Magnetic properties of solid 3He: What do we know and what do we learn?

1796

Physica 109 & IlOB (1982) 179fS1804 North-Holland Publishing Company

MAGNETIC PROPERTIES OF SOLID 3He: WHAT DO WE KNOW AND WHAT DO WE LEARN?

M.C. CROSS Bell Laboratories, Murray Hill, New Jersey 07974, USA

A review is given of recent advances in understanding the magnetic properties of solid ‘He and their relationship to quantum exchange processes, Three topics are emphasized: the informal from recent nuclear magnetic resonance experiments; a simple model Hamiltonian that qualitatively accounts for much of the data; and some suggestions for future experimental and theoretical work.

1. Introduction

ith site (sf = f h’). At low temperatures mK) all other

One

of the

crystals,

intriguing

features

such as the solid heliums,

on nearby

lattice

of quantum is that atoms

sites may coherently

exchange

positions leading to ground state energiesand other properties-that depend on the symmetry of the spatial

wavefunction

For 4He, a spin zero boson, tinguishable, and directly observable

under

permutations.

the atoms

are indis-

the exchange process is not - although dramatic changes

degrees

of freedom

(tens of

are frozen

out,

and the weak exchange process dominates the thermodynamics. Note that He, has nothing to do with the magnetic moment of the nuclei: the spins enter as a convenient way of labelling the symmetry of the spatial wavefunction. The flipping of the spins at neighboring sites (TJ goes to Jf ) is accomplished places,

by the two atoms changing

not by dynamic

coupling

to the magnetic

moment.

in properties would result if solid 4He became superfluid due to this process. On the other

apparently

hand, 3He has the additional degree of freedom of the nuclear spin, which allows some labelling

“He showed antiferromagnetic trends consistent with a value of J - 2 mK, not too far away from

of the

the theoretical estimates, ness of these calculations. As data accumulated, clear that the nearest

atoms

so that

become observable. under premutation

the

results

of exchange

In fact, since the symmetry of the spatial wavefunction is,

via the Pauli principle, the symmetry of the

uniquely determined by spin state, the energy

differences resulting from atomic exchange may be described by an effective spin Hamiltonian. For simple two-particle nearest neighbor exchange this Hamiltonian H,, = - 2J c

leads

s,

9

to the familiar

s, ,

iy

with J a positive

constant

Heisenberg

(1) and Si the spin at the

Work

in the

Hamiltonian

late

sixties

confirmed

was

too

and

early

the general

seventies

picture:

solid

considering

the crude-

however, neighbor

it became Heisenberg

simple.

The

qualitative

picture remains: 3He does indeed undergo phase transition at 1 mK to an antiferromagnetic

a

state; but quantitatively the behavior is definitely inconsistent with eq. (1). The great challenge in solid ‘He is to discover the true effective Hamiltonian replacing eq. (1) that governs the properties at low temperatures, and to understand its origin. In solid 3He the spins provide a good handle on the exchange process: a better under-

1797

M.C. Cross i Magnetic properties of solid ‘He

standing

in this system

should

systems where quantum important feature. In this

paper

since

review

by Landesman

In

my

LT

view

is still

15 (for

the

earlier

work

no

reliable or

exchange

rates,

although

qualitative

even

arguments

for

level.

to

reasonably

[2] suggest

results; least

understand a simple

qualitatively

the

model

the

the spin Hamiltonian

for

with

Curie-Weiss going

convincing

much

be imtopics:

that

is at

all the

data;

of information

along: included

state vacancies [3] or spin-polaron type [4] that have some qualitatively appeal-

ing features, but have not been seriously against all the experimental data.

tested

2.1. Measurements

above the phase transition

Well above the sition temperature,

antiferromagnetic (a.f.) tranbut where phonon contribu-

tions are not important, the thermodynamic properties can be expanded in powers of T-l. There are essentially two independent series that may be looked

S(T) -=log2-~-~...,

at: the entropy

s,

NKB

and the susceptibility

s2

or specific heat

and

involve

convincingly

terms,

and so are

One qualitative

piece

by the probably

a four-spin

measurements,

reli-

One might scopic strong

exchange

must

as spin

be

relaxation

of the micro-

in the a.f. state

constraints

on

the

experiments

scopic

state

must

are be

would

until

inferred

effective

neutron

performed

measurements.

provide

possible

Unfortunately,

tering

are in

of the a.f. phase

hope that knowledge

ordering

Hamiltonian.

such

further information, but and difficult to understand

2.2. Measurements

macroscopic 2. What is known from experiment?

known.

measured

terms

[7].

