The 2‐matrix of the spin‐polarized electron gas: contraction sum rules and spectral resolutions

The spin‐polarized homogeneous electron gas with densities ρ_↑ and ρ_↓ for electrons with spin ‘up’ (↑) and spin ‘down’ (↓), respectively, is systematically analyzed with respect to its lowest‐order reduced densities and density matrices and their mutual relations. The three 2‐body reduced density matrices γ_↑↑, γ_↓↓, γ_a are 4‐point functions for electron pairs with spins ↑↑, ↓↓, and antiparallel, respectively. From them, three functions G_↑↑(x,y), G_↓↓(x,y), G_a(x,y), depending on only two variables, are derived. These functions contain not only the pair densities according to g_↑↑(r) = G_↑uarr;(0,r), g_↓↓(r) = G_↓↓(0,r), g_a(r) = G_a(0,r) with r = | r ₁ ‐ r ₂|, but also the 1‐body reduced density matrices γ_↑ and γ_↓ being 2‐point functions according to γ_s = ρ_sf_s and f_s(r) = G_ss(r, ∞) with s = ↑,↓ and r = | r ₁ ‐ r ^′₁|. The contraction properties of the 2‐body reduced density matrices lead to three sum rules to be obeyed by the three key functions G_ss, G_a. These contraction sum rules contain corresponding normalization sum rules as special cases. The momentum distributions n_↑(k) and n_↓(k), following from f_↑(r) and f_↓(r) by Fourier transform, are correctly normalized through f_s(0) = 1. In addition to the non‐negativity conditions n_s(k),g_ss(r),g_a(r) ≥ 0 [these quantities are probabilities], it holds n_s(k) ≤ 1 and g_ss(0) = 0 due to the Pauli principle and g_a(0) ≤ 1 due to the Coulomb repulsion. Recent parametrizations of the pair densities of the spin‐unpolarized homogeneous electron gas in terms of 2‐body wave functions (geminals) and corresponding occupancies are generalized (i) to the spin‐polarized case and (ii) to the 2‐body reduced density matrix giving thus its spectral resolutions.     

Electron momentum density and Compton profile by a semi-empirical approach

Here we propose a semi-empirical approach to describe with good accuracy the electron momentum densities and Compton profiles for a wide range of pure crystalline metals. In the present approach, we use an experimental Compton profile to fit an analytical expression for the momentum densities of the valence electrons. This expression is similar to a Fermi–Dirac distribution function with two parameters, one of which coincides with the ground state kinetic energy of the free-electron gas and the other resembles the electron–electron interaction energy. In the proposed scheme conduction electrons are neither completely free nor completely bound to the atomic nucleus. This procedure allows us to include correlation effects.We tested the approach for all metals with Z=3–50 and showed the results for three representative elements: Li, Be and Al from high-resolution experiments.     

约化密度矩阵及其在量子信息处理中的应用

密度矩阵在量子力学中主要是扩展了态矢量的概念.约化密度矩阵是通过对整个系统中的密度矩阵的某一子系求部分迹得到的.本文首先介绍了约化密度矩阵的定义,然后指出约化密度矩阵在纠缠态的定义、两体系统中量子纠缠的度量、系统与环境耦合造成的退相干的物理机制等量子信息前沿领域方面的应用.     

Electron density distribution in Si and Ge using multipole, maximum entropy method and pair distribution function analysis

The local, average and electronic structure of the semiconducting materials Si and Ge has been studied using multipole, maximum entropy method (MEM) and pair distribution function (PDF) analyses, using X-ray powder data. The covalent nature of bonding and the interaction between the atoms are clearly revealed by the two-dimensional MEM maps plotted on (100) and (110) planes and one-dimensional density along [100], [110] and [111] directions. The mid-bond electron densities between the atoms are 0.554 e/?³ and 0.187 e/?³ for Si and Ge respectively. In this work, the local structural information has also been obtained by analyzing the atomic pair distribution function. An attempt has been made in the present work to utilize the X-ray powder data sets to refine the structure and electron density distribution using the currently available versatile methods, MEM, multipole analysis and determination of pair distribution function for these two systems.      

