A satellite peak has been observed in the NMR spectrum of rotating 3He-A; the peak intensity depends linearly on Ω at the high angular velocities, Ω=0.6−1.5 rad/s, needed to resolve it. The frequency shift of the satellite is independent of Ω. These results strongly suggest the existence of vortices in rotating 3He-A with the vortex density proportional to Ω. Another satellite peak also has been observed which probably is due to solitons.Peer reviewe

Hakonen, Pertti J.

Ikkala, Olli T.

Islander, S. T.

Physical Review Letters

Aaltodoc

This is an electronic reprint of the original article.
This reprint may differ from the original in pagination and typographic detail.
Author(s): Hakonen, Pertti J. & Ikkala, O. T. & Islander, S. T.
Title: Experiments on Vortices in Rotating Superfluid 3He-A
Year: 1982
Version: Final published version
Please cite the original version:
Hakonen, Pertti J. & Ikkala, O. T. & Islander, S. T. 1982. Experiments on Vortices in
Rotating Superfluid 3He-A. Physical Review Letters. Volume 49, Issue 17. 1258-1261.
ISSN 0031-9007 (printed). DOI: 10.1103/physrevlett.49.1258
Rights: © 1982 American Physical Society (APS). This is the accepted version of the following article: Hakonen,
Pertti J. & Ikkala, O. T. & Islander, S. T. 1982. Experiments on Vortices in Rotating Superfluid 3He-A.
Physical Review Letters. Volume 49, Issue 17. 1258-1261. ISSN 0031-9007 (printed). DOI:
10.1103/physrevlett.49.1258, which has been published in final form at
http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.49.1258.
All material supplied via Aaltodoc is protected by copyright and other intellectual property rights, and
duplication or sale of all or part of any of the repository collections is not permitted, except that material may
be duplicated by you for your research use or educational purposes in electronic or print form. You must
obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or
otherwise to anyone who is not an authorised user.
Powered by TCPDF (www.tcpdf.org)
VOLUME 49, NUMBER 17 PHYSICAL REVIEW LETTERS 25 OCTOBER 1982
because of the uncertainties in Z,«.
Analysis of the waveguide coupler shows that
the lower-hybrid waves should interact primar-
ily with electrons with energies greater than 30
keV. This prediction is consistent with the x-ray
and microwave radiation spectra. How the elec-
trons from the 1-keV background plasma are
raised to this energy has yet to be explained in
this experiment and indeed is one of the principal
unresolved problems in rf current drive.
This work represents a very significant step in
the development of a steady-state current drive
for tokamak plasmas. Discharges have been pro-
duced in which essentially all of the power to
maintain a high current is derived from the rf
fields for periods up to 3 sec. There is strong
evidence that the current is carried by high-en-
ergy (-100-keV) electrons, which are concentrat-
ed in the inner core of the discharge.
It is a pleasure to acknowledge the invaluable
contribution of the High Power RF Group, the
PLT technical staff, and the Data Acquisition
staff. We are grateful for a critical reading of
the manuscript by T. K. Chu and for continued
support from Dr. D. M. Meade, Dr. P. H. Ruther-
ford, and Dr. H. P. Furth.
This work was supported by the U. S. Depart-
ment of Energy under Contract No. DE-AC02-76-
C H03073.
'T. Ohkawa, Nucl. Fusion 10, 185 (1970).
P. C. Thoneman et a l., Nature (London) 169
(1952).
3N. J. Fisch, Phys. Rev. Lett. 41, 873 {1978).
~P. Lallja, in Proceedings of the Second ToPical Con-
ference on Q Plasma Heating, L,ubbock, Texas, 1974
(Texas Tech. Univ. , Lubbock, Tex. , 1974).
'K. L. Wong, Phys. Rev. Lett. 43, 438 (1979).
6R. McWilliams et al. , Phys. Rev. Lett. 44, 245{1980).
K. L. Wong et a2. , Phys. Rev. Lett. 45, 117 (1980).
R. J. LaHaye etal. , Nucl. Fusion 20, 218 {1980).
T. Yamamoto et al. , Phys. Rev. Lett. 45, 716 (1980).J. L. Luxon et al. , General Atomic Co. Report No.
QA-A15820, 1980 (unpublished) .
"M. Nakamura et al. , Phys. Rev. Lett. 47, 1902
(1981).
S. C. Luckhardt etal. , Phys. Rev. Lett. 48, 152(1982).
'3K. Ohkubo et a l., Nucl. Fusion 22, 203 (1982).
' S. C. Luckhardt et al. , in Proceedings of the Third
Joint Varenna-Qrenoble International Symposium,
Commission of the European Communities, Brussels,
1982 (to be published).
K. Ohkumbo, S. Takamura, and JIPP T-11 Group,
in Proceedings of the Third Joint Varenna-Grenoble
International Symposium, Commission of the European
Communities, Brussels, 1982 (to be published).
'6D. Grove et al. , in Proceedings of the Conference on
Plasma Physics and Ãuclear Fusion Research {Interna-
tional Atomic Energy Agency, Vienna, 1977), Vol. I,
