Residual stress measurements with portable X-ray devices using diffraction techniques were first demonstrated in the 1960s. Since then, the measurement precision and portability of these devices have improved significantly because of advances in high-resolution detectors and analysis methods. However, questions regarding accuracy have persisted because of the intrinsic limitations of such small devices and the unknown true stress states in materials of interest. Previous studies of precision and accuracy have suggested that precision is mainly a function of the instrument and measurement environments and that accuracy depends not only on instrumental parameters, but also involves sample conditions such as anisotropy, heterogeneity, and processing history. Even though sample-dependent accuracy characterization is not trivial, it is worthwhile to quantify measurement errors from a nonideal sample when general assumptions (isotropic and homogeneous ensembles under a biaxial stress state) are not satisfied. Therefore, we carried out in situ uniaxial tension and torsion experiments with a round 1010 carbon steel tube, and then compared the measured normal and shear stresses with the applied (true) values for accuracy assessment. Based on the obtained uncertainties, we propose practical measurement and analysis protocols for precise and accurate stress measurements.

	X-ray diffraction (XRD) techniques have been used to measure the residual stress of polycrystalline materials since the 1920s (1), and continuous advances have been made in both hardware development and analysis methods to determine the stress profile with increased accuracy and precision (2). However, measurement speed and portability have also become important criteria for industrial applications. In the 1960s, Boeing and Northrop Grumman led the development of portable X-ray systems to survey their aircraft parts, using the Laue back-reflection technique on photographic film (3,4). After the 1970s, most portable X-ray devices were equipped with a position-sensitive detector (PSD)-that is, a linear detector instead of a point detector or film-to avoid sample rotation or nondigitized film reading, which causes considerable errors because of the short sample-to-detector distance (SD) of such small devices (5).

	The advent of two dimensional (2D) detectors such as the multiwire PSD, the charge coupled device (CCD, in the 1970s), and the image plate (IP, in the 1980s) greatly improved the performance of stress measurements because these detectors collect signals from more grains. To utilize the partial or full Debye ring contour from a 2D detector, relevant analysis methods such as cosα (6), XRD

 (7), or the full-ring fitting method (8) had to be incorporated. Regardless of detector types, the traditional sin

ψ method was dominant until the 1980s; since the 1990s, the 2D method has become more available along with the development of high-resolution 2D detectors. 

	The X-ray stress analyzer in our study was based on an IP detector and used a cosα analysis method, and we performed a series of tests to answer the following questions:

	·      What is the intrinsic precision limit (smallest error) that an instrument can provide (9)? 

	·      How is the 2D analysis (cosαor full-ring fitting) different from the sin

	·      What are the necessary conditions for an accurate stress measurement (11)? 

	We have answered those questions by using stress-free ferrite powders and 1018 mild steel bar samples through in situ uniaxial tensile loading tests in the past. Based on our previous understanding, we introduced two more complications to extend the use of this technique to more practical applications: Is there any sample dependency in terms of shape or chemical composition, even with the same elastic properties? And, can we measure shear stress as well as normal stress? In this article, we answer those questions by uniaxial tension and torsion tests with 1010 mild steel tube samples.

	The basic principle of the stress measurement is to determine the interplanar spacing 

 from the diffraction peak position in a lab coordinate system (Ï

), as a built-in strain gauge, followed by the transformation to the sample coordinate system (Ï, ψ), and to relate this strain to the stress via the stiffness (or compliance) tensor. If the isotropic assumption is satisfied, strain or stress can be expressed in a simple form as equation 1:

ψ method with a point detector requires a set of measurements at different sample tilt angles (ψ

measured at angles (Ï, ψ) can be written as shown in equation 2 if the material’s elastic behavior is isotropic and homogeneous:

	If the in-plane stress is not dependent on the angle Ï, the equation can be further simplified as shown in equation 3, which is the well-known sin

ψ equation. The biaxial in-plane stress can then be determined from the slope of a 

	The cosα method was first proposed in the late 1970s using photographic film (6), and was applied to the image plate in the early 1990s (12). Since each point on the Debye ring comes from a different orientation of the scattering vector (not only from Ï, but also ψ in the sin

ψ geometry), one can utilize such information with a single measurement without any sample rotation. A strain transformation for a given geometry, leads to the final stress, as shown in equations 4 and 5:

 are the α dependent strain components and angles η and α are defined in Figure 1b. It should be emphasized that the stress equations 3–5 above are valid for isotropic, homogeneous, and biaxial-stress conditions. If not, or if a steep stress gradient exists near the surface, the equations must be modified as reported for the sin

					Figure 1: (a) Experimental setup with a portable X-ray device next to the tube sample in the electromechanical testing machine. Insets display strain gauges and the beam spot on the sample surface. (b) Measured back-diffracted Fe (211) peak. The black dashed lines show the region that the cosα method uses, and black circles indicate data used for the two-tilt sin

	An Instron 5984 electromechanical testing machine and an Instron 1321 axial torsional test frame were used to apply uniaxial tension and torsion to the tube sample, conforming to ASTM International 179 and SAE International J524 standards. The tube material was very low carbon steel meeting the American Iron and Steel Institute (AISI) 1010 specifications, which was cold drawn and annealed at 650 °C. The microstructure shows a primary ferrite phase (grain size 12 µm) with small islands of pearlite. In addition, no significant texture was detected from the microscopes (scanning electron microscope and electron backscatter diffraction) and a laboratory diffractometer. The tube outer diameter, inner diameter, wall thickness, and length were 1 in. (25.4 mm), 0.902 in. (22.9 mm), 0.049 in. (1.24 mm), and 12 in. (305 mm), respectively. 

