The quest for ultralow temperatures: What are the limitations?

Physica 109 & IlOB (1982) 1485-1498 North-Holland Publishing Company

THE QUEST FOR ULTRALOW Frank

TEMPERATURES:

WHAT ARE THE LIMITATIONS?

POBELL

Inst. f. Fesrkiirpetforschung,

Kernforschungsanlage

Jiilich, D-5170 Jiilich, Fed. Rep. Germany

The recent progress of nuclear refrigeration, and in particular the performance of the Jiilich double-stage nuclear refrigerator which has refrigerated samples to 38pK, is discussed. Currently, nuclear refrigeration is limited by the electronic thermal resistance of pure metals, by heat leaks, and by the long time scale typical for physics at ultralow temperatures. Especially annoying is the heat released from the cold parts of refrigerators which decays on the time scale of many days and whose origin is unknown. Eventually cosmic ray heating may limit the performance, and 10 p K seems to be a brick wall for refrigeration. Last, but not least, it is argued that for presently known problems in condensed matter physics at ultralow temperatures, of which helium, superconductivity and nuclear magnetism are discussed, it does not seem meaningful to try to refrigerate to temperatures below 10 PK.

1. Development

and status of nuclear refrigera-

tion The only known way to enter the submillikelvin regime is refrigeration by adiabatic nuclear

demagnetization.

1956 by Kurti nuclear

It was first applied

et al. [l] who demagnetized

spins to a few microkelvin.

electrons

and the lattice

12 mK, and the nuclei ing temperature

within

However,

of the sample

warmed

stayed

in

about long times, we mean many hours or perhaps days because characteristic times become very long at very low temperatures. The experimental conditions for nuclear

Cu

filamentary

the

produce

at

back to this start-

wire the

3He-4He

necessary

low starting for

Already this pioneering experiment made it obvious that the problem in nuclear refrigeration

measurements.

is not

interest

the

achievement

of a very

low

nuclear

superconducting

necessary

continuous SQUID

a few minutes.

refri-

geration became more easily accessible in the sixties or early seventies by the development of

superfluidity

high

dilution

solenoids

starting refrigeration

temperatures,

high

sensitivity,

In addition, of 3He in

fields,

for the

and of the low-power

the detection

1972 greatly

to of

of the

enhanced

in very low temperatures.

temperature. In refrigeration it is the temperature of the sample to be refrigerated rather

Goodkind and his coworkers were the first to refrigerate 3He by nuclear demagnetization of Cu

than the temperature

[2]. The technique has steadily been improved since then, particularly by the impressive advances achieved by Lounasmaa’s group who

of the refrigerant

which

is

of importance. The problems are therefore to achieve a sufficiently large cooling capacity at very low temperatures, a very low heat leak in a rather complicated adequate thermal

cryogenic apparatus, and contact between refrigerant

and the sample to be refrigerated. Only then can the low nuclear temperature be maintained for a long enough time and be transferred by the hyperfine interactions to electrons and lattice of the refrigerant, and eventually to samples and thermometers attached to it. And if we talk 0378-4363/82/0000-0000/$02.75

@ 1982 North-Holland

reached When

0.38mK in a “He sample in 1978 [3]. the situation was summarized by

Varoquaux in 1978 at LT 1.5 [4], the lowest temperature to which 3He was refrigerated by nuclear demagnetization of Cu was 0.3 mK, obtained by Osheroff, who eventually lowered his minimum temperature to 0.21 mK in 1980 [5]. Another milestone was the Cu cascade nuclear refrigerator at Helsinki which reached in 1979 a

F. Pobell / The quest for ultralow temperatures

1486

nuclear

spin

polarization

nuclear

temperature

tronic

temperature

development netization review

corresponding

of about of

0.1 PK

0.25mK

of refrigeration by Andres

All present

a

This demag-

in a forthcoming

and Lounasmaa

cryostats

(see fig. l(a)) use a 3He-4He dilution for precooling the nuclear stage.

refrigerator The latter

consists

magnetized viding

of lo-30

moles

of thin Cu wire

by a superconducting

maximum

solenoid

fields of around

increased

the cooling

gerators; important

and as it turned out, it was even more to decrease the heat leaks. These

requirements

pro-

8 T. Connected

l(b)),

troduced gerator

and

conducting

example,

results

using

precooling

copper to

in a reduction

in a field

r = 15 mK,

of for

of only 4% of the

nuclear

entropy

(see fig. 2). For further

it was ditions,

necessary to improve the starting conBi and r, for the demagnetization which

]

progress

refrigeration

the

the final

“He-4He nuclear

heat switches.

to remove magnetized

because

by recent

stage dilution

stage.

refri-

design (fig. is inrefri-

Of course,

one then needs two superconducting magnets for magnetizing the nuclear stages and two super-

refrigerators decreases as T* and usually drops to values below 1 PW at around 10 mK. This is and

nuclear

between

must have temperature

unfortunate

of the nuclear

have been fulfilled

another

to the nuclear stage is the sample to be refrigerated. But the refrigeration power of dilution

Bi = ST

power

of double-stage nuclear refrigerators. In a double-stage nuclear refrigerator

[7].

demagnetization

typically

nuclear

[6].

by nuclear

of Cu is summarized

article

to

at an elec-

materials Vleck

stage

the large heat of magnetization of the second nuclear stage. The only suited

for this task

paramagnets

with

nuclear moments. In these materials, nucleus applied

The first nuclear

a very large cooling capacity at a of only a few millikelvin to be able

the

are metallic

hyperfine effective

can be increased considerably external field by polarization

Van

enhanced field

at the

over the of the 4f

Dilution

3mm@Cu Rods (10 moles1 \

/

8T Solenoid

05mmPCu Wires (IO moles)

Single-Stage Demagnetization

Nuclear Refrigerator

Double - Stage Demagnetization

Nuclear Refrigerator

Fig. 1. Schematic of a typical single-stage and of the Jiilich double-stage nuclear demagmetization stage decreases the starting temperature for demagnetizing the Cu in the double-stage design. transmitted by thermal conductance through the superconducting heat switch and the mechanical

refrigerator. The extra PrNis It also decreases the heat leak supports.

