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Nucleus as an Open System: Continuum Shell Model and New Challenges Vladimir Zelevinsky NSCL/ Michigan State University Supported by NSF Caen, GANIL May 30, 2014. OUTLINE. From closed to open many-body systems

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as an Open System:

Continuum Shell Model and

New Challenges

Vladimir Zelevinsky

NSCL/ Michigan State University

Supported by NSF

Caen, GANIL May 30, 2014



  • From closed to open many-body systems
  • Effective non – Hermitian Hamiltonian
  • Doorways and phenomenon of super-radiance
  • Continuum shell model
  • Statistics of complex energies
  • Cross sections, resonances, correlations and fluctuations
  • Quantum signal transmission


  • NaftaliAuerbach (Tel Aviv University)
  • Luca Celardo (University of Breschia)
  • Felix Izrailev (University of Puebla)
  • Lev Kaplan (Tulane University)
  • GavriilShchedrin (MSU, TAMU)
  • Valentin Sokolov (BudkerInstutute)
  • SurenSorathia (University of Puebla)
  • Alexander Volya(Florida State University)
NSCL and FRIB Laboratory 543 employees, incl. 38 faculty, 59 graduate and 82 undergraduate studentsas of April 21, 2014
  • NSCL is funded by the U.S. National Science Foundation to operate a flagship user facility for rare isotope research and education in nuclear science, nuclear astrophysics, accelerator physics, and societal applications
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User group of over 1300 members with approx. 20 working groups


NSCL Science Is Aligned with National Priorities Articulated by National Research Council RISAC Report (2006), NSAC LRP (2007),NRC Decadal Survey of Nuclear Physics (2012), “Tribble Report” (2013)

Properties of nuclei – UNEDF SciDAC, FRIB Theory Center (?)

  • Develop a predictive model of nuclei and their interactions
  • Many-body quantum problem: intellectual overlap to mesoscopic science, quantum dots, atomic clusters, etc. – Mesoscopic Theory

Astrophysical processes – JINA

  • Origin of the elements in the cosmos
  • Explosive environments: novae, supernovae, X-ray bursts …
  • Properties of neutron stars

Tests of fundamental symmetries

  • Effects of symmetry violations are amplified in certain nuclei

Societal applications and benefits

  • Bio-medicine, energy, material sciences – Varian, isotope harvesting, …
  • National security – NNSA

Reaping benefits from recent investments while creating future opportunities

frib science is transformational
FRIB science answers big questionsFRIB Science is Transformational
  • FRIB physics is at the core of nuclear science:“To understand, predict, and use” (David Dean)
  • FRIB provides access to a vast unexplored terrain in the chart of nuclides
examples for cross disciplinary and applied research topics
Examples for Cross-Disciplinary and Applied Research Topics

Medical research

Examples: 47Sc, 62Zn, 64Cu, 67Cu, 68Ge, 149Tb, 153Gd, 168Ho, 177Lu, 188Re, 211At, 212Bi, 213Bi, 223Ra (DOE expert panel)

MSU Radiology Dept. interested in 60,61Cu

-emitters 149Tb, 211At: potential treatment of metastatic cancer

Plant biology: role of metals in plant metabolism

Environmental and geosciences: ground water, role of metals as catalysts

Engineering: advanced materials, radiation damage, diffusion studies

Toxicology: toxicology of metals

Biochemistry: role of metals in biological process and correlations to disease

Fisheries and Wildlife Sciences: movement of pollutants through environmental and biological systems

Reaction rates important for stockpile stewardship – non-classified research

Determination of extremely high neutron fluxes by activation analysis

Rare-isotope samples for (n,g), (n,n’), (n,2n), (n,f) e.g. 88,89Zr

Same technique important for astrophysics

Far from stability: surrogate reactions (d,p), (3He,a xn) …

Vision: Up to 10 Faculty Positions for Cross-Disciplinary and Applied Research


From closed to open (or marginally stable)

many-body system


Bound states

Mean field, quasiparticles


Residual interactions

Pairing, superfluidity

Collective modes

Quantum many-body chaos (GOE type)

Open systems:

Continuum energy spectrum

Unstable states, lifetimes

Decay channels (E,c)

Energy thresholds

Cross sections

Resonances, isolated or overlapping

Statistics of resonances and cross sections

Unified approach? (Many…)



From giant resonances to superradiance

The doorway state is connected directly to external world, other states (next level) only through the doorway.

