Proposal of Simplified Modified Williamson-Hall Equation

Abstract

Williamson-Hall (WH) plots are characterized by irregular arrangement of data due to the elastic anisotropy in each {hkl} plane. In order to correct the effect of elastic anisotropy, Ungár developed a unique methodology using the contrast factor C, so called the modified Williamson-Hall (mWH) method. When X-ray with the wave length λ was used for diffraction analysis and diffraction angle θ and integral breadth β was obtained in each diffraction peak, the following mWH equation is constructed as functions of the parameter K (=2sinθ/λ) and ΔK (=βcosθ/λ).   

Here, the parameter α is dependent on the crystallite size. The parameter φ and O are constants but the O-value is much smaller than the φ-value. In the mWH plots in 60% cold rolled ferrite (Fe-0.0056%C), ultra low carbon martensite (Fe-18%Ni) and 20% cold rolled austenite (SUS316L), it was confirmed that the value of OK²C is negligibly small. As a result, the following simplified equation is applicable for the analysis by mWH method.   

1. Introduction

In general, radiation diffraction analysis is applied for highly dislocated materials to evaluate dislocation characterization. Williamson proposed a basic approach to evaluate the micro-strain ε which is produced by dislocations. When X-ray with the wave length λ was used for diffraction analysis and diffraction angle θ and integral breadth β was obtained in each diffraction peak, the following Williamson-Hall (WH) equation is constructed as functions of the parameter K (=2sinθ/λ) and ΔK (=βcosθ/λ).¹⁾ In this study, full width at half maximum (FWHM) is used instead of integral breadth as with some researchers.²⁾   

$$\Delta K=\alpha +\epsilon K$$(1)

Here, the parameter α is dependent on the crystallite size. Various values are obtained in several diffraction peaks, thus the values of parameter α and ε are determined in the relation between K and ΔK. However, linear relation is not obtained usually due to the elastic anisotropy in each {hkl} crystal plane. Such an elastic anisotropy is usually discussed in connection with the orientation parameter Γ which is expresses by the following equation as a function of Miller index {hkl}.   

$$\begin{array}{l}\Gamma =\left(h^2{k}^2+k^2{l}^2+l^2{h}^2\right)/{\left(h^2+k^2+l^2\right)}^2\\ \left(0\text{≦}\Gamma \text{≦}1/3\right)\end{array}$$(2)

Ungár proposed a method to correct the elastic anisotropy in WH plots by the contrast factor C which is given by the following equation,³⁾   

$$C=C_{\text{h}00}\left(1-q\Gamma \right)$$(3)

where C_h00 is the contrast factor in the crystal plane {h00} and q is a constant, which depend on the screw component of dislocation S (0≦S≦1) as follows.   

$$C_{\text{h}00}=\left(1-\text{S}\right)C_{\text{h}00}^E+S{C}_{\text{h}00}^S$$(4)

  

$$q=\left(1-S\right)q^E+S{q}^S$$(5)

$C_{\text{h}00}^E$ and $C_{\text{h}00}^S$ are the contrast factor of edge dislocation and screw dislocation, q^E and q^S are the q-value of edge dislocation and screw dislocation, respectively. They are material constants which depend on the elastic constants: $C_{\text{h}00}^E$=0.259, $C_{\text{h}00}^S$=0.295, q^E=1.30, q^S=2.67 in bcc iron and $C_{\text{h}00}^E$=$C_{\text{h}00}^S$=0.291, q^E=1.61, q^S=2.32 in fcc iron.⁴⁾ Therefore, once a certain value is given for the parameter S, the contrast factor C in Eq. (3) is identically determined.

Ungár reconstructed the WH equation as follows applying the contrast factor C, so called modified Williamson-Hall (mWH) equation and the plots of ΔK vs. $K\sqrt{C}$ are called mWH plots.   

$$\Delta K=\alpha +\phi K\sqrt{C}+O{K}^{2}C$$(6)

Here, φ and O are the coefficients depending on the microstructural factors. In the mWH method, the C-value is determined to minimize the fitting error of mWH plots against Eq. (6) and then the value of parameter S is introduced from the determined C-value.

The optimal C-value is obtained in the plots of ΔK vs. $K\sqrt{C}$ but the analyzing accuracy is greatly heightened adding the value of parameter α. In order to correct the elastic anisotropy in WH plots, author has developed a new method, so called the direct-fitting (DF) method.⁵⁾ Applying the DF method to WH plots, the elastic anisotropy of each crystal plane has been accurately corrected and linear relation is realized. Thus, the α-value is precisely determined from the fitting line.

