Second sound in spin-polarized 3He4He solutions

1575

Physica 109 & 1lOB (1982) 1575-1582 North-Holland Publishing Company

SECOND SOUND IN SPIN-POLARIZED Dennis S. GREYWALL Bell Laboratories,

3He-4He SOLUTIONS

and Mikko A. PAALANEN

Murray Hill, New Jersey 07974, USA

The second sound velocity of several dilute 3He-4He mixtures has been measured down to a temperature of 10 mK and in a magnetic field of 93 kOe. We discuss these results and show that they are quantitatively consistent with a nuclear-spin polarization of the ‘He atoms which reaches a maximum of 36%.

1. Introduction The low-temperature properties of dilute 3He in 4He mixtures can be altered appreciably by the application of large magnetic fields. The effect of the field is to produce a net nuclear spin polarization of the 3He atoms which dominate the thermodynamic and transport properties of the mixture. A detailed theoretical analysis of this system, based only on general quantum mechanical considerations and which applies in the limit of very low-temperatures, has recently been put forth by Bashkin and Meyerovich [l]. The most dramatic effect that they predict for full polarization of the ‘He spins is an increase in the 3He quasiparticle mean free path by several orders of magnitude, with the mean free paths becoming as large as tens of centimeters. This would correspond to tremendous changes in, for example, the viscosity and the thermal conductivity. The effect is a direct consequence of the very large difference in the S-wave and P-wave cross sections. If the spins are scattering antiparallel, a pair of 3He atoms will interact via S-wave scattering, for which there is a large cross section; for the fully polarized system the spins are necessarily parallel and the interaction must be of the P-wave type, in order that the total wave function be antisymmetric. However, it is not only the transport properties that are modified by a magnetic field; the thermodynamic and hydrodynamic properties will also be

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@ 1982 North-Holland

altered, to changes in systems. It probed by tion [2].

a much smaller extent, due to the the Fermi momenta of the two-spin was mainly this latter effect that was our study of second sound propaga-

2. Experimental details The measurements were performed using a technique (2 kHz 5 f 5 7 kHz) on resonance mixtures with nominal 3He molar concentrations X of 0.001, 0.003, and 0.010 in zero field and in a magnetic field H of 93 kOe. The temperature range of the measurements was lo-100mK. Other details, including a brief description of the apparatus, are given in ref. 2.

3. Results and discussion 3.1. Second sound velocity, H = 0 The thermodynamic and hydrodynamic properties of dilute 3He-4He mixtures can be determined quite accurately if, as Landau and Pomeranchuk [3] proposed, one assumes that the 3He quasiparticles constitute an independent excitation system which exists in addition to the phonons and rotons. At low temperatures (Ts lOOmK), however, the phonon and roton contributions are extremely small and one needs to

1576

D.S. Greywall

be concerned The

mainly

excitation

of these

but the deviations

for momenta

greater

than

about

is about

100 mK.

solutions

‘“r---

quasiparticles significant

roughly

only

0.6 A-‘.

We

of 0.01, the Fermi

0.2 A-’ and the Fermi energy

Working

less than 0.01 100mK it can

/ Second sound in 3He-4He

from the free particle

become

note that for a 3He concentration momentum

Paalanen

with the 3He quasiparticles.

spectrum

has been shown [4] to depart form,

and M.A.

with

concentrations

of

and at temperatures less than then be assumed, to be good

approximation,

that

the

spectrum

is

purely

quadratic,

i.e. that the 3He quasiparticles

make up

a nearly

ideal

with

Fermi

gas of particles

an

effective mass m T. For small X and T, Khalatnikov’s [5] general expression for the second sound velocity in 3He4He mixtures can be shown to reduce to a very simple equation which takes the limiting forms:

,

where

R is the gas constant

mass.

These

(1) MZ is the molar

are just the asymptotic

for the first sound So,

and

second

sound

velocity

mixtures,

experiment,

under

conditions

of our

first sound

in the gas of 3He quasiparticles.

crepancies, are a clear the

3He

of

an

ideal

rn: = 2.341m3

Fig.

to

second

corresponds

Fermi gas computed for all X. The dis-

cannot

I

60

I )O

be

Fig. 1. Velocity of second sound as a function of temperature for three sample concentrations. The squares are data from ref. 4. The solid curves were computed under the assumption that the ‘He quasiparticles constitute an ideal Fermi gas.

the

which increases with increasing X, indication that interactions between quasiparticles

I

60

gas.

