Systematical calculation on alpha decay of superheavy nuclei Zhongzhou Ren 1,2 ( 任中洲 ), Chang Xu 1 ( 许昌 ) 1 Department of Physics, Nanjing University, Nanjing, China 2 Center of Theoretical Nuclear Physics, National Laboratory of Heavy-Ion Accelerator, Lanzhou, China

Outline 1. Introduction 2. Density-dependent cluster model 3. Numeral results and discussions 4. Summary

1. Introduction Becquerel discovered a kind of unknown radiation from Uranium in M. Curie and P. Curie identified two chemical elements (polonium and radium) by their strong radioactivity. In 1908 Rutherford found that this unknown radiation consists of 4 He nuclei and named it as the alpha decay for convenience.

Gamow: Quantum 1928 In 1910s alpha scattering from natural radioactivity on target nuclei provided first information on the size of a nucleus and on the range of nuclear force. In 1928 Gamow tried to apply quantum mechanics to alpha decay and explained it as a quantum tunnelling effect.

Various models Theoretical approaches : shell model, cluster model, fission-like model, a mixture of shell and cluster model configurations…. Microscopic description of alpha decay is difficult due to: 1. The complexity of the nuclear many- body problem 2. The uncertainty of nuclear potential.

Important problem: New element To date alpha decay is still a reliable way to identify new elements (Z>104). GSI: Z= ; Dubna: Z= ,118 Berkeley: Z= ; RIKEN: Z=113. Therefore an accurate and microscopic model of alpha decay is very useful for current researches of superheavy nuclei.

Density-dependent cluster model To simplify the many-body problem into a few-body problem: new cluster model The effective potential between alpha cluster and daughter-nucleus: double folded integral of the renormalized M3Y potential with the density distributions of the alpha particle and daughter nucleus.

In Density-dependent cluster model, the cluster-core potential is the sum of the nuclear, Coulomb and centrifugal potentials. R is the separation between cluster and core. L is the angular momentum of the cluster. 2. The density-dependent cluster model

is the renormalized factor.  1,  2 are the density distributions of cluster particle and core (a standard Fermi-form). Or  1 is a Gaussian distribution for alpha particle (electron scattering).  0 is fixed by integrating the density distribution equivalent to mass number of nucleus. 2.1 Details of the alpha-core potential

Double-folded nuclear potential

Where c i =1.07A i 1/3 fm; a=0.54 fm; R rms  1.2A 1/3 (fm). The M3Y nucleon-nucleon interaction: two direct terms with different ranges, and an exchange term with a delta interaction. The renormalized factor in the nuclear potential is determined separately for each decay by applying the Bohr-Sommerfeld quantization condition. 2.2 Details of standard parameters

For the Coulomb potential between daughter nucleus and cluster, a uniform charge distribution of nuclei is assumed R C =1.2A d 1/3 (fm) and A d is mass number of daughter nucleus. Z 1 and Z 2 are charge numbers of cluster and daughter nucleus, respectively. 2.3 Details of Coulomb potential

In quasiclassical approximation the decay width  is P  is the preformation probability of the cluster in a parent nucleus. The normalization factor F is 2.4 Decay width

The wave number K(R) is given by The decay half-life is then related to the width by 2.5 decay half-life

For the preformation probability of  -decay we use P  = 1.0 for even-even nuclei; P  =0.6 for odd-A nuclei; P  =0.35 for odd-odd nuclei These values agree approximately with the experimental data of open-shell nuclei. They are also supported by a microscopic model. 2.6 Preformation probability

2.7 Density-dependent cluster model The Reid nucleon-nucleon potential Nuclear Matter : G-Matrix M3Y Bertsch et al. Satchler et al. Alpha Scattering RM3Y 1/3  0 DDCM Electron Scattering Nuclear Matter Alpha Clustering (1/3  0) Alpha Clustering Brink et al PRL Decay Model Tonozuka et al. Hofstadter et al.

3. Numeral results and discussions 1. We discuss the details of realistic M3Y potential used in DDCM. 2. We give the theoretical half-lives of alpha decay for heavy and superheavy nuclei.

The variation of the nuclear alpha-core potential with distance R(fm) in the density-dependent cluster model and in Buck's model for 232 Th.

The variation of the sum of nuclear alpha-core and Coulomb potential with distance R (fm) in DDCM and in Buck's model for 232 Th.

The variation of the hindrance factor for Z=70, 80, 90, 100, and 110 isotopes.

The variation of the hindrance factor with mass number for Z= isotopes.

The variation of the hindrance factor with mass number for Z= isotopes.

The variation of the hindrance factor with mass number for Z= isotopes.

