Rotating Superfluid 3He in Aerogel

Takao Mizusaki

Department of Physics, Graduate School of Science, Kyoto University

Collaborators:

Kyoto University, M. Yamashita, A. Matsubara, R. Ishiguro# and Y. SasakiOsaka City University, O. IshikawaISSP, Univ. Tokyo, Y. Kataoka and M. KubotaCNTB-CNRS, Yu. M. Bunkov

#ENS-Paris

Outline

Rotating Superfluid 3He in Aerogel

(1) Comparison with other data without rotation

The sample is 98 % arogel (Bunkov’s sample)

(2) Singular core cortex and the l-texture is strongly pinned

in A-like Phase

(3) Critical velocity for vortex penetration and persistent current

in B-Phase

PurposeP-wave superfluidity in aerogel:

Impurity effect of p-wave superfluid in aerogel

Rotation experiment of 3He superfluid in aerogel:non-uniformities of superfluid in aerogel or amorphous superfluid

1. Extremely hard type II superfluidityB → φ

　Hc １ → Ωc １〜

　　Hc ２ → Ωc ２ ~ 　

2. What kind of vortices?3. Texture in aerogel and its coupling with flow

The texture is pinned strongly in A-like phase and weakly in B-phase.

4. Vortex and pinning effect Amorphous superfluidity → flux creep model

€

h2m

⎛

⎝ ⎜

⎞

⎠ ⎟1

R 2

€

h2m

⎛

⎝ ⎜

⎞

⎠ ⎟

1

ξ (T )2

§1. Phase diagram of superfluid in aerogel (cooling process)

€

Tcbulk = 2.51 mK

Tcaerogel = 2.07 mK

TCA→B = 1.75 ~ 1.80mK

0

0.2

0.4

0.6

0.8

1

1.2

0

0.5

1

1.5

2

2.5

3

1 1.5 2 2.5

dH0/dt > 0

dH0/dt < 0

NormalA-phaseB-phase

2.07mK1.73mK 1.80mK

Frequency shift

Magnetization

Cooling

T (mK)

Frequency shift

(kHz)

ML

iqui

d/M

tota

l

Pressure = 3.0 MPa,

H0 = 22 mT

Two Phases:

1) A-like phase (ESP)

2) B-phase

Porosity 98 %

Phase diagram of superfluid in aerogel (warming process)

T (mK)

0

0.2

0.4

0.6

0.8

1

1.2

0

0.5

1

1.5

2

2.5

3

1 1.5 2 2.5

NormalB-phase

2.07mK

Frequency shift

Magnetization

Warming

Frequency shift

(kHz)

ML

iqui

d/M

tota

l

-phase is superheated up to Tcaero

§2. A-phase under rotation Cooling conditions through Tc

CASE 1: 0 rad/s, 2 K/min.

CASE 2: 0 rad/s, 20 K/min.

CASE 3: +0.10 rad/s, 3 K/min.

CASE 4: -0.01 rad/s, 1 K/min.

CASE 5: +6.28 rad/s, 1 K/min.

CASE 6: -6.28 rad/s, 1 K/min.

Result for a bulk sample

　　　 (JLTP 60, 187 (1985) )

Ω×= 065.0total

vortexI

I

Ω×≤ 0048.0total

vortexI

I

-0.5 0 0.5 1 1.5 2

case 1case 2case 3case 4case 5case 6

Frequency shift (kHz)

T = 0.83 Tc (1.75 mK)Ω = Ω (T = Tc)P = 3.4 MPa

Results:• No change for cooling conditions nor with rotation• No signal for spin-wave vortex

signal

NMR in A phase under rotation (continuous vortex)

€

Rt2 = 1

Rt2 <1 Spin Wave attached to

the soft core vortex

Result for a bulk sample

　　　

(JLTP 60, 187 (1985) )

Ω×= 065.0total

vortexI

I

( Without rotation)

Change of the A-phase Texture due to Rotation

0.75

0.8

0.85

0.9

0.95

1

1.05

-6 -4 -2 0 2 4 6

case1case2case3case4case5case6

Rotation speed (rad/s)-0.5 0 0.5 1 1.5 2

0.00(rad/s)

-6.28(rad/s)

0.00(rad/s)

T = 0.83 TcP = 3.4 MPa

Frequency shift (kHz)

Nor

mal

ized

Pea

k H

eigh

t

The main peak height decreased and the spectrum becomes slightly broader to higher frequency : ( 0→-6.26 rad/s→0)

1) The peak height deceases for any change of rotation speed and direction.2) The A-phase texture is strongly pinned by aero

gel and is deformed elastically by rotation. (Annealing effects)

Summary for A-phase under rotaion

1) A-phase texture is strongly pinned by aerogel

2) The texture is slightly and elastically deformed by rotation

3) No signal for a soft core vortex even when it is cooled through Tc under 6.28 rad/s

● 　 Singular core vortex exits since the l-texture is strongly pinned

or ● The life time of spin-wave is short in aerogel and NMR spectrum for spin wave is broadened.

