1 systematic calculations of alpha decay half-lives of well- deformed nuclei zhongzhou ren (...
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Systematic calculations of alpha decay half-lives of well-
deformed nuclei
Zhongzhou REN (任中洲 )
• Department of Physics, Nanjing University, Nanjing, China
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Outline
• Introduction
• Multi-channel cluster model (MCCM):
(1) to solve coupled-channel Schrödinger equations for quasi-bound states
(2) both alpha-decay half-lives and branching ratios of deformed nuclei are obtained
• • Summary
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Introduction
Proton radioactivity (Z≥51)
Alpha decay (Z≥52)
Cluster radioactivity (Z≥87)
Spontaneous fission (Z ≥90)
α decay: back to the early days of nuclear physics (1896---).
Rutherford: three kinds of radioactivity, alpha, beta, gamma; existence of nucleus by alpha scattering.
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There are more than 400 nuclei that exhibit the alpha-decay phenomenon (yellow one).
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neutron number
111112113
114
117
115
118
116
160 162
164 166 168 170 172 174
176 178 180 182 184
152 158156154
Mt 266
Db 262 Db 263
Sg 266
Db 258Db 256 Db 260Db 257
Rf 260 Rf 261 Rf 262 Rf 263Rf 259Rf 256Rf 255 Rf 258
Bh 261 Bh 262
Rf 257
Db 261
Sg 260 Sg 261 Sg 263Sg 259
Bh 264BhHs
Ds
Sg 258
Lr 259
No 258
Lr 260
No 259
Lr 261 Lr 262
No 262No 260
Lr 258
No 257
Lr 255
No 254
Lr 254
No 253
Lr 257
No 256
Lr 256
No 255
Md 257
Fm 256
Md 258
Fm 257
Md 259 Md 260
Fm 258 Fm 259
Md 256
Fm 255
Md 253
Fm 252
Md 252
Fm 251
Md 255
Fm 254
Md 254
Fm 253
Es 255 Es 256Es 254Es 251Es 250 Es 253Es 252
Cf 255 Cf 256Cf 253Cf 250Cf 249 Cf 251 Cf 252 Cf 254
110/273110/271
111/272
CHART OF THE NUCLIDES
NoMdFmEsCf
pro
ton
n
um
ber
150
DbRfLrNoMdFmEsCf
Z = 114
108Hs 267Hs 265Hs 264
a
aa
aa
a
110/270
Hs 266
Sg 262
112/2859.1539 s
Z/A
T1/2
E (MeV)
110/269
Mt 268
EC
-
SF
112/277
110/267
MtHs 269 Hs 270
Sg 265Sg
aaa
aa
aa
aa
aa
a
aa
a aa
aaaa
108/275
110/279
106/271
112/284112/282
114/286114/28710.01
114/2889.95
116/290
115/288115/287
113/284113/283
111/280
109/276
107/272
111/279
109/257
116/29110.85 10.74
112/285
110/281
114/2899.82
9.169.54
9.30
8.53
10.00
10.4610.59
10.12
9.75
9.71
9.02
10.37
10.33
105/268
15 ms
32 ms 87 m s
6.3 m s
0.1 s
0.15 s
0.17 s
0.72 s
9.8 s
16 h
9.7 m s
0.48 s
0.1 s0.5 m s
3.6 s
0.18 s
2.4 m in
9.6 s
34 s
0.56 s 0.63 s 2.7 s0.16 s10.20
112/2834.0 s
a116/29210.6616 ms
107/217
116/29353 ms
1.8 ms118/29411.65
105/2671.2 h
10.53
9.70
104/268104/2672.3 h
48 238 249Ca + U.... Cf
208 50 70Pb + Ti.... Zn
It has been used as a reliable way to identify new synthesized elements and isomeric states.
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Theoretical description of alpha decay
• Phenomenological description(1) the Geiger-Nuttall law(2) the Viola-Seaborg formula(3) ……
• Semiclassical approximation (WKB)(1) the cluster model(2) the density-dependent cluster model (DDCM)(3) the generalized liquid drop model (GLDM)(4) the super asymmetric fission model (SAFM)(5) ……
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alpha decay and quantum mechanics
• Quantum mechanics: originated from atomic physics. Two kinds of states in textbook: bound, scattering
1928 , Gomov: quantum tunnel • Unstable nuclei (238U): finite lifetime: Quasi-Bound
State (QBS)
• Old models: WKB, Bohr-Sommerfeld quantization, semi-classical approximation
• alpha-decay : pure quantum effect. To solve Schroedinger-eq. for QBS
• Generalized density-dependent cluster model• Multi-channel cluster model (MCCM)
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V0 is determined by the characteristic of the alpha-cluster quasibound state.
Woods-Saxon shape nuclear potentials
QBS: wave function of Woods-Saxon potential, tail
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Multi-channel cluster model (MCCM): alpha-decay of deformed nuclei 2010-2011
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Five-channel calculation of fine structure in the alpha decay of well-deformed nuclei
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Deformed system
We consider a spherical alpha-particle interacts with a deformed core nucleus which has an axially symmetric nuclear shape.
The decay process is described by the tunneling of the alpha particle through a deformed potential barrier, which is approximated by an axially deformed Woods-Saxon field.
