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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

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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


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  • 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

    • 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

    • 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 experimentation

    • A unified description of nuclear structure and nuclear reaction aspects

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Diagonalization Shell Model

(medium-mass nuclei reached;dimensions 109!)

Honma, Otsuka et al., PRC69, 034335 (2004)

and ENAM’04




Configuration space

Effective Interactions

Open channels

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Coupling of nuclear structure and reaction theory

Thomas/Ehrman shift

  • ab-initio description

  • continuum shell model

    • Real-energy CSM (Hilbert space formalism)

    • Gamow Shell Model (Rigged Hilbert space)

  • cluster models

  • 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

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Proton and Alpha


Narrow resonances

A. Arima and S. Yoshida,

Nucl. Phys. A219, 475 (1974)

Non-adiabatic approaches

See references in

A.T. Kruppa and W. Nazarewicz,

Phys. Rev. C69, 054311 (2004)

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exotic decay

heavy ion coll.









Fission and Fusion

nuclear collective dynamics

  • 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

  • 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

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Ab-initio, EFT,…

talk by Barrett

  • 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


K. Nollett, Phys.Rev. C63, 05400 (2001)

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

  • 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)

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

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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:

  • 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

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Resonant (Gamow) states

Also true in many-channel case!



complex pole

of the S-matrix

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

  • Siegert, Phys. Rev. 36, 750 (1939)

  • Gamow, Z. Phys. 51, 204 (1928)

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One-body basis

Contour is discretized

GSM Hamiltonian matrix

is complex symmetric

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[email protected] 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?

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Spectroscopic factors in GSM

One-nucleon radial overlap integral:

One-nucleon radial overlap integral in GSM:

usually approximated by

a WS wave function at

properly adjusted


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 extracted from experimental cross section

Prone to significant errors close to particle


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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.

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Coupling between analog states

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

C.F. Moore et al.

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

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N. Michel et al., GSM


J. Rotureau et al., DMRG


The GSM Hamiltonian

For the O-isotopes:

zero surface delta interaction

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

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

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  • 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

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The continuum in reaction theory channels


Spectroscopic information

comes from reactions

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

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

J. Tostevin PRC66, 024607 (2002)

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Why is the shell structure changing at extreme isospins? channels






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talk by Ogata channels

Kyoto (postdoc)

talk by Nakatsukasa

Kyoto (vis. prof.)

Coupling of nuclear structure and reaction theory

(microscopic treatment of open channels)

Current US effort (RIATG)

  • Ab initio reaction theory

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

  • CDCC

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

  • Continuum Shell Model

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

  • Narrow resonances (proton emitters, SF)

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

  • Fission

    • LANL, LLNL, ORNL, Seattle

  • Fusion

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

  • Continuum QRPA

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

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Conclusions channels

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!

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120 channelsSn











Gamow states of finite potential

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Resonance states, properties channels

Can one calculate G with sufficient accuracy?

For narrow resonances,

explicit time

propagation difficult!

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Resonant states, properties channels

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

S can be taken as a sphere of radius R:

Also true in many-channel case!

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“Spin-orbit splitting” in GSM channels

Influence of configuration mixing and continuum coupling

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S channels1n(6He)=0