Systematic study of fusion reactions leading to super-heavy nuclei

Ning Wang

Guangxi Normal University

www.ImQMD.com/wangning/

BLTP/JINR-KLFTP/CAS Joint Workshop on Nuclear Physics, Aug. 2-6, 2012, Dubna

1. Introduction

2. capture cross sections

3. Survival probablity Wsur

4. Fusion probabilty PCN

5. Conclusion

I. Capture cross sections with the Skyrme energy-density functional

Density distributions of the reaction partners

Entrance-channel fusion barrier

Fusion 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

(f

m-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 fusion barrier

8 12 1630

40

50

60

70

80

Vb (

MeV

)

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.

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)

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 (

MeV

-1)

B (MeV)

Dder

Deff

16O+144Sm

(b)

D (

MeV

-1)

B (MeV)

Dder

Deff

16O+154Sm

(d)

D (

MeV

-1)

B (MeV)

Dder

with E=2.5

Dder

with E=4.0

Deff

19F+208Pb

(c)

D (

MeV

-1)

B (MeV)

Dder

Deff

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 1600.01

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

About 70% systems are less than 0.005, which gives the system error 18%.

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

rms deviation for (E>B)

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

WS*3). Masses of super-heavy nuclei

Alpha decay energies of super-heavy nuclei have been predicted

rms ~ 248 keV to 46 Qa of SHN

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

N=178

WS*

N=178

WS*

N=162 N=178

WS*

1) quasi-fission barrier

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

III. Fusion probability

Yu. Oganessian

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 WangPRC(2011) ~ 35 fb ~ 20 fb ~ 5 fb ~ 3 fb

Nan WangPRC(2012) ~ 110 fb ~ 50 fb ~ 6 fb ~ 4 fb

Siwek-WilczynskaPRC(2012)

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

Conclusion and discussion

Methods for calculations of capture cross sections, survival probability of compound nucleus and the fusion probability in fusion reactions leading to super-heavy nuclei have been established and checked 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）

Thanks for your attention

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*

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 …

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