A~70区原子核基态和激发态的
三轴形变
沈水法1 许甫荣2 郑世界2
1(东华理工大学核工程技术学院)
2(中国科学院高能物理研究所)
• Experimentally, it is difficult to determine
the value of the triaxial parameter for a
nucleus. When the axial symmetry breaks,
one cannot determine at the same time
the quadrupole deformation parameters of
both 2 and  from the experimental B(E2)
value.
1. Approximations in the sum-rule
method
2
g .s.
Q
  0 E2 2


r
2
(1)
where r denotes the different 2+ excitations. In
practice, the restriction to r=1, 2 turns out to be
sufficient. The quantity Q is a measure of the
symmetric quadrupole deformation rms which
includes both static (rigid) and dynamic (soft)
contributions:
r
2
g .s .
 rms 
4
3ZR02
Qg2.s.
1/ 2
7
  
10 
3/ 2
  0 E2 2 2 E2 2  2 E2 0
•
(2)
• Furthermore, in the estimate of cos 3 g .s. one can again
involve only the first and second 2+ excitations (r=1, 2)

 2
0
E
2
2
22 E 2 22 :
but also additionally neglect the term 1
2
cos3 g .s.
Qg2.s.

r

r

t

t

r ,t
1/ 2
cos 3 g .s.

7
  
10 
Qg2.s.
3/ 2
[ 0 E 2 21
2
21 E 2 21
 2 0 E 2 21 21 E 2 22 22 E 2 0 ]
(3)
The value of
1
3
 eff  arccos( cos 3 g .s. )
corresponds (up to higher order terms) to the collectivemodel asymmetry angle .
An overall correlation between the asymmetry
parameter  and quadrupole deformation emerges
from this presentation
• It should be stressed that the present analysis based on
the third-order products of matrix elements (Eq. (3)) can
not determine whether the triaxiality is soft (dynamic) or
rigid (static)[1]. The softness in  can be determined from
sixth-order products of experimental E2 matrix elements
available at present only in very few cases [6].
• According to the works of Refs.[20, 21], one should not
expect rigid triaxiality in the ground state of any nucleus.
• [1]W. Andrejtscheff, P. Petkov, Phys. Lett. B329(1994)1
• [20]S. Åberg, H. Flocard, and W. Nazarewicz, Annu. Rev.
Nucl. Part. Sci. 40,439(1990)
• [21]N. V. Zamfir and R. F. Casten, Phys. Lett. B260,
265(1991)
• In this work, we investigate the possible
maximum triaxiality of 30 in the ground
states of even-even nuclei in the A70
region. The total-Routhian-surface (TRS)
method has been used to determine the
stability of the triaxiality with rotational
frequency. The TRS calculations are
carried out by means of the pairingdeformation-frequency
self-consistent
cranked shell model.
Fig. 1. The shape evolution with A with respect to the 2 (upper panel)
and -deformation (lower panel) deduced from total Routhian surfaces
(TRS) diagrams for ground states in even-A 64-80Ge. The error bars
display the deformation values within an energy range of less than
100keV above the minimum, giving an indication of the softness of the
nucleus with respect to the corresponding shape parameter.
Our result:
• For the ground states of germanium
isotopes, we see the shape transitions
from a triaxial shape in 64Ge to nearly
oblate shapes in 66-72Ge, and to a =-30
triaxial shape again in 74Ge, and toward
weakly deformed prolate shapes in 78,80Ge.
Fig. 2. Similar to Fig. 1, but for ground states in even-A 68-82Se.
Fig. 3. Calculated total Routian surfaces for 74Ge positive-parity states,
at ħ=0.3 and 0.7MeV (upper panels) and 1.0 and 1.2MeV (lower
panels) corresponding to I(2-24)ħ. A black dot indicates the lowest
minimum in each case, and the energy difference between neighboring
contours is 200keV.
Fig. 4. The calculated Woods-Saxon single-particle
diagrams against the triaxial deformation .
• 我们看到，在Z = 32 和N = 32处有一形变的壳能
隙(shell gap)。 TRS计算显示， 核64Ge有一不太
软的三轴形状 (参见图1)。 但是， 在N=34出现一
个扁椭球壳能隙， 其结果导致在66Ge中的扁椭球
形状。 随着中子数的增加， 扁椭球中子能隙的效
应减小， 因此更重的锗同位素的形变向三轴(或长
椭球)形状变化。 再者对质子，Z=34处也存在扁
椭球能隙。 这(结合N = 34扁椭球中子能隙的效应)
就是为什么比 78Ge更轻的硒同位素具有大形变扁
椭球形状的的原因(参见图2)。
图4(a). 对74Ge核由实验结果提取出
的转动惯量与TRS计算值的比较
25
experimental
74
theoretical
Ge
20
J
(1)
15
10
5
0
0.0
0.5
1.0
h
1.5
图4(b). 对74Se核由实验结果提取出
的转动惯量与TRS计算值的比较
30
25
15
J
(1)
20
10
5
0
0.0
0.5
1.0
h
1.5
实 验 上 ħ( 单 位 :MeV) 和  (1)/ħ2 ( 单
位:MeV-1)可用以下公式提取：
 
I
(1)

(由  I ( I  1) 得到)
H
dE

（按正则方程，q= , 我们有＝
得到）
2
p
d I ( I  1)
E
理 论 上 ħ( 单 位 :MeV) 和 
位:MeV-1)可用以下公式提取：
是自变量
Ix 

 
, 0
 ˆjx   
  Ix 
(1)

 
, 0
 ˆjx  
(1)/ħ2(
单
• It needs to be mentioned that the present TRS
calculations cannot reasonably reproduce the
experimental data of observed excited states. The data
[25,26] show strongly vibrational effect that is not
included in the TRS model. On the other hand, the
present model does one-dimensional principal-axis
cranking. For a triaxial shape, in principle, one should
run three-axis cranking. However, the onedimensional cranking model should be able to give a
right description of deformation.
• As a supplement, in our present work, the triaxiality
parameter  is calculated using the following formula
•
2
9
8 2


E1, 2 ( I  2 ) 
[1  1  sin 3 ]
2
•
, (4)
0 sin 3
9
• which is the result of solution of the Bohr Hamiltonian for
soft triaxial nuclei[21, 22], and the  value is carried out to
E (2 )
be 28.869 by the ratio E (2 ) , where the experimental
values of these two levels were adopted from recent
work. It shows that despite the different technique used
to obtain those values they agree fairly well with our
results.
• [26]L. Fortunato, S. De Baerdemacker, and K. Heyde,
Phys. Rev. C74, 014310 (2006)
• [27]A. S. Davydov and G. F. Filippov, Nucl. Phys. 8,
237(1958)

2

1
Summary
• Total-Routhian-Surface calculations by means
of the pairing-deformation-frequency selfconsistent cranked shell model have been
carried out for germanium and selenium
isotopes, in order to search for possible stable
triaxial deformations of nuclear states. The
maximum triaxiality of |γ|∼30 is found in the
ground and excited rotational states of the
nuclei 64,74Ge. The triaxiality has its origin
from triaxial shell gaps at Z, N=32.
• 该工作发表在:
S.F. Shen, S.J. Zheng, F.R. Xu, and R.
Wyss, Phys. Rev. C84, 044315(2011)