14 Feb. 2013@RIKEN

Skyrme EDF for collective modes ofexcitation in deformed nuclei with HPC

Kenichi YoshidaOutline
Skyrme energy-density functional method for collectivedynamics
Deformation effects in low-energy and giant dipole
excitations
Large-amplitude collective motion
Shape-phase transition in neutron-rich Cr isotopes
Mass parameters of a fissioning nucleus

Skyrme EDF for collective dynamics
starting point: Skyrme EDF
variation w.r.t densities:
The coordinate-space Kohn-Sham-Bogoliubov eq. for ground states
J. Dobaczewski et al., NPA422(1984)103“s.p.” hamiltonian and pair potential:

Skyrme EDF for collective dynamics
starting point: Skyrme EDF
variation w.r.t densities:
The coordinate-space Kohn-Sham-Bogoliubov eq. for ground states
J. Dobaczewski et al., NPA422(1984)103

Coupling to the continuum is appropriately taken into account.
Deformation is described easily.

“s.p.” hamiltonian and pair potential:

Skyrme EDF for collective dynamics
starting point: Skyrme EDF
variation w.r.t densities:
The coordinate-space Kohn-Sham-Bogoliubov eq. for ground states
J. Dobaczewski et al., NPA422(1984)103“s.p.” hamiltonian and pair potential:
quasiparticle basis

KY, N. V. Giai, PRC78(2008)064316

The quasiparticle RPA eq. for excited states
Collective excitation = coherent superposition of 2qp excitations:residual interactions:

KSB calculation: MPI parallelization
use of N processors
neutrons
proc.
1

proc.
2protons
proc.
N/2

diagonalization w/ lapack:
dgeev

proc.
1

proc.
2

proc.
N/2+1

proc.
1

proc.
N/2

proc.
N

densities, hamiltonians
proc.
N/2+1

iterations: Broyden

proc.
NA. Baran et al.,
PRC78(2008)014318

QRPA calculation on parallel computer: MPI and BLACSBLACS:
Basic Linear Algebra Communication Subprograms•linear algebra oriented message passing interface•communication layer of ScaLAPACKScalable Linear Algebra PACKageQRPA calculation on parallel computer: MPI and BLACSQRPA: use of N processors
matrix elements of the QRPA eq.:
Ex.: use of 4 processors

QRPA calculation on parallel computer: MPI and BLACSQRPA: use of N processors
matrix elements of the QRPA eq.:
Ex.: use of 4 processors
block matrix

QRPA calculation on parallel computer: MPI and BLACSQRPA: use of N processors
matrix elements of the QRPA eq.:
Ex.: use of 4 processors

ScaLAPACK
2D-block cyclic distribution

for load balancing

function: indxl2g for distribution
subroutine: pdsyev for diagonalization

Giant resonance in a heavy systemmabs (mb)400

HFB cal. (64 CPUs)

238

U

300

Box size:Cut-off:

200

QRPA cal. (512 CPUs)

100

Cut-off:

0240

400

Pu

300
200

For each Kπ# of 2qp excitation: ~50,000
matrix elements: 590 CPU hours

100

diagonalization: 330 CPU hours

0510 15 20
E (MeV)

25about 200 min for total strengths !

Deformation effect in photo absorption cross sections
KY, T. Nakatsukasa, PRC83(2011)021304R
146

Potential energy (MeV)Sm

144-1180-1200

148-1220150-1240

152

154-1260-0.4 -0.2

0

β0.2 0.4

B(E1) (e2 fm2)

Low-lying dipole excitations in Sm
0.14 144
0.12
0.1
0.08
0.06
0.04
0.02
002

SmKY, T. Nakatsukasa,PRC83(2011)021304R

154Sm

In 144Sm46

 (MeV)no strength below 8 MeV

80.14
0.12
0.1
0.08
0.06
0.04
0.02
0✓consistent with other QRPA cal.
N. Parr et al., PRC67(2003)034312✓inconsistent with exp.
A. Zilges et al., PLB542(2002)430246810

 (MeV)

144Sm

154Sm

enhancement of the low-lying strength
154Sm

5% of EWSR

a part of the mode coupling effects is inthe deformed QRPA

Transition densities in 154Sm

isoscalar structure
IV strength (e2 fm2)

0.8

K=0
K=1

0.6

proton

0.4

0.2

002

4E (MeV)

68neutron

Deformation effect in IS dipole excitation
5000

Strength (fm6)