Residual‐density mapping and site‐selective determination of anomalous scattering factors to examine the origin of the Fe K pre‐edge peak of magnetite

The electron‐density distribution and the contribution to anomalous scattering factors for Fe ions in magnetite have been analyzed by X‐ray resonant scattering at the pre‐edge of Fe K absorption. Synchrotron X‐ray experiments were carried out using a conventional four‐circle diffractometer in the right‐handed circular polarization. Difference‐Fourier synthesis was applied with a difference in structure factors measured on and off the pre‐edge (E_on = 7.1082 keV, E_off = 7.1051 keV). Electron‐density peaks due to X‐ray resonant scattering were clearly observed for both A and B sites. The real part of the anomalous scattering factor f′ has been determined site‐independently, based on the crystal‐structure refinements, to minimize the squared residuals at the Fe K pre‐edge. The f′ values obtained at E_on and E_off are ?7.063 and ?6.682 for the A site and ?6.971 and ?6.709 for the B site, which are significantly smaller than the values of ?6.206 and ?5.844, respectively, estimated from the Kramers–Kronig transform. The f′ values at E_on are reasonably smaller than those at E_off. Our results using a symmetry‐based consideration suggest that the origin of the pre‐edge peak is Fe ions occupying both A and B sites, where p–d mixing is needed with hybridized electrons of Fe in both sites overlapping the neighbouring O atoms.     

The spectral density of the QCD Dirac operator and patterns of chiral symmetry breaking

We study the spectrum of the QCD Dirac operator for two colors with fermions in the fundamental representation and for two or more colors with adjoint fermions. For N_f flavors, the chiral flavor symmetry of these theories is spontaneously broken according to SU (2N_f → Sp (2N_f) and SU (N_f → O (N_f), respectively, rather than the symmetry breaking pattern SU (N_f) × SU (N_f) → SU (N_f) for QCD with three or more colors and fundamental fermions. In this paper we study the Dirac spectrum for the first two symmetry breaking patterns. Following previous work for the third case we find the Dirac spectrum in the domain λ Λ_QCD by means of partially quenched chiral perturbation theory. In particular, this result allows us to calculate the slope of the Dirac spectrum at λ = 0. We also show that for λ 1/L₂ Λ_QCD (wing L the linear size fo the system) the Dirac spectrum is given by a chiral Random Matrix Theory with the symmetries of the Dirac operator.     

On the statistical mechanics approach in the random matrix theory: Integrated density of states

We consider the ensemble of random symmetricn×n matrices specified by an orthogonal invariant probability distribution. We treat this distribution as a Gibbs measure of a mean-field-type model. This allows us to show that the normalized eigenvalue counting function of this ensemble converges in probability to a nonrandom limit asn and that this limiting distribution is the solution of a certain self-consistent equation.     

Single-particle spectral density of the unitary Fermi gas: Novel approach based on the operator product expansion,sum rules and the maximum entropy method

Making use of the operator product expansion, we derive a general class of sum rules for the imaginary part of the single-particle self-energy of the unitary Fermi gas. The sum rules are analyzed numerically with the help of the maximum entropy method, which allows us to extract the single-particle spectral density as a function of both energy and momentum. These spectral densities contain basic information on the properties of the unitary Fermi gas, such as the dispersion relation and the superfluid pairing gap, for which we obtain reasonable agreement with the available results based on quantum Monte-Carlo simulations.     