Other

detail.

ground theories

temperatures,

at least

selection

in the past few

be

order

is provided

processes, give model-dependent

from all the work done

by Guyer

able result [6] B > 0. Such a result is impossible to obtain with a two- or three-spin exchange

and useful things to do to push the understanding we might have further. This is obviously a years, and many interesting papers that do not directly bear on these subjects will be overlooked. In particular I will not discuss theories more exotic than multiple exchange, such as

may

the effect of further

less reliably

of

(given by S,, and 0 the

well. The higher

to lower

eliminating

by moments

(see the review

constant)

reasonably

of various

experimental

Hamiltonian

consistent

is

that few parti-

recent

S,, S2, 8 and B are given

[.5]). The first corrections

There

calculation ratios

cle multiple exchange processes should portant. I will therefore discuss three work

where

in understanding

microscopic

rates

see

1.

T-O++

X

ad-

proceedings).

to the phenomenological

exchange

is an

on the

[l] in those

increase

to all

of atoms

I will concentrate

vances

restricted

be relevant

tunnelling

scat-

the

micro-

indirectly

from

It is important

at

this stage to be clear how much is known and how much simply guessed about the ordering. To date, the only information on the microscopic ordering that is independent of assumptions of a model Hamiltonian comes from nuclear magnetic resonance (NMR) experiments by Adams

et al. [8] and Osheroff

measures

the

resonant

frequency

et al. [9]. NMR of

the

spin

system precessing in a magnetic field. Above the a.f. transition this occurs at the Larmor frequency yH. The exciting result observed was a large frequency shift (characterized by a frequency -800 kHz) in the a.f. phase. To understand the importance of this result one must realize that NMR probes the energy

179x

M.C. Cross I Magnetic properties

dependence

of spatially

uniform

spin

of solid ‘He

the spin and orientation

rotations,

and so the exchange Hamiltonian produce shifts of the resonance

cannot by itself away from the

energy:

Larmor

leaves,

E= E&/)+;y’Sx-‘S-S-H,

frequency.

energy

leading

the nuclear

In 3He this

to shifts,

magnetic

as the

the anisotropic

dipole

part

9 minimizing

the total

(5)

of

interaction

with x the susceptibility

tensor.

The results

of a

comparison of such solutions with the data (fig. 1) leads to the following conclusions (for more I

i-j

details (3)

modes This

is

a

much

0.1 ~K/atom.

To produce

must

lead

energy the large

of

order

shifts obser-

must be first order in HD: the without perturbations due to

ved, the anisotropy a.f. state, calculated H,,

smaller

to a non-zero

expectation

value

(HD) = ED. The shift is then of order J”2(ED)1’2 -

see Fisher

(i) Experiment per

suggests

crystal

dependence three

and Cross [12]).

modes

the

domain,

presence

with

to the

spectra,

with

complicated

a simple

in contrast

low field to high field behavior

of two field to the

cross-over suggested

from by eq.

(4) for a general a.f. state. This suggests that the NMR motion may be characterized by a single axis d (two variables) and that ED and x must be

105Hz, as observed. Similarly, lattice strain not by itself lead to these large effects.

canThis

uniaxial.

immediately

must

x = x0(1 + a),

(6)

E,=f&A.&

(7)

requires

that

the

a.f.

state

break the original cubic symmetry and rules out the classic a.f. states of a b.c.c. lattice [lo] (NAF or type I, SCAF]l and SCAFl give no shift, as may arguments alone.

be

or type II) which seen

by

symmetry

the ordering Fisher detail the complete

(w, H) spectrum

on single

Osheroff not

(see ref. 9). The approach

require

sublattice

any

structure,

o, H 4 J to write of motion

a priori but

rank tensor

A, ?A2 Z- h3) reflecting

To learn more about and I analyzed in some measured

where A is a second

crystals

we used did

assumption instead

quasihydrodynamic

by

used

about

the

the

fact

equations

the sublattice

the

arrangement,

(with eigenvalues

spatial

ceptibility anisotropy. (ii) The zero frequency requires A to have planar

mode

at zero

of

symmetry,

field

i.e. A, = A2

so that ED = ih (i - d)” + constant

.

