Scales of critically stable few-body halo system
-
Upload
fulton-larson -
Category
Documents
-
view
21 -
download
0
description
Transcript of Scales of critically stable few-body halo system
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:
Next: