1. Continuum Shell Model and New Challenges
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

2. OUTLINE 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

3. THANKS Naftali Auerbach (Tel Aviv University)
Luca Celardo (University of Breschia)
Felix Izrailev (University of Puebla)
Lev Kaplan (Tulane University)
Gavriil Shchedrin (MSU, TAMU)
Valentin Sokolov (Budker Instutute)
Suren Sorathia (University of Puebla)
Alexander Volya (Florida State University)

4. NSCL and FRIB Laboratory 543 employees, incl
NSCL and FRIB Laboratory employees, incl. 38 faculty, 59 graduate and 82 undergraduate students as 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
FRIB will be a national user facility for the U.S. Department of Energy Office of Science – when FRIB becomes operational, NSCL will transition into FRIB
2011
2009
2003
User group of over 1300 members with approx. 20 working groups

5. The Evolution of Nuclear Science at MSU

6. 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
UNEDF: Unified Nuclear Energy Density Functional
SciDAC: Scientific Discovery through Advanced Computing
SC proton-therapy cyclotrons (ACCEL/VARIAN): PSI - Switzerland, St. Petersburg - Russia, RPTC Munich - Germany, Scripps - San Diego, Riyadh - Saudi Arabia
Reaping benefits from recent investments while creating future opportunities

7. FRIB 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
FRIB science answers big questions

8. 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

9. From closed to open (or marginally stable) many-body system
CLOSED SYSTEMS:
Bound states
Mean field, quasiparticles
Symmetries
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…)

10. Examples: IAS, single-particle resonance, giant resonances
DooRWAY STATES
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

12. 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

13. 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 et.al. Eur. Phys. J. A 4, 209 (1999)
N. Auerbach, V.Z.. Phys. Lett. B590, 45 (2004)

15. Physics and mathematics of coupling to continuum
New part of Hamiltonian: coupling through continuum
[1] C. Mahaux and H. Weidenmüller, Shell-model approach to nuclear reactions,
North-Holland Publishing, Amsterdam 1969

16. Two parts of coupling to continuum
Integration region involves no poles
State embedded in the continuum
Form of the wave function and probability

17. (+) means + i0 (Eigenchannels in P-space) (off-shell) (on-shell)
Factorization (unitarity), energy dependence
(kinematic thresholds) , coupling through continuum

18. Self energy, interaction with continuum
Correction to Harmonic Oscillator Wave Function
s,p, and d waves (red, blue, black)
17O
momentum
Gamow shell model
N Michel, J. Phys. G: Nucl. Part. Phys.
36 (2009)
Notes:
Wave functions are not HO
Phenomenological SM is adjusted to observation
No corrections for properly solved mean field
A. Volya, EPJ Web of Conf. 38, (2012).

19. The nuclear many-body problem
Single-particles state (particle in the well)
Many-body states (slater determinants)
Hamiltonian and Hamiltonian matrix
Matrix diagonalization
Traditional shell-model
Effective non-Hermitian and
energy-dependent Hamiltonian
Channels (parent-daughter structure)
Bound states and resonances
Matrix inversion at all energies (time dependent approach)
Continuum physics
Formally exact approach
Limit of the traditional shell model
Unitarity of the scattering matrix

20. Ingredients Intrinsic states: Q-space Continuum states: P-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

21. EFFECTIVE HAMILTONIAN
One open channel

24. Interaction between resonances
Real V
Energy repulsion
Width attraction
Imaginary W
Energy attraction
Width repulsion

25. Dynamics of states coupled to a common decay channel
11Li model
Dynamics of states coupled to a common decay channel
Model
Mechanism of binding

26. 11Li model Dynamics of two states coupled to a common decay channel
A1 and A2
opposite signs
Model H

27. Continuum Shell Model Calculation sd space, HBUSD interaction
A. Volya and V. Zelevinsky,
Phys. Rev. Lett. 94, (2005);
Phys. Rev. C 67, (2003);
Phys. Rev. C 74, (2006).
Oxygen Isotopes
Continuum Shell Model Calculation
sd space, HBUSD interaction
single-nucleon reactions

28. Predictive power of theory
Continuum Shell Model prediction
Measured
[1] C. R. Hoffman et al., Phys. Lett. B 672, 17 (2009); Phys.Rev.Lett.102,152501(2009); Phys.Rev.C 83,031303(R)(2011); E. Lunderberg et al., Phys. Rev. Lett. 108, (2012).
[2] A. Volya and V. Zelevinsky, Phys. Rev. Lett. 94, (2005); Phys. Rev. C 67, (2003); 74, (2006).
[3] G. Hagen et.al Phys. Rev. Lett. 108, (2012)

29. [2] A. Volya and V. Z. Phys. Rev. C 74 (2006) 064314, [3] G
[2] A. Volya and V.Z. Phys. Rev. C 74 (2006) , [3] G. Hagen et al. Phys. Rev. Lett. 108 (2012)

30. Continuum shell model:
Detailed predictions
For Oxygen isotopes;
Color code - for widths
[A. Volya]

31. VirVirtual excitations into continuum
Figure: 23O(n,n)23O Effect of self-energy term (red curve). Shaded areas show experimental values with uncertainties.
experiment 2+ 1+
Experimental data from:
C. Hoffman, et.al. Phys. Lett. B672, 17 (2009)

32. Two-neutron sequential decay of 26O
A. Volya and V. Zelevinsky, Continuum shell model, Phys. Rev. C 74, (2006).
Predicted Q-value: 21 keV
Z. Kohley, et.al PRL 110, (2013) (experiment)

33. CSM calculation of 18O
States marked with longer lines correspond to sd-shell model; only l=0,2 partial waves
included in theoretical results.

