Existence, uniqueness, and calculation of the physically acceptable solution of a particular case of the Sherman equations

When a chemical sample made of N elements is analyzed by using sequential selective excitation by monochromatic X-ray beams and selective measurement of the characteristic X-rays, the production of secondary fluorescence does not interfere with the measurements. This experimental situation leads to a particular simple case of the Sherman equations which can be written in this instance as linear equations. The linear equations thus obtained are shown to be very similar to the equations appearing in the classical models of Beattie and Brissey and of Lachance and Traill. The linear algebra proves the existence of N different sets of solutions, but the Perron Frobenius theorem ensures that there is one and only one physically feasible solution, and also leads to the method for obtaining it. This equation solution method can be extended to the equations appearing when standard samples of pure elements are also measured.The propagation of the errors in the measurements to the errors in the sample concentrations has been calculated and simulated, and the results have shown that the solution is well conditioned.     

Application limits of the double irradiation technique in the detection of nuclear interference reactions

The double irradiation technique, which is used to detect the production of a given nuclide from different chemical elements, by two different reactions in a polyenergetic neutron flux, cannot be generally applied. The application limits have been defined and calculated, based on the statistical fluctuation of the measured activities. The experimental verification and the practical applications of the calculated limits for activation analysis, and transmutation reaction cross section studies are discussed.     

Grazing-exit and micro X-ray fluorescence analyses for chemical microchips.

Grazing-exit x-ray fluorescence (GE-XRF) and micro x-ray fluorescence (micro-XRF) methods were applied to chemical microchips as a detection method. Since an energy-dispersive x-ray detector was used, the simultaneous detection of multiple elements was possible. An analyzing region was especially designed on the microchip so that a sample solution could be dried and concentrated in a suitable area corresponding to the size of the primary x-ray beam. Finally, it was confirmed that both analytical methods could be combined well for use with a microchip. In GE-XRF, the background intensity in the XRF spectrum was reduced at grazing-exit angles. In addition, a good relationship between the x-ray fluorescence intensities and the concentrations of standard solutions that were introduced into the microchip was obtained. This indicates that the GE-XRF method is feasible for trace elemental analysis in chemical microchip systems. In micro-XRF, an attempt was made to concentrate and dry the analyte within a small analyzing region. The preliminary results indicated that the micro-XRF method could be applied for the analysis of microchips.     

An stable approach to a kind of problems involving the inversion of a Volterra integral equation of the first kind: application to x-ray fluorescence analysis

In some physical problems it is necessary to obtain a function coming from the inversion of an unstable problem, and use it to calculate some global quantities by integrating it weighted by the appropriate weighting functions. When the desired function comes from a first kind Volterra integral equation, the explicit inversion can be avoided by integrating by parts in the integrals in which the above mentioned function appears. That is the case of the fundamental parameters method of x-ray fluorescence analysis. To obtain the concentrations of chemical elements in the sample which is analyzed it is necessary to calculate some integrals of the spectral distribution of the fluorescence exciting x-ray beam multiplied by a weighting function which depends on the concrete analysis to be done. The spectral distribution of the fluorescence exciting beam is related to the experimental measurements of the fluorescence excited on a set of targets made up of pure elements by a Volterra integral equation of the first kind, and it can be obtained by inverting the Volterra equation. By integrating by parts in the integrals in which the spectral distribution appears we avoid the unstable reconstruction of the spectrum of the fluorescence exciting x-ray beam and the concentrations can be calculated in a stable fashion.     

Potentiometric characterization of weak acids by multiple sample addition II. The effect of chemical interferences and the practical performance of linearization methods

Potentiometric multiple addition of a sample containing a pure weak acid to a solution of supporting electrolyte has been previously shown [C. Maccà and A. Merkoci, Talanta, 41 (1994) 2033] to be formally suitable for the determination of the dissociation constant and concentration of the sample acid. Linear equations have been developed for the treatment of experimental data to yield, simultaneously or separately, the chemical parameters of the acid solution. These equations are now tested on real samples together with analogous equations for titrations, and the results are compared with those obtained with rigorous statistical methods. For the determination of the acidity constant with samples of known concentration, multiple samples addition is comparable with titration. When the sample concentration is unknown and must be determined simultaneously to the acidity constant, the results obtained by linearized multiple sample addition can be seriously affected by impurities present, even at low level, either in the sample or in the supporting solution. Linear equations accounting for the effects of basic or acidic impurities in the sample or in the supporting solution are developed. Sample addition is confirmed to be a useful complement to pH-metric titration for the determination of acidity constants of moderately weak acids by non-linear regression; linearization of data is a convenient technique for screening purposes and a powerful means of detecting and correcting some common pitfalls, interferences and contaminations, whose effects are enhanced in linearized sample addition.     