(8)

[ll]: with A = Al - A3 > 0 and i a vector

S=

symmetry

and 6 gives the sus-

(iii) Solution

ySxH+aEDla7),

(4)

of the equations

in the lattice.

of motion

for S

and d [9] then gives frequencies

7j = yH - y2x-‘S, for the spin density S and the variable 3 specifying changes in orientation of the spin ordering in terms of small rotations about the three coordinate axes, with ED(q), the dependence of the dipole energy on these rotations, leading to an additional torque on the total spin. These equations are to be solved for small oscillations about

(9) with o,_= yH and fii= y*xo’A, provided the susceptibility anisotropy 6 < 0. (The opposite case S > 0 leads to quite different high field behavior.) Eq. (9) contains two parameters n,(T), and then for each pair of spectra 8, the

M.C.

angle

i and H, which

between

Cross 1 Magnetic

will be different

for each domain. It then fits very well all measured spectra (e.g. fig. 1). It should be pointed out that the simple form of the spectra flop effects for any orientation from the special

result

(i, perpendicular energy

(e.g. no spin

of the field) arises

that a single orientation i and H, minimizes

to both

learned

from

the

to

reconstruct

arrangement! can

only

hand. quite which

a

Clearly,

any

be an educated

we are fortunate a lot about must

detailed such

energy to the simple much that the energy

(ii) The

group

microscopic the

NMR

uniaxiality

in real

space

through

(i) requires

axes of the original

cubic structure,

and then i is

this axes. In fact we know that i must be a 4-fold axis since for each single crystal three pairs spectra were seen, fit by values of 8 satisfying

of

other

tells

us

3

C COS28i= 1 )

(l(J)

i=l

of the a.f. state,

to reduce

the

dipole

form eq. (8) but not is isotropic. The form

[13] to contain

i moves

to d as the index

the point symmetry of the spatial arrangement of spins to retain one 3-fold or one 4-fold symmetry

so of

the expression for dipole energy (H,), eq. (3) leads to the following conclusions. (i) The uniaxiality in spin space requires the spin point

normal

the

fit to the

On

the

the symmetry

be sufficient

plane

reconstruction

guess. that

I 700

metry of order 23 (and then 2 is this axis). This permits not only the obviously uniaxial structures with (s,) all up or down along d, but others, such as helicoidal states in which (s,) spirals in the the lattice.

data is the form of the dipole energy, eq. (8) and the susceptibility, eq. (6). This is all we have to try

of solid ‘He

of

eq. (5) for all fields, and no reorientation

occurs. The information

properties

an axis of sym-

as expected each crystal

for the three possible domains in with i along [loo], [OlO] or [OlO].

Note that these conclusions follow from the observation that (Hu) is the contraction of two second rank tensors, one in real space and one in spin space. Low order symmetry axes are then sufficient to guarantee complete these tensors about the axes. Any further spin tween

state

reconstruction

requires

the

experimental expectation

value Y’x;‘A,

in the ground made

difficult,

state

of of

of

of the microscopic

quantitative

extrapolation

invariance

comparison

be-

to

T = 0,

the

and

the

with A calculated

of

theoretical from (Hu)

(eq. (3)). This comparison

since

the

correlation

is

function

required (sp$)/(i h)2 cannot be quantitatively calculated independently of a model spin Hamiltonian. For the classical Neel description of each state zero

approximately taking

/ ’

OO

this may be done, point corrections

but in a spin l/2 system may be significant. We

investigated

this

question

by

-I

_~ LAxI_.

A

500

LARMOR

~~Y?k---FREQUENCY

1500

[kHz]

Fig. 1. NMR spectra for a single crystal of solid ‘He [9] at 0.49mK. The solid lines are theoretical curves generated from eq. (9) with &/2~ = 777.7 kHz and values of co& chosen to obtain a best fit to the data.