34. Continuum Shell Model He isotopes
Cross section and structure within the same formalism
Reaction l=1 polarized elastic channel
References
[1] A. Volya and V. Zelevinsky
Phys. Rev. C 74 (2006)
[2] A. Volya and V. Zelevinsky
Phys. Rev. Lett. 94 (2005)
[3] A. Volya and V. Zelevinsky
Phys. Rev. C 67 (2003)

35. Specific features of the continuum shell model
Remnants of traditional shell model
Non-Hermitian Hamiltonian
Energy-dependent Hamiltonian
Decay chains
New effective interaction – unknown…
(self – made recipes) …

36. 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

37. 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 )

38. 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)

39. Green’s function calculation
Advantages of the method
-No need for full diagonalization or inversion at different E
-Only matrix-vector multiplications
-Numerical stability

40. Interplay of collectivities
Definitions
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

41. Interplay of collectivities
Definitions
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

44. Most effective excitation
Pygmy resonance
Orthogonal:
GDR not seen
Parallel:
Most effective excitation
of GDR from continuum
At angle:
excitation of GDR
and pigmy
Parallel case:
Delta-resonance
and particle-hole
states with pion
quantum numbers
A model of 20 equally
distant levels is used

45. 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, (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

47. Resonance width distribution
(chaotic closed system, single open channel)
G. Shchedrin, V.Z., PRC (2012)

48. Adding many “gamma” - channels

49. 0.1 0.5 1.0 5.0 No level repulsion at short distances!
Level spacing distribution
in an open system with
a single decay channel:
No level repulsion in
the intermediate region
0.1
0.5
1.0
5.0
No level repulsion at short distances!
(Energy of an unstable state is not well defined)

50. Super-radiant transition
in Random Matrix Ensemble
N= 1000, m=M/N=0.25

51. 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

52. Typical Example N=1000 e=0 and v=1 Critical decay strength g about 2
Decay width as a function of energy
Location of particle

53. Disordered problem

54. Disordered problem
Localization
of a particle
(or signal
transmission)

55. 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

56. Transmission picture T(12) for M=4;
Blue dashed lines – very strong continuum coupling;
All equal branches
Non-equal branches
Critical disorder parameter

57. EPL 88 (2009) 27003 Many – Body One-Body
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
One-Body

58. 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, (2005); Phys. Rev. C 74, (2006).
Continuum shell model (CSM)
5. N. Michel, W. Nazarewicz, M. Ploszajczak, and T. Vertse, J. Phys. G 36, (2009).
Alternative approach: Gamow shell model
6. G.L. Celardo et al. Phys. Rev. E 76, (2007); Phys. Lett. B 659, 170 (2008);
EPL 88, (2009); A. Ziletti et al. Phys. Rev. B 851, (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, (2002). Pions and Delta-resonance
11. N. Auerbach and V.Z. Rep. Prog. Phys. 74, (2011). Review - Effective Hamiltonian
12. A. Volya. EPJ Web of Conf. 38, (2012). From structure to sequential decays.
13. A. Volya and V.Z. Phys. At. Nucl. 77, 1 (2014). Nuclear physics at the edge of stability.
10. A. Volya, Phys. Rev. C 79, (2009) Modern development of CSM

59. CHALLENGES:
No harmonic oscillator
Correlated decays
Cluster decays
Transfer reactions
Microscopic derivation of
the Hamiltonian
Collectivity in continuum
New applications
>>>>>>

60. 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?
long-time

61. Quantum mechanics of decay
Why exponential decay?
Time evolution and decay in quantum mechanics
Survival amplitude and probability
Resonance wave function E

62. Discussion continues: Is radioactive decay exponential?
The GSI oscillations Mystery (2008)
Periodic modulation of the expected exponential law in EC-decays of different highly charged ions –
Litvinov et al. Phys. Lett. B 664, 162 (2008);
P. Kienle et al. Phys. Lett. B 726, 638 (2013).
Period = 7 sec !
Half life 5,730 ± 40 years
mean-life time 8,033 years
Carbon dating and non-exponential decay (2012)
“If the decay of 14C is indeed non-exponential... this would remove a foundation stone of modern dating methods."
Aston EPL 97, (2012).

63. Why and when decay cannot be exponential
Initial state “memory” time
Internal motion in quasi-bound state
Remote power-law
There are “free” slow-moving non-resonant particles, they escape slowly
Example 14C decay: E0=0.157 MeV t2=10-21 s
=73

64. Time dependence of decay, Winter’s model
Winter, Phys. Rev., 123,

65. Dynamics at remote times
Winter’s model:
Dynamics at remote times
background
resonance

66. Internal dynamics in decaying system Winter’s model

67. 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

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

69. CSM L. V. Grigorenko, et al. Phys. Rev. C 84, 021303 (2011)
V. Zelevinsky, A. Volya, Yad. Fiz. 77, issue 7, 1-14 (2014).
CSM

70. Low-energy phase-space decay laws
Decay and nuclear mean field
At low energies amplitudes are
defined by penetrability which is
given by channel radius
R-Matrix expressions
e(MeV)
γ(keV)
r(fm)
5He
0.895
648
4.5*
17O
0.941 98
3.8
19O
1.540
310
3.9

71. 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, (2009).

73. 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
One-Body

74. Continuum Shell Model and New Challenges
Nucleus
as an Open System:
Continuum Shell Model and
New Challenges
Vladimir Zelevinsky
NSCL/ Michigan State University
Supported by NSF
Bruyères-le-Châtel, May 2014