In quest of 4He supersolid

a work with J. Peter Toennies (MPI-DSO Göttingen), Franco Dalfovo (Uni Trento),

Robert Grisenti & Manuel Käsz (Uni Frankfurt), Pablo Nieto (Automoma Madrid)

History of a conjecture: BEC in a quantum solid ?

Vacancy diffusivity and solid 4He Poisson ratio

The Geyser effect in solid 4He vacuum expansion

Bernoulli flow of a nominal 4He solid

Suppression of flow anomalies by 1% 3He

4He vacuum expansion from low -T sources

Firenze 2005 - 1

History of a conjecture:

BEC in a quantum solid?

1969

Andreev $ Lifshitz

1970

Chester Leggett

1977

Greywall

2004

Kim & Chan

2004

Ceperley & Bernu

Firenze 2005 - 2

Kim & Chan

2004

measurements of non-classical rotational inertia

Firenze 2005 - 3

no trend ?

Kim

& C

han

Firenze 2005 - 4

Galli & Reatto

2001

(a) no ground state vacancies but only thermal vacancies

(b-d) ground state + thermal vacancies (for different vacancy formation energies)

what about injected (non-equilibrium) vacancies?

Firenze 2005 - 5

Vacuum expansion of solid 4He

)/(4 2dmkTSPu detdet

Firenze 2005 - 6

2/1/

200

)/2(

)/4(

s

s

P

dAuu

continuity

Bernoulli

Firenze 2005 - 7

4He phase diagram

Firenze 2005 - 8

The Geyser effect

Firenze 2005 - 9

Period vs. T at constant pressure

40.7 bar

35.0 bar

32.0 bar

TTm 0

Firenze 2005 - 10

Period versus P0 at constant temperature

3

2

2

1

)(0

mPP

Bernoulli

Firenze 2005 - 11

1)( min,/0 lsPPP

Ps/l information on dynamical processes inside solid 4He

P information on Poisson ratio of solid 4He

Firenze 2005 - 12

Poisson ratio of solid 4He

Firenze 2005 - 13

Plastic flowmotion of dislocation

motion of vacancies dominant in solid He(high diffusivity!)

Polturak et al experiment (PRL 1998)

vacancy injection at s/l

interface + sweeping by

pressure gradient

Firenze 2005 - 14

PVa F

kTD vvvv 0uFu

v

P

u

v

P

u

Vacancy drift

solid 4He p-type SC

Firenze 2005 - 15

Va = V* - Va Va = 35.15 Å3 (atomic volume)

V* 0.45Va (vacancy isobaric formation

volume)

A0

As/l

L

Virtual volume to be filled by vacancies

in the time L/u0

u0

a

lsv

VX

AA

u

u

0

0/

0 2

/1

The vacancy mechanism

poise1016

8

0

/0

aav

lssolid VXV

AA

Firenze 2005 - 16

accumulation of vacancies up to a critical concentration Xc

drift + diffusion

diffusion

Pre

ssur

e

distance from s/l interface

0 L

COLLAPSE!

Geyser mechanism

vacancy bleaching &

resetting of initial conditions

Data on vacancy diffusivity and concentration in 4He

Firenze 2005 - 17

Transport theory

),(2

2txG

vvC

x

vu

x

vD

t

v

rvvv

PVC vvv2

),()()()()(),( 00 txuXxLxtXtxG s

0),(),( XtxXtxv

v

uvuvuu v

vvv

)(

vCvuvPVvu

vx

v

v

uvuvvu

vvvvvionlinearizat

vvv

')('

'')(

2

vreffrr C211,

1

Generation function

surface generation velocity

Firenze 2005 - 18

]'4

'4

erfc[),(0

'4/)'(/'/21

2

t

tDxtuts

to

vvrr eetvD

dtutvD

xtvueXtxv

*erf

*

2erf

),0(),0(')(

/21*/

41

0

t

u

utee

tuX

tvutvDtj

vv

s

v

ttvv

vvosc

v

)/(* rvrv

FukTuD vvv /4/4 2

Solution for L

Excess vacancies

Current at the s/l interface (x = 0) due to excess vacancies

rvs uu /2

= surface depletion layer thickness

Firenze 2005 - 19

- the shape of the current depends on 2 parameters (, )

