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RIBLL1合作会议, 兰州， 2013.8
形变弱束缚核的理论进展
Junchen Pei (裴俊琛)
School of Physics, Peking University
State Key Laboratory of Nuclear Physics and Technology
Contents
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Recent progress: weakly-bound deformed nuclei
Method progress: deformed coordinate-space HFB approach
Results: Exotic shapes and continuum effects and MADNESS
Summary
HFB solvers and Continuum effects-------J.C. Pei
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Principles of theory progress
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
The descriptive precision of theoretical approaches depends very much the degrees
of freedom we assumed in solving nuclear many-body problems
The history we observed is that in what perspective to chose the degree of freedom
in developing theoretical tools is a topic closely related to the development of
computing capabilities.
For example:
ab inito no-core full configuration (A~14)
For heavier nuclei, we have to freeze out
more degrees of freedom: HFB

Progress:
the continuum degree of freedom;
from 2D to 3D without restriction on
symmetries
from single configuration to configuration
mixing

HFB solvers and Continuum effects-------J.C. Pei
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Expected exotic structures
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Exotic nuclear structures appear as a threshold effect: halo and cluster
Pairing anti-halo versus continuum coupling effect
Continuum resulted in phase-space decoupling in quasiparticle spectrum
2-n halos still have puzzles due to 3-body and continuum effects
Spherical halo
extensive studies
pygmy dipole res.
Not likely exist
Pei, Zhang, Xu, 2013
Molecular-like
F. Nunes, NPA, 2005
Egg-like Halo
recently predicted
dipole, quardpole res.
Misu, Nazarewicz, NPA, 1997
Zhou SG et al, PRC(R)2010
Prof. Ye’s group
Rotational band
Various core-halo deformation decoupling situations
HFB solvers and Continuum effects-------J.C. Pei
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Continuum effects
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Continuum coupling: pairing induced; depend on different angularmomentum
Weakly-bound states coupled with continuum: resonances
Near-threshold non-resonant continuum is important for halos
High energy continuum: Thomas-Fermi approximation (Pei et al. 2011)
Enhance stability; enhance halo
HFB solvers and Continuum effects-------J.C. Pei
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Continuum effects
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6He: momentum distribution with continuum
Breakup, elastic scattering, transfer…
S.N. Ershov, et al, PRL 1998
J. Al-Khalili & F. Nunes, JPG 2003
HFB solvers and Continuum effects-------J.C. Pei
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Continuum coupling in HFB theory
Hartree-Fock-Bogoliubov includes generalized quasi-particle correlations;
while BCS is a special quasiparticle transformation only on conjugate states.

The general HFB-DFT equation (or BdG) is the
theoretical tool of choice for describing
medium-mass and heavier nuclei and also for
cold atomic gases. (ab inito for A≤14)
HFB is superior to BCS for describing weakly-bound
systems where continuum coupling becomes essential
Pei et al. PRC, 2011
HFB solvers and Continuum effects-------J.C. Pei
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Continuum coupling in HFB theory

Current progress: particularly difficult for deformed nuclei
Diagonalization on single-particle basis
 Direct diagonalization on coordinate-space lattice
 Outgoing boundary condition: difficult for deformed cases
※H. Oba, M. Matsuo, PRC, 2009, self-consistent calculations are still missing
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Coordinate-space HFB takes an advantage for describing weakly-bound systems
and large deformations
Bound states, continuum and embedded resonances are treated on an
equal footing; L2 discretization leads to a very large configuration space
 Computing resources and capabilities are increasing exponentially
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HFB solvers and Continuum effects-------J.C. Pei
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Deformed coordinate-space HFB approach
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Breaking geometric symmetry is very expensive: parallel calculations
2D axial symmetric HFB based on B-splines, finite-difference method
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J.C.Pei., M. V. Stoitsov, G. I. Fann, W. Nazarewicz, N. Schunck, and F. R. Xu,
PRC, 2008 (HFB-AX: Much faster and be able to calculate heavy nuclei and cold atoms)
3D MADNESS-HFB with Multi-wavelets techniques and
sophisticated parallel techniques

