Part III Matter and methods at low temperatures
The importance of low temperatures
The range of temperatures in which biological life exists is extremely small with respect to the
temperature range in the Universe. More than 50% of the temperatures in the Universe, seen in the log
scale, fall into the “low temperatures” see fig. 29 a)
Low temperature ranges:
I) Kelvin range:
Technique
Universe
He4 evaporation
He3 evaporation
II) Mili-Kelvin range :
Technique
He4 / He3 dilution
Pomeranchuk
Electron magnetic
moments
Available
Since
Typical
Min. Record
2.74K
1.3K
0.3K
2.74K
0.4K
0.3K
Available
Since
1965
1965
1934
Typical
Min. Record
10 mK
3 mK
3 mK
2 mK
2 mK
2 mK
Available
Since
1956
Typical
Min. Record
50 K
10 nK
1908
1950
III) Micro-Kelvin range:
Technique
Nuclear magnetic
moments
For an overview see: K. Gloos et al J. Low Temp. Phys., 73, 101 (1988)
Low temperature progress history – see fig. 29 b
The most typical cooling technique uses cryogenic liquids as cooling agents. Properties of cryoliquids:
see 29c.
Liquid air
(mostly Oxygen + Nitrogen) - not in common use. Converts into liquid oxygen which is explosive!
Explosion triggered by a contact of lq. O2 with organic liquids (e.g.,– vacuum pumps oil) or with solids
with developed surface (powders).
Tboil O2 = 90.2 K
TboilN2 = 77 K
Nitrogen evaporates first and liquid air becomes enriched in explosive liquid Oxygen.
Liquid N2
Commonly used cryogen – typically obtained by distillation of lq. air in industrial plants, or as a side
product in the artificial fertilizer production line.
Problems:

Aggressive NO gases.

Displaces oxygen in closed volumes. To avoid suffocation provide for adequate ventilation in
the working area!
Advantage: Liquid N2 is fairly cheap and easily available.
prices:
Europe 1 NIS/l
Il 2.
BGU
1 - 2.7 NIS/l
4 NIS/l (promised to get to the IL level soon)
Liquid Hydrogen
Undergoes exothermic reaction with O2 to form water - explosive!
Once used to cover the range between solid N2 and liquid He4, now covered by closed cycle and
variable temperature flow and bath helium cryostats.
Problem excessive heat releases due to orto-para transition of H2 in very low temperature experiments:
H proton has a spin I=1/2. Therefore, H2 molecule can have total spin equal to 1 or 0.
Case I = 1: – symmetric nuclear state orto H2
Case I = 0: - para – H2
Orto-para transition occurs in small gas bubbles inside the solid (bubble diameter  0.1). Orto- para
transition is a spontaneous conversion associated with heat release  1nW/g. (para state has lower
energy of molecule rotational states)
Liquid Helium –
The most important cryogen and the most interesting substance by itself.

Discovered in 1868 in spectra of solar pertubances

In 1895 found on Earth in minerals, since than a race to liquefy the gas.

Invention of a devar – by Devar (for P.R. purposes!)

First liquefied by Kammerling-Ones in Leiden 1908

First commercial liquefier – Collins in 1947 – opens new era in experimental physics.

