SOLID HELIUM THREE AT VERY LOW TEMPERATURES

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SOLID HELIUM THREE AT VERY LOW TEMPERATURES

R. Richardson

To cite this version:

R. Richardson. SOLID HELIUM THREE AT VERY LOW TEMPERATURES. Journal de Physique

Colloques, 1970, 31 (C3), pp.C3-79-C3-89. �10.1051/jphyscol:1970307�. �jpa-00213850�

JOURNAL DE PHYSIQUE

Colloque C 3, supplément au n° 10, Tome 31, Octobre 1970, page C 3 - 79

SOLID HELIUM THREE AT VERY LOW TEMPERATURE S(*)

R. C. RICHARDSON

Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York, 148 50

Résumé. — Cet article passe en revue les propriétés de PHe

solide aux très basses températures.

Au-dessous de 20 mK la contribution des spins domine complètement l'entropie ohr solide. A la pression de fusion et à une température de plusieurs millikelvins, le solide a une interaction d'échange J/k = — 0,7 mK ; cette interaction produira un alignement antiferromagnétique des spins à 2 mK. Des mesures du paramètre d'échange à l'aide des isochores d'expansion l-j^,) et de la susceptibilité magnétique nucléaire sont présentées. La courbe de fusion de l'He

aux basses températures est intéressante pour des raisons à la fois théoriques et techniques. La pente de la courbe de fusion donne des informations directes sur îa différence d'entropie entre les phases liquide et solide. Un mélange des deux phases liquide et solide peut être comprimé pour former un solide au voisinage de la température d'alignement des spins (effet Pomeranchuk).

Des mesures de la courbe de fusion et des expériences basées sur l'effet Pomeranchuk sont passées en revue.

Abstract. — The properties of solid

He at very low temperatures are reviewed. Below 20 m °K the spin contribution completely dominates the solid entropy. The solid at the melting pressure at several millikelvin has an exchange interaction Jjk = — 0.7 m °K which will cause antiferroma- gnetic ordering of the spins at 2 m °K. Measurements of the exchange parameter by expansion isochores (dP/d7% and nuclear susceptibility are reviewed. The melting curve of

He at low temperatures is interesting for both theoretical and technical reasons. The slope of the melting curve gives direct information about the difference in entropy between the liquid and solid. A two phase mixture of liquid and solid may be compressed to form a solid near the spin ordering tem- perature (the Pomeranchuk effect). Measurements of the melting curve and experiments using the Pomeranchuk effect are reviewed.

I. Introduction. — Solid helium is in many ways similar to the dielectric crystals formed by the other inert rare gases. Many-body calculations using the Lennard-Jones potential between the atoms have been successfully used to describe the gross thermal properties of the solid, the approximate magnitude of the binding energy of atoms in the solid, the Debye temperature, and the elastic constants. The remarkable difference that exists in solid helium is that the large amplitude zero point motion permits the atoms in adjacent lattice sites to undergo an exchange, or tunneling-like interchange, through the mutual elec- trostatic potential barrier. Even though the magnitude of the tunneling energy is quite small, of order 1 m °K, compared with the Debye temperature of the crystal, 20 °K, the existence of the tunneling motion in solid

3

He is easily verified in nuclear magnetic resonance experiments at relatively high temperatures near 1 °K.

At the lower temperatures, below 20 m°K, the exchange motion completely dominates the thermal and magnetic properties of the solid. There is strong (*) This work was supported by the Advanced Research Projects Agency through the Materials Science Center at Cornell University.

evidence in the least dense crystals of

He, which have the largest exchange energy, that the nuclear spins have an antiferromagnetic ordering transition near 2 m °K.

In the following article we shall be concerned primarily with the properties of

He related to the spin interaction of the solid. No attempt will be made to review the theory of quantum crystals or the experiments related to the harmonic crystalline effects in the solid, such as the heat capacity at high tempe- ratures, the thermal conductivity, the sound velocity, etc. Here, we shall summarize the reasons for the expected spin ordering and review the experimental progress to date.

II. The Exchange Interaction. — An early quali- tative discussion of solid

He was given by Pome- ranchuk [1]. He recognized that at very low tempera- tures only the nuclear spins would contribute to the disorder of the solid so that the solid entropy is Rln

per mole until temperatures low enough that the energy of thermal disorder, kT, is as small as the energy tending to align the spins. In his discussion he ignored the exchange interaction and considered only

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1970307

C 3 - 8 0

R.

the coupling of the nuclear magnetic dipoles, of order p/a3, where p is the magnetic moment, and a is the interatomic distance. Considering only the dipole- dipole interaction and in zero external magnetic field, the temperature predicted for spin alignment is lop7 OK. Subsequent work has led us t o understand that the characteristic spin interaction is much larger.

Nevertheless, Pomeranchuk predicted many of the basic thermodynamic features of the system and suggested the technique for cooling 3He by adiabatic compression which we will discuss later in more detail.

