Experiments on rotating superfluid 3He

22

Physica 126B ( 1 9 8 4 ) , _ - _ . N o r t h - l l o l l a n d Anlsterdam

EXPERIMENTS M.Krusius,

ON R O T A T I N G S U P E R F L U I D P.J.Hakonen

Low T e m p e r a t u r e Helsinki 02150

The The

from a series

~He s u p e r f l u i d s

implications

the quantized in the ligth

parameter

[

1.

of T e c h n o l o g y

Finland

Espoo,

the

and J . T . S i m o l a

Laboratory

University

results

on

3He

of

of transverse

ow N MR m e a s u r e m e n t s

in the r o t a t i n g

these

state

experiments

are reviewed.

on the s t r u c t u r e

or'

v o r t i c e s in the A and B phases are d i s c u s s e d of recent t h e o r e t i c a l work on the order

distribution

in the

vortex

lattice.

iNTRODUCTION

Introduetory

remarks

Customarily terized

by frictionless,

tier o f very

pantie!los

nature.

tat}on rices

The

by

in which

the

uniformity thereby

against years

the singular Of t h e

the first

become

amended

a

in

in this short formation we focus

a

the

simple

shall

of

o£

first

cores

veLoc,~ity,

examine

the

;

- -

[I] q

i hill;

[2,< (rc)iu/~,:~

'['I']IS

?,]hdaIflcnh;~!

also

expressed

been

!ow

performed It now h~n

<)nn~ e t e d around

a

:L

i}b

i,::, the)

!'

:;i i - r

£radient

, r

X[i<[ ; ] V,';:J [;~

: .~'.

icrL)?
p!',cp¢:vLy by

r'e~, bz;n closo(J

~= .'

'~ ~,i

,

+ ( i;

nonstraknt

ti~,~ [~st

i~' ;~r~ ,

the

-

",

)

bt]~qt'f]t/L,i

":~,,:t

that

(' [l~e/l [ a L i ( , n

th,:

loop

~

in ,d

~Z(rV. : ~ :!',impl},' ",]i pt r' [' l :')}d

vari J S!i
aiiY!p ] ~:

}lave

to

i)!:

V • dz,

=

,~ays.

~n

<,SU[)~2i~'flilJ~JS

%he 3He results,

structure vortex

'h

:

:

<),',

propor'tion:i!

[;iN',~[:i~t ,:['

)rd!a

J,

!C

,cOrXJeHis/it ~ Jn ,;f ~mgy ,

w<~veFun,:t /,:)r

the e x p e r i m e n t a l

oenbral

the

have

!zrlpl j t j d (

break

eonvent, iona i

in

introduttion phase

:'o

:~t,r i k i n g

review

about brief

to

vortices

on the vortices

question

accommodates

During

the

on the most

t,[:~

in rotation.

quantized number

t [uid

~.::

topological

flow.

that

of

.h~-

SI]~)PPIiHi(]

mo

vortex

measurements

('.lear'

~hi:

b/

correlated

:up
the

3He s u p e r f l u i d ~

concepts

:

dlt. ioe of q~l~iql izt,J "+~:~'-

rotational

the

After

a

hu'a

is

irret.{tt(:,n~]

Ls

superfluid

remove

st¢~

highly

whieh

forming

and

on

]I

th~ suplrtiuid

namely

!h~

it,

order'

rctt£tiOP~

aboub

t¢, WiJ[h

{~ , ;()[T1mOdl] t ,

,=ll ! n ~ L l i , ~ l

an a x i s

~ the

i'

J

hrlmogeT-nt< 4::

!ri~i>rr;cr ? ~;[r' I~ p o f

;,uperi'[

p,~rt i ,

.d

of the vortices.

to

the

of

superfluid

core

pr'operhie~, 8U,

structum~

:;.i

)t

s

w<

m 14r"

~;'.~

bet'om~:5

qua~'liizcd

to

dependent

flow

:

qHant!z;~<

v
x

~.it{

, L

siRguiar

vortices

the continuous

in ~}{e-B and

vortex

textures

chert procued

in 3He-A.

formed, [

2.

Vortices The order

presented

in r o t a t i n g parameter

in terms

superfluid

of s u p e r f l u i d

of a complex

4He

order

ORe

scalar

is

re-

fun(t ion

i[i

[ ~ [ 2• )

the

~(?)

=

,,/+01

e

1¢(~,:

0378-4363/84/$03.00 © Elsevier Science Publishers (North-Holland Physics Publishing Division)

~o

such

of in

f.l".at

,rod

::or~ t h c

t,he

s~ngui:~r

i r l t . ~ r p t r t. i c l e

,)I' t } i ~

around , irculat,

, cr,,

wt i

~(~

h

Lc , : t in

: p}i~ i n g .

t h< i m

i:: ,

d,msity

: r',~dius

:_:oh~r(:ii<:e l < : n z t h

superflow units

~

::sh,p e r f h J i d

wit?in

su[)erfluid

O irculat ion him 4

B.V,

the

vanJsh~'b

eomparab/e

quart}zeal

f'Je[d

<,on

"~[ie

/'it<

i:-

qu,arttu:

M. Krusius et al. / Experiments on rotating superfluid 3He

1.5

,

,

,

,

23

i QR

o

/

Ps 1.0

I/

~n

/I

0.5

/I 0.[}

'

0

'

'

10

5

r/a

0

R0 R

Fig.

I.

Superfluid

density p s/p

and

flow

velocity Vs= h/m~r as a function of the distance r

from

the

superfluid length

axis of the simple phase vortex in 4He.

The

characteristic

is

the

interparticle distance in

(a)

scale

of

Fig. in

2. a

Schematic diagram of the flow velocity

rotating

radius

cylindrical

R 0 in

the

container

presence

of

a

with

lattice

a of

quantized vortex lines.

superfluid 4He. t

K : ~ ~ • d~ = n ~ , where n=0,±I,±2 m4 J s In

the

bulk

particular rotates

liquid the singly quantized state

n=1

predominates since the kinetic energy ~PsVs I 2 is proportional to n 2. The

first

erties

of

measurement

exploring

bucket

ducted

E.L. Andronikashvili

by

both

first conand D.V.

in the late forties. The experiment was

performed

with

essentially the

experiment

the

flat

expectation

to

find

an

meniscus of the superfluid in

rotating container since the parabolic free

surface

contour

in

gravitational

the

reduced

of a classical liquid rotating field was expected to be

by Pn /p ' the fraction of normal fluid

present. However, the

meniscus

the

early

instead the classical shape of

was verified. Some years later in fifties

this

and

other

related

been

In

1977

the of

a

At

using

the total circulation NK matches the solid

body rotation, sity n

v velocity the

becomes of

whole

rotation

Thus the

homogeneous vortex den-

proportional

to

the

angular

rotation, the

in

i.e. n = 22/K. Hence on v superfluid performs solid body

the rotating container and behaves

superficially

like

average

velocity

flow

a

classical 2R

(see

was

O.V.

of the 3He

put

forth by E.L

Lounasmaa to explore supefluids.

time

the

series of

and

hydrodynamic 2

detection

Many new vortex phenomena have been

discovered

which

illuminate

superfluid

3He phases and in retrospect present

convincing

justification

The

of

by

four

the

Academy. whose

this research

phases of

the

interpretation.

different

names

various in

major

is a

institutions

USSR Academy of Sciences and

as coauthors,

1981

for

two

A great number of friends and

included

construction

the

rotating 3He work in Helsinki

project

colleagues,

in

complex

of

a

vortices

the

structure

of the

this

hydrodynamic

work

cryostat

different Also

should

rightfully be

have participated first

in the in

the

and since early

measurements and their

at the Cornell University

fluid with an

rotating

performed by now in superfluid 3He.

