Phase transitions of spin polarized 3He : a thermodynamical nuclear orientation technique ?

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Phase transitions of spin polarized 3He : a thermodynamical nuclear orientation technique ?

B. Castaing, P. Nozières

To cite this version:

B. Castaing, P. Nozières. Phase transitions of spin polarized 3He : a thermodynamical nuclear orienta- tion technique ?. Journal de Physique, 1979, 40 (3), pp.257-268. �10.1051/jphys:01979004003025700�.

�jpa-00209106�

Phase transitions of spin polarized 3He : ^a thermodynamical

nuclear orientation technique ?

B. Castaing ^{and P. Nozières}

Institut Laue-Langevin, 156X, 38042 Grenoble Cedex, ^France (Reçu ^{le 24} août 1978, accepté ^{le 24 novembre} 1978)

Résumé.

Sous bien des aspects, l’3He complètement polarisé ^se présente comme un nouveau corps quantique,

différent de l’3He normal. Cet article rappelle qu’on peut l’obtenir de façon purement thermodynamique, ^en partant du solide fortement polarisé par ^un champ magnétique à très basse température, ^à condition de travailler sur une

échelle de temps inférieure à T1, ^le temps ^de relaxation de l’aimantation. Les points cruciaux du problème

dépen-

dance de l’énergie ^du liquide ^avec l’aimantation, ^{valeur de} T1

^sont longuement discutés, ^ainsi que les nouvelles

possibilités offertes. En particulier ^l’3He complètement polarisé ^se présente ^comme la meilleure chance d’observer

une phase ^de solide lacunaire telle que l’ont discutée A. Andreev ^et I. M. Lifshitz.

Abstract.

Completely polarized ^{3He behaves} in many respects very differently from normal 3He. In principle

it can be obtained by purely thermodynamical methods, starting ^{from a} strongly polarized ^solid (by ^a magnetic

field at very low temperature), ^as long ^as the overall experimental time is less than T1, ^the magnetic relaxation time.

We discuss the crucial points ^{of such an} experiment

dependence ^of liquid ^3He energy ^on magnetization, ^value

of T1

^{and the new} experimental possibilities ^it opens. In particular completely polarized 3He appears ^as the best chance to observe a vacancy solid phase, first described by A. Andreev and I. M. Lifshitz.

Classification Physics ^Abstracts

67.00

1. Introduction.

Nuclear spin relaxation in

liquid ^{’He is a} compromise ^{between two} mechanisms.

(i) Intrinsic relaxation in the bulk, ^{due to collisions} of two particles ^{via the} dipole-dipole interactions.

Because of motional narrowing, ^the corresponding

relaxation rate is inversely proportional ^{to the} spin

diffusion coefficient D.

(ii) Extrinsic relaxation on the walls, ^{either on} magnetic impurities ^or indirectly ^{because the walls}

are coated with a layer ^{of solid ’He where} Tl ^{is short.}

Such a mechanism requires ^{diffusion to the} wall, and the corresponding ^{rate is}

^D.

Thus, altogether

where A depends ^{on the wall, B on the bulk.}

Around 1 K, ^{the intrinsic} Tl is of order a few minutes ; ^unless special precautions ^are taken, ^wall relaxation usually predominates. ^It is however _pos- sible to observe the intrinsic Tl by appropriate ^clean- ing of the walls [1].

Recently, ^{it has been} found that appropriate coating

of the walls could reduce ^A drastically. Barbé, ^Laloë and Brossel [2] showed that a solid hydrogen ^coat- ting practically stopped ^wall relaxation around 1 K

- a result confirmed by ^recent experiments ^of

Taber [3]. ^Neon coating ^{was studied} by Chapman

and Bloom, ^who argue that A is controlled by ^the

adsorbed layer ^{of 3He. Even more} simply preferential adsorption ^{of ’He can} provide ^an insulating layer [5]

which prevents formation of a further solid layer ^of

’He. From all these results, ^it _appears possible ^to stop relaxation at the walls - at least at temperatures

> 0.1 K at which the diffusion coefficient is reasonably

small (when D --> _oo, wall relaxations would always

take over).

If that is _so, Tl ^is long, ⁱⁿ the range 1-10 min. It is then possible ^{to achieve non} equilibrium states, with ^a very large spin polarization, which would be unattai- nable in _any reasonable magnetic field. More speci- fically, ^{F. Laloë and C. Lhuillier} [6] ^have proposed ^to polarize ^{the atoms} by optical pumping ⁱⁿ the vapour

phase, ^{and to observe the} thermodynamical properties

of such an artificially polarized system : transport properties, liquid vapour equilibrium, ^etc...

The present paper follows the reverse path : ^{if the}

relaxation time Tl ^is long, up and down spins ^{are like}

different chemical species, ^{and one} may apply ^to

them the usual concepts ^of phase equilibrium, ^frac-

tionate distillation, ^etc... Hence the possibility ^of polarizing ^the spins by purely thermal methods.

Let M be the magnetization, F(M) ^{the free} energy

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01979004003025700

density. ^The system ^is subject ^{to an external} magnetic

field H,,,,. ^{If M} is allowed to relax, thermodynamical equilibrium corresponds ^{to a} minimum of the function G

F - MHeX,. We may define an internal magnetic

field

M will relax spontaneously ^{until H is} equal ^to Hext.

