Scales of critically stable few-body halo systemScales of critically stable few-body halo system

Tobias Frederico

Instituto Tecnológico de AeronáuticaSão José dos Campos - Brazil

Marcelo T. Yamashita – Itapeva /UnespLauro Tomio – IFT/Unesp/São PauloAntonio Delfino – UFF/Niterói Sadhan K. Adhikari - IFT/Unesp/São Paulo

Collaborators

FB18, Santos, Aug.21-26, 2006

FB18, Aug. 2006

Nuclear and Atomic weakly bound three-body halo systemsHow to study weakly bound three-body systems?Thomas-Efimov effectThomas-Efimov effect Scaling limit & limit cycle, scaling functions and correlations between observablesScaling limit & limit cycle, scaling functions and correlations between observablesGeneral classification scheme: n-n-c or A-A-BGeneral classification scheme: n-n-c or A-A-B Threshold conditions for an excited N+1Threshold conditions for an excited N+1 Efimov stateEfimov state Three-bosons: analytic structure & Efimov state trajectoryThree-bosons: analytic structure & Efimov state trajectory Root mean square radiiFour boson systems: new scale?Summary and perspectives

OUTLINEOUTLINE

FB18, Aug. 2006Two-neutron halo nucleus

First observation

6He T. Bjerge, Nature 138, 400 (1936)

11Li colliding with some targets growth of the cross section

Tanihata et al., Phys. Rev. Lett. 55, 2676 (1985)

T. Kobayashi et al. Phys. Lett. B 232, 51 (1989)Li-9

neutrons

FB18, Aug. 2006 Nuclear weakly bound three-body halo systems

core

n n

core-neutron-neutron halo nuclei

11Li 14Be 20C

Binding energy ~ MeV or < MeV

Rnn(Exp) ~ 6 - 8 fm (11Li)

F. M. Marqués et al. Phys. Rev. C 64, 061301 (2001)

M. Petrascu et al. Nucl. Phys. A 738, 503 (2004)

FB18, Aug. 2006Atomic weakly bound three-body systems

A

B B

A-B-B weakly bound moleculesA-B-B weakly bound molecules ultra-low binding ~ mK or < mK

133Cs3 (trapped ultracold gas near a Feshbach resonance)

4He3 4He2 – 7Li 4He2 – 6Li 4He2 – 23Na R4He-4He~ 10 A

o dimer R4He-4He~ 50 A

o

FB18, Aug. 2006How to study weakly bound three-body systems?

Use a realistic interaction and calculate the Hamiltonian eigenstates....

What details of the interaction are important for the results?

Large systems are peculiar: size >> interaction range!

....and the eigenfunction of the Hamiltonian satisfies a free Schrödinger equation almost everywhere for nonzero interparticle distances!

asymptotic wf behaviour & universalityZero-range

interaction

FB18, Aug. 2006How to study weakly bound three-body systems?

Charateristic phenomena:

Thomas collapse (1935) and Efimov effect (1970)

ro 0 |a|

??? infinitely many weakly bound states |a|/ro

Thomas-Efimov effect!

8

8

FB18, Aug. 2006How to study weakly bound three-body systems?Thomas-Efimov effectThomas-Efimov effect

Skorniakov and Ter-Martirosian equations (1956)

Thomas collapse: 8

Efimov effect: 0

0

Adhikari,TF,Goldman, PRL74 (1995) 487

= /

= /

3 3 = /

FB18, Aug. 2006

Scaling limit:Frederico et al PRA60 (1999)R9Yamashita et al PRA66(2003)052702

Scaling limit & limit cycleScaling limit & limit cycle

Limit cycle:Mohr et al Ann.Phys. 321 (2006)225

Efimov 1970Scaling function

FB18, Aug. 2006Scaling functions: Correlation between observablesScaling functions: Correlation between observables

Correlation between S-wave observables

Phillips plot: triton B.E. X doublet scattering length

2nd order neutron-deuteron polarization observables X triton B.E.

