two-step model

沈彩万浙江 · 湖州师范学院

in the synthesis of superheavy elements

Two-step Model

第十三届全国核结构研讨会 · 赤峰2010 年 7 月 27 日

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Collaborators

Y. Abe (RCNP, Japan)

D. Boilley (GANIL, France)

E. G. Zhao (ITP, CAS)

G. Kosenko (Omsk Univ., Russia)

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Outlook

Introduction Fusion hindrance Two-step model in the fusion:

Sticking process Formation process

Calculations Conclusion

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Commonly used model:

Compound Nucleus Theory

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Binary Processes(DIC)

Reseparation(Quasi-Fission)

C. N.

SHESpontaneous decays(, fission)

n

J

JMC

J EPEPJk

*)()()12( surv..fusion2res

p

k

/2

/2

Sketch map of the process

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Parameters for the description of formation

A1A2

q1 = R/R0 q2 = p1 = pR/R0 p2 = p

R12

12

AA

AA

: asymmetric parameter ，R0: spherical radius of the compound nucleus

7/38

100100Mo+Mo+100100MoMo

8/38

110110Pd + Pd + 110110PdPd

9/38

Average value of the neck parameter

10/38

For fixed and R/R0, )(VV

After equilibrium, the distribution probability of

]/)([Exp0 TVww

1

0

1

0

dw

dw

cbaV 2)(

1.0erf)1(erf

])1([Exp)(Exp1

0000

200

200

0

0

aea

aa

a

where: Taaab / ),2/( 00

数值计算

22* )10/(' TATaE

Approximately:

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Criteria for fusion hindrancein radial evolution

1.0

2.0 3.04.0

5.0

6.0

7.0

8.0

9.0

10

11

12

13

14

15

16

17

18

19

20

21

22

23

8.0

7.0

24

6.0

4.0

3.0

2.0

251.0

0

-2.0

-3.0

-5.0

26

-10

-11

-13

-16

-18

-19

-20

-21

-25

-26

27

-32

-33

-38

-39

-40

27

-42

28

29

-49-50

31

-54

-58

34

-59

28

29

30

31

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

= 9.2028x10-4 MeV2J

c

2

Vq = 27.4 MeV

+

= 1.0, L = 0

48Ca+238U

R/R0

If system evolves to spherical case: without fusion hindrance.

If system evolves to two fragments: with fusion hindrance.

(F.H)(no F.H.)

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Fusion hindrance area: (radial evolution)

100Mo+100Mo

110Pd+110Pd

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Features in the synthesis of SHE

1. Double barrier penetrations

Coulomb barrier; Liquid drop barrier

48Ca+238URCB = 14.14fmRC = 11.86fmRLB = 9.5fm

R

VCoulombEnergy

Liquid-dropEnergy

RLB

RC

RCB

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Features of the SHE synthesis

2. Shell correction takes very important role

Fission barrier: Bf = Bf(LD) E(shell)

Bf (liquid-drop fission barrier) : ～ MeV

E(shell) (shell correction energy): ～ MeV

E(s

hell

)

Bf

(LD)

R

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Fusion Probability = Psticking* Pform

Sticking probability: Psticking

R

V

VB

Contact Point = Rp+Rt

Ec.m.

Coulomb Potential

Liquid Drop Potential

PSticking

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Sticking probability:

(1) Surface friction model

(2) Empirical formula by Swiatecki

[Swiatecki et al., PRC 71, 014602(2005)]

(parameters are slightly changed to fit the experimental capture cross section for 48Ca+238U, 244Pu, 248Cm )

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在超重核区对 B0 和 C 进行重新拟合

B0 = Bswiat + B

experimental capture cross section: M. G. Itkis et al., Nuovo Cimento A111, 783 (1998).

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Extrapolation of parameter B

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Formation Probability

Formation probability: Pform (Using LD model)

R

V

VB

Rc

Contact Point = Rp + Rt

Ec.m.

Coulomb Potential

Liquid Drop Potential

PStickingPform

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Equation of motion for R and

Langevin equaiton:

jiji

jijkjkijkjjkii

i

pmdt

dq

tRgpmppmqq

V

dt

dp

)(

)()()(2

1

1

11

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Tracks of motion with random force

1.0 3.0

5.0

7.0

9.0

11

13

15 17

19

2123

25

27

5.0

3.0

27

1.0

-1.0

29

31

-7.0

-11

-13

29

31

0.0 0.5 1.0 1.5

0.0

0.2

0.4

0.6

0.8

R/R0

+

48Ca+238U

with random force

Ek=50MeV

N

NpP k

')(

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Formation probability

According to the friction model ， the relative momentums are distributed in Gaussian form ：

mT

pp

mTpf k

k 2

)(exp

2

1)(

2

Then we get formation probability ：

kk

k

kkkform

dpN

pNpf

dppPpfP

)(')(

)()(

For the fusion of heavy systems, 0p

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Example

48Ca + 247Bk

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Evaporation probability (HIVAP)

Statistical evaporation model !

(P.Moller) 45.0 shellshell EE

(factor: fit to the experimental data for 48Ca+248Cm )Yu. Ts. Oganessian et al., PRC70, (2004)064609

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(1) Repeat 48Ca+249Cf

Application

Experimental data: Yu. Ts. Oganessian, PRC70, (2004)064609

2n

3n 4n

5n

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(2) 48Ca + Bk isotopes ( Z = 117)

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2n

3n4n

5n

48Ca + 249Bk

2009 年 7 月 27-10 月 23(70 天 )

2.4×1019 dose

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22 mg 249Bk transport

Prices per 1 mg

197Au ≈ 0.03 US$239Pu ≈ 4 US$ 48Ca ≈ 80 US$249Cf ≈ 60,000 US$

Bk(NO3)3

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(3) 48Ca +Es Z = 119, A = 300

T1/2=472d

2n

3n 4n

5n

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Z = 119, A = 302 T1/2=276d

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Z = 120, A = 305 T1/2 = 100.5d

Last chance for 48Ca to synthesize SHE

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小结 根据重核融合的特点，融合过程分为两步 ： 粘连过程和形成过程。 融合阻止 (fusion hindrance) 起源于重核融合过程中的液滴能位垒。形成过程的郎之万模拟自动考虑了这一影响。 计算了 48Ca 引起的系列反应，与实验较好符合。计算的 48Ca+249Bk 被实验所证实。 剩余截面 ( 通过裂变位垒 ) 对壳修正严重依赖。后者需要更好的理论计算。

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Thanks !