§3. B-phase under rotation

0

2 10-5

4 10-5

6 10-5

8 10-5

0.0001

-4 -2 0 2 4 6 8 10

Frequency shift(kHz)

T=0.77Tc,Ω=0

B-phase spectrum at rest

The cw-NMR spectrum is broader than that of the flare-out texture in bulk

-2 0 2 4 6 8

2.00 rad/s1.00 rad/s0.00 rad/s

2 - 0 rad/s: The spectrum shifted again.

-2 0 2 4 6 8

2.00 rad/s3.00 rad/s

2 and 3 rad/s: The absorption shifted to the higher frequency region.

-2 0 2 4 6 8

4.00 rad/s5.00 rad/s6.28 rad/s

4-6.28 rad/s: This change stopped.

P = 3.0 MPa, T = 0.59 TC ( in B-like Phase)

-2 0 2 4 6 8

6.28 rad/s5.50 rad/s4.00 rad/s3.00 rad/s

6.28 and 5.5 rad/s: The spectrum changed in a reverse way.

4 and 3 rad/s: NMR spectrum is almost the same as that taken before rotation.

Frequency shift (kHz)-2 0 2 4 6 8

0.00 rad/s1.00 rad/s

0 and 1 rad/s: No change

Frequency shift (kHz)

NM

R a

bsor

ptio

n (a

rb. u

nit) Acceleration Deceleration

NMR Spectra in Rotation

B-phase NMR under flow B-phase NMR

1) For small velocity

DSN VVV <−o0=θ

DSN VVV >−

2) For large velocityoo 4.630 <<θ

€

Δf =ΩB

2

2 fL

sin 2θ

=ΩB

2

2 fL

×4

51−

VD

VN − VS

⎛

⎝ ⎜

⎞

⎠ ⎟

2 ⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

: Relative velocity

€

Δf =ΩB

2

2 fL

sin 2θ

€

Δf = 0

Counter flow peak

Counter flow peaks for a bulk sample

Note:Flare-put texture for Ω=0

€

F1(Ω) ≡1

ITotal

I( f ,Ω)

1− f − fL( ) / f0fL

fL + f0∫ df

=1

πR 2

vN (r,Ω)− vS (r,Ω)

vdD∫ dS

Assume that some part is completely pinned and the other part is completely free.

( )df

fff

fI

IFFF

ff

fLTotal

L

L∫

+

−−

Ω=−Ω=Ω

0

0

111/1

),(1)0()()(

δδ

( )∫ Ω−=ΩR

dSrfffI0

),(),( δ

• Counterflow vs. Frequency shift f (r,Ω):

•NMR intensity I( f , Ω) vs. f (r,Ω): (Local Approx.)

0/)),((1

1),(),(

ffrfv

rvrv

Ld

SN

−Ω−=

Ω−Ω

-2 0 2 4 6 8

0.00 rad/s

6.28 rad/s

L

B

ff

2

2

0

Ω=

54

Å~

Frequency shift (kHz)

Analysis for counter flow peaks under rotation

Intensity of Counterflow

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

-6 -4 -2 0 2 4 6

1st

2nd3rd

cw-NMR absorption by flow

Ω (rad/s)

T = 0.68 Tc

€

VD∫

VN − VS

VD

dS

Ωc : critical velocity for creeping

　 of vortex 　

ΩD : critical velocity for n-texture

deformation

Note: no deformation until (VN-VS) > VD

Hysteresis curve of (VN-VS)

Hysteresis curve due to vortex pinning

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-6 -4 -2 0 2 4 6

Vns > 0Vns < 0

Rotation speed (rad/s)

T = 0.68 Tc

＝Moment of the relative velocity

/2 (rot/s)

(n –

s)

/2

Superfluid experiment in Al2O3

Detection of Persistent current

(H. Kojima et. al. P.R.L.27, 714 (1971) )

€

VN − VS

VD

dS0

∫

Flow pattern to explain the hysteresis curve for

Acceleration

CΩ<Ω CΩ>Ω

deceleration

€

VN − VS

VD

dS0

∫

Hysteresis curve for the flow pattern

∫−

dSV

VV

D

SN

(rad/s) Ω

C = 2.5 rad/s

Vortex is pinned until (VN-VS) exceeds the de-pinning critical velocity Vc

T=0.59 Tc

0

-6 -4 -2 0 2 4 6

dataVortex Pi nni ng ModelBul k l i qui d

In bulk liquid, vortices can move freely.