In this paper, the α-value obtained by the DF method was applied to the mWH method using Eq. (6) and then the coefficient φ and O were obtained in different type of steels; 60% cold rolled ferrite (Fe-0.0056%C), ultra low carbon martensite (Fe-18%Ni) and 20% cold rolled austenite (SUS316L). If the parameter O is much smaller than φ in every steel, the following simplified equation may be applicable in the analysis by mWH method.   

$$\Delta K=\alpha +\phi K\sqrt{C}$$(7)

2. Determination of α-value by the Direct-fitting Method

Figure 1 shows the WH plots in 60% cold rolled ferrite, as an example. The data of {220} is not plotted here because the diffraction intensity was very weak. The original plots (solid circles) are characterized by irregular arrangement of data due to the elastic anisotropy in each crystal plane. In the DF method, such an elastic anisotropy is corrected by the parameter ω which is identified by the ratio of diffraction Young’s modulus; ω=$E_{\text{hkl}}^{*}$/E₀.⁵⁾ Here, $E_{\text{hkl}}^{*}$ and E₀ are the diffraction Young’s modulus in each {hkl} plane and the standard Young’s modulus respectively. The value of ω is identically determined to minimize the fitting error of ΔK vs. (K/ω) against linear relation. We confirmed that the ω value obtained by DF method is almost the same as that calculated by Kröner model.⁵⁾ This indicates the reasonability of DF method on the correction of anisotropy in WH plot. The corrected data are plotted by open squares. In the case of bcc iron, the $E_{\text{hkl}}^{*}$-value of {110} and {211} plane is very close to E₀ so that the effect of correction is small, while it is found that the elastic anisotropy in the other crystal planes has been accurately corrected by the DF method. The slope gives the ε-value in Eq. (1) and the α-value is given by the star mark (0.00158 nm⁻¹). Authors have confirmed that the α-value can be accurately determined in austenite (fcc iron) by the DF method.

Fig. 1.

Original Williamson-Hall plots and Williamson-Hall plots corrected by direct-fitting method in 60% cold rolled ferrite.

3. Application of the Parameter α to the Modified Williamson-Hall Equation

As mentioned above, the reliability of parameter α is so high that it can be fixed in Eq. (6). On the other hand, the relation between ΔK and $K\sqrt{C}$ is changeable depending on the value of parameter S which governs the C-value. Therefore, the S-value was determined to give the best fitting against Eq. (6). Firstly, the result in 60% cold rolled ferrite is shown in Fig. 2. In this case, the best fitting was realized at S=0.260. It is found that the O-value is much smaller than φ-value.

Fig. 2.

Modified Williamson-Hall plots in 60% cold rolled ferrite, which were determined to give the maximum fitting index.

Figure 3 shows the cases of ultra low carbon martensite (a) and 20% cold rolled austenite (b). The best fitting was realized at S=0.850 in the former and S=0.486 in the latter. It is also confirmed that the O-value is much smaller than φ-value. These results indicate that the right third term in Eq. (6) can be neglected and the simplified equation; Eq. (7) can be applicable for the mWH method.

Fig. 3.

Modified Williamson-Hall plots in as-quenched ultra low carbon martensite (a) and austenite (b), which were determined to give the maximum fitting index.

Finally, Fig. 4 shows the comparison as to the values of parameter φ, which were obtained by the quadratic fitting against Eq. (6) (φ2) and the linear fitting against Eq. (7) (φ1). Regarding ferrite and austenite, several data are added to check the correlation between φ1 and φ2. It is found that almost same values are obtained in both fitting methods (φ1=φ2). It should be noted here that negative values are obtained for the parameter O in the case; φ2>φ1, although the O-value should be positive. In such a case, the φ-value tends to be overestimated. As a result, it is recommended to apply the simplified Eq. (7) is used for the analysis by mWH method. Authors have confirmed that similar result is obtained for the other metals; nickel, copper and aluminum.

Fig. 4.

Comparison of the values of parameter φ, which were obtained by linear fitting φ1 and quadratic fitting φ2.

Acknowledgement

This research was performed by the support of JSPS, KAKENHI Grant number JP15H05768. This work was also partially supported by the Research Society for quantum-beam analysis of microstructures and properties of steels.

References

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