	The miniature portable X-ray device selected for this study was equipped with an image plate 2D detector (spatial resolution: 50 µm) and analysis software (cosα method) for instantaneous results. In addition to portability, stress measurements are fast (less than 2 min) and convenient (single exposure) compared to the sin

ψ method. The specifications of the instrument are as follows: tube: 20 keV, 1.0 mA; target: chromium; beam energy: λ = 2.2909 Å, 5.4 keV; collimator size: 1 mm diameter; spot size: 2 mm diameter; sample-to-detector distance: approximately 35–40 mm; data acquisition time: 40 s; data read or analysis time: 50 s.

	X-ray diffraction profiles of the Fe (211) peak were measured at each loading point under load control with various tilt angles (ψ

) and sample-to-detector distances (SDs) for the uniaxial tensile tests and with ψ

 = 35°, SD = 35 mm for the torsional tests. The goal of the in situ loading tests is to quantify accuracy by comparing stresses measured by diffraction with the electromechanical testing machine values since an operator knows exactly how much stress is applied to the sample. Recently, we demonstrated such uncertainties using the 1018 steel bar with the X-ray beam on a flat surface via uniaxial tensile loading (11). However, engineering components are not necessarily flat, and also are not likely to have only normal stress components. Therefore, we chose a round 1010 mild carbon steel tube (McMaster, Inc.) for the torsion test. A tubular sample is ideal for this study because it is more representative of actual applications with alignment difficulty and is convenient to use in twisting experiments. The differences between the old (1018) and new (1010) steel are shown in the Table I. 

	In our previous study, we showed that the intrinsic precision limit of this device is 9 µε, corresponding to 2 MPa for steel (9), and that cosα and sin

ψ are equivalent and generate the same results within the error (10). In addition, we showed that the sample (machine) tilt angle, ψ

, is one of the most sensitive parameters for measurement accuracy of the cosα method analyzing the back-reflected Debye ring (11) by stress-free reference powders and a solid bar with flat geometry. 

	In this study, we have confirmed these phenomena with some differences. Figure 2a shows the measured stress versus applied stress using three different analysis methods from four independent measurements. Loading was applied up to 275 MPa with an interval of 25 MPa. Because a ψ

 angle of 45° was known to be the most accurate condition from the 1018 steel bar measurements, the same conditions were applied to the 1010 steel tube. First of all, this plot shows that the accuracy does not depend on the analysis method in the elastic regime even though each method is sensitive to different parameters (11). However, the measured stress values were around 75% of the correct ones (25% off) for all cases. Figure 2b displays measured shear stress versus applied shear stress. Because of the capacity limit of the axial torsional test frame, shear loading was only applied up to 50 MPa. It demonstrates that shear stress can be traced with reasonable accuracy. However, note that shear stress (σ

) error bars (±10–20 MPa) in Figure 2b are the fitting errors for the slope of 

 versus sin α as shown in equation 5 from a single measurement while the error bars (±10 MPa) in Figure 2a are the precision determined from multiple measurements. Therefore, shear stress measurements seem marginally acceptable at this point with greater errors than normal stress, and need to be tested with higher loading and multiple measurements.

					Figure 2: (a) Measured stress versus applied stress at ψ

 = 45° under uniaxial tensile load with three different analysis methods and (b) under torsional loading at ψ

 = 35°. Error bars in (a) are the standard deviation from four measurements, that is, precision, and the error bar in (b) comes from the fitting error of the single dataset from equation 5.

	To better understand the inaccurate normal stress measurements of the 1010 tube sample and discrepancy from the previous 1018 bar sample, a tilt angle (ψ

)-dependent accuracy plot (Figure 3) was generated from the five measurements at each value of ψ

 with a reduced number of tensile loading steps. In other words, the cosα slope of Figure 2a is marked as a blue circle at ψ

 = 45° in Figure 3. Note that the manufacturer recommends that measurements be made at ψ

 = 35° because it provides the most precise and accurate results from the steel reference powders. However, for solid steel samples, both 1018 and 1010 do not show the best results at or near 35°. Measurements with stress-free powders show reasonable accuracy at 25° < ψ

 < 45° (9), while our steel sample measurements were the most inaccurate in that range. The error at low tilt angles (ψ

 < 15°) is an inevitable problem regardless of analysis methods (cosα or sin

ψ) because of the narrow range of the scattering vectors.

					Figure 3: Measured/applied stress ratio (slope in Figure 2a) versus sample (or machine) tilt angle (ψ

) with 1018 (11) and 1010 carbon steel from five independent measurements. Accuracy depends not only on the instrumental parameters (ψ

), but also on the sample itself. The green dashed line indicates 100% accurate results.

	Two problems in Figure 3 need to be addressed: inaccurate results from both the 1018 and 1010 samples, and the discrepancy between them. It was known that X-ray measurements do not give macro-stress values because of residual pseudo-macro-stress formation if the specimens have been stretched, compressed, bent, rolled, or drawn (16). This fictitious stress depends not only on the magnitude of deformation, but also the crystallographic orientation. For example, the Fe (211) peak generates a lower magnitude of stress than the (310) peaks, even when X-ray elastic constants are applied, presumably because of plastic anisotropy (17,18). Despite the unknown processing history, these long bar and tube samples are suspected to have been subjected to rolling or drawing processes. Since this 

 dependency is well-known behavior in plain carbon steel, lower stresses from a proper instrumental setting can be ascribed to the pseudo-macro-stress from the prior plastic deformation. 

 dependency of pseudo-macro-stress, Taira and colleagues reported that the magnitude of the measured stress increases with the amount of carbon content up to 0.5% (18). This stress increase with increased carbon content can be one possible explanation of why 1010 steel with lower carbon content (C: 0.08–0.13%) shows a lower stress level than 1018 steel (C: 0.14–0.20%). Secondly, texture can be another reason. The discrepancy between steel samples becomes larger at higher tilt angles. Based on the electron backscatter diffraction and full range diffraction profile, the 1018 steel bar shows some texture along the vertical (axial) direction whereas the 1010 steel does not. Since diffraction measurements with higher tilt angles probe grains more aligned in the axial direction, texture developed from previous plastic deformation may cause this discrepancy.

