Clustering Aspects in Nuclei， From 2013-04-01 To 2013-04-26 , Beijing
Proton radioactivity, alpha decay, cluster emission and
spontaneous fission in a generalized liquid drop model
Hongfei Zhang (张鸿飞)
School of Nuclear Science and Technology, Lanzhou
University, China
April 16, 2013 KITPC Beijing
Outline
Model
Applications
Proton radioactivity
Alpha decay
Spontaneous fission
Cluster emission
Summary and outlook
Model
The potential barriers are constructed by a generalized liquid
drop model (GLDM) . (G. Royer and B. Remaud, NPA 444 (1985) 477)
E  EV  ES  EC  EN  Eshell  EPair
For one body shapes, assuming volume conservation and constant density,
the volume, surface and Coulomb 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 shifts the
peak 2.1 fm towards a more external position for 264Hs.
(c) Shell 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.
Ning Wang and Min Liu, 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 deformed surface area to the spherical surface area.
So the pairing energies vary with Bs along the shapes evolution.
Applications
1 Proton radioactivity
Decay constant
  S p 0 P
Spectroscopic factor
(D.S. Delion, R.J. Liottab, R. Wyssb, Phys. Rep. 424(2006)113
BCS approximation for pairing correlation
Blocking method for the unpaired odd nucleon
NL3 parameters
Junqing Li, Yong Zhou, Zhongyu Ma, Baoqiu Chen, PRC65 (2002)064305
205
83
Bi
2 alpha decay
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.
Calculations on alpha-decay half-live
Viola-Seaborg formulae with
Sobiczewski constants (VSS)
H.F. Zhang, G. Royer, J.Q. Li, PRC 84 (2011) 027303
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 half-life of
alpha decay.
The preformation factor and
assault frequency reflect the
nuclear structure features of
the mother nucleus.
(Zhang and Royer, PRC 77, (2008), 054318)
(Zhang and Royer, PRC 77, (2008), 054318)
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
Which decay mode should be the actual alpha-decay process,
Fission-like mode or cluster-like mode?
An approach to deal with the assault frequency
Zhang, Royer and Li, PRC 84 (2011) 027303
G is global quantum number.
 4, for( N , Z )  82

g i  5, for82  ( N , Z )  126
 6, for( N , Z )  126

Zhang, Royer and Li, PRC 84 (2011) 027303
The alpha decay process is rather a radioactivity emission process of an alpha-cluster
preformed on the surface of the mother nucleus but before the potential penetration.
Alpha-decay properties of Superheavy nuclei
To find out the reasonable
alpha-decay energy and
The new features of SHN.
MMM results form
Wang et al., PRC 81 (2010)
044322, PRC 82 (2010) 044304.
Experimental data from Audi 2012.
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 )
H.F. Zhang, Y. Gao, N. Wang, J.Q.Li, E.G Zhao, G. Royer, PRC85(2012)014325
One proton separation energies and alpha decay energies of 162 isotones
and 184 isotones.
H.F. Zhang, Y. Gao, N. Wang, J.Q.Li, E.G Zhao, G. Royer, PRC85 (2012) 014325
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.
H.F. Zhang, Y. Gao, N. Wang, J.Q.Li, E.G Zhao, G. Royer, PRC85 (2012) 014325
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 270Hs 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) with the MMM Q.
H.F. Zhang, Y. Gao, N. Wang, J.Q.Li, E.G Zhao, G. Royer, PRC85 (2012) 014325
3 spontaneous fission
Ellipsoidal deformation in spontaneous shape sequence
Spontaneous fission potential barrier
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.
Spontaneous fission 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.
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 higher than experimental ones. 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.
X.J. Bao, H.F. Zhang, G. Royer, J.Q. Li, Nucl. Phys. A 906 (2013) 1
X.J. Bao, H.F. Zhang, G. Royer,
J.Q. Li, Nucl. Phys. A 906 (2013) 1
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.
X.J. Bao, H.F. Zhang, G. Royer, J.Q. Li, Nucl. Phys. A 906 (2013) 1
X.J. Bao, H.F. Zhang, G. Royer, J.Q. Li, Nucl. Phys. A 906 (2013) 1
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.
X.J. Bao, H.F. Zhang, G. Royer, J.Q. Li, Nucl. Phys. A 906 (2013) 1
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.
4 Cluster radioactivity
Within the preformed cluster model approach, the values of
the preformation factors have been deduced from
the experimental cluster decay half-lives.
Blendowske rule
for the spectroscopic
factor
( AC 1) / 3
S  S
Blendowske, Walliser, Phys. Rev. Lett. 61 (1998) 1930
The law according to which the preformation
factors follow a simple dependence on the mass
of the cluster was confirmed by our calculations.
Zhang, Dong, Royer, Zuo, Li, Phys. Rev. C 80 (2009) 037307
Our calculations are consistent with the experimental
data and the results from DDM3Y interaction.
New possible islands of cluster emitters around the
doubly magic nucleus 100Sn and in the proton and
neutron ranges, Z = 56–64 and N = 58–72, respectively,
have been predicted.
.
The first experiment concluded the nonobservation of 12C
emission by 114Ba [26].
Blendowske, Walliser, Phys. Rev. Lett. 61 (1998) 1930
S  S( AC 1) / 3
This rule is not valid for the heavy
particle radioactivity for the superheavy
nuclei !
Why?
Summary and outlook
•
The proton radioactivity, alpha-decay, spontaneous fission and cluster
emission are described by a generalized liquid drop model successfully.
•
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 and spontaneous fission are competitive decay modes for
superheavy nuclei.
•
Refitting the mass formulae, not only include the Volume, Surface, Coulomb
energies, but also include the microscopic shell energy, and pairing correlation,
then adopt the quasi-molecular shape sequence and proximity to describe the
decay properties of the heavy and superheavy nuclei more reasonably.
Collaborators
G. Royer (Nantes University/IN2P3,France)
Junqing Li (李君清), Wei Zuo (左维) (兰州近物所)
Zhon-yu Ma (马中玉)、Bao-qiu Chen (陈宝秋)（中国原子能科学研究院）
En-guang Zhao (赵恩广) (中科院理论物理研究所)
Ning Wang (王宁) (广西师范大学)
Yuan Gao (高远) (兰州大学/杭州电子科技大学)
董建敏、王艳召、苏昕宁、张海飞、李晓恒、包小军、王永佳、王佳眉、黄银、
王莎 (兰州大学)。
Thank you !