Bernardes and Primakoff [2] suggested that, because of the exchange interaction, the characteristic spin ordering temperature would be much larger than that predicted by Pomeranchuk. In fact, the dipolar contribution is essentially negligible and in zero external field the spins may be described by the Hamiltonian

where 1' means the sum is only over nearest neighbors and J is the exchange energy parameter between spins i and j. The magnitude of J is maximum near the melting pressure where 1 J/k 1 is of order I m OK.

The intrinsic sign of J determines the type of order- ing ; a negative sign permits a pair of atoms in the singlet state to have the lowest energy and thus produces antiferromagnetic ordering.

A great deal of theoretical attention has been given to calculations of the magnitude, sign and volume dependence of J. Bernardes and Primakoff [2]

made the first attempt t o calculate directly the exchange constant of pairs of 3He atoms interacting through the van der Waals potential.

V(.

4

[(p)

^(;)

where &/k

10.2 OK and o

2.56 A.

Their inability to treat the pair wave functions in the region of strong repulsion, due to the term ( ~ / r ) ~ ~ as r < a , called the hard core, led to an over- estimate of the magnitude of the exchange parameter.

Subsequent workers have : added a correlation function g(r) to exclude the atoms from the hard core region [3, 41, introduced a cluster expansion of the ground state energy to account explicitly for the various magnitudes of the two and three body interactions 15-71, and accounted for the lattice motional effects on J by using an Einstein-oscillator approximation to the phonon spectrum [8]. Thou- less [9] demonstrated that the sign of J must be inherently negative for two particle exchange regardless of the explicit details of the hard core interaction.

The present state of the theory has been recently reviewed by Guyer [lo]. The essential results predicted by the modern state of the theory are the following

(1) J is negative. This is because the Pauli principle allows a pair of atoms in the singlet state to have single particle wave functions which are broader, and hence have less kinetic energy, than those in the triplet state.

(2) The magnitude of J depends upon the amount of overlap between the neighboring atoms. J is proportional to the overlap integral, P, given by

where

is the two particle wave function in which pi(i) is the single particle wave function and g12(12) is the correlation function which cuts off pI2(12) as the distance between the atoms r I 2 < o. Using Gaussian single particle wave functions of the form

where R , is the position of the lattice site and cr2 is a measure of the zero point kinetic motion such that

< K. E. > A2 cr2/m kgD, Guyer and Zane [8]

have obtained values of J of order 1 m OK at the melting pressure which decrease rapidly with decreas- ing molar volume. This is essentially because

in the single particle wave function increases with the Debye temperature, OD, so that the amount of atomic overlap in the crucial region near r12

o, where the major contribution to the overlap, integral arises, decreases rapidly with decreasing volume in spite of the fact that the average distance between the atoms is also decreasing (see figure 1).

FIG.

The maximum overlap occurs when particle

1

+

and particle 2 is near 412 - 012. The particles must be a away from one another. The crucial region for determining the magni- tude of the exchange interaction is shaded (after Guyer [lo]).

SOLID HELIUM THREE AT VERY LOW TEMPERATURES

3 - 81 The magnitude of the exchange parameter in solid

3He was first determined [11, 121 through studies of nuclear magnetic resonance relaxation at relatively high temperatures between 1 OK and 0.3 OK. Although thermodynamic quantities such as the entropy and magnetic susceptibility are affected very little by such a small interaction at these temperatures, the nuclear resonance relaxation times which depend upon the efficiency of the magnetic dipole-dipole interaction are influenced very much by the exchange motion of the atoms. This may be seen directly by considering the transverse relaxation time T,. In a rigid lattice at T

0 OK so that there is no thermal motion in the spin system the typical dielectric solid may be expected to have a resonance line width of order Am, - ^1/T2 - ^,u2/fia3 - ^{5 x} lo4 rdn/s where ,u is the magnetic moment and a is the lattice constant.

The linewidth may be expected to vary as 1/V so that the typical change in the width in varying the volume from 24 cm3 to 16 cm3 would be about 40 %.

In dielectric systems in which there is motion in the position of the spins, such as liquids or solids with thermally activated spin diffusion, the line beco- mes narrowed because of the random fluctuation of the local field AwmOtion

T , ( A ~ , ) ~ where zc is the characteristic time of the motion. For ~ e at ~ ,

V

23 cm3 for instance, with z, -- fi/J ^--5 x lo-%, we have a linewidth which does not vary with tempe- rature below 1 OK and

AwmOtion - ^{5 x lo-' x (5 x lo4)' -} 10' rdn/s so that the line is much narrower than that expected for the rigid lattice and further, varies as (JV2)-I.

The magnitude of the linewidth increases by about lo3 in varying the volume from -- 10 cm3/mole to 24 cm3/mole. The magnitude of the exchange motion can also be seen in measurements of the spin diffu- sion coefficient [ l l , 131 which has a constant value below about 1 OK of order a2 (~/fi), and from studies of the spectral efficiency of the spin lattice relaxation rates in the appropriate temperature range, where, l/T1

K1/T2 ( J,(o,) + 2 J, (2 w,) ) and where J,(w,) is the spectral density function which has approximately the form

o, is the Larmor frequency and Kl and K, are cons- tants [12, 141.