J.C.Pei, G.I. Fann, R.J. Harrison, W. Nazarewicz, J. Hill, D. Galindo, J. Phys. Conf. Ser., 2012
HFB solvers and Continuum effects-------J.C. Pei
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Hybrid parallel calculations for large boxes
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MPI+OpenMP hybrid parallel calculations
Large boxes calculations are
crucial for describing density
diffuseness, large deformation
and discretized continuum
From 20 fm to 30 fm, the
estimated computing cost
increased by 40 times.
J.C. Pei et al., JPCS 402, 012035(2012)
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Works well in Tianhe-1A; Tianhe-2 is coming next year!
HFB solvers and Continuum effects-------J.C. Pei
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Egg-like halo structure
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38Ne, (a) neutron density; (b) neutron pairing density
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About 2 neutrons in the halo
beta2 = 0.24, beta2_pair=0.48
Mainly contributed by near-threshold continuum
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r

25
(a)
38
Ne
-8.0
20
-6.0
15
-5.0
-4.0
10
-3.0
-2.0
5
-1.0
-0.90
0
0
5
10
15
20
25
z
25
(b)
20
r
15
10
5
Isovector deformation: pygmy quardpole resonances
0
0
5
10
15
20
25
z
New exotic “egg”-like halo structure found; accurate approach
is essential
J.C. Pei, Y.N. Zhang, F.R. Xu, PRC (rapid communications) 87, 051302(2013)
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Near-Threshold Continuum
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Different box calculations to distinguish resonances and continuum states
Near threshold non-resonant continuum is responsible for halo and surface
deformations
No halo in 40Mg since no near-threshold continuum contributions (N=28)
Smoothed neutron quasiparticle spectrum of Ω𝜋=1/2±
HFB solvers and Continuum effects-------J.C. Pei
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Phase-space decoupling
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Non-resonant continuum states gradually grows and decouples in heavy nuclei
sparse negative-parity level density is crucial; deformed halos in medium-mass
nuclei is possible, e.g. in 110Ge
Core-halo decoupling is related to the phase space decoupling in quasiptaricle
spectrum
HFB solvers and Continuum effects-------J.C. Pei
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Coordinate-space vs. Basis calculations
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Surface asympotics; energy spectrum
Basis HFB has problem in broad resonances
Y.N. Zhang, J.C. Pei, F.R. Xu, in preparation, 2013
HFB solvers and Continuum effects-------J.C. Pei
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Density and pairing density
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38Ne
HFB solvers and Continuum effects-------J.C. Pei
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Quasiparticle spectrum near thershold
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particle
pairing
HFB solvers and Continuum effects-------J.C. Pei
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Box size dependence of energies

Total energy is not sensitive to box size, however….
Y.N. Zhang, J.C. Pei, F.R. Xu, in preparation, 2013
HFB solvers and Continuum effects-------J.C. Pei
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Stability Peninsula (continuum and deformation)
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Interesting!
Y.N. Zhang, J.C. Pei, F.R. Xu, in preparation, 2013
HFB solvers and Continuum effects-------J.C. Pei
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3D MADNESS-HFB
Multi-resolution ADaptive Numerical Scientific Simulation
• Multi-resolution:
 CS infrastructure:
Task-oriented: MPI, Global arrays, multi-threaded, futures (asynchronous computation),
loadbalance
API
1. do A
2. do B if A
3. do C
4. do D if A
Task dependencies:
managed by Futures
WorldTaskQueue
ThreadPool
Main
MPI
1.
3.
2.
4.
HFB solvers and Continuum effects-------J.C. Pei
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Multi-Wavelets
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Decomposition
Wavelet space at level n
Scaling function expansion: expensive
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Transformation
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HFB solvers and Continuum effects-------J.C. Pei
Truncation
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MADNESS-HFB strategy
Step-1:
Step-2:
HFB solvers and Continuum effects-------J.C. Pei
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MADNESS-HFB performance
J.C. Pei et al, JPCS, 2012
S. Bonger et al., CPC, 2013
HFB solvers and Continuum effects-------J.C. Pei
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MADNESS-SHF+BCS solver
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Benchmark for the triaxial deformed nucleus Mo110:
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Next step: HFB calculations is very expensive.
J.C. Pei, G. Fann, W. Nazarewicz, et al. in preparation, 2013
HFB solvers and Continuum effects-------J.C. Pei
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Summary
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Coordinate-space HFB has a very good opportunity to explore new exotic
structures and continuum effects, with the development of supercomputing
facilities
To be done:
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It will be interesting to look for continuum effects in excited states, such as Pygmy
dipole and quardpole resonances based on deformed coordinate-space QRPA, and
it is on track.
MADNESS-HFB for nuclear calculations
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