Correction of Elastic Anisotropy in Williamson-Hall Plots by Diffraction Young’s Modulus and Direct Fitting Method

Abstract

It is known that the micro-strain in cold worked iron can be evaluated by the classical Williamson-Hall method using the three data of diffraction peaks: {110}, {211} and {220}. It is not clarified that the obtained value gives the true micro-strain or not. In addition, the accuracy of analysis is not so high because the diffraction strength from {220} plane is generally very weak. In this paper, three methods, i.e. classical Williamson-Hall method, Diffraction Young's Modulus Correction method and Direct Fitting method, ware attempted to reconfirm the reasonability of the classical Williamson-Hall method and to estimate accurate values of the parameter α and the micro-strain ε in the Williamson-Hall equation. The results obtained are as follows: 1) Elastic anisotropy in the Williamson-Hall plots is corrected using the parameter ω which relates to the values of diffraction Young's modulus. 2) The optimal values of parameter ω can be determined by the Direct Fitting method, which can be used to determine the timely orientation-dependent diffraction Young's modulus (E * hkl) in cold worked specimens. 3) It was confirmed that the classical Williamson-Hall method can generally give reliable values for the parameter α and the microstrain ε. 4) No large difference is found for the values of micro-strain ε from the three methods. 5) There is a clear linearity between the micro-strain ε and yield stress in cold rolled iron specimens.

Simultaneous Broadening Analysis of Multiple Bragg Edges Observed by Wavelength-resolved Neutron Transmission Imaging of Deformed Low-carbon Ferritic Steel

“…In this section, we show that the Bragg-edge broadenings of diffraction indices 110, 200, 211, 220, 310 and 321 are consistent with those of the corresponding diffraction peaks. Then, we also show that the corrected classical Williamson-Hall (ccWH) plots 25) of the Bragg-edge data are proportional to the magnitude of the scattering vector corrected by the ratio of the diffraction Young's modulus. Finally, we discuss the validity of parameters obtained from the ccWH plot, by comparing with the results of previous diffraction studies.…”

“…22) On the other hand, Takaki et al have reported that the corrected classical WH method, the ccWH method, in which the crystal elastic anisotropy is corrected by the ratio of diffraction Young's modulus, is usable for dislocation density evaluation of a low-carbon ferritic steel. [23][24][25][26] We applied the ccWH method in our feasibility study because the sample condition was similar to that of Takaki's study, although the goal in the future is to apply the mWH/mWA method. In any case, various evaluation methods for micro-strain, dislocation density/arrangement/ character and crystallite size, using the diffraction peak line-broadening/profile analysis, have been proposed and applied in materials science and engineering.…”

“…Thus, a mWH plot has been proposed as a correction using the dislocation contrast factor. 21) On the other hand, another approach, the corrected cWH (ccWH) plot, was proposed by Takaki et al 25) They have suggested that this method was usable for dislocation density evaluation of a deformed low-carbon ferritic steel. 23,24,26) For a low-carbon ferritic steel with a grain size of severaltens of micrometres, the calculation procedure of dislocation density from the ccWH plot has been investigated by experimental studies using both TEM (transmission electron microscope) and X-ray diffraction.…”

“…For these reasons, we applied Takaki Here, ω hkl is the ratio of the diffraction Young's modulus. 25) Table 1 shows theoretical values of ω hkl of each diffraction index. 25) These were calculated by the Krӧner model 32) and the elastic stiffness coefficients.…”

“…25) Table 1 shows theoretical values of ω hkl of each diffraction index. 25) These were calculated by the Krӧner model 32) and the elastic stiffness coefficients. 33) Takaki et al tested the ccWH plots using Eq.…”