The importance of the exchange interaction to nuclear magnetic relaxation was first shown by Reich [ l l ] and by Garwin and Landesman [12].

The explicit analysis of these experiments is rather complex and has been reviewed in detail by Meyer [15].

For our discussion here, we are primarily interested in the magnitude and variation of J with molar volume deduced from these experiments which are shown in figure (2). The lines labeled RHG and RHM respectively, are the values obtained by Richards, Hatton and Gifford [16] and by Richardson, Hunt and

0.04 ^{1 I I}

21 22 23 24

v

FIG. 2.

The Variation of the Exchange Integral with Volume in Solid JHe.

The open circles are the values of J determined from the (dP/dT)v measurements of Panczyk and Adams [25], the line labelled PSSA is from similar measurements by Panczyk, Scribner Straty and Adams [23], the lines RHM and RHG are determi- nations of (Jlk) from nuclear relaxation experiments by Richard- son, Hunt and Meyer [17] and Richards, Hatton and Giffard

[16] (after Panczyk and Adams [25]).

Meyer [17] from the relaxation studies. The discre- pancy between the two curves arises in part from a difference in the analysis to obtain J from TI data and in part from a difference in the measured values of Tl at the larger molar volumes. The calculations are, in fact, still undergoing refinement [18]. The spectral density function Jl(w) used to deduce the value of J in the relaxation measurements apparently depends upon the number of neighbors in the sur- rounding lattice and has not yet been determined theoretically. Moreover, the precise deriviation of this function is required in order to determine the coefficient relating T, to J and V2 and the o, depen- dence of TI upon J. The values of Jl(o) have been obtained experimentally [19] so that the volume variation of J can be specified from relaxation mea- surements and, further, the magnitude of J was fixed by the agreement with the value determined in Tl measurements at lower temperatures where

TI - ^{TEL (1 + (CJC,,))} where TEL is the charac- teristic time for the exchange systems to equilibrate with the lattice and C,/C,, is the ratio of the Zeeman heat capacity to the exchange heat capacity and varies as (wo/J/fi)2. The important information available in this work is the variation of the exchange energy with volume. It has been determined even for 16 cm3 < V < 20 cm3 where J < 0.1 mOK is so small that it is essentially impossible for it to be determined in thermostatic measurements at the

lowest experimental temperatures available.

C 3 - 8 2 R.

C.

111. Low Temperature Properties of Solid He3.

-

A. THE

He3.

At tempera- tures below about 0.2 OK most of the thermodyna- mic properties of solid He3 are dominated by the disorder of the nuclear spin system. The Heisenberg Hamiltonian for the spin system in an external field Ho is given by

where

is the gyromagnetic ratio. The partition func- tion for the Hamiltonian is given by,

Baker, Gilbert, Eve and Rushbrooke [20] have eva- luated the high temperature series expansion of the partition function (7) in the form

The functions Fo, and F, are given by

and

The coefficients en and a, have been evaluated through the first 10 terms for several cubic symmetries and the similar coefficients in F,, F3 and F., have been calculated for the first 8 terms in the series. We will primarily be interested in the thermodynamic quanti- ties which have been measured in the low field limit as p H / k T + 0. The entropy in zero field obtained from (8a) and (8b) is found in the usual way :

where the e, and el are In 2 and 0 respectively and e,, e,, and e, are 12, - 24, and 168 for the body centered cubic structure. The corresponding specific heat in zero magnetic field is

The NCel temperature for the spin 112 antiferro- magnet in a bcc lattice has been computed by Rush- brooke and Wood [21]. Using a two sublattice model with a Hamiltonian of the form

Je

- 2 Z' ~ 6 . f ~ ^- y f r [ T Ti.g

&.H (11)

i < J i

"1

with the i and j atoms belonging to the two sublattices, they estimate the Ntel point with a high temperature

series expansion of the magnetic susceptibility and obtain T,,

2.75 (JIkT) for the bcc lattice.

The entropy of solid 3He below the NCel temperature is expected to be dominated by spin wave excitation.

There is no experimental data at this time for any properties of 3He below the NCel point. Bernardes and Primakoff [2] have, however, applied the spin wave theory of Van Kranendonk and Van Vleck [22] to bcc 3He and obtain

for the leading term in the entropy as T

0.

B. THE

3He.

The usual direct determination of the entropy is by means of heat capacity measurements. In 3He such measurements at very low temperatures are techni- cally quite difficult because of the long time constant for thermal equilibrium due to the boundary resistance.

Panczyk, Scribner, Straty, and Adams [23] have obtai- ned essentially equivalent information from elegant measurements of the pressure changes with tempera- ture in the solid at constant volume using a very sensitive strain gauge [24]. Using the MaxwelI rela- tion :

we obtain

where once again we have ignored the lattice contri- bution to the entropy and consider the magnetic field to be zero. Eq. (13a) can be rewritten in the form

is a Gruneisen constant for the spin system. (A similar relation is well known for the lattice contribution to (BP/~T), where

is the usual Gruneisen constant and the specific heat is that of the phonons). We note that since y, is negative the pressure is expected to increase with decreasing temperature.