Fig. 1 compares our zero-field second sound velocities measured for three different 3He concentrations with the corresponding first sound velocities assuming

,

40

expressions

of an ideal Fermi

in the

I

20

T(M)

T -=sTF , TsTF,

,

:RT,/M:

2

‘2= 1 $RTIM;

0

completely

neglected. Indeed, without interactions a second sound wave could not exist. Using a concentration-dependent effective mass the calculated curves can be made to nearly coincide with the data. In other words, the effect of interactions can be considered to cause only a renormalization of mf. A very good estimate of the small relative effects of applying a magnetic field should therefore be possible using the perfect gas assumption.

2 shows sound

the

measured

velocity

resulting

plication of a 93 kOe field. will discuss these data. 3.2.

increase from

in the the

ap-

In the following

we

General comments, T = 0, Hf 0.

The zero-temperature density-of-states diagram from the 3He atoms is shown in fig. 3 for the case of non-zero magnetic field. The difference

in

the

total

energy

of

equivalent

momentum states is equal to 2pH. p is the magnetic moment of a 3He atom. Clearly, if the magnetic field is large enough all of the spins will be oriented parallel to the field. The condition for this to occur is 2/3H 2 ~r,~utied = 22’3e~.~=~ which can be rewritten

PH ---&

2

2-1’3

=

0.79 .

(2)

D.S. Greywall

and M.A.

i

Paalanen

/ Second sound in ‘He-4He

however,

I

177

so1ution.s

can be polarized

56 kOe.

But

note

with

a field of only

TF for this

that

6;:,:1:,

very

dilute

sample is only 6mK. This means that to achieve a high polarization, very low temperatures must

be achieved

(section

3.4).

For smaller fields (and at T = 0) the fraction spins parallel to H is determined by

N

of

,”

0.003

0.0 I

(3)

1

0

which results from equating the expressions the chemical potentials of the up and down

3 I

‘**. 00

0 c t-

0

. .

O.Olo”

0 0 .

systems.

0 o

. 1

-~

in

I

0.5

I.0

1.5

fig.

The solution 4 as

PHI&,,.

The

responding

T’ TF,O Fig. 2. Relative change in the second sound velocity resulting from the application of a 93 kOe magnetic field. The solid curves were determined numerically with the use of eq. (6) and correspond to ~H/EF,o = 0.285, 0.135, and 0.061 for X = 0.001, 0.003, and 0.010, respectively.

Using the relations TF,()= 2.6X213K and @Hike = 7.8 X IO-‘H K . kOe-‘, eq. (2) implies that H must be greater than 2.6 x 104X2’3kOe. Therefore to completely polarize a 1% sample (at T = 0) requires a field of 1200 kOe! A 100 ppm sample,

applied tion

to this equation

a function scale

of the

at the

concentration field

is 100 kOe.

experiments

top

of the

must

be done

is plotted

reduced gives

field

the

mixture

Clearly

for spin

corif the

high-polarizaon very

dilute

samples. 3.3. Sound velocity of an ideal Fermi gas, T = 0, H# 0. The general expression for the adiabatic sound velocity of a one-component system is c* = (aP/dp)s.

For

our

two-component

X (ppm)

system

(up

H = IOOkOe

IF

TOTAL ENERGY

200 I

I

I

I

0.9 -

0.70.8C \

FERMI LEVEL

-J

I

0

0.2

(

1

I

0.4

/ 0.6

/

I 0.8

I I.0

PH “F.0

Fig. 3. Density of states diagram subjected to a magnetic field.

for

an ideal

Fermi

gas

Fig. 4. Fraction of spins parallel versus the reduced field.

to the field at T ==0 plotted

1578

D.S. Greywall

and M.A.

