Diffraction Line-Profile Shape by Synchrotron and Laboratory X-ray Sources

INTRODUCTION

Precise knowledge of diffraction line-profile shape is of utmost importance in x-ray powder diffraction, especially in line-broadening analysis, Rietveld refinement, and other whole-powderpattern-fitting programs. In this regard, laboratory x-ray sources were researched extensively in the past, but synchrotron radiation remains inadequately characterized, despite its increasingly frequent recent use. Most of the line-profile models rely on a milestone study of Caglioti, Paoletti, and Ricci' that was developed for neutron diffraction and later adapted in the synchrotron case.2 Basic studies of synchrotron powder diffraction were undertaken by Cox et aZ.3w,h o also gave a comprehensive review4 of the field. Synchrotron radiation is inherently advantageous to laboratory sources for line-broadeningstudies for many reasons: naturally high beam collimation provides a superior resolution, the wavelength of a monochromatic beam can be easily tuned, and line shape is generally simpler and controlled to our preference. Most important, however, is the high resolution, that is, the narrow instrumental line profile implies a high sensitivity to the small physical broadening.

For the laboratory measurements, we used a horizontal goniometer in divergent Bragg- Brentano flat-plate geometry with both incident and diffracted Soller slits to minimize axial beam divergence, 2 mm divergent and 0.2 mm receiving slits. Cu Kq, radiation was scanned with a cooled germanium solid-state detector. Synchrotron-radiation measurements were performed on the X3B 1 beamline at the National Synchrotron Light Source (NSLS), Brookhaven National Laboratory. The triple-axis parallel geometry included Si channel- 111 -cut monochromator, flat specimen, Ge 111 -cut analyzer crystal, and proportional detector (Figure 1). Typical NIST SRM LaB, diffraction lineprofiles are presented in Figure 2. At this diEaction angle, synchrotron radiation gives four times smaller line width and 2.5 times larger peak-to-background ratio, despite twice as large a background count. Both line profiles are closely approximated with the Voigt function or its pseudo-Voigt and Pearson VII approximations.4 However, it is still a matter of debate5 why the line profiles tend to be almost pure Lorentz functions at high angles, the same effect that is observed for laboratory sources. Therefore, it is desirable to study the overall effect of geometrical aberrations on the difiaction-line shape.

Figure 1: Schematic view of X3Bl NSLS beamline in the (vertical) equatorial plane. M: monochromator crystal; ES: equatorial slit; S: specimen; A: analyzer crystal; D: detector.

Figure 2 Diffraction-line profiles of MST SRM LaE16 obtained at laboratory and synchrotron (NSLS) x-ray sources. P/B denotes the peak-to-background ratio.

SYNCHROTRON DIFFRACTION-LINE SHAPE

The main equatorial instrumental factors affecting the diffraction-line profile and/or position are the following:
(i) Source height (vertical angular distribution of the polychromatic beam) is approximated with the Gauss function at the bending magnet. It depends on the storage-ring electron (positron) relativistic factor y, the photon energy c, and the critical photon energy ec (5.04 keV at NSLS):

where the vertical (equatorial, for it is in the scattering plane) divergence

Here, v, E, and m, are the electron (positron) speed, energy, and rest mass, respectively, and c is the speed of light.
(ii) Equatorial slit width

(iii) Normalized Darwin Bragg-reflection shape6 of the monochromator and analyzer (perfect) crystals (rocking curve):

Here, s defines the region for a perfect reflection (without absorption) from a crystal.
(iv) Specimen effects that cause important aberrations in laboratory divergent geometry, such as transparency, flat surface, and its missetting, are negligible in synchrotron parallel geometry with the analyzer crystal.
The most important axial aberration is a divergence, which sometimes causes severe asymmetry at low angles. The effect on powder lime shapes was considered by van Laar and Yelon7 and recently applied to high-resolution synchrotron diffractometers by Finger, Cox, and Jephcoat.
The total diiaction-line profile results from a convolution of all the contributions, which has to be accomplished numerically. However, for most purposes, a simple estimation of line widths as a fimction of diffraction angle may suffice. Wavelength dispersion follows from the Bragg law:

Here, the shape of perfect Bragg reflection is approximated with the Gauss function. They depend on the structure factor, polarization, absorption, and temperature.

To recognize the relative importance of various contributions, we estimate the angular resolution at the X3B 1 NSLS beamline with 8 keV photon energy, that is, the approximate Cu Ka wavelength: