1. 第十四届全国核结构大会暨第十次全国核结构专题讨论会
湖州，2012年
Energy Density Functional Description of Nuclear Spectrum and Shape Transition
李志攀
西南大学物理科学与技术学院
大家早上好！非常荣幸能有这次机会来到美丽的湖州，和大家一起交流讨论我们最近的一些工作：我今天报告的题目是能量密度泛函对原子核激发谱及形状演化的研究

2. Outline 1 Introduction 2 Theoretical framework 3
Results and discussion 3
The talk includes these four parts, and I will mainly focus on the results and discussion
Summary 4

3. Nuclear Low-lying Spectrum
Nuclear low-lying spectrum is an important physical quantity that can reveal rich structure information of atomic nuclei
Shape and shape transition 0+ 2+ 4+ 6+ 8+ 3- 5- 7- 9- 1-
原子核低激发谱是一个重要的物理量，能反映丰富的原子核结构信息，
包括形状及形状演化，如简谐能谱对应于球形原子核，而在well-deformed的原子核中，
可观测到明显的基态带、beta振动及gamma振动带等，当然介于二者之间，还存在着大量的过渡核；
另一方面，观测到的低能负宇称带指示了原子核中可能存在稳定八极形变。

4. Nuclear Low-lying Spectrum
Nuclear low-lying spectrum is an important physics quantity that can reveal rich structure information of atomic nuclei
Shape and shape transition
Evolution of the shell structure
T. Baumann
Nature06213(2007) Z N
N=20
N=28
N=16
其次，原子核低激发谱可作为原子核壳结构演化，特别是奇特原子核中幻数变化的“探针”。
如该丰中子核区中N=20，28壳的消失以及N=16新幻数的产生，这些都可由第一个2+激发态的能量
随核子数的演化清晰展现出来。如在O同位素链中N=16处观测到突然增加的E(2+)，
而在Ne，Mg同位素链中N=20处却给出了很小的2+能量，

5. Nuclear Low-lying Spectrum
Nuclear low-lying spectrum is an important physics quantity that can reveal rich structure information of atomic nuclei
Shape and shape transition
Evolution of the shell structure
Evidence for pairing correlation
原子核低激发谱还可为对关联的存在提供佐证：即偶偶核的低激发态密度远小于相邻的奇A核。
此外，原子核低激发谱还可反映许多其他的核结构信息，此处不再一一赘述。
因此，鉴于以上原因，有必要来微观系统地研究原子核低激发谱及其相关物理

6. Covariant Energy Density Functional (CEDF)
CEDF: nuclear structure over almost the whole nuclide chart
Scalar and vector fields: nuclear saturation properties
Spin-orbit splitting
Origin of the pseudo-spin symmetry
Spin symmetry in anti-nucleon spectrum ……
Spectrum: beyond the mean-field approximation
Restoration of broken symmetry, e.g. rotational
Mixing of different shape configurations
Ring96, Vretenar2005, Meng2006
PES
为系统微观研究原子核低激发谱，协变密度泛函可作为一个合适的出发点。由于以下优点，协变密度泛函可成功描述核素
版图中绝大部分原子核的结构信息。
这些优点有：同时包含标量和矢量场，能自洽给出核的饱和性质；其次，自然包括自旋-轨道劈裂等等。
然而，为描述原子核的谱学性质，我们必须超越平均场近似，即恢复破缺的对称性，如转动对称性等；
对于过渡核，还必须考虑不同形状组态的混合。
最近，角动量投影及生成坐标方法被成功推广至轴对称进而三轴的协变密度泛函理论，并应用于一些轻核低激发谱的研究
然而，该模型计算十分耗时。为系统研究原子核低激发谱，我们采用了另一个思路：
即基于协变密度泛函构建5维集体Hamiltonian的方法。
However, to describe the spectrum, one has to go beyond the mean-field approximation:
Namely, to restore the broken symmetry, particularly the rotational symmetry,
and mix the different shape configurations, especially for the transitional nuclei with the soft potential.
Recently, the AMP+GCM method has been extended to the axial-symmetry and further triaxial deformed CEDF.
However, these model will consume lots of time. To perform the systematic investigation,
Here, we will adopt an alternative method, namely to construct the 5-dimentional collective Hamiltonian
based on the CEDF.
AMP+GCM: Niksic2006, Yao2010
5D Collective Hamiltonian based on CEDF