Elemental distribution in breast tissue samples using X-ray fluorescence microtomography

Summary The X-ray fluorescence microtomography (XRFMT) is a non-destructive technique, based on the detection of X-ray fluorescence emitted by the elements in the sample, and it is used to complement other techniques for sample characterization. In this work some tissues of human breast (healthy tissue, benign tumor and malignant tumor) have been analyzed in order to verify the efficiency of the system in the determination of the elemental distribution in these types of samples. The experiments were performed at the X-ray fluorescence beam line (D09B-XRF) of the Brazilian Synchrotron Light Source (LNLS), Campinas, Brazil. A white beam was used for the excitation of the elements and an HPGe detector detected the fluorescence photons. The incident beam was monitored by an ionization chamber and a fast scintillator detector was used to detect the transmitted radiation. All the tomographies have been reconstructed using a filtered-back projection algorithm. In the tissues samples, the elements of higher concentration were Zn, Cu and Fe.     

X-ray fluorescence analysis in the nanogram region with a total reflected and a Bragg polarized primary beam

Improvements of the detection limit in photon induced X-ray fluorescence analysis are obtained by the reduction of the radiation background at constant excitation conditions. Two methods are presented: 1) Bragg-reflection at 2 ν=90° results in a beam linearly polarized and monchromatic. Positioning the detector at an angle 90° to the direction of the polarized beam suppresses the production of scattered photons from sample and sample carrier into the detector. The use of single crystals instead of amorphous scattering materials increases the intensity of the polarized beam, in some cases up to a factor 10⁴. Detection limits and actually used crystal materials and lattice planes for Braggreflection are given in tables. Samples may be of either form, liquid or solid, no special sample preparation is required. 2) The application of X-ray total reflection on the polished and plane surface of a reflector which serves as the sample carrier reduces the radiation background. In the case of X-ray total reflection there is scarcely a penetration of the X-rays into the reflector material. The penetration depth of some 100 Å in the reflector-substrate can be compared to an extreme thin foil of same thickness. Therefore the reflector-substrate combines its mechanical strenght, its compact and constant geometric form, its chemical resistance and the possibility of easy handling-with the low background properties of ultra-thin foils. Samples should be liquids where the liquid part of the matrix is evaporated and the remaining trace elements are investigated. Both method are capable of detecting elements in the nanogram region.     

Investigation of element distribution and homogeneity of TXRF samples using SR-micro-XRF to validate the use of an internal standard and improve external standard quantification

Total reflection X-ray fluorescence analysis (TXRF) offers a nondestructive qualitative and quantitative analysis of trace elements. Due to its outstanding properties TXRF is widely used in the semiconductor industry for the analysis of silicon wafer surfaces and in the chemical analysis of liquid samples. Two problems occur in quantification: the large statistical uncertainty in wafer surface analysis and the validity of using an internal standard in chemical analysis. In general TXRF is known to allow for linear calibration. For small sample amounts (low nanogram (ng) region) the thin film approximation is valid neglecting absorption effects of the exciting and the detected radiation. For higher total amounts of samples deviations from the linear relation between fluorescence intensity and sample amount can be observed. This could be caused by the sample itself because inhomogeneities and different sample shapes can lead to differences of the emitted fluorescence intensities and high statistical errors. The aim of the study was to investigate the elemental distribution inside a sample. Single and multi-element samples were investigated with Synchrotron-radiation-induced micro X-ray Fluorescence Analysis (SR-μ-XRF) and with an optical microscope. It could be proven that the microscope images are all based on the investigated elements. This allows the determination of the sample shape and potential inhomogeneities using only light microscope images. For the multi-element samples, it was furthermore shown that the elemental distribution inside the samples is homogeneous. This justifies internal standard quantification.     

