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

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

On affinity relating two positive measures and the connection coefficients between polynomials orthogonalized by these measures

We consider two positive, normalized measures dA(x) and dB(x) related by the relationship dA(x)=(C/(x+D))dB(x) or by dA(x) = (C/(x^2+E))dB(x) and dB(x) is symmetric. We show that then the polynomial sequences {a_{n}(x)}, {b_{n}(x)} orthogonal with respect to these measures are related by the relationship a_{n}(x)=b_{n}(x)+{\kappa}_{n}b_{n-1}(x) or by a_{n}(x) = b_{n}(x) + {\lambda}_{n}b_{n-2}(x) for some sequences {{\kappa}_{n}} and {{\lambda}_{n}}. We present several examples illustrating this fact and also present some attempts for extensions and generalizations. We also give some universal identities involving polynomials {b_{n}(x)} and the sequence {{\kappa}_{n}} that have a form of Fourier series expansion of the Radon--Nikodym derivative of one measure with respect to the other

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

A high order qq-difference equation for qq-Hahn multiple orthogonal polynomials

A high order linear $q$-difference equation with polynomial coefficients having $q$-Hahn multiple orthogonal polynomials as eigenfunctions is given. The order of the equation is related to the number of orthogonality conditions that these polynomials satisfy. Some limiting situations when $q\to1$ are studied. Indeed, the difference equation for Hahn multiple orthogonal polynomials given in \cite{Lee} is corrected and obtained as a limiting case

Transformation design and nonlinear Hamiltonians

We study a class of nonlinear Hamiltonians, with applications in quantum optics. The interaction terms of these Hamiltonians are generated by taking a linear combination of powers of a simple `beam splitter' Hamiltonian. The entanglement properties of the eigenstates are studied. Finally, we show how to use this class of Hamiltonians to perform special tasks such as conditional state swapping, which can be used to generate optical cat states and to sort photons.Comment: Accepted for publication in Journal of Modern Optic

Interaction of photons with plasmas and liquid metals: photoabsorption and scattering

Formulas to describe the photoabsorption and the photon scattering by a plasma or a liquid metal are derived in a unified manner with each other. It is shown how the nuclear motion, the free-electron motion and the core-electron behaviour in each ion in the system determine the structure of photoabsorption and scattering in an electron-ion mixture. The absorption cross section in the dipole approximation consists of three terms which represent the absorption caused by the nuclear motion, the absorption owing to the free-electron motion producing optical conductivity or inverse Bremsstrahlung, and the absorption ascribed to the core-electron behaviour in each ion with the Doppler correction. Also, the photon scattering formula provides an analysis method for experiments observing the ion-ion dynamical structure factor (DSF), the electron-electron DSF giving plasma oscillations, and the core-electron DSF yielding the X-ray Raman (Compton) scattering with a clear definition of the background scattering for each experiment, in a unified manner. A formula for anomalous X-ray scattering is also derived for a liquid metal. At the same time, Thomson scattering in plasma physics is discussed from this general point of view.Comment: LaTeX file: 18 pages without figur

Exact limiting relation between the structure factors in neutron and x-ray scattering

The ratio of the static matter structure factor measured in experiments on coherent X-ray scattering to the static structure factor measured in experiments on neutron scattering is considered. It is shown theoretically that this ratio in the long-wavelength limit is equal to the nucleus charge at arbitrary thermodynamic parameters of a pure substance (the system of nuclei and electrons, where interaction between particles is pure Coulomb) in a disordered equilibrium state. This result is the exact relation of the quantum statistical mechanics. The experimental verification of this relation can be done in the long wavelength X-ray and neutron experiments.Comment: 7 pages, no figure

Wigner quantization of some one-dimensional Hamiltonians

Recently, several papers have been dedicated to the Wigner quantization of different Hamiltonians. In these examples, many interesting mathematical and physical properties have been shown. Among those we have the ubiquitous relation with Lie superalgebras and their representations. In this paper, we study two one-dimensional Hamiltonians for which the Wigner quantization is related with the orthosymplectic Lie superalgebra osp(1|2). One of them, the Hamiltonian H = xp, is popular due to its connection with the Riemann zeros, discovered by Berry and Keating on the one hand and Connes on the other. The Hamiltonian of the free particle, H_f = p^2/2, is the second Hamiltonian we will examine. Wigner quantization introduces an extra representation parameter for both of these Hamiltonians. Canonical quantization is recovered by restricting to a specific representation of the Lie superalgebra osp(1|2)