	Therefore, the measurement accuracy is highly dependent on crystallographic plane, chemical composition, texture, and processing history of the sample. Even though it is not possible to know all the sources of error, one can calibrate the sample and instrument together with a single parameter, such as ψ0 in this case. After the calibration chart is made, stress measurements with diffraction techniques can be quite useful in terms of speed, convenience, and accuracy. 

	The effect of sample geometry and measurement environment also should not be neglected. The average precision error bars (±16 MPa at 200 MPa, 8%) for 1010 steel are larger than those for 1018 (±6 MPa at 200 MPa, 3%) between 25° and 45° in Figure 3. This is likely because of the round geometry causing more problems of beam focusing and alignment, although some error may have come from sample heterogeneity since measurements took place at five different locations. Note that the best stress precision achievable from stress-free steel powders is 2 MPa; it was observed that the usual environment in New York City where subways pass nearby easily increased the error up to 10% (±20 MPa at 200 MPa applied load) on the 1010 tube sample.

	We can reach the following conclusions from this work:

	·      Precision error by the instrument has been reduced significantly from the past because of improvements in technology.

	·      The cosα method is fast, precise, and convenient, but accuracy is not guaranteed. 

ψ and other 2D analysis methods (cosα or full-ring fitting) yield statistically indistinguishable results in the elastic regime, but they may deviate after yielding because of the information volume difference and heterogeneous plastic flow.

	·      Precision depends on the instrument parameters, measurement environment, and sample geometry.

	·      Accuracy depends not only on the measurement parameters but also on sample conditions such as elastic anisotropy, processing history, chemical compositions, and texture.

	·      Shear stress can be measured, but the error is much larger than for normal stress.

	The above conclusions allow some recommendations for stress measurements with a portable X-ray device. This general idea can be applied to any portable system, regardless of detector type (IP, CCD, semiconductor) or detection geometry (point detector, one-dimensional, linear PSD, 2D).

	·      Be aware that most commercial devices use the stress equation for isotropic and homogeneous materials under biaxial stress. The biaxial stress condition can be justified if the beam penetration depth is shallow using a low X-ray energy. Try to measure different spots to check sample heterogeneity and use the X-ray elastic constant (XEC) to reduce the error arising from the elastic anisotropy.

	·      If XEC is not available and only a single reflection is used, the (211) for body-centered cubic (BCC) and (311) for face-centered cubic (FCC) materials are good choices because XEC, 

/2, for each reflection is close to the (1+υ)/

 that the stress equation uses. However, this is not valid if the specimen has undergone plastic deformation as demonstrated in this study. 

	·      If a 2D detector is used, be sure to have a continuous Debye ring with sufficient intensity. Also check the linearity of strain against sin

ψ or cosα. If strain is not linear, the standard equations for sin

ψ or cosα cannot be used, and a more advanced treatment is required.

ψ and cosα methods can be used with the 2D data. For the sin

ψ method, make sure to place the specimen at the exact center of rotation. The cosα method is insensitive to the sample location and sample-to-detector distance, but beam center calibration is normally not adjustable by users. 

	·      Repeat the measurement at the same and different spots to separate instrument precision and sample heterogeneity.

	·      If only relative measurements are important, use the instrumental conditions that provide the best precision. For this specific device, ψ

	·      If the absolute values are important, create a calibration chart for the sample of interest by applying the known stresses, using one single adjustable instrumental parameter. For example, the tile angle, ψ

, is the calibration parameter, and the ψ

 of 45° gives the most accurate results for the 1018 carbon steel.

	Discussions with Professor I.C. Noyan, Professor Pat Mooney, and Mr. Hazar Seren at Columbia University are gratefully appreciated. The authors acknowledge Dr. Adrian Brügger for providing the Instron 5984 at the Robert W. Carleton Strength of Materials Laboratory. We also thank Mr. Toshikazu Suzuki for providing the portable X-ray machine and software.

		D.A. Bolstad, “Application of Portable X-Ray Stress Techniques at the Commercial Airplane Division of the Boeing Co,” Automotive Engineering Congress, Detroit, Michigan (1967), pp. 1–4.

		R. Homicz, “Fundamentals and Basic Techniques of Residual Stress Measurements with a Portable X-Ray Diffraction Unit,” Technical Report No. 0148-7191, SAE Technical Paper, 670151 (1967).

 (John Wiley & Sons, Hoboken, New Jersey, 2011).

		A. Kampfe, B. Kampfe, S. Goldenbogen, B. Eigenmann, E. Macherauch, and D. Lohe, 

		J. Ramirez-Rico, S.-Y. Lee, J. Ling, and I.C. Noyan, 

		S.-Y. Lee, J. Ling, S. Wang, and J. Ramirez-Rico

Articles

To test the accuracy of residual stress measurements made with portable X-ray devices, measured normal and shear stresses were compared with the applied (true) values for accuracy assessment. From those results, practical measurement and analysis protocols for precise and accurate stress measurements are proposed.

Advances in Stress Measurement with Portable X-Ray Diffraction: Uncertainties in Normal and Shear Stress Determination on Mild Steel Pipe

With the availability of high intensity synchrotron sources, fast electronic detectors, and computational resources, there has been a resurgence in the application of small-angle X-ray scattering for a large range of problems in materials science. Some of these developments are discussed using examples drawn from porous structures, protein solution, and polymers and fibers. New analytical techniques such as Monte Carlo simulations of solution scattering data and full pattern analysis of the diffraction data from lamellar structures are also described. Finally, some of the online resources available for such analysis are provided.