Table 1 : Half-lives of superheavy nuclei AZAZ AZ AZQ  (MeV)T  (exp.)T  (cal.) ± (+8.4/-0.8)ms0.8ms ± (+155/-15)ms64ms ± (+140/-33)ms38ms ± (±X)s5.5s ± (+3.3/-0.8)s1.4s ± (+10/-2)s0.1s ± (±X)min37.6min

Table 2 : Half-lives of superheavy nuclei AZAZ AZ AZQ  (MeV)T  (exp.)T  (cal.) ± (+18/-3.8)s30.1ms ± (±X)  s53  s ± (+2.0/-0.5)ms1.4ms ± (±X)min2.0min ± (±X)  s93  s ± (±X)ms0.58ms ± (+140/-40)  s78  s

Table 3 : Half-lives of superheavy nuclei AZAZ AZ AZQ  (MeV)T  (exp.)T  (cal.) ± (+1300/-120)  s79  s 268 Mt 264 Bh10.299± (+100/-30)ms22ms 269 Hs 265 Sg9.354± (±X)s2.3s 267 Hs 263 Sg10.076± (±X)ms22ms 266 Hs 262 Sg10.381± (+1.3/-0.6)ms2.2ms 265 Hs 261 Sg10.777± (±X)  s401  s 264 Hs 260 Sg10.590± (±0.30)ms0.71ms

Table 4 : Half-lives of superheavy nuclei AZAZ AZ AZQ  (MeV)T  (exp.)T  (cal.) 267 Bh 263 Db9.009± (+14/-6)s12s 266 Bh 262 Db9.477±0.020~1s1s 264 Bh 260 Db9.671± (+600/-160)ms237ms 266 Sg 262 Rf8.836± (±X)s10.6s 265 Sg 261 Rf8.949± (±X)s8.0s 263 Sg 259 Rf9.447± (±X)ms266ms 261 Sg 257 Rf9.773± (±X)ms34ms

Cluster radioactivity: Nature 307 (1984) 245.

Nature 307 (1984) 245.

Phys. Rev. Lett. 1984

Phys. Rev. Lett.

Dubna experiment for cluster decay

Although the data of cluster radioactivity from 14 C to 34 Si have been accumulated in past years, systematic analysis on the data has not been completed. We systematically investigated the experimental data of cluster radioactivity with the microscopic density-dependent cluster model (DDCM) where the realistic M3Y nucleon-nucleon interaction is used. DDCM for cluster radioactivity

Half-lives of cluster radioactivity (1) Decay Q/MeV Log 10 T expt Log 10 T Formula Log 10 RM3Y 221 Fr— 207 Tl+ 14 C Ra— 207 Pb+ 14 C Ra— 208 Pb+ 14 C Ra— 209 Pb+ 14 C Ra— 210 Pb+ 14 C Ra— 212 Pb+ 14 C Th— 208 Pb+ 20 O Th— 206 Hg+ 24 Ne

Half-lives of cluster radioactivity (2) Decay Q/MeV Log 10 T expt Log 10 T Formula Log 10 RM3Y(2) 231 Pa— 207 Tl+ 24 Ne U— 208 Pb+ 24 Ne U— 209 Pb+ 24 Ne U— 206 Hg+ 28 Mg Pu— 208 Pb+ 28 Mg Pu— 206 Hg+ 32 Si Cm— 208 Pb+ 34 Si

The small figure in the box is the Geiger-Nuttall law for the radioactivity of 14 C in even-even Ra isotopic chain.

Let us focus the box of above figure where the half-lives of 14 C radioactivity for even-even Ra isotopes is plotted for decay energies Q -1/2. It is found that there is a linear relationship between the decay half-lives of 14 C and decay energies. It can be described by the following expression New formula for cluster decay half-life

Cluster decay and spontaneous fission Half-live of cluster radioactivity New formula of half-lives of spontaneous fission log 10 (T 1/2 )=21.08+c 1 (Z-90)/A+c 2 (Z-90) 2 /A +c 3 (Z-90) 3 /A+c 4 (Z-90)/A(N-Z-52) 2

DDCM for alpha decay

Further development of DDCM

DDCM of cluster radioactivity

New formula of half-life of fission

Spontaneous fission half-lives in g.s. and i.s.

4. Summary We calculate half-lives of alpha decay by density-dependent cluster model (new few- body model). The model agrees with the data of heavy nuclei within a factor of 3. The model will have a good predicting ability for the half-lives of unknown mass range by combining it with any reliable structure model or nuclear mass model. Cluster decay and spontaneous fission

Thanks Thanks for the organizer of this conference