7. 5-dimensional Hamiltonian
Brief Review of the model
Construct
5-dimensional Hamiltonian
(vib + rot)
E(Jπ), BE2 …
Cal Exp.
3D covariant Density Functional
ph + pp
Coll. Potential
Moments of inertia
Mass parameters
Diagonalize:
Nuclear spectroscopy
接下来我们简单介绍一下模型的基本思想，
It starts from the five-dimensional collective Hamiltonian, which describes the vib, rot, and the coupling between them.
Such Hamiltonian will be mapped to the triaxial covariant density functional, including both ph and pp channel.
It means that from the microscopic calculation, one can determine the collective parameters for the 5D Hamiltonian.
Diagonalize the Hamiltonian, we can obtain the nuclear spectrum, like the excited energies, EM transitions, and so on.
T. Niksic, Z. P. Li, D. Vretenar, L. Prochniak, J. Meng, and P. Ring 79, (2009)

8. Interesting topics Microscopic Analysis of nuclear QPT
Fission barrier and SD band
Shape evolution in N=28 isotones
Effect of the time-odd mean-field
Next part, we will apply the developed model to the microscopic analysis of nuclear QPT.

9. Nuclear Quantum Phase Transition(QPT) - 1st order
Nuclear QPT
Two approaches to study
QPT
Spherical
Deformed E
Critical β
Potential
Order par.
量子相变描述量子多体系统随非热的控制参量变化时，基态性质的突变行为。在原子核核中可以表现为基态形状的突变行为，如该caton中展示的：
对于满壳原子核其为球形，半满壳为稳定形变，而介于二者之间可能存在临界过渡状态。
为描述这种演化过程，特别是其中存在的临界行为，Iachello建议了两种方法：1，2
因此，我们将两种方法结合，基于前面发展的模型微观研究原子核量子相变。
此处以Nd同位素链中的一阶相变为例进行了计算，图中展示了偶偶的Nd同位素在beta-gamma平面内的位能曲面,
从Nd146至Nd152，原子核从近球形演化为稳定长椭形变。其中在Nd150中，在0.2<beta<0.4
时位能曲面很软，且在gamma<30度时，gamma方向上位能曲面具有抛物线特征，与X(5)临界点
对称性对应的位能曲面一致。
基于该位能曲面及计算得到的惯量参量，即可构造原子核的集体哈密顿量，进而得到原子核低激发谱，
Iachello, PRL2004
Method of Landau based on PES
Computation of order parameters

10. First order QPT ... detailed spectroscopy has been reproduced well !!
Spectrum
图中给出了计算得到的Nd150低激发谱，并与实验相比较。模型很好地再现了激发态能量、带内及带间BE2跃迁几率。
特别是对于0_2->2_1的带间跃迁。
... detailed spectroscopy has been reproduced well !!

11. Li, Niksic, Vretenar, Meng. Lalazissis & Ring, PRC79,054301(2009)
First order QPT
Characteristic features:
X(5)
Sharp increase of R42=E(41)/E(21) and B(E2; 21→01) in the yrast band
对其他原子核做类似计算，即可来研究集体特征量随同位素链的演化规律。
此处以E(4_1)/E(2_1)及BE2跃迁为例，计算很好地再现了实验，且二者在Nd150
附近存在快速增加行为。另外，Nd150的E(4_1)/E(2_1)与X(5)相一致。
相关细节及对其他类型相变的描述参见这些文献
Li, Niksic, Vretenar, Meng. Lalazissis & Ring, PRC79,054301(2009)
Li, Niksic, Vretenar, &Meng. PRC80,061301(R) (2009)
Li, Niksic, Vretenar, & Meng. PRC81, (2010)

12. Fission barrier and SD band
Extended 3DRMF+BCS to 3DRHB
随后，我们将模型进一步推广至基于三轴相对论Hartree-Bogoliubov理论，
并以Pu240为例，研究了其位能曲面、裂变位垒高度、形状Isomer
及基于Isomer的超形变带等。在考虑了三轴自由度之后，理论计算很好地再现了
实验的裂变位垒高度；同时计算得到的基态形变转动带及基于shape Isomer的超形变带
与实验相符。
Li, Niksic, Vretenar, Ring & Meng. PRC81,064321(2010)

