Estimation of the coherently scattering domain size in alloys from neutron diffraction data

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Abstract

An analysis of the diffraction spectra of the Fe74Al26 alloy was conducted using a high-resolution neutron diffractometer to determine the size distribution of structurally ordered clusters dispersed within the structurally disordered matrix of the alloy. The Scherrer method was generalized for this purpose, based on the analysis of diffraction peak profiles, determining peak widths at heights of 1/5 and 4/5 of the maximum, and assuming the validity of the gamma distribution for cluster sizes (Pielaszek method). A comparison of results obtained using the Scherrer, Williamson–Hall, and Pielaszek methods was carried out, demonstrating good agreement between them. An algorithm for calculating the log-normal distribution function of cluster/particle sizes is proposed. The experimental data were obtained using a time-of-flight neutron diffractometer, and the analysis was performed for two variants of variable scanning: in crystallographic (direct) (d-scale) and reciprocal (H-scale) spaces, with estimates of possible systematic errors. It was concluded that the determined average sizes possess the necessary degree of stability, meaning they weakly depend on the applied variable scanning and the total number of experimental data points.

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About the authors

B. Yerzhanov

Joint Institute for Nuclear Research; Kazan (Volga Region) Federal University

Author for correspondence.
Email: bekarys@jinr.ru
Russian Federation, Dubna, 141980; Kazan, 420008

I. A. Bobrikov

Centre for Cooperative Research on Alternative Energies (CIC energiGUNE), Basque Research and Technology Alliance (BRTA)

Email: ibobrikov@cicenergigune.com
Spain, Alava Technology Park, Albert Einstein 48, Vitoria-Gasteiz 01510

A. M. Balagurov

Joint Institute for Nuclear Research; Lomonosov Moscow State University

Email: bala@nf.jinr.ru
Russian Federation, Dubna, 141980; Moscow, 119991

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Supplementary files

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2. Fig. 1. Neutron diffraction pattern of the Fe74Al26 alloy in the d scale — the interplanar distance (a) and in the H = 1/d scale — the length of the vector in the reciprocal lattice (b). The experimental points, calculated profile, difference curve and Miller indices of some peaks are shown. Vertical dashes are the positions of the diffraction peaks for the D03 structural phase (cubic, Fmm, a = 5.795 Å).

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3. Fig. 2. Dependencies corresponding to formulas (2) for two variants of the scanning variable d (a) and H (b) for TOF diffractometers on sources with a short (HRPD, ISIS) and long (HRFD, IBR-2) pulse, as well as for a diffractometer with a monochromatic beam (HRPT, SINQ) on a stationary neutron source.

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4. Fig. 3. The sizes of the RCS of the D03 phase clusters in the Fe74Al26 alloy, obtained using the Scherrer formulas (3) from the widths of the first few superstructural peaks in the diffraction patterns presented in the d (a) and H (b) scales. The errors of the points are statistical, the horizontal lines correspond to the average RCS size obtained using the least squares method.

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5. Fig. 4. Williamson-Hall plots for the first main and superstructural (odd Miller indices) peaks from the Fe74Al26 alloy, presented in the d (a) and H (b) scales. Solid lines are drawn using the least squares method, the dashed line corresponds to the HRFD resolution function. The d-scale plot additionally shows the widths of the superstructural peaks measured after heating and cooling the sample with = 145 nm. The widths of the main peaks remained virtually unchanged.

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6. Fig. 5. Comparison of lognormal (L(R), dashed line) and gamma (G(R), solid line) distributions calculated from the characteristics of peak 111 from the Fe74Al26 alloy after casting (a) and slow heating and cooling (b) and having the same first and second moments.

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7. Fig. 6. Lognormal distributions calculated by the Pelaszek method for the characteristics of several superstructural peaks of the Fe74Al26 alloy. Distributions are normalized by amplitude. Dashed lines show the average volume sizes of the RCS.

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8. Fig. 7. Gamma distributions calculated by the Pelaszek method for the characteristics of several superstructural peaks of the Fe74Al26 alloy, for the initial state (solid lines) and after heating and cooling of the alloy (dashed lines). Distributions are normalized by area. The vertical lines show the average volume sizes of the RCS.

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