Examples: IAS, single-particle resonance, giant resonances

at high excitation energy, intermediate structures.

Feshbach resonance in traps, superradiance

single particle decay in many body system
Single-particle decay in many-body system

Evolution of complex energies

  • 8 s.p. levels, 3 particles
  • One s.p. level in continuum

Total states 8!/(3! 5!)=56; states that decay fast 7!/(2! 5!)=21 – superradiant doorways

examples of superradiance
Examples of superradiance

Mechanism of superradiance

Interaction via continuum

Trapped states - self-organization

Narrow resonances and broad superradiant state in 12C

in the region of Delta

  • Optics
  • Molecules
  • Microwave cavities
  • Nuclei
  • Hadrons
  • Quantum computing
  • Measurement theory

Bartsch Eur. Phys. J. A 4, 209 (1999)

N. Auerbach, V.Z.. Phys. Lett. B590, 45 (2004)

physics and mathematics of coupling to continuum
Physics and mathematics of coupling to continuum
  • New partof Hamiltonian: coupling through continuum

[1] C. Mahaux and H. Weidenmüller, Shell-model approach to nuclear reactions,

North-Holland Publishing, Amsterdam 1969

two parts of coupling to continuum

State embedded in the continuum

  • Integration region involves no poles
Two parts of coupling to continuum
  • Form of the wave function and probability

(+) means + i0

(Eigenchannels in P-space)



Factorization (unitarity), energy dependence

(kinematic thresholds) , coupling through continuum

self energy interaction with continuum
Self energy, interaction with continuum
  • Gamow shell model


Correction to Harmonic Oscillator Wave Function

s,p, and d waves (red, blue, black)

N Michel, J. Phys. G: Nucl. Part. Phys.

36 (2009) 013101

  • Notes:
  • Wave functions are not HO
  • Phenomenological SM is adjusted to observation
  • No corrections for properly solved mean field
  • momentum

A. Volya, EPJ Web of Conf. 38, 03003 (2012).


Traditional shell-model

  • Continuum physics
  • The nuclear many-body problem
  • Effective non-Hermitianand
  • energy-dependent Hamiltonian
  • Channels (parent-daughter structure)
  • Bound states and resonances
  • Matrix inversion at all energies (time dependent approach)
  • Single-particles state (particle in the well)
  • Many-body states (slater determinants)
  • Hamiltonian and Hamiltonian matrix
  • Matrix diagonalization

Formally exact approach

Limit of the traditional shell model

Unitarity of the scattering matrix

  • Intrinsic states: Q-space
    • States of fixed symmetry
    • Unperturbed energies e1; some e1>0
    • Hermitian interaction V
  • Continuum states: P-space
    • Channels and their thresholds Ecth
    • Scattering matrix Sab(E)
  • Coupling with continuum
    • Decay amplitudes Ac1(E) - thresholds
    • Typical partial width =|A|2
    • Resonance overlaps: level spacing vs. width

“kappa” parameter

No approximations until now



One open channel

interaction between resonances
Interaction between resonances
  • Real V
    • Energy repulsion
    • Width attraction
  • Imaginary W
    • Energy attraction
    • Width repulsion

11Li model

Dynamics of states coupled to a common decay channel

  • Model
  • Mechanism of binding
11 l i model dynamics of two states coupled to a common decay channel
11LimodelDynamics of two states coupled to a common decay channel
  • A1 and A2
  • opposite signs
  • Model H

A. Volya and V. Zelevinsky,

Phys. Rev. Lett. 94, 052501 (2005);

Phys. Rev. C 67, 054322 (2003);

Phys. Rev. C 74, 064314 (2006).