13. E(21+) B(E2) Shape evolution & coexistence in N=28 isotones
近年来，人们对丰中子核中N＝28幻数的消失开展了一系列的研究工作。图中展示了实验测得的Ca，Ar，S，Si同位素链21＋低激发能及BE2跃迁的变化规律，
其中，N＝28 Ar，S，Si同中子素具有较低的21＋及显著增强的BE2跃迁，指明其可能存在形变基态且相应的球形能隙在逐渐消失
In recent years a number of studies have been devoted to the investigation of the fragility of the N = 28 magic number in neutron-rich nuclei.
Especially, the low-energy spectra display an evidence of ground-state deformation and/or shape coexistence in the neutron rich nuclei below 48Ca.
And correspondingly the spherical shell gaps of these nuclei are progressively reduced.
B(E2)

14. Shape evolution & coexistence in N=28 isotones
在展示我们的结果之前，这里给出了采用Gogny相互作用的HFB计算得到的轻核区形变Contour图
Before our results, I would like to present the calculations given by HFB theory with Gogny D1S effective interaction.
Here is the contour map for the calculated beta deformation.

15. Shape evolution & coexistence in N=28 isotones
N=28
48Ca
44S
42Si
在Ca48以下我们发现这些同中子素表现出明显的形变，且随质子数减少形变增大
And in this N=28 isotone, it is interesting to find below 48Ca, particularly for 44S, 42Si, and 40Mg, pronounced deformation is predicted.
40Mg

16. Shape evolution & coexistence in N=28 isotones
这里展示了我们采用三轴RHB理论计算得到的N＝28同中子素链的位能曲面变化趋势，
随质子数减少，原子核形状快速变化。由球形双幻核Ca48到长椭扁椭共存的S44,
再到扁椭的Si42,最后至稳定长椭极小的Mg40.同时，理论模型还给出了逐渐减弱的球形能隙
特别是这两个值和目前已有的实验结果吻合
Here we show the self-consistent triaxial quadrupole binding energy maps of 𝑁 = 28
isotones in the 𝛽 − 𝛾 plane.
The binding energy maps display a rich variety of rapidly evolving shapes.
Only four protons away from the doubly magic 48Ca, a coexistence of prolate and oblate minima in 44S is predicted.
Two proton less, Si42 is well oblate deformed, but in Mg40, a well prolate minimum appears.
The RHB model also predicts a reducing spherical shell gap.
最近，我们还对N=28的同中子链做了研究，随质子数减少，形状从球形的
Ca48，经长椭和扁椭共存的S44及扁椭形变的Si42，至长椭形变的Mg40。
图中给出了S44的单粒子能级随形变beta的变化，蓝色点划线为费米面。扁椭极小对应于形变
的中子28壳，而长椭极小则基于形变的质子Z=16的子壳形成。
计算的激发谱与实验相符

17. Shape evolution & coexistence in N=28 isotones
The variation of shapes in an isotopic, or isotonic, chain is governed by the
evolution of the underlying shell structure of single-nucleon orbitals. Here we show the evolution of the single particle levels along this prolate-triaxial-oblate line for neutron and protons. Obviously, the shape coexistent phenomenon in 44S can be attributed to the combined effect of both the prolate proton gap and oblate neutron gap near the Fermi surface.
Prolate: Δp = 5.05 MeV
Oblate: Δn = 3.91 MeV

18. Shape evolution & coexistence in N=28 isotones
类似的，在42Si中，质子和中子均具有显著的扁椭能隙，从而给出了其稳定的扁椭极小。
Similarly in 42Si but both proton and neutron gaps appear in the oblate side, which produce a well oblate deformed 42Si.