Paalanen

spins, down spins) another contraint has to be specified. There are two cases to consider: (i) the polarization remains constant, and (ii) the chemical potentials of the two-spin systems remain in equilibrium. That is to say, either the two-spin systems are nearly independent and take a long time to reach thermal equilibrium, or, in the opposite extreme, they come into equilibrium very rapidly compared to the frequency of the sound wave. Case i. The pressure of an ideal gas is given by

I Second sound in 3He-4He

is also the thermodynamic p=-

av> (au

s,ng+ in

(4)

3v’

where nk is the kinetic part of the internal energy per particle and v is the volume per particle. This expression is a direct consequence of using a purely quadratic energy spectrum and so is valid even if the Fermi gas is degenerate. There

5r

identity:

.

Because the polarization is magnetic contribution to the unchanged. Eq. (5) can then replaced by uk. From eqs. follows quite directly that -02

C3P

held constant, the total energy is also be rewritten with u (4) and (5) it then

aP

&= (ap> =m:(av> s,“3t,n

p=2uk

so1ution.s

S,“3 + in3

=10x 9 m3*'

(6)

This relation is valid at all temperatures and for all polarizations. The solid curves in fig. 1 were determined using this relation. The results for H = 0 are also shown in fig. 5, but in reduced form. Stoner’s [6] values for the internal energy were used. Note that the vertical axis also corresponds to the reduced pressure of the gas (i.e. osmotic pressure of the mixture). At T = 0, HZ 0, eq. (6) can also be evaluated quite easily. The quantity uk is then giVetI by

4-

Eliminating the chemical potential p from eq. (7) using the relation 3-

-_ 2-

(8)

it follows-that

I-

/ / / / OL 0

/ 0.5

I 15

I

IO T’TF.

o

Fig. 5. Sound velocity determined by eq. (6) or the pressure determined by eq. (4) plotted as a function of temperature. H = 0.

Case ii. We now consider the case in which the two-spin systems rapidly come to equilibrium. The velocity we now wish to compute is

(10)

D.S.

If V is the

total

Greywall and M.A. Paalanen

volume

and

N3 is the

by

The

At

last equality

is via a Maxwell

the expression

relation.

solutions

only of n3 = N3JV and so

for the velocity

becomes 1.0

0

0.2

(12) Finally,

0.6

0.4

0.6

3

PH/EF,o

Fig. 6. Sound velocities determined by eq. (9) (solid curve) and by eq. (1.5) (dashed curve) as a function of reduced field. T = 0.

using

(F + PH) K (n3 f )“’

(13)

Anderson

et al. [7]; we find

W, = 2 x 10L’X2’“T2

and

(14)

Under 0.001, that

we obtain

This result

is identical

and Meyerovich The

sound

to that found

(15)

by Bashkin

[ 11. velocities

determined

by eqs.

(9)

and (15) are plotted

in fig. 6 as solid and dashed

curves,

Since the approach

respectively.

towards

thermal equilibrium is governed by the spin diffusion equation, these two velocities are analogous to adiabatic and isothermal propagain

the

an

ordinary

material

where

it is the

thermal diffusion which is important. The characteristic frequency separating the high and low frequency regimes is then given by w, = 7,-’ = c*ID,. D, is the spin diffusion constant. The solid curve in fig. 6 then corresponds to the low frequency regime wr, 4 1 and the dashed curve to WT, 9 1 (but w’T~,,,,4 1). The actual value of ws can be estimated using the spin diffusion data of

conditions

06) of our

T > 0.010 K) o, > 2 our

experiment

are well in the or,+

1 regime.

explains why we observed field-induced in u2 considerably larger than predicted 3.4. Sound

Tf 0, Hf

velocity

(X >

x

10’s_’ which means performed at low kHz

measurements

frequencies

(G!zi)I= 2[(2!$)“~+ (&$)1’3]1.

tion

1570

X (ppm)IF H=lOOkOe

total

number of “He atoms, the density is given N,m :lV, and so eq. (10) can be written

T = 0, p is a function

/ Second sound in ‘He-“He

at

constant

This

increases in ref. 1.

polarization,

0.

In order to make a direct comparison with the experimental data, the sound velocity determined by eq. (6) was computed at finite temperature. We list the equations form the numerical calculations: quasiparticle

needed

energy

ii=k* e=2mj*; Fermi-Dirac

to per-

(17) function

fI = (1 + exp[(s + /3H - p))lkeT]}-’ .

m9

1580

D.S. Greywall

and M.A.

Paalanen

number density

09)

internal energy kc

I

ut=ypH+&

1

-1

0

af ,k2dk.