In

The first

first

Fig.

2).

numerous

cryostat for the mK-temperature

NMR I

sponsored

that

and

present

techniques.

Finnish

vortices adjusts itself such

by

rotating experiments have already been performed

singly quantized rectilinear vortices. The total the

proposal

rotating

range.

number

of

a

vortices

joint

N

confirmed

step of this project called for the construction

superfluid

by a lattice of

thoroughly

Andronikashvili

effort.

perforated

that the vortex lattice

normal fluid. These features

3. Rotating superfluid 3He experiments

observations were explained by the fact that the becomes

means

the

experiments.

superfluid 4He in the rotating state

was the rotating Osborne

the prop-

have

this

with

experiments 3'4 have been

M. Kn~sius et al. ,/K~'periments on rotating supe<[luid 31A,

24

ii

RESULTS

CENTRAL

I.

Vortex

core

The

3He

paired

much

4He

the

than

the

the effect

Cooper

momentum

are

In

state

L

interparticle

in

the

and s u p e r f l u i d

correlation

crystal

extented

Because

at

types

closer

of vortex

multitude

of

structures

out

reveal

fascinating

in

the

in

lar phase

vortices

core

which

observed ently

also

persisting remanent mapped in

become

result

exist

these

is

with

superflow

56 '

pairlng

Lion

structure

and t e m p e r a t u r e s pressure

This

in the q u a s i - i s o t r o p i quantity

of'

interest

the frequency

discow~ry

~xis

~ with

field

}~. T h e

differline

clearly

measurements 6

oi the

of trapped

transition,

as

is shown

independent

of

shift.

I

I

I

~0-

k APHASE/

~f

~!£fcr,mt

interactions,

context, the

state,

in

the

of'

by

large

and

rotating of

insert

bulk

liquid

frequency related

axis to

Fig. NHR

density

rotatin~

from

and

~ii~ned

n.

P<

in uh~

io~:~tion peak

state

!nd

Thi:

shift

il [ustr'kte;

the

absorption~

in the rotating

the

aff(ct{<

measured

Thus

ot

unitorr;,iv

frequency

states .

/~ is

away

maxima

~-

n-C]ee~

}~ l s

: tare

amounts

the

the NMR a b s o r p t i o n

the

textunai

in t h e

~:

rib.r,,

th<, absent,

[s d i r e c t l y

v( r t l ( e s

incre~:ing

by increasing

riagnet l

the,

wIli<.il we sh~]l

stationary

along H while with

!:: %y

r(SL~lt { [[r)I

in

liquid

the texture

appearance

imimary

;[,~ ~pp]i~m

aevera~

the

L_ o

t<

distributiorl

In the bulk

stationary

",~

:w

is eontrolipd

('f

surface

irl

ahit't

interplay

tween

SOLID

~h{~

"~xtur'(,

iS r e f l e c t e d

40

tip

of th,: magneti<, anisotrop~

r esp(}~:~ spatial

P phases

.

B-liquid

tilted

!ht

pressure:

~s

the

this

wi!ib

in the NHR m e a s u r e m e n t

The

orientation

teraetions.

by

~<

Of ~,

rations the

vortex

ih<

traT] 2

vortex.

therefore

singu-

in ~h~>

the

~iat} ~it low

ref
is

]i;'.o b~

idmnt ify

above

pressuYe

observed

pi~il.cc

transition

in the rc£ion

line as the high

second

coupiin£

:he ! i iqL~id, must

iri

observed

to

par'aJL~:ix

to the symm0tri<

vortex

is also

techniques,

r~ ~Fect

Cooper

and appar-

in the p r e s e n c e

3. It is of first order,

J-L+S=O

with

line

tii~ axially

it" ~rind we

is very

by the two d i f f e r e n t

A-liquid

It i:"

field.

th< strong

thls

transition

phase

L=S=I

Thus

.stabilize

With

vortices

The

which

cone.

The phase

in h y d r o d y n a m i c

boundary.

interactions,

vortex

it turns

distinctly

tr<~.nsiticn

the A-B phase

the local the

th<

magnetic

angular

two d i f f e r e n t

in NHR m e a s u r e m e n t s

vortieity.

Fig.

which

that

rlea!'iy

rotdtJoIt ;~nd w n y

pn~se

rise

states

remarkable

ot

the a p p l i e d

the

possible.

vortex

of'

for

parameter

complexity

velocity

independent

responsible

new properties.

structures.

both

on

that many d i f f e r e n t

bulk B - l i q u i d

separates

gives

order

experiments

striking

that even

ent

this

different

borne

The most

of

examination

f

distance,

orbital

liquid

textures.

iength

= I . The a n i s o t r o p i c

like

lo

core r e p u l s i o n

pairing

out

p-state

eontnast

coherence

of hard

pairing

3He-B

orbital

condensates.

the s u p e r f l u i d

longer

reflecting

in

transition

superfluids

fermion

superfluid is

phase

the angular

H

~n t h<

is qir~et,y

structure

of the.

vortices.

~20

Fig.

C3-

L illustrates

shifts

as

through

10

29.3

0

I

1.5 Fig.

3-

vortex the

Flrst

order

phase

core s t r u c t u r e

measurement

transition

3.0 line of the

as d e t e r m i n e d

frequency

shifts

) and p e r s i s t e n t

techniques

( dashed

the

bar. shift

the

vortex

from

( solid

superflow line

) .

i%

density the is and

The

striking

feature

density.

o Clearly

discontinuity hence

only

an

velocity

the

at o!

increase

<,t the

a given angular' shill

the

the

magnitude

based

whir,~

of tile vortex

the vortex

explanation

at

of the B-fluid.

irrespective

with

in

on

by the s t r u c t u r e

is the d i s c o n t i n u i t y

also

scales

sean

pressure

reflecting

the properties

= 0.60

Frequency

while

dependence

vortices

at T/T

rotation

controlled

the

liquid

a temperature

2t a
increasing

increases

temperature

occurs

the bulk during

B-phase

With

NMR

velocity

Z5

in3He-B

of NMR

line with data points with a e - g y r o s e o p e

I

2.0 T (inK)

measured

ol

density on

the, ind

d phas~

M. Krusius et al. I Experiments on rotating superfluid 3He

25

i

,~o ',f~ 01s I-

"-".