Nowadays we obtain He from natural gas sources. Most reach sources in USA and Poland, up
to 10% He.
Two common stable isotopes He4 and He3.
He4 I=0  boson.
He3 I=1/2  fermion.
He3 constitutes a 10-7 fraction of He gas from natural sources and 10 -6 of the He in the atmosphere.
For commercial use He3 is obtained for as a by-product of tritium manufacture in a nuclear reactor:
6
3Li
+ 10n  31T + 42He
,
3
3
1T 2He
+ 0-1e +  (electron anti-neutrino)
He3 is separated from tritium by diffusion processes. This method of production made He3 available
only in the late 1950’s (byproduct of a cold war). He 3 is very expensive! Europe 1000 NIS/l of gas at
standard temp. and pressure.
There are two more unstable Helium isotopes He6 (1/2 =0.82s) and He8 (1/2 =0.12s) they have not
yet been liquefied.
Properties of liquid Helium:
isotope
He3
He4
Boiling Temp. (K)
3.19
4.21
Critical Temp.(K)
3.32
5.20
Superfluid Tc (K)
Density g/cm3
0.0025
2.177
0.082
0.145
Classical molar volume cm3/mole
12
12
Actual
36.84
27.58
Phase diagrams of He3 and He4 , see figs 29 d,e
Liquid Helium is most exotic and interesting liquid exhibiting many unique properties due to the fact
that it is a quantum liquid.
Features:
 low boiling points and low Tc.
 remains liquid at vapor pressure even at 0 K! One needs at least 25/34 bar at T0 to get He to
the solid state.
 He4 melting line is constant within 10-4 K below 1K.
 He3 melting curve has pronounced minimum at 0.32 K.
 small density and large molar volume.
These are consequences of:
1. Weak binding forces between the atoms. They are Van der-Waals forces (closed electron sshells). For the same reason no static dipole moment and smallest known atomic polarizability
 = 0.1232 cm3/mole The resulting dielectric constants:
3 =1.0426
4 =1.0572
Therefore, He liquid is transparent.
2.
Due to small atomic mass helium isotopes have huge quantum mechanical zero-point energy.
1
 Vm  3
h2

 is the distance between the atoms.
,
where
a

E0 
2 ma 2
 NA 
The large zero point energy causes vibrations of the amplitude of the order of 1/3 of the separation
between the atoms in the liquid state. Zero point energy causes that helium, in contrast to all other
known substances, remains liquid at 0K.
Figures: (energy vs. molar volume)
Of course He3 has smaller mass therefore the influence of zero point energy is stronger, therefore it has
lower Tc, smaller latent heat of evaporation, and larger vapor pressure then He 4.
Criterion for quantum liquid is the ratio:
Ekin/Epot
E kin E pot  1
Xe
Kr
Ar
N2
Ne
H2
He4
He3
0.06
0.1
0.19
0.23
0.59
1.73
2.69
3.05
*** He3, He4 and H2 are quantum liquids.
Latent heat of evaporation and vapor pressure.
These are essential parameters for cryo-cooling applications.
He4 as a quantum liquid has latent heat of only 25% of a corresponding classical liquid, see 29f. As a
result He bath has a small cooling power and easily evaporates. For this reason a pre-cooling with
liquid nitrogen and He gas is required, and careful shielding and protecting from heat leaks. How to do
that in practice? This is the whole art and knowledge of the cryogenics.
Vapor pressure – see 29 g
Clausius–Clapeyron equation:
Sgas -Sliq
 dp 
,
  =
 dt  vap Vm.gas -Vm.liq
where S is entropy, Vm is the molar volume. Noting that: S gas  Sliq  L T ,
Vm.liq  Vm. gas , and
Vm. gas  R T P , we have:
L T  p
 dp 
  
RT 2
 dt  vap
Assuming that:
He
L
T  =
const., see fig. 29f, we have that p vap
e
-
L
RT
Consequences of exponential decay of pvap with temperature:


cryogenic pumping
lowering the temperature of a bath by pumping on the vapor above the liquid.
Limits: solidification and superfluidity.
Cooling power of a pumped bath:
Q = n H lq -H vap  = nL ,
where n is the number of particles pumped out in a unit of time. We pump the “hottest” particles and
lower the mean energy of the liquid. Usually a pump with constant volume pumping speed is used,
therefore:
n  Pvap T  ,
and the cooling power:
Q  LPvap  e