Through a set of measurements of the temperature

coefficient of the pressure at various molar volumes,

Panczyk, Scribner, Straty and Adams [23], and Panc-

zyk and Adams [25] were able to obtain both C and

yJ. Figure (3) shows the most recent of these measu-

rements. The linear increase in pressure as a function

of 1/T corresponds to the leading term in the specific

heat Equation (10) so that the slopes of the curves

equal 3 y, RJ2/k2 V. The sharp initial decrease in

A P at high temperatures, shown for V

24.02 cm3,

is due to the Iattice contribution to the heat capacity.

SOLID HELIUM THREE AT VERY LOW TEMPERATURES

the extremely slow spin-lattice relaxation rates in the more dense samples and by the isotopic phase separa-

6

tion of He4 impurities [27] which upsets the thermal

equilibrium at the lowest temperatures.

5

In order to compute the low field susceptibility in

the temperature regime defined by

z .-

^{pH/kT < J/kT < 1 ,}

m

lo -

we again use an expansion of the partition function calculated by Baker et al. [20] and obtain

2

_x m2

=

--F,

kT (;TI (1 5a)

I

for the zero field susceptibility, where F, is given in

(8c). The values of a, through a, for the bcc lattice

o are 1, 8, 96, 1 664, 36 800, and 1 008 768 and explicit

o

evaluation of F,(pH/kT) yields

l / T ( K - ' )

FIG.

-The Change in Pressure versus Molar Volumes of

xT

1 +

+ 12

+ 34.7 - +

Solid 3He. c [ 4 k i (I&)' ^(kJT13

The value of

is determined from this set of curves by iden- tifying the slopes of the lines with the coefficient

,

.

For V

cm3/mole the high temperature phonon contri- where C

Np2/k is the Curie constant. Equation (15) bution is shown (after Panczyk and Adams

can be approximated by a Curie-Weiss law in the form The value of

can be deduced from the set of curves

provided that one has the knowledge that J becomes quite small as the volume is decreased. The values of J obtained from this analysis are included in figure 2.

The agreement between the values of J found in the expansion isochore measurements and the nuclear magnetic relaxation measurements is excellent.

If the term of order (J/kT)3 in the specific heatexpan- sion (10) is included in the expression for the pressure coefficient so that

we see that the low temperature departures from the asymptotic l/TZ behavior would indicate the sign of J. In the solid with the largest volume, where

I J/k I -- 0.8 m°K, the pressure coefficient would be expected to show large departures below the asymp- totic 1/T2 value for a negative J if the measurements were extended below 10 m

C. MEASUREMENTS

- The earliest efforts 1261 to deter- mine the ordering temperature of solid 3He were through measurements of the nuclear magnetic sus- ceptibility. The experiments were quite difficult because of the low temperatures required in the simultaneous presence of a magnetic field and paramagnetic coo- ling salts. The systematic study of the susceptibility as a function of the molar volume was hindered by

where 8

- 4 J/k is the Weiss constant. Determina- tion of the 8 in susceptibility measurements can thus give both the magnitude and the sign of the exchange constant.

The have been recent reports of a number of suscep- tibility measurements of solid 'He which are accurate enough and extend to sufficiently low temperatures to determine the sign of J. Johnson, Rosenbaum, Symko and Wheatley [28] fitted the earlier suscep- tibility measurements of the solid down to 20 m OK by Anderson, Reese, and Wheatley 1291 to the Curie- Weiss equation and obtained 6

3.5

0.5 m OK at V

24.0 cm3. Pipes and Fairbank [30] measured the susceptibility between 23.3 and 24.2 cm3 down to 40 mOK. Kirk, Osgood and Garber [31] measured the Weiss constant over a wide range of volumes, 21 cm3 to 24 cm3, down to 5 m OK. Sites, Osheroff, Richardson and Lee [32] measured the susceptibility down to 6 m OK for the solid at melting (V

24.1 cm3) and obtained 0

3.0 + 0.3. The results of all these experiments are listed in Table. I. In every case the results are in good agreement with the values of J obtained in (dP/aT), measurements by Panczyk, Scribner, Straty and Adams [23] and by Panczyk and Adams [25], and with the average J from T, and Tz measurements in the review by Meyer [15].

It may be further concluded that J is unambiguously

negative and that the NBel temperature of the solid

at melting lies between 1.8 and 2.4 m OK.

R. C. RICHARDSON TABLE I

Summary of Weiss Constants Obtained in Nuclear Susceptibility Measurements of Solid 3He Molar Volume

Work cm3

- -

Anderson, Reese and Wheatley [29]. ... 24.0 Sites, Osheroff, Richardson and Lee [32]. .... 24.1 Pipes and Fairbank [30]. ... 24.2 23.6 23.3 Kirk, Osgood and Garber [31]. ... 24.0 23.1 22.0 21.0

8 Minimum Temperature m OK of Measurements

-

3.5 + ^{0.5 20 m OK}

3.0 + ^{0.3 7 m OK}

4.89 + ^{2 40 m OK}

3.98 + ²

1.5 -t 2

2.9 -1 0.7 5 m OK

1.3 + ^0.3

0.48 + ^0.16

0.44 + ^0.30

Figure 4 shows the measurements by Sites et a1 [32]

obtained in compression cooling experiments. The temperature was monitored by measuring the nuclear magnetic susceptibility of 6 3 C ~ in a bundle of fine copper wires imbedded in the solid. Similar thermo- metry was also used by Pipes and Fairbank [30] and by Kirk, Osgood and Garber [31]. The latter also

FIG. 4. - Susceptibility of Solid 3He.