	With the availability of synchrotron X-ray sources, fast two-dimensional (2D) electronic detectors, and enhanced computational capabilities, there has been a resurgence in the use of small-angle X-ray scattering (SAXS) in biology and materials science. The technique is now more accessible and has become an indispensable analytical tool for studying statistically representative microstructures of heterogeneous materials at 10–1000 nm length scales (1). 

	The ability of X-rays to be absorbed, fluoresce, and diffract is used to examine materials by imaging, spectroscopy, and crystallography, respectively. SAXS, on the other hand, measures the coherent scattering of X-rays at small angles to the primary beam (<2° with copper Kα radiation; 1.54 Å wavelength) to probe the structure at the meso-length scales that commonly occur in biology and materials science. This capability includes measuring the shapes and sizes of nanoparticles and large molecules, domains and voids, organized structures at large length scales, and, in general, electron density fluctuations and inhomogeneities with characteristic dimensions between 10 and a few hundred nanometers (2). SAXS complements other techniques such as microtomography, electron microscopy, and atomic force microscopy (AFM). When neutrons are substituted for X-rays, the resulting small-angle neutron scattering (SANS) data can extend the utility of small-angle scattering (SAS) by being able to adjust the contrast based on the nuclear structure rather than the electron density differences as in X-rays (3).

	The areas in biology and materials science that can benefit by using SAXS fall into two broad categories: particulate systems (dilute and crowded particles) and nonparticulate systems (random two-phase and ordered structures). Systems with dilute particles include proteins and polymers in solution, colloids, defects in crystals, and inclusions in solids and voids (porous materials). Examples of dense particulate systems with long-range organization are colloidal aggregates, clusters of nanoparticles, microemulsions, and fractal objects. Solid two-phase systems include polymers, catalysts, nanocomposites, porous media, metallic alloys, and other microphase separated systems. Finally, examples of ordered structures (self-assembled systems) can be found in biological structures, semicrystalline polymers, block copolymers, fibers, liquid crystalline materials, and nanostructures. This article highlights experimental requirements and applications, with examples drawn from protein solutions, porous structures, and polymers.

	The SAXS experiment is conceptually simple. The sample is illuminated by X-rays, and the scattered radiation is registered by a detector. Because the SAXS measurements are done very close to the primary beam, at small angles, collimation of the beam is stringent and the instrument requires long flight paths (on the order of meters). Consequently, essential elements of the instrument are proper alignment of the numerous slits to define the beam, containment of the sample in a cell with minimum scatter, the smallest possible beam stop that does not contribute to stray scattering, and an evaluated beam path.

	Until the 1980s, SAXS experiments were carried out using instruments equipped with laboratory X-ray tubes. But higher intensity sources are required to compensate for the loss in intensity resulting from fine collimation and the long flight path. Synchrotron sources reduce the time of SAXS data collection from as long as 24 h to seconds, or even shorter. Additionally, these sources make it possible to develop hyphenated techniques by combining X-ray scattering with differential scanning calorimetry (DSC) (4), infrared (IR) or Raman spectroscopy, electrochemistry, fluorescence, X-ray spectroscopy (5), dynamic mechanical analysis, and, more recently, computed tomography (6). It is now possible to routinely carry out time-resolved experiments to study the influence of temperature and deformation to understand the physical mechanisms behind macroscopic phenomena such as thermal expansion and physical deformation by monitoring the motion of the molecular chains. Microdiffraction with beams of ~0.5 µm (7,8) can be used to characterize spatial distribution of defects at polymer interfaces, nanostructures in fiber-reinforced composites and high-performance fibers, interfaces, and skin-core morphologies.

	Scattering from unoriented structures are analyzed by reducing the observed data into one-dimensional (1D) scans such as the one shown in Figure 1. The various regions in the patterns are indicated in the figure. Such 1D data are typically analyzed by using approximations (such as Guinier and Porod) that are valid over a limited angular range, correlation, or pair-wise distribution functions, and by modeling (2). Scattering from oriented structures needs to be analyzed using 2D data of the type shown in Figure 2. Such patterns can be used to gain valuable insight into the relation between structure and deformation. These 2D patterns can be analyzed by full pattern fitting. In many instances, such 2D patterns fall on an elliptical grid, and therefore are best analyzed in elliptical coordinates (9,10). 

					Figure 1: Schematic of 1D SAXS scan from a particulate system. The intensity is plotted as a function of scattering vector 

, which is related to the scattering angle 2Ï´ by the relation 

 = (4π sinÏ´/λ), where λ is the wavelength of the X-rays. The three commonly recognized regions of the SAXS regions are indicated in the figure.

					Figure 2: SAXS data in the form of contour plots from an ordered condensed phase. The data were fitted in elliptical coordinates to two functions corresponding to the lamellar peak and the equatorial streak (11). A schematic of the tilted lamellae that could give rise to such a pattern is shown on the right.

	The central diffuse scattering that arises from dilute systems (Figure 1) is analyzed to obtain particle parameters such as the radius of gyration (

) from Guinier plots, molecular weight from 

(0), compactness of the molecule from Kratky plots, surface area, volume, and surface-to-volume ratio and fractal dimensions (

) from Porod plots (2,12). Immense activity to probe the behavior of proteins in solution has led to the advent of robot-controlled, stream-lined data collection at the synchrotron sources and automated high-throughput analysis (13,14). More recently, biomolecule structures are being modeled with one-bead-per-residue representation (course-grain model), as a set of spherical structures, or composed of several candidate structures that are aligned and subsequently clustered to reproduce the observed data. In these modeling efforts, SAXS data is used to choose a subset of conformations from a large pool of candidate structures generated from prior knowledge of the interactions between the various domains or residues within the molecule. Alternatively, the scattering body is expressed as a series of spherical harmonics, and the scattered intensity is calculated as the sum of the contributions from these substructures (15). Note that fitting high-resolution structural models to low-resolution data is susceptible to overfitting.
	 

