沈彩万湖州师范学院

8 月 11 日 ▪ 兰州大学

准裂变与融合过程的两步模型描述

合作者： Y. Abe, D. Boilley, 沈军杰

p.2

Content

Introduction of the model Quasi-fission stage Fusion stage Summary

p.3

Binary Processes(DIC)

Reseparation(Quasi-Fission)

C. N.

SHESpontaneous decays(a, fission)

n

Sketch map of the process

p.4

Theories to describe the fusion stage

Fluctuation-Dissipation theory

DNS (di-nuclear system) model

ImQMD model

…

p.5

Two-step Model

(1) Coulomb barrier; (2) Liquid drop barrier

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

R

VCoulombEnergy

Liquid-dropEnergy

RLB

RC

RCB

including two-consecutive steps overcoming the barriers

Pfusion = Psticking* Pform

p.6

(1) Sticking probability Psticking

(a) Surface friction model

(b) Empirical formula by Swiatecki

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

(c) Quantum tunneling

(d) …

p.7

(2) Formation Probability: Pform

Using liquid drop model

R

V

VB

Rc

Contact Point = Rp + Rt

Ec.m.

Coulomb Potential

Liquid Drop Potential

PStickingPform

p.8

Parameters for the description of formation

A1A2

q1 = R/R0 q2 = ap1 = pR/R0 p2 = pa

R12

12

AA

AA

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

p.9

Average value of the neck parameter

]/)([Exp0 TVww

1.01

0

1

0

dw

dw

p.10

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.)

p.11

Equation of motion for R and a

Langevin equaiton:

jiji

jijkjkijkjjkii

i

pmdt

dq

tRgpmppmqq

V

dt

dp

)(

)()()(2

1

1

11

p.12

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

p.13

Formation and fragment mass-distribution

NNpP k /')(

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

formation

quasi-fission

initial point with pk

N

NpP k

''),(

p.14

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

mT

pp

mTpf k

k 2

)(exp

2

1)(

2

For the fusion of heavy systems, 0p

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

Initial radial momentum distribution at contact point

p.15

(A)

Fragment mass distribution of Quasi-fission

p.16k

kkkkk

Jqf dp

N

pNpfdppPpfP

)(''

)(),()()(

Probability distribution of fragment after sticking:

N

NpP k

''),(

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

quasi-fission

p.17

The cross section for mass distribution of quasi-fission

2

)()()( TJqf

Jqf

Jqf

APPAP

dAA

dA

AA

AA

AA

TT

T 2 ,

2

2

2

11:

p.18

Mass-distribution probability in the formation stage

)()12(2 APPJ Jqf

Jstick

)(AP Jqf

238U + 26Mg

p.19

Exp: W.Q. Shen, PRC (1987)

238U+16O 238U+26Mg 238U+32S

p.20

238U+35Cl 238U+40Ca 238U+65Zn

Properties: the larger Elab and heavier target, the wider fragment mass-distribution of quasi-fission.

p.21

Heavier target, wider mass distribution

Difference in the formation stage Difference in the sticking stage

p.22

Larger Elab, wider mass distribution

Lighter target Heavier target

p.2323

(B) Fusion process

p.24

(1) Formation probability

Then we get formation probability ：

NNpP k /')( 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

formation

p.25

Survival Probability (statistical evaporation model) [HIVAP program]

J

JMC

J EPEPJ *)()()12( surv..fusion2

res

(2) Fusion cross section

Residue cross section

p.26

Key parameters:

(i) re-adjust the parameters in Swiatecki’s formula in the calculation of Psticking (DB, C)

(ii) Shell correction factor fshell = 0.48

Application to the 50Ti induced reaction to synthesize SHN

p.27

Adjusting DB and C to fit experimental data

The two reactions are not hindered[Gaggerler et al., Z. Phys. A 316, 291(1984)]

and

thus the fusion cross sections are used to adjust the parameter DB and C.

p.28

Z = 120

: ~fb

Comparison with others:

(a) Feng, Adamin, Nasirov, Liu, Nan Wang, Zagrebaev: ~0.1pb

(b) Ning Wang et al.: ~20 fb

p.29

1. The experimental data of quasi-fission is reproduced by two-step model. However more detailed aspects still should still be considered.

2. The residue cross section for 50Ti+250Cf is calculated.

The predicted cross section is still far away from the

current facilities.

3. Different method to calculate the capture cross section

should be considered in near future.

Summary

p.30

Thank you !