Diffraction Correction Research Articles

Diffraction peak identification and correction in EDXRF spectroscopy

Energy dispersive X-ray fluorescence (EDXRF) spectroscopy is an analytical technique often used to create maps of the elemental chemical composition. The fluorescence signal can be influenced by x-ray diffraction peaks, a phenomenon that can lead to difficulties or errors in interpreting spectral peaks. However, diffraction can also be useful if we successfully separate it from the fluorescence signal. In this paper, we propose different methods to deal with the diffraction peaks by using the signals obtained from a spectrometer equipped with two detectors. These methods were tested on a thin section of a rock sample to show their effectiveness in processing diffraction peaks. Each method showed its ability to deal with the diffraction peaks in a more or less effective way and each has its own limitations. The latter have to be taken into account when analyzing the EDXRF signal, whether in a geological context or in any other context.

Diffraction correction in high-precision pulse-echo and multiple-reflection ultrasonic measurement systems for fluids.

In high-precision ultrasonic measurement systems, diffraction correction models accounting for electrical and mechanical boundary conditions may be needed, as shown in prior work using a finite element diffraction correction (FEDC) model for one-way transmit-receive systems. Such descriptions may also be needed for pulse-echo and multiple-reflection ultrasonic measurement applications. The FEDC model is here generalized to n-way measurement systems (n = 1, 2, 3,…) using coaxially aligned piezoelectric transducers in a fluid medium. Comparisons are made with existing diffraction correction models, based on baffled-piston theory combined with (i) specular reflection or (ii) reflection modeled as radiation from a "new source." Numerical results are given for an example system with two identical cylindrical piezoelectric disks, operating in a fluid medium at ambient conditions. The piston-type diffraction correction models deviate notably from the FEDC model both in the near- and far-fields, and also from each other. The deviations are expected to be application-specific and depend, e.g., on the reflector-to-sound-beam diameter ratio, distance, frequency, and the transducers' vibration patterns. The results show that accurate description of the diffraction effects, such as the one provided by the FEDC model, may be needed in high-precision ultrasonic measurement systems.

Qualitative validation on a numerical model for the direct stability assessment of surf-riding and broaching

A previously developed numerical model for surf-riding and broaching is qualitatively validated based on the criteria of direct stability assessment (DSA) proposed in SDC 7. Roll decay tests are conducted to obtain roll backbone curves, which are compared with calculated GZ curve to demonstrate their consistency. Then, roll response curves under different wave frequencies in regular beam wave are obtained. It is verified that roll response curve “fold around” the newly defined “peak backbone curve”. Cases with fixed yaw under different Fn are chosen to demonstrate the numerical model's capability of reproducing surf-riding equilibrium. Turning circle test is conduced to demonstrate correct modelling of maneuvering forces and the capability to reproduce heel due to ship turning. Wave surging forces are calculated under different wave numbers and amplitudes. The correct tendency of phase difference between wave surging force and the incident wave is observed, which reveals the correct modelling of wave forces in the model. Moreover, a preliminary quantitative validation is conducted under various cases. Through comparison with experimental results, it is recommended that the wave diffraction correction of the wave surging forces should be included in the simulation model in order to avoid overestimation of surf-riding and broaching.

Longitudinal ultrasound measurements of polyethylene oxide across dilute to concentrated aqueous solutions

Measuring the properties of polymer solutions is crucial for understanding fundamental polymer physics. The longitudinal ultrasound measurement method offers a promising technique for characterizing these properties. In this study, we develop a mathematical model to correlate longitudinal ultrasonic signals with the material properties of polyethylene oxide (PEO) from dilute to concentrated aqueous solutions. After applying reflection and diffraction corrections, the accuracy of the measurement results improves to over 97 %. Our findings reveal a positive correlation between the speed of sound, density, and longitudinal modulus with the mass concentration of PEO. Additionally, ultrasonic relaxation analysis reveals that PEO solutions exhibit at least one relaxation process in the frequency range of 3–7 MHz, independent of concentration and molecular weight. Compared to dilute solutions, the relaxation amplitude in concentrated solutions is larger and accompanied by strong nonlinearity, attributed to conformational changes caused by molecular entanglement in concentrated solutions.

