2012超重核合成与性质研讨会（SHN2012），8月9-13, 2012, 兰州
推广的液滴模型及其应用简介
张鸿飞
兰州大学核科学与技术学院
内 容
一
物理目标
二
模型简介
三
模型应用
四
总结与展望
一
物理目标
采用改进的宏观模型定量描述：
•
1 原子核质量（结合能）与形变
•
2 alpha衰变
•
3 质子放射性
•
4 集团放射性
•
5 自发裂变
•
6 诱发裂变
•
7 各种衰变模型竞争，较好确定原子核寿命
•
8 熔合反应
自发裂变与阿尔法衰变的竞争
Alpha deacy is the main decay mode for SHN, spontaneous
fission is also competitive decay for some heavy nuclei.
The known heaviest even-even nucleus decays by alpha emission. For the products Z=114,
Z=112, 106,104, spontaneous fission will be competitive with alpha-decay.
Phys. Rev. Lett. 104, 142502 (2010)
The known heaviest odd-A and odd-odd nuclei are also detected by alpha decay. For
the products, the decay mode is still alpha emission up to Rg and Db. It seems that the
odd-even effect plays an important role on selecting the decay mode for SHN.
The decay mode of 114 isotopes is also isospin dependent.
Confirm the odd-even effect is very important for determination of the decay mode.
An experimental indication that Z=114 shell closure predicted around N=184.
It is an interesting topic to study the competition between alpha-decay and spontaneous fission.
二
模型简介
(1) Potential barrier ( from GLDM)
The spontaneous fission and alpha-decay potential barriers are constructed
by a generalized liquid drop model (GLDM).
E  EV  ES  EC  EN  Eshell  EPair
For one body shapes, assuming volume conservation and constant density,
the volume and surface energies in this model read:
EV (def )  15.494(1 1.8I 2 ) A(1  0.00377T 2 )
2
3
ES (def )  17.9439(1  2.6I ) A (S / 4R0 )(1  1.5T / 17)(1  T / 17)
1 3 2 Z2
1 V( )  R( )  3
EC 
e (
)
sin d
R0 2  V 
R0 
4 0 5
0
2
2
3
2
For two separated spherical nuclei,
EV (def )  15.494[(1 1.8I12 ) A1  (1 1.8I 22 ) A2 ](1  0.00337T 2 )
2
3
1
2
3
2
3
1.5
T
ES (def )  17.9439[(1  2.6 I ) A  (1  2.6 I ) A ]  (1 
T )(1  ) 2
17
17
2
1
EC 
1
4 0
0.6e Z
2
2
1
2
2
/ R1  0.6e 2 Z 22 / R2  e 2 Z1 Z 2 / r

The improvements were made for the conventional liquid drop model
(a) Quasi-molecular shapes
Two parameters s1=a/c1, s2=a/c2
are used to describe the shape evolution
The most impressively feature of the quasi-molecular shapes is that it can describe the process
of the shapes evolution from one body to two separated fragments in a unified way
(b) Proximity energy
The nuclear energy EN has been introduced to take into account
the finite-range effects of the nucleon-nucleon force between the
close surfaces . (Royer, NPA 444, (1985) 477)
  0.9517 (1  2.6 I d2 ) is surface parameter.
The introduction of proximity force lowers the barrier of 7.3 MeV, and shift the
peak 2.1 fm towards a more external position for 264Hs.
(c) Shell correction energy
W.D.Myers, Droplet Model of Atomic Nuclei
(Plenum,New-York, 1977).
The Strutinsky method was adopted to calculate
Single particle levels ei are calculated in an axially Woods-Saxon potential.
Wang Ning and Liu Min Phys Rev C 81,067302(2010).
(d) Pairing energy
The pairing energy has been calculated with the following expressions used in a
recent version of the Thomas–Fermi model
(W.D. Myers and W.J. Swiatecki, Nucl. Phys. A601 (1996) 141)
is the ratio of the surface area of the nucleus at the actual shape
to the surface area of the nucleus at the spherical shape.
(2) Half-life
J.Randrup, S.E.Larsson, P.Moller,
S.G.Nilsson,K.Pomorski and
A.Sobiczewski, Phys.Rev.C 13
229(1976).
Where k is 14.8.
Only several channels (in the bottom of every curve) play the key role to determine
the spontaneous fission half-life. The decay constant of the mother nucleus is the
summation of all possible spontaneous fission decay constant.
三
模型应用
1 质子放射性
衰变常数
谱因子
  S p 0 P
1 阿尔法衰变
Alpha decay process
There are two different decay modes for alpha decay
in the market, cluster-like and fission-like modes.
Cluster-like mode
Fission-like mode
The experimental investigation can not unambiguously
distinguish these two decay modes up to now.
In the framework of cluster-like mode
(Zhang and Royer, PRC 77, (2008), 054318)
(1)
(2)
(3)
(4)
The preformation factor P0 of an alpha cluster inside
the mother nucleus can be extracted from Eq. (1).
The penetration probability
plays the most important role
to determine the life-time of
alpha decay.
The preformation factor and
assault frequency reflect the
nuclear structure features of
the mother nucleus.
Clearly the closed shell structures
play the key role for the preformation
mechanism, and more the nucleon
number is close to the magic nucleon
numbers, the more the preformation
of alpha cluster is difficult inside the
mother nucleus.
Zhang et al., PRC 80, (2009) 057301
In the framework of fission-like mode
(Wang,Zhang, Zuo and Li, CPL 27, (2010), 062103)
log10v0
An approach to deal with the assault frequency
Zhang,Royer and Li, PRC 84 (2011) 027303
G is globle quantum number.
 4, for( N , Z )  82

g i  5, for82  ( N , Z )  126
 6, for( N , Z )  126

The alpha decay process is rather a radioactivity emission process of a cluster
preformed on the surface of the nucleus but before the potential penetration.
Alpha decay properties of SHN
To find out the reasonable
alpha decay energy and
the features of SHN.
MMM results form
Wang et al., PRC 81 (2010)
044322, PRC 82 (2010) 044304.
Experimental data from Audi 2013.
The RMS deviation with respect
to 2149 measured nuclear masses
Is 0.441 MeV ( the corresponding
results with FRDM is 0.566 MeV).
Another impressed improvement is
the RMS deviation of 46 SHN is
reduced 0.263 MeV ( 0.566 MeV
for FRDM )
One proton separation energies and alpha decay energies of 162 isotones
and 184 isotones.
Potential energy surface of 270Hs and 298114 by Constrained Relativistic Mean Field
theory with the parameters NL3
The nucleus 270Hs is a double sub-magic nucleus and 298114 is a spherical
double magic nucleus.
Alpha decay life-times of Hs and Z=114 isotopes with WKB penetration
probability, and the potential barrier is constructed by the GLDM.
The calculated alpha decay half-live of 264Hs is 23.33 second with MMM Q,
15.14 second with experimental Q. For 298114, the calculated alpha decay
half-live is 1537588.07 seconds ( about 18 days).
3 Cluster radioactivity
4 自发裂变
(1) Spontaneous fission of heavy and superheavy nuclei
Spontaneous fission potential barrier for 235U
It is evident the ellipsoidal deformation will decrease the potential barrier, and the
Shell corrections of the fragment will produce the second and third humps.
Comparison between theoretical (t) and experimental (e) barrier characteristics for
actinide nuclei. Ea, Eb and Ec are the first, second and third peak heights while E2 and E3
are the energies of the second and third potential minima relatively to the ground state
energy (in MeV).
Ea and Eb are consistent with the experimental data. The predicted values of the second
Minimum energy is a little too high. The still sparse but exciting data for the third barrier are
Correctly reproduced.
The external barrier disappears since the attractive proximity forces can no more
compensate for the repulsive Coulomb forces.
The spontaneous fission half-life is very sensitive with the nucleon number of the mother nucleus.
The logarithm of average deviations of a total of 47 spontaneous fission nuclei is:
The average deviation between theoretical spontaneous fission half-life and the
experimental ones is less than 100 times.
The trend of the theoretical results form [11] follows the experimental ones well, but the values
are systematically larger than the experimental data. Our calculated results can reproduce the
experimental spontaneous fission half-life better.
It is evident, for many superheavy nuclei, the spontaneous fission half-lives are
long enough to be measured by the present experimental setups, if the spontaneous
fission is the main decay mode.
(2) Competition between spontaneous fission and
Alpha-decay for superheavy nuclei
Our calculations can reasonably reproduce the experimental results, and the spontaneous
can compete with alpha-decay. The magic neutron number N=184 is clear, and the half-lives
decrease quickly after this neutron number.
From neutron number N=174 to 186, the alpha-decay half-lives are shorter than spontaneous
fission half-lives. These SHN can be identified by their alpha-decay properties. The neutron
magic number N=184 is confirmed.
四
总结与展望
•
采用推广的液滴模型计算了原子核的质子放射性、阿尔法衰变、重离子放射
性以及自发裂变，结果与实验符合很好，并作了大量预言.
•
推广的液滴模型的计算结果表明，原子核的阿尔法衰变具有预形成过程.
•
超重核的自发裂变与阿尔法衰变具有竞争性.
•
下一步的研究将考虑对重核和超重核，质子放射性、阿尔法衰变、重离子放
射性以及自发裂变的竞争，较好预言原子核的半寿命.
•
采用推广的液滴模型研究熔合反应，并将准分子形状机制引入到与双核模型。
Collaborators
G. Royer (Nantes University/IN2P3,France)
李君清, 左维 (兰州近物所)
马中玉、陈宝秋（中国原子能科学研究院）
赵恩广、周善贵(中科院理论物理研究所)
王宁 (广西师范大学)
高远 (兰州大学/杭州电子科技大学)
董建敏、包小军、王永佳、王佳眉、黄银、王莎 (兰州大学)
Thank you !