理论物理中若干前沿问题的研究—Bose-Einstein凝结的转变温度与关联函数、各向异性电阻率测量理论、电声超导模型的渐近严格精确解
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摘要
本文研究了若干理论问题,包括谐振子势约束Bose-Einstein凝结的转变温度和关联函数、基于线电流电极模型的各向异性电阻率测量理论、如何解决变量分离法中遇到的非齐次边界条件的普遍难题、以及超导电声子模型的渐近精确严格解。
通常的谐振子势约束BEC转变温度表达式基于化学势为零的假设,在物理与数学上有一定的不自洽性,本文从化学势的角度,仔细研究了这一问题和给予合理的论述,并且利用Bose-Einstein函数推导与计算了二维与三维体系的化学势与转变温度,然后给出了数值结果拟合的转变温度公式。我们发展了Bose-Einsten函数的负系数低温展开式,其高精度和良好的收敛性有助于改进数值计算。利用这一函数,无外场理想情形的问题也从新的角度得到了讨论。我们还得到了二维与三维有外场情形Tc的拟合表达式。
关联函数可以反映体系的量子统计关联特性。我们将从严格定义出发,得到了谐振子势约束BEC的关联函数的表达式。由于高阶Hermite函数高速剧烈振荡,无法对长距离的关联函数进行分析和计算,我们克服了重重困难,成功地获得一个双变量函数级数的严格求和公式,将结果用初等函数表示。它是著名的Bloch单变量和式的推广。我们进而给出大粒子数体系的关联函数,它具有简洁、平稳、收敛性良好的Gauss型级数表式。然后顺利地完成了高精度的数值计算。它们能够反映整体的量子关联效应。包括转变温度Tc附近的量子关联的渐近行为。
超导的理论与实验研究一直在不断的发展,就实验而言,高温超导体的各向异性薄膜的电阻率测量是一个十分有趣和重要的问题。因为垂直于薄膜表面的c轴电阻率测量存在困难,所以我们发展了线电流电极模型,设计了三种测量组态,其好处在于设计简单、利于实现,并且在理论上可以严格讨论.我们利用源像法得到了电势的级数求和表示,然后根据Ω-级数理论将其化为形式简单的表达式。该方法有利于克服c轴电阻率测量不灵敏的困难。
为了满足实验需要,我们还进一步提出基于一个样品,一个平面测出电阻率三个分量的方法与理论,并精确地考虑电极的边缘效应。由于利用源像法和Ω-级数理论,我们得到了新的组态的严格结果。这一理论好处在于有利于避免多样品测量可能存在的不一致性,而且设置简易。同时,我们给出了严格结果的不同解析表示(即第Ⅰ类和第Ⅱ类严格解),它们在不同的情形下具有不同的收敛性和用处。
在相关研究中,一些工作(例如知名学术刊物(如J. Applied Physics)上发表的论文)在分离变量法中遇到了非齐次边界条件的普遍难题。我们获得一批严格求和公式,进而提出利用虚设边界条件的方法和解析开拓一致性定理,在相当普遍的框架下解决了这类难题和困扰。
寻求一个高维(高于一维)的、存在相变的、同时具有实际的物理背景的量子统计问题的严格解,是个挑战性的问题。
我们从电声相互作用模型出发,运用严格的推广的Stratonovich算子等式,在虚拟空间、在热力学极限下,将有效哈密顿量线性化,进而将Fermi二次型哈密顿量对角化定理推广到虚拟空间,并严格实现对角化,从而获得在热力学极限下渐近准确的配分函数,以及各种热力学的表达式,并就唯一性给出论证。
我们还利用鞍点法,获得了新的能隙方程,它与BCS理论中的能隙方程相似,但又不相同.它具有一些自己的特点。我们对新的能隙方程作了详细研究。结果表明,新的严格解理论不但自身是渐近准确的,而且可以说明广为实验支持的BCS理论为什么是正确的,其理论误差是可以忽略的,从而也给出BCS理论以支持,同时BCS理论反过来为严格解提供例证。
There exists certain self-inconsistency in some common-sensed expression of tran-sition temperature (Tc) of Bose-Einstein condensation (BEC) in harmonic traps. The principle of assuming μ=0at Tc is reexamined and a possible solution is proposed for2D and3D cases considering the derivatives of chemical potential. An effective approx-imation tool, the Bose-Einstein function, is studied and then applied to the solving of Tc because of this function's excellent convergency. The expansions of BE functions at low temperatures, including those with negative indexes, are developed. These expan-sions are useful not only for the field-free case, but also for BEC in harmonic traps. Analytic fitting expressions of Tc in2D and3D cases are obtained.