Total-reflection X-ray fluorescence imaging

A new experimental technique for surface imaging using total-reflection X-ray fluorescence (TXRF) is described. Although TXRF has so far been used to analyze the average chemical composition of rather large sample areas in the order of centimeters squared, a new opportunity to obtain spatial information has arisen through the combination of conventional TXRF and position-sensitive measurement using a collimator and a CCD camera. The most significant point here is that the extremely close detector sample geometry of TXRF measurement fits very well with the present imaging procedure. Scanning of the sample and/or incident beam is not necessary, and therefore the exposure time is reasonably short, typically 3–10 min. The number of pixels is approximately 1 million, and the spatial resolution obtained was several tens of microns in the present preliminary case. The selective-excitation capability of tunable monochromatic synchrotron radiation enhances the present imaging technique. Changing the energy of incident photons makes it possible to distinguish the elements, and one can obtain a surface image of the specific elements.     

X射线荧光光谱法测定电解锰中锰、硅、磷和铁含量

熔融制样X射线荧光光谱法测定电解锰中锰、硅、磷和铁含量。用熔融后的四硼酸锂制作铂金坩埚保护层,以BaO2做氧化剂,在马弗炉内通过逐渐升温来氧化电解锰,然后熔融制取玻璃熔片,用X射线荧光(XRF)光谱法分析电解锰中锰、硅、磷和铁含量。锰、硅、磷和铁的相对标准偏差RSD分别为0.23%、2.82%、0.31%和0.53%。与其它分析方法比较,其结果更稳定。有效消除了电解锰熔融制样过程中的坩埚腐蚀问题,分析误差可完全控制在国家相关标准允许的范围内,实现了电解锰中各元素的快速准确测定。     

Hydrodynamics of nonlinear viscoplastic fluid in cylindrical hydrocyclone

A flow of non-linear viscoplastic fluid film in a cylindrical hydrocyclone was simulated. The distribution of the velocity component and also a dependence of a fluid film thickness on an axial coordinate were determined by numerical solution of a set of rheodynamic model equations for various rheological properties of the liquid and dimensionless quantities. An influence of plastic properties of the liquid on damping of the circumferential component of the velocity was examined. The results were physically substantiated.     

Determination of bromine, chlorine, sulphur and phosphorus in peat by X-ray fluorescence spectrometry combined with single-element and multi-element standard addition

Bromine (20-40 ppm), chlorine (200-500 ppm), sulphur (0.2-3%) and phosphorus (300-1000 ppm) in peat have been determined by X-ray fluorescence spectrometry (XRFS) combined with the standard-addition method. Chlorine, sulphur and phosphorus have also been determined by other methods and agreement between the results is good. Theoretical calculations based on the Sherman equation were made to validate the linearity of the standard-addition curves. A multi-element standard-addition technique has been tested with addition of all elements at the same time. The results for chlorine were high but after correction for the difference in attenuation coefficient between the sample and added compound the results agreed with those from single-element standard addition.     

Analysis of archaeological ceramics by total-reflection X-ray fluorescence: Quantitative approaches

This paper reports the quantitative methodologies developed for the compositional characterization of archaeological ceramics by total-reflection X-ray fluorescence at two levels. A first quantitative level which comprises an acid leaching procedure, and a second selective level, which seeks to increase the number of detectable elements by eliminating the iron present in the acid leaching procedure. Total-reflection X-ray fluorescence spectrometry has been compared, at a quantitative level, with Instrumental Neutron Activation Analysis in order to test its applicability to the study of this kind of materials. The combination of a solid chemical homogenization procedure previously reported with the quantitative methodologies here presented allows the total-reflection X-ray fluorescence to analyze 29 elements with acceptable analytical recoveries and accuracies.     

Simulating systematic errors in X-ray absorption spectroscopy experiments: Sample and beam effects

The article presents an analytical model to simulate experimental imperfections in the realization of an X-ray absorption spectroscopy experiment, performed in transmission or fluorescence mode. Distinction is made between sources of systematic errors on a time-scale basis, to select the more appropriate model for their handling. For short time-scale, statistical models are the most suited. For large time-scale, the model is developed for sample and beam imperfections: mainly sample inhomogeneity, sample self-absorption, beam achromaticity. The ability of this model to reproduce the effects of these imperfections is exemplified, and the model is validated on real samples. Various potential application fields of the model are then presented.     