Let us now assume that the spin relaxation is blocked.

The relevant thermodynamical variable is M instead of Hext8 ^We may still define a field H through (1)

-

but it bears no relation to the physical applied ^field Hertz ^{Rather, H} represents the difference in chemical

potential between i and 1 spins (’)

(Jl(1

DFIDN,, being defined in the absence of Hext).

H is related to M by ^a magnetic equation ^{of state,} H(M). ^{On a time scale «} Tl, ^it may take _any value.

Only ^over times > Ti will it relax to Hertz

Consider now the equilibrium ^{of two} phases, ¹

and 2. Since the free energy is stationary ^under exchange ^{of either} i or 1 ^atoms between 1 and 2,

the chemical potentials ^{are the same in the two} phases,

and thus Hl

H2 ^{= H -} irrespective ^{of the} applied

field Hext’ which does not influence the equilibrium (its contribution to Jla is the same in both phases). ^The phase equilibrium ^is parametrized by (H, T), ^the magnetizations Mi ^and M2 being usually ^different.

In the (M, T) plane, ^the phase diagram has the usual

liquidus-solidus shape ^{shown on} figure ^{1 - a} figure

which clearly displays ^the different behaviour on

times « or > Tl. ^{Assume that we start with} phase (1)

Fig. ^1.

^A typical phase diagram ^{in a} magnetic field. At a given temperature, phase (1) ^with magnetization MA ^{is in} equilibrium ^with phase (2) ^at magnetization MB. ^{If one} transforms (1) ^into (2) ^at

constant H, ^the points ^{A and B do not move} (spin relaxation making

for the change ⁱⁿ M). If the transformation proceeds ^{at constant M,}

AB moves up to CD : the effective magnetic field increases in the transition.

(’) Throughout the paper, ^{we set} the magnetic ^{moment of the 3He}

atom equal ^{to 1 : H is an} energy, and the magnetization

at point A, ^{and transform it to} phase (2) ^at point ^{B :}

we thus deplete ^the magnetization ^of phase (1).

-

For a slow _process (» Tl), relaxation is avai- lable to restore any missing magnetization. ^{Since H}

must remain equal ^to Hext’ MA ^and MB ^are unchanged

as the transformation proceeds : ^the points ^{A and B}

do not move.

-

For a fast _process ( Tl), ^{M must} be conserved :

as the transformation proceeds, ^{AB moves} up, until it reaches CD when (1) ^is completely ^{turned into} (2).

The effective field H is thus increased by ^the phase transformation, ^well beyond ^its original equilibrium

value Hext.

As an example, consider solid ’He, ^{at a} temperature T much higher ^{than the} magnetic ordering tempera-

ture : it obeys ^{a Curie law, and thus} M1 ⁼ CH/T.

After melting, it becomes a Pauli paramagnet, ^with

magnetization M2

CHITF. ^Fast melting ^{must con-}

serve M. If H was equal ^to Hext at ^the start, it becomes

HTF/T ^{at the end}

^a spectacular ^{increase if T is}

small !

In this _paper, we develop ^these thermodynamical

arguments ^{in some} detail, ^{and we} propose experiments

which might ^{allow the} production ^of highly polarized 3He matter. In section 2 we first discuss the properties

of homogeneous liquid ^{or solid 3He. The} melting phase diagram ^of polarized ^3He is described in sec-

tion 3. Section 4 deals briefly with other phase equili-

bria (liquid-vapour, ^or dissolution in 4He). ^{In sec-}

tion 5, ^we try ^{to estimate the} spin relaxation time Ti, ^a

central feature in our problem. Specific experimental proposals ^{are made} in section 6, together ^with specu- lations on new physical ^{effects that could} emerge at such high effective ^fields (such ^as the vacancy solid

proposed long ^ago by ^{Andreev and Lifshitz} [3]).

2. Properties ^{of the} homogeneous phases.

^Solid 3He cristallizes in the b.c.c. lattice at moderate _pres-

sures. Its molar volume is large (vS

^24.2 cm3 jmole

on the melting ^{curve at T =} 0). ^Its compressibility

is unusually large, Ks - 6 x 10-3 ^{bar-1. It behaves}

as a Curie paramagnet ^{down to} TN

¹ mK, ^{at which} temperature ^{it turns into an} antiferromagnetic ^state [8- 10]. ^The exchange energy J responsible ^{for such a Néel}

transition is probably ^{due to} rotational tunnelling

of a pair of atoms around each other, ^{in the} anisotropic

cage formed by ^the surrounding ^medium (2). ^Recent experimental ^data [9] suggest ^{that a moderate} magnetic field,

⁴ kG, ^{can turn the} antiferromagnetic phase

into a new state, apparently ^{a weak} ferromagnet [11].