A new expression of correlation function of BEC in harmonic traps is presented and calculated. A new Gaussian type expression with two coordinates for different dimensions is obtained for the main part of correlation function. It is simple, well convergent and can asymptotically represent quantum correlation when N→∞. We hope this expression may be useful in the study of correlation behavior near transition temperature for Bose system in traps.
Superconductivity continuously receives intensive attention both in theoretical and experimental studies. Resistivity, especially that of anisotropic films, is an interesting property in the study of superconductors. A line current (LC) method is developed for solving the difficulty in measuring anisotropic resistivity's c component in the vertical direction of films. The insensitivity difficulty is solved through the arrangement of probes on the crystal and the application of Ω-series'theory. The extraction of c component is based on three measurements of two films with different orientations. The theory can also be applied to bulk samples of high Tc superconductor or semiconductor.
A forgoing solution for LC method is presented for measuring resistivity's three components of films based one sample. The exact solutions of type I and type II, with the edge effects taken into account, are obtained for the theory. Some exact solutions of the related simultaneous equation systems are obtained and expressed by elementary functions analytically, which will be useful for calculation.
The condition of variable separation is discussed since some common mistakes are related this method. The uniqueness of analytic continuation are emphasized for its application in such problems. For some work has applied incorrect boundary condition but obtained formally correct result, the reason is analyzed and the correct form of result is obtained through different method.
Starting from an electron-phonon model Hamiltonian (BCS type), we try to obtain an asymptotically exact solution. This is done under thermodynamic limit, through Stratonovich identity and the diagonalization theorem. With saddle point method, the partition function and a new gap equation are obtained. The gap equation is similar to that of BCS, but has an additional term. It causes the energy gap to be non-zero near the defined critical temperature. It offers us a different viewpoint towards the basic theory of superconductivity.
通常的谐振子势约束BEC转变温度表达式基于化学势为零的假设,在物理与数学上有一定的不自洽性,本文从化学势的角度,仔细研究了这一问题和给予合理的论述,并且利用Bose-Einstein函数推导与计算了二维与三维体系的化学势与转变温度,然后给出了数值结果拟合的转变温度公式。我们发展了Bose-Einsten函数的负系数低温展开式,其高精度和良好的收敛性有助于改进数值计算。利用这一函数,无外场理想情形的问题也从新的角度得到了讨论。我们还得到了二维与三维有外场情形Tc的拟合表达式。
关联函数可以反映体系的量子统计关联特性。我们将从严格定义出发,得到了谐振子势约束BEC的关联函数的表达式。由于高阶Hermite函数高速剧烈振荡,无法对长距离的关联函数进行分析和计算,我们克服了重重困难,成功地获得一个双变量函数级数的严格求和公式,将结果用初等函数表示。它是著名的Bloch单变量和式的推广。我们进而给出大粒子数体系的关联函数,它具有简洁、平稳、收敛性良好的Gauss型级数表式。然后顺利地完成了高精度的数值计算。它们能够反映整体的量子关联效应。包括转变温度Tc附近的量子关联的渐近行为。
超导的理论与实验研究一直在不断的发展,就实验而言,高温超导体的各向异性薄膜的电阻率测量是一个十分有趣和重要的问题。因为垂直于薄膜表面的c轴电阻率测量存在困难,所以我们发展了线电流电极模型,设计了三种测量组态,其好处在于设计简单、利于实现,并且在理论上可以严格讨论.我们利用源像法得到了电势的级数求和表示,然后根据Ω-级数理论将其化为形式简单的表达式。该方法有利于克服c轴电阻率测量不灵敏的困难。
为了满足实验需要,我们还进一步提出基于一个样品,一个平面测出电阻率三个分量的方法与理论,并精确地考虑电极的边缘效应。由于利用源像法和Ω-级数理论,我们得到了新的组态的严格结果。这一理论好处在于有利于避免多样品测量可能存在的不一致性,而且设置简易。同时,我们给出了严格结果的不同解析表示(即第Ⅰ类和第Ⅱ类严格解),它们在不同的情形下具有不同的收敛性和用处。
在相关研究中,一些工作(例如知名学术刊物(如J. Applied Physics)上发表的论文)在分离变量法中遇到了非齐次边界条件的普遍难题。我们获得一批严格求和公式,进而提出利用虚设边界条件的方法和解析开拓一致性定理,在相当普遍的框架下解决了这类难题和困扰。
寻求一个高维(高于一维)的、存在相变的、同时具有实际的物理背景的量子统计问题的严格解,是个挑战性的问题。