(a) xTvsT-1 in units of unity Curie constant. Data from five runs are shown.

(b) T*

versus

showing upper bound on Nel temperature. Circles and triangles represent two different runs.

(Every other point is plotted.) Open points are nonequilibrium points, and their abscissas have no significance (after Sites,

Osheroff, Richardson and Lee

used a powder of platinum and found the susceptibi- lities of lg5Pt and 6 3 ~ u to agree within 5 % down to 5.3 m OK. The open points in the lower graph of figure 4 represent measurements of the 3He magnetiza- tion taken while the system was under compression, when the copper thermometer was not in equilibrium with the 3He. The abscissas in the graph represent the measurements of the copper susceptibility thermo- meter and the ordinates, labeled T*, are the reciprocal of the 3He susceptibility normalized with a unity Curie Constant. Apparently, during the mechanics of the compressional cooling the 3He surrounding the copper thermometer was warmer than the 3He loca- ted within the 3He susceptibility coil located about 1 cm above the copper thermometer. The abscissas for the open points have no real significance ; however, the magnitude of the 3He susceptibility measured can be used to place an upper bound on the NCel temperature. The straight line represents the asymp- totic susceptibility predicted by the Curie-Weiss law. The actual susceptibility of an antiferromagnet must always [33] fall on points above and to the left of the Curie-Weiss curve so that the minimum T*

for 3He corresponding to the Nee1 point, occurs at temperatures lower than that of the Curie-Weiss law.

The temperature, 2 m OK, obtained by projection of the largest measured 3Ae susceptibilities on the Curie- Weiss curve thus represents a directly measured upper bound for the Ntel point which is essentially indepen- dent of the form of the theory. It is, of course, possible that measurements at lower temperatures would produce larger vaIues of the 3He susceptibility, thus forcing the Ntel temperature to lower values. The experiment was performed in a magnetic field of 2 kilogauss so that the Ntel temperature was proba- bly depressed about 5 % below that in zero field if one assumes a parabolic dependence of Tn on the applied field in the form

Johnson, Symko, and Wheatley [34] made measure-

ments of the nuclear spin diffusion coefficient, D,

SOLID HELIUM THREE AT VERY LOW TEMPERATURES C 3 - 8 5 in preliminary work on the adiabatic compression

experiments to be discussed in the next section. They found that at the lowest temperatures obtained in the compression experiments, less than 3 m OK, the value of D decreased to about 213 of the essentially constant value it has between 10 m OK and 0.5 OK [15].

Such a decrease corresponds to an increase in the correlation time of the spin motion

-- a2/D) and is expected in the vicinity of the magnetic ordering transition. It will be most exciting to see both the measurements of the nuclear magnetic susceptibility and the spin diffusion coefficient extended through the ordering transition itself, as they probably will be in the near future.

D. THE

POMERANCHUK EFFECT.

The calculations by Pomeranchuk [1] in

1950 stimulated great interest in the melting proper- ties of 3He. Through comparison of the entropies of the Fermi liquid and the disordered spin system of the solid he predicted a minimum in the melting pressure which exists at 0.3 OK. He further suggested that adiabatic compression of the liquid at melting at temperatures lower than that of the minimum mel- ting pressure would produce cooling to the spin ordering temperature of the solid.

The thermodynamic quantities necessary to obtain the entropy of liquid 3He have been investigated extensively. The most comprehensive and systematic measurements are those by Wheatley and cowor- kers 1351. The liquid entropy at melting at low tempe- ratures may be calculated by extending the entropy obtained in the heat capacity data of Abel, Anderson, Black and Wheatley at 27.0 atm. 1361 to the melting pressure by using the expansion coefficient and compressibility of the liquid at higher pressures obtained by Anderson, Reese, and Wheatley [37].

At the lowest temperatures the liquid is highly ordered and behaves essentially as the Fermi liquid theory of Landau 1381. The entropy has only small departures from a linear increase with temperature.

The entropies of the liquid and solid are represen- ted in figure 5. The curve labelled Ss ,/R represents the solid entropy at melting in zero magnetic field and the one labelled S, ,/R is the entropy of the liquid. We may construct the melting curve by using the Clausius-Clapeyron equation

having a knowledge of the difference in entropy between the two phases as well as the difference in the volume. We see immediately that dP/dT

0 at the points of intersection of the two entropy curves which correspond to a minimum in the melting pres- sure at approximately 0.3 OK and a maximum at some very low temperature yet to be determined. Below 0.3 OK the entropy of the liquid is less than that of

I

I

0.0 0. I 0 . 2 0.3 0.4 0.5

Temperature ( K )

FIG.