  • Oxygen Isotopes
  • Continuum Shell Model Calculation
  • sd space, HBUSD interaction
  • single-nucleon reactions

Measured 2009-2013

  • Predictive power of theory
  • Continuum Shell Model prediction 2003-2006

[1] C. R. Hoffman et al., Phys. Lett. B 672, 17 (2009); Phys.Rev.Lett.102,152501(2009); Phys.Rev.C83,031303(R)(2011); E. Lunderberg et al., Phys. Rev. Lett. 108, 142503 (2012).

[2] A. Volya and V. Zelevinsky, Phys. Rev. Lett. 94, 052501 (2005); Phys. Rev. C 67, 054322 (2003); 74, 064314 (2006).

[3] G. Hagen Phys. Rev. Lett. 108, 242501 (2012)


[2] A. Volya and V.Z. Phys. Rev. C 74 (2006) 064314, [3] G. Hagen et al. Phys. Rev. Lett. 108 (2012) 242501


Continuum shell model:

Detailed predictions

For Oxygen isotopes;

Color code - for widths

[A. Volya]

vir vir tual excitations into continuum
VirVirtualexcitations into continuum


experiment 2+

Figure: 23O(n,n)23O Effect of self-energy term (red curve). Shaded areas show experimental values with uncertainties.

Experimental data from:

C. Hoffman, Phys. Lett. B672, 17 (2009)


Two-neutron sequential decay of 26O

A. Volya and V. Zelevinsky, Continuum shell model, Phys. Rev. C 74, 064314 (2006).

  • Predicted Q-value: 21 keV
  • Z. Kohley, PRL 110, 152501 (2013) (experiment)

CSM calculation of 18O

States marked with longer lines correspond to sd-shell model; only l=0,2 partial waves

included in theoretical results.

continuum shell model he isotopes
Continuum Shell Model He isotopes
  • Cross section and structure within the same formalism
  • Reaction l=1 polarized elastic channel


[1] A. Volya and V. Zelevinsky

Phys. Rev. C 74 (2006) 064314

[2] A. Volya and V. Zelevinsky

Phys. Rev. Lett. 94 (2005) 052501

[3] A. Volya and V. Zelevinsky

Phys. Rev. C 67 (2003) 054322

specific features of the continuum shell model
Specific features of thecontinuum shell model
  • Remnants of traditional shell model
  • Non-Hermitian Hamiltonian
  • Energy-dependent Hamiltonian
  • Decay chains
  • New effective interaction – unknown…

(self – made recipes) …

energy dependent hamiltonian
Energy-dependent Hamiltonian
  • Form of energy-dependence
    • Consistency with thresholds
    • Appropriate near-threshold behavior
  • How to solve energy-dependent H
  • Consistency in solution
    • Determination of energies
    • Determination of open channels
interpretation of complex energies
Interpretation of complex energies
  • For isolated narrow resonances all definitions agree
  • Real Situation
    • Many-body complexity
    • High density of states
    • Large decay widths
  • Result:
    • Overlapping, interference, width redistribution
    • Resonance and width are definition dependent
    • Non-exponential decay
  • Solution: Cross section is a true observable (S-matrix )
calculation details time dependent
Calculation Details, Time – Dependent
  • Scale Hamiltonian so that eigenvalues are in [-1 1]
  • Expand evolution operator in Chebyshev polynomials
  • Use iterative relation and matrix-vector multiplication to generate
  • Use FFT to find return to energy representation

*W.Press, S. Teukolsky, W. Vetterling, B. Flannery, Numerical Recipes in C++

the art of scientific computing, Cambrige University Press, 2002

T. Ikegami and S. Iwata, J. of Comp. Chem. 23 (2002) 310-318

green s function calculation
Green’s function calculation
  • Advantages of the method
  • -No need for full diagonalization or inversion at different E
  • -Only matrix-vector multiplications
  • -Numerical stability
interplay of collectivities
Interplay of collectivities


n - labels particle-hole state

n – excitation energy of state n

dn - dipole operator

An – decay amplitude of n

  • Two doorway states of different nature
  • Real energy: multipole resonance
  • Imaginary energy: super-radiant state

