Systematical calculation on alpha decay of superheavy nuclei

Zhongzhou Ren1,2 (任中洲 ), Chang Xu1 (许昌 )

1Department of Physics, Nanjing University, Nanjing, China

2Center 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 1896.

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 4He 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=110-112; Dubna: Z=114-116,118 Berkeley: Z=110-111; 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 2 2N C

1V(R) = V (R) + V (R) + (L + ) h /(2μR )

2

2. The density-dependent cluster model

is the renormalized factor. 1 , 2 are the density distributions of cluster p

article 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.

N 1 2 1 1 2 2V (R) = λ dr dr ρ (r )ρ (r )g(E,| S |)

i i 0 i iρ (r ) = ρ /[1+ exp((r - c )/a)]

2.1 Details of the alpha-core potential

Double-folded nuclear potential

Where ci =1.07Ai1/3 fm; a=0.54 fm; Rrms1.2A1/3 (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.

α α

exp(-4s) exp(-2.5s)g(E,| S |) = 7999 - 2134 - 276(1- 0.005E /A )δ(S)

4s 2.5s

2.2 Details of standard parameters

For the Coulomb potential between daughter nucleus and cluster, a uniform charge distribution of nuclei is assumed

RC=1.2Ad1/3 (fm) and Ad is mass number of

daughter nucleus.

Z1 and Z2 are charge numbers of cluster and daughter nucleus, respectively.

221 2

C CC C

21 2

C

Z Z e RV (R) = [3 - ( ) ] (R R )

2R R

Z Z e= (R R )

R

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

3

2

R2

α

R

hΓ = P F exp[-2 dRK(R)]

4μ

R2 R

2

R1 R1

1 πF dR cos ( dR'K(R') - ) = 1

K(R) 4

2.4 Decay width

The wave number K(R) is given by

The decay half-life is then related to the width by

2

2μK(R) = | Q - V(R) |

h

1/2T = hln2/Γ

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 e

xperimental data of open-shell nuclei. They are also supported by a microscopic m

odel.

2.6 Preformation probability

2.7 Density-dependent cluster model

The Reid nucleon-nucleon potential

Nuclear Matter : G-MatrixM3Y

Bertsch et al.

Satchler et al.

Alpha ScatteringRM3Y

1/30DDCMElectron Scattering

Nuclear MatterAlpha Clustering (1/30)

Alpha Clustering

Brink et al.

1987 PRLDecay 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 withdistance R(fm) in the density-dependent cluster model and in Buck's model for 232Th.

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

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= 90-94 isotopes.

The variation of the hindrance factor with mass number for Z= 95-99 isotopes.

The variation of the hindrance factor with mass number for Z= 100-105 isotopes.

Table 1 : Half-lives of superheavy nuclei

AZ AZ Q(MeV) T(exp.) T(cal.)

294118 290116 11.810±0.1501.8(+8.4/-0.8)ms

0.8ms

292116 288114 10.757±0.150 33(+155/-15)ms 64ms

290116 286114 10.860±0.150 29(+140/-33)ms 38ms

289114 285112 9.895±0.020 30.4(±X)s 5.5s

288114 284112 10.028±0.050 1.9(+3.3/-0.8)s 1.4s

287114 283112 10.484±0.020 5.5(+10/-2)s 0.1s

285112 281110 8.841±0.020 15.4(±X)min 37.6min

Table 2 : Half-lives of superheavy nuclei

AZ AZ Q(MeV) T(exp.) T(cal.)

284112 280110 9.349±0.050 9.8(+18/-3.8)s 30.1ms

277112 273110 11.666±0.020 280(±X)s 53s

272111 268109 11.029±0.020 1.5(+2.0/-0.5)ms 1.4ms

281110 277108 9.004±0.020 1.6(±X)min 2.0min

273110 269108 11.291±0.020 110(±X)s 93s

271110 267108 10.958±0.020 0.62(±X)ms 0.58ms

270110 266108 11.242±0.050 100(+140/-40)s 78s

Table 3 : Half-lives of superheavy nuclei

AZ AZ Q(MeV) T(exp.) T(cal.)

269110 265108 11.345±0.020 270(+1300/-120)s 79s

268Mt 264Bh 10.299±0.020 70(+100/-30)ms 22ms

269Hs 265Sg 9.354±0.020 7.1(±X)s 2.3s

267Hs 263Sg 10.076±0.020 74(±X)ms 22ms

266Hs 262Sg 10.381±0.020 2.3(+1.3/-0.6)ms 2.2ms

265Hs 261Sg 10.777±0.020 583(±X)s 401s

264Hs 260Sg 10.590±0.050 0.54(±0.30)ms 0.71ms

Table 4 : Half-lives of superheavy nuclei

AZ AZ Q(MeV) T(exp.) T(cal.)

267Bh 263Db 9.009±0.030 17(+14/-6)s 12s

266Bh 262Db 9.477±0.020 ~1s 1s

264Bh 260Db 9.671±0.020 440(+600/-160)ms 237ms

266Sg 262Rf 8.836±0.020 25.7(±X)s 10.6s

265Sg 261Rf 8.949±0.020 24.1(±X)s 8.0s

263Sg 259Rf 9.447±0.020 117(±X)ms 266ms

261Sg 257Rf 9.773±0.020 72 (±X)ms 34ms

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 14C to 34Si 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 Log10 T

expt Log10 TFormula Log10

RM3Y

221Fr—207Tl+14C 31.29 14.52 14.43 14.86

221Ra—207Pb+14C 32.40 13.37 13.43 13.79

222Ra—208Pb+14C 33.05 11.10 10.73 11.19

223Ra—209Pb+14C 31.83 15.05 14.60 14.88

224Ra—210Pb+14C 30.54 15.90 15.97 16.02

226Ra—212Pb+14C 28.20 21.29 21.46 21.16

228Th—208Pb+20O 44.72 20.73 20.98 21.09

230Th—206Hg+24Ne 57.76 24.63 24.17 24.38

Half-lives of cluster radioactivity (2)

Decay Q/MeV Log10 Texpt Log10 T

Formula Log10RM3Y(2)

231Pa—207Tl+24Ne 60.41 22.89 23.44 23.91232U—208Pb+24Ne 62.31 20.39 21.00 20.34

233U—209Pb+24Ne 60.49 24.84 24.76 24.24

234U—206Hg+28Mg 74.11 25.74 25.12 25.39

236Pu—208Pb+28Mg 79.67 21.65 21.90 21.20

238Pu—206Hg+32Si 91.19 25.30 25.33 26.04

242Cm—208Pb+34Si 96.51 23.11 23.19 23.04

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

Let us focus the box of above figure where the half-lives of 14C 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 14C and decay energies.

It can be described by the following expression-1/2

10 1/2 1 2 1 2log (T ) = aZ Z Q + cZ Z + d + h

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

log10(T1/2)=21.08+c1(Z-90)/A+c2(Z-90)2/A

+c3(Z-90)3/A+c4(Z-90)/A(N-Z-52)2

-1/210 1/2 1 2 1 2log (T ) = aZ Z Q + cZ Z + d + h

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