Systematic study of fusion reactions leading to super-heavy nuclei

Ning Wang ( 王宁 )

Guangxi Normal University

www.ImQMD.com/wangning/

Workshop on Synthesis of Super-heavy nuclei, August 10-13, 2012, Lanzhou

1. Introduction

2. Capture cross section

3. Survival probablity Wsur

4. Fusion probabilty PCN

5. Summary

1) to study the three parts individually; 2) to estimate the model uncertainty

I. Capture

II. Decay

III. Formation

# Coulomb Barrier (Skyrme EDF)

# Barrier Distribution

# Deformation & Dynamics … (ImQMD)

# Fission Barrier

# Masses & Shell corrections (mass formula)

# Fission Fragment Yields … (DNS)

# Quasi-Fission Barrier

# Potential Energy Surface

# Dynamics …

I. Capture cross sections with the Skyrme energy-density functionalDensity distributions of the reaction partners

Entrance-channel Coulomb barrier

Capture cross sections

Skyrme energy-density functional

Barrier penetration & empirical fusion barrier distribution D(B)

M. Liu, N. Wang, Z. Li, X. Wu and E. Zhao, Nucl. Phys. A 768 (2006) 80

Ning Wang, et al., Phys. Rev. C 74 (2006) 044604

Woods-Saxon form for densities

Search for the minimum of energy by varying densities (R0p, R0n, ap, an)

according to Hohenberg-Kohn theorem

0 2 4 6 8 10 120.00

0.02

0.04

0.06

0.08

0.10

(fm

-3)

r (fm)

p n

208Pb

1. Determination of density distributions

E1E2

Sudden approximation for density

R

V.Yu. Denisov and W. Noerenberg, Eur. Phys. J. A15, 375 (2002).

2. Entrance-channel inter-nucleus potential

8 12 1630

40

50

60

70

80

Vb (M

eV)

R (fm)

28Si+92ZrB0

R0

3. Fusion (capture) cross section

with

D(B) considers the coupling between the relative motion and other degrees of freedom such as dyn. deform. etc.壳效应，形变效应，动力学效应

16O+208Pb, E=80MeV, ImQMD

for reactions with nuclei near the beta-stability line but the neutron-shell is not closed

The fusion excitation functions for a series of reactions with 16O bombarding on medium mass targets.

Wang et al. Sci China G 52, 1554 (2009)

suppression

enhancement

50 55 60 65 70 75

0.0

0.1

0.2

0.3

50 60 70

0.0

0.1

0.2

70 80 90 100 110

0.0

0.1

0.2

24 28 32 36 40

0.0

0.2

0.4

D (M

eV -1

)

B (MeV)

Dder

D

eff

16O+144Sm

(b)

D (M

eV -1

)

B (MeV)

Dder

D

eff

16O+154Sm

(d)

D (M

eV -1

)

B (MeV)

Dder

with E=2.5 D

der with E=4.0

Deff

19F+208Pb

(c)

D (M

eV -1

)

B (MeV)

Dder

D

eff

12C+92Zr

(a)

24 30 36 42 480.01

0.1

1

10

100

1000

fus (

mb)

Ec.m. (MeV)

exp. calc.

12C+92Zr

(a)40 50 60 70

0.01

0.1

1

10

100

1000

fus (

mb)

Ec.m. (MeV)

exp. calc.

16O+92Zr

(b)75 80 85 90 95 100

0.01

0.1

1

10

100

1000

fus (

mb)

Ec.m. (MeV)

exp. calc.

35Cl+92Zr

(d)70 80 90 100

0.01

0.1

1

10

100

1000

fus (

mb)

Ec.m. (MeV)

exp. calc.

33S+92Zr

(c)

40 48 56 640.01

0.1

1

10

100

1000

fus (

mb)

Ec.m. (MeV)

exp. calc.

16O+112Cd

(e)50 60 70 80 90

0.01

0.1

1

10

100

1000

fus (

mb)

Ec.m. (MeV)

exp. calc.

16O+144Nd

(f)50 60 70 80 90 100

0.01

0.1

1

10

100

1000

exp. calc.

fus (

mb)

Ec.m. (MeV)

16O+166Er

(g)60 80 100 120 140 160

0.01

0.1

1

10

100

1000

10000

fus (

mb)

Ec.m. (MeV)

exp. B.B.Back exp. H.Q.Zhang calc.

16O+232Th

(h)

75 90 105 1200.01

0.1

1

10

100

1000

10000

20 30 40 50 601

10

100

1000

100 110 120 130 140 1500.01

0.1

1

10

100

1000

100 120 140 160 1800.01

0.1

1

10

100

1000

10000

100 110 120 130 140 1500.01

0.1

1

10

100

1000

105 120 135 150 1650.01

0.1

1

10

100

1000

110 120 130 140 1500.01

0.1

1

10

100

1000

120 140 160 180 2000.1

1

10

100

1000

120 140 160 180 2000.01

0.1

1

10

100

1000

fus (m

b)

Ec.m.

(MeV)

exp. calc.

19F+197Au

(a)

fus (m

b)

Ec.m.

(MeV)

exp. calc.

28Si+28Si

(b)

fus (m

b)

Ec.m.

(MeV)

exp. calc.

28Si+178Hf

(c)

fus (m

b)

Ec.m. (MeV)

exp. calc.