In aerogel

Counterflow decreases for Ω > Ωc.

vortices are strongly pinned

large counter flow velocity is needed for vortex creeping.

Moment of Counter flow

0.50.60.70.80.910.511.522.533.5

CTT /

Cri

tica

l ang

ular

vel

ocit

y (r

ad/s

) ΩC vs. reduced temperature

( )C

SSNC T

TFFV −∝−∝ 1/ ρ

Depinning Mechanism

d = 10.5 m

⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

4

1

)(

4ln

2 T

d

dVC ξπ

κ

Glaberson Donnely Instability

d: the average distance between pinning centers

Critical Velocity

W.I. Glaberson and R.J. Donnelly Phys. Rev. 141, 208 (1966)

:Counterflow :Self-induced velocity

⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

4

1

)(

4ln

2 T

d

dVC ξπ

κ

Critical velocity is determined by the average distance d

d = 10.5 m

Pinning is infinitely strong.

Glaberson Donnelly Instability

Critical Velocity for de-pinning

€

ρ SVS2

2= δ (FN − FS )∝ δΔ ∝ Δ

ΩC ∝ 1−T

TC

Vortex Pinning may occur due to a local inhomogeneities of the condensation energy Δ

This model has a mild temperature dependence, which should be observable in the experiment.

What determines d ?

Small-angle x-ray scattering(J. V. Porto and J. M. Parpia, P.R.B. 59, 14583 (1999) )

130 nm 5 nm

d = 10.5 m?

Summary for rotation experiment for superfluid 3He in aerogel

B-phase: The n-texture was deformed by flow and the counter-flow peak appeared.

The hysteresis was appeared when the relative flow velocity exceeded above Vc. The critical velocity did not depend on temperature This was caused by expansion of vortex(G-D instability) from the pinning center and the creeping of vortex started .

The average distance of the pinning centers was about 10 m

A-phase ： No vortex signal was observed in aerogel　　 ( this is different from bulk sample)

The l-texture is pinned to aerogel

Rotating Ultra-low Temperature Cryostat built at ISSP.

Nuclear StageRRR=500 (not well-annealed)Residual horizontal-field cancellation coil(No magnetic material near the cryostat)

○ Sub-mK temperature under 1 rot/sec○ Excess heat input due to a rotation of 1 rot/sec < 1 nW ○ Continuous run for one month after a demagnetization

Rotating ULT Cryostat and Experimental Set-up

Structure of vortex in bulk liquid−array of vortex

~ 100 nm

~ 10 m

A phase 4 typesB phase 3 types

Continuous

vortex

Singular Vortex

Analysis for counter flow peaks under rotation

€

f =ΩB

2

2 fL

×4

51−

VD

VN − VS

⎛

⎝ ⎜

⎞

⎠ ⎟

2 ⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

Derivation of (Vn-Vs) from cw-NMR spectrum

62

2

m

m

mV

f

DD

L

B

∗

=

Ω

ξh

€

ΔI( f )1− f / f3

dff1

f2∫ = VD∫

VN − VS

VD

dS

-1 0 1 2 3 4 5 6 7

0.00 rad/s

6.28 rad/s

€

ΔI(Δf )€

f1

€

f2

Frequency shift (kHz)

： Dipole frequency in B-phase

： Larmor frequency

:Critical velocity for Fredericks Transition

€

1

1− f / f2

=VN − VS

VD, where

€

f2 =ΩB

2

2 fL

×4

5

fL

-0.1

0

0.1

0.2

0.3

0.4

-6 -4 -2 0 2 4 6

1st ac1st dec2nd ac2nd dec3rd ac3rd dec

1st

2nd 3rd

Ωc

Ωd

Ωv

F1(Ω)

( / )Rotation speed rad s

-Ωv

-0.17

This pinned superflow at 0 rad/s is so stable that the dissipation was not observed within 40 hours.

1. 0 ~ D: No change due to insensitivity of n vector for |VN - VS| < VD

3. C < : Decrease of counterflow from the linear behavior of normal flow, it is due to appearance of superfluid velocity created by vortices.

rdrrnr

rzV VS ′′′×= ∫

κ

2)(ˆ

ˆ2

nV (r): vortex density

The curve showed the hysteresis behavior once exceeds C.

4. < V: The counterflow |VN - VS| increased again even in deceleration and remained at 0 rad/s, which shows the superflow remained at 0 rad/s by pinning of vortices.

2. D ~ C: Linear increase due to the solid body rotation of Normal fluid velocity

Ω×=rVN

T = 0.59 TC

dSv

rvrv

R Dd

SN∫Ω−Ω ),(),(1

2

Counterflow Intensity vs. Ω

Small-angle x-ray scattering(J. V. Porto and J. M. Parpia, P.R.B. 59, 14583 (1999) )

130 nm 5 nm

d = 10.5 m?