- the time scale implies another parameter (v)

- the ratio of the oscillation amplitude to the constant

background is measured by X0Vauv/u0 and is of the order

of a few percent (as seen in experiment)

fitting

reduced form:

1*//2/ vvsv uuty

]erferf2[)( 041 yye

euXtj y

y

vosc

Firenze 2005 - 20

Theory vs. experimentDv = 1.3·10-5 cm2/s

v = 5.4·1010 s/g

uv = 2.0·10-3 cm/s

us = 2uv

s = 60 s

v = 13 s

* = 10.7 s

0 = 82 s

P0 = 31 bar T0 = 1.74 K

best fit with = 4 = 1.214

Firenze 2005 - 21

better fits are obtained with finite

L (one more parameter)

large means fast recombination

Firenze 2005 - 22

Period 0 vs. diffusivity

finite L approximate solution by Green’s function method

2

021

010 )*

(*

v

c

c

XX

XXerf Xc = critical concentration

v

vc

X

X

*1

*1)( 21

0

L

D

XX

v

c

)(0

0

2

LL

L

Firenze 2005 - 23

mm3.00

2

L

LvD

64.05.0

Firenze 2005 - 24

Anomalies below the ’

point!

Firenze 2005 - 25

a sharp transition in the flow regime at 1.58 K !

Firenze 2005 - 26

Effects of 3He

on the anomalies

from R. Richardson et al

Firenze 2005 - 27

3 He-vacancy binding energy

Firenze 2005 - 28

normal behaviour induced by less than

1% 3He !

Firenze 2005 - 29

CONCLUSIONS

1. The geyser effect indicates (via Bernoulli’s law) an oscillation of the s/l (quasi-)equilibrium pressure at a given T: vacancy concentration appears to be the only system variable which can give such effect.

2. Below the ’ temperature flow anomalies are observed:

(a) The most dramatic one is the occurrence of a Bernoulli flow corresponding to pressures > Pm, at which 4He should be solid. (b) Below 1.58 K a sharp drop of the geyser period signals a dramatic change in the flow properties of solid 4He. These anomalies, suggesting superflow conditions, are attributed to injected excess vacancies, and agree with Galli and Reatto predictions for a vacancy-induced (Andreev-Lifshitz) supersolid phase.

3. A 3He concentration of 0.1% is shown to suppress the flow anomalies, suggesting a quantum nature of the superflow.

Firenze 2005 - 30

gA

I

L

PP

Ag

I

LL

PP

mL

L

000

0

2 I = flow (current), assumed approximately constant over a period

A0 = tube section

A = average flow cross section in the s/l interface

region (A is slightly < A0)

g0 = conductivity far away from the s/l interface

due to the equilibrium concentration of

vacancies X0 : g0 = X0v where v is the

vacancy mobility

g = conductivity near the s/l interface: g = Xv

where X is the actual vacancy concentration

near the s/l interface. Immediately after the

collapse (brown and red lines in the figure)

X << X0 and g << g0 whereas just

before the collapse (green line) X >> X0 and

g >> g0 . When X = X0 (purple line) the

gradient is the same between 0 and L0.

The corresponding gradients are inversely

proportional (see figure)!

1 Pressure gradients:

3 Length L of the gradient near the s/l interface (solve the above system for PL and L):

AggA

Ag

I

gA

L

PP

L

L m

00

00

0

0

01

where the term in parenthesis is constant. For A A0 it appears that L grows with g/g0 = X/X0 as qualitatively shown in the figure. Thus the sensor during the period measures a pressure varying from P0 to Ps/l