• |

~'k

'p,~"

'- - Q =1.71rad/s

I"-,

o-~=l.4orQms

I k

/ ~oO
/

0.15 _°°~°° ,,,,: °o

+-Q:1.15red/s - -~

!

°% ~L', \

~

o= ' i

I% jo ~oD,

~ A o It

%\ Ooo~ \ ,.~ oil ~

r~o > (~

: o.oo~ad/s

i

",,\ / " ~

\°%o \9~

~_3 0,10 <3

; It. it. ~ f{kHz)

"%%

.

0.05

/

00sl-/

\ 'O,o-,

i~l t

I

//

i,,.

1"% I

000

/

I

I

t,-% k \.~ o~-. 0% ~I +~ +%

l!i'-,--li~ Li ,

0.6

Fig. 5. M e t a s t a b i l i t y of the NMR frequency shift during continuous rotation across

reduced

Larmor

0.8

0.9

velocities

v 0 measured

at

of rotation.

different

angular

in

the

structure of th@ individual

A second variant of the same the

transition

first

and

corresponding in

Fig.

4

represents recorded

is to

is

shown

the

experiment

character in

of

Fig. 5. The data

here as a solid line:

equilibrium

stopping data

equilibrium

o,

NMR

shift

during

solid line, intermittent

rotation. The insert shows the NMR signal in the Larmor

state,

frequency,

rotation.

~

concentrated

close

to

the

and the shifted signal during

= 1.4 rad/s

, P = 29.3 bar and H :

28.4 mT.

with

the

that

in

equilibrium continuous

structure

the

can

be

response we may conclude stable r o t a t i o n the vortex

supercooled

and superheated

it

behaviour which is

shift the

magnitude

of the

rotation

induced

NMR

can be examined to extract information on structure

textural slight

of

the vortices.

interaction depairing

comes

The dominating

about because of a

of the spin component perpen-

dicular to ~ with S : O. This interaction z

rotation at regular

temperature sweep.

points

The

ilthe

the m e a s u r e m e n t s at 1.4 rad/s

intervals during the trast

order

shown

the by

o,

cool-down;

well beyond the transition temperature.

vortex is acceptable. lustrates

transition:

during

The liquid ~ressure P =

29.3 bar and the magnetic field H = 28.4 mT.

transition

in a temperature scan

phase

rotation during warm-up;

stationary

-

vortex rotation

continuous

frequency ~0 as a function of

temperature

the

continuous

I

__

0.7

Fig. 4. NMR frequency shift A~..= ~ the

0.8

%~_ ~\X

T/Tc from

0.7

T/Tc

,. ,',,,_ _-<

0.5

0.6

~,'-

\

'.It

0.5

In con-

FH = - a(~.~) 2

in Fig. 5 represent the

vortex behaviour during a temperature sweep with

aligns

uninterrupted

are additional m e c h a n i s m s which deflect ~ from

1.4

rad/s

namely

at

temperature

rotation:

in

%his experiment at

the

r o t a t i o n was stopped only once,

the

minimum

sweep

temperature

changed

where the

sign. By comparison

~

along ~.

In the rotating state there

in the bulk B-liquid. S u p e r f l o w ~ brings about s a depopulation asymmetry with respect to the orbital pairing and causes ~ to tilt according to

M. K~,rm.~ ct dl. //:Xl)crimd~w.~ rm r~;laliH}f ,~'up~'
26

' - -K............. ~

.

.

.

.

.

.

.

.

2

In the B - p h a s e clear

dipolar

th~

<.~ t h e

I,~'inimization

spin-orbit

interaction

I

r~

<,f

~i~, i

Coopur

pairs

spaces

to be n o t a t e d

by

requir~}~

the r o t a t i o n

tropy

axis

to

by

orbital

respect

Rsikn,ib)

the a n g l e

track r~ h,

respect

to the

oratory

space

R i. T h e

of

ametrized

executing energy

free

in

the

more

otm',r

.~s -

:*n ¢ ir

the

rotations

~F

l{ [;', . s

155 ° ~.~

i~

/

'

~K 10[

&",

~e :

of

th< wit',

ti,.~,

a 5"

<~

I

orienting

continuous of

the

the

ane

p~r

\%

];

material

NHR s h i f t .

In

from

few

vortex

the


the

intervortex

superfluid

texture

samples

vortex

the

Since c.

of

an a v e r a g e and

the <,ore ] r d d ~ ~ ('.

fields

t~p to

effect

th<~

ly this high

~

AX

L

lattie6.

tinuous This

NMR

an a s s u m p t i o n

has

axial

of the

low p r e s s u r e

the

remarkable whether

core

the

_] .k

. ..

.

r'u< ]

.o.

]~:~

"o

•

['i-~.

-

,,.t

",~ -,"

~uu C

~l~r _; b ~ l e r

T 1 []

sue

;c -

.~ "

I!

-

]r

P

.

'r,~ d;

r~tDr(.son[

r:~d,:',

.

t I lone

r~rp~,'"

to ~

radius

property

that

applied

h~

i11

.'

parallel

or a n t i p a r a l l e l

to

~.

shows

the NMR

5.

the

the

field

the a n g u l a r

ism

antfpara!lel

!~,

',,jr'tt!x

8

:

tilt

N: <£,lt, i . !

pre:t,-ur :]tui

qlsp[ayz

:',Jbstant

:r.i~

!'er2ar~, i;~;r

....

tk~n

lr

ia] i }

witil " :m,i !] p a r a i 2 , ~as{.

,~, r-.

)!

"!hf~ is attrit, l~t
m a g n e t ie m(me!iL ,:1 ['!i r~,sid, ~ J;~

spontaneous Vortex

,;tore,

ayld give,~t r ' i S ~ ;

energy

centrJ

bution

ii

the

text!Jr.=]

£ ,~

pressure

,orte:<

th~

plans-

shift

In ~ow

is

shift

4

FH

~

velocity

frequency

J;

lOW p r e s s u r e

the


~h~: ' , i t i h

irish.nail

is o b t a i n e d

inside

the NMR

almost

i:~

~ ([r,p~ r ~ 'lr~, rid,:.

shift

the

r : lO,~. core the o t h e r hand,

on

)I

1.~

gyromagnet

in Fig.

very

uf'

able

from a s t r o n g

pairing

vortex,

more

the

as s e e n

result

leads

f u N ( t ]or~

diL~ar-

dis<~on-

spin

the

oore.

whieh

on

once

a

whiie

for

depends

Fig.6

,

~ with

.

]q

}].

resu,t

< urv~;s

to t m

significant

is large,

core.

The

AX i n

instance,

towards

Such

' .

velocity

proportional

for'

vortex

for

estimate

left

the

.

•~,~r

.;

q

. 08

mr.

iS

is m o r e

preference

ible

r'~Latz

1o

.

Lt'l,?

with

Lower

of

NHk

iv~

the

corresponds

rex

~ r~ v core anisotropy

shift

could,

%% o

.