(sss,p>= (sS)(sf) ,

(11)

where for (s?) we take the value resulting from the classical description, reduced by a factor Cc,to take into account the zero point fluctuations. The

M.C. Cross i Magnetic properties of solid ‘He

1800

calculation problem

of (Hn) of dipole

to the following (i) The

then

reduces

to a standard

sums. Studying

these sums led

conclusions.

helicoidal

states

based

on

a

[loo]

wavevector (i.e. (s,) = i A4 Re[(l, i, 0) exp(ik * r;)] with k along [ 1001) g ive a negative A and so are inconsistent with the data. (ii) We

suggested

consisting

of

an

“uudd”

(fig.

ferromagnetically

planes arranged in the sequence down (with respect to d) that consistent

(100)

= 2.42 )

with p the number (iii) Longer

to eq.

m t m J of such planes

larger

a “magnetic temperature” of 5.83 mK [15]). As a first guess for +/Jwe might use the value 0.85, which is typical of estimates for conventional

values

(12) are

of h (values

3.0 (m = 3)

3.3

a.f. states

prediction &/2~the experimental bring

the

two

[16]. This

leads

to the

= 1050 kHz, considerably above value of 825 kHz. In fact, to results

into

agreement

requires

$ - 0.67. a worryingly large renormalization factor. Recently, Iwahashi and Masuda [ 171 have a spin wave calculation

phase

using

a

below.

model

Their

lowest

for the uudd

Hamiltonian

order

described

calculation

gest large renormalizations motion, giving a Cc,= (s:)/fh nitude. doubt

density.

to successively

corresponding

(12)

sequences

[ 141 (giving

performed

up-up-downis qualitatively

with the data, giving

h/3pZ(yh/2)‘@

lead

2) state

aligned

Cross

does sug-

due to zero point of about this mag-

However, such a large effect brings into both the spin wave approximation

scheme,

and the simple

factorization

procedure,

eq. (11). Indeed, at next order these authors find a value of I/J considerably closer to unity (+ -

(m = 4) converging

0.80). This quantitative

Of course

fore be considered an open question. Notice that the longer sequences make the agreement worse, although perhaps not significantly so (e.g. 3 t 3 J

to a limit for large m of 4.2). m = 1 is the NAF state with cubic

symmetry and gives A = 0. Using the result eq. (12) for the uudd structure leads to the estimate 0,,/27r = 1230 x $ kHz ,

(13)

where

we have used the extrapolation to zero of the magnetic susceptibility temperature measured

at low temperatures

by Osheroff

and

comparison

there-

worse by 12%). We can also not rule out updown states with incommensurate wavevectors along [loo]. although such states. with some (s,) very small, seem unlikely at low temperatures. or other

more

found.

exotic

To date,

states

that we have

no other

structures

not even consistent

with the NMR date have been discovered. Roger and Delrieu [18] have suggested an additional reduction dinates

in A due

to fluctuations

r, in the dipole

motions) bringing better agreement

of the coor-

sum (zero

the prediction with experiment.

this is a large correction

Fig. 2. The uudd structure. The spin direction i may point in any direction in the plane normal to i to minimize the dipole energy.

must

point for uudd I doubt

in the uudd

state,

lattice into that since

uncorrelated, spherically symmetric fluctuations in the nuclear positions lead to no change in the dipole sums, and such a description should be good in the uudd state where nearest neighbors. whose motion might be significantly correlated, do not contribute to the dipole sums. Other measurements of the a.f. phase also provide useful information. Measurements of entropy [19], susceptibility [14] and NMR

M.C. Cross / Magnetic properties of solid 3He

resonant frequency [9] temperature dependences linear

long wavelength

a.f. state. cription

provide

power from

spin wave spectrum

measurements

by an order

should posed

These

show the expected

the barrier

of the phase

diagram

are

of McMahan hinderance serious

interactions.

Firstly,

a surprisingly

note that the small corner

of

the H-T plane. From the Curie-Weiss constant might expect a transition temperature of

one

2-3 mK, and a corresponding of

1 mK

to suppose

only very few exchange

estimates of exchange constants. What has become clear in recent years, following the work

particle

values

exchange

processes have comparable rates. As a result of point (i), there are no reliable

details

30 kG.

makes

be reasonable

for competing

of order

in h gives a hundred-

in J. This of course

and

discussed by Osheroff in these proceedings [20]. One notable feature is the considerable evidence occupies

- a 50% change

fold change

to the size of

pro-

2.3. The low temperature phase diagram

a.f. phase

is very sensitive

phase,

the des-

Hamiltonian.

The known

result

of any

low temperature test

in an

(ii) The

rates very hard to calculate. On the other hand, for phenomenological fitting to the data it may

confirm

a quantitative

law the

1801

Compare and

upper these

4.4 kG.

critical

trated

atoms

exchange,

ring exchange by a formula

is

more

for multiple

than

processes.

of Delrieu

much This

is illus-

et al. [2]:

h - 27&lLIF,

(1%

field

with the actual

Furthermore,

the

characterizing the important physics of a variation calculation they propose of the exchange barrier

temperature

distance

magnetization.

question of whether there is any ordering verse to the field remains unanswered.