1
T
Cooling power drops rapidly with temperature. Practical limits:
 He4  1.3K ,
 He3  0.3K
 N2 solidification comes first (see Fig 30a).
Specific Heat
Specific heat of 1g of He at 1.5 K1J/K, whereas for Copper at the same temperature Cv  10-5 J/K very important for cooling procedures! Thermal behavior of a cryogenic apparatus (e.g., response time)
is in most cases determined by the amount and thermal behavior of He it contains. In addition, latent
heat of He, even if small compared to other materials, is large compared to the specific heat of other
low temperature materials. Both properties mean that the temperature of an experiment will rapidly
follow the temperature of its refrigerating He bath.
Quantum nature of He manifests itself in a phase transition at 2.18 K - appearing as a pronounced
maximum in the specific heat
Lambda point, fig 30b.
Discovered experimentally by Dana and Kamerling Ones in 1920 - not published, and mistakenly
attributed it to an artifact!. “Rediscosvered” by Keesom and Clusius 1932 in the same laboratory. They
believed their data and published. Keesom introduced the names He-I and He-II.
At -point He undergoes Bose-Einstein condensation in momentum space. He3 is Fermion, and for a
long time it was believed that the superfluidity in He3 is prohibited by Pauli principle.
In 1972 – superfluid transition was found in He3 at 2.5 mK. Pairing of atoms with opposite spins occurs
end Bose condensation is possible. {D.D. Osheroff et al, Phys.Rev.Lett.29, 920 (1972)}
Transport properties, thermal conductivity and viscosity of He4
In the normal fluid state at temperatures above T =2.18K He4 behaves, due to its low density, almost
like a classical gas (the same applies to He3 at T > 0.1K).
Due to low thermal conductivity at T > 2.18K (about a factor of 104 lower than Cu and factor of 10
lower than stainless steel) liquid He-I always boils with strong bubbling. While pumping on the bath
there will be a strong temperature gradient between the surface and the volume, accompanied by liquid
agitation and strong boiling.
Below -point the thermal conductivity of He-II is infinite (for practical realistic situations it is finite
but very large ~10 W/cmK). As a consequence the liquid does not boil in any conditions (heating,
pumping, etc…)
For more on physics of Superfluidity see e.g.:
D. R Tilley, J. Tilley “Superfluidity and Superconductivity”, Hilger, Bristol 1990.
Because the thermal conductivity is very large, temperature, or entropy, waves propagate in this
quantum liquid. This wave propagation is called the “second sound” to distinguish from the “first
sound”, i.e., from density waves. The velocity of the second sound ~ 0.1 the velocity of the density
waves (“first sound”).
In the superfluid state He-II has zero viscosity, s~0. He-II in the superfluid state can flow through
capillaries and holes with zero viscosity if the flow velocity does not exceed a critical value (note the
analogy to the superconducting current flow). Consequences:
1) Apparatus that seems to be real tight at T > T  may show a leak when filled with He-II.
2) He- II “escapes” from the containers.
Figure
Due to s~0 and Van der Waals interaction with the container walls a thin liquid film (30 nm thickness)
moves along the container walls. This leads to enhanced evaporation of He 4 baths at T<T, as the film
flows to hotter places and evaporates.
Superfluid He film thickness
d[nm] ~ 30h-1/3[cm]
Obviously d strongly depends on the cleanliness and structure of the wall.
The creeping He-II film sets the limit for temperature lowering by pumping on the bath, e.g. for S104
(l/h) at 1 bar, Tmin =1K.
He3 at mK range
Pairing of He3 atoms occurs at 2.5(0.93mK) for p = p melting (pSVP). This was one of the major driving
forces for development of mili-Kelvin cooling techniques.
Properties of He3:
Lower mass and lower zero point energy than He4. Therefore, lower density, behaves like classical gas
at T > 0.1K. At lower temperatures behaves very close to electron Fermi liquid.
As a Fermi liquid it differs pronouncedly from He4. Specific heat is very large (c~T/TF) and down to
mK range decreases linearly with temperature.
Before the transition into superfluid it has very high viscosity but also good conductivity. The viscosity
just before the transition is comparable to the viscosity of light machine oil.
Materials at low temperatures
Specific heat:

Isolators - mostly phonons (Debay model):
3
T 
  for T<0.1 ΘD
 θD 
c ph

Metals:
c = γT+βT 3 (linear term due to electronic specific heat)