F. Pobell / The quest for ultralow temperatures

l-w

10 9 a

1

0.1

10

100

T [mKl Fig. 2. Nuclear entropy of Cu (solid lines), and of PrNiS (dashed lines) in various fields, compared to the rate of entropy reduction, Soa, of a typical dilution refrigerator (dotted line). For Cu, S,,, = 11.5 J/K mole and for PrNiS, S,,, = 14.9 J/K mole. Owing to the hype&e enhancement of the external field, the entropy of PrNiS can be reduced at rather high temperatures at which &R is still large.

shell.

Their

usefulness

was first pointed have

been

since

investigated

1968

hyperfine gerant

for nuclear

out by Altshuler

[9].

The

enhanced

consisting

of

moments,

like

by Andres

and

advantage

of

material

[lo] as compared

refrigeration [8], and they

demagnetized

Bucher using

as a nuclear

a

refri-

to Cu is shown

entropy

perature refrigerators substantially

reduction

where

the

occurs cooling

is still large. larger

at a higher power of

the

The

bare is

nuclear eventually

to reach the low final temperature.

now this double-stage

plied in Leiden

with latter

[12], Jiilich

design

has been

[13]. and Tokyo

ap[14].

2. The Jiilich double-stage nuclear refrigerator

tem-

of dilution

The achievement

reduction

material Cu.

for the

Van Vleck paramagnet PrNi5 in fig. 2. Owing to the large hyperfine field seen by the ‘41Pr nuclei, the

Until

a

of a nuclear

entropy results in a much larger cooling Unfortunately, the hyperfine enhanced moments in most of these compounds

capacity. nuclear are cou-

pled by indirect exchange interaction conduction electrons. This interaction spontaneous nuclear ordering which

via the leads to we have

observed to occur at 0.40 mK in PrNi5 [ 111. This limits the minimum accessible temperature. To take full advantage of the possibilities of a double-stage design one should use a hyperfine enhanced material with large cooling power in the first nuclear stage to precool a second stage

The Jiilich double-stage refrigerator (fig. 3) is at present the most powerful operating nuclear refrigerator, both with respect to minimum electronic temperature and with respect to cooling capacity in the milli- and microkelvin range; details about its design and performance have been published [13]. Sixty rods of PrNis (1.86 kg or 4.29moles) are used in the first nuclear stage [15]. Demagnetizing

this stage alone

from a field

of 6T and a temperature of 10 mk (23 mK) results in a temperature 0.19 mK (0.31 mK) [ll, 131. This minimum temperature makes PrNiS a very effective refrigerant, and indicates that PrNi5 can produce temperatures comparable to those available in presently operating single-

1488

F. Pobell / 7’he quest for ultralow temperatures

Field profile of magnets

5mT space

-Vacuum

space

Liquid helium space Mixing chamber of dilution refrigerator Al

heat switch

I

MC heatshleld ~IK hmtshield Vacuum jocket 6 Teslo i m-1

PTNb demag stag (only 3 of 60 rods drown) Central fherrnal link Thermol path to PrNis

Al heat switch 2 5mT space Experimental space -Them701 path to Cu space (only I of 3 legs drawn)

i

I3 Tes!u magnet

Cu demog stoge

IOcm

E 5

6 6,?,?.

_ getii

fiekl, TL

0

Fig. 3. Schematic of the low-temperature part of the Jiilich double-stage nuclear refrigerator and field profile of its magnets.

stage Cu nuclear refrigerators. The heat which this stage can absorb as it warms in zero field from 0.25 to 1 mK is equivalent to a month of operation in the presence of a heat leak of 5 nW. The second nuclear stage is constructed from 96 rectangular copper rods of 2 x 3 mm2 cross section. Taking into account the variation of the magnetic induction over the length of the rods, the quantity of copper in the field is equivalent to 10 moles in 8 T. The rods form a regular, rigid array with 0.5 mm spacing between them. The

