.4 Joule-Thomson Cooling
areas of H = const.
lines of intersection with H = const.
inversion curve
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IV.1 Generation of Low Temperatures IV.1.4 Joule-Thomson Cooling
Joule-Thomson coefficient (without approx.):
22
121
)1)(2(
VbVRTaC
bVbRTa
p
JT
large volume (p >> a/V2, V >> b) :
b
RT
a
CPJT 2
1
inversion curve: points where µJT = 0
vdW gas: (2a/RT)(1-b/V)2 = b
inversion temperature Tinv: 2
12
V
b
bR
aTinv
equation of state gives Tinv(p,T)
maximum inversion temperature: bR
aTinv
2
maximum inversion
temperature
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IV.1 Generation of Low Temperatures IV.1.4 Joule-Thomson Cooling
experimental data for N2:
H
JTp
T
slope of isenthalps
repulsion
attraction
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gas maximum inversion temperature [K]
Helium-3 (23)
Helium-4 45
Hydrogen 205
Neon 250
Nitrogen 621
Air 603
Carbon monoxide 652
Argon 794
Oxygen 761
Methane 939
Carbon dioxide 1500
Ammonia 1994
IV.1 Generation of Low Temperatures IV.1.4 Joule-Thomson Cooling
vdW gas can be liquefied only for T < Tinv !!!
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IV.1 Generation of Low Temperatures IV.1.4 Joule-Thomson Cooling
closed cycle cooling:
• gas is cooled by JT-expansion until liquid drops out the impedance
Carl von Linde (1842 – 1934)
“Linde-process”
(Source: PTB Braunschweig)
• patent application by Carl von Linde on May 12, 1903 (liquefaction of oxygen)
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IV.1 Generation of Low Temperatures IV.1.4 Joule-Thomson Cooling
Lindesche Gasverflüssigungsanlage (1895)
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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 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 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 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 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 (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 Verflüssigung von Chlor und Kohlensäure war sein Verfahren gefragt. Die Firma florierte, 1890 zog sich Linde aus dem operativen Geschäft in den Aufsichtsrat seiner Aktiengesellschaft zurück. In den Jahren 1892 bis 1910 nahm er seine Professur wieder auf. Auf der Grundlage der Arbeiten von James Prescott Joule, Sir William Thomson (Lord Kelvin of Largs) und der Einführung des Gegenstromverfahrens konnte Linde 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.
IV.1 Generation of Low Temperatures IV.1.4 Joule-Thomson Cooling
http://de.wikipedia.org/wiki/11._Junihttp://de.wikipedia.org/wiki/1842http://de.wikipedia.org/wiki/Berndorfhttp://de.wikipedia.org/wiki/Oberfrankenhttp://de.wikipedia.org/wiki/16._Novemberhttp://de.wikipedia.org/wiki/1934http://de.wikipedia.org/wiki/M%C3%BCnchenhttp://de.wikipedia.org/wiki/Linde_AGhttp://de.wikipedia.org/wiki/Eidgen%C3%B6ssische_Technische_Hochschule_Z%C3%BCrichhttp://de.wikipedia.org/wiki/Eidgen%C3%B6ssische_Technische_Hochschule_Z%C3%BCrichhttp://de.wikipedia.org/wiki/Eidgen%C3%B6ssische_Technische_Hochschule_Z%C3%BCrichhttp://de.wikipedia.org/wiki/Rudolf_Julius_Emanuel_Clausiushttp://de.wikipedia.org/wiki/Rudolf_Julius_Emanuel_Clausiushttp://de.wikipedia.org/wiki/Gustav_Anton_Zeunerhttp://de.wikipedia.org/wiki/Gustav_Anton_Zeunerhttp://de.wikipedia.org/wiki/Franz_Reuleauxhttp://de.wikipedia.org/wiki/Franz_Reuleauxhttp://de.wikipedia.org/w/index.php?title=Kottern&action=edithttp://de.wikipedia.org/wiki/Berlinhttp://de.wikipedia.org/wiki/M%C3%BCnchenhttp://de.wikipedia.org/wiki/Krausshttp://de.wikipedia.org/w/index