Applied Physics
Vol. 09  No. 10 ( 2019 ), Article ID: 32702 , 7 pages
10.12677/APP.2019.910049

Study of Even-Even Nuclei 100Zr by Interacting Boson Model

Hongfei Dong, Yin Wang, Xiaowei Li, Lijun Lv, Tianzhi Wei

Chifeng University, Chifeng Inner Mongolia

Received: Oct. 9th, 2019; accepted: Oct. 21st, 2019; published: Oct. 28th, 2019

ABSTRACT

Even-even nuclei 100Zr were studied within the framework of the interacting boson model. The E-Gamma Over Spin (E-GOS) was drawn, and the analysis of the dynamic symmetry limit found that 100Zr is a transition nuclei from U(5) vibrational limit to SU(3) rotational limit , close to O(6) dynamic symmetry limit. At the same time, the energy spectrum of low-lying states of 100Zr was fitted, the components of the wave function were analyzed, and the B(E2) values of transitions between low-lying states of 100Zr were analyzed respectively. The results show good agreement with the available experimental data.

Keywords:Even-Even Nuclei, Interacting Boson Model, Energy Level, E-GOS Curve, Electromagnetic Transitions

IBM模型对偶–偶核100Zr的理论研究

董鸿飞,王印,李晓伟,吕立君,魏天枝

赤峰学院,内蒙古 赤峰

收稿日期:2019年10月9日;录用日期:2019年10月21日;发布日期:2019年10月28日

摘 要

本文在相互作用玻色子模型框架下对偶–偶核100Zr进行了理论研究。绘制了基态带的E-GOS曲线,讨论其动力学对称性极限性质,数据分析表明100Zr是具有U(5)振动极限到SU(3)转动极限之间的过渡核,趋近于O(6)极限。同时文中也拟合了100Zr核的低能谱的谱带,并对波函数结构进行了理论研究,计算了100Zr核的低能谱部分的电磁跃迁,计算结果表明理论计算与实验值符合较好。

关键词 :偶–偶核,相互作用玻色子模型,能谱,E-GOS曲线,电磁跃迁

Copyright © 2019 by author(s) and Hans Publishers Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

1. 引言

采用唯象的理论模型是原子核结构研究的重要手段。相互作用玻色子模型(interacting boson model,简称为IBM)就是一个十分成功的研究原子核集体运动的代数模型,利用该模型,人们成功地描述了原子核低激发能谱、电磁跃迁以及相变等性质 [1]。在IBM中,假设原子核有一个稳定的双幻核芯,价核子配对成角动量是0或2的核子对,这些核子都被看作是玻色子。角动量L为0的s玻色子和L为2的d玻色子共有六种,这六种玻色子算符构成了IBM模型的哈密顿量,即能谱的生成代数是U(6)。从U(6)开始约化,有U(5)、SU(3)和O(6)三种约化方式。约化的三个子群链为:

U ( 6 ) U ( 5 ) O ( 5 ) O ( 3 ) O ( 2 ) U ( 6 ) SU ( 5 ) O ( 3 ) O ( 2 ) U ( 6 ) O ( 6 ) O ( 5 ) O ( 3 ) O ( 2 ) (1)

这三个子群链分别对应于不同类型的动力学对称性,用来描述原子核的三种集体运动极限:振动、转动和γ-不稳定特性 [2] - [7]。

三个极限的晕态能谱和能级衰变能分别为:

E I = I 2 ω E γ ( I I 2 ) = ω

E I = 2 2 J I ( I + 1 ) E γ ( I I 2 ) = 2 2 J ( 4 I 2 )

E I = I ( I + 6 ) 16 E ( 2 + ) E γ ( I I 2 ) = E ( 2 + ) 4 ( I + 2 )

[8],做I-R曲线既为E-GOS曲线。

将基向量表示为 | Ψ = | n d , n β , n Δ , L d , L ,哈密顿量可写成多极展开形式为:

H = EPS n d + 1 2 ELL ( L L ) + 1 2 QQ ( Q Q ) 5 7 OCT [ ( d d ˜ ) ( 3 ) × ( d d ˜ ) ( 3 ) ] 0 ( 0 ) + 15 HEX [ ( d d ˜ ) ( 4 ) × ( d d ˜ ) ( 4 ) ] 0 ( 0 ) (2)

其中:

L L = 10 3 [ ( d d ˜ ) ( 1 ) × ( d d ˜ ) ( 1 ) ] 0 (0)