p. 21 ~
' C. F. F. Karney and N. J. Fisch, Phys. Fluids 22,
1817 (1979).
1 P. C. Efthimion et al. , Bull. Am. Phys. Soc. 26,
975 (1981).
'~S. Bernabei and D. W. Ignat, Nucl. Fusion 22, 735
(1982).
Kxyeriments on Vortices in Rotating Suyerfluid 3He-A
P. J. Hakonen, 0. T. Ikkala, and S. T. Islander
Io~ Temperature ~hortatory, Helsinki University of Technology, SF-02150 Espoo 15, Finland
(Received 6 July 1982)
A satellite peak has been observed in the NMR spectrum of rotating 3He-A; the peak
intensity depends linearly on 0 at the high angular velocities, 0 = 0.6-1.5 rad/s, needed
to resolve it. The frequency shift of the satellite is independent of Q. These results
strongly suggest the existence of vortices in rotating He-A with the vortex density pro-
portional to Q. Another satellite peak also has been observed which probably is due to
solitons.
PACS numbers: 67.50.Fi
Singularities in superfluid 'He, as well as in
other ordered media, have interested experi-
mentalists and theorists during recent years. ' '
However, because of the experimental difficul-
ties, rotating superfluid 'He has not been ex-
plored until very recently. Rotation is a well-
defined way to produce singularities in super-
fluids and, in addition, creation of singularities
1258 1982 The American Physical Society
VOLUME 49, NUMBER 17 PHYSICAL REVIEW LETTERS 25 OcvosER 1982
oP 0
0 =1.21 radls
Z0
K0
V)
CQ
K
Z
0 2 4 6
U
r
1
Z,'
0
2
0
g
FIG. 1. Transverse NMR absorption line shapes at
1 —T/T, = 0.267 as a function of frequency showing the
satellite peaks as well as the 0-dependent broadening
of the main A-phase peak. Part (b) is a closeup of
part (a), showing the soliton peak at 0 = 0 and the vor-
tex peak at 0 = 1.21 rad/s. In the latter case the soliton
peak has already decayed.
is intimately connected with rotation of the super-
fluid component. Theory predicts that in rotating
'He-A either singular or nonsingular vortices
are formed, depending on external conditions. '
We expected that these structures could be de-
tected by nuclear-magnetic-resonance techniques. '
Our optimism was motivated by experiments in
which characteristic NMR satellite peaks, due to
textural singularities, have been observed. ' '
In this Letter we report the discovery of a sat-
ellite NMR peak in rotating 'He-A. The satellite
is observed in addition to the 0-dependent broad-
ening of the main peak described earlier. ' This
satellite remained unobserved in our previous
experiments because its amplitude is very small.
The measurements were performed in a rotat-
ing nuclear demagnetization cryostat. ' The ex-
perimental space was cylindrical, 5 mm in diam-
eter and 30 mm in length. We performed trans-
verse cw NMR measurements by modulating the
axial magnetic field around Hp 284 Pe. The
pressure was 29.3 bars. The A phase was en-
tered from the Fermi-liquid region with the cry-
ostat either at rest or under rotation at a fixed
Q. A high velocity, between 0.6 and 1.5 rad/s,
3-
Nx
2-
@(P
normal peak
~e
r z
~ ~OqOy
~Q
~4
~
yQC
~t»
'r$
0
0
ag
/
/
/
/
/
/
/
~
Io
~g
~
~
/
vortex
peak
/0' I
0 O. t
soliton
peak
I
03
I
0.2 0.4
1-T/Tc
FIG. 2. Frequency shift of the satellites as a function
of T measured with 0 = 0.6-1.43 rad/s after cooldown
to the A. phase either at rest or under rotation. For
experimental reasons the figure shows the frequency
shifts of the satellites and the position of the normal
liquid peak relative to the bulk A-phase peak. The
dashed line is interpolated from the result by Ahonen,
Krusius, and Paalanen (Ref. 10).
was needed to resolve the satellite.
Actually, we observed two different satellite
peaks between the Larmor frequency f, and the
bulk A-phase frequency f~ =(fo'+f~')'/', where
f~ is the longitudinal resonance frequency (see
Fig. 1).
One satellite was observed only during rota-
tion; its frequency shift and intensity behaved
reproducibly. Because this satellite is broad, it
can be seen as a separate peak only sufficiently
far from T„mainly in supercooled 'He-A, where
the bulk A-phase frequency shift is large. We
call this satellite the vortex peak for reasons
which will become clear later.
The other satellite also has a well-defined fre-
quency shift but its intensity drastically changes
from one experiment to another. This satellite
appears most clearly after a quick cooldown at
rest to the A phase. The peak intensity decreas-
es with time, most of it disappearing in about
half an hour during rotation. We call this satel-
lite the soliton peak for reasons to be explained
later.
The frequency shifts of the satellites were
studied as functions of T and Q; experimentally
it was easiest to measure Af =f~-f„,. Figure
2 shows af for the vortex peak and for the soli-
