He3/He4 DILUTION REFRIGERATOR
Until about 1950 the only way to cool an experiment below 1 K was magnetic refrigeration unit with a
paramagnetic salt. This has been replaced by He3 cryostats for operation down to 0.3 K. In 1962 H.
London proposed to use the mixing heat of He3/He4 mixtures for refrigeration. 1965 the first
refrigerator operates down to 0.22K. The fierce competition between Leiden (NL) and Dubna group
(USSR). Soon the mK range becomes accessible. Today commercial devices operate continuously
between 2 mK and 1K.
Properties of He3/He4 mixtures.
Phase diagram of He3/He4 at the saturated vapor pressure, see fig. 38a
x  x3 
n3
n3  n4
Pure liquids: He4 superfluid, He3 not (in this temperature range). Mixing depresses superfluid phase
transition. Eventually superfluidity ceases to exist at x>67 %. At x=0.67 and T=0.87 the  line meets
with the phase separation line. Below this temperature He3 and He4 are only miscible for certain
limiting concentration which depends on temperature. The hashed region is a non-accessible range of
temperatures and concentrations for helium mixtures. If one cools a mixture with x>0.67 below 0.87K
the liquid separates into two phases; one rich in He3 and the other rich in He4. He3 rich phase will float
on the top of He4 rich phase.
At T→0: He3 rich phase will become pure He3.
On the He4 rich side - a new phenomenon! Constant concentration of 6.5% of He 3 in He4 at saturated
vapor pressure even at 0K! This is finite solubility phenomenon. The limiting concentration depends on
pressure, see fig. 38 b.
Cooling in He3/He4 dilution refrigerator is achieved by transferring He3 atoms from He3 rich phase to
the diluted (mostly He4) phase. The specific heat of He3 is larger in the diluted phase giving rise to a
cooling effect.
Importance of the finite solubility
Cooling power of an evaporating cryogen
Q= n L ~ p(T) ~ e-1/T
In the dilution the corresponding quantities (for L and
n ) are the entropy of mixing, ΔH   ΔcdT ,
(c is the difference in the specific heat), and the concentration of the diluted phase.
Finite solubility allows for a high He3 molar flow rate even at low temperatures. Because specific heat
of both phases is proportional to T we have
Q  xH  T 2
Advantage of the dilution cooling:
the cooling power does not decay exponentially with decreasing temperature.
(graph , page 142)
Physics behind the finite solubility of He3 in He4
Case x  0 and one atom of He3 at the phase separation line has to decide where to go:
Van der Waals forces in the He3 and He4 are identical (full electron- shells in both atoms). But He3 has
a smaller mass and larger zero quantum fluctuations E 0. Thus, He4 atom occupies smaller volume than
He3. Consequently, He3 atom can be closer to He4 than to He3  there is stronger binding in the dilute
phase  this leads to a finite solubility.
Many He3 atoms in liquid. He4 (x>0)
Situation is now different changed due to two effects:
1. Attractive interaction between He3 atoms due to nuclear magnetic moments of He3.
2. Density effect: He3 needs more space than He4 → liquid near He3 is less dense than that close to
He4. This low-density region is felt by other He3 atoms that want to combine with the former one
because there it has to push less in order to have a space for itself.
Due to this binding interactions the binding energy of He3 in He4 should increase with increasing x. But
there is Pauli principle: new atoms are required to occupy higher and higher energy levels. Therefore,
the binding energy of He3 will decrease due to their Fermi character when it reaches the binding energy
of He3 in pure He3. The limit concentration of 6.5% corresponds to the concentration at which the
binding energy in He3/He4 mixture equals to the binding energy in pure He3.
He4 atoms at the phase separation line: He4 atoms prefer to stay in He4 phase for the same reason as
He3 atom does. But He4 is a boson and there is no limiting concentration as the binding energy does not
decrease with increasing concentration. As a result the concentration of He 4 rapidly reaches zero in the
upper He3 rich phase if the temperature drops to zero.
Cooling power of the dilution process:
Q = n 3 [H d (T)-H c (T)]
Hc(T) – enthalpy of concentrated phase and Hd(T) – enthalpy of diluted phase. In general,
T
H (T )  H (0)   c(T )dT 
0
(we have neglected irrelevant pV terms here). To have a cooling effect we must have the following
relation of the molar heat capacities: c3d(T) > c3c(T) ()
Liquid He3 is a strongly interacting Fermi liquid and theoretical calculations of c are not available.
Semi-empirical expressions, valid for T<40 mK, reads:
C3 = 2.7 RT = 22T [ J/mol K]
H3(T) = H3(0)+11T2 [ J/mol]
The diluted phase is weakly interacting Fermi liquid and one mole of mixture has
C3d = 0.75 m*/m3 (Vm/x)2/3T,
where m* - effective mass, reflects the influence of a neighboring He4 atoms, m3 – pure He3 atom mass.
The molar volume of the mixtures
Vm = Vm,4 (1+0.284x)
Where Vm,4 is molecular volume of a pure He4. Observe that cv per He3 atom increases with decreasing
x!
c3,d 
x1/3T
, thus, c3d (6.5%)  106 T [J/mol He3 K]
m
When two phases are in equilibrium the chemical potentials of both phases are equal 3c = 3d.
With  = H-TS we have
H3c-TS3c = H3d-TS3d
and
T
H 3d (T) = H 3 (0)+11T 2 + T  (
0
c3d c3
J
- ) dT = H 3 (0)+95T 2 [
]
T T
mol He3
The cooling power resulting from transferring n3 moles of He3 through the separation line
Q (T )  n3[ H 3d (T )  H 3 (T )]  84n3T 2
with n3=100 mol/s and T=10 (30) mK we find Q  1 (10) W
For more details on physics of mixture and thermodynamics of mixing process see:
O. V. Loynasmaa: “Experimental Principles and Methods Below 1K” , Academic Press, London 1974.
Realization of a He3/He4 dilution refrigerator
Gedankenexperiment for comparison of a helium evaporation refrigerator and a He 3-He4 dilution
refrigerator:
The real thing cannot operate in this mode because the heavier mixture cannot float on the top of He 3
rich phase. The real design, see fig.38 c.
Critical elements:

