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 manybody systems
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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
THANKS
2011
2009
2003
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From closed to open (or marginally stable)
manybody system
CLOSED SYSTEMS:
Bound states
Mean field, quasiparticles
Symmetries
Residual interactions
Pairing, superfluidity
Collective modes
Quantum manybody 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…)
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, singleparticle resonance, giant resonances
at high excitation energy, intermediate structures.
Feshbach resonance in traps, superradiance
Evolution of complex energies
Total states 8!/(3! 5!)=56; states that decay fast 7!/(2! 5!)=21 – superradiant doorways
Mechanism of superradiance
Interaction via continuum
Trapped states  selforganization
Narrow resonances and broad superradiant state in 12C
in the region of Delta
Bartsch et.al. Eur. Phys. J. A 4, 209 (1999)
N. Auerbach, V.Z.. Phys. Lett. B590, 45 (2004)
[1] C. Mahaux and H. Weidenmüller, Shellmodel approach to nuclear reactions,
NorthHolland Publishing, Amsterdam 1969
(+) means + i0
(Eigenchannels in Pspace)
(offshell)
(onshell)
Factorization (unitarity), energy dependence
(kinematic thresholds) , coupling through continuum
17O
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
A. Volya, EPJ Web of Conf. 38, 03003 (2012).
Formally exact approach
Limit of the traditional shell model
Unitarity of the scattering matrix
“kappa” parameter
No approximations until now
EFFECTIVE HAMILTONIAN
One open channel
11Li model
Dynamics of states coupled to a common decay channel
A. Volya and V. Zelevinsky,
Phys. Rev. Lett. 94, 052501 (2005);
Phys. Rev. C 67, 054322 (2003);
Phys. Rev. C 74, 064314 (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 et.al 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]
1+
experiment 2+
Figure: 23O(n,n)23O Effect of selfenergy term (red curve). Shaded areas show experimental values with uncertainties.
Experimental data from:
C. Hoffman, et.al. Phys. Lett. B672, 17 (2009)
Twoneutron sequential decay of 26O
A. Volya and V. Zelevinsky, Continuum shell model, Phys. Rev. C 74, 064314 (2006).
CSM calculation of 18O
States marked with longer lines correspond to sdshell model; only l=0,2 partial waves
included in theoretical results.
References
[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
(self – made recipes) …
*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) 310318
Definitions
n  labels particlehole 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 multidimensional vectors
A and d
Definitions
n  labels particlehole 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
Orthogonal:
GDR not seen
Parallel:
Most effective excitation
of GDR from continuum
At angle:
excitation of GDR
and pigmy
Parallel case:
Deltaresonance
and particlehole
states with pion
quantum numbers
A model of 20 equally
distant levels is used
Loosely stated, the PTD is based on the assumptions that
swave neutron scattering is a singlechannel process, the
widths are statistical, and timereversal 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)

(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)
Adding many “gamma”  channels
Level spacing distribution
in an open system with
a single decay channel:
No level repulsion in
the intermediate region
0.5
0.1
1.0
5.0
No level repulsion at short distances!
(Energy of an unstable state is not well defined)
Superradiant transition
in Random Matrix Ensemble
N= 1000, m=M/N=0.25
Hamiltonian Matrix:
N=1000
e=0 and v=1
Critical decay strength g about 2
Decay width as a function of energy
Location of particle
Disordered problem
Localization
of a particle
(or signal
transmission)
Star graph
Ziletti et al. Phys. Rev. B 85, 052201 (2012)
Manybranch (M) junction coupled at the origin
Longlived quasibound states at the junction
Average width of all widths or of (allM) widths, M=4
Universal “phase transition”
SIMILAR SYSTEMS: inserted qubit
sequence of twolevel atoms
coupled oscillators
heatbath environment
realistic reservoirs
biological molecules
Transmission picture T(12) for M=4;
Blue dashed lines – very strong continuum coupling;
All equal branches
Nonequal 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 manybody states, 20 open channels
Cross section (conductance) fluctuations
as a function of openness.
No dependence on the character of chaos,
onebody (disorder) or
manybody (interactions).
Transition to superadiance: kappa=1
(‘’perfect coupling”)
Many – Body
OneBody
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)
Superradiance in quantum optics
3. V.V. Sokolov and V.G. Zelevinsky, Nucl. Phys. A504, 562 (1989); Ann. Phys. 216, 323 (1992).
Superradiance in open manybody 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. MayerKuckuk, Ann. Rev. Nucl. Sci. 16, 183 (1966). ”Ericson fluctuations”
9. N. Auerbach and V.Z. Phys. Rev. C 65, 034601 (2002). Pions and Deltaresonance
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
Quantum Decay: exponential versus nonexponential
* [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: shorttime
main (exponential)Oscillations?
longtime
Remote powerlaw
Initial state “memory” time
Internal motion in quasibound state
Example 14C decay: E0=0.157 MeV t2=1021 s
Winter, Phys. Rev., 123,1503 1961.
Winter’s model:
Dynamics at remote times
t2
t1
24O
Time evolution of several SM states in 24O. The absolute value of the survival overlap is shown
A. Volya, Timedependent 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 manybody
chaos (coming from interactions)
and onebody disorder
as a function of degree of
openness (coupling to continuum);
Kappa=1 is “perfect coupling”
(phase transition to superradiance)
Many – Body
OneBody
Nucleus
as an Open System:
Continuum Shell Model and
New Challenges
Vladimir Zelevinsky
NSCL/ Michigan State University
Supported by NSF
BruyèresleChâtel, May 2014