ton peak. Within our accuracy Df for the vortex
peak depends neither on ~~ nor on whether the
1259
VOLUME 49, NUMBER 17 PHYSICAL REVIEW LETTERS 25 OCTOBER 1982
10
0
5
(1/2) wi
0-a (i/2) Istic
1.5 2.00 05 10
Q (rad/s)
FIG. 3. Averaged relative intensity of the vortex
peak as a function of O. The closed symbols corre-
spond to the case when the rotation was started in the A
phase, while the open symbols are data from experi-
ments where the sample was cooled to the A. phase
while rotating at a fixed O. Error bars are roughly
of the size of the symbols and refer to the relative in-
tensities at different 0 calculated by the method shown
in the inset.
sample was cooled to the A phase at rest or
while rotating.
We have also investigated the intensity of the
vortex peak as a function of Q. The vortex peak
and the main peak partially overlap. Since the
main peak is also affected by rotation' and it is
nonsymmetric, we estimated the satellite inten-
sity I„& by the method illustrated in the inset of
Fig. 3. It seems to us that this method of analy-
sis provides a consistent way to compare inten-
sities at different 0's and it should, moreover,
give a reasonable estimate of the satellite inten-
sity because at low temperatures, where the
peaks are well separted, the satellite appears
roughly symmetric. The measured intensities
are scattered by + 10% at 0= 1.4 rad/s and by
+ 30% at 0=0.6 rad/s. These error limits in-
clude a slight increase in I„&with temperature
which might be due to the increasing overlap with
the main peak. Therefore, the intensity analysis
was restricted to low enough T, mainly 1 —T/T,)0.2.
Figure 3 shows averaged values of vortex peak
intensities normalized to the total NMR absorp-
tion, as functions of Q. The closed and open
symbols represent experiments during which the
A phase was entered from the Fermi liquid either
at rest or while rotating at a fixed 0, respective-
ly. In the first case a linear dependence on 0 is
observed, with I«& /I„, =0.0585~ /(rad/s). In the
second case the points seem to fall systematical-
ly above this line. The half width at half maxi-
mum of the vortex peak (see the inset of Fig, 3)
is about 1 kHz, independent of Q.
The time scale for changes in the NMR signal
is quite rapid; with our acceleration and deceler-
ation, 0.03 rad/s', the vortex peak height followed
roughly the instantaneous speed of the cryostat,
with a few seconds time lag at most.
In stationary 'He-A Gould and Lee' have re-
ported a transverse NMR satellite in an open cyl-
inder, with the magnetic field perpendicular to its
axis. The frequency shift could be described by
the parameter 8, = (f„,-f,')' '/f i, having a tem-
perature-independent value of R, = 0.835. Maki
and Kumar' explained this by composite solitons
which are textural domain walls. Such a dipole-
unlocked" region forms a potential well which
can trap localized spin waves, thus producing a
well-defined NMR satellite frequency. The value
of R, is characteristic of different solitons and
of dipole-unlocked textures; it can be used to
classify them. '
We find for our soliton peak A, =0.846& 9.015
which is close to the value of Gould and Lee.
This suggests that the satellite is caused by soli-
tons which might be created by our quick cool-
down to the A phase; it has been observed that
rapid temperature changes increase the intensity
of the soliton satellite. ' However, large textural
changes brought about by flow during rotation ap-
pear to destabilize the solitons.
For vortex peaks the R, value is substantially
smaller. A, changes from 0.53 at 1 —T/T, —= 0.30
to 0.70 at 1 —T/T, —= 0.15, the error estimates
being +0.01 and +0.03, —0.05, respectively. It is
not likely that these satellites are due to solitons,
because the types of solitons expected in an open
cylindrical vessel should produce larger values
of R, .' Solitons might be formed irregularly
during a sharp acceleration but they would in
that case not be manifested in the regular fashion
shown in Fig. 3.
Bruinsma and Maki" showed that dipole-locked
linear textures are not capable of causing la, rge
negative frequency shifts from f~. Pointlike
singularities probably cannot produce a satellite
peak as large as we observe. Consequently, we
think that there must be dipole-unlocked linear
textures in our rotating sample, their number
and thus also their combined volume increasing
1260
VOLUME 49, NUMBER 17 PHYSICAL REVIEW LETTERS 25 OGTQBER 1982
linearly with 0; this, in turn, would cause the
satellite intensity to be proportional to Q. It
seems to us that the dependence shown in Fig. 3,
combined with the result that the satellite fre-
quency does not depend on A, is strong evidence