The use of the ratios of intensities of spectral lines for X-ray fluorescence analysis of metal alloys and oxide materials

An X-ray fluorescence analysis technique is proposed, which is based on using ratios of intensities of spectral lines. The technique includes performing calculations for evaluation of calibration equations, which allows using few reference samples or carrying out standardless analysis, if necessary. That parameters of the calibration equations depend linearly on concentrations of disturbing elements allows one to simplify taking their influence into account. To apply the developed technique to analysis of samples containing a significant amount of undetectable light elements, a use of a dependence of intensity ratio of the characteristic radiation of the X-ray tube’s anode, coherently and incoherently scattered by a sample, on a total content of undetectable elements is proposed. The technique’s adequacy is demonstrated by analysis of standard steel samples, metal cuttings and iron-ore materials.     

Simple analytic solution of the rate equations for the early-stage aggregation kinetics of colloidal particles

The potential energy of the interaction between two approaching colloidal particles obtained by the DLVO theory can exhibit a maximum, a primary minimum, and a secondary minimum on the potential curve of the interparticle interaction energy. Behrens and Borkovec (J Colloid Interface Sci 225: 460, 2000) considered a set of coupled nonlinear differential rate equations for the early-stage aggregation kinetics of colloidal particles by taking into account the influence of the secondary minimum and derived an approximate solution to the rate equations as well as their exact numerical solutions. In the present article, an improved simple analytic solution is derived for these rate equations. The obtained solution, which involves two distinct (fast and slow) exponentially decay constants, is found to be in excellent agreement with numerical solutions to the rate equations with negligible errors.     

Multielement determination of the composition of sulfide ores using X-ray fluorescence analysis

The potentials of the multielement X-ray fluorescence analysis of sulfide ores from one sample are studied. Using cluster analysis, sulfide ores are divided into four groups; within each group, the ores are similar in chemical composition and matrix effects. Calibration equations are obtained for each group. The performance characteristics of the determination results are evaluated.     

Analytical characterization of CuInS(2) semiconductor material

Systematic analytical procedures have been developed for determination of the stoichiometry of CuInS(2) and estimation of trace elements, including dopants and impurities, in the material. Samples of CuInS(2) are digested in an oxidizing acid to ensure completely transformation into Cu(2+), In(3+) and SO(4)(2-) ions. The stoichiometry determination is made sequentially by controlled potential electro-deposition of copper, followed by its EDTA titration, titrimetric determination of indium and gravimetric determination of sulphate, in a single sample solution. The relative errors for the determination of Cu and In are found to be -0.08% and +0.11% respectively, fulfilling the requirement for accurate stoichiometry assessment; that for S is -0.66%, which though rather high is still acceptable. For the determination of trace elements in CuInS(2), multistage combined procedures are employed. Cu in the sample solution is removed by electro-deposition and In by extraction of HInBr(4) with isopropyl ether, then most of the trace elements are finally determined by atomic-absorption spectrometry, and the rest by neutron-activation analysis. All the steps involved in the procedures have been optimized by using radioisotopes as tracers. By the procedures developed, a wide range of trace elements in CuInS(2), down to submicrogram level, can be determined.     

Weak-anchoring effects on a Freedericksz transition in an annulus

This paper investigates the effect of weak anchoring on static orientation patterns in a sample of nematic liquid crystal confined to a cylindrical annular geometry, in the presence of a magnetic field. The particular arrangement considered in detail is that in which a magnetic field is applied in the azimuthal direction to a sample of liquid crystal that is initially uniformly aligned parallel to the cylindrical axis and is weakly anchored at the inner cylinder, but strongly anchored at the outer cylinder. A static solution of the non-linear continuum equations is presented which suggests a possible experiment for evaluating the surface elastic coefficient k₂₄. In addition, a linear stability analysis based on the dynamic theory yields results for critical phenomena in agreement with those derived from the non-linear solution.     

Non-linear optical behavior in atomic fluorescence flame spectrometry

The non-linear dependence of the atomic fluorescence radiance upon the irradiance of the excitation source is discussed. Theoretical equations based on the assumption of steady state conditions are derived for a two-level atomic system and for a continuum source of excitation both on a relative as well as on an absolute basis. The theoretical results show that the approach of saturation sets a limit to the fluorescence radiance. The experimental results obtained with the use of a pulsed, tunable dye laser by nebulizing different elements in analytical flames are shown to be in satisfactory agreement with these theoretical results. The theory also predicts that, for a two-level system, the proportional dependence of the fluorescence signal upon the quantum efficiency observed at low irradiances is removed under saturation conditions.