Q Q = 5 [ { ( s d ˜ + d s ) ( 2 ) + CHQ 5 ( d d ˜ ) ( 2 ) } × { ( s d ˜ + d s ) ( 2 ) + CHQ 5 ( d d ˜ ) ( 2 ) } ] 0 (0)

式中的EPS、ELL、QQ、OCT、HEX、CHQ为模型的可调参数 [9]。本文工作是在合理的范围内调节参数值,使计算结果符合实验数据。

2. 计算结果

本文研究的是100Zr,它有10个价质子(空穴)和10个价中子,共组成10个玻色子。

2.1. 各级限值和E-GOS曲线

根据其实验能谱可以计算出R值并做E-GOS曲线,R值见表1。相应的E-GOS曲线见图1。本文选取了实验数据 20 1 + 以下的角动量为偶数的能级。

Table 1. Experimental data and the dynamic symmetry limit of 100Zr

表1. 100Zr核的实验及各动力学极限值

Figure 1. Curve: the E-Gamma over spin of 100Zr

图1. 100Zr核的E-GOS曲线

2.2. 模型参数

通过拟合实验的能级,确定了模型的参数,见表2

Table 2. Hamiltonian matrix of 100Zr

表2. 100Zr的哈密顿参数

2.3. 能谱结果

在选定的参数下,理论计算的能级与实验能级的对比图见图2。可见所选参数较好地拟合了低激发态能谱,其中Band 1和Band 2的符合程度均很好,只是在较合理的范围内存在一定的误差。

Figure 2. Experimental energy states and theoretical energy states of 100Zr

图2. 100Zr的实验能谱与理论能谱

2.4. 波函数

确定了模型参数,我们就可以给出每条能级具体的波函数,本文主要用到的波函数的结构为:

| 0 1 + 0.5248 | s 10 d 0 + 0.4457 | s 8 d 2 + 0.0292 | s 6 d 4 + 0.0003 | s 4 d 6

| Ψ 0 1 0.724 | 0 , 0 , 0 , 0 , 0 + + 0.668 | 2 , 1 , 0 , 0 , 0 + + 0.171 | 4 , 2 , 0 , 0 , 0 + + 0.018 | 6 , 3 , 0 , 0 , 0 + + 0.001 | 8 , 4 , 0 , 0 , 0 +

| 0 2 + 0.4704 | s 10 d 0 + 0.4528 | s 8 d 2 + 0.0756 | s 6 d 4 + 0.0012 | s 4 d 6

| Ψ 0 2 0.686 | 0 , 0 , 0 , 0 , 0 + 0.673 | 2 , 1 , 0 , 0 , 0 + 0.275 | 4 , 2 , 0 , 0 , 0 + 0.035 | 6 , 3 , 0 , 0 , 0 + 0.002 | 8 , 4 , 0 , 0 , 0 +

| 0 3 + 0.8624 | s 6 d 4 + 0.1010 | s 8 d 2 + 0.0317 | s 4 d 6 + 0.0048 | s 10 d 0

| Ψ 0 3 0.069 | 0 , 0 , 0 , 0 , 0 + + 0.318 | 2 , 1 , 0 , 0 , 0 + 0.929 | 4 , 2 , 0 , 0 , 0 + 0.178 | 6 , 3 , 0 , 0 , 0 + 0.011 | 8 , 4 , 0 , 0 , 0 +

| 2 1 + 0.8335 | s 9 d 1 + 0.1519 | s 7 d 3 + 0.0105 | s 8 d 2 + 0.0002 | s 6 d 4

| Ψ 2 1 0.913 | 1 , 0 , 0 , 2 , 2 + 0.103 | 2 , 0 , 0 , 2 , 2 + + 0.390 | 3 , 1 , 0 , 2 , 2 + 0.015 | 4 , 1 , 0 , 2 , 2 + + 0.062 | 5 , 2 , 0 , 2 , 2 + 0.001 | 6 , 2 , 0 , 2 , 2 + + 0.004 | 7 , 3 , 0 , 2 , 2 +

| 2 2 + 0.7410 | s 7 d 3 + 0.1203 | s 9 d 1 + 0.0899 | s 8 d 2 + 0.0434 | s 5 d 5 + 0.0051 | s 6 d 4 + 0.0003 | s 2 d 8 + 0.0001 | s 3 d 7

| Ψ 2 2 0.347 | 1 , 0 , 0 , 2 , 2 + + 0.300 | 2 , 0 , 0 , 2 , 2 + + 0.861 | 3 , 1 , 0 , 2 , 2 + + 0.071 | 4 , 1 , 0 , 2 , 2 + + 0.208 | 5 , 2 , 0 , 2 , 2 + + 0.007 | 6 , 2 , 0 , 2 , 2 + + 0.018 | 7 , 3 , 0 , 2 , 2 + + 0.001 | 9 , 4 , 0 , 2 , 2 +