- The Entropies of Liquid and Solid H3e and the Melting Curve.

(a) The entropies of the solid at melting, SS,M/R and liquid at melting SL,M/R. The dashed line represents the change in temperature that occurs in an adiabatic compression process.

An ideal compression begun with liquid at melting at

m

will completely solidify at -

^m

(b) The Melting Curve. The curve may be constructed with knowledge of the entropies in (a) and the difference in volume at melting. The curve has a negative slope below

because the liquid entropy is less than that of the solid. The arrows indicate the initial and final pressures associated with the corn-

pression described in (a).

the solid and the volume of the liquid is always about 1.25 cm3/mole greater than that of the solid [39], so that the slope of the melting curve is negative.

Calculations of the melting properties of 3He below 10 m OK have been made by Thompson and Meyer [40], Goldstein 141-431, Scribner, Panczyk, and Adams [39], and by Johnson, Symko and Wheatley [34].

Using the value of (Jlk),

0,72 mOK (from

NMR and dP/dT measurements), an inflection point

in the melting curve is expected near 7.5 m OK and

the maximum in the melting pressure should be

near 0.5 m OK. The inflection point corresponds to

the temperature at which the liquid and solid have

equal heat capacities. The temperature of the maxi-

mum pressure is located by using an expression such

as Equation (12) for the solid entropy below the NCel

point. The total increase in melting pressure below the

NCel point is expected to be only of order atmo-

spheres and the decrease between the maximum pres-

sure at 0.5 m OK and absolute zero to be of order

atmospheres. Goldstein [42] has also used the

series expansions of Baker et al. [20] to calculate the

melting properties of 3He in a magnetic field. In the

C 3 - 8 6 R. C. RICHARDSON

limit of very strong fields such that pH 9 J, the Zeeman interaction in the spins dominates the Hamil- tonian of the system and the resultant polarization of the spins permits the entropy of the solid to decrease below Rln 2 at higher temperatures.

The melting curve has been measured below 0.1 OK by Anderson, Reese and Wheatley [37], by Zeiss [44], by Scribner, Panczyk and Adams [39] and recently down to very low temperatures by Johnson, Symko and Wheatley [34]. The work by Scribner et al. [39]

extends down to 16 m OK and has the greatest preci- sion in the pressure measurements.

The inflection point in the melting curve has been measured by Johnson, Syrnko, and Wheatley [34].

The data of Johnson et al., shown in figure 6 were obtained in compressional cooling experiments which we will describe later. The solid lines are calculated derivatives of the melting pressure using the values of Jlk as labelled for calculating the solid entropy.

The authors use the Hamiltonian, X

J Ti

to define J/k so that the values of J / k should be divi- ded by 2 for comparison with those in figure 2. The temperature T* is that obtained in susceptibility measurements of powdered cerium magnesium nitrate (CMN). Equilibrium pressure measurements were

FIG. 6.

-

The circles are values

of

T*.

X

C x.6

in

made at even lower temperatures than those shown in the figure but the corresponding temperatures below 3 m OK were uncertain due to the magnetic ordering of the salt thermometer. Further progress in any of the measurements below 3 m OK must await the develop- ment of new thermometry techniques.

Compressional cooling of 3He, the Pomeranchuk effect, was first demonstrated by Anufriyev [45].

We can understand why a two phase mixture of liquid and solid 3He cools when compressed by examining the entropy diagrams in figure 5. Below 0.318 OK [39]

the liquid becomes more ordered than the solid as a

result of the Fermi degeneracy of the liquid. If the system, initially in the liquid phase, is compressed, solid begins to form ; and with further decrease in the volume the mixture cools following the melting curve until it is completely solidified. The dotted line in the figure represents a process in which there are no irre- versible increases in the entropy. At the end of the compression the solid must display the same amount of order as the initial liquid and hence the adiabatic process permits cooling to a temperature near that of the spin ordering in the solid. The amount of heat removed at constant temperature in the compression process, T AS, is large and is equivalent to that which may be extracted from paramagnetic cooling salts with similar volumes. The technique has the additional considerable advantage that it will still work even in quite large magnetic fields, although the limiting low temperature will be raised as the field increases [42].

Experimenters were discouraged from attempting the scheme by the observation that the mechanicalwork which must be done to compress the mixture, P, _AV, is much largcr than the heat which may be extracted at constant temperature - TAS. The system might not be cooled at all if there are serious irreversible heat losses due to friction in the compression process either from heating of the wall of the cell while it is being deformed to decrease the volume of the mixture or from friction in the solid 3He itself as it is being formed. Anufriyev [45] demonstrated that the frictional heat losses need not be too severe. He successfully cooled a sample of 3He from an initial temperature of 50 m OK, obtained by adiabatic demagnetization of a paramagnetic salt, to below 20 m OK by increa- sing the pressure of liquid 4He in a chamber surroun- ding a flexible metal cell that contained the liquid 3He. The final temperature in Anufriyev's experiment was probably much colder than 20 m OK but he was prevented from measuring lower temperatures by insufficient thermal contact between the 3He and his susceptibility thermometer (CMN).

Johnson, Rosenbaum, Symko and Wheatley [28]

showed that temperatures near the NCel point of solid 3He could indeed be achieved in mechanical compres- sion experiments. The cell used in the melting pressure experiment by Johnson, Symko and Wheatley [34]

is shown in figure 7. The 3He is contained in a chamber made in two sections. The upper section has flexible walls made from a flattened 70-30 cupronickel (CuNi) tube that is reinforced by two smaller CuNi tubes which act as springs to support the chamber so that it may withstand pressure differences as large as 30 atmospheres. The lower section is made of an epoxy resin and contains a susceptibility thermometer, powdered CMN, and a capacitance strain gauge to measure the pressure of the 3He. The strain gauge capacitor is similar to those used by Adams [24]

and consists of two beryllium copper plates separated

by a thin mylar ring. The capacitance between the

SOLID HELIUM THREE AT VERY LOW TEMPERATURES C 3 - 8 7

FIG.

7.

The Compression Cell of Johnson, Symko and Wheatley.

Cell is mounted in the mixing chamber at a continuous dilu- tion refrigerator. A, inlet to mixing chamber. B, outlet of mixing chamber. C, 4He inlet. D, 3He inlet. E, heatcuring epoxy resin.

F, nylon. G, flexible part of 3He cell consisting of a flattened 9.52 mm

d.

0.41

mm

wall 70

30 cupronickel (CuNi) tube with slit 4.76

o. d.

0.25 mm

wall CuNi tubes for springs. H, outer wall of 4He space consisting of a 15.9 mm

o. d.

0.41 mm

wall CuNi tube. CMN, powdered cerium magnesium nitrate salt thermometer. Capacitor, beryllium copper plates separated by 0.025 mm - thick Mylar ring.

Initially cell is cooled by mixing chamber to 25 m

with 3He liquid under pressure. Compressional cooling is achieved by forcing additional liquid 4He through inlet C to distort the walls,

of the 3He cell. The temperature is measuredwiththe CMN susceptibility and the 3He pressure is measured with the

capacitor (after Johnson, Symko and Wheatley

two plates changes with variations in the 3He pressure which deforms the beryllium copper wall on the bottom of the cell.

The 3He is compressed by pressurizing liquid 4He contained in a CuNi cylinder which surrounds the flexible upper portion of the 3He chamber. The heat content of the entering superfluid 4He is removed by heat exchange with portions of the apparatus, not shown, at higher temperatures. Initially there is no pressure on the liquid 4He and the liquid 3He is precooled under pressure by the mixing chamber of at continuousiy operating dilution refrigerator [46]

contained witbin the outer epoxy cylinder of the cell.

At temperatutes below that of the minimum melting pressure a plug of solid 3He forms in the capillary leading to the 3He cell if the pressure of the 3He liquid is greater than 29 atmospheres, thus sealing a constant volume of 3He within the chamber. The thermal boundary resistance [47] between the 3He and the mixing chamber is large so that the initial

cooling to 25 mOK takes on the order of a day;

however, this same resistance assures good thermal insulation for the 3He once the compression has begun.

After the 3He has been cooled to the desired initial temperature, additional 4He is forced into the inter- mediate cylinder to produce the desired decrease of the volume available for the 3He. The pressure of the liquid 4He may be raised to 25 atmospheres before the 4He is solidified and the maximum required change in the 3He pressure is only 5 atmo- pheres. The fractional change in volume required to completely crystallize the 3He, AV,/V,,, -- 1.25124, is only about 5 % so that a relatively large fraction of the 3 ~ e may be physically separated from the place where the actual deformation of the chamber is taking place. This is useful in experiments where the measurements such as magnetic susceptibility of CMN or NMR measurements require the absence of large quantities of metals.

By starting the compression at a temperature of 23.6 m OK, Johnson, Rosenbaum, Symko and Whea- tley cooled a two phase mixture to a magnetic tem- perature of the CMN thermometer of 2.15 m OK.

In the subsequent experiment by Johnson, Symko and Wheatley 1341 a maximum equilibrium pressure of 33.910 atmospheres was obtained at the end of a compression. Using the value Jlk

0.72 m OK to calculate the solid entropy, the melting temperature corresponding to 33.910 atmospheres is 1.5 m OK and the entropy of the solid is 0. I7 Rln 2.

The cell used by Sites, Osheroff, Richardson, and Lee [32] is shown in figure (8). The principle of its operation is, of course, the same as that in the pre- viously described experiments. The compression appa- ratus consists of three chambers formed by a pair of concentric phosphor bronze (95 % copper, 5 % tin) bellows placed inside a brass cylinder. The 3He sample is contained in the innermost chamber and 3He in the outer two chambers. Initially all three chambers are pressurized to approximately the appropriate melting pressure of their contents. The sample cell is then precooled to about 25 m OK with a dilution refrigerator in contact with the outside walls of the cell. The pressure of the 3He in the intermediate cham- ber is then slowly released giving rise to a net down- ward force on the top plate of the bellows assembly.