On aberrations in saw‐tooth refractive X‐ray lenses and on their removal

The X‐ray lens, which is composed of opposing canted saw‐tooth structures, originally assembled from cut‐out pieces from long‐playing records, is understood by recognizing that an incident plane X‐ray wave will traverse a varying number of triangular prisms in them. The refraction will deflect any beam towards the prism tips and the variation of the deflection angle, which grows linearly with the number of traversed prisms, can result in X‐ray focusing. The structure offers focusing flexibility by simply changing the taper angle. This report will discuss the aberrations arising in the saw‐tooth structure in its simplest form with identical prisms. It is found that the saw‐tooth structures in low‐Z materials with focal length below 1 m provide less flux density in the focal spot than stacks of one‐dimensionally focusing refractive lenses with identical transmission function. This is due to excessive aberrations in the regular structure, which are absent in stacks of concave lenses, and which limit the focusing to spot sizes of just submicrometre dimensions, as measured experimentally for some lenses. It will be shown that this limitation can be overcome by appropriately modifying the prism shape. Then the image size could be reduced by about an order of magnitude to the diffraction limit with competitive numbers even below 0.1 µm. Microfabrication techniques are identified as the appropriate means for producing the structures.     

A plug‐and‐measure diagnostic tool for flux,resolution and beam size of a soft X‐ray beamline

A compact multi‐functional diagnostic tool has been installed for commissioning beamlines at the Taiwan Light Source. For a photon beam, the instrument can measure the photon flux, energy resolution and beam size, consecutively with a photodiode or gold mesh, a windowless gas cell and a movable fluorescence screen viewport. Two gratings with ruling densities of 350 and 1000 lines mm^?1 and dispersing photons of energies from 80 to 1200 eV were calibrated with a photon flux of 10¹¹ photon s^?1 at slit openings of 50 µm × 50 µm; a maximum resolving power of greater than 10000 was certified with an absorption spectra of gaseous samples. Pressure differences over four orders of magnitude were achieved between the ion chamber and the flux measurement chamber with a single capillary differential pumping stage. A sequence of beam profiles was measured by moving continuously in the vicinity of the nominal focal positions. This tool is useful in commissioning or trouble‐shooting at most beamlines in a synchrotron facility.     

First‐principles X‐ray absorption dose calculation for time‐dependent mass and optical density

A dose integral of time‐dependent X‐ray absorption under conditions of variable photon energy and changing sample mass is derived from first principles starting with the Beer–Lambert (BL) absorption model. For a given photon energy the BL dose integral D(e, t) reduces to the product of an effective time integral T(t) and a dose rate R(e). Two approximations of the time‐dependent optical density, i.e. exponential A(t) = c + aexp(?bt) for first‐order kinetics and hyperbolic A(t) = c + a/(b + t) for second‐order kinetics, were considered for BL dose evaluation. For both models three methods of evaluating the effective time integral are considered: analytical integration, approximation by a function, and calculation of the asymptotic behaviour at large times. Data for poly(methyl methacrylate) and perfluorosulfonic acid polymers measured by scanning transmission soft X‐ray microscopy were used to test the BL dose calculation. It was found that a previous method to calculate time‐dependent dose underestimates the dose in mass loss situations, depending on the applied exposure time. All these methods here show that the BL dose is proportional to the exposure time D(e, t) ? K(e)t.     

空气密度与气体普适常量测量实验的改进

对测量空气密度和气体普适常量实验装置进行改进,用圆柱形有机玻璃密度瓶代替玻璃密度瓶,用高精度指针式真空表代替热偶真空规式真空计,利用橡胶管和密度瓶接口处的缓慢增气特性来改变空气压强,用电子物理天平称量气体质量,改进后的装置较适合学生实验.     

Atomic versus ionized states in many-particle systems and the spectra of reduced density matrices: A model study

We study the spectrum of appropriate reduced density matrices for a model consisting of one quantum particle (electron) in a classical fluid (of protons) at thermal equilibrium. The quantum and classical particles interact by a shortrange, attractive potential such that the quantum particle can form atomic bound states with a single classical particle. We consider two models for the classical component: an ideal gas and the cell model of a fluid. We find that when the system is at low density the spectrum of the electron-proton pair density matrix has, in addition to a continuous part, a discrete part that is associated with atomic bound states. In the high-density limit the discrete eigenvalues disappear in the case of the cell model, indicating the existence of pressure ionization or a Mott effect according to a general criterion for characterizing bound and ionized electron-proton pairs in a plasma proposed recently by M. Girardeau. For the ideal gas model, on the other hand, eigenvalues remain even at high density.     