Infrared emissivity of icy surfaces

Context. Most analyses of the infrared emission of Saturn’s rings and icy satellites have considered pure water ice as the constituent of regolith and particle surfaces. Visual and near-infrared observations have shown, however, that darkening and reddening contaminants are present at a fraction level of a few percent. In the spectral domain 10–2000 cm−1, water ice becomes transparent in a few windows, which in particular causes the roll-off of emissivity of icy surfaces that is observed below 50 cm−1. Their emissivity there may be affected by these contaminants. Aims. We present a quantitative global sensitivity analysis of a hybrid Mie-Hapke model to evaluate the influence of regolith properties and contaminant fraction on the infrared emissivity of icy rings or moons over this spectral range. Methods. A hybrid Mie–Hapke model of the hemispherical emissivity ε*h(Wn) was made, including various diffraction correction and mixing types with tholins or amorphous carbon grains, or grain size distributions and some anisotropy in emission. A Sobol global sensitivity analysis provided quantitative levels of importance for these factors versus wave number wn. Results. Given the a priori uncertainties, the most important factor acting on ε*h(Wn) remains the size distribution of regolith grains and the average anisotropy factor ξ. For wn> 50 cm−1, ξ, the power-law index p and the minimum amin of the size distribution are most influential. In windows of water-ice transparency (10–50, 300–600, and 900–1300 cm−1), the emissivity is also sensitive, but to a lesser extent, to the maximum grain size amax and the fraction f of contaminants, if mixed at the molecular level. Conclusions. This model provides a self-consistent tool for interpreting multi-modal observations of the thermal emission from icy surfaces. It also offers interesting insights into recent mid-infrared observations of Saturn’s rings and Jupiter’s moon Ganymede by the JWST-MIRI instrument.

EDIC intensity correction of electron diffraction

Transmission electron microscopy (TEM) has recently become indispensable in determining crystal structures. The location of atoms in crystals can be determined using electron diffraction (ED) intensity data series if the diffracted intensities are directly proportional to the square of the structure factor (|Fhkl|2). However, due to the crystal thickness, the used electron wavelength and the potential misalignment of the measured crystal the detected intensities differ from the ideal values. A method, Electron Diffraction Intensity Correction (EDIC), and a computer program have been developed to recover the ideal |Fhkl|2 proportional intensities from experimental data for kinematic scattering, for further structure studies.

Numerical Integration and Angular Spectrum Modeling of Resonant and Non-Resonant Wave Interaction with a Solid Plate

The angular spectrum (AS) model is widely used to calculate the pressure transmitted through, and reflected from, a fluid-immersed solid plate, using a baffled piston as source. However, narrow peaks (infinite peaks for the lossless case) in the piston aperture function and in the plane wave transmission and reflection coefficients, in addition to a highly oscillatory integrand, cause challenges in the numerical integration. To overcome the challenges related to the aperture function peak, the AS model is reformulated using integration by parts. The peaks in the transmission and reflection coefficients are identified as symmetric and antisymmetric Scholte modes, and a numerical integration method is provided enabling integration over these peaks. The AS-model is also used to obtain the full-wave diffraction correction for a pulse-train transmitted through, or reflected from, the plate, which is highly relevant in pulse-transmit or pulse-receive measurements. The diffraction effects caused by the plate are calculated by suppressing resonance effects in the plate and obtaining the corresponding plane wave pressure transmission and reflection coefficients. A hybrid Gauss-Filon adaptive numerical integration method is used to calculate the pressure. These calculations are conducted in APAS, a MATLAB program developed in the current work and provided as open-source supplementary material.

A Phantom-Free Approach for Estimating the Backscatter Coefficient of Aggregated Red Blood Cells applied to COVID-19 Patients.

The ultrasound backscatter coefficient is a frequency-dependent quantity intrinsic to biological tissues that can be recovered from backscattered radiofrequency signals, granted acquisitions on a reference phantom are available under the same system's settings. A phantom-free backscatter coefficient estimation method is proposed based on Gaussian-shaped approximation of the point spread function (electronics and piezoelectric characteristics of the scanner's probe) and the effective medium theory combined with the structure factor model, albeit the proposed approach is amenable to other models. Meanwhile, the total attenuation due to intervening tissues is refined from its theoretical value, which is based on reported average behaviors of tissues, while allowing correction for diffraction due to the probe's geometry. The reference phantom method adapted to a similar approach except for the Gaussian approximation is also presented. The proposed phantom-free and reference phantom methods were compared on ten COVID-19 positive patients and twelve control subjects with measures on femoral veins and arteries. In this context, red blood cells are viewed as scatterers that form aggregates increasing the backscatter under the COVID-19 inflammatory condition. The considered model comprises five parameters, including the mean aggregate size estimated according to polydispersity of aggregates' radii, and anisotropy of their shape. The mean aggregate size over the two proposed methods presented an intraclass correlation coefficient of 0.964 for consistency. The aggregate size presented a significant difference between the two groups with either two methods, despite the confounding effect of the maximum Doppler velocity within the blood vessel and its diameter.

