Happy Birthday JUSTIPEN!!! Happy Birthday JUSTIPEN!!!

Happy Birthday JUSTIPEN!!!

Happy Birthday JUSTIPEN!!!

Nuclei as open quantum many-body systems

Witold Nazarewicz (Tennessee)

JUSTIPEN Meeting, RIKEN July 10-11, 2006

Introduction and motivation Some recent examples

US effort and potential for JUSTIPEN activities

Summary

• Interdisciplinary science Interdisciplinary science

Much interest devoted to the study of small quantum systems that are coupled to an environment of scattering wave functions (quantum dots, molecules, clusters, nuclei, hadrons)

• New regime New regime

Weakly bound or unbound states cannot be treated in a Closed

Quantum System formalism. A consistent description of the interplay between scattering states, resonances, and bound states in the many- body wave function requires an Open Quantum System formulation.

• No escape No escape

Properties of unbound states lying above the particle (or cluster)

threshold directly impact properties of bound states and the continuum structure.

• Splendid opportunity for theory of exotic nuclei and RNB Splendid opportunity for theory of exotic nuclei and RNB experimentation

experimentation

A unified description of nuclear structure and nuclear reaction aspects

Diagonalization Shell Model

(medium-mass nuclei reached;dimensions 10⁹!)

Martinez-Pinedo ENAM’04

Honma, Otsuka et al., PRC69, 034335 (2004) and ENAM’04

Challenges:

Configuration space Effective Interactions Open channels

Challenges:

Configuration space Effective Interactions Open channels

Coupling of nuclear structure and reaction theory

ab-initio description continuum shell model

Real-energy CSM (Hilbert space formalism) Gamow Shell Model (Rigged Hilbert space)

cluster models

Thomas/Ehrman shift

Challenges:

•Treatment of continuum in ab initio

•How to optimize CSM configurations spaces?

•Effective forces in CSM

•Multi- channel reaction theory

•Halo nuclei: an ultimate challenge!

•virtual state

•center of mass

•cross-shell effects

Proton and Alpha Emitters

Non-adiabatic approaches

See references in

A.T. Kruppa and W. Nazarewicz, Phys. Rev. C69, 054311 (2004)

Narrow resonances

A. Arima and S. Yoshida, Nucl. Phys. A219, 475 (1974)

Fission and Fusion

nuclear collective dynamics

Challenges:

•selection of appropriate degrees of freedom

•simultaneous treatment of symmetry

•coupling to continuum in weakly bound systems

•dynamical corrections; fundamental theoretical problems.

•rotational, vibrational, translational

•particle number

•isospin

Variety of phenomena:

•symmetry breaking and quantum corrections

•LACM: fission, fusion, coexistence

•phase transitional behavior

•new kinds of deformations

Significant computational resources required:

•Generator Coordinate Method

•Projection techniques

•Imaginary time method (instanton techniques)

•QRPA and related methods

•TDHFB, ATDHF, and related methods

Q

Q

E

coexistence shape

coexistence

Q

Q

Q E

exotic decay heavy ion coll.

fission/fusion exotic decay heavy ion coll.

Challenges:

• Treatment of continuum in VMC and GFMC

• NCSM calculations with A=4 projectiles

• Low-energy reactions in NCSM

P. Navratil, Phys. Rev. C70, 054324 (2004)

Ab-initio, EFT,…

ab-initio description (GFMC, NCSM, Faddeyev, …)

realistic wave functions and electroweak currents applications to radiative capture reactions

cluster form factors, spectroscopic factors

EFT description

reactions on deuterium, halo nuclei ^EXAMPLES:K. Nollett, Phys.Rev. C63, 05400 (2001) P. Navratil, Phys.Rev. C70, 054324 (2004)

P. Navratil et al., Phys. Lett. B634, 191 (2006)

talk by Barrett talk by Barrett

Continuum Shell Model -an old tool!

Continuum Shell Model -an old tool!