Heat exchanges.

Flow impedances.
In many systems attempts were made to reduce room temperature gas handling circuits to the internal
charcoal pumping.
Still design with film burner 38 d)
Heat exchangers:
Concentric 38 e)
Sintered powders 38 f),g).
Examples of the real devices figs.39,40.
Pomeranchuk cooling
Solidification of the matter at a rate dn/dt usually results in the production of heat
dQ/dt = dn/dtT (Ssolid – Sliquid) < 0
In 1950 Pomeranchuk predicted that for He 3 below  0.3K Sliq < Ssolid (!) due to the quantum nature of
He3.
In the liquid state He3 atoms are indistinguishable and we have to apply Fermi-Dirac statistics as for
electrons- in metals. He3 atoms in the liquid state move rather freely and strongly interact and arrange
themselves in opposite nuclear spin couples. In the solid state atoms are constrained to vibrate around
their lattice sites. Again, at mK temperatures phonons are negligible and only nuclear spin
contributions to the entropy count. (Solid He3 is a bbc structure nuclear paramagnet). The Pauli
principle does not apply to the solid since atoms are already distinguished by their spatial coordinates
 magnetic weak ordering is possible in the liquid state  Sliq < Ssolid.
The melting curve of solid He3
 See fig. 40a.
According to Clausius-Clapeyron (at melting)
dpm ( Sl  S s )m

dT
(Vl  Vs )m
dpm
Clearly, since below 0.3K Ss>Sl and Vm,l>Vms therefore
< 0 and we have a cooling effect. The
dT
cooling power
dQ/dt = dnsol /dt T(Ssol-Sliq) m
where (Ssol-Sliq )– the latent heat of freezing and dnsol /dt the rate of solidification
(graph, page 149)
Pomeranchuk process is a one shot cooling process. Bears no practical importance today but have
brought enormous progress in He3 physics – discovery of superfluidity at 2.5mK and antiferromagnetic
phase transition at 1mK. Cornell University Pomeranchuk cooler used for these experiments –
schematics, see fig. 40b. The cooler starts with He3 pre-cooled to T<0.32 K.
Magnetic cooling
Proposed by P. Debay and (independently) by W. F Giauque in 1926. (Kamerling Onnes pumped He4
down to 0.7 K and declared this as a limit of possible low temperatures). Practical implementation only
in 1933 in Berkeley – down to 0.53K and soon 0.27K (Leiden). Since the discovery of He 3/He4 dilution
techniques – no practical importance.
The principle – Adiabatic demagnetization of a paramagnetic salt.
Consider paramagnetic ions with electron magnetic moment  in a solid. Let the energy of the
interaction between the moments themselves as well as with an externally applied magnetic field be
small compared to kBT. This means that we are considering free paramagnetic ions with magnetic
moment  and a total angular momentum J with entropy contribution.
S = R ln(2J+1)
If they are completely disordered in 2J+1 possible orientations in the temperature range of interest,
their magnetic disorder entropy (~ J/mol of refrigerant) is always large compared to other entropies of
the system.
Fig. 41a – Entropy diagram
If the temperature is decreased eventually, the interactions between magnetic moments will become
comparable to the thermal energy. This will lead to to spontaneous magnetic order with ferro-, or
antiferromagnetic orientation of electron magnetic moments. As a result the entropy will decrease to
zero as required by the third law of thermodynamics. Externally applied magnetic field will interact
with magnetic moments and the ordering will occur at higher temperatures.
Procedures:






Precool to the starting temperature T i.
Apply magnetic field Bi to perform isothermal magnetization at T i from B=0 to B=Bi (AB`). During this process the heat of magnetization is absorbed by the precooling bath.
Thermally isolate the paramagnetic salt from the bath.
Adiabatically demagnetize from B=Bi to B=Bf ~ mT (B–C).
This is a one shot technique.
Spontaneous ordering of magnetic moments is the lower limit for refrigeration.
Paramagnetic salts:
“High” temperature salts:

MAS: Mn2+ SO4 (NH4)2SO4 . 6H20

FAA: Fe22+ (SO4)3(NH4)2SO4 . 24H20
“Low” temperature salts:

CPA: Cr23+ SO4 K2SO4 . 24H20

CMN: 2Ce3+ (NO3)3 3Mg(NO3)2 . 24H20
Tc = 0.17 K
Tc = 0.03 K
Tc=0.01 K
Tc=0.002 K
The working salt must contain ions with only partially filled electron- shells, i.e., 3d transition elements
or 4d rare earth elements. The salts contain water of crystallization which assures a large distance (~
10Å in CMN) between the ions thus providing low magnetic ordering temperature.
Entropies of these salts – Fig. 41b solid line: zero field, dashed: 2T.
A typical refrigerator based on paramagnetic cooling – fig. 41c.
(Page 41, pictures)
Paramagnetic salts are not used for cooling but they are still important today as they serve as low
temperature thermometers ( 
  ).
T
Nuclear Adiabatic Demagnetization
1934- C. J. Gorter and independently N. Kurti. Practical implementation in 1956 by Kurti at Oxford.
The magnetic cooling limit is set by spontaneous ordering of magnetic moments. In the approximation
of dipole-dipole interaction only we get
Tc 
2
r3
Nuclear moments ~ nuclear magneton n = 5.1 x 10-27 J/T. (compare to Bohr magneton: B = 9.3 x 1024
J/T). Therefore, the ordering temperature for nuclear moment is much lower then that of electronic
moments, typically ~ 0.1 K or below.
Paramagnetic salts were poor thermal conductors (problems of heat exchange). Here we can use pure
metals with high thermal conductivity. Moreover, the density of magnetic moments can be higher than
the density of electronic moments in diluted paramagnetic salts.
Advantages are counterattacked by experimental problems resulting from the small size of nuclear
moments. To achieve a reduction of nuclear magnetic entropy of at least few percent we get into
demanding conditions for magnetic field and starting temperature. For example, for the most
commonly used cooling agent, Cu, with starting Bi=8 T and Ti=10 mK we get the reduction of entropy
of only ~ 10% - see Fig. 42a.
Final problem: The transfer of the spin temperature. Nuclear moments may have very low temperature,
but one has to transfer it to the electrons and lattice vibrations. In 1956 Kurti performed the first
successful magnetic cooling down to microKelvin range but at the same time the lattice temperature
stayed at 12 mK and in few minutes the spin system warmed up to this temperature. Only more than a
decade latter Lounasmaa in Helsinki managed to cool the lattice as well.
Realizations:
Single stage Cu and double stage PrNi5/Cu magnetic refrigerators: fig. 42b.
Problems of heat links and coupling to the external world, fig.42e.
 External: rf – screened room 120 dB up to 1 GHz needed!.
 mechanical – seismic, building vibrations, pumps.
 Eddy currents.
 Internal load – orto-para conversion of Hz rate tunneling reconfigurations (avoid noncrystalline solids!)
 Strain relaxation – wait!
Low temperature part of the University of Bayeruth nuclear refrigerators: Fig.42c.
The view of the first stage: Fig. 42d.
(Page 42, pictures)
LOW TEMPERATURE THERMOMETRY
Thermodynamic temperature:
Definition based on the Carnot cycle (II law of Thermodynamics): Reversibility means that