19. Shape evolution & coexistence in N=28 isotones
Low-lying spectrum
C. Force et al., PRL105
更进一步，我们研究了N＝28同中子素链的低激发谱，特别是共存核44S的低激发能级及电磁跃迁性质。
低的21＋及增强的BE2跃迁指明44S具有形变的基态，最近法国GANIL课题组测量了与02＋有关的两个跃迁，
通过与壳模型结果比较，指出44S为长椭－球形共存。
我们的理论计算很好的再现了21＋ 22＋激发态及两个跃迁几率。而偏高的02＋及过强的单极跃迁指明理论计算得到的02＋与基态01＋混合过强。
We can also investigate the low-lying spectrum. Here we particularly focus on the spectrum of 44S. The lower 2^+_1 state and
Enhenced BE2 values indicates S44 has a deformed g.s.. More recently, these two transitions are measured, and by comparing with the shell model calculation,
A prolate-spherical shape coexistence was inferred.
The model nicely reproduces both the excitation energy and the reduced transition probability B (E 2 2-0) for the first excited
state 2+ and the theoretical value for B (E 2; 0_2+-2_1) is also in good agreement with data. The excitation energy of the second 0+ state, however,
is calculated higher than the experimental counterpart. Together with the fact that the calculated monopole transition strength is also larger than the corresponding
experimental value of 8.7(7), this result indicates that there is more mixing between the theoretical 0+ states than what can be inferred from the data.

20. Shape evolution & coexistence in N=28 isotones
Low-lying spectrum
这可在集体波函数分布中得到证实，可以看出二者在这片区域具有较大重叠。
This can be seen more clearly in the probablity density distribution in the beta-gamma plane for these two 0+ states. Where a strong overlap is found between each other.
Li, Yao, Vretenar, Niksic, Chen & Meng, PRC84, (2011)

21. Effect of time-odd mean-field
The ATDHFB mass tensor
P. Ring and P. Schuck, “The Nuclear Many-Body Problem”
此前的模型中惯量参量的计算均采用了Inglis－Bealyea公式，无法包含奇时间平均场的贡献。
为自恰包含其贡献，可从ATDHFB推导出的惯量参量表达式出发，然而这里包含相互作用矩阵求逆运算，
无法用于实际的计算。最近，我们发现其相互作用矩阵的逆刚好等价于原子核对外界集体振动或转动扰动的响应函数，
该响应函数满足Bethe－Spleter方程，进而我们对方程做高阶展开，同时采用可分离的
点耦合有效相互作用，即可得到自恰包含了奇时间平均场贡献的惯量参量。该方法十分有效，可用于中重核
的计算

22. Effect of time-odd mean-field
The ATDHFB mass tensor
这里以Ne20的转动惯量Ix为例，给出了对该方法中高阶展开的收敛性检验，发现展开到二阶即可给出和推转程序计算
得到的Ix很接近的结果，因此，在实际计算中我们将展开到二阶。

23. Effect of time-odd mean-field
128Xe MOI
此后我们以gamma软的Xe128为例，计算了包含奇时间场贡献的转动惯量和集体质量，记为I(TV) 和 B(TV)，并与相应的IB结果相比较，
图中给出了二者比值的Contour分布，整体而言，奇时间场增大了原子核的惯量参量。使转动惯量增加10～25％不等，
而使集体质量最大增至为IB结果的1.5倍。
128Xe Mass

24. Z.P. Li et al. in preparation
Effect of time-odd mean-field
Effect of time-odd mean-field on the spectrum
更进一步，研究了奇时间场对对低激发谱的影响，左列对应于IB给出的激发谱，中间的结果考虑了奇时间场贡献，右边为实验值。
显然，奇时间场增大了惯量参量，从而使能谱整体压缩，与实验更好的符合。特别是02＋的位置。
Z.P. Li et al. in preparation

25. Summary Microscopic Collective Hamiltonian based on
CEDF has been developed
Application to the interesting topics
Nuclear QPT
Fission barrier and SD band in 240Pu
Shape evolution & coexistence in N=28 isotones
Effect of time-odd mean-field
总结一下，首先我们发展了基于协变密度泛函的集体哈密顿量模型，并用于研究了以下有趣的前沿问题，包括。。。。。。

26. Thank You ! J. Meng & JCNP group D. Vretenar & T. Niksic P. Ring
L. Prochniak
G. A. Lalazissis
J. M. Yao

27. Collective Hamiltonian

28. Collective Parameter

29. Collective Parameter

30. Collective Parameter