少体系统束缚态性质研究
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摘要
通常11Li,14Be和17B被认为是具有三体结构的核素,其核芯外存在两个束缚得非常弱的价中子。这些核素具有Borromean性质,即当它们作为三体系统束缚在一起时,只有一个弱的束缚态,而其中的两体子系统则都没有束缚态存在。在本文中,我们仅研究了晕核束缚态的性质并且文中采用的数值计算算法侧重于变分法。基于三体模型,采用唯象的两体相互作用势,如汤川势能和指数势能形式,变分计算给出了束缚态能量以及其它一些晕核特性如其异常大的均方根半径等,同时也给出了晕中子密度分布的一个解析表达式。所有的理论计算结果与实验数据一致。
对于两体系统,本文研究了非相对论和相对论两种情况。对于非相对论情况,我们讨论了如何直接数值求解Schrodinger方程。因为我们知道只有少数几个势能形式存在解析解,而大部分一般形式的势必须求助于数值方法。该数值方法在三体模型中也是非常有用。我们是采用唯象的核力模型来描述三体系统中两体子系统之间的相互作用的。因为是唯象的,所以势能形式中的许多参数必须利用实验数据进行拟合。该方法将在拟合时起重要作用。对于相对论情形,我们找到了某些可以解析求解的系统,如Dirac振子系统和由非球谐振子势能1/2r2+A/2r2描述的系统。对于前者我们获得了动力学算符矩阵元的一些有用的解析表达式并对它们的性质加以讨论。对于后者,在矢量势和标量势相等的情况下,我们给出了其Klein-Gordon方程和Dirac方程的s波束缚态解。
关于QCD,本文主要讨论了Nambu-Jona-Lasinio模型和Dyson-Schwinger方程。我们介绍了前者在两味夸克模型中的应用并且阐明了夸克质量的动力学产生是手征对称性破缺的一个特征。对于后者,当存在非零流夸克质量情况下,夸克传播子所满足的DS方程存在着两个定性不同的解。在手征极限下其中一个相应于Nambu-Goldstone相,而另一个则相应于Wigner相。另外,我们还提出了一种在有限化学势下计算QCD配分函数的普适方法。我们发现配分函数完全由dressed夸克传播子决定直到有限化学势到达某个常数为止。从该点出发,我们获得了判别Nambu相和Wigner相之间发生相变的一个判据。
The nuclei11Li,14Be and17B are pronounced halo-structures which have two loosely bound valence neutrons. These nuclei are Borromean, i.e. while they are bound and there is only one bound state, they are considered as three-body system, with on bound states in the binary subsystems. We restrict this thesis to bound state properties and our procedures emphasize on the variational approach. Based on the three-body model, with the two-body phenomenological potential, such as Yukawa potential or exponential potential, our variational computation reproduce the binding energy and such halo characteristics as the abnormally large matter r.m.s. radius and give one analytical expression for the halo neutron distribution of matter density. All the results we get are accordant with the experimental data.
For two-body system, there are two cases, the non-relative and the relative. With regard to the non-relative case, we discussed the numerical approach to solve Schrodinger equation directly, for the reason that there are only several potentials which can be solved analytically. This approach is also very useful in the three-body system, because in our three-body model, the interactions between the binary subsytems are described by phenomenological potentials, all the parameters of which must be fitted with ex-periment data. And this approach will play an important role in this process. For the relative case, we found certain systems that could be analytically studied, such as the Dirac oscillator and three-dimensional non-harmonic oscillator potential1/2r2+A/2r2For the former some analytical expressions of the matrix elements for dynamical operators are obtained and their behaviour is discussed in details. For the later, the s-wave bound solutions of both Dirac equation and Klein-Gordon equation are given when the scalar potential is equal to the vector potential.
For QCD, what we discussed are about the Nambu-Jona-Lasinio model and the Dyson-Schwinger equation. The former is reviewed in its flavor SU(2) versions applied to quarks. The dynamical generation of quark masses is demonstrated as a feature of chiral symmetry breaking. For the later, it is shown on general ground that there exist two qualitatively distinct solutions for the quark propagator in the case of non-zero current quark mass. One solution corresponds to the Nambu-Goldstone phase and the other one corresponds to the Wigner phase in the chiral limit. Additionally, we propose a general method for calculation the partition function of QCD at finite chemical potential. It is found that the partition function is totally determined by the dressed quark propagator at finite chemical potential up to a multiplicative constant. From this a criterion for the phase transition between the Nambu and the Wigner phase is obtained.