At the moment there exists _many interpretations

of such a magnetic ^behaviour [12] ^: multiple

(2) ^{Such a} tunnelling ^is certainly ^assisted by ^a transient deforma- tion of the cage. Note that a modest decrease in molar volume raises the potential barrier : hence a drastic drop ^{in the} already ^small J, leading ^{to the observed} large magnetic ^{Gruneisen constant :}

d Log J

d Log y

’

20.

exchange [13], spin polarons around vacancies [14],

stabilization of the ferromagnetic ^state by ^zero point

vacancies [15], ^etc... Experiment will decide which of these models is correct (actually, they ^{do not exclude}

each other, ^and reality may partake ^of several). ^Here

we call attention to the model of Andreev et al. [15],

based on the spontaneous appearance of vacancies in

a magnetic field : this possibility will appear later in

our discussion.

In any case, ^we shall not be concerned with such low temperatures. ^{We thus} disregard completely ^the magnetic transition. As long ^{as T exceeds a few mK}

the magnetization ^{is well} represented by ^{a Curie law.}

Under typical conditions, ^T

⁵ mK, H

⁸⁰ kG,

the spin polarization ms reaches 85 %.

Liquid ^{’He above a few mK is a} standard Fermi

liquid, ^{well described} by ^{the Landau} theory ^{at low} enough temperature ^and magnetic ^{field. The Landau}

parameters m*/m, Fi, Fô ^are well known as a function of _pressure [16]. ^{We are} especially concerned with the low field spin susceptibility, ^{which we} may write as

the spin temperature TF being ^{defined as}

The Landau theory applies ^as long ^as T, H TF.

The energy TF ^{decreases from 0.54 K at} vapour pressure, down to 0.30 K at melting pressure. It cha- racterizes the liquid features of the system : ^{flow of}

quasiparticles, ^etc... (which become harder and harder

as the density increases).

Besides the plastic ^flow typical ^{of a} liquid, ^{’He also} displays ^acoustic oscillations, ^{in which the atoms} vibrate in the cages formed by ^the instantaneous

configuration ^{of their} neighbours. ^The corresponding

characteristic energy is that of a phonon with atomic

wave length

0D plays ^{the same role as the} Debye temperature ^{in the} solid

and indeed it is quite comparable : longitu-

dinal phonons ^{are much the same in} liquid ^{and solid}

’He. 0, ^increases with pressure, from 11 K at vapour pressure up ^to 25 K ^at melting pressure.

Traditionally, ^{one considers} liquid ^{’He as a} nearly ferromagnetic system, ^{to which one can} apply ^the

usual paramagnon concept [17]. ^{Indeed the} exchange

enhancement factor, (1 ⁺ Fô)-1, ^{is not small, - 3}

throughout the relevant pressure range. The effect

is sizeable

yet ^not dramatic. What is really striking

is the huge ^ratio OD/TF, ^which reaches 80 near melting

pressure. From such a vantage point, ^{one tends to}

view liquid ^{’He as} nearly ^{solid. Thére exist two}

distinct _energy scales. On the fast scale, - n/ (JD,

the atoms vibrate in their own cages. The system ^is locally rigid, ^{like a} glass ^which, despite ^{the lack of a} crystal lattice, ^{can resist a stress.} On the much slower scale li/TF, ^the cages ^start drifting around each other, leading ^{to the} plastic deformation expected ^{in a} liquid.

According ^to (4), magnetic properties involve the slow time scale ni TF, ^{That can} be understood easily

on physical grounds. (2 XL) - 1 ^is just ^the coefficient of m2 in the usual Landau expansion ^{of the} ground ^state

energy

A finite _XL is thus tantamount to an m-dependent Eo.

Now the magnetization dependence ^of Eo ^can only

arise from the exchange ^{between the atoms of the} liquid, which allows them to sense their relative spins (if they ^cannot exchange, they ^{are de} facto, ^{if not de} jure, distinguishable). Finally, ^{hard core atoms can} only exchange by rotating around each other. In the solid, ^the surrounding ^{matter can at best} give way

slightly, ^{but it cannot} participate ^{in the rotation}

hence _{the very} small tunnelling probability. ^{In the} liquid, ^the vicinity ^{can follow the} rotating pair (back- flow), ^{and thus J is much} larger. Nevertheless, exchange ^{involves a} real motion of the fluid, ^{hence the} slow plastic time scale.

To put ^it shortly, ^the large ^measured susceptibility of liquid 3He ^can arise from two causes :

(i) ^The liquid ^is nearly ferromagnetic. ^{Then the}

coefficient a in (7) ^is accidentally very small

but the higher ^terms b, ^..., may be much larger.

(ii) ^The liquid ^is nearly solid. The whole _energy scale for magnetic properties ^{is then -} TF, ^limited

as it is by ^{the slow} exchange ^{of two atoms} (their ^rota-

tion being ^hindered by ^the nearby matter).

Fig. ^2.

^{A sketch of} _Mi and MI ^{as a} function of m in liquid ^3He

at zero temperature.

In order to answer the dilemma we must examine the behaviour of Eo(m) ^{for large m. Let us} for instance

plot ^{the chemical} potentials ,uT and ,u ^for up and down

spins ^{as a} function of m (Fig. 2). ^{The initial} slopes ^are

simply 1 ^{= 2} T,13. ^{The m2 term in the} expansion of J1.

XL

may be found by integrating ^the Gibbs-Duhem

relationship

(s, v, m are quantities per particle). ^We thus obtain

near the origin, ^{and for T}

^{0 :}

When m - 1, _J.1t ^and _y, go ^to well defined limits.