.php?title=Polytechnische_Schule_M%C3%BCnchen&action=edithttp://de.wikipedia.org/wiki/Maschinenbauhttp://de.wikipedia.org/wiki/Rudolf_Dieselhttp://de.wikipedia.org/wiki/K%C3%A4ltemaschinehttp://de.wikipedia.org/wiki/K%C3%A4ltetechnikhttp://de.wikipedia.org/wiki/MAN_AGhttp://de.wikipedia.org/wiki/K%C3%BChlschrankhttp://de.wikipedia.org/wiki/Ammoniakhttp://de.wikipedia.org/wiki/Gashttp://de.wikipedia.org/wiki/Arbeit_(Physik)http://de.wikipedia.org/wiki/Paraffinhttp://de.wikipedia.org/wiki/1873http://de.wikipedia.org/wiki/G%C3%A4runghttp://de.wikipedia.org/wiki/Mainzer_Aktien_Bierbrauerei_AGhttp://de.wikipedia.org/wiki/Mainzer_Aktien_Bierbrauerei_AGhttp://de.wikipedia.org/wiki/Mainzer_Aktien_Bierbrauerei_AGhttp://de.wikipedia.org/wiki/Mainzer_Aktien_Bierbrauerei_AGhttp://de.wikipedia.org/wiki/Heinekenhttp://de.wikipedia.org/wiki/Carlsberghttp://de.wikipedia.org/wiki/21._Junihttp://de.wikipedia.org/wiki/1879http://de.wikipedia.org/wiki/Linde_AGhttp://de.wikipedia.org/wiki/K%C3%A4ltetechnikhttp://de.wikipedia.org/wiki/Chlorhttp://de.wikipedia.org/wiki/Kohlens%C3%A4urehttp://de.wikipedia.org/wiki/1890http://de.wikipedia.org/wiki/Operatives_Gesch%C3%A4fthttp://de.wikipedia.org/wiki/Aufsichtsrathttp://de.wikipedia.org/wiki/1892http://de.wikipedia.org/wiki/1910http://de.wikipedia.org/wiki/James_Prescott_Joulehttp://de.wikipedia.org/wiki/William_Thomson,_1._Baron_Kelvinhttp://de.wikipedia.org/w/index.php?title=Gegenstromverfahren&action=edithttp://de.wikipedia.org/wiki/Linde-Verfahrenhttp://de.wikipedia.org/wiki/Linde-Verfahrenhttp://de.wikipedia.org/wiki/Linde-Verfahrenhttp://de.wikipedia.org/wiki/Tieftemperaturphysikhttp://de.wikipedia.org/wiki/Fraktionierte_Destillationhttp://de.wikipedia.org/wiki/Sauerstoffhttp://de.wikipedia.org/wiki/Stickstoffhttp://de.wikipedia.org/wiki/Physikalisch-Technische_Bundesanstalthttp://de.wikipedia.org/wiki/Physikalisch-Technische_Bundesanstalthttp://de.wikipedia.org/wiki/Physikalisch-Technische_Bundesanstalthttp://de.wikipedia.org/wiki/Bayerische_Akademie_der_Wissenschaftenhttp://de.wikipedia.org/wiki/Bayernhttp://de.wikipedia.org/wiki/Ludwig_II._(Bayern)http://de.wikipedia.org/wiki/Adelhttp://de.wikipedia.org/wiki/1916http://de.wikipedia.org/wiki/Werner-von-Siemens-Ringhttp://de.wikipedia.org/wiki/Werner-von-Siemens-Ringhttp://de.wikipedia.org/wiki/Werner-von-Siemens-Ringhttp://de.wikipedia.org/wiki/1910http://de.wikipedia.org/wiki/Weltwirtschaftskrisehttp://de.wikipedia.org/wiki/1929http://de.wikipedia.org/wiki/1934
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IV.1 Generation of Low Temperatures IV.1.4 Joule-Thomson Cooling
schematics of a Helium liquefier
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Chapt. IV - 57
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IV.1 Generation of Low Temperatures IV.1.5 Summary
-
Chapt. IV - 58
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IV.1 Generation of Low Temperatures IV.1.5 Summary
-
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specification for cryocooler: • 1 Watt of cooling @ 80 K, rejecting heat at 300 K • 10 year life • 230 K to 340 K survival temperature • survival of launch vibration (non-operating) • low exported vibration • high efficiency • no maintenance possible oil-free
Northrop Grumman's HEC cryocooler
IV.1 Generation of Low Temperatures IV.1.5 Summary
Stirling cycle miniature cryocooler: - lightweight cooler, ideal for cooling of sensors and other electronics when low power consumption is important - mean time before failure of 24,000 hours - cooling capacity of 1 W @ 80 K - power consumption of only 55 W.
