Coupled-channel study of fine
structure in the alpha decay of
well-deformed nuclei
Zhongzhou REN (任中洲)
• Department of Physics, Nanjing University,
Nanjing, China
Outline
• Introduction
• Coupled-channel Schrödinger equations
(1) Evaluation of interaction matrix elements
(2) Pauli exclusion principle
(3) Alpha-preformation factors
(4) Hypothesis of Boltzmann distributions for daughter
states …
• Summary
Introduction
Proton radioactivity (Z≥51)
Alpha decay (Z≥52)
Cluster radioactivity (Z≥87)
Spontaneous fission (Z ≥90)
The study on α decay
dates back to the early
days of nuclear physics. It,
although one of the oldest
objects of study in nuclear
physics, remains an
attractive decay mode.
There are more than 400 nuclei in the periodic
table that exhibit the alpha-decay phenomenon.
proton number
It has been used as a reliable way to identify new
synthesized elements and isomeric states.
CHART OF THE NUCLIDES
1.8 ms
11.65
116
116/290
116/291
15 ms
6.3 m s
10.85
115
Pb +
50
70
115/287
115/288
32 ms
87 m s
10.59
Ti.... Zn
a
114
113/283
113
113/284
0.1 s
10.12
112/277
a
112
0.48 s
114/286
a
111
0.17 s
10.37
110/269
110/267
110/270
a
110/273
110/271
a
Ds
a
Mt 268
Mt 266
Mt
109/2 75
109/276
9.7 m s
0.72 s
9.71
10.33
Hs 264
Hs
Bh 261
H s 265
Hs 266
Hs 267
Bh 262
Sg 259
Sg 260
Hs 270
Sg 261
Sg
Sg 262
Sg 263
Sg 265
a
108
a
Bh 264
Bh
Sg 258
Hs 269
a
107/272
9.8 s
9.02
a
a
105/267
105/268
1.2 h
16 h
a
Sg 266
a
3.6 s
a
116/293
16 ms 53 ms
10.66
10.53
a
0.56 s
10.01
a
a
114/289
114/288
Z = 114
2.7 s
0.63 s
9.82
9.95
a
10.00
112/282
a
0.5 m s
112/283
112/284
4.0 s
0.1 s
112/285
34 s
9.16
a
a
a
9.75
110/279
110/281
0.18 s
9.70
9.6 s
a
108/275
a
0.15 s
9.30
107/2 71
a
111/280
114/287
0.16 s
9.54
111/279
10.74
a
116/292
10.46
10.20
111/272
Ca + 238U.... 249Cf
a
117
208
48
118/294
118
a
106/271
2.4 m in
8.53
Db 256
D b 257
Db 258
Db 260
D b 261
Db 262
D b 263
Db
Rf 255
Rf 256
Rf 257
Rf 258
Rf 259
R f 260
Rf 261
Rf 262
R f 263
Lr 254
Lr 255
Lr 256
Lr 257
Lr 258
Lr 259
Lr 260
Lr 261
Lr 262
N o 253
N o 254
N o 255
N o 256
N o 257
N o 258
N o 259
N o 260
Md 252
Md 253
Md 254
Md 255
Md 256
Md 257
Md 258
Md 259
Fm 251
Fm 252
Fm 253
Fm 254
Fm 255
Fm 256
Fm 257
Fm 258
Es 250
Es 251
Es 252
Es 253
Es 254
Es 255
Es 256
Cf 249
Cf 250
C f 251
Cf 252
Cf 253
C f 254
Cf 255
176
178
180
182
184
104/267 104/268
Rf
2.3 h
Lr
No
164
166
168
170
Fm 259
154
156
174
112/285
EC
39 s
9.15
Cf 256
E (MeV)
Cf
152
172
T1/2
Fm
150

Md 260
Md
Es
Z/A
No 262
158
160
162
neutron number
SF
国内超重新核素实验265Bh (Z=107)
265Bh的实验结果与理论预言一致
Theoretical description
• Phenomenological description
(1) the Geiger-Nuttall law
(2) the Viola-Seaborg formula
(3) ……
• Semiclassical approximation (WKB)
(1) the shell model
(2) the cluster model
(3) the density-dependent cluster model (DDCM)
(4) the generalized liquid drop model (GLDM)
(5) the super asymmetric fission model (SAFM)
(6) ……
核alpha衰变寿命计算(2006-2010)，
2006-2008 建立了原子核alpha衰变的一个新模型形变的密度依赖结团模型(2006)，并系统计算了从
轻核区到 超重区原子核alpha衰变寿命 (DDCM---发表了一系列论文)。
在此基础上，2009年建立了新版本的密度依赖结
团模型：纯量子模型，解准束缚态耦合薛定谔方程
(GDDCM)。
2010, 建立了多道结团模型(MCCM)
问题来源
• 量子力学源于原子物理：束缚态，散射态（教科书）
• 1928，Gomov用量子力学定性解释原子核α衰变
• 不稳定原子核的特点：有限寿命—准束缚态（Q-BS）
• 已有理论模型：半经典或准经典近似（WKB, BohrSommerfeld quantization）
• α衰变是一个纯量子效应，应解准束缚态薛定谔方程
• GDDCM is a new version of DDCM:
• 1) pure quantum version of decay
• 2) wave functions are obtained by S-eq. for Q-BS.
Woods-Saxon势球形核准束缚态波函数
Woods-Saxon shape
nuclear potentials
V0 is determined by the
characteristic of the alphacluster quasibound state.
形变核alpha衰变新模型：多道结团模型(MCCM)
Five-channel calculation of fine structure in the
alpha decay of well-deformed nuclei
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.
Schematic diagram of the alpha decay of welldeformed even-even nuclei
EI
 I ( I  1)
Key points (五耦合道计算)
• 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.
The total wave function of the system
 JM   ( )r 1  unJ I (r ) Y (rˆ)   I JM
I
The set of coupled equations for the radial components
 2  d2

 (   1) 

 2
   Q0  EI   u (r )
2
r

 2  dr

V ' (r )u ' (r )  0, [  (n I )]
'
The multipole expansion of the interaction potential
max
V (r )   (r )    Y 00
 0
The coupling potential between channels α and α’
(1) 
V , ' (r )   (r )
4


(2 ' 1)(2 I  1)(2  1)
'  00 0 W ( '  JI ; I ')  I    I '
For rotational nuclei, the reduced matrix elements are
assumed as
 I   I '
(2  1)(2I ' 1)

I '  K 0 IK
4 (2I  1)
Coupled-channel wave functions
(1) The potential depth V0 is adjusted to make all
channels reproduce the experimental QJd values.
(2) The Wildermuth condition G  2n  
4
g
i 1
i
(3) Boundary conditions for different channels
un j (r  0)  0;
un j (r  )  N j G (k J d r )  iF (k J d r )  .
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.
The total decay width representing the tunneling
through the deformed barrier
  { I } P  ( EI ) I
The partial decay width corresponding to the decay
into a core state I
| un I ( R) |2
I 
 G ( k I R) 2  F ( k I R) 2
2
kI
The alpha-decay half-lives and branching ratios (BR)
are expressed as
T1/ 2  ln 2 / 
BR  P  ( EI ) I 100%
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.
Sensitivity of the calculated branching ratios to the
energy spectrum of daughter nuclei
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 and halflives to the deformation β2 values of daughter nuclei
The comparison of experimental alpha-decay half-lives
with theoretical ones for well-deformed emitters
2
1
i
i
  0.19
   log10 Texpt
Tcalc


i 1 34
35
Calculated results for two isotopes of Pu
240Pu
Exp.
(%)
Cal.
(%)
242Pu
4.6×10-5 4.6×10-6 +
8
0.00106 0.00147
0.084
0.048
27.1
27.73
6+
4+
Exp.
(%)
Cal.
(%)
---
2.6×10-6
0.00086 0.00232
0.0307
0.0341
23.48
23.85
72.22
0+
T1/2(s) 2.07×1011 2.74×1011
6+
4+
2+
72.8
8+
2+
76.49
76.12
0+
T1/2(s) 1.18×1013 1.93×1013
Calculated results for two isotopes of Cm
242Cm
Exp.
(%)
Cal.
(%)
244Cm
2.0×10-5 3.8×10-5 +
8
0.0046
0.0053
0.035
0.077
25.92
31.04
6+
4+
Exp.
(%)
Cal.
(%)
4.0×10-5 2.8×10-5 +
8
0.00352 0.00733
0.0204
0.0479
23.1
28.60
4+
2+
74.08
68.87
0+
T1/2(s) 1.41×107 1.32×107
6+
2+
76.9
71.34
0+
T1/2(s) 5.72×108 5.68×108
Calculated results for two isotopes of Cf
250Cf
Exp.
(%)
Cal.
(%)
---
5.8×10-5
~0.01
0.010
0.3
0.66
15.0
22.73
252Cf
8+
6+
4+
Exp.
(%)
Cal.
(%)
6.0×10-5 7.9×10-5 +
8
0.002
0.24
15.7
0.0089
0.95
4+
19.76
2+
84.7
76.60
0+
T1/2(s) 4.13×108 3.09×108
6+
2+
84.2
79.29
0+
T1/2(s) 8.61×107 8.87×107
Calculated results for two isotopes of Fm
252Fm
Exp.
(%)
Cal.
(%)
---
3.8×10-4
0.023
0.022
0.97
1.45
15.0
21.60
254Fm
8+
6+
Exp.
(%)
Cal.
(%)
---
4.8×10-4
0.0066 0.0126
0.82
4+
20.30
2+
2+
84.0
76.93
0+
T1/2(s) 9.14×104 4.70×104
6+
1.41
4+
14.2
8+
85.0
78.28
0+
T1/2(s) 1.17×104 7.95×103
The comparison of experimental branching ratios with
theoretical ones for well-deformed emitters
Summary
We present a five-channel calculation of fine
structure in the alpha decay of deformed nuclei
within the multichannel cluster model.
• Well consider 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.
致谢
• 感谢各位同行对我们工作的支持
• 感谢shen, Li, Zhang, Yu …老师对本次会
议召开的辛勤工作