• U. Fano, Phys. Rev. 124, 1866 (1961)

• C. Mahaux and H. Weidenmüller: “Shell Model Approach to Nuclear Reactions” 1969

• H. W. Bartz et al., Nucl. Phys. A275, 111 (1977)

• D. Halderson and R.J. Philpott, Nucl. Phys. A345, 141

• …• J. Okolowicz, M. Ploszajczak, I. Rotter, Phys. Rep. 374, 271 (2003)

Recent Developments:

Recent Developments:

SMEC•K. Bennaceur et al., Nucl. Phys. A651, 289 (1999)

•K. Bennaceur et al., Nucl. Phys. A671, 203 (2000)

•N. Michel et al., Nucl. Phys. A703, 202 (2002)

•Y. Luo et al., nucl-th/0201073 Gamow Shell Model

•N. Michel et al., Phys. Rev. Lett. 89, 042502 (2002)

•N. Michel et al., Phys. Rev. C67, 054311 (2003)

•N. Michel et al., Phys. Rev. C70, 064311 (2004)

•R. Id Betan et al., Phys. Rev. Lett. 89, 042501 (2002)

•R. Id Betan et al., Phys. Rev. C67, 014322 (2003)

•G. Hagen et al, Phys. Rev. C71, 044314 (2005) Other approaches

Resonant (Gamow) states Resonant (Gamow) states

H ˆ Y = e - i ^G 2 Ê Ë ˆ

¯ Y

Y 0, ₍ k _{) = 0, Y (} r ^r , k ₎ æ æ æ

Æ O

_{( )} kr

k

= 2 m

h

e

- i ^G

2 Ê Ë ˆ

¯ complex pole

of the S-matrix outgoi

ng soluti

on

•Humblet and Rosenfeld, Nucl. Phys. 26, 529 (1961)

•Siegert, Phys. Rev. 36, 750 (1939)

•Gamow, Z. Phys. 51, 204 (1928)

Also true in many-channel case!

Contour is discretized

GSM Hamiltonian matrix is complex symmetric

One-body basis One-body basis

NSCL@MSU 2005

• How well does nuclear theory describe the energy and spatial structure of the single particle wave functions?

• Can the parameterized N-N force adequately describe the short-range correlations among the nucleons?

Electron-induced proton knock-out

• What is the role of long-range correlations and open channels?

• How well does nuclear theory describe the energy and spatial structure of the single particle wave functions?

• Can the parameterized N-N force adequately describe the short-range correlations among the nucleons?

Electron-induced proton knock-out

• What is the role of long-range correlations and open channels?

Spectroscopic factors in GSM

One-nucleon radial overlap integral:

One-nucleon radial overlap integral in GSM:

Spectroscopic factor in GSM:

In contrast to the standard CQS SM,

the final result is independent of the s.p.~basis.

In usual applications, only one term remains usually approximated by a WS wave function at properly adjusted

energy

Usually extracted from experimental cross section

Prone to significant errors close to particle thresholds

Threshold anomaly

E.P. Wigner, Phys. Rev. 73, 1002 (1948), the Wigner cusp G. Breit, Phys. Rev. 107, 923 (1957)

A.I. Baz’, JETP 33, 923 (1957)

A.I. Baz', Ya.B. Zel'dovich, and A.M. Perelomov, Scattering Reactions and Decay in Nonrelativistic Quantum Mechanics, Nauka 1966

A.M. Lane, Phys. Lett. 32B, 159 (1970)

S.N. Abramovich, B.Ya. Guzhovskii, and L.M. Lazarev, Part. and Nucl. 23, 305 (1992).

• The threshold is a branching point.

• The threshold effects originate in conservation of the flux.

• If a new channel opens, a redistribution of the flux in other open

channels appears, i.e. a modification of their reaction cross-sections.

• The shape of the cusp depends strongly on the orbital angular momentum.

Y(b,a)X

X(a,b)Y

C.F. Moore et al.

Phys. Rev. Lett. 17, 926 (1966)

Coupling between analog states

in (d,p) and (d,n)

C.F. Moore et al., Phys. Rev. Lett. 17, 926 (1966)

For the He-isotopes: finite-range surface Gaussian interaction For the O-isotopes:

zero surface delta interaction

The GSM Hamiltonian

N. Michel et al., GSM

nucl-th/0601055

J. Rotureau et al., DMRG

nucl-th/0603021

non-perturbative behavior

scattering continuum essential

WS potential depth decreased to bind ⁷He. Monopole SGI strength varied

WS potential depth varied

•The new paradigm is born: shell-model treatment of open channels

•The non-resonant continuum is important for the

spectroscopy of weakly bound nuclei (energy shifts of excited states, additional binding,…)

•SFs, cross sections, etc., exhibit a non-perturbative and

non-analytic behavior (cusp effects) close to the particle-

emission thresholds. These anomalies strongly depend on

orbital angular momentum

Spectroscopic information comes from reactions

The continuum in reaction theory

Parallel momentum distributions for nucleon removal from ¹⁵C + ⁹Be

J. Tostevin PRC66, 024607 (2002)

Parallel momentum distributions for nucleon removal from ¹⁵C + ⁹Be

J. Tostevin PRC66, 024607 (2002)

Experiments with exotic nuclei will require a firm understanding of reaction theory in order to

interpret the meaning of the data Experiments with exotic nuclei will require a firm understanding of reaction theory in order to

interpret the meaning of the data

CDCC

Why is the shell structure changing at extreme isospins?