Since H = Mt - J.1 p ^the magnetization ^curve m(H)

has the form sketched on figure ^{3. The} central issue is now the following :

-

either the extrapolation ^{of the low m} expansion

is qualitatively ^{correct. The relevant} energy scale remains - TF ^for any value of m. Then model (ii) ^is

correct,

-

or extrapolation ^to large m grossly ^underesti-

mates the actual variation of J.1t ^and y,. ^{Then we have} the same situation as in a second order phase ^transi-

tion. ’He is of type (i) (nearly ferromagnetic).

Fig. ^3.

^The magnetization ^{as a} function of H for liquid ^{’He at}

zero temperature (full curve) ^and at finite T (dotted curve).

We believe that the former case holds : liquid 3He ^is fundamentally ^a nearly rigid system, ^and extrapolat- ing the low field susceptibility ^to large m ^{is a reaso-}

nable rough approximation. ^{As it stands, this is an act}

of faith. Only ^a microscopic calculation of Eo(m)

could provide ^an unambiguous proof. ^{Such a calcula-}

tion is _very hard. Low density expansions ^{are of no}

avail for ’He : Oné must resort to molecular dyna-

mics [18]. Preliminary calculations by Levesque [19]

seem to support ^our contention

one must await

more detailed results before concluding. ^{As of now,}

it is better to rely ^on experiment : ^{we shall see that the} phase diagram ^{of ’He in} very high fields should _pro- vide an unambiguous ^answer.

Let us assume that extrapolation ^of (9) ^to large m

is qualitatively ^{correct. Then the} variation of _M, between m

0 and m

1 is

7p/3

^{0.2 K at}

low pressure. This is to be compared ^to po itself

obtained from the heat of evaporation. ^At p ---> 0, po

2.5 K. The magnetic ^variation of pi ^{is thus}

a small correction : liquid ^{’He remains} firmly bound,

even when its spins ^are fully polarized. ^{Such a conclu-}

sion is markedly different from what one would expect for a free Fermi _gas (in ^which spin alignment multiplies

the Fermi _energy by 2213).

A few more words on volume variations of the

liquid. ^{The molar} volume vL ^is quite large, going ^from

37 cm’ at zero pressure down to 25.4 cm3 at melting

pressure (T

0). (Note ^the large compressibility.)

The change ^{of volume on} melting ^{is small :}

In large magnetic fields, ^{one must} worry about

magnetostriction. ^From (8), it follows that

The solid is a pure Curie magnet, ^and thus vs is field

independent. ^{In the} liquid, integration ^of (4) yields ^at

once

If we extrapolate this formula to m

1, ^{we find a} variation AVL - 0.1 ^{cm’ near} melting : magnetostric-

tion is a small correction (at vapour pressure, OvL ^is larger, - ^0.8 cm’).

3. The liquid-solid equilibrium.

^{In zero field,}

the melting ^{curve has the} shape ^{shown on} figure ^4.

It displays ^a minimum around 0.3 K. As explained long ago by Pomeranchuk this minimum arises from the spin entropy ^{of the} solid, ss ⁼ Log ^{2, which}

overcomes the entropy ^{of the} liquid sL

yT ^{at low}

Fig. ^4.

^The melting ^{curve in zero field} (full curve), ^{in a small}

field H TF (dotted curve) ^{and in a} large ^field H > TF (dot-dash

curve).

enough temperature. ^The liquid-solid equilibrium

curve obeys ^{the Clausius} Clapeyron relationship

In zero field, ^and neglecting ^the variation of (VL - vs),

we obtain the melting pressure

Hence the minimum in p(T), ^{which occurs around}

T = TF.

In finite fields, ^the melting pressure decreases. ^{For a}

given temperature, the variation Ap(H) is obtained

from(ll)

(Here again, ^{we have} neglected magnétostriction ; Ap ^is 0, ^{since the solid is more} susceptible ^{than the} liquid.) ^The integral ⁱⁿ (13) ^is represented graphically

in figure ^{5. In moderate} fields, H « TF ^the magnetiza-

tion of the liquid ^is negligible. ^The drop ⁱⁿ melting

pressure is controlled by ms, and it occurs mostly ^for

T % ^{H -} hence the dotted curve in figure ^{4. For} large ^fields H it TF, ms reaches saturation, ^and Ap ^is

instead controlled by mL. dp ^{extends to} higher tempe- ratures, and ^it eventually ^{washes out the minimum in}

the melting ^curve (dash-dot ^{curve in} Fig. 4). Indeed, for fully polarized matter, there is no ^{spin entropy}

left in the solid

hence’no Pomeranchuk minimum.

Fig. ^5.

^The magnetization ^{curves of} liquid and solid ’He : (a) ^{at a} small finite temperature, ^T TF, (b) ^{at T}

0. The lower-

ing ^{of the} melting pressure involves the shaded area.

The resulting melting pressure drop is sketched ^as

a function of H on figure 6, ^{both at T = 0} (full curve)

and at finite T (dotted curve). For H « T, Ap ^starts quadratically

Fig. ^6.

^The lowering ^{of the} melting pressure ^{as a} function of field at T

0 (full curve) ^{and a finite} T (dotted curve).

and it is _very small. For H > T, ^{it becomes} linear, going ^{to a finite limit when H »} TF. ^{The result is} especially simple ^{if T = 0. Then}

We thus know the beginning ^{of the curve of} figure ^6.

In estimating ^{the limit} Opmax ^for high fields, ^{we meet} again the dilemma of the preceding ^{section. If the fluid}

is not close to a ferromagnetic instability, extrapola-

tion of (15) ^{to m}

^{1 is not} grossly unreasonable. We thus find

In such a case, the melting pressure remains positive

even for an infinite H. The phase diagram ^of fully polarized ^{’He has the} shape ^{shown on} figure ^7a.

If instead the large XL is due to the proximity ^{of a} ferromagnetic instability, ^the m(H) ^curve flattens, ^and

Fig. ^7.

^Possible phase diagrams ^for fully polarized ^{3He :}

(a) ^If I1Pm ^is small, ^a liquid phase persists ^{down to T}

^0. (b) ^If

A/?m ^is large, ^a triple point is restored. (c) ^For incomplete polariza-

tion, ^case (b) ^should yield ^two triple points t, ^and t2.

it takes a much larger ^field to saturate m. The shaded

area in figure ⁵ increases, ^and f1p Max is ^much larger.

A correction by ^{a factor 5} would make the melting

pressure negative. ^{Then the} melting ^{curve meets the} evaporation ^{curve, and we recover the usual} phase diagram ^{with a} triple point ^{shown on} figure ^{7b. Note}

that in such a case, there must exist a range of magnetic

fields for which the minimum of the melting pressure is

negative, white/?(0) ^{is still} positive. ^The phase diagram

should then display the unusual shape ^of figure 7c, with two triple points.

It thus _appears that a study ^{of the} phase diagram

should answer unambiguously ^the question ^raised

in section 2

namely ^the behaviour of strongly polarized liquid ^{’He. Of course, if we} could achieve

a field H - TF, ^{there would be no need for a} phase diagram

^{we could} study ^{the curve} ML(H) directly.

Unfortunately, ^{0.3 K - 4 MG ! The reason for} study- ing melting ^{is that one} generates effective fields of that order of magnitude by working faster than the relaxa- tion time Tl.

Our argument is illustrated in figure 8, ^the equivalent

of a liquidus-solidus ^{curve. For a} given temperature T,

we plot ^the magnetizations mL and _ms of the two

phases ^{which are in} equilibrium ^{at a} pressure p.

Figures ^{8a and 8b} correspond respectively ^{to the} phase diagrams ^of figure ^7a and 7b. Consider for instance

case (a), ^{and start from a solid at} point ^{B. The first} drops ^of liquid correspond ^to point ^{A. In the absence}

of spin relaxation, ^{AB moves down as} melting pro- ceeds. Melting ^is completed ^when the pressure reaches pc; by ^that time the effective magnetic field is much

higher ^than Hext. ^{If we} stop ^{at a} pressure intermediate between _PA and _pc, melting ^is incomplète ; ^subse- quently, ^{it is} triggered by spin relaxation, ^which reduces H and moves CD back toward AB. (Note

the unusual situation in which melting ^is completed only ^{thanks to} magnetic relaxation.)

Fig. ^8.

^The liquidus-solidus diagram ^for melting ^of polarized

’He, ^{as a} function of _pressure for a given ^{T. Cases} (a) ^and (b) ^cor- respond ^{to the} phase diagrams ^of figures ^{7a and 7b.}

In case (b), melting ^{cannot be} completed ^if the initial solid magnetization exceeds the critical value _mc, however small the pressure. There again, ^{it is} only spin relaxation which will allow full melting.

Melting ^{the solid} at constant m increases the field H

(the liquid goes from A to C). Conversely, freezing ^the liquid by ^a pressure increase decreases H, ^{the solid}

going ^{from B to E in} figure ^8a. Subsequently, spin

relaxation will take the solid back to B - but in a

transient regime, ^{the solid is} underpolarized - ^a

feature which bears on the efficiency of Pomeranchuk

cooling.

All these figures ^{have been} plotted at constant T, conceptually ^the simplest ^case. Actually, ^{the heat}

transfers at such low temperatures ^are extremely difficult, because of the Kapitza resistance at the walls.

Consequently, any transformation will by necessity

be adiabatic rather than isothermal. Let us assume

that the transformation is slow (and ^thus reversible)

for _every internal variable but m. Then, ^the picture

should be modified in two ways. First the liquidus ^and

solidus curves of figure 8 should be drawn at constant

entropy S, instead of constant T. Second, ^{we should}

worry about entropy production (and ^concurrent temperature increase) during the process of spin

relaxation.

Strictly speaking, ^{since there are now two conserved}

extensive quantities ^{in the} transformation, ^{M and} S,

we should plot liquidus-solidus surface ^{in the} (p, M, S)

space. For ^a given initial condition however

say

point ^{A in} figure ^8a

^the temperature ^{is a well} defined function of the percentage ^of liquid phase

and we can thus plot equilibrium ^{curves in the} (p, m) plane

keeping in mind that these curves would

change for another initial state A. The corresponding diagram ^is qualitatively ^{the same as in} figure ^{8. Let us}

start for instance from a solid at very low temperature To ; ^{its initial} entropy ^is purely magnetic, Ss(ms).

