systematical calculation on alpha decay of superheavy nuclei zhongzhou ren 1,2 ( 任中洲 ), chang...
TRANSCRIPT
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
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.
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
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
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