Model Hamiltonian

Driving GDR externally

(doing scattering)

Everything depends on

angle between multi-dimensional vectors

A and d

interplay of collectivities1
Interplay of collectivities


n - labels particle-hole state

n – excitation energy of state n

dn - dipole operator

An – decay amplitude of n

Model Hamiltonian

Driving GDR externally

(doing scattering)

Everything depends on

angle between multi dimensional vectors

A and d

pygmy resonance
Pygmy resonance


GDR not seen


Most effective excitation

of GDR from continuum

At angle:

excitation of GDR

and pigmy

Parallel case:


and particle-hole

states with pion

quantum numbers

A model of 20 equally

distant levels is used


Loosely stated, the PTD is based on the assumptions that

s-wave neutron scattering is a single-channel process, the

widths are statistical, and time-reversal invariance holds;

hence, an observed departure from the PTD implies that

one or more of these assumptions is violated

P.E. Koehler et al.

PRL 105, 072502 (2010)


  • Time-reversal invariance holds
  • Single-channel process
  • Widths are statistical (whatever it means…)
  • Intrinsic “chaotic” states are uncorrelated
  • Energy dependence of widths is uniform
  • No doorway states
  • No structure pecularities

(b) and (d) are wrong; (c), (e), (f), (g) depend on the nucleus


Resonance width distribution

(chaotic closed system, single open channel)

G. Shchedrin, V.Z., PRC (2012)


Level spacing distribution

in an open system with

a single decay channel:

No level repulsion in

the intermediate region





No level repulsion at short distances!

(Energy of an unstable state is not well defined)


Super-radiant transition

in Random Matrix Ensemble

N= 1000, m=M/N=0.25

particle in many well potential
Particle in Many-Well Potential

Hamiltonian Matrix:

  • Solutions:
  • No continuum coupling - analytic solution
  • Weak decay - perturbative treatment of decay
  • Strong decay – localization of decaying states at the edges
typical example
Typical Example


e=0 and v=1

Critical decay strength g about 2

Decay width as a function of energy

Location of particle


Disordered problem


of a particle

(or signal



Star graph

Ziletti et al. Phys. Rev. B 85, 052201 (2012)

Many-branch (M) junction coupled at the origin

Long-lived quasibound states at the junction

Average width of all widths or of (all-M) widths, M=4

Universal “phase transition”

SIMILAR SYSTEMS: inserted qubit

sequence of two-level atoms

coupled oscillators

heat-bath environment

realistic reservoirs

biological molecules


Transmission picture T(12) for M=4;

Blue dashed lines – very strong continuum coupling;

All equal branches

Non-equal branches

Critical disorder parameter


EPL 88 (2009) 27003

Cross section (conductance) fluctuations

in a system of randomly interacting

fermions, similarly to the shell model,

as a function of the intrinsic interaction

strength. Transition (lambda =1) –

onset of chaos, exactly as in the theory

of universal conductance fluctuations

in quantum wires

7 particles, 14 orbitals,

3432 many-body states, 20 open channels

Cross section (conductance) fluctuations

as a function of openness.

No dependence on the character of chaos,

one-body (disorder) or

many-body (interactions).

Transition to superadiance: kappa=1

(‘’perfect coupling”)

Many – Body



1.C. Mahaux and H.A. Weidenmueller, Shell Model Approach to Nuclear Reactions (1969)

Formalism of effective Hamiltonian

2. R.H. Dicke, Phys. Rev. 93, 99 (1954)

Super-radiance in quantum optics

3. V.V. Sokolov and V.G. Zelevinsky, Nucl. Phys. A504, 562 (1989); Ann. Phys. 216, 323 (1992).

Super-radiance in open many-body systems

4. A. Volya and V. Zelevinsky, Phys. Rev. Lett. 94, 052501(2005); Phys. Rev. C 74, 064314 (2006).

Continuum shell model (CSM)

5. N. Michel, W. Nazarewicz, M. Ploszajczak, and T. Vertse, J. Phys. G 36, 013101 (2009).

Alternative approach: Gamow shell model

6. G.L. Celardo et al. Phys. Rev. E 76, 031119 (2007); Phys. Lett. B 659, 170 (2008);

EPL 88, 27003 (2009); A. Ziletti et al. Phys. Rev. B 851, 052201 (2012).; Y. Greenberg et al. EPJ

B86, 368 (2013).