Sumitomo Heavy Industries
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Chapt. IV - 60
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IV.1 Generation of Low Temperatures IV.1.6 Evaporation Cooling
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Chapt. IV - 61
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IV.1 Generation of Low Temperatures IV.1.6 Evaporation Cooling
• everyday experience: sweating, wind direction, cooling of coffee, …
moisten finger, evaporation cooling
• microscopically:
• evaporation: work required to overcome binding forces
only the fastest molecules will do it
high-energy particles are lost
liquid cools down
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Chapt. IV - 62
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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)
• Hvap should be small to reach large cooling power at low temperatures
• numbers: about 1 K can be reached with 4He, about 0.3 K with 3He
• boiling point can be calculated by using the Clausius-Clapeyron equation, if heat of vaporization and the vapor pressure of the liquid at a certain temperature is known
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Chapt. IV - 63
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IV.1 Generation of Low Temperatures IV.1.6 Evaporation Cooling
Clausius-Clapeyron equation:
Hvap: molar latent heat [J/mole]
• approximate expression using pV = RT (ideal gas):
• integration yields (assuming that Hvap is constant over the considered T range):
≈ 90 J/mole
for 4He
• normal boiling temperature: pressure above liquid
boiling temperature at p0
(boiling point corresponds to the temperature at which the vapor pressure of the liquid equals the surrounding environmental pressure)
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Chapt. IV - 64
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IV.1 Generation of Low Temperatures IV.1.6 Evaporation Cooling
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Chapt. IV - 65
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IV.1 Generation of Low Temperatures IV.1.6 Evaporation Cooling
www.mallister.com/graphics/vapor3.jpg
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liquid 4He (c.f. chapter II)
• boson
• liquid down to 0 K (@ 1 atm)
• superfluid 4Helium at 2.17 K
– Bose condensation: macroscopic number of atoms in ground state
– very low viscosity
– very high heat conduction
– strange thermomechanical effects
– creeping on vertical surfaces
– vortex core with radius 0.8Å @ 0.6K
– explained by a two-fluid model
• density 125 kg/m3
IV.1 Generation of Low Temperatures IV.1.6 Evaporation Cooling
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Chapt. IV - 68
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Liquid Helium cryostats:
• LHe has small latent heat
good thermal insulation by vacuum
LHe container of poor thermal conductivity, glass or stainless steel
thermal radiation shield at liquid Nitrogen temperature to reduce black-body radiation
IV.1 Generation of Low Temperatures IV.1.6 Evaporation Cooling
bath cryostat - sample is immersed in the LHe
gas flow cryostat - sample is located in cold He gas
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Chapt. IV - 69
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LHe
IV.1 Generation of Low Temperatures IV.1.6 Evaporation Cooling
Liquid Helium container
vacuum radiation shields
narrow neck to minimize - heating by radiation - heating by thermal conduction
typical losses
- 1 l of LHe / day
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Chapt. IV - 70
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IV.1 Generation of Low Temperatures IV.1.6 Evaporation Cooling
He-bath cryostat
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IV.1 Generation of Low Temperatures IV.1.6 Evaporation Cooling
He-gas flow cryostat
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Chapt. IV - 72
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• no liquid Nitrogen required • radiation shield cooled by cold helium return gas
IV.1 Generation of Low Temperatures IV.1.6 Evaporation Cooling
He-gas flow cryostat (II)
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3) • reducing the vapour pressure over
bath of 4Helium
• temperature down to 1.2 K at pumping speed of 10 m3/h
Hvap: molar latent heat [J/mole]
≈ NA∙Ebinding: 90 J/mole for 4He
IV.1 Generation of Low Temperatures IV.1.6 Evaporation Cooling
Liquid 4He temperature < 4.2 K Clausius-Clapeyron equation:
cooling power:
RT
HpHnQ
vapvapgas
exp
rate of atoms going to gas phase up to 10 mW cooling power @ 1.2K
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Liquid 3Helium (c.f. chapter II):
• fermion
• superfluid at 2.5mK
– formation of weakly bound fermions: Cooper pairs
• density 59 kg/m3
• higher vapour pressure than 4He due to smaller latent heat: Hvap = 40 J/mole cooling power ≈ 80mW @ 1.2K and 10 m
3/h pumping speed
• 0.3 K by pumping 3He vapour
– some cm3
– 0.1mW cooling power @ 0.3K
• 3He obtained by nuclear reactions
• extremely expensive
• 1 liter of 3He gas costs about US $5.000 (2012)
IV.1 Generation of Low Temperatures IV.1.6 Evaporation Cooling
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IV.1 Generation of Low Temperatures IV.1.6 Evaporation Cooling
Liquid 3He cryostat
4He
3He ≈ 0.3 K
4He pump
3He pump
3He backflow
4He impedance 4He bath 4.2 K
vacuum
condensation of backflowing 3He gas
latent heat of 3He: Hvap = 40 J/mole as compared to 90 J/mole for 4He larger cooling power
≈ 80mW @ 1.2 K for 3He as compared to ≈ 10mW @ 1.2 K for 4He
minimum temperature: ≈ 300 mK (cooling power ≈ 0.1 mW)
≈ 1.2K
flow restriction for condensed 3He
RT
HpHnQ
vapvapgas
exp
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IV.1 Generation of Low Temperatures IV.1.7 Dilution Cooling
makes use of miscibility gap of 3He/4He mixtures (compare section I.6)
revision of some facts on 3He/4He mixtures cf. chapter I.6
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• bonding:
• 𝑉𝐴𝐵 >1
2𝑉𝐴𝐴 + 𝑉𝐵𝐵 complete miscibility (e.g. water and alcohol)
• 1
2𝑉𝐴𝐴 + 𝑉𝐵𝐵 > 𝑉𝐴𝐵 phase separation (e.g. water and petrol)
A A A B B B
VAA VAB VBB
critical point
miscibility gap
increasing mixing with increasing T:
F = U – TS min
binding energy
thermal motion
optimization of binding energy complete phase separation @ T = 0
minimization of free energy
IV.1 Generation of Low Temperatures IV.1.7 Dilution Cooling – revision of I.6
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• binding energy of 3He in 3He (V33) and 4He (V34):
(i) 3He in 3He: binding energy is given by the latent heat of evaporation L3:
binding energy for single 3He atom: 𝜖3,𝑐 = −𝐿3
𝑁𝐴=
𝜇3,𝑐
𝑁𝐴
𝜇3,𝑐 = chemical potential of pure (concentrated) liquid
(ii) single 3He atom in liquid 4He:
binding energy for single 3He atom: 𝜖3,𝑑(𝑥3 → 0) =𝜇3,𝑑 𝑥3→0
𝑁𝐴
𝜇3,𝑑 = chemical potential of dilute phase
• 𝜖3,𝑑 0 > 𝜖3,𝑐 or vice versa ? 3He has smaller mass larger zero point fluctuations occupies larger volume binding of 3He is larger in 4He than in 3He due to larger density of 4He |𝜖3,𝑑 0 | > |𝜖3,𝑐|
corresponds to case 𝑉𝐴𝐵 >1
2𝑉𝐴𝐴 + 𝑉𝐵𝐵 complete miscibility expected,
why miscibility only up to 6.5% ??
IV.1 Generation of Low Temperatures IV.1.7 Dilution Cooling – revision of I.6
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Questions:
• why can‘t we dissolve more than 6.5% of 3He in 4He at T = 0 ? two effects: (i) 3He forms degenerate Fermi liquid 𝑇𝐹 increases with 𝑥3 for 𝑥3 > 6.5%, the Fermi energy exceeds the gain in binding energy
(ii) 𝜖3,𝑑 𝑥3 > 𝜖3,𝑑(𝑥3 = 0) effective attraction between two ³He atoms (magnetic and volume effect)
• why don‘t we have a complete phase separation into 3He and 4He at T = 0 ? results in finite disorder, violation of 3. law of thermodynamic ? no: we have degenerate Fermi gas, ordering in k-space
IV.1 Generation of Low Temperatures IV.1.7 Dilution Cooling – revision of I.6
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x3
0.065
−𝜖3,𝑑(0)
−𝜖3,𝑐
𝑬𝐩𝐨𝐭 𝒙𝟑 + 𝒌𝐁𝑻𝐅 𝒙𝟑
0 gaseous ³He
pure liquid ³He
³He diluted in 4He
𝑬𝐩𝐨𝐭 𝒙𝟑 = 𝟏 + 𝒌𝐁𝑻𝐅 𝒙𝟑 = 𝟏
for 𝑥3 > 6.5%: 𝑬𝐩𝐨𝐭 𝒙𝟑 + 𝒌𝐁𝑻𝐅 𝒙𝟑 > 𝝐𝟑,𝒄 = 𝑬𝐩𝐨𝐭 𝒙𝟑 = 𝟏 + 𝒌𝑩𝑻𝐅 𝒙𝟑 = 𝟏
separation of pure ³He
energy
𝑬𝐩𝐨𝐭 𝒙𝟑
IV.1 Generation of Low Temperatures IV.1.7 Dilution Cooling – revision of I.6
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• for Fermi liquid: Cconcentrated < Cdiluted (x3 = 0.065) (𝐶𝑉 ∝ 𝑇/𝑇𝐹, 𝑇𝐹 ∝ 𝑛
2/3)
• with we therefore obtain:
Uconcentrated (T) < Udiluted (T) • on transition across phase boundary: F ≈ U = const increase of U results in decrease of temperature !