28Si+198Pt

(d)

fus (m

b)

Ec.m. (MeV)

exp. calc.

29Si+178Hf

(e)

fus (m

b)

Ec.m. (MeV)

exp. calc.

30Si+186W

(f)

fus (m

b)

Ec.m.

(MeV)

exp. calc.

31P+175Lu

(g)

fus (m

b)

Ec.m.

(MeV)

exp. calc.

32S+181Ta

(h)

fus (m

b)

Ec.m.

(MeV)

exp. calc.

32S+182W

(i)

60 70 80 900.01

0.1

1

10

100

1000

fus (

mb)

Ec.m.

(MeV)

exp. calc.

16O+186WDeviations from exp. data for 120 reactions

For about 70% systems, the deviations are smaller than 0.005, estimated systematic error 18%.

N. Wang et al., J. Phys. G: 34 (2007) 1935

rms deviation for (E>B0)

II. Survival probability Wsur with HIVAP

The sensitive parameters:

1. fission barriers (Liquid-drop barriers, Sierk’s barriers…)

2. level density parameters (Fermi gas model, angular-momentum and shape-dependent)

3. masses shell corrections and particle separation energies

In the standard HIVAP code: ra=1.153fm

Wang, Zhao, Scheid, Wu, PRC 77 (2008) 014603

Fusion-fission ： EDF ＋ HIVAP

For 68% reactions, the deviations are smaller than 0.0714,

Estimated systematic errors of the HIVAP code: 1.85Wsur and Wsur /1.85

Deviations of calculated evaporation (and fission) cross sections from exp. data for 51 fusion-fission reactions

A reliable nuclear mass formula is crucial for a description of the properties and production cross sections of super-heavy nuclei

WS : PRC 81 (2010) 044322

WS*: PRC 82 (2010) 044304

WS3: PRC 84 (2011) 014333

3). Masses of super-heavy nuclei

An improved nuclear mass formula

WS*

152

Alpha-decay energies of super-heavy nuclei have been predicted

rms ~ 248 keV to 46 Qa of SHN

H.F. Zhang, et al., Phys. Rev. C 85, 014325 (2012)

N=178

WS*

N=178

WS*

N=162 N=178

WS*

B0

BfBqf

III. Fusion probability

E* also influences the results

水坝

1) quasi-fission barrier

Wang, Tian, Scheid, PRC84, 061601(R) (2011)

Yu. Oganessian

Wang, et al, PRC77, 014603 (2008)

Fusion probability

2) Evaporation residual cross sections

Mean barrier height PRC84, 061601(R) (2011)

Uncertainty at E>Bm : 1.18 (capture) x 1.85 (Wsur) x 2 (PCN) = 4.4

Opt. 50Ti+249Bk 50Ti+249Cf 54Cr+248Cm 58Fe+244Pu

ZagrebaevPRC(2008) ~ 50 fb ~ 40 fb ~ 20 fb ~ 5 fb

Liu-BaoPRC(2011) ~ 600 fb ~ 100 fb

NasirovPRC(2011) ~ 100 fb ~ 70 fb

Ning Wang（王宁）PRC(2011)

~ 35 fb ~ 20 fb ~ 5 fb ~ 3 fb

Nan Wang （王楠）PRC(2012)

~ 110 fb ~ 50 fb ~ 6 fb ~ 4 fb

Siwek-WilczynskaPRC(2012)

~ 30 fb ~ 6 fb ~ 1 fb ~ 0.1 fb

exp. (GSI) < 70 fb < 200 fb (from talk of D. Ackermann)

Summary Models for calculations of capture cross sections, survival

probability of compound nucleus and the fusion probability in fusion reactions leading to SHN have been tested step by step.

Coulomb barrier, fission barrier and quasi-fission barrier play important roles for the calculations of three parts.

More precise calculations for masses, fission barriers, fission fragment yields and the study of fusion dynamics are still required.

China Institute of Atomic energy ：Zhu-Xia Li 、 Xi-Zhen Wu 、 Kai Zhao （李祝霞） （吴锡真） （赵凯）Institute of Theoretical Physics (CAS) ： En-Guang Zhao （赵恩广）Justus-Liebig-Univ. Giessen ： Werner Scheid

Guangxi Normal Univ. Min Liu （刘敏）Anyang Normal Univ. Jun-Long Tian （田俊龙）

Fission fragment yields with driving potentialDNS:

En-Guang ZhaoShan-Gui ZhouJun-Qing LiNan Wang

Nuclear engineering

Nuclear structure

Fission cycle in nuclearastrophysics

Thanks for your attention

Some codes and data are available at ： www.ImQMD.com

Kinetic Nuclear Coulomb

Skyrme energy-density functional

M. Brack, C. Guet, H.-B. Hakanson, Phys. Rep. 123, 275 (1985).

Skyrme force SkM*

1). Fission barrier

Nuclei Cohen-Swiatecki Sierk Dahlinger MWS 244Pu 4.16 5.17 3.95 4.13 256No 1.74 1.44 1.02 1.19

2). Level density parameters

In the standard HIVAP code: Ed=18.5MeV, ra=1.153fm

Large-angle quasi-elastic scattering

PRC78, 014607 (2008)

Tail of the barrier distribution influences the large-angle quasi-elastic cross sections

S. G. ZhouTail of barrier distribution influences the large-angle quasi-elastic cross sections