Th:

i ~ - •i2

Huh

n

contribution

pressure

I o

.

frorr

p

cone susceptibility

re, lat

measured

s.

. . O-

Oyromagn
nt

curw-

.

in ,my

^

A

06

.

interaction

a c~ I c o r e F C = -~ with

05

r~rspect

dnglus

ex eeds

•

b.

<~ntr'gy ['}i

osoillat~

vortex

I~ l g d J ff'ert,

tad/'

FG,

e'~o~

•

13

cohc~rcn<,~,

conditions

orienting

over' the

is

shift

:,<,re

energy

,lees nob

t.h~r,,

the

the magnet, i,

in s u c h

i ,n

{n

to

~'

is

.t'i'~oI

!,n<

magneti(

gradient

amount

core

to

B-liqui

distance,

cores

appreciable resulting

addition

in m a g n e t i c

gauss

' a measur~

the

{ m}~j<;r

localized

texture,

to

superFlow

cone.

from

hundred

length

the

the

contribution

influences

the

e o n t r ibut ion

differs

magnetic

from for

measured

"\

=

witt

lisua=/~ form"

responsible

discontinuous

originating

a

effect

,

I,R,-

115 ° ~1

The

-,

0=

c,n~, h:~s

spat<,

spin

con'¢enic~nt

2

to e a c h

about

directions

in the

counterflow by

flow

sp~li

inter~ction~

t I~e p r o p e r

operatin~I

mq

the, m i L u I C = {re; (os( Z

in the o r i e n t i n g

keep

vector

with

matrix

~

Therefore

the

bT- d ~

comparison pressure

a larger cating

L i

to

R

the

vertex

spontaneous ~

preference

....

j

high is

~hus

dJs!inf~l~i:sh~d

i!}'

,sore

m a g n e t i z a t icn

indi-

towards

ferromagn~ti

:
27

M. Krusius et al. / Experiments on rotating superfluid 3He

pairing.

The

magnetization some

10 -5

UN

Nevertheless,

of

per

ence

of

smaller

high

pressure

the

spontaneous

atom

in

the vortex suggests

spontaneous vortex

the

core

involve

gauge

transformation e i~ by the phase angle ~ ,

result

from

the

of the order parameter

region,

whether

occupied

by

or normal liquid.

that

the relative magnitudes and symmetry

In summary we note

properties of the different contributions to the NMR

shift in the rotating

bulk B-liquid infor-

the parity transformation P, the

time inversion and complex

perpendicular symmetric

to

the

vortex

is

not

obtained

nonzero. totic

These

bulk

with

only

line.

B-liquid

In

core

superfluid 4He

conductor

in

structure

only

the

or

3He-B

in

the s-wave super-

simple

axially

phase vortex with a singular

core

while in the p-wave superfluid lex order parameter matrix sibilities

become

exists only parameter

respect

is

possible

with a 3x3 comp-

many

available:

with

symmetric

different posthe

singularity

to the B-phase Order

components and superfluidity does not

necessarily vanish in the core. M.M. Salomaa and G.E. Volovik have worked out try classification

ameter configurations duce for the

the general symme-

of the possible

order

tex,

has

with

roughly

the

singly quantized axially symmetric

upper

half

core

Tsuneto

and may on

u-vortex

incorporinter-

designated as the o-vor-

of Fig. 7. The o-vortex with a

be

broken

which

has

the

symmetric

in

different

ways

symmetry is preserved: same five C

with

respect

The

(r) nonzero but

to PI only and has

similarly a normal core. The v-vortex has all nine C amplitudes nonzero and real, is symmet~ ric under P2 and has a superfluid core a.s.o. 9 Salomaa mized in

and

terms

and,

Volovik

have numerically mini-

the Ginzburg-Landau free energy expansion of the different vortex states. 9 They

conclude

case the vortex order parameter

have to be

was first discussed by T. Ohmi, T. T. Fujita. I0 The symmetry of the

depending is

CO0 and C+_

thus 5 nonzero real amplitudes C P~ (r) the radial distributions shown in

o-vortex

par-

in the core. 9 They intro-

PI' P2and P3

C_+,

they obtain nonvanishing values at

normal

most

when C_+= CO0 = C+_= I

mediate r. This vortex,

2. V o r t e x

The

components represent the asymp-

ated,

transition

axis.

is invariant under the operation by

on both sides

phase

denoted

solution in which the order parameter

Instead, also C++ and C__

the vortex

conjugation

by T, and a rotation 0 J of the spin and orY,w bital spaces by an angle ~ around an axis y

mation can be retrieved about the core structure of

P3:PIP2

which

superfluid from

P2 = T O J y,~,

core.

magnetization of the

could

distribution

PI = P e i~ ,

the pres-

a superfluid core while presumably the

nonunitary

1,

extremely small, only

its magnitude

much

in

magnitude is, however,

that

the minimum energy configuration

in fact, the only stable solution among the

singly

quantized

axisymmetric

vortices at low

pressures is the v-vortex with nine nonvanishing [a(n=1)]

= A(T)

C00 el*

C0_e2i* I.

real amplitudes C

(r).

These are

shown

as

a

function of r in Fig. 7. Among these CO+ and C+0

[C_+e i*

C_oe2i¢

C__e31~

with zero circulation do not vanish on the axis. CO+

The nine amplitudes C

(r) are complex functions

describes

responsible

for

the

A-phase

pairing

and

is

the susceptibility anisotropy.

with the first index specifying the spin and the

represents a ferromagnetic pairing state, +0 referred to as the B-phase , which produces the

second the orbital quantization state. Note that

main share of the spontaneous magnetization.

of

the

radial

distance r from the vortex axis

C

This

in this representation we simply assume @ = 0 in

vortex

the R . in B-phase. The existence tion

between

implies to

of a first order phase transitwo

different

core

structures

a transition from one discrete symmetry

another.

The

symmetry

group

of the above

identification is

experimental ive

that a

the

looses

the four symmetry elements

the

in

number

resolution

frequency

shifts

pressure

A rigorous quantitat-

is still missing, of on

partly due to

poorly

ameters are involved. Also the

order parameter with discrete axial symmetry has

low

in agreement with the

information.

comparison

the fact

of

qualitatively

known par-

NMR measurement

approaching T

vanish.

c Therefore

where the

M. Kntsius et al. / Experiments ott rotating supe
28

F

I

I

I

la)

I

~.

10 z

,tOo%

\[%'oJ 1.0 _ " ~ -

1.0-

"°0% ~,, °°'oo-c~ o

•~

C

Oo.,

v vortex

6=00

-

o 18.1 b a r • 17.1 bar [] 15.5 bar

q2

I0

0

,Cn.

O.7

u'laqzmmmmmEim---_

I

I

0.8

0.9

05 Fig.

H,.

Cc

-0s : "


O0

215

as

T

three

and at

c minimum

7. R a d i a l

five

real

order

parameter

v-vortex.

amplitudes

which

also

for

10

~/~

amplitudes

the

magnitude

5

of

T

21~-~

the o r d e r

The upper

parameter

half

are nonzero

for

shows

and

in

the o - v o r t e x .