low-field region (55 mT) for experiments and thermometers begins at the copper plate above the Cu stage. The plate has three tapered holes into which coldfingers, holding experiments and thermometers, can be mounted by pressing matching cones into them. The total amount of Cu in the nuclear stage and in the experimental region is about 2 kg. After precooling the magnetized nuclear stages to about 25 mK, the PrNiS is demagnetized from 6 to 0.2 T. It precools the 10 moles of Cu in a field of 8 T to about 5 mK. At 5 mK and in 8 T, 27% of the nuclear entropy of the Cu has been removed, giving it a very large cooling power. After operating the lower heat switch but before demagnetizing the Cu stage, the PrNiS is further demagnetized to below 2 mK to serve as a thermal guard. For demagnetization of the copper stage, the field is decreased from 8 to 0.01 T exponentially in time with a time constant between 40 min and 2 h. In the design of our refrigerator we have tolerated only very small amounts of insulating materials (~0.1 g), and special care has been taken to obtain a very rigid construction to reduce vibrational heating. Our main thermometer is a platinum wire NMR thermometer [16] mounted in the same was as the samples to be investigated in the experimental region about 200 mm above the center of the Cu bundle. As shown in fig. 4, the Pt thermometer indicates that the refrigeration proceeds without noticeable losses or deviations from the ideal B/T, = constant line to about 0.4 mK; only as the field on the Cu refrigerant is further decreased do deviations become apparent. The temperatures down to 38 PK recorded in the experimental region are the lowest electronic temperatures ever measured. By remagnetizing we could demonstrate that the temperature T, of the Cu nuclei must have followed the full line in fig. 4 without noticeable deviation and must have reached a nuclear spin temperature of about 5 /_LK [13]. Consequently, the electronic temperature in the center of the Cu nuclear stage must have reached tem-

F. Pobell I The quest for ultralow temperatures

We

have

mance

l-w

recently

of this

investigated

refrigerator

in

performing repeated re- and over a period of two months. are shown

the more

perfordetail

by

demagnetizations The main results

in figs. 5 and 6. The heat leak is about

10nW during the first days after cooldown. and decreases as 0 = 38 nW e-“7idaysor as 0 = C&t- ‘; the data

are not accurate

between

these

and

0.2

0

0.1 0

0.6

difference nuclear

down

to about

between stage

0.8

temperature

caused by temperature gradients from the heat leak (see below).

heat

OLK) @K)

(nW)

?’

@K)

T,

OLK)

0.1 nW/(mole

refrigerant)

about

after

cooldown.

and 0.1 nw or about 10”


refrigerant)

three

weeks

is the most advantageous

calculated

from

thermal

the

measured

conductivity

(determined

from

of their

heat

the electrical

conductivity),

and the nuclear temperature of the bundle (see fig. 6). Also, the measured temperature does not be expected

I). The

We can exclude

of

the actual

which

because heat

leak.

heat leak is larger then

the apparatus

as long as it does.

result

28

10

4: 116 (246) 116 (249)

1;

1 5 6.4 (&

the used

of the heat

is

fea-

leak,

materials

as much as would

the

or W/g

ture of our apparatus. The temperature measured by the Pt thermometer is always higher than the temperature

decrease

region

(1::)

(see figs. 7 of I nW or

leak

leak

flows

Assuming from

the

0.5 5 5.7

0.2 5 5.3

(Z)

(f:,

(S)

(if)

0. 1 5 5. I (1:) 10 (39)

0.1 10 10 (ii)

from the that

than that measured would that

not stay cold the measured

experimental

Table I (T& at the Cu plate in the Calculated temperatures at the Pt NMR thermometer experimental region (Tp), and of the electrons in the center of the Cu nuclear bundle (T,) in the Jiilich double-stage nuclear refrigerator [13], as a function of the total heat T, of the Cu stage. T, has been calculated in leak 0 and of the nuclear temperature the approximation PCLB < kaT.. In the calculation, the parameters used were n = 10 moles, B, = 11.2 mT. ~c” = 1.1 K sec. (ycU= 10 W/K2 cm (determined from the residual resistivity ratio), a~ = 0.3 W/K2 cm, and 6, = 1O-‘3 W as the heat leak into the F’t thermometer from r.f. excitation used in the measurements. The values in parentheses are calculated under the assumption that the conductivity of the Cu stage is a factor of five smaller and & is a factor of 30 larger: these values agree well with the measured ones (see fig. 6) Q T” T,

to distinguish

decrease 6 /.LK (see table

the

small

after seven weeks,

[Tl

and of the experimental

rather

0.01 nW/(mole

Fig. -1. The measured temperature of a sample in the experimental region of the Jiilich double-stage nuclear refrigerator as a function of the magnetic field applied to the copper refrigerant. for two different temperature scales. The solid lines mark the temperature of the copper nuclei during the demagnetizations. In (a) the measured electronic temperature of the sample is indistinguishable from the nuclear temperature of the refrigerant. In (b), below 400p.K, deviations from the ideal B/T = constant line appear; they are caused by temperature gradients due to the heat leak.

peratures

8). The

enough

two time dependences

region

1490

F. Pobell / The quest for ultralow temperatures

8 O/. Ii,,,

0.01 Tesla

I

r

,

1

,

I

,

I

,

I

- 20 10

bW1

t_

;z 200 1 - 100

5

- 2 1

-

t-

5c

-

-o--____

2c

a- 0.2

\ 10I

0

along the Cu rods to the center of the nuclear bundle, we find the results given in table I and fig. 6. We should have observed a temperature below 12pK in the experimental region instead of the measured 38,uK. The measured values and behavior of the temperature at the Pt NMR thermometer in the experimental region could be explained (see fig. 6 and table I) if the thermal conductivity of the Cu (and of its connections) is smaller by a factor of five than determined from the residual electrical resistivity (700) of the Cu rods in the nuclear stage, and if the heat leak into the Pt thermometer is larger by a factor of 30 than the heat from the r.f. excitation. The assumption that the heat leak into the Pt thermometer is only coming from the r.f. excitation (lo-l3 pW), and is therefore constant, certainly is

1

I

10

1

1

,

20

,

30

. ,

,

LO

,

03 50

t [days1

7.l.O

Fig. 5. The left part shows the decrease of temperature, starting at 3.8mK, of a sample in the Jtilich double-stage nuclear refrigerator when the magnetic field on the Cu refrigerant is decreased exponentially in time for 10 h from 8 to 0.01 T. The right part shows, in an expanded temperature scale and a compressed time scale (one digit is one day), the development of the sample temperature after demagnetization. It takes about 4 days until the apparatus has relaxed to the minimum temperature. The sample was kept below 50hK for ten days. On 5 January an additional heatinput of 1 nW was supplied to the sample to accelerate the warm-up.