dQ
 0  T / Q  const.
T
Carnot process gives the temperature only in terms of ratios or to within a multiplicative constant; the
absolute values in a temperature scale have to be fixed by a definition.
First temperature scale: Celsius 1742: T (H2O boiling)–T (H2O melting) = 100 degrees
Celsius scale is not very appropriate as we have a thermodynamic absolute zero. In 1854 Lord Kelvin
proposes to establish a thermodynamic temperature scale. Only 100 year later, an international
convention in 1954 has been adopted:
1 degree = 1K = Ttriple (H2O)/273.16
Having established the zero point on the scale one can use an experiment related to a Carnot cycle to
establish a temperature scale. Such procedure is however not practical. For practical measurements
fixed temperature points that can be used to calibrate the temperature scale have been established by
defining international temperature scales.
International temperature scales
IPTS–68 – official scale for temperatures above 14K (amended in 1975).
This scale superseded the earlier scales: ITS – 27 and IPTS – 48.
EPTS–76 Provisional temperature scale for 0.5 – 30 K range established in 1976.
Unfortunately, all above scales have significant errors ~ 10 -4K.
ITS–90 agreed in 1990 and approved by Comite` International des Poids at Messures. This is current
official scale. Ranges from 0.65K up to highest measurable temperatures in terms of Planck`s
monochromatic radiation.
 Scale – Fig. 44.
 Fixed points: 44a)
In the overlapping ranges ITS – 90 is defined at follows:
1) 0.65-5K: vapor pressure/temperature relations of He3 (0.65-3.2K) and of He4 (1.25-5K). The
agreed equation is
9
2)
T   Ai [
i 0
ln p  B i
]
C
Constants Ai,Bi and C are tabulated see 44b. Tabulated temperature vs. pressure relations are give
in 44c), 44d).
(Page 44, temperatures)
3) 3.0-24.5561 K (Triple point of Ne) use constant volume the gas thermometer calibrated at fixed
points given in table 44a).
4) 13.8033 (Triple point of Hg) – 1234.93 K (freezing point of Ag) – Electrical resistance of pure Pt,
calibrated at fixed points.
5) Above 1234.93 K – Planck`s radiation law.
ITS – 90 contains detailed instructions on how to calibrate a thermometer in various temperature
ranges.
Other practical but not officially accepted low temperature fixed points.
1) From the superseded EPT 76 scale.
Tb (H2) – 20.27 K at 1 bar
Ttr (H2) – 13.80 K
Tc(Pb) – 7.2 K
Tb(He) – 4.21 K
Tc(In) – 3.415 K
T(He) – 2.1768 K
Tc(Al) – 1.18 K
Tc(Zn) – 0.1851 K
Tc(Cd) – 0.519 K
2) NBS Superconducting Fixed Point Device: Fig. 45a) and some of the fixed superconducting
critical points: Fig.45b)
3) Fixed points of liquid and solid He3: Fig.45c).
Currently there is a proposed for establishing a mK scale based on He 3.
(Page 45, Figs.)
Thermometers
Primary thermometer: our theoretical knowledge about the measured property of mater is good
enough to calculate the thermodynamic temperature without any calibration.
Example – gas thermometer using a relation between pressure, volume and temperature for gases.
Using this type of thermometers is difficult and usually restricted to national standards laboratories.
Secondary thermometers: are the “workhorses” of laboratory measurements. These thermometers
have to be calibrated against a primary thermometer or at fixed points. Secondary thermometers are
usually more sensitive and more convenient to use.
1)
Gas thermometry
Based on the gas equation:
pV = nRT
Possible two setups:
For real gases corrections to the equation have to be introduced by “virial coefficients”. Use only in
standards labs.
2)
The vapor-pressure thermometry
Measure the vapor pressure above the liquid and
use ITS – 90 scale for calibration. To avoid
gradients and hydrostatic pressure differences use a
small cell with a vacuum jacket capillary instead of
measuring the pressure above the surface.
3)
He3 melting pressure thermometry – use melting curve Fig.40a) and Greywell scale
5
p  p A   aiTi
i 3
pA – pressure at which transition from normal to superfluid He3 occurs– this is the reference.
Coefficients are tabularized in the Greywell scale.
4)
Thermoelectricity
Thermoelectric power between two different metals measured with a reference function at a reference
temperature (usually at 00C). N.B. the thermoelectric power vanishes for T→0.
Usual thermocouples (e.g. constantan – Cu) become insensitive below 10K. – see 46a). With new
specially developed couples (highly diluted magnetic alloys – Kondo alloys) one can extend the
technique down to mK range, see 46a).
In the cryogenic range the reference junction at reference bath of He4 at 4.21 K. With Au-Fe system it
is possible to obtain resolution of 1K at 1K.
Practical limit of thermocouples: 100 mK.
5)
Resistance thermometry