³He/4He dilution refrigerator
• operation principle: remove ³He atoms from the dilute phase below Tk = 0.87 K
transport of ³He atoms across phase boundary to maintain equilibrium concentration corresponds to evaporation of ³He from concentrated phase cools as the latent heat of evaporation is removed
concentrated (lighter)
diluted,6.5% (heavier)
³He
U
T
Uconcentrated
Udiluted
³He
IV.1 Generation of Low Temperatures IV.1.7 Dilution Cooling – revision of I.6
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• assumption: one mole of ³He crosses boundary
• removed heat depends on enthalpy difference between concentrated and dilute phase
• cooling power
• since there is no volume change
• with and
we obtain the entropy
(standard expressions for Fermi liquid)
IV.1 Generation of Low Temperatures IV.1.7 Dilution Cooling – revision of I.6
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IV.1 Generation of Low Temperatures IV.1.7 Dilution Cooling – revision of I.6
3He, concentrated
3He, diluted
Fermi sphere
large 3He density large Fermi sphere high TF
small 3He density small Fermi sphere low TF
fraction of thermally excited 3He atoms increases ( T/TF) entropy increases going from concentrated to diluted phase removed heat: dQ = TdS
thermally excited 3He atoms
plausibility consideration kBT
kBT
Fermi sphere
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• large cooling power requires large throuphput
IV.1 Generation of Low Temperatures IV.1.7 Dilution Cooling
23 84 TnQ
> 90% 3He
3He pump
from ≈ 1.2 K 3He condenser
heat exchangers < 1% 3He
still (≈ 0.7 K)
flow impedance
concentrated phase
dilute phase
6.5% 3He
100% 3He
heater to allow for effective pumping of 3He
mixing chamber (≈ 0.01 K)
dilute phase
concentrated phase
pumping of 3He generates osmotic pressure 3He flows from mixing chamber to still only possible if 3He atoms cross phase boundary cooling
minimum temperature: ≈ 1.5 mK
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• numbers: vapor pressure of 3He at 0.7 K: 0.0828 mbar = 8.28 Pa @ 300 K we obtain:
large 3He pump is required
• what is the required pumping speed ??
we assume that 3He is an ideal gas (R = 8.31 J / mole K)
IV.1 Generation of Low Temperatures IV.1.7 Dilution Cooling
• example: desired cooling power: 10-5 W still temperature: 0.7 K mixing chamber temperature: 10 mK
s /mole 0012.0)10(84
1022
5
3
n
pRTnV /3
l/s 360 s/m 0.363 28.8/30031.80012.0 3 V
23 84 TnQ
𝑅 = 8.31 J/mole K
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mixing
chamber
heat
exchangers
still
IV.1 Generation of Low Temperatures IV.1.7 Dilution Cooling
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JANIS Model JDR-500 Dilution Refrigerator
IV.1 Generation of Low Temperatures IV.1.7 Dilution Cooling
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3He shows minimum in melting curve at T = 0.32 K (compare I.5.2) can be used for cooling of ³He Pomeranchuk effect
Isaak Jakowlewitsch Pomeranschuk
(Исаак Яковлевич Померанчук;
born: 20. Mai 1913 at Warschau;
died: 14. Juli 1966 at Moscow)
Russian physicist.
IV.1 Generation of Low Temperatures IV.1.8 Pomeranchuk Cooling
0.1 1 1010
1
102
103
104
p (
bar
)
T (K)
phase diagram of 4He and 3He
3He
4He hcp
bcc
liquid
hcp
liquid
bcc
fcc
-
Chapt. IV - 91
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IV.1 Generation of Low Temperatures IV.1.8 Pomeranchuk Cooling
F
liquidsolidT
TTRSSTnQ
2-ln2) -(
2
3
• explanation: solid phase: atoms are ordered,
spins are disordered and determine entropy: Ssolid = R ln2 at low T: antiparallel ordering of spins, S decreases towards zero liquid phase: atoms are spatially disordered, but ordering in k-space (Fermi liquid) entropy of Fermi liquid:
• Clausius-Clapeyron equation:
• always: Vsol < Vliq
• for T < 0.32 K: disorder larger in solid than in liquid phase !!