K=0|K|=1

4000

5000

148Sm

3000

3000

2000

2000

1000

100005000

05000

4000
3000

146Sm

3000
2000

1000

1000

05000

05000

144Sm

3000

2000

2000

1000

1000
24150Sm

4000

3000

00

152Sm

4000

2000

4000

154Sm

4000

6E (MeV)

81000

246E (MeV)

810

Deformation effect in IS octupole excitation

Strength (fm6)

K=0|K|=1|K|=2|K|=3

160000140000120000100000
80000
60000
40000
20000
0
160000140000120000100000
80000
60000
40000
20000
0
160000140000120000100000
80000
60000
40000
20000
00

2

4

6

E (MeV)

148Sm

160000140000120000100000
80000
60000
40000
20000
0

154Sm

146Sm

160000140000120000

152Sm

144Sm

100000
80000
60000
40000
20000
0
1600001400001200008100000
80000
60000
40000
20000
00

10

150Sm

246E (MeV)810

Deformation effect: coupling between IS dipole and LEOR

5000

dip

4000

5000

150Sm

5000

152Sm

4000

3000

3000

3000

2000

2000

2000

1000

1000

1000

0
160000140000120000100000
80000
60000
40000
20000
000
1600001400001200000160000140000120000100000
80000
60000
40000
20000
00

oct

2

4

6E (MeV)

810

100000
80000
60000
40000
20000
00

2

4

6E (MeV)8154Sm

4000

102

4

6E (MeV)810

Indication of the shape fluctuation around N=84-1180

Quadrupole moment (e fm2)

-1200

800
600

NdSm-1220

400
KSBExp.
QRPA

200082

848688
Neutron number

90928284868890
Neutron number-1240

92-1260-0.4 -0.2

00.2 0.4

Overestimation of the quadrupole transition strength w/ QRPA
close to the critical point of the shape-phase transition
from spherical to quadrupole deformed shapeLECTIVITY AT N = 40 IN NEUTRON-RICH 64 Cr

PHYSICAL REVIEW C 81,

KY, N. HInohara,
Quadrupole collectivity around N=40
PSM calculations provide
an adequate descrip
PRC83(2011)061302R(a)

CrRAPID(b)the PSM results match the R4/2 results for Fe r
1.5A. Gade COLLECTIVITY
ATFe
N = 40 IN NEUTRON-RICH Cr
PHYSICALREVIEW
C 81, 05t
et al., PRC81(2010)051304R
while the spherical fpg
shell model
predicts
58 but deviates by 0.4–0.5 for the lighte
for 64 Cr,
Crcalculations provide an adequate descripti
PSM(a)(b)difference
Crin results
performanceof
thetwoformoderesults
Fe rethePSM
matchtheR-500
4/2
1.0
1.5Fe
attributedomissionof theνd5/2orbit theinwhileto
thethesphericalfpg shellmodel
predicts
64 its implicit inclusion in the PSM aspaceforandCr, but deviates by 0.4–0.5 for the lighter C60
Cr
differencein performanceof themodels
indeedshown
byCaurier et al.
[12]twoand
Sorl-510
1.00.5attributed
to theplay
omissionof the νd5/2role
orbitinintht
the d5/2
orbit may
an important
space and its implicit inclusion in the PSM apSph. SM fpg (Cr)quadrupole
collectivity
in thisregion.
62
indeedshown
byCaurieretal.
[12] and Sorlin
PSM(Fe)90.5CrThereactionchosen,Be-induced
inelast-520
thedorbit
mayplay
animportantrolein the5/20
Sph. SM fpg
GXPF1A (Cr)fast, quadrupole
exotic projectiles,
offersadvantages a(c)(d)(Cr)collectivity inthis region.
PSM (Fe)GXPF1A (Fe)9The
reactionchosen,Be-induced
inelastic
advantages
compared
tothemore tradition
3.0
640
GXPF1A (Cr)fast,
offersexcitation.
advantages In
asCrexotic projectiles,(c)(d)
intermediate-energy
Coulomb-530
GXPF1A (Fe)advantages
comparedthe more traditiona
3.0jectiles
are excitedin thetoelectromagneticfi
intermediate-energy
Coulomb excitation. In t
66 for example, Au,nucleus,
and in medium-h
jectilesfiel
Cr are excited in the electromagnetic+
nucleinucleus,
typically
only theAu,
firstand2 instateis exc-540
for example,medium-hea+
step[22].
The
measured
excitation
cross
sect
nucleitypicallyonly
thefirst2stateis excit
68
2.0
stepCr[22].
measured
excitation
cross sectio
translated
into Thea B(E2)transition
strengththa
2.0translated
into a B(E2)andtransition
strengththatcoi-550
quadrupole
collectivity
can bereadily
quadrupole collectivity and can be readily comdictionsof nuclear models [23]. However, prodictions of nuclear models [23]. However, promin-beamγ -raybackground,in-beamγ -raybackground,suchsuchas
as bremsstrah
bremsstrahlu
28 32 36 284032 36 28
32364040
28 32 36 40 -560
the interaction
withthethehigh-Z
canobscuobs
the interaction
withhigh-Ztarget,target, can-0.4-0.2transitions
0 0.20.6keV).
γ -ray(Eγ(E0.4<600
In 99Be-ind
Be-in
γ transitions-ray
600keV). In
Neutron Number
Neutron Number
γ <scattering,
Z-dependent, low-ener` stronglyZ-dependent,scattering,
the thestrongly
low-en++FIG. 4. Experimental(Color online) Experimental
21 excitationandthe in-beambackgroundisisavoided,avoided, leading
IG. 4. (Color online)
21 excitationenergies energies
andof theofin-beambackgroundleadingto
-490