18.1

phase

settled

region and

conclusions

are

have

based to

feature

is

be

different pressure

the

presence

is of

transition above

line

the

fall

a

vicinity

the

first

transition

0.95T e

could

At

minimum since

at

18.1

bar o n l y

In the r e g i o n determined

and

conceivably

T

2

thus still

phase

immediate

line

ot

the

,/or

~owards

T i~: still e m(asurements

':he NMR

the

are

(:d

the

vortex

input

is n e e d e d

sensitive

q u a n t i z a t ion

measurement negative

the

~ligl

pressur~

Finally,

packed

VsPs/p

of in

a

with

20

superflow

~m p o w d e r

trapped

~ ~* that that

rotation velocity

velocity

v

c

the

vortex

phas{ in
superflow

~:

path

i:

angular

which

the

r,,orma

momentum

persistent

L ,

currenln

i:.:

t e c h n i q u e ,v If the

at a s u f f i c i e n t l y

velocity v

vort, ex c o r e ~

displayed

flow

,, t'<,r

~:ind. 12

for d a m p i n g

the ac g y r o s c o p e

umd

tnstane< ,

persistent

is p r e c i p i t a t e d

preparation

The

the

structure

different

torus-shaped

the

with

a critical

note

:Fore

p o t e n t ia]s

dramatically

The m e c h a n i c a l

from

measured

tier.

we m i g h t is a l s o

component.

two

not f< ~"

states

For

trapping

experirqent

perhaps

o[! suci~ prop;rt i~::

state.

in the

one

acre

to the c o r e

the

of ions

is c l e a r l y

,m,~"

4.

structure

experimental

<~f

vicinity

~xtr~polation

of

transition.

Obviously,

measured

appear e from b e f o r e

the

t<

sJos<, to th~

yet

properties

to T

vortex

t~]~:

clot<,

has

the

the d a t a

pressures

a satisfactory candidate been identi l i e d• ,. 9 , 1 0 , I I

the

in the

the

from

temperaturu

vortex,

15.5

bar

of

in Fig.

transition

indicating

close

Finally,

of

three

minimum

17.1

extrapol'ation

be r e l i a b l y

transition

c the

at

resolved

is o b s e r v e d .

at;

transition:

twice

k =

function T

to

while

transition

8 where

a

T

differen~

the b a s i s

As to the

which

c This

caution.

of

, in p a r t i c u l a r ,

transition.

k cannot

second

phase

crossed well

as

close

is s e e n

a smooth

one

a

shown

vortex

towards

Fig.

is

second on

Ginzburg-Landau

with

in

the

no t r a n s i t i o n

transition

to

considered

pressures of

the

on e x t r a p o l a t i o n s

demonstrated

+ leers Iflow temperature in

bar

for

of

bar" in the

identifying experimental

fumtion

transition

on

indicated

the

similar

a

pressure

at

~, :: kflow + k<,ore ,:,~ ~be contribution

energy

. Consequently,

e rex

distribution

parameter

free

vortices

exist

of

The

textural

O0

nine

10

T/Tc Co

all

0.5 -

0.5

15

matrix

05

....

_4-

C..

Z

Fig.

__,.-

°

'"~x C

z; O0 . . . .

1.0

I

high

[] it p e r s i s t s at P s t o p p i n g the r o t a

after c experiences

a

pronounced

M. Krusius et al. / Experiments on rotating superfluid 3He discontinuity at a pressure ture. the

Simultaneously temperature

appearance

a

dependent

tempera-

plateau is observed in

drift with time indicating the

of

a

latent

heat

during

a

slow

29

able in the spin dynamics, the A-phase possesses pronounced order

uniaxial anisotropy. Customarily the

parameter

represented

texture

in

the

warm-up across the vortex phase transition. This

i and the magnetic unit vector d.

observation

vector d lies in the plane

becomes

possible

due

to the fact

that the gyroscope ring is only weakly thermally

spin

coupled

field

thin

to

the

torsion

the

refrigerator

capillaries.

resonant

frequency

proportional

to

In this measurement

of

Ps(T)/p

via two pairs of the

and

gyroscope

employed as a thermometer. The

anomalies in

superflow

and

the

saturated

temperature

persistent

drift

of

the

quantization

The

magnetic

perpendicular to the

axis fixed along the applied

direction,

while

the dipolar spin-orbit

In the A-liquid superflow satisfy the

the

liquid is

interaction in turn aligns I and d parallel.

is

can therefore be

A

in terms of the orbital unit vector

the

requirement

does

not

need to

of potential flow, on

contrary vorticity is supported by way of a

continuous

winding

of the 1-field according to

the Mermin-Ho relation

gyroscopic measurements produce as a function of pressure and temperature the critical line shown in

Fig.

3

result.

which

very

( V × ~ )z

gyroscopic

First, the minimum pressure of the critical line is at a lower pressure

indicating

that

all of the displacement

In the mid seventies continuous which

need

the

temperature

singular

the two ex-

vortex

techniques

of

measurement which were employed periments. turns

in

Second, the gyroscopic critical line

steeply

towards

higher

approaching T . c Consequently, it can be

pressures

on

and

vortex

provide

required

between the two results cannot be traced back to different

to

a

circulation

connection has yet to be established between the

integrating

two transition lines. For this it should be kept

lattice.

mind that the restricted geometry within the

voids

of

the

represents the

flow the is and

The

flow

persisting

of

velocity

are

gyroscope

ring

volume that is

in

of

a

NMR

discontinuous

observed

the

the for

the

stationary state

generally zero

is entirely different: superflow not

persistent but highly damped

magnetic

field,

in

particular,

which

all

very

which

similar

yet

on zero

NMR d

experiments is

rigidly

plane, eg. along x,

vortex

soft

is

concentrated.

This

is illustrated in Fig. 9 , is to

of

the

the

prevail:

vorticity

situation,

the

to

exist

textures. In the large

by the appearance of soft vortex cores inside of

structure

have

by

vorticity V×~ ~ 0 . In the rotating s ^ however, the uniform i texture is broken

throughout the flow volume.

Let us next turn

full

because of dipole locking^gives

irrotational

vortices

the

obtained

uniform planar 1 field along x with

vorticity appears to be continuously distributed

3- Vortex textures in 3He-A

only

transverse

turn a

state,

superfluid 3He-B. In the case of 3He-A

is

vortex

the

to

vanishing

also in the

vortex at zero

for which

along a cell boundary of the vortex

locked

in

a

Chechetkin. It

4~

conditions

to

with

continuous

V.R.

conditions

different

rise

state

first

quantized

axial polarizing field of

important

properties

The

However, no measurements

which

that

rotation without the

by

doubly

field continuous

vortex pinning and remanent vortex

situation at

fact

superflow

that

strings

the

cylindrical

measurement.

proves

in

a different environment from that of

open

critical

powder

realized

suggested by P.W. Anderson

and

of

]

can be constructed

superfluid

was

magnetic field

was

core.

