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

(1) where r denotes the different 2+ excitations. In

practice, the restriction to r=1, 2 turns out to be sufficient. The quantity is a measure of the symmetric quadrupole deformation rms which includes both static (rigid) and dynamic (soft) contributions:

2..sgQ

2..2

034

sgrms QZR

r

rsg EQ22

.. 220

• (2)• Furthermore, in the estimate of one can again inv

olve only the first and second 2+ excitations (r=1, 2) but also additionally neglect the term :

(3) The value of

corresponds (up to higher order terms) to the collective-model asymmetry angle .

1/ 23/ 22

. . . .,

7cos3 0 2 2 2 2 2 2 2 010g s g s r r t t

r t

Q E E E

1/ 23/ 2 22

. . . . 1 1 1

1 1 2 2

7cos3 [ 0 2 2 2 2 210

2 0 2 2 2 2 2 2 2 0 ]

g s g sQ E E

E E E

. .1 arccos( cos3 )3eff g s

. .cos3 g s

2

1 2 2 20 2 2 2 2 2E E

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 A70 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 pairing-deformation-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

0.0 0.5 1.0 1.50

5

10

15

20

25

0.0 0.5 1.0 1.50

5

10

15

20

25

J(1)

h

experimental

theoretical74Ge

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

0.0 0.5 1.0 1.50

5

10

15

20

25

30

0.0 0.5 1.0 1.50

5

10

15

20

25

30

J(1)

h

实验上 ħ(单位 :MeV)和 (1)/ħ2 ( 单位 :MeV-1)可用以下公式提取：(1) ( ( 1) )

2 ( 1)

I I I

Ep I I

由 得到

H dE（按正则方程，q= ,我们有 ＝ 得到）

d

理论上 ħ(单位 :MeV)和 (1)/ħ2( 单位 :MeV-1)可用以下公式提取： 是自变量

, 0 , 0

ˆ ˆx x xI j j

(1)xI

• 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 one-dimensional 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

• • , (4)• which is the result of solution of the Bohr Hamiltonian for soft triaxi

al nuclei[21, 22], and the value is carried out to be 28.869 by the ratio , 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)

]3sin9811[

3sin9)2( 22

0

2

2,1

IE

2

1

(2 )(2 )

EE

Summary

• Total-Routhian-Surface calculations by means of the pairing-deformation-frequency self-consistent 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. W

yss, Phys. Rev. C84, 044315(2011)