	Central diffuse scattering is also present in catalysts and ceramics, where it is used to obtain void volume fraction, pore size distributions, internal surface areas, and pore morphologies (16). Examples include zeolites, solution-mediated colloids, gels and suspensions, and for nanoparticle assemblies of soot or silica particles in flames (17). Ultra-SAXS (USAXS) is used to examine sintering, microcracking, cavitation, or creep damage on the micrometer scale. At synchrotron sources, it is possible to carry out anomalous SAXS studies by varying the energy close to an X-ray absorption edge of a specific element and thus vary the scattering contrast of one microstructure component against another.

	Mesoscale ordered structures give rise to discrete reflections rather than diffuse scattering. These Bragg-like features form a 2D scattering pattern such as the one shown in Figure 2. Many biological structures such as membranes and collagen have been extensively investigated using SAXS, and continue to provide new insights into these structures (18). Ordered structures in synthetic polymers include lamellae and micelles from many polymers that crystallize, and a large number of very interesting ordered nanostructures, such as spherical micelles and cylindrical (worm-like) micelles, bilayers and vesicles, and bicontinuous structures in block copolymers. The 2D patterns of the type shown Figure 2 that arise from lamellar structures are analyzed to obtain characteristics of the lamellae such as height, width, spacing, and the orientation, which are used to study the crystallization and deformation (11,19). The results obtained using a nano-calorimeter have been especially revealing about the nature of the thermal transitions and polymer. Similarly, nanobeam diffraction has been used to study interfaces and spherulitic morphology. In addition to the lamellar reflections, there is a streak along the equator and the central diffuse scattering that contains information about the larger length scale structures such as fibrils and lamellar stacks. Both the lamellar structure and the equatorial streak are best analyzed by full-pattern fitting of the image in elliptical coordinates (9,10,20).

	Many software packages are available online for processing, analyzing, and modeling SAXS data. Some commonly used software includes ATSAS from the European Molecular Biology Laboratory (EMBL), FIT2D and BIOSYS from the European Synchrotron Radiation Facility (ESRF), and IRENA from the Advanced Photon Source (APS). Additional information about the software and other small-angle scattering resources is available at 

	SAXS, in contrast to the more common imaging techniques such as tomography and microscopy, can provide a view of the structure that is statistically averaged over a large sampling area. However, SAXS data, unlike the imaging techniques, appears in reciprocal-space and thus cannot provide a direct view of the structure in real-space. To address this problem, computational tools that model the structure are being developed to make the technique accessible to nonexperts. Because of these advances, coupled with remarkable developments in instrumentation, SAXS has a great potential to be widely used for investigating mesoscopic structures in diverse fields including biology, soft materials, colloids, catalysts, glasses, ceramics, and metallurgy. 

	This article was adapted from the material that was presented at the Polymer Diffraction workshop held as a part of the 2017 Denver X-ray conference (July 31–August 4, 2017).

, Vol. 1076 (Springer-Verlag New York, New York, 2007).

		W. Bras, S. Nikitenko, G. Portale, A. Beale, A. van der Eerden, and D. Detollenaere, 

 1742–6596 (2010). doi:10.1088/1742-6596/247/1/012047.

		F. Schaff, M. Bech, P. Zaslansky, C. Jud, M. Liebi, M. Guizar-Sicairos, and F. Pfeiffer, 

		C. Riekel, M. Burghammer, and M. Muller, 

		M.H. Koch, P. Vachette, and D.I. Svergun, 

		J. Hyeon-Lee, G. Beaucage, S.E. Pratsinis, and S. Vemury, 

		A. Angelova, B. Angelov, V.M. Garamus, P. Couvreur, and S. Lesieur, 

 is an Associate Research Professor with the Center for Biomaterials at Rutgers, the State University of New Jersey, in New Brunswick, New Jersey. Direct correspondence to: 

There has been a resurgence in the application of small-angle X-ray scattering for a large range of problems in materials science. This article highlights experimental requirements and applications, with examples drawn from protein solutions, porous structures, and polymers.

Recent Developments in Small-Angle X-Ray Scattering

Fatty acid methyl esters (FAMEs) are usually derived from vegetable oil and are typically blended with regular diesel fuel in the 5–20% range. An elemental analysis of pure FAMEs, used vegetable oil, or the reprocessed vegetable oil as well as biofuel blends is required to show compliance with the relevant specifications. Several test methods are available that describe the use of energy-dispersive X-ray fluorescence (EDXRF) for the analysis of sulfur in fuel, such as International Organization for Standardization (ISO) 13032 and ASTM International D7220. The current test methods, using EDXRF, are limited to fuels with a discrete maximum amount of FAME or oxygen. This article describes a quick and economical procedure to check samples for their oxygen content to ensure that the material used for diesel fuel blending is really FAME-based and not mineral-oil based. The oxygen content determined by this approach is used to apply the required matrix correction without a priori knowledge (which is typically required using other approaches), improving sulfur content determination accuracy. In addition to the analysis of the sulfur content, this article describes EDXRF as a technique to monitor other elements, including P, Cl, Na, K, Mg, and Ca, among others, that can have detrimental effects on combustion engines.

	These days, renewable fuels and other biofuels play a growing role in the offer of automotive fuels. Currently, fatty acid methyl esters (FAMEs), mostly derived from vegetable oil, are typically blended with regular diesel fuel in the 5–10% or 20% range. An alternative for the fairly expensive fresh vegetable oil is recycled vegetable oil derived from the food industry. An elemental analysis of the pure FAMEs, the used vegetable oil, the reprocessed vegetable oil, and the biofuel blends is required to show compliance with actual specifications.