After completion of adiabatic melting, ^that entropy

must be recovered in the liquid, ^whose temperature is such that

In zero field, Tl ^would just ^{be the} temperature ^{of the} minimum in the melting ^curve (which precisely ^cor- responds ^{to a zero} entropy change). ^{In a} large field, Ss ^{is reduced ; if we} neglect ^the dependence ^of SL ^on magnetization, Ti should be smaller - say ^{0.1 to}

0.2 K. The pressure drop pmaX required ^to complete melting ^will consequently ^be higher than in the iso- thermal case (heating ^and magnetization ^{act in the}

same direction). ^{Note that the} temperature ^increase

on adiabatic melting ^is just ^{the reverse of Pome-}

ranchuk cooling by ^adiabatic freezing.

Let us finally ^{consider the heat} dissipation ^{due to} spin relaxation after the fast melting ^{at constant m}

has been completed. ^Let S(Je, M) ^{be the} entropy ^at

constant pressure, Je being ^the enthalpy. ^{When M}

varies by ôM, ^the magnetic ^work provided ^{to the fluid}

is ô W

Hext ^{ôM. The} entropy production ^{in an}

adiabatic _process is thus

Hence the irreversible entropy production ^{due to}

relaxation from _ml to m2 (per particle) :

From (19), ^one may infer the temperature change

(note ^{that in a slow} process, H remains equal ^to Hext : ^then àSi,,

0). Actually ^one may get ^it directly by ^an energy conservation argument (treating ^the

relaxation as a magnetic Joule Kelvin process). ^The

total magnetic ^{work furnished} in the transformation is Hext(m2 - mi) ; in the absence of heat exchange,

one must have

Consider first melting, ^{in which} ml » m2’ The last term of (20) ^is negligible (equivalently, ^{H »} Hext ⁱⁿ (19)). ^Since

it follows that (T2 - Tl) is of order Fp - ^{0.3 K. In}

the reverse case of freezing, M2 » ml ; then the last

term of (20) is instead dominant. The enthalpy ^{of the}

solid has the form

where OD ^{is the} Debye temperature (Jeg ^{is m} indepen-

dent if we neglect exchange). ^{In that} case,

Thus the final temperature T2 ^{should be}

Irreversible behaviour ^{in a} magnetic field makes Pomeranchuk cooling very inefficient.

4. Other phase equilibria ^of polarized ^’He.

^The’

Clausius Clapeyron equation (11), ^{can also be} applied

to the liquid-gas equilibrium. ^{Since the} gas is again

a Curie paramagnet, ^more susceptible ^{than the} liquid,

the _vapour curve will move upward - ^{as the} magnetic

field is increased. The displacement, however, ^should

not be spectacular.

First of all, it is known that the critical point ^{is not}

sensitive to the statistics. There exist two types ^of quantum ^{effects :}

-

Effects due to the finite spread of thermal wave

packets, ^which average the effective interaction bet-

ween atoms. Such effects are important

^but they

are the same for bosons or fermions.

-

Effects due to the symmetrization ^{of the wave} function, through ^the exchange ^of two atoms. There the statistics is crucial.

The latter corrections are known to be small near

Tc

^{it is} just ^{for that reason that de Boer et al. were}

so successful in predicting ^{the critical} point ^{of 3He}

from that of ’He by a ^mere scaling ^{of the mass.} Now,

the spin ^structure of the fluid can only ^{enter in the} exchange correction. If the latter are small, ^the critical

point ^{should be} essentially independent ^of magnetiza-

tion.

At low temperature, quantum effects become

important ^{in the} liquid, ^{and the} vapour pressure ^curve is affected by ^the magnetic field. Rather than using (11),

we note that the _up and down spin vapour pressure of a

perfect gas ^at temperature ^{T are} proportional ^to

(from ^{which it follows} Pt/p¡

^{e HIT :} the vapour is a

Curie paramagnet). ^{As soon as} H > T, the vapour is

fully polarized ^{and its} pressure is controlled by ,ut.

If we extrapolate ^the low field behaviour (9), ^the change ^of y ^{T on} going ^from mL ⁼ 0 to mL ⁼ 1 is

TF/3 1" ^0.1 K, much smaller than Mo = - 2.5 K :

the liquid vapour ^curve is not changed drastically.

If the extrapolation ^were quite wrong, ^one could conceive that _1À, would become > 0 when m ⁼ 1 : the liquid ^{would cease to be found.} (It ^{would take}

a minimum _pressure to make it stable.) ^{We need}

not consider that case : for such a large variation of y,, ^the melting ^{curve would meet the} evaporation

curve well before y, changes sign.

A similar discussion applies ^{to the} equilibrium

between polarized ^’He and its dilute solution in ’He.

In fact, ^{at low} temperature, the 4He ^{substrate acts as a}

vacuum, and the dilute solution is like a dense _gas.

