高温稠密物质结构及状态方程的研究
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摘要
在密度泛函理论(DFT)框架下,针对高温稠密、中温稠密、低温常态等不同的物理条件,采用Thomas-Fermi模型(TF)、平均原子模型(AA)、广义梯度近似模型(GGA)和局域密度近似模型(LDA)等不同能量密度泛函形式,描述了物质的电子结构。在适当描述电子结构的基础上,计算原子(或者离子)之间的相互作用,根据计算得到的相互作用,利用经典分子动力学方法分别对物质的离子结构及其物态方程进行描述;在过渡区域采用经典的分子动力学方法研究固体的熔化和等离子体的成核过程。
首先对交换作用分别采用广义梯度近似(GGA)和局域密度近似(LDA),用缀加平面波加局域轨道方法(APW+lo),对金和铝的面心立方晶格结构(fcc)、体心立方晶格结构(bcc)和六角密堆积晶格结构(hcp)在固体状态下的电子结构及固-固结构相变进行了计算,并分别给出了金和铝在绝对零度(不考虑晶格振动的贡献)时的物态方程数据。对金在不同晶格结构(fcc、bcc、hcp)、不同体积下电子态密度的特征进行了深入研究,找出发生固-固结构相变时费米面附近电子态密度变化规律,然后对发生这种固-固结构相变的物理原因进行了定性的分析。最后采用平均场理论,讨论了在固体状态下晶格振动对系统能量的贡献。
随着体系温度升高、密度变大,在碰撞电离效应和压致电离效应共同作用下,原子将会发生电离。在这样的外部条件下,离子或原子之间具有较强的相互作用,物质处于固体时的高对称性晶格结构受到破坏,离子或原子的能级简并被解除,能级发生分裂。这时体系既不能采用固体理论来描述,又不能把离子当成单个原子进行描述。但离子或原子的分裂能级之差较小,不同离子的同一能级就形成一个准连续的带,原子或离子的同一轨道电子将在带中服从Fermi-Dirac分布。在平均原子模型中,考虑了这种物理过程后,模型更接近于物理实际。以金和铝为例,分别在温度为1、10、100、1000eV时对物态方程和平均电离度进行了计算,结果表明:考虑了能级展宽效应后计算出的电子态密度与第一原理的结果吻合较好,而且考虑了能级展宽效应后,计算结果可以得到较大的改进,与实验和其他理论结果符合较好,并消除了由于压致电离导致物态方程和电离度随物质密度变化产生跳跃的台阶,这也充分证明了考虑这种物理效应的必要性。
在平均原子模型中,束缚电子的径向分布通过求解Dirac径向方程得到,把能量大于离子球边界上势能的电子认为是自由电子,自由电子的径向分布采用Thomas-Fermi统计模型得到。事实上,在等离子体中由于有带正电的原子核存在,自由电子还处于局域化状态,特别是在中低温状态下,采用Thomas-Fermi统计模型来描述能量刚刚大于零的自由电子将与物理事实产生较大差别。因此,在平均原子模型的基础上,我们通过求解能量大于零的Dirac径向方程来描述这些局域的自由电子。
随着温度的降低,离子之间的关联效应变得越来越重要。为了探索在高温稠密等离子体条件下原子或离子间强的关联效应,我们首先采用已经修正的平均原子模型计算单个原子或离子周围电子的径向分布,在含温的密度泛函理论框架下,计算离子-离子之间的相互作用势;然后将其应用到经典的分子动力学程序中,计算了Al和Fe等离子体在高温稠密状态下的离子结构及离子对总压强的贡献,计算结果与实验和其它理论计算结果符合较好。同时也检验了我们提出的计算离子间相互作用势模型的合理性。
随着温度的进一步降低,离子的平均动能减少,当离子的动能小于离子间相互作用的势能时,离子将发生聚集,所以在低温等离子体中将发生成核现象;或者从另一角度来说,当物质处于固体时离子在平衡位置附近振动,随着温度的升高,离子的动能增加,当离子的动能大于相互作用的势能时,物质将发生熔化。因此,在得到离子间相互作用势的基础上,运用分子动力学研究在这一区域微观粒子的运动规律、空间位形分布以及随着外部环境(温度和密度)变化导致的固相、液相、等离子体状态和团簇之间的相变。
In the framework of density functional theory (DFT), electronic structures are described using the different energy density functions in the different temperature regions. The Thomas-Fermi model (TF), the average atom model (AA), the local density approximation (LDA) and the generalized gradient approaximation (GGA) are used in the hot dense matter, warm dense matter and low temperature region, respectively. The ion-ion pair potentials are calculated based on the electronic structures. And the classical molecular dynamics simulations are performed for the ion motion on the basis of the calculated pair potentials. In the middle region, microcosmic characters of melting and nucleation are discussed by using classical molecular dynamics.