Trapped atomic trimer B.E. X recombination rate

FB18, Aug. 2006

Three-boson wave function:

Weakly bound system wave function & contact interactionWeakly bound system wave function & contact interaction

+ (12) + (13)

q1

R1

(1)

(2) (3)

FB18, Aug. 2006General classification scheme: n-n-c or A-A-BGeneral classification scheme: n-n-c or A-A-B

BORROMEANBORROMEAN TANGOTANGO

SAMBASAMBA ALL-BOUNDALL-BOUND

bound statevirtual state

Yamashita, Tomio and T. F. Nucl. Phys. A 735, 40 (2004)

FB18, Aug. 2006General classification scheme: n-n-c or A-A-BGeneral classification scheme: n-n-c or A-A-B

Scales: Scales:

Energy of the bound/virtual nn system

Energy of the bound/virtual nc system

Energy of the Nth state of the nnc system

A = mass of the core

FB18, Aug. 2006

Amorim,TF,Tomio PRC56(1997)2378

BorromeanBorromean

SambaSamba

TangoTango

All-boundAll-bound

Halo-nuclei: Threshold for an excited N+1Halo-nuclei: Threshold for an excited N+1 Efimov state Efimov state

Knn=(Bnn)1/2

Knc=(Bnc)1/2

nn virtual nn bound

nc virtual

nc bound

FB18, Aug. 2006Weakly bound molecules: Weakly bound molecules: Threshold for an excited N+1Threshold for an excited N+1 Efimov stateEfimov state

Delfino,TF,Tomio JCP 113 (2000) 7874

All-boundAll-bound

TangoTango

SambaSamba

BorromeanBorromean

Kaa=(Baa)1/2

Kab=(Bab)1/2

FB18, Aug. 2006

Bound 3-body state

-E2-E3

Virtual 3-body state

Three-body cutTwo-body cut

Three-bosons: analytic structure & Efimov state trajectoryThree-bosons: analytic structure & Efimov state trajectory

-E3 (N) Three-body cut

Bound 2-body stateBound 2-body state

x

x

x

-E2

Virtual 3-body state

x-E3

(N+1)

x

FB18, Aug. 2006 Efimov state trajectory: 2-body boundEfimov state trajectory: 2-body bound

FB18, Aug. 2006

-E3 (N)

3-body Resonance

Three-body cut

x

x

Three-bosons: analytic structure & Efimov state trajectoryThree-bosons: analytic structure & Efimov state trajectory

Bound 3-body state

-E3 Three-body cut

Virtual 2-body stateVirtual 2-body state

x

x

3-body Resonance

x-E3

(N+1)

FB18, Aug. 2006Efimov state trajectory: 2-body virtualEfimov state trajectory: 2-body virtual

S-wave three-boson resonance S-wave three-boson resonance

Evidence of continuum resonances in recombination of ultracold Cs atoms

FB18, Aug. 2006Evidence of continuum resonances in ultracold cesium gasEvidence of continuum resonances in ultracold cesium gas

M.T. Yamashita, “Triatomic states in ultracold gases”Parallel session R6-16, Friday

FB18, Aug. 2006

Threshold for an excited N+1Threshold for an excited N+1 Efimov state Efimov state

Arora, Mazumdar, Bhasin, PRC69(2004)061301(R)Mazuumdar, Rao, Bhasin, PRL97(2006)062503 Resonance in n+19C

FB18, Aug. 2006Root mean square radii

CM<r2

AB>

<r2A>

<r2B>

<r2BB>

A

B B

FB18, Aug. 2006Root mean square radiiRoot mean square radii

Scaling functions for the radii

,,

,,

333

2

333

2

E

E

E

EREr

E

E

E

EREr

ABAACM

ABAAAA

= A or B+ two-body bound state- two-body virtual state

FB18, Aug. 2006

Caroço E3 (MeV) EnA (MeV) √<rnn2> (fm) √<rnn

2>EXP (fm)

0 5.14He 0.973 0.3 (v) 4.6 5.9 ± 1.2

4.0 (v) 3.69Li 0.32 0 9.2 6.6 ± 1.5

0.8 (v) 5.9

0 9.79Li 0.29 0.05 (v) 8.5 6.6 ± 1.5

0.8 (v) 6.7

0 8.69Li 0.37 0.05 (v) 7.7 6.6 ± 1.5

0.8 (v) 6.212Be 1.337 0 4.6 5.4 ± 1.0

0.2 (v) 4.218C 3.50 0.16 3.0 -

0.53 4.4

Root mean square radiiRoot mean square radii

Yamashita, Tomio and T. F. Nucl. Phys. A 735, 40 (2004)

Core

Exp:

FB18, Aug. 2006Root mean square radiiRoot mean square radii

33

2

33

2

E

EK

E

EK

E

EK

E

EK

nnnn

nnnn

nAnA

nAnA

1.02 nAK

200A

nA boundnA virtual

nA boundnA virtual

-1.0 -0.5 0.0 0.5 1.00.7

0.8

0.9

1.0

0.6

0.7

0.8

Samba Borromean All-Bound

Tango

Samba Borromean

All-Bound Tango

Knn

/|KnA

|

(<r nn

2 >|E

3|)1/

2(<

r nA

2 >|E

3|)1/

2

FB18, Aug. 2006Root mean square radii

BORROMEAN

TANGO

SAMBA

ALL-BOUND

bound statevirtual state

For a fixed EFor a fixed E33

>

>

>

FB18, Aug. 2006Neutron-neutron correlation functionNeutron-neutron correlation function