Toulouse

represents

it

body

break the

texture

3~ -~ × -~

quantization of circulation

solid

vortex

G.

• [

textures

the

for

argued that a direct

in

~ i = 2m3r

similar to the NMR

Nevertheless, there are two qualitative

differences. value

is

that

in

the

isotropic

superfluid except for the size and the

vortex core. The diameter of core

is

of the order of the

dipolar length SD = 10 um which sets the scale on A-phase

where the

proved to be no less intriguing.

which

the dipolar energy starts to win over the

gradient

energy

and

on

which 1 can bend away

^

In

contrast

to

the

B-phase with weak biaxial

anisotropy, which generally only becomes observ-

from

d.

At a typical rotation speed of I rad/s

the spacing between the vortices is about 0.3 mm

30

/' l£x'perin

M. Knzsius et al.

/

A ~

-

-

7 ('# tf ~;s rotating s u p e d l u i d " lfc

a}

/ _

~

A

~_!/

i

//

(~D) Q=O

/

I[ .B

Fig.

9

•

The

transverse

discribution

of v o r t i c i t y

plane

at h i g h

axial

The

voPticity

is

in the

magnetic

fields

confined

inside

v

I

I

I

I

I

I

I

1

I

I

I

i

^

( RII~II= ). the

soft

vortex

is u n i f o r m l y

and

thus

soft

in

presence

addition,

comparable 0.01

to the

generally

the

assumed

to

d field,

mope

uniformly

the

less

soft

enePgy

and

(H/HD)2 value

4.

cope the

is o n l y

The

The the

HD=

with

of

reduced

the

the

the

of

Furthermore, of

the

/

2

z

1

:~s

i't~ l<;

o,,<,

O I

I

0

I

2 f-fo

p{ lap

scale

Fig.

]O.

iransvers~.

[1[

Ii

vortex

aRC

part

I

4

6

signaJ

~ ~,i :; s J ~ [

l)*-w,

wi tfi

I

(kHz)


1.

satellite

i:

in I: : h phtn:,

K

< .%."~

d!si,i,~yed

[rl

r[,

t!!,,

v e t 'e i C~ ]

iRt:r~a~ ( <

['},

beta

,m

. ! r]:

amplif'ieal ion.

A liquid

in l a r g e

i tex%ur<

response

in the

shift

satellite

from

sate]iit~

the

thus

and

is s h o w n ill Fig.

1C, the

is less

than

{s a 1] .

frequency the

bulk

structur(:

does


:~bou!

li)xtL]r{

~,

depend


or/

1 emp
,i

iH

i' ,

i<~termJn~.,J :< I t

r'otat

as

Z,14

is

imiivJ,u~.:l

not

:,

: o r l l p o : : ! t ~;

imprint]'! shift

ii:

whi)l

CS

N~4R

I }.

':,<

<) r ] t e x t

,

~ .

meohQi]!;',m J ' ( f

Hn_'oeked

frequen,:y ef

kiriK,

P. ::on,~rK}e m o d e s

! hl

tile

~'~.~

unl

[: f c,ufsh[

t!niP

oC

depen(]s

but

lh< t'rom

!,,~ o r i g i r ~ t e

di~t]~ ?h, =

:

wave

:nd

it,

and

in

solitons

by

of

spin

w e l 1-knowP..

lia:

with

or,r-.

s~li~ t

of

line

'!he

vortex

loealiz{)d

t.hu~

t
NHH

,Jxitation

!en

:~s : . w,lo,-ity

+h'

A

~,Ni

parameters. Sev(ral

pt'ak is speed

soft

reduced

magnitude

It displ~{ys

velocity Fig.

I0.

t,h,~

tni]

[n th{

with

satellite

in Fig.

density.

dependence

from

a f'~w pcr('ent

to the r o t a t i o n

angular

NHR

li~c: N H R

is a s i g n a l a smal~

by

or

appreeiabi(,

shown

vortex

iH t~

be s e t t l e d

models,

is o n l y

as

as n o t e d the

J

is

remains

attenuation

texture

which

temperature

function

the

dipolar

only

core

absorption

to

z i

L_ 3

I rr

lh~

and

partieuK~r,

addition

proportional

therefore

can

and

In

resonance

integrated

little

<:ore

roughly

calculated

vortex

is o b s e r v e d main

a J

I

x throughout

the

in

sound

amplitude

directly

shift

cope

possible

b r o a d e n i n g . 1'I~

the

by

]He-A

of

and,

zero

different

O rn

region

a hard

measurements

peak

of

structure

eg.

signature

the

m a g n u t ie anisot ropy

enePgy

~ortiees

region

of

comparing

the

self

fields

cope

presence

the

by

to z e r o

eharacteristi<

!'

[-.5 mT.

detailed

soft

the

O = 1.21 r a d / s

Length

voPtex

along

l-

;= d i a m e t e P

the other" hand,

dipolar' the

continuous

magnetic

hard

pinned

!

vorti 'try

coherence

be r e p l a c e d

since

where

in the

with

tends

on

than

augmented

cope

parameter

phase. 13 The or

be

super'fluid

cor(~s

b)

n-

mope

distPibuted

vortex

Within

order

tt~e

^

d - x.

i textuPe

may

a hard

~m.

A-phase

The

continuously

of

outside

along

of m a g n i t u d e

diameter.

produces

which,

since

locked

order

one

cope

CONe

copes

dipole

z 0

the

,ah:ulit

soft.

i< i]

core

i::v,

7eer'.

sir'u,:tur{z

propertt~s.13,18,15,1('/r

~]l

with

.inguLm

proven rotation HMR

,2

singly to

quantizh'd

be

enorgetioa2

spee.ds.

signal

its

Ly

',r',

!laP<

£or'~:

preferable

11

::i~ar'ly

!~]e:

:ilru

{~r'e

,:;

NH

ras<~s ~ s o Y !

" ~ { ) v l r r bh(} r e [ ; ,

properties

ptmf:)Fr'K
r;L )v;

N,.d

witr:

M. Krusius et aL/ Experiments on rotating superfluid 3He

I

I/

/

I

I

1.0 ,o--

I

31

I

I

I

singular

singular

o8 5

w

""~='~ _~ ~

o.B 0.05 o

0.4

0°

a 25°

A 90 ° O l ~ ,'B

I 1

i

I

0.2

2

I

o

I

o.1

(rad/sec) Fig.

11 .

NMR

satellite

absorption

intensity

measured

total resonance absorption,

function

of

the

of

28.~

illustrate and

w

the

mT.

the

shown as a

angular velocity of rotation.

The m e a s u r e m e n t s have been performed field

of

peak n o r m a l i z e d to the s i m u l t a n e o u s l y

The additional solid lines

calculated

vortices

in an axial

from

vortex model of Ref.

Ref.

absorption of the v

Fig.

12.

Temperature

RT

from

the

corresponding

only

v

comparison The

is

normalized

vortex

structure. with

P2

good

shown

and

find

ten

structure.

in Figs.

absorption Fig.