Quantum signal transmission

7. C.W.J. Beenakker, Rev. Mod. Phys. 69, 731 (1997). Universal conductance fluctuations

8. T. Ericson and T. Mayer-Kuckuk, Ann. Rev. Nucl. Sci. 16, 183 (1966). ”Ericson fluctuations”

9. N. Auerbach and V.Z. Phys. Rev. C 65, 034601 (2002). Pions and Delta-resonance

11. N. Auerbach and V.Z. Rep. Prog. Phys. 74, 106301 (2011). Review - Effective Hamiltonian

12. A. Volya. EPJ Web of Conf. 38, 03003 (2012). From structure to sequential decays.

13. A. Volyaand V.Z. Phys. At. Nucl. 77, 1 (2014). Nuclear physics at the edge of stability.

10. A. Volya, Phys. Rev. C 79, 044308 (2009). Modern development of CSM



  • No harmonic oscillator
  • Correlated decays
  • Cluster decays
  • Transfer reactions
  • Microscopic derivation of
  • the Hamiltonian
  • Collectivity in continuum
  • New applications
  • >>>>>>
Quantum Decay: exponential versus non-exponential

* [Kubo] - exponential decay corresponds to the condition for

a physical process to be approximated as a Markovian process

* [Silverman] - indeed a random process, no “cosmic force”

* [Merzbacher] - result of “delicate” approximations

Three stages: short-time

main (exponential)Oscillations?


why and when decay cannot be exponential

Remote power-law

Initial state “memory” time

  • There are “free” slow-moving non-resonant particles, they escape slowly
Why and when decay cannot be exponential

Internal motion in quasi-bound state

Example 14C decay: E0=0.157 MeV t2=10-21 s

  • =73
time dependence of decay winter s model
Time dependence of decay, Winter’s model

Winter, Phys. Rev., 123,1503 1961.


Winter’s model:

Dynamics at remote times

  • resonance
  • background
is it possible to have oscillatory decay
Is it possible to have oscillatory decay?
  • Decay oscillations are possible
  • Kinetic energy - mass eigenstates
  • Interaction (barrier)- flavor eigenstates
  • Fast and slow decaying modes
  • Current
  • oscillations
  • Survival probability
  • [1] A Volya, M. Peshkin, and V. Zelevinsky, work in progress

Oxygen Isotopes

  • Continuum Shell Model Calculation
  • sd space, HBUSD interaction
  • single-nucleon reactions

Time-dependent approach


  • Reflects time-dependent physics of unstable systems
  • Direct relation to observables
  • Linearity of QM equations maintained
  • No matrix diagonalization
  • New many-body numerical techniques
  • Stability for broad and narrow resonances
  • Ability to work with experimental data

Time evolution of several SM states in 24O. The absolute value of the survival overlap is shown

A. Volya, Time-dependent approach to the continuum shell model, Phys. Rev. C 79, 044308 (2009).


EPL 88 (2009) 27003

Variance of cross section fluctuations

for a system of randomly interacting

fermions similarly to the nuclear shell

model as a function of the strength

of internal chaotic interaction:

In the transition to chaos (lambda=1),

we see precisely the same evolution

from 2/15 to 1/8 as predicted by theory of universal conductance fluctuations in quantum wires.

Identical results for many-body

chaos (coming from interactions)

and one-body disorder

as a function of degree of

openness (coupling to continuum);

Kappa=1 is “perfect coupling”

(phase transition to super-radiance)

Many – Body




as an Open System:

Continuum Shell Model and

New Challenges

Vladimir Zelevinsky

NSCL/ Michigan State University

Supported by NSF

Bruyères-le-Châtel, May 2014