	Typically, this type of analysis is done using inductively coupled plasma–optical emission spectrometry (ICP-OES) using the following test methods:

	·      EN14538-2006 (1) describes the analysis of Ca, K, Mg, and Na in FAME, samples require a 1:1 dilution (50%) in kerosene (to reduce matrix effects) and no internal standard is used.

	·      EN16294-2012 (2) describes the analysis of P in FAME, samples require a 1:3 dilution (25%) in kerosene and the use of an internal standard is mandatory.

	Sulfur is not part of these methods and is determined using different technologies. The analysis requires some sample preparation and investment in purchase and operating cost of the analyzer.

	This report describes a quick and economic alternative to quantify the sulfur content in FAMEs using energy-dispersive X-ray fluorescence (EDXRF).

	Besides the analysis of the sulfur content, EDXRF equipment (depending on the analyzer performance), is capable of monitoring additional elements such as P, Cl, Na, K, Mg, Ca, and many more that can be harmful to car engines.

	Recycled vegetable oils can be added into diesel fuel and used as secondary burner fuel. An elemental analysis to show the absence of toxic elements is part of the quality control. This can also be done with the XRF equipment.

	In addition, it would be of interest to at least check the samples for their oxygen content to make sure that the material used for blending in the diesel fuel is really based on FAMEs and not based on mineral oil.

	Several test methods are available that describe the use of EDXRF for the analysis of sulfur in fuel. The International Organization for Standardization (ISO) method 13032 (3) and ASTM International method D7220 (4) are two of them.

	The currently available test methods using EDXRF are limited to fuels with a discrete maximum amount of FAME or oxygen. In the example of ISO 13032, the maximum content of FAME is 10% or 3.7% of oxygen. The reason is that the oxygen has a matrix effect on the sulfur signal measured by XRF. To be able to use XRF for this type of analysis, various approaches are possible:

	·      Correction using defined correction values depending on the known oxygen content, which requires a priori knowledge of the oxygen content.

	·      Correction using fundamental parameters based on the matrix, which has to be predefined, and which also requires a priori knowledge of the oxygen content.

	·      Correction using backscatter information from the sample. This approach does not require any a priori knowledge of the sample matrix, but the backscatter correction must be based on a signal, which should be close to the analyte’s X-ray fluorescence energy because otherwise additional effects occur.

	·      Additional references to test methods are also available (5–7).

	For other elements of interest, the same effect is visible. The influence on the results varies depending on the element.

	Our procedure used a Spectro Xepos analyzer in which samples were excited by a forced-air-cooled 50 W end-window X-ray tube combined with a doubly curved HAPG crystal for monochromatization and polarization of the primary tube spectrum. In addition, direct excitation using Pd-K and Co-K radiation can be used to optimize the excitation conditions for groups of elements. A high-resolution silicon drift detector (SDD) was used to collect fluorescence radiation from the sample. The highly stable spectral resolution of the SDD was <130 eV for Mn Kα.

	The measurement parameters are given in Table I.

	The sample preparation for the analysis is straightforward: 4 g of sample material is poured into a disposable XRF sample cup with an outer diameter of 32 mm. The analytical side of the sample cup is closed with a 4-µm-thick polypropylene film. For additional safety, the sample cups are placed into a sample holder, which is closed with a second polypropylene film of the same thickness.

	As explained above, biodiesel blends according to specification EN 590 typically can be analyzed with a calibration prepared for diesel fuel. Using the same calibration for the analysis of a biofuel sample with 12.5% oxygen would lead to a significant bias in the analysis result as can be seen in Figure 1.

					Figure 1: Calibration for S in mineral oil and biofuel with 12.5% oxygen.

	In many cases the oxygen content of the sample being analyzed is not known, and therefore the best option for an appropriate matrix correction is to determine the oxygen content of the sample. Using XRF, the direct determination of the oxygen content in fuel is not possible because the fluorescence energy of oxygen is very low and is not detectable.

	The second approach is to use the backscatter information of the sample to indirectly determine the content of oxygen. The so-called Compton scatter signal will decrease with increasing O-content. To be able to achieve this accurately, the energy of the Compton scatter signal should be at relatively low energy because otherwise volume effects will affect the determination.

	Since the EDXRF system used in our procedure is equipped with an X-ray tube with a thick binary Pd-Co alloy anode, in principle, there are two options to use backscatter information: one from the excitation using the characteristic radiation of Pd and the second using the characteristic radiation of Co.

	Because of the reasons listed above, the cobalt excitation of the EDXRF system can be used for this determination with good accuracy. Figure 2 shows the calibration of the measurements of samples with known oxygen concentration.

					Figure 2: Calibration for O in fuels.

	The figure demonstrates the sensitivity and accuracy of this indirect determination of the oxygen content. Based on this determination, the differentiation between materials based on FAMEs or on mineral oil is easily possible.

	For the accurate determination of the trace element content, the oxygen content determined with this procedure is used within a fundamental parameters program. Table II shows some examples.

	As previously listed, other trace elements can be determined simultaneously using EDXRF. Table III gives detection limits for additional trace elements.

	Although there is currently no specified test method using XRF for the determination of these elements in biofuels, the listed limits of detection (LODs) clearly demonstrate that at least a screening for the content of these elements is possible using the EDXRF, even at low levels.

	EDXRF clearly can be used for the analysis of S and other trace elements in biofuel. The determination of the oxygen content helps to compensate for matrix effects efficiently, and a priori knowledge of the oxygen content is not required.

		EN14538-2006: Fat and oil derivatives. Fatty acid methyl ester (FAME). Determination of Ca, K, Mg and Na content by optical emission spectral analysis with inductively coupled plasma (ICP-OES). 

		EN16294-2012: Petroleum products and fat and oil derivatives. Determination of phosphorus content in fatty acid methyl esters (FAME). Optical emission spectral analysis with inductively coupled plasma (ICP-OES). 

		International Organization for Standardization, ISO 13032, “Petroleum products – Determination of low concentration of sulfur in automotive fuels – Energy-dispersive X-ray fluorescence spectrometric method” (ISO, Geneva, Switzerland, 2012).

		ASTM International, D7220, “Standard Test Method for Sulfur in Automotive, Heating, and Jet Fuels by Monochromatic Energy Dispersive X-Ray Fluorescence Spectrometry” (ASTM International, West Conshohocken, Pennsylvania, 2017).

		European Committee for Standardization (CEN, French: Comité Européen de Normalisation) 

 is the senior product manager and product manager for XRF at Spectro Analytical Instruments GmbH in Kleve, Germany. Direct correspondence to: 

The EDXRF method described here is a quick and economical procedure to check samples for their oxygen content to ensure that the material used for diesel fuel blending is really FAME-based and not mineral-oil based.

EDXRF Analysis of Sulfur and Trace Element Content in FAMEs

In recent years, researchers have been making important developments to advance the effectiveness of spectroscopic techniques for biomedical uses ranging from the identification of infectious agents to measuring the edges of cancerous tumors. X-ray fluorescence (XRF) spectroscopy is among the techniques that can have useful medical applications. David R. Chettle, a professor in the Department of Physics and Astronomy at McMaster University in Hamilton, Ontario, Canada, uses XRF for the in vivo measurement of toxic elements in human subjects, with the goal of developing devices that can be used to investigate the possible health effects of toxin exposure. He recently spoke to us about his research.

Your research has focused on the use of XRF for noninvasive in vivo measurements of several elements in human subjects (1–4). In vivo bone lead measurements were first performed in the mid-1970s. What has been the main motivation for developing in vivo bone lead measurement techniques with XRF?

	Lead (Pb) is toxic to humans. Nearly all Pb in the adult human body is stored in bone, and Pb stays in bone a long time, with biological half lives ranging between 5 years and 30 years. Taken together, these facts mean that bone Pb reflects long-term Pb exposure and measurements of bone Pb provide insight into the long-term human metabolism of Pb. Several research groups have therefore used in vivo XRF bone Pb measurements to study the metabolism of Pb and to investigate the relationship between health effects in chronic Pb exposure, both in the workplace and in the general environment.

Can you briefly describe the major advances in the technique since the 1970s with respect to instrumentation, method, and minimum detectable level?

	The first in vivo bone Pb measurements, carried out by Ahlgren and Mattsson in the early 1970s (5), used the 122-keV γ-rays from 

Co to excite Pb K-shell X-rays. They used a 90° scattering geometry and a Ge(Li) detector. Later Somervaille and colleagues (6) used the 88-keV γ-rays from 

Co, together with a near 180° scattering geometry and, by that time, a HpGe detector. There were two main advantages: The energy was very close to the K-shell absorption edge (88 keV) in Pb, so the photoelectric cross section was larger than for 

Co. Secondly, the dominant Compton scattering feature was lower in energy than the Pb K X-rays for the 

Cd system, whereas the Compton scatter feature was higher in energy than the Pb K X-rays for the 

Co system. This meant that the continuum underlying the Pb X-ray signal was much smaller for 

	There have been subsequent developments of the 

Cd bone Pb system. The first-generation system used an annular 

Cd source with an inner diameter of 22 mm, with a 16-mm-diameter HpGe detector, so the source surrounded the detector. The second-generation system, dating from the early 1990s, inverted this geometry; a small spot source of 

Cd was mounted centrally in front of a larger (51-mm-diameter, 20-mm-thick) HpGe detector. This configuration has been most widely adopted in different research laboratories. The limitation on the second-generation system was the maximum pulse throughput rate. So the third-generation system replaced the single detector with an array of four smaller detectors in a “clover leaf” configuration. So, the total throughput could be four times larger and the detector resolution was also improved, resulting in a significant improvement in minimum detectable level. These successive improvements are summarized in Table I.

What other techniques have been used to perform these measurements? What are the advantages of the in vivo XRF approach compared with those methods?

	The main technique, apart from X-ray fluorescence, that is used for in vivo elemental analysis is neutron activation analysis. This technique either counts γ-rays emitted following the decay of a radioisotope induced by the neutron irradiation, or γ-rays emitted promptly (~10

 s) after neutron absorption. For most elements either X-ray fluorescence or neutron activation analysis is clearly preferable, so the two techniques should be seen as largely complementary. For example, for Pb, X-ray fluorescence is preferred. The Pb nuclei are unusually stable, because of their nuclear structure, so cross sections for neutron absorption are small compared to the photoelectric cross section in Pb. On the other hand, for aluminum, X-ray energies are so low that they are very unlikely to escape from the body, whereas the radioisotope 

Al, formed by neutron absorption in natural 

Al, emits a γ-ray of energy of 1.79 MeV, so neutron activation is to be preferred for Al. For some elements, either technique is viable. For cadmium, the photoelectric cross section at 60 keV is 49.0 cm

. The cross section for neutron absorption in the isotope 

Cd is 20,000 barn, which is equivalent to 931 cm

Cd is only 0.1222 abundant, so across the element Cd, the neutron absorption cross section is 114 cm

. There are other factors to be taken into account, but effectively the two techniques are comparable in the ability to detect Cd in the human body.