0.5

Fig. 6. Time dependence of the heat leak (0) and of the temperature in the experimental region (0) in the Jiilich double-stage nuclear refrigerator. For r 5 30 days, the heat leak behaves like Q = 28 nW e-‘/7dayS.The dash-dot line is the temperature expected from the heat leak and the properties of the materials used. The dashed line is the expected temperature if the conductivity of the Cu and of its connections is a factor of five worse than that calculated from the residual resistivity of the Cu rods in the nuclear stage, and if the heat input into the NMR thermometer is a factor of thirty larger than just the r.f. excitation (see table I).

optimistic. In addition, preliminary measurements of the thermal conductivity of the Cu stage at T < 1 mK gave values which are indeed roughly a factor of five smaller than that calculated from the electrical conductivity of the Cu rods alone at 4.2 K. The data in table I and fig. 6 demonstrate that the performance of the refrigerator is limited by thermal gradients rather than by the spin lattice relaxation, the nuclear temperature, the cooling capacity, or losses during demagnetization. Nevertheless, this refrigerator is a valuable tool for a variety of experiments which have not been possible until now. It allows one to perform experiments on large samples with a high specific heat down to 38 /*K. And above all, electronic and lattice temperatures of macroscopic samples can be maintained below 50 PK for 10 days (see fig. 5). In addition to its operation as a contoo

F. Pobell / The quest for ultralow temperatures

ventional

double-stage

nuclear

refrigerator,

the

required

mechanical

design of the two stages allows repeated doublestage operation or entropy pumping between the

stretch

two nuclear

their mechanical

stages.

In the following

two sections

presently known technology achievement of substantially temperatures

than

whether

those

required

for

phenomena

obtained

lower

presently

may lower

allow the electronic

bundle. Another

now,

actually

reduce nuclear

expected

lematic.

till

are

known

of condensed

their residual current

up

matter

or

and

stability.

resistivity

properties

losses increase possible

tors

could

discussed large

of existing

be improved features,

Assuming be based

improvement

For example,

gradients,

that future on presently

eventually

discussed

of the abovethe

excessively

and the origin

and

nuclear known

refrigerators technology,

will and

the same design as shown in of the main obstacles which

limit

further

progress

will

be

in the following.

The results a severe tronic heat gerant)

shown

bottleneck

resistance leak

is the

I demonstrate finite

of the pure metals

of 1 nW (only

is enough

experimental calculation

in table

that

thermal

elec-

used. A total

0.1 nW/mole

to keep the temperature

region above 20pK and above 50 PK

pared

Cu refriin the

according according

to to

experimental results; even though the Cu nuclei sit at 5 p K and the electrons in the center of the bundle should be at about 7 ELK, the residual resistivity ratio of the Cu stage is about 700, and its ratio (area/length) is 0.5 cm. Of course, the use of purer material with higher thermal conductivity and better joints is possible. But purer material will be quite soft and the nuclear stage has to be designed differently to achieve the

to

of the

superconducting

magnet

steep when experiments

the

values

making

of

about

additional progress tion of heat leaks.

and region,

But both the conductivity of the bundle may be

but not substantially,

to

possibly

improved

existing 10 PK

heat

com-

refrigerators. accessible.

will require

3.2. Time-dependent

a further

Any reduc-

leaks from

internal

sources As shown

in fig. 6, the main

part of the heat dependent.

leak

in our

apparatus

total

energy

corresponding

during

3.1. Properties of the nuclear stage

be

have to be in a “field-free”

somewhat,

refrigera-

might

the compensation

as is usually required. and the dimensions

of heat leaks were understood.

will have essentially figs. 1 and 3, some may

if some

particularly

temperature

time dependence

nuclear

eddy of the

the distance between experiments and stage; but this requirement, too, is prob-

thermometers

The performance

[13]. In addition,

with the conductivity

has to be extremely

3. Is 10 mK the brick wall for refrigeration?

had to

by about 20%) to improve

field of the magnetizing

physics.

We already

our Cu rods by about 3% (which increased

I discuss whether

temperatures

1491

a two

months

is time

to the heat running

time

The

released is about

20 mJ. This corresponds

to the relaxation

of lo-”

(10e4) mole

tunneling

if they

of two-level

are separated bute

states

by 3 K (30 K). If we were to attri-

the observation

to tunneling

transitions

of

molecules adsorbed on the cold surfaces of the refrigerator, we would need about 100 (10) adsorbed atomic layers if each of them makes a tunneling transition. This seems to be too much, and we have to search for other origins. Time-dependent heat leaks with comparable time constants

have also been

observed

in other

apparatuses [5, 17-221; some examples are shown in figs. 7 and 8. Fig. 7 demonstrates that one has to wait for about a month or two until this part of the heat leak has decayed so that it becomes comparable to the time-independent part of the heat leak in some nuclear refrigerators. But after