R4/2

Total Energy (MeV)

E(2+1 ) (MeV)

64

SkM*

Collective Hamiltonian approach
The classical collective Hamiltonian in (1+2)D:

axially symmetric rotor

Requantization with the Pauli’s prescription
The collective Schrödinger equation:
Collective wf in the lab. frame

The collective Schrödinger equation in the intrinsic frame:

Normalization:

Collective wf in the intrinsic framewith the volume element

Mass parameters of collective modes in the QRPA
Generalized momentum and coordinate operators,
Hamiltonian in the RPA orderCollective kinetic energy for the beta vibration:The most collectivemode,projection onto a coordinate

cf. Ring and Schuck

Local QRPA mass: extension to the non-equilibrium point
N. Hinohara et al., PRC82(2010)064313
Kohn-Sham-Bogoliubov eq with a constraint:

QRPA on the constrained-HFB state

Vibratonal mass along the collective coordinateCrcritical point at 60Cr3R4/2

2.8

0.7

2.6

0.6

2.4

0.5

2.2

0.4

20.3
0.2

1.4
1.2
1
0.8
0.6
0.4
0.2
02500

E(2+) (MeV)

0.1
008

0.1

0.2

0.3

0.4

0.5
58Cr
60Cr
62Cr
64Cr
66Cr
68Cr

64
2B(E2) (e2 fm4)

2000
00

1500
1000
500
0Exp.:A. Gade et al., PRC81(2010)051304R
58

60

6264
Mass number

KY, N. HInohara, PRC83(2011)061302R

66

68

0.1

0.2

0.3

0.4

0.5

20αk = C1 /C2Toward a microscopice = r k − αk ydescription of fission dynamics
v = AeC3 = (e, v)/(v, v)
xk+1 = xk + αk pk + C3 e
r k+1 = e − C3 v
C1 = (r ∗0 , r k+1 )βk = C1 /(C2 C3 )pk+1 = r k+1 + βk (pk − C3 y)
end do

UNEDF

Collective Hamiltonian for fission

1!
H=Mkl (q)q̇k q̇l + Vcoll (q)2kl

q = (q1 , · · · , qn )Parameters we need to evaluate microscopicallyVibrational mass parameter of the fissioning 256Fm
Mass parameter (MeV<1b<2)

0.06

LQRPACrankingATDHFB-C0.05

256

Fmcomparison with

0.04
0.03

ATDHFB-cranking

0.02

SkM* + mixed-type pairing

0.01

A. Baran et al., PRC84(2011)054321

Etotal (MeV)

2t2 (MeV )050-5-10-15-1875

Cranking

-1880
-1885
-1890
-1895
010203040

Quadrupole moment Q20 (b)5060Summary
Skyrme-EDF based QRPA for deformed nuclei
enhanced dipole strength in low-energy region
deformation: coupling between the IS dipole and LEOR
Skyrme-EDF based Collective Hamiltonian
shape-phase transition in neutron-rich Cr isotopes
critical point from spherical to quadrupole deformed
shape at around 60Cr
Toward a microscopic description of fission dynamics based on
Skyrme EDF