Ultrasonic Attenuation of Carbon-Fiber Reinforced Composites

Ultrasonic attenuation measurements were conducted on cross-ply and quasi-isotropic lay-ups of eight types of carbon-fiber reinforced composites (CFRPs) using through-transmission methods with diffraction correction. Attenuation values were substantially higher than those of unidirectional composites and other structural materials. Wave modes, fiber distributions, matrix resins, and consolidation methods affected total attenuation. Transverse mode, quasi-isotropic lay-up, and polyimide and thermoplastic resins generally produced higher attenuation. No clear trends from the fiber distribution were revealed, indicating that it is not feasible presently to predict the attenuation of various lay-ups from the unidirectional values. That is, direct attenuation tests for different laminate lay-ups are needed. This work expanded the existing attenuation database by properly determining the attenuation coefficients of two additional layup types of CFRP laminates. Results showed the merit of ultrasonic attenuation measurements for quality control and structural health monitoring applications. A crucial benefit of the through-transmission methods is that they enable the prediction of Lamb wave attenuation in combination with software like Disperse (ver. 2.0.20a, Imperial College, London, UK, 2013).

Finite element-based diffraction correction for piezoelectric transducers accounting for diffraction at transmission, propagation, and reception.

Existing diffraction correction models for ultrasonic transmit-receive measurement systems rely on simplifying assumptions with respect to the boundary conditions at the transmitter or receiver. Common simplifications include approximating the sound field radiated by a piezoelectric transducer using a baffled piston model and assuming that the receiver's electrical response is proportional to the spatially averaged free-field pressure over its front surface. In many applications, such simplifications may be adequate, but their validity and accuracy need to be evaluated and quantified. Here, a diffraction correction model utilizing the full set of electrical and mechanical boundary conditions at the transmitter and receiver is presented, avoiding these simplifications. The model is based on finite element modeling of coaxially aligned piezoelectric transducers in a fluid medium. Comparison is made with existing models for an example case of cylindrical piezoelectric ceramic disk transducers operating in air at 50-300 kHz and 0.03-2 m apart, relevant for, e.g., sound velocity and absorption measurements in fluids and ultrasonic gas flow metering. In the near-field, errors introduced by the simplifications are up to 3 dB and 47° for the first radial resonance. Generally, such errors are application-specific and depend on distance, frequency, transducer construction, vibration pattern, and medium properties.

Measurement and In-Depth Analysis of Higher Harmonic Generation in Aluminum Alloys with Consideration of Source Nonlinearity.

Harmonic generation measurement is recognized as a promising tool for inspecting material state or micro-damage and is an ongoing research topic. Second harmonic generation is most frequently employed and provides the quadratic nonlinearity parameter (β) that is calculated by the measurement of fundamental and second harmonic amplitudes. The cubic nonlinearity parameter (β2), which dominates the third harmonic amplitude and is obtained by third harmonic generation, is often used as a more sensitive parameter in many applications. This paper presents a detailed procedure for determining the correct β2 of ductile polycrystalline metal samples such as aluminum alloys when there exists source nonlinearity. The procedure includes receiver calibration, diffraction, and attenuation correction and, more importantly, source nonlinearity correction for third harmonic amplitudes. The effect of these corrections on the measurement of β2 is presented for aluminum specimens of various thicknesses at various input power levels. By correcting the source nonlinearity of the third harmonic and further verifying the approximate relationship between the cubic nonlinearity parameter and the square of the quadratic nonlinearity parameter (β∗β), β2≈β∗β, the cubic nonlinearity parameters could be accurately determined even with thinner samples and lower input voltages.

Parabolic equation simulation of diffraction effects in a sound beam propagating through a flowing fluid

Ultrasonic transit time difference flow meters are today industrially accepted for custody transfer measurements of oil and for natural gas. Such meters are currently planned to be used also subsea and in remote operations, where the calibration possibilities are few. In such applications the speed of sound measured by these meters will be a powerful input for estimation of density and calorific value of the flowing oil or gas. The ultrasonic transit time measurements in such meters are carried out in a flowing oil or gas, over a range typically between 6 and 40 in. For precise transit time measurements over such ranges, diffraction corrections may be of high importance. Diffraction effects for an acoustic beam generated by a uniform piston source and propagating through a flowing fluid are therefore studied numerically. The flow direction will be perpendicular to the propagation direction of the acoustic beam. The investigation is based on a narrow-angle three-dimensional parabolic equation. Effects both on amplitude and on phase will be presented.