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Schematic diagram of the alpha decay of well-deformed even-even nuclei
( 1)IE I I
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The comparison of experimental alpha-decay half-lives with theoretical ones for well-deformed emitters
235
10 expt calc1
1log 0.19
34i i
i
T T
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Calculated results for two isotopes of Pu
0+
2+
4+
6+
Exp. (%)
Cal. (%)
240Pu
72.8 72.22
27.1 27.73
0.084 0.048
0.001470.00106
T1/2(s) 2.07×1011 2.74×1011
8+4.6×10-5 4.6×10-6
0+
2+
4+
6+
Exp. (%)
Cal. (%)
242Pu
76.49 76.12
23.48 23.85
0.0307 0.0341
0.002320.00086
T1/2(s) 1.18×1013 1.93×1013
8+--- 2.6×10-6
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Calculated results for two isotopes of Cm
0+
2+
4+
6+
Exp. (%)
Cal. (%)
242Cm
74.08 68.87
25.92 31.04
0.035 0.077
0.00530.0046
T1/2(s) 1.41×107 1.32×107
8+2.0×10-5 3.8×10-5
0+
2+
4+
6+
Exp. (%)
Cal. (%)
244Cm
76.9 71.34
23.1 28.60
0.0204 0.0479
0.007330.00352
T1/2(s) 5.72×108 5.68×108
8+4.0×10-5 2.8×10-5
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Calculated results for two isotopes of Cf
0+
2+
4+
6+
Exp. (%)
Cal. (%)
250Cf
84.7 76.60
15.0 22.73
0.3 0.66
0.010~0.01
T1/2(s) 4.13×108 3.09×108
8+--- 5.8×10-5
0+
2+
4+
6+
Exp. (%)
Cal. (%)
252Cf
84.2 79.29
15.7 19.76
0.24 0.95
0.00890.002
T1/2(s) 8.61×107 8.87×107
8+6.0×10-5 7.9×10-5
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Calculated results for two isotopes of Fm
0+
2+
4+
6+
Exp. (%)
Cal. (%)
252Fm
84.0 76.93
15.0 21.60
0.97 1.45
0.0220.023
T1/2(s) 9.14×104 4.70×104
8+--- 3.8×10-4
0+
2+
4+
6+
Exp. (%)
Cal. (%)
254Fm
85.0 78.28
14.2 20.30
0.82 1.41
0.01260.0066
T1/2(s) 1.17×104 7.95×103
8+--- 4.8×10-4
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The comparison of experimental branching ratios with theoretical ones for well-deformed emitters
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Summary
We present a five-channel calculation of fine structure in the alpha decay of deformed nuclei
within the multi-channel cluster model.
• Include the internal effect of daughter states, the coupling effect of various channels, Pauli exclusion principle, nuclear deformation.
• Agree well with the experimental data; good prediction for half-lives and branching ratios.
• The alpha transition to high-spin states is a powerful tool to probe nuclear structure properties of daughter nuclei.
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Thanks
• Thanks for the support of colleagues
• Thanks for organizers of this conference
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• The deformed potential V is expanded in spherical multipoles to order 12.
• The dynamics of the core is included in evaluating the interaction matrix elements.
• The Boltzmann distribution hypothesis is proposed for daughter states to simulate the internal effect of nuclear states on alpha-cluster formation.
• A more realistic description of alpha decay has been achieved.
Key points ( five channels)
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The total wave function of the system
1 ˆ( ) ( ) ( )JJM n I I JM
I
r u r Y r
The set of coupled equations for the radial components
2 2
02 2
' ''
( 1)( )
2
( ) ( ) 0, [ ( )]
I
dQ E u r
dr r
V r u r n I
max
000
( ) ( )V r r Y
The multipole expansion of the interaction potential
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The coupling potential between channels α and α’
, '
'
( 1)( ) ( ) (2 ' 1)(2 1)(2 1)
4
' 00 0 ( ' ; ') I I
V r r I
W JI I
For rotational nuclei, the reduced matrix elements are assumed as
'
(2 1)(2 ' 1)' 0
4 (2 1)I I
II K IK
I
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(2) The Wildermuth condition
(3) Boundary conditions for different channels
Coupled-channel wave functions
( 0) 0;
( ) ( ) ( ) .d d
n j
n j j J J
u r
u r N G k r iF k r
4
1
2 ii
G n g
(1) The potential depth V0 is adjusted to make all channels reproduce the experimental QJd values.
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Alpha-cluster formation
• A constant preformation factor is used for all even-even nuclei (Pα =0.36).
This value is not only consistent with the experimental data of open-shell nuclei but also supported by the microscopic calculation.
• The hypothesis of Boltzmann distributions ρ(EI) is proposed for daughter states, as Einstein did for molecules with a set of discrete states.
This implies that there is a gradual decline in the Pα factor with increasing daughter spins.
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The total decay width representing the tunneling through the deformed barrier
{ }( )I II
P E
The partial decay width corresponding to the decay into a core state I
22
2 2
| ( ) |
( ) ( )n II
II I
u Rk
G k R F k R
The alpha-decay half-lives and branching ratios (BR) are expressed as
1/ 2 ln 2 /
BR ( ) 100%I I
T
P E
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Sensitivity of the calculated half-lives and branching ratios to the decay Q0 value for the alpha decay of 244Cm, showing the crucial effect on half-lives.
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The decrease of BR with increasing the E2 value is more evident as we proceed to higher-spin states.
There is an increase in the half-life by about 28% as the E2 value is varied from 40 to 80 keV.
Sensitivity of the calculated branching ratios to the energy spectrum of daughter nuclei
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Sensitivity of the calculated branching ratios and half-lives to the deformation β2 values of daughter nuclei