You used the in vivo XRF technique in a four-year study of the levels of bone strontium in individuals suffering from osteoporosis or osteopenia who have self-administered with strontium citrate supplements (2). Can you briefly describe the method used for collecting the data in this study?

I emits γ-rays at a little over 35 keV, but the predominant emission is of tellurium X-rays. In addition, the 

I was in the form of seeds used for brachytherapy, in which the iodine is adsorbed onto silver beads, so the source emits silver X-rays, as well as the direct emissions from 

I. The detector was a Si(Li) (16 mm diameter, 10 mm thick) and a backscatter (~180°) geometry was used. The Sr was measured in a finger and at the ankle, and each measurement took 30 min.

You and your group have also studied the bone lead concentration of a cross section of the population of Toronto, Ontario, Canada (3). What basic approach was used for those measurements? What was the age range of your test subjects, and what general results did you obtain?

	The bone Pb study was conducted in close collaboration with Health Canada and colleagues at other institutions. A third-generation system (four HpGe detectors, each 16 mm diameter, 10 mm thick) was used for bone Pb measurements at the tibia (shin) and at the calcaneus (heel). A questionnaire was administered and blood samples were also collected. One of the objectives for Health Canada was to test how much information could be collected in a 1-h visit by each participant; so each bone Pb measurement lasted 22 min. The youngest participant was one year old and the oldest 82. The participants were roughly equally distributed between the sexes and across the age range. The lowest bone Pb levels were in people in their 20s or early 30s, and there was little difference between women and men. The bone Pb levels were about a factor of two lower than they had been in the nearby city of Hamilton in studies conducted in the 1990s.

Gadolinium, which is used in magnetic resonance imaging (MRI) contrast reagents, was the focus of another of your in vivo XRF bone studies (4). Why was gadolinium an important element for monitoring by XRF in this study? How did your experimental setup and method for this study differ from that used in the bone lead research?

	Gadolinium is a very valuable contrast agent for MRI. It is administered in a bound form, and it is generally presumed that virtually all the Gd is eliminated from the body over a short period of time. However, evidence is growing that a small proportion of the Gd can be retained and the question is being raised as to whether this minor amount of residual Gd could in any way be harmful. This is the context in which the in vivo measurement of this residual Gd was developed. The system used is very nearly the same as that used for bone Pb, using the four-detector array. The spectral analysis obviously focuses on the Gd X-rays, rather than the Pb X-rays; we have also attempted to minimize the amount of tungsten associated with the source and the source collimator, because W K X-rays Compton scattered through about 180° raise the continuum level under the Gd X-rays and so worsen the detection limit.

What are the next steps in your research?

	In terms of improving detection limit for Pb or Gd, we are exploring the ultrahigh-throughput pulse processing electronics that have been successfully used with silicon drift detectors, primarily at synchrotrons. There are challenges with using these electronics with the relatively large HpGe detectors needed for the 40–100 keV energy range, so we are actively working with the companies concerned to improve detector resolution while still being able to take advantage of the dramatically large count rate throughput that can be achieved.

	In terms of applications, there is evidence that parameters that have been treated as fixed-such as the relationship between cumulative blood Pb and uptake of Pb into bone, or the half life governing the release of Pb from bone back into blood-may not be constant, but may depend on intensity of exposure or cumulative past exposure, or they may vary with a person’s age. This theory is controversial and does have potential public health consequences, so we wish to explore it further.

	There is some evidence that levels of Sr in western people may be less than is best for their bone health, but there is also evidence that supplementing with Sr may have harmful side effects. Our data showed that bone Sr levels increased by at least 10 times, and as much as 100 times in people who were self supplementing, and this increase occurred during a relatively short period of up to four years. It is possible that this is too much, in too short a time, and too late-that is, the supplementation was done after the damage to bone had already been sustained. We would like to investigate the hypothesis that much more modest supplementation, administered very gradually and earlier in life, could help to diminish the prevalence of osteoporosis. This would be a multistep, multidisciplinary, and highly collaborative program of research.

	We have only just begun to make Gd measurements. As well as trying to improve the detection limit, we need to find out how Gd retention varies with the type of contrast agent used, number of times administered, time since administration, and the varieties of health or disease of the people concerned.

		H. Moise, D.R. Chettle, and A. Pejovi´c-Mili´c, 

		M.L. Lord, F.E. McNeill, J.L. Gräfe, A.L. Galusha, P.J. Parsons, M.D. Noseworthy, L. Howard, and D.R. Chettle, 

		S. Behinaein, D.R. Chettle, M. Fisher, W.I. Manton, L. Marro, D.E.B. Fleming, N. Healey, M. Inskip, T.E. Arbuckle, and F.E. McNeill, 

		L. Ahlgren, K. Lidén, S. Mattsson, and S. Tejning, 

		L.J. Somervaille, D.R. Chettle, and M.C. Scott, 

David R. Chettle, a professor at McMaster University in Hamilton, Ontario, Canada, uses X-ray fluorescence spectroscopy for the in vivo measurement of toxic elements in human subjects, with the goal of developing devices that can be used to investigate the possible health effects of toxin exposure. He recently spoke to us about his research.

In Vivo XRF Analysis of Toxic Elements

Issue PDF

Click the title above to open the Spectroscopy November 2017 X-Ray Spectroscopy Methods & Applications Supplement, Vol 32 No s11, in an interactive PDF format.

Special Issues-11-01-2017

Advances in Stress Measurement with Portable X-Ray Diffraction: Uncertainties in Normal and Shear Stress Determination on Mild Steel Pipe

Recent Developments in Small-Angle X-Ray Scattering

EDXRF Analysis of Sulfur and Trace Element Content in FAMEs

In Vivo XRF Analysis of Toxic Elements

Vol 32 No s11 Spectroscopy November 2017 X-Ray Spectroscopy Methods and Applications Supplement PDF