Pure metals: Platinum – Fig. 46b) – set points PT–100 scale, Fig.46c)
Semiconductors: For intrinsic semiconductors resistance increases with increasing
temperature.
R (T )   e
E
2 k bT
Typically, the resistance for materials used in thermometry do not agree with this equation and
empirical relations between R and T have to be used, fig. 46d. Two most widely used materials are
specifically doped Ge and carbon resistors.
Germanium:
 n-type 1015-1019 As doped
 p-type 1015-1019 Ga doped
Below 100K conductance is due to carriers delivered by donors and acceptors. Possible operation range
0.3K < T < 40K. At lower temperatures R becomes too large.
R(T) for Ge – see 46e), thermometer design– see 46f).
Ge thermometers are very stable, within 0.1%. Therefore, can be bought calibrated with computer
polynomial fitting equation and coefficient.
Carbon resistors:
“Allan Bradley” (AB) and “Speer”– cheap radioelectronic elements. Turned out to be excellent
thermometers. Consequently their price was raised but is still lower than Ge thermometer price.
(Page 46, figs.)
Best Carbon thermometers:
A-B – 1/8W good at T>1K.
Matushita: ERC18GK, 18SG (not produced any more).
Speer:1/2W – type 1002
Elements which are not produced anymore are available from stocks of “cryogenic” companies for
high price.
R(T) for a Speer resistor : Fig.46g). Magnetoresistance of an AB resistor Fig.47h).
A way to reduce thermal capacity of the resistor see Fig. 47i). Resistor trimmed-down to thin flake Fig.
47j). Proper thermal fixing of the resistors Fig. 47k).
Thick film chip RuO2 resistorsEither fabricated in the lab or commercially available – metal ceramic consisting of a mixture of
conducting RuO2, Bi2RuO2 and a lead-silicate glass PbO-B2O3-SiO2 that has been heated above its
glass point. Accuracy within 0.5% at the range 800mK  T  25mK.
R(T) = R0 exp(T0/T)0.345
Very small size (few mg), reproducible, cheep, low magnetoresistance (! For high field applications).
6)
Noise thermometry
<V2>t = 4kB TR f
<V2> = ff+fS(f) df
for R=1k, f = 1kHz,
 at T= 4K noise level 10-8V
 at T=10mK noise level 10-10V
Measurements using Superconductivity Quantum Interference Devices – SQUID`s.
(Page47, Figs.)
7)
Magnetic Thermometry with Electronic Paramagnets.
Basis:
M
B
M 
 0 
0T
B
T
Curie constant. This is in fact a primary thermometer.
CMN thermometer – problematic high time constant: Fig.47a).
To improve time response - CMN pressed with Ag powder into a foil: Fig.47b).
Entire CMN thermometer: Fig. 47c). Measurement limit down to 5-8mK,  ~ 10-40s.
8)
Magnetic Thermometry with Nuclear Paramagnets.
Smaller magnetic moments – more sensitive measurements:  n = n/Tn Fig.47d)
Measurements techniques:

Non-resonant – static measurements of magnetization: Fig. 47e).

Resonant NMR techniques (pulsed): Fig. 47f).
Problem of the relation between the spin and lattice temperatures: Fig. 47g).
Example of a design of Pt wire NMR thermometer: Fig. 47h).