for vortices in rotating 'He-A, the number of
vortices being proportional to O. We believe
that the satellite is produced by localized spin-
wave modes trapped on each vortex core.
There are, to our knowledge, no detailed pre-
dictions of the effect that singular or nonsingular
vortices would have on the 'He-A NMR spectrum
in a high magnetic field. Therefore, we crudely
estimate the relative satellite intensity due to
either singular or nonsingular vortices, I„,/I„„
by the ratio S„/S„where S„ is the effective
cross-sectional area of the vortex core and S, is
the primitive cell area of the vortex lattice. " We
write S„=m()t.g~)', where $~= 6 pm is the dipolar
coherence length and A. is of the order of unity,
with its actual value depending on the core struc-
ture. Therefore, we obtain the experimentally
verified linear fl dependence I„,/I„, = m(Xg~)'
x 4m, Q/iVh, where R is the circulation quantum
per vortex. Comparing with our experimental
date we find A. = 4 for singular vortices with N = 1
and A. = 6 for nonsingular vortices with A =2.
These values of A. are quite large but, taking into
account our crude model and the exponential heal-
ing of the core texture, they are still plausible.
Calculations with some model textures of ro-
tating 'He-A in an axial II suggest"" dipole-un-
locking near each vortex. On the other hand,
stationary 'He-A confined to long and narrow (x( $~) cylindrical pores must have singular or non-
singular dipole-unlocked linear structures but,
in this case, caused by boundary conditions. In
such experiments"' two satellite peaks were ob-
served between f, and f~, corresponding to two
different linear textures. One of these satellites
has the A, value of the same order as our vortex
peak, also supporting linear singularities in our
case.
The fact that cooldown to the A phase under ro-
tation produces a systematically larger satellite
than cooldown at rest can be evidence of different
kinds of vortices in these two cases, or evidence
of nonequilibrium effects for singular vortices. "
The latter structures should be energetically
favored in a high axial field, 4 although their for-
mation might be inhibited by a topological energy
barrier. However, recent calculations by Sep-
pala and Volovik" suggest the identification with
nonsingular vortices, because the investigated
singular vortices were not able to produce such
large NMB frequency shifts as were observed.
We wish to express our gratitude to G. E. Vol-
ovik for his advice and suggestions at the early
stages of this work. We also thank M. Krusius
and O. V. Lounasmaa for several useful discus-
sions, and A. I. Fetter, G. A. Kharadze, P. Ku-
mar, V. P. Mineev, and R. E. Packard for crit-
ically reading the manuscript. The experimental
help by T. K. Markkula and H. K. Seppala is
gratefully acknowledged. This work was finan-
cially supported by the Academy of Finland.
K. Maki and P. Kumar, Phys. Rev. 8 17, 1088
(1978).
V. P. Mineev and G. E. Volovik, Phys. Rev. B 18,
3197 {1978}.
36. E. Volovik and V. P. Mineev, Pis'ma Zh. Eksp.
Teor. Fiz. 24, 605 {1976) I JETP Lett. 24, 561 (1976)].
4G. E. Volovik and P. J. Hakonen, J. Low Temp.
Phys. 42, 503 (1981).
J. Saunders, D. S. Betts, D. F. Brewer, S. J. Swithen-
by, and W. S. Truscott, Phys. Rev. Lett. 40, 1278
(1978).
6C. M. Gould and D. M. Lee, Phys. Rev. Lett. 41,
967 (1978).
'C. M. Gould and D. M. Lee, Phys. Rev. Lett. 37,
1223 {1976).
'P. J. Hakonen, O. T. Ikkala, S. T. Islander, O. V.
Lounasmaa, T. K. Markkula, P. Roubeau, K. M. Salo-
heimo, G. E. Volovik, E. L. Andronikashvili, D. I.
Garibashvili, and J. S. Tsakadze, Phys. Rev. Lett.
48, 1838 (1982).
'P. J. Hakonen, O. T. Ikkala, S. T. Islander, T. K.
Markkula, P. Roubeau, K. Saloheimo, D. I. Garibash-
vili, and J. S. Tsakadze, Physica (utrecht) 107B, 567
(1981).
' A. I. Ahonen, M. Krusius, and M. A. Paalanen, J.
Low Temp. Phys. 25, 421 (1976).
"In a dipole-unlocked texture the 1 —and d -vectors
describing the orbital and the spin parts of the order
parameter are not parallel, whereby the dipole energy
is not minimized.
' R. Bruinsma and K. Maki, Phys. Rev. B 18, 1101
(1978).
'3Solid body rotation demands 2P. = Vx (v~). Integra-
tion over cross sections S=nS p containing n vortices
gives
n
2QS = fs ds ~ (V&(v,)) = Z fdo; ~ v, = noh/2m 3,i-"i
yielding Sp, Stokes s law and quantization of circulation
have been employed.
M. Nakahara, T. Ohmi, T. Tsuneto, and T. Fujita,
Prog. Theor. Phys. 62, 874 (1979).
'5T. Passvogel, N. Schopohl, M. Warnke, and L. Te-
wordt, J. Low Temp. Phys. 46, 161 (1982).
' E. J. Yarmchuk and R. E. Packard, J. Low Temp.
Phys. 46, 479 (1982).
'~H. K. Seppala and G. E. Volovik, to be published.
1261