Superconductors e.g. Aluminum
Figure
II-order phase transition with no latent heat.
cph is not influenced by the transition and follows the Debay’s law. All change is due to electronic
specific heat. At Tc the jump reflects a new “degree of freedom” of entering into the state of
superconductivity. The jump value is (from the BCS theory) c=1.43Tc, where Tc is cv. Of normal
electrons at T=Tc .Below Tc:
ce. s  e
b
Tc
T
Such exponential temperature dependence of c is a signature of an energy gap E in the density of
states, i.e. this reflects the number of electrons thermally excited across the energy gap.
Non-crystalline solids
Possibility of re-arranging structure through atom tunneling between two configurations. Existence of
Two-Level-Fluctuator (TLF) structures. This new degree of freedom allows for performing structural
re-arrangements. An example: SrO2 in vitreous state At low
temperatures the associated contribution:
ca= aTn , n1
Because the specific heat contributed by disorder decreases
linearly with decreasing temperature instead of T 3, as for
lattice vibrations, the specific heat of a dielectric in a glassy
state is much higher than in its crystalline state at low
temperatures. Often even lager than a metal.
Magnetic materials
For materials possessing magnetic moment at low temperatures there are (2I+1) ways in which
magnetic moments can orient themselves with respect to the external magnetic field. These form
“additional degrees” of freedom.
For example: a simple ½ spin system (two possible spin orientations).
Transitions from the lower to the upper level can be thermally activated and will contribute
2
E
k BT
 E 
e
cm  

E
 k BT  
k BT
 1  e

This contribution is called “Schottky anomaly”. If the splitting



2
E  kBT , then
 E 
cm  

 2 k BT 
2
For a metal with such a T-2 contribution we have, at T<1K, where the lattice cv is negligible,
c   T   T 2
N.B: At T<1K the specific heat of some commercial alloys containing paramagnetic atoms (often used
for wiring the low temperature apparatus) can be strongly enhanced. e.g:
 Manganin: 87% Cu, 13% Mn, or
 Constantan: 57% Cu, 43%Ni.
Recommended wires for low temperature systems are those with no magnetic impurities.
Figure
Thermal Expansion
Atoms are displaced from their equilibrium positions by thermal excitations. In the harmonic
approximation the potential:
V  r-r0    r-r0 
2
In this approximation there is no thermal expansion. However, in reality the potential that atom
experiences due to interaction with neighbors is asymmetric.
V
r/a
The non-harmonic components give rise to thermal expansion. Thermal expansion coefficient 
 
1  


 T  p
that does vanish for T0, as with decreasing amplitudes of oscillations one gets closer to the harmonic
approximation.
Thermal expansion of various materials, see Fig: 30d. There are 3 groups of materials:
1. Special alloys manufactured to have extremely low .
2. Metals - contract by about 0.2%-0.4% when cooled from 300K to low temperatures.
3.
Organic materials 1-2 % contraction.
Joining different materials in a cryogenic apparatus one has to take into account the difference in their
thermal expansion coefficients. Examples:
Joining tubes:
A tube with the largest expansion
should be on the outside so that the
solder point is not pulled open during
the cooldown
O’ring seal:
The screw should have a larger expansion coefficient than the flange. Best practice to use washers with
a very small expansion coefficient to provide further tightening of a seal. Brass base, S.S. flange, brass
screw, W washer
Epoxy feed through for leads: Epoxy with large  should contract on a thin wall metal tube. Use epoxy
with fillers to match for .
Thermal conductivity
Rate of heat flow per unit area (A = area):
q
Q
 T
A
Heat can be carried out by electrons and/or by phonons. Heat carriers are scattered — to calculate κ
apply transport theory, which in its simplest form is a kinetic gas theory.
Treating heat carriers as gas diffusing through the material we obtain in the simplified model:
κ =
1 c
3 Vm
vλ
Vm is molar volume, v is velocity of particles,  is mean free path, c is specific heat. The factor 1/3
comes from considering 1-D heat flow in 3-D space.
Lattice thermal conductivity: Phonons
κ ph =
1 cph
v sλ ph  T 3λ ph
3 Vm
for T below 0.1 D.
Here v s is the velocity of phonons (close to the velocity of sound). In this temperature range the
photon-photon scattering is dominant and (T) increases with decreasing temperature.
At low temperatures, T<<QD, the number of phonons is small. The dominant scattering mechanism is
phonon-crystal defects or crystal boundaries.  is temperature independent and κ(T) follows the c(T)
dependence and decreases with decreasing T as:
 ph  c ph  T 3 ,
resulting in a maximum in thermal conductivity, indeed see fig: 30 d
Electronic Thermal Conductivity
e 
1 ce
vF e  T e (T )
3 Vm
Dominates in metals where:
At high temperatures – phonon scattering dominates.
At low temperatures – defect and impurity scattering dominates (see Fig: 31b).
 e  ce  T
Combination of two mechanisms also results in a maximum (fig: 30e, 31a).
Position of the maximum depends on the perfection of the metal.
Superconductors:
κ e,s  Te
-
ΔE
k bT
E - energy gap, E=1.76 kBTC (in BCS)
κs decreases rapidly with temperature as Cooper pairs cannot leave the ground state. At T < 0.1T C κs
approaches that of an insulator κ~T3.
Figure
These features are used for wiring ( and κ) and for thermal switches.
Figure
Thermal contact and isolation
Good materials:
Cu (soft!) problems with nuclear specific heat at T<0.1 K
Ag (expensive! soft)
Al (soft, good down to 1.2K, soldering problems)
Pure metals: as above, one can reach up to κ ~10T [W/Kcm]. Typical κ ~T.
Isolation