| 2 3 + 0.8338 | s 8 d 2 + 0.0599 | s 5 d 5 + 0.0564 | s 7 d 3 + 0.0451 | s 9 d 1 + 0.0040 | s 6 d 4 + 0.0007 | s 3 d 7

| Ψ 2 3 0.212 | 1 , 0 , 0 , 2 , 2 + 0.913 | 2 , 0 , 0 , 2 , 2 + + 0.238 | 3 , 1 , 0 , 2 , 2 + 0.245 | 4 , 1 , 0 , 2 , 2 + + 0.063 | 5 , 2 , 0 , 2 , 2 + 0.026 | 6 , 2 , 0 , 2 , 2 + 0.001 | 7 , 3 , 0 , 2 , 2 +

| 2 4 + 0.9079 | s 5 d 5 + 0.0656 | s 8 d 2 + 0.0262 | s 7 d 3 + 0.0002 | s 9 d 1 + 0.0001 | s 2 d 8

| Ψ 2 4 0.014 | 1 , 0 , 0 , 2 , 2 + + 0.256 | 2 , 0 , 0 , 2 , 2 + 0.002 | 3 , 1 , 0 , 2 , 2 + 0.953 | 4 , 1 , 0 , 2 , 2 + 0.004 | 5 , 2 , 0 , 2 , 2 + 0.162 | 6 , 2 , 0 , 2 , 2 + 0.009 | 7 , 3 , 0 , 2 , 2 +

| 4 1 + 0.9348 | s 8 d 2 + 0.0646 | s 6 d 4 + 0.0007 | s 4 d 4

| Ψ 4 1 0.967 | 2 , 0 , 0 , 4 , 4 + + 0.254 | 4 , 1 , 0 , 4 , 4 + + 0.027 | 6 , 2 , 0 , 4 , 4 + + 0.001 | 8 , 3 , 0 , 4 , 4 +

| 4 2 + 0.9087 | s 6 d 4 + 0.0650 | s 8 d 2 + 0.0263 | s 4 d 6 + 0.0001 | s 2 d 8

| Ψ 4 2 - 0.225 | 2 , 0 , 0 , 4 , 4 + + 0.953 | 4 , 1 , 0 , 4 , 4 + + 0.162 | 6 , 2 , 0 , 4 , 4 + + 0.009 | 8 , 3 , 0 , 4 , 4 +

| 4 3 + 0.9689 | s 7 d 3 + 0.0309 | s 5 d 5 + 0.0002 | s 3 d 7

| Ψ 4 3 0.984 | 3 , 0 , 0 , 4 , 4 + + 0.176 | 5 , 1 , 0 , 4 , 4 + + 0.013 | 7 , 2 , 0 , 4 , 4 +

| 4 4 + 0.9549 | s 5 d 5 + 0.0310 | s 7 d 3 + 0.0140 | s 3 d 7

| Ψ 4 4 0.176 | 3 , 0 , 0 , 4 , 4 + + 0.977 | 5 , 1 , 0 , 4 , 4 + + 0.118 | 7 , 2 , 0 , 4 , 4 + + 0.004 | 9 , 3 , 0 , 4 , 4 +

2.5. 电磁跃迁

利用波函数我们可以进一步研究原子核的电磁性质,本文计算了低激发能级的B(E2)值,见表3

Table 3. The B(E2) of electromagnetic transitions of 100Zr

表3. 100Zr电磁跃迁的B(E2) 值

3. 结论

本文用IBM模型对偶–偶核100Zr进行了研究,在模型所选的参数下拟合了低激发能级,计算结果在一定的误差允许范围内是合理的。同时也用能级的对应的波函数计算了约化跃迁几率。100Zr核素的E-GOS曲线结果表明100Zr是具有U(5)振动极限到SU(3)转动极限之间的过渡核,趋近于O(6)极限,具有较明显的γ-不稳定特性。

基金项目

内蒙古自治区教育厅自然科学重点项目(NJZZ17296)。

文章引用

董鸿飞,王 印,李晓伟,吕立君,魏天枝. IBM模型对偶–偶核100Zr的理论研究
Study of Even-Even Nuclei 100Zr by Interacting Boson Model[J]. 应用物理, 2019, 09(10): 403-409. https://doi.org/10.12677/APP.2019.910049

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