S
lnT
R ln2
0
liquid
solid
320 mK
• cooling power:
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Chapt. IV - 92
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IV.1 Generation of Low Temperatures IV.1.8 Pomeranchuk Cooling
stamp stainless steel bellow
liquid
solid pressure cell
• precooling to T < Tmin
• adiabatic compression solidification and cooling • lowest T: ≈ 1.5 mK limitation due to antiparallel
spin ordering in solid 3He
10-3
10-2
10-1
100
2.8
2.9
3.0
3.1
3.2
3.3
3.4
p (
MP
a)
10-3
10-2
10-1
100
0.0
0.2
0.4
0.6
0.8
S /
R
T (K)
liquid
solid
liquid
solid
Tmin
ln 2
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Chapt. IV - 93
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IV.1 Generation of Low Temperatures IV.1.9 Adiabatic demagnetization
magnetic refrigeration: based on the magnetocaloric effect
magnetocaloric effect: - magneto-thermodynamic phenomenon - reversible change in temperature is caused by exposing a material to a changing magnetic field - also known as adiabatic demagnetization
http://upload.wikimedia.org/wikipedia/commons/0/08/Magnetocaloric_effect1_04a.svg
-
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IV.1 Generation of Low Temperatures IV.1.9 Adiabatic demagnetization
Adiabatic magnetization: - 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 reduced (T + ΔTad). Isomagnetic enthalpic transfer: - added heat is removed by coupling to heat sink (-Q) - magnetic field is held constant - after heat removal, magnetocaloric material and the coolant are separated (H = 0). Adiabatic demagnetization: - the substance is decoupled from heat sink no heat exchange with environment entropy stays constant - magnetic field is decreased, the thermal energy causes the magnetic moments to overcome the field, and thus the sample cools energy (and entropy) transfers from thermal entropy 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)
thermodynamic cycle:
http://upload.wikimedia.org/wikipedia/en/c/cb/MCE.gif
-
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• adiabatic 𝑆𝐵𝑖
𝑇𝑖= 𝑆
𝐵𝑓
𝑇𝑓 𝑇𝑓 = 𝑇𝑖 ⋅
𝐵𝑓
𝐵𝑖
• experiment: use of copper: Bi = 3 T, Bf = 0.3 mT, Ti = 10 mK Tf ≈ 1μK
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)
• adiabatic cooling: dQ = 0 dU = B dM
– step1: magnetize sample with field B, work must be done on sample and heat is released to thermal bath at Ti
– step2: thermally isolated sample and remove magnetic field B, sample uses internal energy to demagnetize and temperature falls to Tf < Ti
-
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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 ?
i
f
i
f
T
TB
spinT
Tspinspin dT
T
STdTCBQ )0(
2
2int
2
2
22
6
)1()12ln(
T
BB
k
JJgJNkS
B
BB
2int
2
2int
2
BB
BBTT
i
f
if
remaining internal field due to finite magnetic interactions (should be as small as possible)
(cooling capacity)
• entropy of spin system with spin quantum number J for gµBB
-
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IV.1 Generation of Low Temperatures IV.1.9 Adiabatic demagnetization
Ti Tf
Bf = 0 Bi = 5T en
tro
py
S
1
2
1 switch on magnetic field at constant Ti (coupling to heat sink)
2 switch off magnetic field for thermally isolated sample cooling to Tf
i
f
i
f
T
TB
spinT
Tspinspin dT
T
STdTCBQ )0(
)12ln( JNkB
0 0
medium
heat sink
heat switch
T
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IV.1 Generation of Low Temperatures IV.1.9 Adiabatic demagnetization
• paramagnetic salts: e.g MAS = MnSO4 · (NH4)2SO4 · 6H2O cooling of electron spins material with large entropy S/R but large Bint lowest temperatures Tf ≈ 100 mK large cooling capacity
e.g CMN = 2Ce(NO3)3 · 2Mg(NO3)2 · 24H2O cooling of electron spins material with small entropy S/R but small Bint lowest temperatures Tf ≈ 2 mK small cooling capacity • nuclear demagnetization: e.g 63Cu (L = 3/2) or 65Cu (L = 3/2) (Bint ≈ 0.3 mT, Ti ≈ 10 mK, Bi ≈ 3 T) cooling of nuclear spins Tf (Bf = 0) ≈ 1 µK problem: transfer of spin temperature to lattice long spin-lattice relaxation time other materials: 141PrNi5 (L=5/2),
195Pt (L=1/2)
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• spin-lattice coupling cools the lattice
– metals: ≈ 1000 s (hyperfine interaction: t = CKorringa/Te, Korringa relation) – non-metals: ≈ hours to several days
• spin-lattice coupling: spin temperature increases and lattice temperature decreases
in thermal equilibrium
• calculation of equilibrium temperature for copper: (Bint ≈ 0.3 mT, Ti ≈ 10 mK, Bi ≈ 3 T)