English

Experiments on Vortices in Rotating Superfluid 3He-A

P. J. Hakonen

O. T. Ikkala

S. T. Islander

Crossref

Experiments on Vortices in Rotating Superfluid<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mmultiscripts><mml:mrow><mml:mi mathvariant="normal">He</mml:mi></mml:mrow><mml:mprescripts /><mml:mrow /><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow /><mml:mrow /></mml:mmultiscripts></mml:mrow></mml:math>-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>A</mml:mi></mml:math>

A satellite peak has been observed in the NMR spectrum of rotating He3-A; the peak intensity depends linearly on Ω at the high angular velocities, Ω=0.6-1.5 rad/s, needed to resolve it. The frequency shift of the satellite is independent of Ω. These results strongly suggest the existence of vortices in rotating He3-A with the vortex density proportional to Ω. Another satellite peak also has been observed which probably is due to solitons.Peer reviewe

Hakonen, P. J.

Ikkala, O. T.

Aaltodoc Publication Archive

This is an electronic reprint of the original article.This reprint may differ from the original in pagination and typographic detail.Author(s): Hakonen, Pertti J. & Ikkala, O. T. & Islander, S. T.Title: Experiments on Vortices in Rotating Superfluid 3He-AYear: 1982Version: Final published versionPlease cite the original version:Hakonen, Pertti J. & Ikkala, O. T. & Islander, S. T. 1982. Experiments on Vortices inRotating Superfluid 3He-A. Physical Review Letters. Volume 49, Issue 17. 1258-1261.ISSN 0031-9007 (printed). DOI: 10.1103/physrevlett.49.1258Rights: © 1982 American Physical Society (APS). This is the accepted version of the following article: Hakonen,Pertti J. & Ikkala, O. T. & Islander, S. T. 1982. Experiments on Vortices in Rotating Superfluid 3He-A.Physical Review Letters. Volume 49, Issue 17. 1258-1261. ISSN 0031-9007 (printed). DOI:10.1103/physrevlett.49.1258, which has been published in final form athttp://journals.aps.org/prl/abstract/10.1103/PhysRevLett.49.1258.All material supplied via Aaltodoc is protected by copyright and other intellectual property rights, andduplication or sale of all or part of any of the repository collections is not permitted, except that material maybe duplicated by you for your research use or educational purposes in electronic or print form. You mustobtain permission for any other use. Electronic or print copies may not be offered, whether for sale orotherwise to anyone who is not an authorised user.Powered by TCPDF (www.tcpdf.org)VOLUME 49, NUMBER 17 PHYSICAL REVIEW LETTERS 25 OCTOBER 1982because of the uncertainties in Z,«.Analysis of the waveguide coupler shows thatthe lower-hybrid waves should interact primar-ily with electrons with energies greater than 30keV. This prediction is consistent with the x-rayand microwave radiation spectra. How the elec-trons from the 1-keV background plasma areraised to this energy has yet to be explained inthis experiment and indeed is one of the principalunresolved problems in rf current drive.This work represents a very significant step inthe development of a steady-state current drivefor tokamak plasmas. Discharges have been pro-duced in which essentially all of the power tomaintain a high current is derived from the rffields for periods up to 3 sec. There is strongevidence that the current is carried by high-en-ergy (-100-keV) electrons, which are concentrat-ed in the inner core of the discharge.It is a pleasure to acknowledge the invaluablecontribution of the High Power RF Group, thePLT technical staff, and the Data Acquisitionstaff. We are grateful for a critical reading ofthe manuscript by T. K. Chu and for continuedsupport from Dr. D. M. Meade, Dr. P. H. Ruther-ford, and Dr. H. P. Furth.This work was supported by the U. S. Depart-ment of Energy under Contract No. DE-AC02-76-C H03073.'T. Ohkawa, Nucl. Fusion 10, 185 (1970).P. C. Thoneman et a l., Nature (London) 169(1952).3N. J. Fisch, Phys. Rev. Lett. 41, 873 {1978).~P. Lallja, in Proceedings of the Second ToPical Con-ference on Q Plasma Heating, L,ubbock, Texas, 1974(Texas Tech. Univ. , Lubbock, Tex. , 1974).'K. L. Wong, Phys. Rev. Lett. 43, 438 (1979).6R. McWilliams et al. , Phys. Rev. Lett. 44, 245{1980).K. L. Wong et a2. , Phys. Rev. Lett. 45, 117 (1980).R. J. LaHaye etal. , Nucl. Fusion 20, 218 {1980).T. Yamamoto et al. , Phys. Rev. Lett. 45, 716 (1980).J. L. Luxon et al. , General Atomic Co. Report No.QA-A15820, 1980 (unpublished) ."M. Nakamura et al. , Phys. Rev. Lett. 47, 1902(1981).S. C. Luckhardt etal. , Phys. Rev. Lett. 48, 152(1982).'3K. Ohkubo et a l., Nucl. Fusion 22, 203 (1982).' S. C. Luckhardt et al. , in Proceedings of the ThirdJoint Varenna-Qrenoble International Symposium,Commission of the European Communities, Brussels,1982 (to be published).K. Ohkumbo, S. Takamura, and JIPP T-11 Group,in Proceedings of the Third Joint Varenna-GrenobleInternational Symposium, Commission of the EuropeanCommunities, Brussels, 1982 (to be published).'6D. Grove et al. , in Proceedings of the Conference onPlasma Physics and Ãuclear Fusion Research {Interna-tional Atomic Energy Agency, Vienna, 1977), Vol. I,p. 21 ~' C. F. F. Karney and N. J. Fisch, Phys. Fluids 22,1817 (1979).1 P. C. Efthimion et al. , Bull. Am. Phys. Soc. 26,975 (1981).'~S. Bernabei and D. W. Ignat, Nucl. Fusion 22, 735(1982).Kxyeriments on Vortices in Rotating Suyerfluid 3He-AP. J. Hakonen, 0. T. Ikkala, and S. T. IslanderIo~ Temperature ~hortatory, Helsinki University of Technology, SF-02150 Espoo 15, Finland(Received 6 July 1982)A satellite peak has been observed in the NMR spectrum of rotating 3He-A; the peakintensity depends linearly on 0 at the high angular velocities, 0 = 0.6-1.5 rad/s, neededto resolve it. The frequency shift of the satellite is independent of Q. These resultsstrongly suggest the existence of vortices in rotating He-A with the vortex density pro-portional to Q. Another satellite peak also has been observed which probably is due tosolitons.PACS numbers: 67.50.FiSingularities in superfluid 'He, as well as inother ordered media, have interested experi-mentalists and theorists during recent years. ' 'However, because of the experimental difficul-ties, rotating superfluid 'He has not been ex-plored until very recently. Rotation is a well-defined way to produce singularities in super-fluids and, in addition, creation of singularities1258 1982 The American Physical SocietyVOLUME 49, NUMBER 17 PHYSICAL REVIEW LETTERS 25 OcvosER 1982oP 00 =1.21 radlsZ0K0V)CQKZ0 2 4 6Ur1Z,'020gFIG. 1. Transverse NMR absorption line shapes at1 —T/T, = 0.267 as a function of frequency showing thesatellite peaks as well as the 0-dependent broadeningof the main A-phase peak. Part (b) is a closeup ofpart (a), showing the soliton peak at 0 = 0 and the vor-tex peak at 0 = 1.21 rad/s. In the latter case the solitonpeak has already decayed.is intimately connected with rotation of the super-fluid component. Theory predicts that in rotating'He-A either singular or nonsingular vorticesare formed, depending on external conditions. 'We expected that these structures could be de-tected by nuclear-magnetic-resonance techniques. 'Our optimism was motivated by experiments inwhich characteristic NMR satellite peaks, due totextural singularities, have been observed. ' 'In this Letter we report the discovery of a sat-ellite NMR peak in rotating 'He-A. The satelliteis observed in addition to the 0-dependent broad-ening of the main peak described earlier. ' Thissatellite remained unobserved in our previousexperiments because its amplitude is very small.The measurements were performed in a rotat-ing nuclear demagnetization cryostat. ' The ex-perimental space was cylindrical, 5 mm in diam-eter and 30 mm in length. We performed trans-verse cw NMR measurements by modulating theaxial magnetic field around Hp 284 Pe. Thepressure was 29.3 bars. The A phase was en-tered from the Fermi-liquid region with the cry-ostat either at rest or under rotation at a fixedQ. A high velocity, between 0.6 and 1.5 rad/s,3-Nx2-@(Pnormal peak~er z~ ~OqOy~Q~4~yQC~t»'r$00ag///////~Io~g~~/vortexpeak/0' I0 O. tsolitonpeakI03I0.2 0.41-T/TcFIG. 2. Frequency shift of the satellites as a functionof T measured with 0 = 0.6-1.43 rad/s after cooldownto the A. phase either at rest or under rotation. Forexperimental reasons the figure shows the frequencyshifts of the satellites and the position of the normalliquid peak relative to the bulk A-phase peak. Thedashed line is interpolated from the result by Ahonen,Krusius, and Paalanen (Ref. 10).was needed to resolve the satellite.Actually, we observed two different satellitepeaks between the Larmor frequency f, and thebulk A-phase frequency f~ =(fo'+f~')'/', wheref~ is the longitudinal resonance frequency (seeFig. 1).One satellite was observed only during rota-tion; its frequency shift and intensity behavedreproducibly. Because this satellite is broad, itcan be seen as a separate peak only sufficientlyfar from T„mainly in supercooled 'He-A, wherethe bulk A-phase frequency shift is large. Wecall this satellite the vortex peak for reasonswhich will become clear later.The other satellite also has a well-defined fre-quency shift but its intensity drastically changesfrom one experiment to another. This satelliteappears most clearly after a quick cooldown atrest to the A phase. The peak intensity decreas-es with time, most of it disappearing in abouthalf an hour during rotation. We call this satel-lite the soliton peak for reasons to be