1492

F. Pobell i The quest for ultralow temperatures

t

[days1

-

0.1 0

10

20

30

LO

50

,II,l

60

Fig. 7. Time-dependent heat release in the nuclear refrigerators at Kyoto ([a]; 01 Cu stage with epoxy, 0 = 130 e-m.‘; 0: CU stage without epoxy, d- 65 e-“5.3); at Nagoya ([21]; d = 28 e-“7 for t 5 10 days), and at Julich ([13]; 0 = 28 emri7 for t ~30 days) as well as of 15g Plexiglas ([18]; d = 11 e-‘14), and of 2.6g amorphous Laa.raZnosr ([19]; d = 20 e-“’ 3), Time starts when the apparatus were first cooled to temperatures T c 4.2 K. Whereas the total heat release 16 dt is about 1 mJ/g for amorphous Lao.rsZna.r2 and Plexiglass, it is of order 20 pJ/g for the nuclear refrigerators which contain 1 to 2 kg Cu.

each warm-up to room temperature, the heat leak at low temperatures starts at the same high value, whereas warming to 50 K or 80 K does not seem to influence it very much [5, 19,20,22]. Possible mechanisms for this energy release with time constants of several days are tunneling transitions of protons in insulators, of structural rearrangements in noncrystalline substances, or the relaxation of mechanical strain [S, 17-221. The heat stored at high temperatures is released at low temperature with an extremely long relaxation time. For example, all connections between different materials require glueing, soldering, alloying, squeezing, or welding. They therefore almost unavoidably introduce one or more of the potential internal heat sources mentioned. We find our observation particularly remarkable because of the small amount of in-

1

Fig. tors a). and

10

I / l,,u

100

8. Time-dependent heat release in the nuclear refrigeraat Bell ([5]; A), in Jtihch ([13]; l), and in Kyoto ([20]; The data behave like 0 = C&Lx with x = 2 for the Bell Jiilich data.

sulating material (SO.1 g) in our apparatus [13]. But it contains strained metals to which Osheroff [5] could relate the heat leak of their refrigerator (see fig. 8); in particular, the mentioned stretching of the Cu rods of our nuclear stage by 3% may partly be the origin of the observed d(t). Neganov et al. [22] have recently observed that rolled Cu as well as rapidly crystallized Cu cooled to 1 K releases heat according to d = QOe- 1/*, with d,= 5OOpWlg and T 5 3 days. Annealing at 900°C for 5 h reduced d0 to 5 pW/g without influencing T (we observe d, = 15 pW/g). The process did not depend on temperature for T ~50 K. The authors attribute their observation to the relaxation of thermo-elastic tension at grain boundaries; the latter is supposed to result from varying expansion thermal coefficients in polycrystalline material. It is suggested that a quantum relaxation process occurs via the tunneling transition of two elastic dipoles formed by pairs of point defects between nonequivalent orientations in the thermo-elastic strain field [22]. But if the heat is indeed released

lW)3

F. Pobell 1 7’he quest for ultralow temperatures

by strain

relaxation,

an important Ravex et

dislocations

role. al. [ 191 have

may

also

also play

observed

an

athermal heat release of 0 = 20 nW er/2.3daypfrom 2.6g of amorphous L+.75Zr0.22 at 250 mK; the effect ple!

vanished

after

A detailed

these

investigation

mechanisms

pensable

crystallization of the

[23] seems

requirement

Besides these

this technological storage

progress aspect,

mechanisms

the correct

slow relaxation. possible mal)

relation

time

time There

slow

the physics

the

heat

of

inter-

we do not even

dependence

of the very

discussed

leak,

course,

and

of a (ather-

the

long-

the

transfer

rate

could

successful

operation.

A known to influence

is that caused

by cos-

experimental

through

region

by ionization

the nuclear

stage or the

[26]. It is responsible

fraction

of the minimum

for a

heat

leak in

our apparatus. on the ground

With a mean flux of cosmic rays of 2 cmm2 min-‘, and an average

energy

of

loss

each ionizing particle of find d = 0.07 MeV/sg = we

has been observed with characteristic times of 10ms s [24] to IO3 s [2.5]; the latter is likely not the

cles

upper

collision

rate could be reduced

bundle,

but

heat leaks from external

in

difficult

mic rays passing significant

that vibra-

contribution

heat leak which is extremely

2 MeVlgcm-‘,

3.3. Time-independent

somewhat

and we suspect

tional heating is a significant some other refrigerators.

time tail of the time dependent specific heat of noncrystalline materials. Up till now the latter

limit.

be

reduced, but not, for instance. by an order of magnitude. Nevertheless, the mechanical rigidity of our apparatus is one of the reasons for its

is the aim.

is also the question

between

dependent

indis-

is of great

est. As figs. 7 and 8 demonstrate, know

an

of

the observed

its annoyingly

with time if further

heat

physics

to be

for reducing

heat leaks and for eliminating decrease

of the sam-

result in a temperature increase of one to a few microkelvin corresponding to an energy of about 2pJ/transfer, or a heating of about 15 pW. Of

1.1 x 10ei4 W/g or about passing

through

experimental

apparatus

region. then

would

20 pW for cosmic our

nuclear

Of course, the

be

parti-

bundle

and

the cosmic

ray

by using a smaller

cooling

capacity

of the

reduced

correspondingly.

