The NIST Vacuum Double-Crystal Spectrometer: A Tool for SI-Traceable Measurement of X-Ray Emission Spectra.

The NIST Vacuum Double-Crystal Spectrometer (VDCS) has been modernized and is now capable of recording reference-free wavelength-dispersive spectra in the 2 keV to 12 keV x-ray energy range. The VDCS employs crystals in which the lattice spacings are traceable to the definition of the meter through x-ray optical interferometry with a relative uncertainty ﹤10-8. VDCS wavelength determination relies upon precision angle difference measurements for which the encoders of the rotation stages have been calibrated using the circle closure method for accurate, absolute angle measurement. The new vacuum-compatible area detector allows quantification of the aberration functions contributing to the observed line shape and in situ alignment of the crystal optics. This latter procedure is augmented with the use of a thin lamella as the first crystal. With these new techniques, x-ray spectra are registered with the VDCS on an absolute energy scale with a relative uncertainty of 10-6.

The Certification of Standard Reference Material 1979: Powder Diffraction Line Profile Standard for Crystallite Size Analysis.

This rather long-standing project has resulted in a National Institute of Standards and Technology (NIST) Standard Reference Material (SRM) for the analysis of crystallite size from a consideration of powder diffraction line profile broadening. It consists of two zinc oxide powders, one with a crystallite size distribution centered at approximately 15 nm, and a second centered at about 60 nm. These materials display the effects of stacking faults that broaden specific hkl reflections and a slight amount of microstrain broadening. Certification data were collected on the high-resolution powder diffractometer located at beamline 11-BM of the Advanced Photon Source, and on a NIST-built laboratory diffractometer equipped with a Johansson incident beam monochromator and position sensitive detector. Fourier transforms were extracted from the raw data using a modified, two-step profile fitting procedure that addressed the issue of accurate background determination. The mean column lengths, 〈L〉area and 〈L〉vol, were then computed from the Fourier transforms of the specimen contribution for each reflection. Data were also analyzed with fundamental parameters approach refinements using broadening models to yield 〈L〉area and 〈L〉vol values. These values were consistent with the model-independent Fourier transform results; however, small discrepancies were noted for the 〈L〉area values from both machines and both crystallite size ranges. The fundamental parameters approach fits to the laboratory data yielded the certified lattice parameters.

Certification of Standard Reference Material 660c for powder diffraction.

The National Institute of Standards and Technology (NIST) certifies a suite of Standard Reference Materials (SRMs) to evaluate specific aspects of instrument performance of both X-ray and neutron powder diffractometers. This report describes SRM 660c, the fourth generation of this powder diffraction SRM, which is used primarily for calibrating powder diffractometers with respect to line position and line shape for the determination of the instrument profile function (IPF). It is certified with respect to lattice parameter and consists of approximately 6 g of lanthanum hexaboride (LaB6) powder. So that this SRM would be applicable for the neutron diffraction community, the powder was prepared from an isotopically enriched 11B precursor material. The microstructure of the LaB6 powder was engineered specifically to yield a crystallite size above that where size broadening is typically observed and to minimize the crystallographic defects that lead to strain broadening. A NIST-built diffractometer, incorporating many advanced design features, was used to certify the lattice parameter of the LaB6 powder. Both Type A, statistical, and Type B, systematic, uncertainties have been assigned to yield a certified value for the lattice parameter at 22.5 °C of a = 0.415 682 6 ± 0.000 008 nm (95% confidence).

Certification of SRM 640f line position and line shape standard for powder diffraction.

The National Institute of Standards and Technology (NIST) certifies a suite of Standard Reference Materials (SRMs) to be used to evaluate specific aspects of the instrument performance of both X-ray and neutron powder diffractometers. This report describes SRM 640f, the seventh generation of this powder diffraction SRM, which is designed to be used primarily for calibrating powder diffractometers with respect to line position; it also can be used for the determination of the instrument profile function. It is certified with respect to the lattice parameter and consists of approximately 7.5 g of silicon powder prepared to minimize line broadening. A NIST-built diffractometer, incorporating many advanced design features, was used to certify the lattice parameter of the Si powder. Both statistical and systematic uncertainties have been assigned to yield a certified value for the lattice parameter at 22.5 °C of a = 0.5431144 ± 0.000008 nm.

The Molybdenum K-shell X-ray Emission Spectrum.

We present newly measured spectra of the X-ray emission of a molybdenum metal anode subject to electron bombardment, using a very high dispersion silicon double-crystal spectrometer. The measurement includes the dipole-allowed KL, KM, and KN emission lines, based on an energy scale traceable to the Système International (SI) definition of the meter with a systematic uncertainty below ΔE/E = 10-6. The data are presented as parametrized multi-Lorentzian fits to the results, and as supplementary data with the complete spectrum of each line group, corrected for instrumental effects. The MoKL 3 (Kα 1) line energy was in complete statistical agreement with published measurements, and it showed no asymmetry. Other lines showed varying discrepancies with the literature which lie outside the bounds of probable experimental errors.

Certification of Standard Reference Material 1879b Respirable Cristobalite.

The National Institute of Standards and Technology (NIST) certifies a suite of Standard Reference Materials (SRMs) to address specific aspects of the performance of X-ray powder diffraction instruments. This report describes SRM 1879b, the third generation of this powder diffraction SRM. SRM 1879b is intended for use in the preparation of calibration standards for the quantitative analyses of cristobalite by X-ray powder diffraction in accordance with National Institute for Occupational Safety and Health (NIOSH) Analytical Method 7500, or equivalent. A unit of SRM 1879b consists of approximately 5 g of cristobalite powder bottled in an argon atmosphere. It is certified with respect to crystalline phase purity, or amorphous phase content, and lattice parameter. Neutron powder diffraction, both time-of-flight and constant-wavelength, was used to certify the phase purity using SRM 676a as an internal standard. A NIST-built diffractometer, incorporating many advanced design features was used for certification measurements for lattice parameters.

High-precision measurement of the X-ray Cu Kα spectrum.

The structure of the X-ray emission lines of the Cu Kα complex has been remeasured on a newly commissioned instrument, in a manner directly traceable to the Système Internationale definition of the meter. In this measurement, the region from 8000 eV to 8100 eV has been covered with a highly precise angular scale, and well-defined system efficiency, providing accurate wavelengths and relative intensities. This measurement updates the standard multi-Lorentzian-fit parameters from Härtwig, Hölzer, et al., and is in modest disagreement with their results for the wavelength of the Kα1 line when compared via quadratic fitting of the peak top; the intensity ratio of Kα1 to Kα2 agrees within the combined error bounds. However, the position of the fitted top of Kα1 is very sensitive to the fit parameters, so it is not believed to be a robust value to quote without further qualification. We also provide accurate intensity and wavelength information for the so-called Kα3,4 "satellite" complex. Supplementary data is provided which gives the entire shape of the spectrum in this region, allowing it to be used directly in cases where simplified, multi-Lorentzian fits to it are not sufficiently accurate.

The Lattice Spacing Variability of Intrinsic Float-Zone Silicon.

Precision lattice spacing comparison measurements at the National Institute of Standards and Technology (NIST) provide traceability of X-ray wavelength and powder diffraction standards to the international system of units (SI). Here, we both summarize and document key measurements from the last two decades on six lots of intrinsic float-zone silicon, including unpublished results and recent internal-consistency checks. The comparison measurements link the unknown lattice spacing of a test crystal to a standard crystal for which the lattice spacing has been accurately determined by X-ray/optical interferometry in units traceable to the definition of the meter. The crystal that serves as the standard in all the comparisons is WASO 04, for which the lattice spacing is known with a relative uncertainty of 5 × 10-9. Individual lattice spacing comparison results have typical uncertainties of 1 ×10-8; taking material variability into account, measurements yield relative uncertainties for the test materials of a few tens of nanometers. It is observed that in the case of nearly perfect modern intrinsic float-zone silicon, the variability of the lattice spacing is sufficiently small that for most diffraction applications, a recommended reference value may be used.

Characterization of a self-calibrating, high-precision, stacked-stage, vertical dual-axis goniometer.

We present details on the alignment and calibration of a goniometer assembly consisting two stacked, optically encoded, vertical axis rotation stages. A technique for its calibration is presented that utilizes a stable, uncalibrated, third stage to position a mirror in conjunction with a nulling autocollimator. Such a system provides a self-calibrating set of angular stages with absolute accuracy of ±0.1 second of plane angle (k=2 expanded uncertainty) around the full circle, suitable for laboratory application. This calibration technique permits in situ, absolute angular calibration of an operational goniometer assembly that is requisite for fully traceable angle measurement, as the installation of the encoder is known to change its performance from the angular calibration data provided by the manufacturer.

An Implementation of the Fundamental Parameters Approach for Analysis of X-ray Powder Diffraction Line Profiles.

This work presents an open implementation of the Fundamental Parameters Approach (FPA) models for analysis of X-ray powder diffraction line profiles. The original literature describing these models was examined and code was developed to allow for their use within a Python based least squares refinement algorithm. The NIST interest in the FPA method is specific to its ability to account for the optical aberrations of the powder diffraction experiment allowing for an accurate assessment of lattice parameter values. Lattice parameters are one of the primary certified measurands of NIST Standard Reference Materials (SRMs) for powder diffraction. Lattice parameter values obtained from analysis of data from SRMs 640e and 660c using both the NIST FPA Python code and the proprietary, commercial code Topas, that constitutes the only other actively supported, complete implementation of FPA models within a least-squares data analysis environment, agreed to within 2 fm. This level of agreement demonstrates that both the NIST code and Topas constitute an accurate implementation of published FPA models.

The Optics and Alignment of the Divergent Beam Laboratory X-ray Powder Diffractometer and its Calibration Using NIST Standard Reference Materials.

The laboratory X-ray powder diffractometer is one of the primary analytical tools in materials science. It is applicable to nearly any crystalline material, and with advanced data analysis methods, it can provide a wealth of information concerning sample character. Data from these machines, however, are beset by a complex aberration function that can be addressed through calibration with the use of NIST Standard Reference Materials (SRMs). Laboratory diffractometers can be set up in a range of optical geometries; considered herein are those of Bragg-Brentano divergent beam configuration using both incident and diffracted beam monochromators. We review the origin of the various aberrations affecting instruments of this geometry and the methods developed at NIST to align these machines in a first principles context. Data analysis methods are considered as being in two distinct categories: those that use empirical methods to parameterize the nature of the data for subsequent analysis, and those that use model functions to link the observation directly to a specific aspect of the experiment. We consider a multifaceted approach to instrument calibration using both the empirical and model based data analysis methods. The particular benefits of the fundamental parameters approach are reviewed.

The NIST silicon lattice comparator upgrade.

The NIST silicon lattice comparator has been in service, in various forms, since the 1970s. It is capable of measuring the difference in lattice spacing between specimens of high-quality float-zone silicon to Δd/d ≈ 6 × 10-9. It has recently undergone a thorough update of its control systems and mechanics. These upgrades result in the ability to collect data with improved stability, less settling time of the instrument, and less operator intervention.

Polarization effects of X-ray monochromators modeled using dynamical scattering theory.

The difference in the diffracted intensity of the σ- and π-polarized components of an X-ray beam in powder diffraction has generally been treated according to equations based on dipole scattering, also known as kinematic X-ray scattering. Although this treatment is correct for powders and post-sample analyzers known to be of high mosaicity, it does not apply to systems configured with nearly perfect crystal incident-beam monochromators. Equations are presented for the polarization effect, based on dynamical diffraction theory applied to the monochromator crystal. The intensity of the π component relative to the σ component then becomes approximately proportional to |cos 2θm| rather than to cos22θm, where θm is the Bragg diffraction angle of the monochromator crystal. This changes the predicted intensities of X-ray powder diffraction patterns produced on instruments with incident-beam monochromators, especially in the regions far from 2θ = 90° in the powder pattern. Experimental data, based on well known standard reference materials, are presented, confirming that the dynamical polarization correction is required when a Ge 111 incident-beam monochromator is used. The dynamical correction is absent as an option in the Rietveld analysis codes with which the authors are familiar.

Model-independent extraction of the shapes and Fourier transforms from patterns of partially overlapped peaks with extended tails.

This work presents a technique for extracting the detailed shape of peaks with extended, overlapping tails in an X-ray powder diffraction pattern. The application discussed here concerns crystallite size broadening, though the technique can be applied to spectra of any origin and without regard to how the profiles are to be subsequently analyzed. Historically, the extraction of profile shapes has been difficult due to the complexity of determining the background under the peak, resulting in an offset of the low-frequency components of the Fourier transform of the peak known as the `hook' problem. The use of a carefully considered statistical weighting function in a non-linear least-squares fit, followed by summing the residuals from such a fit with the fit itself, allows one to extract the full shape of an isolated peak, without contributions from either the background or adjacent peaks. The extracted shape, consisting of the fit function recombined with the residuals, is not dependent on any specific shape model. The application of this to analysis of microstructure is performed independently of global parametric models, which would reduce the number of refined parameters; therefore the technique requires high-quality data to produce results of interest. The effectiveness of the technique is demonstrated by extraction of Fourier transforms of peaks from two sets of size-broadened materials with two differing pieces of equipment.

The optics of focusing bent-crystal monochromators on X-ray powder diffractometers with application to lattice parameter determination and microstructure analysis.

The use of an Incident Beam Monochromator (IBM) in an X-ray powder diffractometer modifies both the shape of the spectrum from the X-ray source, and the relation between the apparent diffracted angle and the actual wavelength of the X-ray. For high-accuracy work, the traditional assumption of a narrow line of typically Gaussian shape does not suffice. Both the shape of the tails of peaks, and their width, can be described by a new model which couples the dispersion from the optic to the dispersion from the powder sample, and to its transport to a detector. This work presents such a model, and demonstrates that it produces excellent fits via the Fundamental Parameter Approach, and requires few free parameters to achieve this. Further, the parameters used are directly relatable to physical characteristics of the diffractometer optics. This agreement is critical for the evaluation of high-precision lattice parameters and crystal microstructural parameters by powder diffraction.

Bayesian Inference of Nanoparticle-Broadened X-Ray Line Profiles.

A single-step, self-contained method for determining the crystallite-size distribution and shape from experimental x-ray line profile data is presented. It is shown that the crystallite-size distribution can be determined without invoking a functional form for the size distribution, determining instead the size distribution with the least assumptions by applying the Bayesian/MaxEnt method. The Bayesian/MaxEnt method is tested using both simulated and experimental CeO2 data, the results comparing favourably with experimental CeO2 data from TEM measurements.

Addressing the amorphous content issue in quantitative phase analysis: the certification of NIST standard reference material 676a.

A non-diffracting surface layer exists at any boundary of a crystal and can comprise a mass fraction of several percent in a finely divided solid. This has led to the long-standing issue of amorphous content in standards for quantitative phase analysis (QPA). NIST standard reference material (SRM) 676a is a corundum (α-Al(2)O(3)) powder, certified with respect to phase purity for use as an internal standard in powder diffraction QPA. The amorphous content of SRM 676a is determined by comparing diffraction data from mixtures with samples of silicon powders that were engineered to vary their specific surface area. Under the (supported) assumption that the thickness of an amorphous surface layer on Si was invariant, this provided a method to control the crystalline/amorphous ratio of the silicon components of 50/50 weight mixtures of SRM 676a with silicon. Powder diffraction experiments utilizing neutron time-of-flight and 25 keV and 67 keV X-ray energies quantified the crystalline phase fractions from a series of specimens. Results from Rietveld analyses, which included a model for extinction effects in the silicon, of these data were extrapolated to the limit of zero amorphous content of the Si powder. The certified phase purity of SRM 676a is 99.02% ± 1.11% (95% confidence interval). This novel certification method permits quantification of amorphous content for any sample of interest, by spiking with SRM 676a.