This

II and 12. of

the

11 is a more the v-vortex

w-vortex

with

shown which are qualitatively

agreement

with

vortex

the

core

conclusion

with is

ence of R T is core

frequency

shift v Fig.

On the not

expected

measurements.

measurements.

v

- v

12

o

is

A

2

spin

depends used

calculations value

absorption

the soft

distinguish

vortex wave

of

between the

NMR

to

the

satellite

resonance

on the value of

summarized

~D =

absorption ~D w h i c h is

At present time only

6.0 p m

in Figs.

was

of the vortex

used.

In the

11 and 12 the The integrated

satellite

is propor-

to the area of the soft core and thus to

Consequently,

calculated

by

choosing ~ D = 7.5 pm the

intensities

of the v and w vortices

can be made to coincide with the m e a s u r e m e n t s

A is the c h a r a c t e r i s t i c longitudinal where v L resonance frequency of the A-liquid. If the data

Fig.

is

spinwave

extrapolated

calculations

main

is obviously

rough estimates are available of ~D(P,T).

plotted

o

vortex

broad

absorption

the calculations.

of the vortex satellite

2

the

crucially

in

tional

:

the

more quantitative comparison

calculated

in terms of the

singular

sufficiently directly to

The

A similar

the with

to be less for the hard

other hand, NMR

related

~. v

merge

for the continuous core structure. 13

of

fraction R T such that

2_

than

properties

derived from the results for the

which

in

while

would

in

model produces a roughly

the

14 and the singular

resonance line. Moreover the temperature depend-

P3

times more intense satellite and is clearly variance

the

rough agreement with the continuous

models

Two continuous vortices, the

at

13.

the v and w vortices.

and

shift

Also the calculated

w vortices from Ref.

core i texture to allow to

are

singular

in

A-liquid

probe on the details of the soft core symmetry

symmetry,

at

resonance

satellite

sensitive

core

illustrated

total

temperature.

vortex from Ref.

again

the case of a nonsingular

soft

dependence of the vortex

14 and the singular

13. P = 29.3 bar.

in

quantized

o.~

G i n z b u r g - L a n d a u R T -values are indicated for the

satellite

doubly

o.3

satellite frequency when expressed as a fraction

vortex

measurement

I

0.2 (I-T/To)

in

to the

T

for comparison with the c G i n z b u r g - L a n d a u regime we

11.

On

the

other hand,

in

for a given soft

2 I for the lowest core texture the eigenvalue R Tfirst

eigenmode

does

approximation.

not Thus

depend on ~ D in Fig.

12

is

not

32

M. Kntsius ('t al. /' Exp4"riments on rotati~L~ .SUl~e
!iiiii!/:

.......

.- . . t - ,

Fig.

13.

soft

core

by

The

term

potential

context core

it

is

structure

center

has

potential shallow range

well

lustrated

bhis

ous

It

state.

is a s s o c i a t e d

the

loss

emanated these

repent ful, alien

from

whether

it

with

moaes,

the

No a t t e m p t s

varying

arid d e c e l e r a t i o n

;

w~

irl

rlote

of tee

,

{or£

structure

['he

magneti~

trier Both

the

as a recta-

barrier and

in, bp

Perhaps

to p r o d u c e proved

hard

caused

has

surface.

by c o o l i n g

bulk

the

(.,f

"~, dif

i

s~mp]~-

2<:It,


L}

: tile

,~:

f

~ba

P i e < N ,.

~:

L,}'<* i * ~ : ' i t

, ~

s:,,

: 1('¢, by

:t

N',

!t :,;J t<'IlZ:

Net

TS(

,;

tirs

it

f

X

thrq?u~{hout

i

texture,,'

the

sam,-

til~t

on

point; <3 L o n g

io{'abions of

whorl

,)an

[ ~

Idic

!i

:

L,r

:

[e

t ror~ ~ii< ,<;~\

The

, XJL

sort

f

! ] ]

1.i~e

t't;;;r,

i !:'

wh,:''(

~x ~.

['k~

tent

rb ~ , i

, ~rt u];itli)n. ;

t;;

:,1~ ~t

] hr,4][)i

~exLur~s

~:

] /{di,{ [Nir;ii],

J

!iiV~

bGt

:on:oqu~-nt]},,

~ r

;.

~p<,r

topclo£],>~i

iriu'(Jus

r :>{,M i<,n

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r'£;ii<

:,;','r'~

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ben

witii

,O, p } -,

1,.

or'[~:T

erlergy .

spn{r.:.

p~kLl

l!

[),~ )e t i
uatlorrr:

Z~ <

:err

bl;.

in

,

L, e p p S [ ~ i

,)["1

sym[qetri,,: ^ [

~;i : .

t< 1,<, , L , > ~ [ 7

' ii<.l

consiour',d

ps tiles

[1:

ir

t iie-

be

vorti~

< ,k
t extur,i

th6 unit

two

3wav

'!

r
!h~

L i ~ x t [~r L'

il

1?: {[1, [)~r p e n d J,L; lit' '

Futtl

nonaxi~l

]

:howr

:u t N i
th(

~

oriented

p~lr

i;hRr

f~: ~J'

i:

',, n s i

r','gi'!;

{x[<.

on

texture

/-v(;[';~ex

n

I ,r

,dour ! x

roughly

requirement

acceler

i

.1 t[~

[ oFi~tnttit~[£)rl l

different

,i~

< ; [

t h,'

;ilong of

'

J[i~

[,he llransv~!Ys

fieJd

exist

t im~

~i]~ , rrR)w3 r'q>t,?;;0[t ~t]

the

success

the

to

in

poir]ts

!

sa]

i'ot/!

i}('

L,!S

it.';t a[lt

!,,,

vd £ c h

or',

rotation

d

:,:r'm~1

~L)rH'tgur,j!

o N

almost

tO

n i i i ,~

st!

H~[Li

v,'Gu~ .

V o ] o v^J k .

hard

th(-

possibit:

,i
i~i F i g . ; ] .

precipitation

have

Two

ti~ pp~ ~r::

'~

S

Hi!

expcrlrn~/it

[., i ) h ~ s u

until

cquiiibrium

l}m

via d < o n t i n u -

the r o t a t i o n a l or

is

of a s i n g u l a r

container

seconds

(~xtended

within

energy

of NHR r e s p o n s e by

more

core

an e n e r g y

prevent

tM

a shallows4'

persists

nucleation

the

vortices.

type

0

at h i g h

~ texture

in c o n d e n s a t i o n

features

singular

into

in

th~

'CS

i~<,

r,,st

advantageou:

formed

i

in

iars~

a singular

soft

vort

SU} ( : r I ~] N i c

,;~l'.st

mcasur*m~n:,

~'rom

core

and

with

even

readily

thereafter' The

<:w NMR

the

th6:

,~t

within

with

wave

more

pre-dominate

and

pl,-n~.

t C

rbtNiori

Moreover

, ~:tq

iN};

m.

iK)rrrla !