Teq = 1.03 μK
• lowest reported experimental temperature
demagnetization of Pt: Teq = 2 μK (F. Pobell et al., (1996))
0 nT
T
nucleare
T
T
electron dTcdTceq
f
eq
i
IV.1 Generation of Low Temperatures IV.1.9 Adiabatic demagnetization
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IV.1 Generation of Low Temperatures IV.1.9 Adiabatic demagnetization
R. Gloss et al.,
J. Low Temp. Phys. 73, 101 (1988)
Cu demagnetization stage (length: 525 mm, diameter: 78 mm)
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µK facility of PTB Berlin: Lattice temperatures measured on the 105-mol-copper stage of the Berlin microkelvin facility with Pt-NMR. The achieved minimal temperature was 23.3 µK. The red line depicts the calculated course of temperature for the thermodynamically optimized demagnetization function. heat leak: below 1.5 nW.
IV.1 Generation of Low Temperatures IV.1.9 Adiabatic demagnetization
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Cryogen-free Two Stage Adiabatic Demagnetization Refrigerator from Janis
A cryogen-free two stage adiabatic demagnetization refrigerator using a 4 K pulse tube cryocooler. 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.
IV.1 Generation of Low Temperatures IV.1.9 Adiabatic demagnetization
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IV.1 Generation of Low Temperatures IV.1.9 Adiabatic demagnetization
Continuous Adiabatic Demagnetization Refrigerator (CADR) under development at NASA´s Goddard Space Flight Center - CADR to cool from below 5K to ≈ 35 mK - advantage: no stored cryogens maximizing the lifetime/mass ratio for the instrument
-
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Contents: IV.1 Generation of Low Temperatures IV.1.1 Introduction IV.1.2 Expansion Machine IV.1.3 Regenerative Machine IV.1.4 Joule-Thomson Cooling IV.1.5 Summary IV.1.6 Evaporation Cooling IV.1.7 Dilution Cooling IV.1.8 Pomeranchuk Cooling IV.1.9 Adiabatic Demagnetization
IV.2 Thermometry IV.2.1 Introduction IV.2.2 Primary Thermometers IV.2.3 Secondary Thermometers
-
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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 an infinitesimal quasi-static transformation, and the variation δS of its entropy during this transformation: for reversible Carnot process (dS = 0):
S
QT
T
Q0
Kelvin scale Celsius scale (1742)
see http://www.its-90.com
• Lord Kelvin (1854): there is an absolute zero of temperature scale T0 = 0 K = - 273.15°C 1 K = 1°C
http://www.its-90.com/http://www.its-90.com/http://www.its-90.com/http://www.its-90.com/
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William Thomson (Lord Kelvin)
http://br.geocities.com/saladefisica3/fotos/kelvin.gif
Lord Kelvin by Hubert von Herkomer
Born 26 June 1824(1824-06-26) Belfast, Co. Antrim, Ireland
Died 17 December 1907 (aged 83)[1] Largs, Ayrshire, Scotland [1]
Residence Cambridge, England Glasgow, Scotland
Nationality United Kingdom of Great Britain and Ireland
Institutions University of Glasgow
Alma mater Glasgow University Peterhouse, Cambridge
A variety of physical phenomena and concepts with which Thomson is associated are named Kelvin: • Kelvin material • Kelvin water dropper • Kelvin wave • Kelvin-Helmholtz instability • Kelvin-Helmholtz mechanism • 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)
IV.2 Thermometry IV.2.1 Introduction
http://en.wikipedia.org/wiki/June_26http://en.wikipedia.org/wiki/1824http://en.wikipedia.org/wiki/Belfasthttp://en.wikipedia.org/wiki/County_Antrimhttp://en.wikipedia.org/wiki/Irelandhttp://en.wikipedia.org/wiki/December_17http://en.wikipedia.org/wiki/1907http://en.wikipedia.org/wiki/Largshttp://en.wikipedia.org/wiki/Ayrshirehttp://en.wikipedia.org/wiki/Scotlandhttp://en.wikipedia.org/wiki/Cambridgehttp://en.wikipedia.org/wiki/Englandhttp://en.wikipedia.org/wiki/Glasgowhttp://en.wikipedia.org/wiki/Scotlandhttp://en.wikipedia.org/wiki/United_Kingdom_of_Great_Britain_and_Irelandhttp://en.wikipedia.org/wiki/United_Kingdom_of_Great_Britain_and_Irelandhttp://en.wikipedia.org/wiki/University_of_Glasgowhttp://en.wikipedia.org/wiki/Alma_materhttp://en.wikipedia.org/wiki/Glasgow_Universityhttp://en.wikipedia.org/wiki/Peterhouse,_Cambridgehttp://en.wikipedia.org/wiki/Peterhouse,_Cambridge
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Chapt. IV - 108
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• 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).