explainedlater.The frequency shifts of the satellites werestudied as functions of T and Q; experimentallyit was easiest to measure Af =f~-f„,. Figure2 shows af for the vortex peak and for the soli-ton peak. Within our accuracy Df for the vortexpeak depends neither on ~~ nor on whether the1259VOLUME 49, NUMBER 17 PHYSICAL REVIEW LETTERS 25 OCTOBER 19821005(1/2) wi0-a (i/2) Istic1.5 2.00 05 10Q (rad/s)FIG. 3. Averaged relative intensity of the vortexpeak as a function of O. The closed symbols corre-spond to the case when the rotation was started in the Aphase, while the open symbols are data from experi-ments where the sample was cooled to the A. phasewhile rotating at a fixed O. Error bars are roughlyof the size of the symbols and refer to the relative in-tensities at different 0 calculated by the method shownin the inset.sample was cooled to the A phase at rest orwhile rotating.We have also investigated the intensity of thevortex peak as a function of Q. The vortex peakand the main peak partially overlap. Since themain peak is also affected by rotation' and it isnonsymmetric, we estimated the satellite inten-sity I„& by the method illustrated in the inset ofFig. 3. It seems to us that this method of analy-sis provides a consistent way to compare inten-sities at different 0's and it should, moreover,give a reasonable estimate of the satellite inten-sity because at low temperatures, where thepeaks are well separted, the satellite appearsroughly symmetric. The measured intensitiesare scattered by + 10% at 0= 1.4 rad/s and by+ 30% at 0=0.6 rad/s. These error limits in-clude a slight increase in I„&with temperaturewhich might be due to the increasing overlap withthe main peak. Therefore, the intensity analysiswas restricted to low enough T, mainly 1 —T/T,)0.2.Figure 3 shows averaged values of vortex peakintensities normalized to the total NMR absorp-tion, as functions of Q. The closed and opensymbols represent experiments during which theA phase was entered from the Fermi liquid eitherat rest or while rotating at a fixed 0, respective-ly. In the first case a linear dependence on 0 isobserved, with I«& /I„, =0.0585~ /(rad/s). In thesecond case the points seem to fall systematical-ly above this line. The half width at half maxi-mum of the vortex peak (see the inset of Fig, 3)is about 1 kHz, independent of Q.The time scale for changes in the NMR signalis quite rapid; with our acceleration and deceler-ation, 0.03 rad/s', the vortex peak height followedroughly the instantaneous speed of the cryostat,with a few seconds time lag at most.In stationary 'He-A Gould and Lee' have re-ported a transverse NMR satellite in an open cyl-inder, with the magnetic field perpendicular to itsaxis. The frequency shift could be described bythe parameter 8, = (f„,-f,')' '/f i, having a tem-perature-independent value of R, = 0.835. Makiand Kumar' explained this by composite solitonswhich are textural domain walls. Such a dipole-unlocked" region forms a potential well whichcan trap localized spin waves, thus producing awell-defined NMR satellite frequency. The valueof R, is characteristic of different solitons andof dipole-unlocked textures; it can be used toclassify them. 'We find for our soliton peak A, =0.846& 9.015which is close to the value of Gould and Lee.This suggests that the satellite is caused by soli-tons which might be created by our quick cool-down to the A phase; it has been observed thatrapid temperature changes increase the intensityof the soliton satellite. ' However, large texturalchanges brought about by flow during rotation ap-pear to destabilize the solitons.For vortex peaks the R, value is substantiallysmaller. A, changes from 0.53 at 1 —T/T, —= 0.30to 0.70 at 1 —T/T, —= 0.15, the error estimatesbeing +0.01 and +0.03, —0.05, respectively. It isnot likely that these satellites are due to solitons,because the types of solitons expected in an opencylindrical vessel should produce larger valuesof R, .' Solitons might be formed irregularlyduring a sharp acceleration but they would inthat case not be manifested in the regular fashionshown in Fig. 3.Bruinsma and Maki" showed that dipole-lockedlinear textures are not capable of causing la, rgenegative frequency shifts from f~. Pointlikesingularities probably cannot produce a satellitepeak as large as we observe. Consequently, wethink that there must be dipole-unlocked lineartextures in our rotating sample, their numberand thus also their combined volume increasing1260VOLUME 49, NUMBER 17 PHYSICAL REVIEW LETTERS 25 OGTQBER 1982linearly with 0; this, in turn, would cause thesatellite intensity to be proportional to Q. Itseems to us that the dependence shown in Fig. 3,combined with the result that the satellite fre-quency does not depend