Wave-based numerical investigation on diffraction correction for a low-height barrier in energy-based sound propagation model for road traffic noise

In energy-based noise propagation prediction methods, such as CNOSSOS-EU in Europe and ASJ RTN-Model in Japan, sound propagation is calculated by using the equation of sound attenuation by the geometrical divergence of the sound power from a sound source and adding various attenuation corrections such as diffraction effect and ground effect. In the ASJ RTN-Model, the insertion loss due to a semi-infinite barrier to a free field is generally used for the diffraction correction for a single barrier, the top of which is regarded as a knife wedge, whereas the diffraction correction for a low-height barrier is different from that for a general single barrier. It is given as the insertion loss due to a low-height barrier to a semi-infinite barrier, the top height of which is at the ground surface level, in a free field: the diffraction correction is zero when the low-height barrier is 0 m high. However, its range of application is not clear. In this paper, the diffraction correction for a low-height barrier is discussed by comparing the results of three-dimensional wave-based numerical analysis and propagation calculations using the ASJ RTN-Model.

Specimen-displacement correction for powder X-ray diffraction in Debye-Scherrer geometry with a flat area detector.

The effect of small changes in the speci-men-to-detector distance on the unit-cell parameters is examined for synchrotron powder diffraction in Debye-Scherrer (transmission) geometry with a flat area detector. An analytical correction equationis proposed to fix the shift in 2θ values due to speci-men capillary displacement. This equationdoes not require the use of an internal reference material, is applied during the Rietveld refinement step, and is analogous to the speci-men-displacement correction equations for Bragg-Brentano and curved-detector Debye-Scherrer geometry experiments, but has a different functional form. The 2θ correction equationis compared with another speci-men-displacement correction based on the use of an internal reference material in which new integration and calibration parameters of area-detector images are determined. Example data sets showing the effect of a 3.3 mm speci-men displacement on the unit-cell parameters for 25°C CeO2, including both types of displacement correction, are described. These experiments were performed at powder X-ray diffraction beamlines at the National Synchrotron Light Source II at Brookhaven National Laboratory and the Advanced Photon Source at Argonne National Laboratory.

Simulation of a tissue backscatter coefficient measurement experiment using the finite element method

Ultrasound has been shown to be an effective imaging modality for investigating tissue health. For example, the backscatter coefficient (BSC) has been shown to be a biomarker of tumour response to therapy but is limited in its application clinically due to the difficulty of accurately acquiring attenuation and diffraction corrections. To assess the variability in BSC assessment, we present a finite element tool simulating a singleelement planar reflector substitution method estimate of the BSC of a set of simulated phantoms. BSC estimates were computed with errors <5% of the theoretical value for a range of source apertures and over a set of phantoms with varying scatterer number densities to within 10%. In addition, the firstorder amplitude envelope statistics of backscattered waves were shown to be commensurate with Rayleigh scattering models. These results indicate that the tool is accurate in its replication of soft tissue-like scattering, suggesting that it could be an opportune testing ground for investigating sources of variability in BSC measurements and the testing of algorithms and hypotheses for interrogating the scattering behaviour of soft tissues.

Errors using the spatially averaged free-field pressure approximation for description of the receiving properties of piezoelectric transducers

In modelling of ultrasonic measurement systems and calculation of diffraction corrections for such systems, it is often assumed that the spatially averaged free-field pressure over the front surface of a piezoelectric transducer, ⟨p⟩, can be used to calculate its voltage output. It is of interest to investigate whether errors due to this ⟨p⟩-approximation are acceptable for piezoelectric transducers applied in high-precision ultrasonic measurements in fluids. 3D-axisymmetric finite element (FE) modelling is used to investigate such errors in a transmit-receive measurement setup employing two identical transducers. The approximate output voltage is found by Vout = ⟨p⟩MV and compared to the output voltage from FE modelling of the transmitter-medium-receiver system, Vout,ref, using the appropriate boundary conditions at the two transducers. MV is the open-circuit free-field receiving voltage sensitivity of the receiver for normally incident plane wavefronts, calculated assuming spherical reciprocity. Two examples of piezoelectric transducer pairs are studied, operated in the frequency range of their lower radial modes. It is shown that the ⟨p⟩-approximation may introduce notable deviations between Vout and Vout,ref, of up to approximately 40° for the phase angle and 2 dB for the magnitude. The deviations depend on the fluid medium, separation distance, frequency, and the transducer’s construction and vibrational characteristics.