我们从电声相互作用模型出发,运用严格的推广的Stratonovich算子等式,在虚拟空间、在热力学极限下,将有效哈密顿量线性化,进而将Fermi二次型哈密顿量对角化定理推广到虚拟空间,并严格实现对角化,从而获得在热力学极限下渐近准确的配分函数,以及各种热力学的表达式,并就唯一性给出论证。
我们还利用鞍点法,获得了新的能隙方程,它与BCS理论中的能隙方程相似,但又不相同.它具有一些自己的特点。我们对新的能隙方程作了详细研究。结果表明,新的严格解理论不但自身是渐近准确的,而且可以说明广为实验支持的BCS理论为什么是正确的,其理论误差是可以忽略的,从而也给出BCS理论以支持,同时BCS理论反过来为严格解提供例证。
There exists certain self-inconsistency in some common-sensed expression of tran-sition temperature (Tc) of Bose-Einstein condensation (BEC) in harmonic traps. The principle of assuming μ=0at Tc is reexamined and a possible solution is proposed for2D and3D cases considering the derivatives of chemical potential. An effective approx-imation tool, the Bose-Einstein function, is studied and then applied to the solving of Tc because of this function's excellent convergency. The expansions of BE functions at low temperatures, including those with negative indexes, are developed. These expan-sions are useful not only for the field-free case, but also for BEC in harmonic traps. Analytic fitting expressions of Tc in2D and3D cases are obtained.
A new expression of correlation function of BEC in harmonic traps is presented and calculated. A new Gaussian type expression with two coordinates for different dimensions is obtained for the main part of correlation function. It is simple, well convergent and can asymptotically represent quantum correlation when N→∞. We hope this expression may be useful in the study of correlation behavior near transition temperature for Bose system in traps.
Superconductivity continuously receives intensive attention both in theoretical and experimental studies. Resistivity, especially that of anisotropic films, is an interesting property in the study of superconductors. A line current (LC) method is developed for solving the difficulty in measuring anisotropic resistivity's c component in the vertical direction of films. The insensitivity difficulty is solved through the arrangement of probes on the crystal and the application of Ω-series'theory. The extraction of c component is based on three measurements of two films with different orientations. The theory can also be applied to bulk samples of high Tc superconductor or semiconductor.
A forgoing solution for LC method is presented for measuring resistivity's three components of films based one sample. The exact solutions of type I and type II, with the edge effects taken into account, are obtained for the theory. Some exact solutions of the related simultaneous equation systems are obtained and expressed by elementary functions analytically, which will be useful for calculation.
The condition of variable separation is discussed since some common mistakes are related this method. The uniqueness of analytic continuation are emphasized for its application in such problems. For some work has applied incorrect boundary condition but obtained formally correct result, the reason is analyzed and the correct form of result is obtained through different method.
Starting from an electron-phonon model Hamiltonian (BCS type), we try to obtain an asymptotically exact solution. This is done under thermodynamic limit, through Stratonovich identity and the diagonalization theorem. With saddle point method, the partition function and a new gap equation are obtained. The gap equation is similar to that of BCS, but has an additional term. It causes the energy gap to be non-zero near the defined critical temperature. It offers us a different viewpoint towards the basic theory of superconductivity.
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