Firstly, the electronic structures and phase transitions of Aluminium and Gold have been calculated at the structure of the face-certered cubic (fcc), body-centered cubic (bcc), and hexagonal close-packed (hcp), by using the augmented plane wave plus local orbital method with two distinct exchange-correlation energy functions: the generalized gradient approaximation (GGA) and the local density approximation (LDA). The phase transitions and the equation of state (EOS) at zero temperature are obtained, and possible reasons are disscused by analyzing the electronic density of state at the different phases and different volumes. The contributions from the vibration of crystal lattice are treated by using mean field theory.
With the increase of density and temperature, the matter state will be changed from solid to plasma due to the thermo-ionizaiton and pressure-ionization. In these cases, the periodical boundary condition for the electronic state does not exist any longer and the broadening of the valence electron states does not have the itinerant characteristic either. Furthemore, the electronic energy levels of the atoms and ions will be split into many sublevels due to the interactions among the particles in the hot dense plasmas. However, the splittings are generally so small that the distribution of the sublevels can be treated by broadening the corresponding energy level into energy band with a Gaussian profile, which is normalized to ensure that the integration of the density of state over one hand is equal to the statistical weight of the corresponding atomic level. The distribution of the bound electrons among the energy bands is determined by the continuum Fermi-Dirac distribution. Within a self-consistent field average atom approach, it has been shown that explicit considerations for the electronic energy level broadening have significant effects on the ionization of the atoms and the EOS in hot and dense plasmas. The instability of the pressure induced electronic ionization with density increasing, which occurs often in a normal average atom model and is avoided usually by introducing pseudo-shape resonance states, disappears naturally. As examples, the density dependence of the average ionization and the EOS of Al and Au at 1, 10, 100, and 1000 eV are presented.
In the average atom (AA) model, when the energy of an electron is greater than the potential of the atomic boundary, this electron is considered to be a bound electron, and its radial distribution can be obtained from the radial Dirac equation. If one electron is smaller than the potential of the atomic boundary, it is taken to be a free electron, and the radial distribution can be obtained from Thomas-Fermi statistics. In fact, when electron energy is greater than zero, the electron is localized due to the interaction between electron and nucles in plasmas. In particular, when plasma temperature is very low, the Thomas-Fermi statistics will have big difference. So the description of free electron is modified by solving the radial Dirac equation.
With the temperature decreasing, the ion-ion correlation effects become important in hot dense plasmas. In order to study the ion-ion correlation effects, we develop a model to calculate the ion-ion potentials based on temperature-dependent density functional theory (TDDFT). The electronic structures, including the energy levels and space distributions, are calculated using a modified average-atom model. The calculated electron space number density is divided into two parts: one is a uniformly distributed electronic sea with a density equaling to the total electronic density at the ionic sphere boundary, which is redistributed when space overlap occurs between the interacted ions; the left part of the electronic density represents the dramatic space variations of the electrons due to the nuclear attraction and the shell structure of the bound states, which maintains unchanged during the interactions between the ions. The ion-ion potential is obtained through space integrations for the energy density functions of electron density. Molecular dynamics simulations are performed for the ionic motions on the basis of the calculated potentials in a wide regime of density and temperature. As an example, hot and dense Al and Fe plasmas are simulated to give the EOS and ion-ion pair distribution function.