At low temperatures, ^both phases ^{are Pauli} parama- gnets, but the dilute solution has a lower Fermi tem-

perature and is thus more susceptible. ^{In a moderate} magnetic field the magnetizations ^{of the dense and}

dilute phases ^are in the inverse ratio of their spin

temperatures

At constant magnetization, the effective field H should,

thus decrease _upon dissolution of _pure 3He in 4He.

Conversely, precipitating ^3He from the dilute solu- tion will increase H. Finally, ^{we note that the zero} temperature solubility ^{will increase for} polarized ^3He (like the vapour pressure). ^{It is a} straightforward thermodynamical problem ^to calculate the change ⁱⁿ solubility, etc..., in ^a magnetic ^{field : we shall not do it}

here.

5. The relaxation time Tl.

AU this discussion

relies on the assumption ^that Ti ^is long enough ^that

experiments ^can be carried out on a faster scale. It is

thus of importance ^{to assess its order of} magnitude

and also to make sure that high polarizations ^{will not}

reduce it unduly.

Experimentally, the observed Tl ^{is a} complicated

mixture of extrinsic relaxation at the walls of the _sys- tem (or ^{on the} powder ^{used to cool the} fluid), ^{and of}

intrinsic relaxation in the bulk due to dipole-dipole coupling between the nuclear spins. ^{Hence the} large

scatter in the literature. Here, ^{we shall assume that} extrinsic relaxation can be suppressed completely,

and we shall focus our attention on the intrinsic mechanism.

Since we are mostly concerned with melting, ^we

first consider the Tl ^{of the} liquid (which is also easier to understand). ^{Its low} field behaviour [20] is sketched

on figure ^9. Ti ^{has a minimum at a} temperature

-

0.5 K. The height of the minimum depends ^on

pressure

roughly ^{5 min. at zero} pressure, closer to 2 min. at melting pressure. The data are somewhat uncertain and they represent ^a lower bound. At lower temperatures, Ti ^{increases as} 1/T2. ^The latter increase is easily understood in the framework of the Landau

theory ^{of Fermi} liquids. Relaxation is due to scattering

of two quasiparticles ^{in a} layer ^{of width T around}

the Fermi level. The phase space for such a collision is of order T2 - hence a transition probability ^{of that}

order. The same argument holds for the lifetime r,

of a quasiparticle (controlling transport properties).

The only difference lies in the nature of the interaction.

Fig. ^9.

^{A sketch of the} intrinsic relaxation time of liquid ^{’He as}

a function of temperature.

The scalar potential ^{between two atoms} contributes to Te

but not to Ti, ^{since the} corresponding ^colli-

sions are spin conserving. ^The only way ^to relax spin

conservation is to invoke the dipole interaction.

Since the phase space is the same for the two pro- cesses, the only difference lies in the matrix elements.

As a result

where Hd ^{is the} dipolar interaction and t the scattering

matrix. Let _Wd be the precession frequency ^{in the}

average dipolar ^{field of two} neighbours (- 104) ;

since the interaction is of intermediate strength, ton - TF. ^Thus

At 0.3 K, te is of order 10- i l _s, yielding Tl - ^10’

second, ⁱⁿ qualitative agreement ^{with the observed} value.

The above estimate holds as long ^{as H « T. In the} opposite ^limit H > T, ^the phase ^{space available to a} spin flip ^{collision is much} larger. ^{From the} energy distribution of up ^{and down} spins ^{sketched in} figure 10, it is clear that the particles ^{that can take} part ⁱⁿ spin

relaxation extend over a range H, instead of T. The situation is similar to the lifetime of a quasiparticle

with e » T. In the latter _case, detailed calculation shows that the transition probability ^is proportional

to (n’ ^{T2 +} e’). ^{We can thus} expect ^{that in a field} H » T, the relaxation Tl ^{will be the same as in zero}

field at a temperature ^T’

77/Tr. ^{Since the maximum}

field is of order TF, ^{T’ should be a few tenths of a} K, and the corresponding relaxation time should roughly correspond ^{to the} minimum of figure 9, ^{whatever the} actual temperature ^{T. We conclude that} Tl ^{will remain}

very long

^{several minutes}

^{even in} highly pola-

rized liquid ^3He.

Fig. 10. - The energy distribution of up and down spins in

polarized ^’He.

We now turn to the solid. For fairly high tempera- tures, T > ¹ K, relaxation is supposed ^to result from

a modulation of the dipolar interaction by ^{the random}

motion of vacancies. A recent review of the problem

has been given by ^Landesman [21]. ^The corresponding Tl ^is given by ^the following ^formula

Here ^w is the precession frequency ^{in the} applied magnetic ^field Hext (the only ^one which is relevant in the energy conservation); iL is the hopping ^{time of}

vacancies near a given ^site (which characterizes the

frequency ^{of the} dipolar coupling modulation). M2

is essentially the square of the dipolar precession

frequency wd. (M2

^{5 x} 108 s- 2.) iL depends critically

on the number of vacancies, ^{and thus} Tl ^varies rapidly ^with temperature. ^{For a} given Hext, Tl ^{has a}

minimum _{when W’tL}

1.2, ^{at which} point T1 = W/ M 2’

For an external field of 80 kG, ^W

^{1.5 x} 109, ^and

the minimum T i ^is roughly ^{3 s. Such a} lower bound is

certainly very pessimistic, ^{as at low} temperature ^the number of vacancies is _very small, making iL very long.

At lower temperature, relaxation occurs through

the exchange between nuclear spins, ^and Tl ^is roughly temperature independent. ^The theory ^{of this} regime

is not clearly understood. Experimentally, ^{the corres-} ponding Tl ^increases rapidly ^{with the Larmor fre-}

quency ^w, but decreases when one gets ^{closer to the} melting ^{curve. In 25 kG and at} melting pressure, Tl

is of order 10-30 _{s ;} extrapolating ^{to 80} kG, ^{one should}

find T1 ~ ⁵ min. It thus _appears that at very low temperature the relaxation time of the solid is quite long.

Actually, ^one should be careful in applying ^these

results to a solid in the _process of melting. Very likely, the solid is full of vacancies, ^{and the former} regime ^of vacancy modulation might ^{extend to much}

lower temperature. Thus it is not unconceivable that the minimum value of Ti (a ^few seconds) might apply

even at T N 0.1 K. That is unlikely, though : ^{if there}

are many vacancies, they ^move quickly, andrl ^is short, which again ^makes Tl long.

In any case, the Ti ^{of the solid is not as critical as}

that of the liquid, ^{since the solid} spends ^{a much}

shorter time off equilibrium. ^{Consider an ideal case}

in which the liquid solid interface moves at velocity ^u.

As the solid melts, ^{it enriches} in up spins, ^{since the} liquid ^{is less} polarized. ^If magnetization could diffuse

infinitely fast, this increase in _ms would distribute in the whole sample. ^{With a finite} spin diffusion coeffi- cient Ds, ^the increase in _ms remains localized near the interface, ^{in a diffusion} length ^A

Dslu. ^{The corres-} ponding magnetization profile ^near the interface is sketched on figure ^{11. The} liquid ^{has the same} magne- tization as the bulk solid, ^{the effective} magnetic ^{field H} building up only ^{in the surface sheath} of width Â.

With a typical DS = 3 x 10- 8 cm2/s, ^{and an inter-}

face velocity ^{10 - 3} cm/s, ^{the surface sheath is} very

narrow (3 ⁰⁰⁰ Â). ^{The time a} given ^{atom of the solid} spends ⁱⁿ the surface sheath (i.e. ^off equilibrium) ^is

With the above values, im

0.03 s. The only require-

ment is that the solid Tl ^be longer thanr,,,

^{a condi-}

tion which should be easily satisfied. It may be that

near melting Ds ^{is much} larger ^{due to the} large ^number

of vacancies ; ^{then we could} pull ^{on u} somewhat, ^and anyhow ^{there is a} long way ^to go before _r. reaches Ti.

Finally, ^we might worry about relaxation right ^at

the interface, ^{where the} crystal ^structure breaks down.

There is no theory ^{available there}

^{but a minimum}

condition for such a relaxation would be that the time

an atom spends ^{on the last} crystalline plane ^be larger

Fig. ^11.

^A sketch of the magnetization pattern ^{near a} moving liquid solid interface (melting case).

than the precession period ^{in the} dipolar ^field. Irrespec-

tive of _any narrowing mechanism, ^{one should at least}

have

(where a ^{is the} crystal spacing). ^{For u}

^10- 3, ^this

condition is not met : interface relaxation should not be important.

One may also _worry about fluctuations of the

demagnetizing ^{field in} homogeneous systems, ^with solid and liquid interspersed ^{into a snow like structure.}

Such a situation certainly ^{occurs in} practice. ^These

fluctuations are certainly sufficient to kill T2. However, in a very large ^field Hext, they ^{should not hurt} Tl very

much, ^{as for} macroscopic droplets ^their frequency

spectrum ^{is not} adapted ^to the Larmor frequency ^w.

In conclusion, ^{it seems} that relaxation in the solid should not be the limiting ^{factor in our} problem. ^Of

course, it is hard to conclude, ^{as we} may have forgot-

ten an important ^effect. Still, ^one fact is clear : a large applied magnetic field Hext always helps ⁱⁿ making Tl long.

6. Possible experiments. ^{- The} simplest expe- riment one can think of is simply ^to polarize liquid 3He by ^adiabatic melting of the solid. It amounts to working

a Pomeranchuk cell in reverse in a magnetic ^{field. 3He}

is solidified under _pressure, brought ^{to very, low} temperature

say 5 mK - and placed ^{in a} large magnetic ^field

say H

^{80 kG. We} take it for

granted that the solid is initially ^{in thermal} equili-

brium - that might ^{in fact take} quite ^a long ^time.

The _pressure is then suddenly dropped ^{below the} melting ^curve (there ^{is no heat} exchange ^{because of the} Kapitza resistance). ^If melting ^is completed ^{on a}

time « Tl, ^the resulting liquid ^{retains the same} polarization ^as the initial solid, ² TF/3 ^T bigger ^than

its equilibrium polarization Meq ^{in the applied field.}

The magnetization ^will subsequently ^{relax to} Meq

in a time T,. One should first try ^{to observe that} relaxation. That can be done for instance by applying

a 900 NMR pulse, ^and watching ^the decay ^{of the} spin