Graphite
Plastics (Teflon, Nylon, PMMA, etc)
Ceramics
Thin walled tubing from SS or Monel (Cu0.7 Ni0.3).
Lowest κ found for the nuclear grade graphite.
Thermal boundary resistance
Results from two basic reasons:
1. Kapitza resistance (P. Kapitza 1941).
There is a temperature step between two different cooled materials (see fig 31d)
T  Rk Q
The problem is still not fully understood, in particular for very low temperatures.
2. Generally the actual contact area between two surfaces occupies only 10-6 area of the joint. To
avoid that:

apply pressure (non-practical).

diffusion welding.

grease.

silver paint or powder.

polished surfaces.

soldering. (N.B. regular solders are superconducting!).
The nature of Kapitza resistance between liquid Helium and solids
Acoustic mismatch
Heat transfer occurs via phonons. T at the interface arises from acoustic impedance mismatch. For
solid: sVs~106[g/cm2s]. For helium: hVh~103[g/cm2s]. At the boundary we have different velocity of
phonons. From Snell’s low:
sin( h )
v
 h
sin( s )
vs
Since vh~200m/s, whereas for metals vs~5000m/s, the critical angle of incidence at which phonons from
helium may enter the solid is very small:
c=arcsin(vh/vs)~3
The fraction of phonons hitting the interface within the critical cone is:
1v 
f   h
2  vs 
2
103
Moreover, because of the impedance mismatch not all of these phonons will enter. The transmission
coefficient:
t
4Zh Z s
 Zh  Zs 
2
 2  103
So only a fraction smaller then 10-5 of the phonons will enter the solid. As a consequence:
dT
15 3  s vs

dQ
2 2 k B4T 3 A h vh
3
Rk 
The essential result is that Rk 
1
.
AT 3
The acoustic mismatch model is in a reasonable agreement with most experimental data at
0.01K<T<0.2K, with typical values (for metal-helium interfaces) of:
m2 K 4
Rk AT  10 [
]
W
3
2
However, there are considerable deviations both in the Kelvin range and at T<10mK. Moreover, R k for
He3 and He4 is the same at 1K (no reason why it should be like that) In the K range the coupling is very
good (why?).
Coupling between liquid Helium and metal sinters.
Heat transfer only through phonons – it is even less understood than in the case of metals.
Sintered powders have increased surfaces up to several 100`s of square meters. Vibration modes of
such small particles of the powder are different from that of the bulk material. The lowest allowed
frequency is of the order of 0.1-1 vs/d, where vs is velocity of sound, and d is the particle diameter.
For metal sinters composed of grains of the order of 1 m the fmin~1GHz. So how do phonons pass at
all?!
Particles in the sinter are interconnected by thin bridges. All structure works like a sponge and has a
wide spectrum of soft, low frequency modes – low Rk, large A.
Low temperature heat exchangers (see dilution refrigerators) are build from sintered metal powders. For
He3 – we use dirty sinters containing magnetic moments to facilitate the heat transfer.