Interactions

Many-body Correlations

ChannelsOpen

Coupling of nuclear structure and reaction theory

(microscopic treatment of open channels)

Current US effort (RIATG)

• Ab initio reaction theory Ab initio reaction theory

• LLNL (Navratil, …), Arizona (Bertulani, Barrett,…), ANL (Nollett, …)…

• CDCC CDCC

• Nunes (MSU), Thompson (LLNL), Mukhamedzhanov (Texas A&M)…

• Continuum Shell Model Continuum Shell Model

• Michel, Nazarewicz, Rotureau (ORNL), Volya (FSU), Zelevinsky (MSU).…

• Narrow resonances (proton emitters, SF) Narrow resonances (proton emitters, SF)

• Esbensen (ANL), Nunes (MSU), Mukhamedzhanov (Texas A&M), Nazarewicz (ORNL)

• Fission Fission

• LANL, LLNL, ORNL, Seattle

• Fusion

• Balantekin (Wisconsin), Oberacker, Umar (Vanderbilt)

• Continuum QRPA

• Bertsch (Seattle), Engel (UNC), Nazarewicz, Stoitsov (ORNL), Shlomo (Texas A&M),

talk by Ogata talk by Ogata

Kyoto (postdoc) Kyoto (postdoc)

Kyoto (vis. prof.) Kyoto (vis. prof.) talk by Nakatsukasa

talk by Nakatsukasa

Conclusions

Tying nuclear structure directly to nuclear reactions within a coherent framework applicable throughout the nuclear landscape is an important goal. For light nuclei, ab-initio methods hold the promise of direct calculation of low-energy scattering processes, including those important in nuclear astrophysics, and tests of fundamental symmetries. In nuclear structure for heavier nuclei, the continuum shell model and modern mean-field theories allow for the consistent treatment of open channels, thus linking the description of bound and unbound nuclear states and direct reactions. On the reaction side, better treatment of nuclear structure aspects is equally crucial. The battleground in this task is the newly opening territory of weakly bound nuclei where the structure and reaction aspects are interwoven and where interpretation of future data will require advances in understanding of the reaction mechanism.

(NSAC Nuclear Theory Report)

JUSTIPEN will play important role!

n ^p

120Sn

120Sn

e

e

Unbound states Unbound states Discrete

(bound) states Discrete

(bound) states

V = V

+ V

+ V

Gamow states of finite potential

Gamow states of finite potential

Resonance states, properties Resonance states, properties

F t ( ) = exp - i

h t e - i G 2 Ê Ë ˆ Ï ¯

Ì Ó

¸ ˝

˛ Y ( r r ,k)

F t ( ) F 0 ( ) = exp - G 2h t Ï Ì

Ó

¸ ˝

˛ Y ( r

r ,k) Y( r r ,k)

T

= ln 2 h

G , h = 6.58◊10

MeV ◊sec

Can one calculate

Can one calculate G G with sufficient accuracy? with sufficient accuracy?

€

T

≈ 3 ⋅10

sec = 3babysec. For narrow resonances, For narrow resonances, explicit time

explicit time

propagation difficult!

propagation difficult!

Resonant states, properties Resonant states, properties

Humblet and Rosenfeld: Nucl. Phys. 26, 529 (1961)

h r

j d r S

Ú

^{= G r d}

^r

Ú

r j = h

2 mi ^Y

*

r

— Y - Y r

— Y

( ), r = Y

Y

— r r

j - G

h r = 0 r

— r

j + ∂r

∂t = 0 Ê

Ë ˆ

¯ S S can be taken as a sphere of radius can be taken as a sphere of radius R: R:

G = hR

Ú j

dW r d

r

V

Ú

Also true in many-channel case!

“Spin-orbit splitting” in GSM

Influence of configuration mixing and continuum coupling

S

(

He)=0