(20)

solutions

reduced temperature for several values of the reduced field. The degree of polarization decreases with increasing temperature due to the increasing amount of thermal disorder. The dashed curves in the figure correspond to fixed values of H/T. Moving along these curves in the temperature means a sense of increasing decreasing sample concentration. These curves should tend toward limiting values of e3 t /n3 given by

1

The procedure was to determine by iteration the chemical potential which gave the correct number of particles for given values of H and T and then to evaluate the energy using this value of P. The cut off, k,, in the integration was set at 5kp We show first, fig. 7, the results for the fraction of spins in the up state plotted as a function of I

I Second sound in 3He-4He

I

I

T'~F,o

Fig. 7. Fraction of spins parallel to the field plotted versus the reduced temperature for several values of the reduced field.

F=i[tanh(&)+l],

(21)

which is easily derived [8] assuming Boltzmann statistics. The curve for H/T = 10 kOe 3mK_’ corresponds to our experimental limits of 10 mK and 100 kOe and shows that independent of how dilute we might have made our samples, we would never have been able to force more than 80% of the spins to align with the field. To achieve a higher polarization would require higher fields and/or lower temperatures. Fig. 8 is a very similar type of graph, but now the relative change in the square of the sound velocity is plotted on the vertical axis. We note that since the quantity which is actually computed is the relative change in uk, the vertical axis also gives the relative change in the pressure of the gas. This figure shows that although a zero temperature calculation may indicate a large change in the velocity or pressure, the quantity actually measured at finite temperature will be considerably smaller. The solid curves drawn in fig. 2 and which provide an excellent description of the data are equivalent to those shown in fig. 8. The corresponding values of the reduced field are 0.285, 0.135 and 0.061 for X = 0.001, 0.003 and 0.010, respectively. The fact that the experimental results can be explained quantitatively in this straightforward manner is a clear confirmation that we actually achieved a state (for X = 0.001, T = 0.010 K) in which 68% of the spins were oriented parallel to the field.

D.S. Greywall

and M.A.

0.6 I

Paalanen

i Secorad sound in ‘He-4He

solutions

1581

1

H/T:

SOkOe.mK-

01

0

/ 05

/ IO

I

15

2

T'TF.o

Fig, Al.

L0

1 0.5

I 1.0

/ 1.5

cv = (g),

Specific

heat of an ideal

- (~)&)P/($)T.

Fermi

gas. H = 0

(A3)

T'T~.o

Fig. 8. Relative change in the square of the sound velocity plotted versus the reduced temperature for several values of the reduced field.

which is in a form that can be evaluated numerically. The results for W = 0 shown in fig. Al are in excellent agreement with Stoner’s

Appendix

We include here our numerical results for the specific heat of an ideal Fermi gas in a magnetic field. Although not relevant for the present work, these results may be of value for future experiments. We have u = u(7’,~) and p = E.L[T,its] (see eqs. (17)-(20)), therefore

cv=

(g),,=($), +($), (g),,.

(Al)

Using the identity

642) T'TF,o

the expression for the specific-heat becomes

Fig. A2. Relative changes in the specific heat of an ideal Fermi gas induced by a magnetic field.

1582

D.S. Greywall

and M.A. Paalanen

values. The relative changes in the specific heat are plotted in fig. A2 for several values of the reduced field.

References [l] E.P.

Bashkin

(1981) 1.

and

A.E.

Meyerovich,

Advan.

Phys.

30

I Second sound in 3He-4He

solutions

[2] D.S. Greywall and M.A. Paalanen, Phys. Rev. Lett. 46 (1981) 1292. [3] L.D. Landau and I. Pomeranchuk, Dokl. Akad. Nauk SSSR 59 (1948) 669. [4] D.S. Greywall, Phys. Rev. B20 (1979) 2643. [5] I.M. Khalatnikov, Introduction to the Theory of Superfluidity (Benjamin, New York, 1965). [6] E.C. Stoner, Phils. Mag. 25 (1938) 899. [7] A.C. Anderson, D.O. Edwards, W.R. Roach, R.E. Sarwinski and J.C. Wheatley, Phys. Rev. Lett. 17 (1966) 367. [8] C. Kittel, Introduction to Solid State Physics (Wiley and Sons, New York, 1967).