on A, is strong evidencefor vortices in rotating 'He-A, the number ofvortices being proportional to O. We believethat the satellite is produced by localized spin-wave modes trapped on each vortex core.There are, to our knowledge, no detailed pre-dictions of the effect that singular or nonsingularvortices would have on the 'He-A NMR spectrumin a high magnetic field. Therefore, we crudelyestimate the relative satellite intensity due toeither singular or nonsingular vortices, I„,/I„„by the ratio S„/S„where S„ is the effectivecross-sectional area of the vortex core and S, isthe primitive cell area of the vortex lattice. " Wewrite S„=m()t.g~)', where $~= 6 pm is the dipolarcoherence length and A. is of the order of unity,with its actual value depending on the core struc-ture. Therefore, we obtain the experimentallyverified linear fl dependence I„,/I„, = m(Xg~)'x 4m, Q/iVh, where R is the circulation quantumper vortex. Comparing with our experimentaldate we find A. = 4 for singular vortices with N = 1and A. = 6 for nonsingular vortices with A =2.These values of A. are quite large but, taking intoaccount our crude model and the exponential heal-ing of the core texture, they are still plausible.Calculations with some model textures of ro-tating 'He-A in an axial II suggest"" dipole-un-locking near each vortex. On the other hand,stationary 'He-A confined to long and narrow (x( $~) cylindrical pores must have singular or non-singular dipole-unlocked linear structures but,in this case, caused by boundary conditions. Insuch experiments"' two satellite peaks were ob-served between f, and f~, corresponding to twodifferent linear textures. One of these satelliteshas the A, value of the same order as our vortexpeak, also supporting linear singularities in ourcase.The fact that cooldown to the A phase under ro-tation produces a systematically larger satellitethan cooldown at rest can be evidence of differentkinds of vortices in these two cases, or evidenceof nonequilibrium effects for singular vortices. "The latter structures should be energeticallyfavored in a high axial field, 4 although their for-mation might be inhibited by a topological energybarrier. However, recent calculations by Sep-pala and Volovik" suggest the identification withnonsingular vortices, because the investigatedsingular vortices were not able to produce suchlarge NMB frequency shifts as were observed.We wish to express our gratitude to G. E. Vol-ovik for his advice and suggestions at the earlystages of this work. We also thank M. Krusiusand O. V. Lounasmaa for several useful discus-sions, and A. I. Fetter, G. A. Kharadze, P. Ku-mar, V. P. Mineev, and R. E. Packard for crit-ically reading the manuscript. The experimentalhelp by T. K. Markkula and H. K. Seppala isgratefully acknowledged. This work was finan-cially supported by the Academy of Finland.K. Maki and P. Kumar, Phys. Rev. 8 17, 1088(1978).V. P. Mineev and G. E. Volovik, Phys. Rev. B 18,3197 {1978}.36. E. Volovik and V. P. Mineev, Pis'ma Zh. Eksp.Teor. Fiz. 24, 605 {1976) I JETP Lett. 24, 561 (1976)].4G. E. Volovik and P. J. Hakonen, J. Low Temp.Phys. 42, 503 (1981).J. Saunders, D. S. Betts, D. F. Brewer, S. J. Swithen-by, and W. S. Truscott, Phys. Rev. Lett. 40, 1278(1978).6C. M. Gould and D. M. Lee, Phys. Rev. Lett. 41,967 (1978).'C. M. Gould and D. M. Lee, Phys. Rev. Lett. 37,1223 {1976).'P. J. Hakonen, O. T. Ikkala, S. T. Islander, O. V.Lounasmaa, T. K. Markkula, P. Roubeau, K. M. Salo-heimo, G. E. Volovik, E. L. Andronikashvili, D. I.Garibashvili, and J. S. Tsakadze, Phys. Rev. Lett.48, 1838 (1982).'P. J. Hakonen, O. T. Ikkala, S. T. Islander, T. K.Markkula, P. Roubeau, K. Saloheimo, D. I. Garibash-vili, and J. S. Tsakadze, Physica (utrecht) 107B, 567(1981).' A. I. Ahonen, M. Krusius, and M. A. Paalanen, J.Low Temp. Phys. 25, 421 (1976)."In a dipole-unlocked texture the 1 —and d -vectorsdescribing the orbital and the spin parts of the orderparameter are not parallel, whereby the dipole energyis not minimized.' R. Bruinsma and K. Maki, Phys. Rev. B 18, 1101(1978).'3Solid body rotation demands 2P. = Vx (v~). Integra-tion over cross sections S=nS p containing n vorticesgivesn2QS = fs ds ~ (V&(v,)) = Z fdo; ~ v, = noh/2m 3,i-"iyielding Sp, Stokes s law and quantization of circulationhave been employed.M. Nakahara, T. Ohmi, T. Tsuneto, and T. Fujita,Prog. Theor. Phys. 62, 874 (1979).'5T. Passvogel, N. Schopohl, M. Warnke, and L. Te-wordt, J. Low Temp. Phys. 46, 161 (1982).' E. J. Yarmchuk and R. E. Packard, J. Low Temp.Phys. 46, 479 (1982).'~H. K. Seppala and G. E. Volovik, to be published.1261

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Experiments on Vortices in Rotating Superfluid 3He-A

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