!his

conclusions

structure

the

t%hp

60

• ~,

sf/mfm try

U: f, r

Vs

]OF[=

~ontinuous ~

*

<)

that a sof't

included

the c o n t i n u o u s

stable

by

spin

the a b o v e

liquid

core

hard

diameter

the

the c a l c u l a t i o n s

of

frem

:ontrast

intensity,

deformation

are

Jr:

w vorti,<_~.::;

with

and

otrel~s

phJse

integrated

is m o r e

the

N~e

energetically

to

fields.

of

,rid

vort<,x

the

to n o t e

v

v

i,.< ( f

larger

nevertheless, appears

transv~rse

equal ion

th~

P.~j ~:ymmetry

resonan(:~

with

by

is

the

eigenmodes

wi~ile a c o r e

core

the

witn

,ontinuous

from

has

Summarizing that

of

Y

the

Yor

of %[). 'l]!o

obtained

singular

a

X '

)

essentially measures ^ ^ ] t e x t u r e f r o m x. in

smaller

well

and

of

i

diameters

for

instructive with

a

arc

in

V = -212 - 12 and z y misalignment of the

(i

•

The

iorl

d OUl

form

f l ~ L u . %~ r" o I '

s

in %he v a l u e

(qua1

of

~

: u]

the w v o r t e x

textures.

like

fluctuations

and

for

eigenmodes

SehrOdinger

~

region

(aandu

~

wave

The

(I x,iy)

core

spin

.....

~ . ..........

b)

modified

::::::::i

[

.,<(

[ ype[bo ]

tb

q.

[ J, iv

M. Krusius et al, / Experiments on rotating superfluid 3He

w-vortex with

consists of a circular-hyperbolic

the pair axis oriented perpendicular

With

decreasing

separation low

or

magnetic

field

pa~r to d.

the

pair

is believed to increase until at very

zero

field the pairs dissociate

into a

33

from numerous visitors: G.A.

Kharadze,

particular.

We

understanding

are we

A.L. Fetter,

Maki

and

I.A. Fomin,

V.P. Mineev

deeply

indepted

, in

for

any

have to the patient help from

Salomaa

and

G.E. Volovik. The continuous

lattice of 2~ vortices. The flow fields of the v

encouragement

and

care

and

been indispensable.

w

vortices

approaching

are

almost

identical with v

s in the center and decaying as

zero

I/r outside the soft core.

M.M.

K.

CONCLUSIONS

in

rotating

3He

In

intermediate produce which

a

on quantized vortices

superfluids

centered

vortices.

on

the

particular magnetic

distinct

have

almost

2.

in

the

fields

signature

B-phase the

at

vortices

core

transition

3-

in the ~ texture

structures

behaviour

of

components. possesses

with

ferromagnetic point in

two different

B-liquid low

order

pressure

a

vortex

solution

the

strong

measurements

the

ture with

4~ circulation.

a

continuous struc-

Vortices with a sing-

ular hard core, although lower in energy, do not seem

to

Finally,

be

nucleated

in

the

bulk A-liquid.

it should be pointed out that rotation

has

turned

for

erasing textural defects and singularities.

This

out to be the first reliable method

feature

processes

is of importance for the study of

which

depend on the presence of well

defined ideal textures.

Teor.

J.T. Simola,

O.V.

G.A.

Fiz. 35,

P.J. Hakonen, M.

Lounasmaa,

K.K. Nummila,

R.E. Packard,

A.D. Gongadze,

G.

and G.E. Volovik,

G.E. Gurgenishvili,

Fiz.

P.J. Hakonen,

Nizk.

Temp.

and

7, 821

G.A. (1981)

M. Krusius,

M.M. Salomaa, J.T.

Yu.M. Bunkov, V.P. Mineev, and G.E.

Volovik, 9.

Phys. Rev. Lett. 51,

M.M. Salomaa

and

1362 (1983).

G.E. Volovik,

Phys. Rev.

Lett. 51, 2040 (1983), and Phys. Rev. be published

( to

).

10. T. Ohmi, T. Tsuneto,

and

T. Fujita,

Progr.

Theor. Phys. 70, 647 (1983). 11. T. Passvogel,

L. Tewordt,

and

N. Schopohl,

J. Low Temp. Phys. 56, 383 (1984). 12. V.P. Mineev and M.M. Salomaa, J. Phys. C, 17, L 181

(1984).

13. H.K. Sepp[l[ and G.E. Volovik, 14. H.K.

J. Low Temp.

Sepp[l[,

P.J.

Hakonen,

M.

Krusius,

T. Ohmi, M.M. Salomaa, J.T. Simola, and G.E.

This work has been an inspiring interaction with It is a pleasure to ac-

knowledge the experimental contributions of Yu.M. Ikkala,

Hakonen, and

Phys. 51, 279 (1983).

ACKNOWLEDGEMENTS a large group of people.

P.J.

Islander,

Pis'ma Zh. Eksp.

J.P. Pekola,

Simola,

with the calculated NMR properties

of the orbital texture has

S.T.

[ Soy. J. Low Temp. Phys. 7, 397 (1981) ].

NMR

lead to the conclusion that the soft vortex core

Bunkov,

Kharadze, 8.

of

(1984). Volovik,

Phys. Rev. Lett. 53, 584 (1984). 7.

at high pressures and

intercomparisons

1701 G.E.

Mamniashvili,

case of the second vortex stabilized by

In the A-phase

O.T. Ikkala,

Krusius,

and the first order phase transition

effects

O.V.

Phys. Rev. Lett.

338 (1982) [ JETP Lett. 3-5, 416 (1982) ] . 6.

spin pairing. The allowed symmetry

coupling

K.K. Nummila,

H.E. Hall, P.L. Gammel, and J.D. Reppy, Phys.

Kharadze,

with

temperatures.

J. Low

5-3, 70 (1984). P.L. Gammel, H.E. Hall, and J.D. Reppy, Phys.

Yu.M.

core

to the presence of a superfluid core also

Islander,

Volovik,

and R.E. Packard,

Rev. Lett. 52, 5.

parameter

vortex

S.T.

G.E.

Rev. Lett. 52, 121 (1984). 4.

both involve a singular

spontaneous magnetization which is

consistent properties

which the

The a

between

and

J.P. Pekola, J.T. Simola, Lounasmaa,

B-phase vortex cores are found to suffer a first phase

O.T. Ikkala,

Temp. Phys. 53, 425 (1983).

structure of the

can be resolved from the NMR spectra. The

order

P.J. Hakonen, O.V. Lounasmaa,

The first measurements exclusively

O.V. Lounasmaa has

REFERENCES I.

III

by

Bunkov,

O.T.

Pekola.

Considerable theoretical

S.T.

Islander and

J.P.

input has come

Volovik,

Phys. Rev. Lett. 52, 1802 (1984).

15. K. Maki and X. Zotos, Phys. Rev. B published 16. A.L.

( to

be

) and references therein.

Fetter,

J.A. Sauls,

Phys. Rev. 28B, 5061

(1983).

and

D.L. Stein,