Radii are experimentally extracted from

correlation function

R. Hanbury-Brown and R. Q. Twiss (HBT) - NATURE

177, 27 (1956)178, 1046 (1956)178, 1447 (1956)

First used in astrophysics

Nuclear Physics

FB18, Aug. 2006

nnA

AAA

Annqqqd

pqqdpC

3

23 ,

pA

qA

A

n n' 2A

An

qpq

2

AAn

qpq

One-body density

2

3

2, AnnA

AnnAAnnA

qqqqqdq

AA pq

, Breakup amplitude including the FSI between the neutrons

ipp

pqpd

ipEpq

A

A

Ann

AA 223

2 ,2/1,

is the three-body wave function

Neutron-neutron correlation function

FB18, Aug. 2006

0 20 40 60 800

2

4

6

8

10

12

14

0 100 200 300 4000

2

4

6

8

10

14Be

Cnn

pA[MeV/c]

F. M. Marqués et al.Phys. Rev. C 64, 061301 (2001)

F. M. Marqués et al.Phys. Lett. B 476, 219 (2000)

E3 = 1.337 MeVEnA = 0.2 MeVEnn = 0.143 MeV

asymptotic region ?

x1.425

Neutron-neutron correlation function

M. T. Yamashita, T. Frederico and L. Tomio Phys. Rev. C 72, 011601(R) (2005)

FB18, Aug. 2006

F. M. Marqués et al.Phys. Rev. C 64, 061301 (2001)

M. Petrascu et al.Nucl. Phys. A 738, 503 (2004)

E3 = 0.29 MeVEnA = 0.05 MeV

0 20 40 60 800

2

4

6

8

10

12

0 100 2000

1

2

3

4

5

6

11Li

Cnn

pA[MeV/c]

Enn = 0.143 MeV

E3 = 0.37 MeVEnA = 0.8 MeVE3 = 0.37 MeVEnA = 0.05 MeV

x2.5

Neutron-neutron correlation function

FB18, Aug. 2006

F. M. Marqués et al.Phys. Rev. C 64, 061301 (2001)

E3 = 0.973 MeVEnA = 4 MeV

0 20 40 60 800

2

4

6

8

0 200 400 6000

2

4

6

8

6He

Cnn

pA[MeV/c]

Enn = 0.143 MeV

E3 = 0.973 MeVEnA = 0

x1.12

Neutron-neutron correlation function

FB18, Aug. 2006

A E3 (mK) EBB (mK) EAB (mK) √<rBB2> (Å) √<rAB

2> (Å) √<rB2> (Å) √<rA

2> (Å)4He 106.0 1.31 1.31 9.45 9.45 5.55 5.556Li 31.4 1.31 0.12 16.91 16.38 10.50 8.147Li 45.7 1.31 2.16 14.94 13.88 9.34 6.31

23Na 103.1 1.31 28.98 11.66 9.54 8.12 1.94

Results for different radii of the molecular system ABB

Radii for weakly bound moleculesRadii for weakly bound molecules

Yamashita, Marques de Carvalho, Tomio , T. F., Phys. Rev. A 68, 012506 (2003)

FB18, Aug. 2006

0.0 0.2 0.4 0.6 0.8 1.0

0.4

0.6

0.8

1.0

1.2

(E2/E

3)1/2

(<r H

e

2 >S3)1/

2(<

r He-

He

2 >S3)1/

2

GroundFirst excited

Symbols fromP. Barletta and A. KievskyPhys. Rev. A 64, 042514 (2001)

squares - Ground statecircles - First excited state

Weakly bound molecules

FB18, Aug. 2006Four-boson system: new scale?Four-boson system: new scale?

no new scale

new scale

FB18, Aug. 2006Four-boson system: a new scale?

FB18, Aug. 2006Four-boson system: a new scale?

Tjon line: MeV

FB18, Aug. 2006Summary and perspectives

Zero-range model: classification of weakly-bound systems threshold conditions for excited states and resonances ( evidence of the trajectory of resonance in ultra-cold atoms)

6He, 11Li, 14Be, 20C

4He-4He-(4He, 6Li, 7Li, 23Na)

Neutron-neutron correlation function

Scattering, breakup of halo nuclei and weakly bound molecules: universal properties

Weakly bound & large systems: few scales regime

Exploration of the different possibilities of threshold conditions for resonances

Evidence for a four-boson scale

Four-boson excited states, resonances & scattering

Flexibility:

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