[21], is that the steric

nearby

for pairwise

high field phase shows ferromagnetic properties, e.g. a large (0.6 of saturation), roughly field and independent

and Wilkins of

The trans-

hard

(as opposed sphere

to the wavefunction) for a L measures the total

model.

moved

Here

by all atoms

n/S2 gives the barrier of atoms

exchanging

height,

in the process,

and

with n the number

and S a length

characteriz-

ing the free volume of each exchanging particle at the barrier configuration. Geometrical esti3. What is the exchange Hamiltonian? The value of an exchange be of the form

rate is expected

J - ~,Ie~h . where motion

to

(14)

wg is a typical phonon or zero-point frequency (the attempt frequency), and

the exponential tunnelling. The

factor depicts the barrier to very small (experimental) value

of J/coo-- 10e4 leads to two consequences: (i) The exchange process depends on the exponentially small tails of the wavefunctions, to which straightforward energy calculations are correspondingly insensitive. To reliably calculate J the actual exchange process must be understood and modelled.

mates of eq. (1.5) suggest [2] that a three or four particle exchange may be important because the by a large S, but larger d/nL is compensated higher number exchange less so, because VnL increases

further

Many

authors

but S changes have

little.

investigated

phenomeno-

logical exchange Hamiltonians including four-spin exchange, using high temperature

up to series,

and mean field calculations of the ordered phases (for pre LTlS references see ref. 1). With point (ii) in mind, in a series of papers considered various combinations

[22] Roger et al. of two exchange

processes. Amongst these they suggest two-parameter Hamiltonian

H = J,

3 F;

Plz - Kp

[23] that a

P1234

of4

z

gives a good description

of the data. Here

PIz3 is

a cyclic

three-spin

around

all rings

one

next

spin

cyclic

planar

nearest

permutation of two

neighbors.

favors parallel

of Plzu gives

spins around configuration direction,

and -l/4

quantum

around

From

general

(Note

classical

state

that

for

otherwise cularly

the physical large

represented. papers

exchange This

around

the

single

useful

process

states

for

interpretation mechanism

information

various

exchange

in terms seems

/ uudd

the

: PARAMAGNEl

ring

of a partiwill

done

their on

NAF

-

01 0

a multiple

consequences

[25] and so, although

vides

/

for all

two spin interactions,

has not been

CANTED

arrangement:

in analyzing

[24] the resulting

/j’

0355

the expectation

exchange Hamiltonian it is important to include in the spin representation of the permutation operators

I

of the

(i.e. unfavorable)

best

/’

2 -01

201

with the sign and so three-

for the sequence

(the ground

-1).

J,

Kp =

the ring parallel, 0 for any other of up or down spins along a given

t -+ 1 +

gives

acting

alignment

whereas +l

and

PI234 is a four-

[24] it is known that of eq. (16) K,, J, ~0,

exchange

-

acting

neighbors

and

rings of nearest

spins (ferromagnetism),

ring

nearest

neighbor,

operator

particle value

operator

permutation

arguments convention

the

Cross I Magnetic properties of solid ‘He

MC.

1802

be

analysis

of a dominant

pro-

ground

Hamiltonians,

I

T mK

Fig. 3. Phase diagram of Roger et al. [23] for the model Hamiltonian, eq. (16). The solid lines denote first order transitions, the dashed lines second order transitions.

mis-

in several

possible

I

the

exchange

misguided.)

The resulting phase diagram (fig. 3) calculated by mean field theory and taking J, = -0.1 mK, Kp = -0.355 mK, also consistent with high tem-

there

is in error:

factor

of three

in fact the predicted too

large

value

is a

[12]). It is interesting

that the uudd states arises through the competition between P,2.1 promoting ferromagnetic 3-rings, and P1234which discourages netic 4-rings. Pairs of ferromagnetic the

best

compromise

used-a rather simple complicated proposed

ferromagplanes are

for the

parameter

physical phase.

explanation

values for a

perature series data, is remarkably similar to experiment, except for some qualitative features around