8L.6pK

86.0 I-’K

sources When the time-dependent leak has sufficiently decayed, time-independent

part of the heat we are left with a

contribution

which

is

only 3

about 0.1 nW in our apparatus (see figs. 6 and 7). Heat leaks from obvious external sources, such

5 k

as conduction

F

like supports,

through

mechanical

heat switches

connections

or electrical

leads;

by

residual gases; by radiation; or from r.f. sources, have, in well-designed apparatuses, been reduced to an insignificant level, which means to less than 0.1 nW. But an external source which

is

extremely

difficult

to

calculate

or

to handle experimentally at the required subnanowatt level is heat generated by vibration. Into this category also falls the heat generated by vibration when the helium dewar is refilled. For example, we have to refill our dewar about every 36 h. The associated mechanical disturbances

a E

- b.

c

time

81.6 pK

[h]

Fig. 9. Temperature measured with a pulsed Pt NMR thermometer in the experimental region of the Jiilich doublestage nuclear refrigerator as a function of time. In (a) a heat of about 3 FJ is going into the nuclear stage and experimental region to produce an irreversible temperature increase of 1.5pK. In (b) some substantially smaller heating of the thermometer occurs which then relaxes back to its original temperature due to the uninfluenced temperature of the nuclear stage. Such events are observed about once per day and may be attributed to showers of particles produced by cosmic rays of very high energy.

1494

E Pobell / The quest for ultralow temperatures

And we need a large cooling capacity to cope with the heat leaks for a long enough time, and time constants become extremely long at very low temperatures (see figs. 5-8, and below). In addition, we see about once per day an “instantaneous” temperature increase of order 1 PK with time constants of hours, the typical time constant of the apparatus at these very low temperatures, as shown in fig. 9. These events correspond to energies between 1 and 4 p J, giving an additional heating of order 20 pW. This observation might be attributed to showers of particles produced by very high energy cosmic particles. Therefore about half of the residual heat leak observed after two months’ running time of our apparatus may be attributed to cosmic ray heating and to vibrational heating due to refilling of the helium dewar. Nuclear specific heat and time constants

Yet another severe source of time-dependent heat leaks are nuclear heat capacities. Even though the heat leaking out from them may not be significant, the time to cool a sample can easily become ridiculously long if it is not a metallic element and if it cannot be made very small. Most materials have a large nuclear specific heat C and a large thermal resistivity R’ at very low temperatures, resulting in a long thermal time constant r = R’C, mostly increasing with T-“. For example, noncubic crystal symmetry or charge difference on different atoms even in cubic crystals give rise to a quadrupole nuclear Schottky specific heat at very low temperatures if the nuclear spin I > 4 (which is valid for most popular metals like Au, Cu, Al, etc.). Typical “high” temperature tails of these Schottky contributions have values of C- (1 to 10)/T’ @J/mole K) [27], and they reach maximum values of C,,,,,/nR = 0.44 for a two-level system. Let us assume the following values: CInR = 0.1, a = 5 x 1O-3T II = 0.01 mole (about 1 g),

(W/Kcm) (typical for alloys like brass), and 1 = 8 mm. We then find r = lo6 s 2 12 days at 10 pK! Without question, there will be experiments which can only be performed using materials where all isotopes have 111. In addition, traces of electronic magnetic impurities carrying a localized moment can give rise to a magnetic electronic Schottky specific heat at low temperatures, as observed for only 0.5 ppm Fe in [28]. Only construction and Pt at T ~0.4K measuring techniques which permit the use of very thin samples thermally well linked through a large contact area to the refrigerant can facilitate the reduction of the internal energy of samples and its rapid enough removal. But this method is restricted to only a few special experiments. In addition, time constants for most methods of thermometry will reach unpleasant dimensions in the low microkelvin range. Most widely applied is NMR thermometry using Pt. But T] is about 1 h at 10 PK for Pt, which is one of the elements with the fastest spin lattice relaxation. The use of compounds with shorter relaxation times, like Van Vleck paramagnets, may be impossible because of magnetic ordering and of their low thermal conductivity. Further problems arise from the specific heat and thermal conductivity of connections to thermometers (or of thermometers themselves) which in most cases have to be exposed to a magnetic field and may have a sizeable nuclear magnetic specific heat. Fig. 5 shows the development of temperature after a 10 h demagnetization from 8 to 0.011 T, and from 3.8 mK to 52 PK in our refrigerator. It took more than 4 days until the temperature in the experimental region had relaxed to its minimum value of 41 PK. The observed long time constant may partly result from the establishment of equilibrium between various parts of the Cu bundle exposed to different magnetic fields; the Korringa relaxation time r1 of Cu is 30 h at an electronic temperature of about 10pK. Such long time constants can only be coped with in an apparatus with a very large

F. Pobell / The quest for ultralow temperatures

ratio

of cooling

capacity/heat

leak;

would warm-up too quickly. In general, at 10-100pK many

phenomena

eventually

the

is getting

it may

be too

capacity of refrigerators We have reached

otherwise

time

scale

ridiculously

long

it

for the

for

with

some

helium nuclear

I405

reason.

As

such

liquids and solids, magnetism.

I will

discuss

superconductivity.

the and

long; cooling

and/or human patience. electronic temperatures

4.1. The helium liquids and solids For

superfluid

T = 10e2Tc is 10 /_JK and

“He,

below 10 p K at the nuclear bundle and 38 p K on refrigerated samples [13]. It seems to be possible

higher, so one may not expect much new physics below this temperatures. But a transition to the

to reduce the latter value to about 10 PK. To achieve this goal one has to improve the thermal

anticipated

conductivity

kelvin,

links), the

materials

used

(and

and get rid of the time-dependent

heat

pure

of the leak.

metals

For

should

this

purpose

their part of

well-annealed,

be used for the nuclear

and in the experimental

region.

stage

But on account

of

the problems outlined, it appears to be extremely difficult - if not impossible -for current technology to go beyond the sample

the 10 PK limit in cases where

must be cooled by means of an external

refrigerant.

