Nucleus-Electron Model for States Changing from a Liquid Metal to a Plasma and the Saha Equation

We extend the quantal hypernetted-chain (QHNC) method, which has been proved to yield accurate results for liquid metals, to treat a partially ionized plasma. In a plasma, the electrons change from a quantum to a classical fluid gradually with increasing temperature; the QHNC method applied to the electron gas is in fact able to provide the electron-electron correlation at arbitrary temperature. As an illustrating example of this approach, we investigate how liquid rubidium becomes a plasma by increasing the temperature from 0 to 30 eV at a fixed normal ion-density $1.03 \times 10^{22}/cm^3$. The electron-ion radial distribution function (RDF) in liquid Rb has distinct inner-core and outer-core parts. Even at a temperature of 1 eV, this clear distinction remains as a characteristic of a liquid metal. At a temperature of 3 eV, this distinction disappears, and rubidium becomes a plasma with the ionization 1.21. The temperature variations of bound levels in each ion and the average ionization are calculated in Rb plasmas at the same time. Using the density-functional theory, we also derive the Saha equation applicable even to a high-density plasma at low temperatures. The QHNC method provides a procedure to solve this Saha equation with ease by using a recursive formula; the charge population of differently ionized species are obtained in Rb plasmas at several temperatures. In this way, it is shown that, with the atomic number as the only input, the QHNC method produces the average ionization, the electron-ion and ion-ion RDF's, and the charge population which are consistent with the atomic structure of each ion for a partially ionized plasma.Comment: 28 pages(TeX) and 11 figures (PS

Structure Factor and Electronic Structure of Compressed Liquid Rubidium

We have applied the quantal hypernetted-chain equations in combination with the Rosenfeld bridge-functional to calculate the atomic and the electronic structure of compressed liquid-rubidium under high pressure (0.2, 2.5, 3.9, and 6.1 GPa); the calculated structure factors are in good agreement with experimental results measured by Tsuji et al. along the melting curve. We found that the Rb-pseudoatom remains under these high pressures almost unchanged with respect to the pseudoatom at room pressure; thus, the effective ion-ion interaction is practically the same for all pressure-values. We observe that all structure factors calculated for this pressure-variation coincide almost into a single curve if wavenumbers are scaled in units of the Wigner-Seitz radius $a$ although no corresponding scaling feature is observed in the effective ion-ion interaction.This scaling property of the structure factors signifies that the compression in liquid-rubidium is uniform with increasing pressure; in absolute Q-values this means that the first peak-position ($Q_1$) of the structure factor increases proportionally to $V^{-1/3}$ ($V$ being the specific volume per ion), as was experimentally observed by Tsuji et al.Comment: 18 pages, 11 figure

Combining quantum and classical density functional theory for ion-electron mixtures

We combine techniques from quantum and from classical density functional theory (DFT) to describe electron-ion mixtures. For homogeneous systems, we show how to calculate ion-ion and ion-electron correlation functions within Chihara's quantum hypernetted chain approximation, which we derive within a DFT formulation. We also sketch out how to apply the DFT formulation to inhomogeneous electron-ion mixtures, and use this to study the electron distribution at the liquid-solid interface of Al.Comment: to be published in J. Non-Cryst. Solids, LAM 11 special issu

Pressure formulas for liquid metals and plasmas based on the density-functional theory

At first, pressure formulas for the electrons under the external potential produced by fixed nuclei are derived both in the surface integral and volume integral forms concerning an arbitrary volume chosen in the system; the surface integral form is described by a pressure tensor consisting of a sum of the kinetic and exchange-correlation parts in the density-functional theory, and the volume integral form represents the virial theorem with subtraction of the nuclear virial. Secondly on the basis of these formulas, the thermodynamical pressure of liquid metals and plasmas is represented in the forms of the surface integral and the volume integral including the nuclear contribution. From these results, we obtain a virial pressure formula for liquid metals, which is more accurate and simpler than the standard representation. From the view point of our formulation, some comments are made on pressure formulas derived previously and on a definition of pressure widely used.Comment: 18 pages, no figur

Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials

Nonsymmetric Askey-Wilson polynomials are usually written as Laurent polynomials. We write them equivalently as 2-vector-valued symmetric Laurent polynomials. Then the Dunkl-Cherednik operator of which they are eigenfunctions, is represented as a 2x2 matrix-valued operator. As a new result made possible by this approach we obtain positive definiteness of the inner product in the orthogonality relations, under certain constraints on the parameters. A limit transition to nonsymmetric little q-Jacobi polynomials also becomes possible in this way. Nonsymmetric Jacobi polynomials are considered as limits both of the Askey-Wilson and of the little q-Jacobi case.Comment: 16 pages. Dedicated to Paul Butzer on the occasion of his 80th birthday. v4: minor correction in (4.14

Polynomial solutions of nonlinear integral equations

We analyze the polynomial solutions of a nonlinear integral equation, generalizing the work of C. Bender and E. Ben-Naim. We show that, in some cases, an orthogonal solution exists and we give its general form in terms of kernel polynomials.Comment: 10 page

Generalizations of an integral for Legendre polynomials by Persson and Strang

Persson and Strang (2003) evaluated the integral over [-1,1] of a squared odd degree Legendre polynomial divided by x^2 as being equal to 2. We consider a similar integral for orthogonal polynomials with respect to a general even orthogonality measure, with Gegenbauer and Hermite polynomials as explicit special cases. Next, after a quadratic transformation, we are led to the general nonsymmetric case, with Jacobi and Laguerre polynomials as explicit special cases. Examples of indefinite summation also occur in this context. The paper concludes with a generalization of the earlier results for Hahn polynomials. There some adaptations have to be made in order to arrive at relatively nice explicit evaluations.Comment: 14 pages; minor corrections and reference added; J. Math. Anal. Appl. (2011

How to construct spin chains with perfect state transfer

It is shown how to systematically construct the $XX$ quantum spin chains with nearest-neighbor interactions that allow perfect state transfer (PST). Sets of orthogonal polynomials (OPs) are in correspondence with such systems. The key observation is that for any admissible one-excitation energy spectrum, the weight function of the associated OPs is uniquely prescribed. This entails the complete characterization of these PST models with the mirror symmetry property arising as a corollary. A simple and efficient algorithm to obtain the corresponding Hamiltonians is presented. A new model connected to a special case of the symmetric $q$-Racah polynomials is offered. It is also explained how additional models with PST can be derived from a parent system by removing energy levels from the one-excitation spectrum of the latter. This is achieved through Christoffel transformations and is also completely constructive in regards to the Hamiltonians.Comment: 7 page

Probing Ion-Ion and Electron-Ion Correlations in Liquid Metals within the Quantum Hypernetted Chain Approximation

We use the Quantum Hypernetted Chain Approximation (QHNC) to calculate the ion-ion and electron-ion correlations for liquid metallic Li, Be, Na, Mg, Al, K, Ca, and Ga. We discuss trends in electron-ion structure factors and radial distribution functions, and also calculate the free-atom and metallic-atom form-factors, focusing on how bonding effects affect the interpretation of X-ray scattering experiments, especially experimental measurements of the ion-ion structure factor in the liquid metallic phase.Comment: RevTeX, 19 pages, 7 figure

Phase Diagram for Anderson Disorder: beyond Single-Parameter Scaling

The Anderson model for independent electrons in a disordered potential is transformed analytically and exactly to a basis of random extended states leading to a variant of augmented space. In addition to the widely-accepted phase diagrams in all physical dimensions, a plethora of additional, weaker Anderson transitions are found, characterized by the long-distance behavior of states. Critical disorders are found for Anderson transitions at which the asymptotically dominant sector of augmented space changes for all states at the same disorder. At fixed disorder, critical energies are also found at which the localization properties of states are singular. Under the approximation of single-parameter scaling, this phase diagram reduces to the widely-accepted one in 1, 2 and 3 dimensions. In two dimensions, in addition to the Anderson transition at infinitesimal disorder, there is a transition between two localized states, characterized by a change in the nature of wave function decay.Comment: 51 pages including 4 figures, revised 30 November 200