Off-epicentral measurement of laser-ultrasonic shear-wave velocity and its application to elastic-moduli evaluation

This study investigates the off-epicentral measurement system and methodology of laser-ultrasonic shear waves in the ablative regime, subsequently focusing on its application to the elastic-moduli evaluation. The proposed scheme facilitates the measurement of the shear-wave velocity by using the crosscorrelation of successive shear-wave pulse echoes, ensuring high reliability and reproducibility of the calculation results. Since the geometric diffraction induced by the finite dimensions of laser-ultrasonic source and receiver can increase the measurement error, far-field diffraction correction for the off-epicenter detection has been derived theoretically. Results have shown that the far-field diffraction correction error is of the same order as random error mainly caused by the finite observation time, signal-to-noise ratio and frequency bandwidth, and therefore cannot be neglected. The feasibility of this approach is then demonstrated by inspecting annealed commercial purity titanium plates. Good agreement of experimentally measured and theoretical values of shear-wave velocity confirms the validity of the proposed approach. The ultrasonic measurement-determined shear modulus shows a maximum deviation of ＋3.6% from the standard value of an isotropic material.

Acquisition and evaluation procedure to improve the accuracy of SAED.

The achievement of this work is that fine tuning of experimental and evaluation parameters can improve the absolute accuracy and reproducibility of selected area electron diffraction (SAED) to 0.1% without using internal standard. Due to the proposed procedure it was possible to reach a reproducibility better than 0.03% for camera length between sessions by careful control of specimen height and illumination conditions by monitoring lens currents. We applied a calibration specimen composed of nanocrystalline grains free of texture and providing narrow diffraction rings. Refinements of the centre of the diffraction pattern and corrections for elliptic ring distortions allowed for determining the ring diameters with an accuracy of 0.1%. We analyze the effect of different error sources and reason the achieved absolute accuracy of the measurement. Application of the proposed evaluation procedure is inevitable in case of multicomponent nanocomposites or textured materials and/or having close diffraction rings where application of automated procedures is limited. The achieved accuracy of 0.1% without internal standard is approaching that of routine laboratory XRD, and reduction of instrumental broadening due to the elaborated evaluation procedure allows for separation of close reflections, provides more reliable ring width and thus improved input parameters for further nanostructure analysis as demonstrated on dental enamel bioapatite.

Self-Reduction Mechanism and Luminescence Properties of Eu2+-Eu3+Doped Ca(PO3)2 Phosphor

A series of Ca(PO3)2 phosphors mono-doped with Eu3+ were successfully synthesized by the conventional high-temperature solid-phase method. Rietveld correction of the X-ray diffraction shows that Eu3+ ions occupy Ca2+ positions in the Ca(PO3)2 lattice and have no effect on the matrix structure. In this work, the self-reduction of Eu3+ was observed and verified by photoluminescence (PL), excitation spectroscopy (PLE) and X-ray photoelectron spectroscopy (XPS). Under excitation at 238 nm, Ca(PO3)2: Eu exhibits a broad blue Eu2+ emission band in addition to the usual orange-red emission of Eu3+. The fact that europium ions exist in divalent and trivalent forms was further confirmed, and the corresponding mechanism of the anomalous reduction of Eu3+ to Eu2+ is discussed. On this basis, the effect of Eu concentration on the structure and luminescence properties of Ca(PO3)2 phosphor was investigated in detail. The luminescence of the phosphor exhibited bright tunable emission in the blue region and the red region. These results indicate that the Ca(PO3)2: Eu phosphors have potential applications as a n-UV convertible phosphor for light-emitting diodes.

Measurement of attenuation in a sample with nonparallel surfaces

Ultrasonic attenuation in solids is an important characteristic of structural materials related to their microstructure. It is well known that attenuation measurements in solids require that the front and back surfaces of the sample be parallel. In this paper, an immersion method for attenuation measurement in a sample with nonparallel surfaces is developed. The maximization method is based on independent alignment of the transducer orientations for maximizing each of the multiply reflected signals received. The experiment is accompanied by a measurement model that provides a diffraction correction of the ultrasonic beam. The method is compared with that involvingthe normalization procedure, of the sequence of the reflected pulses, by the transducer directivity function.Those independent measurements are in reasonable agreement for the amplitudes of the first few reflected pulses. In general, the maximization method is advantageous for attenuation measurements by immersion experiments.