Temperatures

below

restricted

to a few examples

be cooled

by self-refrigeration,

new cryogenic

10 /LK might

where specimens

be can

or to the advent

of

mixtures

may

bilities

pairing

of varying

(depending

3He-4He

Now

that

I have

I will discuss

whether

in condensed

matter

summarized

the

progress there physics

for

problems need

temperatures

below 10 PK. Of course, there may be problems in other fields of physics for which it may be necessary to refrigerate to below 10 pK, and there

may be exotic

new phases

resulting

the

‘He

from

ultraweak interactions of which we have not yet thought. But I cannot talk about fields with which I am not familiar or about phenomena of which I am not aware. Therefore, I will restrict the discussion to condensed matter phenomena which are already of present interest at ultralow temperatures or yet unobserved phenomena in this field whose actual existence can be expected

microand there

concentration,

most exciting for ultralow geration

heat

would

the

pairing

theory;

quantum

system.

temperatures

of liquid

and

of

to refrigerate

0.18mK mixtures boundary

be the

So, here the need helium

resistance

and the helium

interest

it might

is obvious. solid

by the boundary exchange

superfluid

be of extreme

Hut refriis severely between

sample. lo-100

the

Surface

rn’

are

areas usually

3He into the submillikel-

vin range [2-5,7]. The minimum which liquid helium has been

of refrigeration,

is any current

hundred

on concentration),

mixtures

for the genera1

required

which may limit further

at a few

magnetic field, pressure, and temperature, and with the expectation of singlet as well as triplet

for below 10 pK?

occur

3He-4He

may be a manifold of different superfluid phases in this quantum system [29]. With the possi-

refrigerant

4. Do we need temperatures

of “He in dilute

or at even lower temperatures,

limited

methods.

BCS-phase

temperatures refrigerated

to are

for 3He [30] and 0.58 mK for “He-4He [S, 301. resistance

Osheroff

has

RK between

measured

the

helium

and

RKAT = 750 (m2 K*/W) for 3He and RKAT2 = 16.2 (m’ K3/W) for an 8% silver powder, mixture

and found

at 0.8 mK 5 T I 4 mK;

a similar

value,

24.8 (m’ K3/W), has been observed by Frossati for a 6.4% mixture at 1.5 mK 5 T 5 7 rnK [31]; A is the surface area. If we assume RKAT” = (Tb:’ - TEt;‘)Al(n + l)d, with n = I for “He and n = 2 for 3He-4He mixtures, and TM the temperature of the refrigerating metal, we have THe_3= (T& + 1.5 x 103@A)“* and TEI-314 = (TL + 60d/A)1’3. The smallest values achieved the ratio d/A are between

up till now for 3 x lo-l2 and

F. Pobell I The quest for ultralow temperatures

1496

lo-”

(W/m”)

can

achieve

[4,5,7,30]. a

example,

1OpW

reduced

to values

and

parentheses

of

These d/A

100m2.

so that

comparable. assumed

Let us assume

the

TM has only

above

in the

equations correspond

scattering

AT:

reduction

become

triplet

superconductor

dual

resistivity

microkelvin

may

there

electron

pairing

mechanisms,

example via paramagnon exchange leading to triplet superconductivity. s-state

lo-‘T

of 8T

are nearby.

for

shielding

has

impurities

But

achieved

T, = 100 pK) be due to pair-breaking

pure

fields

[32], even

changing

the unsuccessful in

a

of order

of magnetic

refrigerator

superconductivity

expected

been

with

a critical field B, of value for the relative

of magnetic

in the nuclear

= of

in

a dream p. achieved

the with for

compounds

ZrZnz

magnitude

higher

Au

fields search

(with

an

down to 38 PK 1321 may by the still present traces

of magnetic impurities. The required level for the concentration of magnetic impurities of below lo-* might be the brick wall for this problem, already at a T, = 100 PK. The most promising candidates

for

or TiBe,

it is many

[34]. Obviously,

superconductivity

orders

of

detection

is not a cryogenic

of but a

problem.

4.3. Nuclear magnetism

for

possibly

superconductor

T, of 10 E.LK would have about 10-‘T and a critical

though

T4

a

lowest

metallurgical

phenomena

at very low temperatures. And more exciting question, whether

to below

with

AT:(K) detection

remain

become

ordering

necessary

that

The

shown

superconducting then, the even

The

shows

will

triplet

lo-‘.

to the resi-

result

range

interest not yet

concentration

The

of a

metallurgy.