• Celsius scale: the Celsius scale has units of °C (degrees Celsius) with the size of the unit equal to 1 Kelvin.
T(°C) = T(K) – 273.15
• agreement of bureaus of standards:
ITS-90 temperature scale for T > 0.65 K (Comité International des Poids et Messures 1990) - the ITS-90 is defined by 17 fixed points and 4 defining instruments. It spans a 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 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.
IV.2 Thermometry IV.2.1 Introduction
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Chapt. IV - 109
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The Water Triple Point The triple point of water is the most important defining thermometric fixed point used in the calibration of thermometers to the International Temperature Scale of 1990 (ITS-90). It is the sole realizable defining fixed point common to the Kelvin Thermodynamic Temperature Scale (KTTS) and the ITS-90; the assigned value on these scales is 273.16 K (0.01°C)
solid
liquid
gaseous
critical point
triple point
(0.01°C, 603 Pa)
273.15 K
(374°C, 21800 kPa)
273.16 K
IV.2 Thermometry IV.2.1 Introduction
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Chapt. IV - 110
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Defining Fixed Points of the ITS-90
Number Temperature Substance a State b Wr (T90)
T90/K t90/°C
1 3 to 5 -270.15 to -268.15 He V
2 13.8033 -259.3467 e-H2 T 0.001 190 07
3 ~17 ~-256.15 e-H2 (or He) V (or G)
4 ~20.3 -252.85 e-H2 (or He) V (or G)
5 24.5561 -248.5939 Ne T 0.008 449 74
6 54.3584 -218.7916 O2 T 0.091 718 04
7 83.8058 -189.3442 Ar T 0.215 859 75
8 234.3156 -38.8344 Hg T 0.844 142 11
9 273.16 0.01 H20 T 1.000 000 00
10 302.9146 29.7646 Ga M 1.118 138 89
11 429.7485 156.5985 In F 1.609 801 85
12 505.078 231.928 Sn F 1.892 797 68
13 692.677 419.527 Zn F 2.568 917 30
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 All substances except 3He 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
IV.2 Thermometry IV.2.1 Introduction
http://www.its-90.com/http://www.its-90.com/http://www.its-90.com/http://www.its-90.com/
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Chapt. IV - 111
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temperature measurement
• definition of temperature via reversible Carnot process is not well suited for establishing useful measuring methods
• in practice: use of fixpoints and interpolation polynoms
• primary thermometers: measured quantity is related directly to temperature (in a theoretically predictably way) no calibration is required
• secondary thermometers: measured quantity varies with temperature in a reproducible way 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
IV.2 Thermometry IV.2.1 Introduction
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Chapt. IV - 112
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T(K)
typical temperature range of some thermometers
IV.2 Thermometry IV.2.1 Introduction
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Chapt. IV - 113
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most common thermometers for 1K < T < 300K
• gas thermometer: p = p(T) – Helium gas ≈ ideal gas down to 10K:
• vapour pressure thermometer: Tliquid = f(pvapor) – pressure of 10 Pa corresponds to 0.4K for 3He
• thermocouples: Vth = Vth(T)
• 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) – p-n junction diode (problem with high bias current self heating)
• capacitance thermometry: C = C(T) – 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
IV.2 Thermometry IV.2.1 Introduction
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Chapt. IV - 114
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3) 1mK ≤ T ≤ 1K:
• magnetic suceptibility thermometer
T
C
B
M 0Curie’s law:
Ce