1 mK and quantitative

as the much

too large value

discrepancies of the upper

such

4. What is in the future?

critical On the theoretical

side certainly

a more quan-

field of the a.f. phase, both of which may be artifacts of the mean field theory. In particular

titative

the a.f. phase is uudd consistent with the symmetry required by the NMR, and the high field phase is a canted pseudoferromagnetic NAF phase (i.e. it would have finite magnetization in zero field) with a magnetization consistent with experiment [ 141. The reader is referred to ref. 23 for a full comparison to the data. (Note, however, that the agreement with the experimental coefficient of the T3 specific heat quoted

model, eq. (16), and experiment is needed-according to the discussion above the model would be less convincing if discrepancies with experiment could only be patched up by adding more exchange processes with comparable rates. Conventional spin wave calculations for the a.f. state at low temperatures [12, 171 suggest large corrections to the present mean field results. The parameters K,, J, used are tantalizingly close to a

comparison

between

predictions

of the

M.C. Cross / Magnetic properties

phase

boundary

(K, = 45, in mean

field

theory

[23]) to another a.f. state, again without cubic symmetry. Associated with this phase boundary

of solid 3He

I803

may disappear at a critical point leaving merely a fictitious line where rapid paramagmetic ordering occurs

(Schottky

is a softening of the spin wave velocity transverse to f-to zero in the mean field approximation.

vestigating

this

In fact, for the parameters

establishing the nature

used the mode

soft that finite temperature to the

consistent rections

with

are very

antiferromagnets,

theory linear

and

cor[17]

The

from calculations

on

result

[17] corrections

spin wave

of

further

including

is to significantly

non-

raise the

soft spin wave velocity, in better agreement with experiments. Careful calculations to compare with experiment certainly seem worthwhile. Many experiments remain to be done. Here I will just suggest a few that may help to elucidate the

microscopic

structure

of the

test the model Hamiltonian. It is well known that exchange very sensitive

to the volume

and

likely

it seems

exchange of steric

that

the

states

or may

constants

are

(a In J/a In V-

20)

ratio

of different

processes, involving varying amounts effects, may also vary with volume.

Since the evidence supports a Hamiltonian of competing effects, application of pressure may lead to interesting results. For the model Hamiltonian,

eq. (16)

the,application

tip the balance between alternate a.f. state. Even

at melting

of pressure

K, and J, leading

pressure,

Direct

point

and this requires

One

in-

shown

Zero

have been

so that linear

is less accurate,

understanding.

large,

metry

experiment.

to the Neel state

to be larger than anticipated other

is so

may to the

tests

NMR

nature

of the

cubic

answered. ment

to distinguish

additional

broken

[14]

show

by the canted

test [26] would specific

heat

at

very

low

of the

transition

higher fields is an interesting question: may continue indefinitely as first order, seems unlikely; critical point

at

the line but this

it may become second order at a (requiring some spontaneously

broken symmetry in the low temperature in addition to the field broken symmetry);

phase or it

tem-

peratures, which decrease exponentially for a paramagnetic phase, but as T’ in the canted NAF phase which wavelengths, More

retains a gapless spin wave even in the field. that

give

mation

on the

experiments

a.f. phase

are

Specific

heats

in magnetic

temperatures since

may

the

fraction

also of

fields

give spin

at long

structural needed at

inforas well.

very

information wave

low here,

modes

that

remain gapless and so continue to contribute the T3 specific heat for kBT < pH is different

to for

various

l/3

possible

for helicoidal

structures

structures).

(e.g. l/2 for uudd, Knowledge

modes would give information sublattices, but such knowledge get a hold Perhaps,

on.

As usual,

experiments

on the number seems difficult

the plea

is made

for the first time

of the optic of to

for neutron

here. in many

coherent exchange

nature

un-

be the measure-

dence that the phase transition line from the disordered phase is probably first order between The

with

et al. [23], remains

review of the understanding of magnetic ties of solid 3He may end on an optimistic

6 kG.

the

to the field consistent

high field, low temperature phase is not known experimentally. Osheroff [14] has presented evi-

4 and

that

but the question

such as suggested

the

sym-

are also needed.

symmetry,

of Roger One

of

any

field phase

transverse

phase

in-

at rela-

-3 mK, but great care in

measurements

this symmetry,

scattering the

for

retains

of ordering NAF

be done

equilibrium is required of the anomaly.

in the high

phase

Experiments

may

tively high temperatures,

spin wave corrections

thermodynamics

anomaly).

question

years,

a

propernote: a

account of the manifestations may well be emerging. A

of

twoparameter model Hamiltonian fits many features. and furthermore there is sufficient data IO provide a quite severe test of any proposed model. Currently, optimism is based on rather poorly known parameters from high temperature series, a suggested a.f. state with a complicated microscopic ordering guessed from a single macro-

I804

scopic

M.C. Cross / Magnetic

measurement,

only qualitatively and much remains

and

the

understand

big

Hamiltonian

tested! Discrepancies remain, to be done to test this picture;

but at least it is a sensible remains

a model

theoretical

exchange

picture.