In superconductivity, it is of current whether metallic elements which have

A BCS singlet,

temperature

Pd is 4 x 10-lo (0 cm) [33]; for the intermetallic

4.2. Superconductivity

are other

the required

by impurities po.

a transition).

its phy-

present

any

of the transition

for

the

[35]. Therefore,

levels of purification and perfection of the crystal are very severe in this case. One can relate the

for 3He-4He

180 PK

may be too high to observe

potential

2 x 108po (0cm) [34,35] triplet superconductivity

to 12 /1 K for 3He (where and

for to be

the two terms

terms

sics may end anyway) (which

that one

d/A = lo-l3 (W/m*),

ratio

triplet

superconductivity are metals which are so strongly paramagnetic that they are almost unstable against ferromagnetism. Among the elements, Pd seems to be the most suited candidate [29,33], and ZrZn2 and TiBez have been discussed among the compounds [29,34]. Unfortunately, triplet pairs are very easily destroyed, not only by magnetic scattering such as are singlet pairs, but more importantly by

The materials which are of interest for investigation of nuclear magnetism can be divided into two groups. In the first group we have the materials with the usual, weak hyperfine interactions, like the insulators CaF, or LiH investigated at Saclay [36] or simple metals like Cu [6]. If the lattice or electronic temperature in these materials is at lo-100 pK, the spin lattice relaxation

time

nuclear

for the duration applies

is long

temperature

of the experiment.

to insulators,

example,

enough

so that

T, will remain

a lower

uninfluenced This certainly

but also to most metals;

or = 30 h for Cu

at 10 PK.

Only

nuclear temperature has then to be reduced below the nuclear ordering transition.

for the to

In the second group the hyperfine enhanced systems, like PrNiS, ordering usually occurs between 0.1 and 10 mK [7,9-111, and refrigeration to below 10 /.LK should not give any new information.

5. Conclusion Low-temperature physics has entered a new and exciting part on the road to absolute zero by making the microkelvin range accessible, and

F. Pobell / 7’he quest for ultralow temperatures

there

are a great

number

portant problems to extension to electronic refrigerated

samples

a formidable dition,

known

in condensed

matter

surpassing

Something

goal.

physics

In ad-

phenomena

do

not

cryogenic

seem

the

outgrowth most

strong

discoveries

the

discoveries

‘He and of nuclear

solid ‘He in the millikelvin I would

be happy

of the refrigeration of course is possible

of

range

is

for such

have often been

of technological examples

such

expectation

justification

recent

are

the

to

barriers.

in a new temperature

Fundamental

The liquid

technology.

or expected

Perhaps

a sufficiently

physics

But on

10 PK seems to be

the present

new phenomena efforts.

and im-

new has to come along to justify

an ambitious itself

to below

task for current

presently

require

of interesting

solve on this part. or lattice temperatures

developments.

in low-temperature of superfluidity

of

antiferromagnetism

of

O.V. Lounasmaa and M.A. Paalanen. Cryogenics 16 (1976) 521: AI. Ahonen, W.J. Gully, O.V. Lounasmaa and M.C. Veuro, J. de Phys. 39 (1978) Ch-1153. J. de Phys. 39 (1978) Ch-I605 [41 E. Varoquaux, Osheroff and W.O. Sprenger. private com]51 D.D. munication (1980); D.D. Osheroff and L.R. Corrucini. Phyx. Lett. 82A (1981) 38. J.P. Ekstriim. J.F. Jacqurnot, M.T. 161 G.J. Ehnholm, Loponen, O.V. Lounasmaa and J.K. Soini. Phys. Rev. Lett. 42 (1979) 1702; J. Low Temp. Phys. 39 (1980) 417. in Progress in Low [71 K. Andres and O.V. Lounasmaa. Temperature Physics, vol. 9. D.W. Brewer. cd. (NorthHolland. 1981). Pl S.A. Al’tshuler. JETP Lett. 19 (1966) 432. [91 K. Andres and E. Bucher, Phys. Rev. Lett 21 (106X) 1221; J. Appl. Phys. 42 (1971) 1522; J. Low Temp. Phys. 9 (1972) 267. 18 (1978) 473. 1101 K. Andres, Cryogenics [Ill M. Kubota, H.R. Folle. C. Buchal, R.M. Mueller and F.

range.

to be wrong

in my forecast

limit at around 10 PK. This according to the maxim that

predictions are very difficult to make, cular if they refer to the future.

[121

in partiCl31

1141

Acknowledgement I am deeply Ch. Buchal, essential

M. Kubota,

R.M.

and H.R.

Folle

[17]

discussions.

they

I also gratefully assistance

by

W.

took

for the [16]

technical

role

[15]

Mueller, in the

the

enthusiastic

to Drs.

development of our refrigerator. Many of the ideas presented in this paper evolved from our daily

and

indebted

acknowledge Bergs

and

J.

Hanssen.

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F. Pobell I The quest for ultralow temperatures

1498

T s 10 mK with a time constant of several hours and a magnitude larger than the nuclear specific heat of the Cu bundle in a field of order 10 mT. Whereas in some cases such excess contributions have been attributed to epoxy (0. Avenel and E. Varoquaux, private communication, 1981), this is not possible for our apparatus. Our observation may be related to the excess specific heat observed for high-purity Cu at T B 30 mK by G. Sellers and A.C. Anderson, Rev. Sci. Instrum. 45 (1974) 1256; E.J. Cotts and A.C. Anderson, J. Low Temp. Phys. 43 (1981) 437; and at T 2 5G mK by D.S. Greywall, Phys. Rev. B 18-(1978) 2127. For highly deformed Cu, G.R. Pickett and coworkers, private communication (1981) observed at about 1 mK an excess contribution equivalent to an internal field of 0.3T! It remains to be shown whether there is a connection between the origin of these observations and the origin of the time-dependent heat release. [24] W.M. Goubau and R.A. Tait, Phys. Rev. Lett. 34 (1975) 1220; M.T. Loponen, R.C. Dynes, V. Narayanamurti and J.P. Garna, Phys. Rev. Lett. 45 (1981) 265. [25] J. Zimmermann and G. Weber, Phys. Rev. Lett. 46 (1981) 661. [26] T.O.

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[33]

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[36]

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