When plasma temperature continues to reducing, the ionic average kinetic energy will decrease. And while the ionic average potential energy is larger than its kinetic energy, the ions will be close to each other and clusters will be formed. On the other hand, with the temperature increasing, the ionic average kinetic energy in the solid state will also increase. And solid state will become liquid state or plasma when the ionic average kinetic energy is larger than its potential energy. In the case, the molecular dynamics are performed to simulate the phase transitions among the matter state of solid, liquid, plasma and cluster with the change of temperature.
首先对交换作用分别采用广义梯度近似(GGA)和局域密度近似(LDA),用缀加平面波加局域轨道方法(APW+lo),对金和铝的面心立方晶格结构(fcc)、体心立方晶格结构(bcc)和六角密堆积晶格结构(hcp)在固体状态下的电子结构及固-固结构相变进行了计算,并分别给出了金和铝在绝对零度(不考虑晶格振动的贡献)时的物态方程数据。对金在不同晶格结构(fcc、bcc、hcp)、不同体积下电子态密度的特征进行了深入研究,找出发生固-固结构相变时费米面附近电子态密度变化规律,然后对发生这种固-固结构相变的物理原因进行了定性的分析。最后采用平均场理论,讨论了在固体状态下晶格振动对系统能量的贡献。
随着体系温度升高、密度变大,在碰撞电离效应和压致电离效应共同作用下,原子将会发生电离。在这样的外部条件下,离子或原子之间具有较强的相互作用,物质处于固体时的高对称性晶格结构受到破坏,离子或原子的能级简并被解除,能级发生分裂。这时体系既不能采用固体理论来描述,又不能把离子当成单个原子进行描述。但离子或原子的分裂能级之差较小,不同离子的同一能级就形成一个准连续的带,原子或离子的同一轨道电子将在带中服从Fermi-Dirac分布。在平均原子模型中,考虑了这种物理过程后,模型更接近于物理实际。以金和铝为例,分别在温度为1、10、100、1000eV时对物态方程和平均电离度进行了计算,结果表明:考虑了能级展宽效应后计算出的电子态密度与第一原理的结果吻合较好,而且考虑了能级展宽效应后,计算结果可以得到较大的改进,与实验和其他理论结果符合较好,并消除了由于压致电离导致物态方程和电离度随物质密度变化产生跳跃的台阶,这也充分证明了考虑这种物理效应的必要性。
在平均原子模型中,束缚电子的径向分布通过求解Dirac径向方程得到,把能量大于离子球边界上势能的电子认为是自由电子,自由电子的径向分布采用Thomas-Fermi统计模型得到。事实上,在等离子体中由于有带正电的原子核存在,自由电子还处于局域化状态,特别是在中低温状态下,采用Thomas-Fermi统计模型来描述能量刚刚大于零的自由电子将与物理事实产生较大差别。因此,在平均原子模型的基础上,我们通过求解能量大于零的Dirac径向方程来描述这些局域的自由电子。
随着温度的降低,离子之间的关联效应变得越来越重要。为了探索在高温稠密等离子体条件下原子或离子间强的关联效应,我们首先采用已经修正的平均原子模型计算单个原子或离子周围电子的径向分布,在含温的密度泛函理论框架下,计算离子-离子之间的相互作用势;然后将其应用到经典的分子动力学程序中,计算了Al和Fe等离子体在高温稠密状态下的离子结构及离子对总压强的贡献,计算结果与实验和其它理论计算结果符合较好。同时也检验了我们提出的计算离子间相互作用势模型的合理性。
随着温度的进一步降低,离子的平均动能减少,当离子的动能小于离子间相互作用的势能时,离子将发生聚集,所以在低温等离子体中将发生成核现象;或者从另一角度来说,当物质处于固体时离子在平衡位置附近振动,随着温度的升高,离子的动能增加,当离子的动能大于相互作用的势能时,物质将发生熔化。因此,在得到离子间相互作用势的基础上,运用分子动力学研究在这一区域微观粒子的运动规律、空间位形分布以及随着外部环境(温度和密度)变化导致的固相、液相、等离子体状态和团簇之间的相变。
In the framework of density functional theory (DFT), electronic structures are described using the different energy density functions in the different temperature regions. The Thomas-Fermi model (TF), the average atom model (AA), the local density approximation (LDA) and the generalized gradient approaximation (GGA) are used in the hot dense matter, warm dense matter and low temperature region, respectively. The ion-ion pair potentials are calculated based on the electronic structures. And the classical molecular dynamics simulations are performed for the ion motion on the basis of the calculated pair potentials. In the middle region, microcosmic characters of melting and nucleation are discussed by using classical molecular dynamics.