Even so, there

question:

on a microscopic

Can

we

level?

Acknowledgements The preparation of this review benefited from many discussions Fisher and D.D. Osheroff, much to any understanding

has greatly with D. S.

who have contributed of solid 3He that

1

have.

References [II A. Landesman, J. de Phys. 39 (1978) C6-1305. PI J.M. Delrieu and M. Roger, J. de Phys. 39 (1978) C6-123; J.M. Delrieu, M. Roger and J.H. Hetherington, J. de Phys. 41 (1978) C7-231 and preprint; M. Roger. Thesis, Universite de Paris&d (unpublished). f31 J.B. Sokoloff and A. Widom. J. Low Temp. Phys. 21 (1973) 463. [41 M. Heritier and P. Lederer, J. Phys. Lett. 3X (1977) L209; M. Papoular. Solid State Commun. 24 (1977) 113: J. Low Temp. Phys. 31 (1978) 595. [51 R.A. Guyer, J. Low Temp. Phys. 30 (1978) 1. Phys. Rev. Lett. 39 [61 T.C. Prewitt and J.M. Goodkind, (1977) 1283; M. Bernier and J.M. Delrieu, Phys. Lett. 6OA (1977) 1.56; D.M. Bakalyar, C.V. Britton. E.D. Adams and Y-C. Hwang, Phys. Lett. 64A (1977) 208.

properties

ofsolid “He

[7] M. Roger and J.M. Delrieu. Phys. Lett. 63A (1977) 309. [8] E.D. Adams, E.A. Schubert, G.E. Haas and D.M. Bakalyar. Phys. Rev. Lett. 44 (1980) 789. [9] D.D. Osheroff, M.C. Cross and D.S. Fisher, Phys. Rev. Lett. 44 (1980) 792. [IO] For descriptions of these phases see M. Roger, J.M. Delrieu and A. Landesman, Phys. Lett. 62A (1977) 449. [ll] B.I. Halperin and W.M. Saslow, Phys. Rev. B16 (1977) 21.54; I.E. Dzyaloshinskii and G.E. Volovick. Ann. Phys. (NY) 125 (1980) 67. [12] D.S. Fisher and M.C. Cross (unpublished). [13] If the state is invarient under (Rrpln]t) with r,,,, a spin then we refer to rotation and t a lattice translation, {R,,,,/O} as the spin point group. [14] D.D. Osheroff and M.C. Cross (unpublished); see D.D. Osheroff, Physica 109/l 1OB (1982) 1461. [15] This value is considerably above the value used in ref. 9. which may be deduced from the measurements near r, of T.C. Prewitt and J.M. Goodkind, Phys. Rev. Lett. 44 (1980) 169’). [16] P.W. Anderson. Phys. Rev. 86 (1952) 694; R. Kubo, Phys. Rev. X7 (1952) 568. 1171 K. Iwahashi and Y. Musuda, preprint, and Physica 108B (19X 1) X.53. [1X] M. Roger and J.M. Delrieu, J. Phys. 41 (1980) C7-241. [19] D.D. Osheroff and C. Yu, Phys. Lett. 77A (1980) 458. [20] D.D. Osheroff. Physica, this volume. [21] A.K. McMahan and J.W. Wilkins, Phys. Rev. Lett. 35 (lY7S) 367. [22] M. Roger, J.M. Delrieu and A. Landesman, Phys. Lett. 62A (1977) 449; M. Roger and J.M. Delrieu. Phys. Lett. 63A (1977) 309; M. Roger, J.M. Delrieu and J.H. Hetherington, J. Phys. Lett. 41 (1980) L139. 1231 M. Roger and J.M. Delrieu and J.H. Hetherington. Phys. Rev. Lett. 45 (1980) 137; and ref. 18. [24] D.J. Thouless, Proc. Phys. Sot. 86 (1%5) 893. [25] I. Okada and K. Ishikawa, Progr. Theor. Phys. 60, 11 (1978); K. Yashida, preprint. [26] D.S. Fisher, private communication.