The resultant motion of the bellows causes the desired

decrease in the volume of the 4He cell. The volume

change is monitored with a cylindrical capacitor

mounted on top of the bellows assembly. The tempe-

rature of the 3He is measured with a copper nuclear

susceptibility thermometer consisting of 0.2 grams

of fine (12 micron diameter) insulated copper wire

inside a nuclear resonance coil in the epoxy tail section

of the 3He cell. The surface area of the wires is suffi-

cient to permit reasonab!~ short equilibrium times

across the thermal boundary between the copper and

3He down to about 1 m OK. There was evidence that

solid 3He, which crystallizes at the warmest places in

C 3 - 8 8 R. C. RICHARDSON

the solid 3He as early as possible during the compres-

To Capacitance

sion process, a small amount of heat in the form of a short burst of rf pulses was applied to the 3He NMR coil to encourage the solid to crystallize in the coil. It was only during the actual compression that the maximum susceptibility in the 3He, discussed earlier, could be observed.

Thermometry is a crucial technical problem in all

Chamber

work near the NCel temperature. Johnson, Symko, and Wheatley [34] have clearly shown that the Pome- ranchuk effect can be used to cool 3He to temperatures beyond the range for which the susceptibility of CMN can be used. Scribner, Panczyk and Adams [39] have suggested that the melting curve of 3He be used as a temperature scale in much the same spirit as the vapor Resistor pressure of 3He and 4He is used at higher tempera- tures. Goldstein [43] has discussed this proposal at length and has published a careful calculation of the melting curve for this purpose. The melting curve thermometry is a particularly valuable idea in expe-

Cu N M R Coil

riments in which the Pomeranchuk cooling principle is employed since a knowledge of the pressure of the two phase mixture can be used to immediately deter- mine the temperature. The difficulties encountered FIG. 8.

Compression Apparatus of Sites, Osheroff, Richardson in CMN thermometry could be overcome by using

and Lee. instead the weak nuclear spin magnetization of metals,

The upper part of the apparatus consists of two concentric such as copper or platinum, which have spin ordering phosphor bronze (95 % copper,

% tin) bellows inside a brass temperatures of order OK [48], calibrated against chamber. The inner bellows (Robertshaw Controls Co., Knox- the melting curve at temperatures low enough to give ville, Tenn.) has o. d. - 1.18 cm., id. - 0.827 cm, relaxed length-

2.24 cm., wall thickness

0.007

cm and 20 convolutions. The good sensitivity. Johnson, Symko and Wheatley [34]

outer bellows (Flexonics Corp., Maywood, Illinois) has o. d.

their in the dP!dT"

1.92

cm., i. d. - 1.28 cm., relaxed length- 1.66 cm, wall thickness IneaSurementS using the melting curve, and Sites, -0.013 8cm, and 10 convolutions. The epoxy tail section has two Osheroff, Richardson, and Lee [32] used the melting nuclearresonancecoils, the upper has 15 turns of No.

AWG pressure to calibrate their copper nuclear resonance copper wire and the lower has 30 turns of No. 32 AWG copper

wire. The lower coil is filled with 0.2

of No. 50 AWG insulated thermometer.

wire (12.5 microns diameter) (REA Magnet Wire Company, Inc. Ft. Wayne, Indiana).

The top plate of the bellows system is free to move vertically. IV. Summary.

There obviously remains much A change in the forces balancing the plate occurs when the to be done in the study of solid

at low tempera- pressure in any of the three chambers changes. Compression

of the 3He in Chamber 1 is accomplished by bleeding the pres- tures. The most interesting problem for the immediate surized liquid 4He out of Chamber I1 (after Sites, Osheroff, future is the examination of the spin ordered state

Richardson and Lee ~321).

the cell during the compression, tended to form around the copper wires in the susceptibility thermometer.

Initial experiments with a larger filling factor of the copper wire, 0.3 grams, did not permit cooling of the copper wires below 12 m OK. Subsequent reduc- tion of the amount of copper in the cell, to permit a more open flow of the compressed 3He, allowed the copper to cool to 5.3 m OK under similar initial condi- tions of the apparatus. Even with this reduction of the amount of copper in the cell, the copper did not cool to as low a temperature as the 3He in the upper coil, not filled with copper. It is very likely that shea- ring friction in solid 3He mixed with wires caused local heating in the 3He within the copper thermo- meter coil. In order to measure the susceptibility of

with measurements of the thermostatic and transport properties below the Ntel temperature. The Pomeran- chuk effect has been demonstrated to be a powerful experimental technique for cooling 3He to very low temperatures and the details of the cooling process itself provide a great deal of information about the solid. The effect can readily be applied to cool other materials of interest to 2 mOK, provided that they can be made in a form with sufficiently large surface area to make the thermal time constant tractable.

The cooling principle may also be used even in large

magnetic fields with the restriction that the minimum

temperature that can be achieved is of the order of

pH!k and will thus be raised above 2 m

for fields

greater than 20 kGs. Studies of the melting curve

in strong magnetic fields should also be quite inte-

resting.

SOLID HELIUM THREE AT VERY

TEMPERATURES C 3 - 8 9

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