Firstly, the electronic structures and phase transitions of Aluminium and Gold have been calculated at the structure of the face-certered cubic (fcc), body-centered cubic (bcc), and hexagonal close-packed (hcp), by using the augmented plane wave plus local orbital method with two distinct exchange-correlation energy functions: the generalized gradient approaximation (GGA) and the local density approximation (LDA). The phase transitions and the equation of state (EOS) at zero temperature are obtained, and possible reasons are disscused by analyzing the electronic density of state at the different phases and different volumes. The contributions from the vibration of crystal lattice are treated by using mean field theory.
With the increase of density and temperature, the matter state will be changed from solid to plasma due to the thermo-ionizaiton and pressure-ionization. In these cases, the periodical boundary condition for the electronic state does not exist any longer and the broadening of the valence electron states does not have the itinerant characteristic either. Furthemore, the electronic energy levels of the atoms and ions will be split into many sublevels due to the interactions among the particles in the hot dense plasmas. However, the splittings are generally so small that the distribution of the sublevels can be treated by broadening the corresponding energy level into energy band with a Gaussian profile, which is normalized to ensure that the integration of the density of state over one hand is equal to the statistical weight of the corresponding atomic level. The distribution of the bound electrons among the energy bands is determined by the continuum Fermi-Dirac distribution. Within a self-consistent field average atom approach, it has been shown that explicit considerations for the electronic energy level broadening have significant effects on the ionization of the atoms and the EOS in hot and dense plasmas. The instability of the pressure induced electronic ionization with density increasing, which occurs often in a normal average atom model and is avoided usually by introducing pseudo-shape resonance states, disappears naturally. As examples, the density dependence of the average ionization and the EOS of Al and Au at 1, 10, 100, and 1000 eV are presented.
In the average atom (AA) model, when the energy of an electron is greater than the potential of the atomic boundary, this electron is considered to be a bound electron, and its radial distribution can be obtained from the radial Dirac equation. If one electron is smaller than the potential of the atomic boundary, it is taken to be a free electron, and the radial distribution can be obtained from Thomas-Fermi statistics. In fact, when electron energy is greater than zero, the electron is localized due to the interaction between electron and nucles in plasmas. In particular, when plasma temperature is very low, the Thomas-Fermi statistics will have big difference. So the description of free electron is modified by solving the radial Dirac equation.
With the temperature decreasing, the ion-ion correlation effects become important in hot dense plasmas. In order to study the ion-ion correlation effects, we develop a model to calculate the ion-ion potentials based on temperature-dependent density functional theory (TDDFT). The electronic structures, including the energy levels and space distributions, are calculated using a modified average-atom model. The calculated electron space number density is divided into two parts: one is a uniformly distributed electronic sea with a density equaling to the total electronic density at the ionic sphere boundary, which is redistributed when space overlap occurs between the interacted ions; the left part of the electronic density represents the dramatic space variations of the electrons due to the nuclear attraction and the shell structure of the bound states, which maintains unchanged during the interactions between the ions. The ion-ion potential is obtained through space integrations for the energy density functions of electron density. Molecular dynamics simulations are performed for the ionic motions on the basis of the calculated potentials in a wide regime of density and temperature. As an example, hot and dense Al and Fe plasmas are simulated to give the EOS and ion-ion pair distribution function.
When plasma temperature continues to reducing, the ionic average kinetic energy will decrease. And while the ionic average potential energy is larger than its kinetic energy, the ions will be close to each other and clusters will be formed. On the other hand, with the temperature increasing, the ionic average kinetic energy in the solid state will also increase. And solid state will become liquid state or plasma when the ionic average kinetic energy is larger than its potential energy. In the case, the molecular dynamics are performed to simulate the phase transitions among the matter state of solid, liquid, plasma and cluster with the change of temperature.
引文
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