对于两体系统,本文研究了非相对论和相对论两种情况。对于非相对论情况,我们讨论了如何直接数值求解Schrodinger方程。因为我们知道只有少数几个势能形式存在解析解,而大部分一般形式的势必须求助于数值方法。该数值方法在三体模型中也是非常有用。我们是采用唯象的核力模型来描述三体系统中两体子系统之间的相互作用的。因为是唯象的,所以势能形式中的许多参数必须利用实验数据进行拟合。该方法将在拟合时起重要作用。对于相对论情形,我们找到了某些可以解析求解的系统,如Dirac振子系统和由非球谐振子势能1/2r2+A/2r2描述的系统。对于前者我们获得了动力学算符矩阵元的一些有用的解析表达式并对它们的性质加以讨论。对于后者,在矢量势和标量势相等的情况下,我们给出了其Klein-Gordon方程和Dirac方程的s波束缚态解。
关于QCD,本文主要讨论了Nambu-Jona-Lasinio模型和Dyson-Schwinger方程。我们介绍了前者在两味夸克模型中的应用并且阐明了夸克质量的动力学产生是手征对称性破缺的一个特征。对于后者,当存在非零流夸克质量情况下,夸克传播子所满足的DS方程存在着两个定性不同的解。在手征极限下其中一个相应于Nambu-Goldstone相,而另一个则相应于Wigner相。另外,我们还提出了一种在有限化学势下计算QCD配分函数的普适方法。我们发现配分函数完全由dressed夸克传播子决定直到有限化学势到达某个常数为止。从该点出发,我们获得了判别Nambu相和Wigner相之间发生相变的一个判据。
The nuclei11Li,14Be and17B are pronounced halo-structures which have two loosely bound valence neutrons. These nuclei are Borromean, i.e. while they are bound and there is only one bound state, they are considered as three-body system, with on bound states in the binary subsystems. We restrict this thesis to bound state properties and our procedures emphasize on the variational approach. Based on the three-body model, with the two-body phenomenological potential, such as Yukawa potential or exponential potential, our variational computation reproduce the binding energy and such halo characteristics as the abnormally large matter r.m.s. radius and give one analytical expression for the halo neutron distribution of matter density. All the results we get are accordant with the experimental data.
For two-body system, there are two cases, the non-relative and the relative. With regard to the non-relative case, we discussed the numerical approach to solve Schrodinger equation directly, for the reason that there are only several potentials which can be solved analytically. This approach is also very useful in the three-body system, because in our three-body model, the interactions between the binary subsytems are described by phenomenological potentials, all the parameters of which must be fitted with ex-periment data. And this approach will play an important role in this process. For the relative case, we found certain systems that could be analytically studied, such as the Dirac oscillator and three-dimensional non-harmonic oscillator potential1/2r2+A/2r2For the former some analytical expressions of the matrix elements for dynamical operators are obtained and their behaviour is discussed in details. For the later, the s-wave bound solutions of both Dirac equation and Klein-Gordon equation are given when the scalar potential is equal to the vector potential.
For QCD, what we discussed are about the Nambu-Jona-Lasinio model and the Dyson-Schwinger equation. The former is reviewed in its flavor SU(2) versions applied to quarks. The dynamical generation of quark masses is demonstrated as a feature of chiral symmetry breaking. For the later, it is shown on general ground that there exist two qualitatively distinct solutions for the quark propagator in the case of non-zero current quark mass. One solution corresponds to the Nambu-Goldstone phase and the other one corresponds to the Wigner phase in the chiral limit. Additionally, we propose a general method for calculation the partition function of QCD at finite chemical potential. It is found that the partition function is totally determined by the dressed quark propagator at finite chemical potential up to a multiplicative constant. From this a criterion for the phase transition between the Nambu and the Wigner phase is obtained.
引文
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