Non‐negative matrix factorization for the near real‐time interpretation of absorption effects in elemental distribution images acquired by X‐ray fluorescence imaging

Elemental distribution images acquired by imaging X‐ray fluorescence analysis can contain high degrees of redundancy and weakly discernible correlations. In this article near real‐time non‐negative matrix factorization (NMF) is described for the analysis of a number of data sets acquired from samples of a bi‐modal α+β Ti‐6Al‐6V‐2Sn alloy. NMF was used for the first time to reveal absorption artefacts in the elemental distribution images of the samples, where two phases of the alloy, namely α and β, were in superposition. The findings and interpretation of the NMF results were confirmed by Monte Carlo simulation of the layered alloy system. Furthermore, it is shown how the simultaneous factorization of several stacks of elemental distribution images provides uniform basis vectors and consequently simplifies the interpretation of the representation.     

A high‐pressure and controlled‐flow gas system for catalysis research

A high‐pressure gas rig for in situ catalytic reactions at X‐ray absorption spectroscopy beamline (BM26A) has been developed. The rig enables catalysts to be studied in a variety of cells under well controlled and industrially relevant operation conditions. A large variety of gas mixtures can be generated and pressures of up to 50 bar with dry gas and 20 bar with wet gas (steam) can be obtained. Analyses of reaction products can be performed using an on‐line mass spectrometer.     

Influence of crystal structure,ligand environment and morphology on Co L‐edge XAS spectral characteristics in cobalt compounds

The electronic structure of a material plays an important role in its functionality for different applications which can be probed using synchrotron‐based spectroscopy techniques. Here, various cobalt‐based compounds, differing in crystal structure, ligands surrounding the central metal ion and morphology, have been studied by soft X‐ray absorption spectroscopy (XAS) at the Co L‐edge in order to measure the effect of these parameters on the electronic structure. A careful qualitative analysis of the spectral branching ratio and relative intensities of the L₃ and L₂ peaks provide useful insight into the electronic properties of compounds such as CoO/Co(OH)₂, CoCl₂.6H₂O/CoF₂.4H₂O, CoCl₂/CoF₂, Co₃O₄ (bulk/nano/micro). For further detailed analysis of the XAS spectra, quantitative analysis has been performed by fitting the spectral profile with simulated spectra for a number of cobalt compounds using crystal field atomic multiplet calculations.     

Characteristics of Friedel pairs and diffraction contrast tomography with non‐perpendicular rotation axis

The characteristics of Friedel pairs in diffraction contrast tomography (DCT) are studied in the condition that the rotation axis of the sample is not exactly perpendicular to the incident X‐ray direction. For the rotation axis approximately aligned along the vertical direction, the Friedel pairs close to the horizontal plane are insensitive to the non‐perpendicularity of the rotation axis, and can be used to refine the sample‐to‐detector distance and X‐ray energy, while the Friedel pairs close to the vertical direction are sensitive to the non‐perpendicularity of the rotation axis, and can be used to determine the rotation axis orientation. The correct matching proportion of Friedel pairs decreases with increasing non‐perpendicularity of the rotation axis. A method of data processing considering rotation axis misalignment is proposed, which significantly increases the correct matching and indexing proportions of the diffraction spots. A pure aluminium polycrystalline sample is investigated using DCT at beamline 4W1A of Beijing Synchrotron Radiation Facility. Based on the analysis of Friedel pairs, the sample‐to‐detector distance and X‐ray energy are refined to be 8.67 mm and 20.04 keV, respectively. The non‐perpendicular angle of the rotation axis is calculated to be 0.10°. With these refined geometric parameters, the matching proportion of the spatial position of diffraction spots is 90.62%. Three‐dimensional reconstruction of the sample with 13 grains is realised using the algebraic reconstruction technique. It is demonstrated that the precise correction of the orientation of the sample rotation axis is effective in DCT suffering from rotation axis misalignment, and the higher accuracy in determining the rotation axis is expected to improve the reconstruction precision of grains.