1. 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

2. 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

3. Introduction Proton radioactivity (Z≥51) Alpha decay (Z≥52)
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.
Proton radioactivity (Z≥51)
Alpha decay (Z≥52)
Cluster radioactivity (Z≥87)
Spontaneous fission (Z ≥90)

4. There are more than 400 nuclei in the periodic table that exhibit the alpha-decay phenomenon.

5. It has been used as a reliable way to identify new synthesized elements and isomeric states.

6. 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

7. Theoretical description
Phenomenological description
(1) the Geiger-Nuttall law
(2) the Viola-Seaborg formula
(3) ……
Semiclassical approximation (WKB)
the shell model
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) ……

8. 核alpha衰变寿命计算( )，
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)

9. 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, Bohr-Sommerfeld 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.

10. 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 alpha-cluster quasibound state.

12. 形变核alpha衰变新模型：多道结团模型(MCCM)

13. Five-channel calculation of fine structure in the alpha decay of well-deformed nuclei

14. 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.

15. Schematic diagram of the alpha decay of well-deformed even-even nuclei

16. 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.

17. The total wave function of the system
The set of coupled equations for the radial components
The multipole expansion of the interaction potential

18. The coupling potential between channels α and α’
For rotational nuclei, the reduced matrix elements are assumed as

19. Coupled-channel wave functions
(1) The potential depth V0 is adjusted to make all channels reproduce the experimental QJd values.
(2) The Wildermuth condition
(3) Boundary conditions for different channels

20. 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.

21. The total decay width representing the tunneling through the deformed barrier
The partial decay width corresponding to the decay into a core state I
The alpha-decay half-lives and branching ratios (BR) are expressed as

22. 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.

23. 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.

24. Sensitivity of the calculated branching ratios and half-lives to the deformation β2 values of daughter nuclei

25. The comparison of experimental alpha-decay half-lives with theoretical ones for well-deformed emitters

26. Calculated results for two isotopes of Pu0+ 2+ 4+ 6+
Exp.
(%)
Cal.
240Pu
72.8
72.22
27.1
27.73
0.084
0.048
T1/2(s) 2.07× ×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
T1/2(s) 1.18× ×1013 8+
---
2.6×10-6

27. Calculated results for two isotopes of Cm0+ 2+ 4+ 6+
Exp.
(%)
Cal.
242Cm
74.08
68.87
25.92
31.04
0.035
0.077
0.0053
0.0046
T1/2(s) 1.41× ×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
T1/2(s) 5.72× ×108 8+
4.0×10-5
2.8×10-5

28. Calculated results for two isotopes of Cf0+ 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 8+
---
5.8×10-5 0+ 2+ 4+ 6+
Exp.
(%)
Cal.
252Cf
84.2
79.29
15.7
19.76
0.24
0.95
0.0089
0.002
T1/2(s) 8.61× ×107 8+
6.0×10-5
7.9×10-5

29. Calculated results for two isotopes of Fm0+ 2+ 4+ 6+
Exp.
(%)
Cal.
252Fm
84.0
76.93
15.0
21.60
0.97
1.45
0.022
0.023
T1/2(s) 9.14× ×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
0.0126
0.0066
T1/2(s) 1.17× ×103 8+
---
4.8×10-4

30. The comparison of experimental branching ratios with theoretical ones for well-deformed emitters

33. 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.

34. 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

36. Thank you!