Characterization of Deformation by Cold Rolling in Ferritic Steel Containing Cu Particles Using Neutron Transmission Analysis

Abstract

Neutron transmission spectra of Fe-2 mass% Cu alloy (Cu steel) were measured to characterize the changes of crystallographic texture of ferrite grains and nanostructure of dispersed Cu particles with cold rolling. Bragg edges appearing in the neutron attenuation coefficient of as-aged Cu steel show a sawteeth pattern corresponding to random texture. With increasing equivalent strain, the 110 Bragg edge changes to a peak and the 200 Bragg edge becomes sharp. These changes indicate the rotation of {110} planes toward a tilt angle of 32° to the rolling plane and the increase in the fraction of the {100} planes in the rolling plane. This can be explained by the evolution of <111>//ND, <322>//ND, and <100>//ND preferred orientations with the cold rolling, where ND denotes the normal direction. In the wavelength range longer than 0.4 nm, the neutron attenuation coefficient increases due to a small-angle neutron scattering (SANS) contribution from dispersed Cu particles in the matrix. Comparing the experimental results with simulation, the change in the SANS contribution indicates that the dispersed Cu particles are elongated with the cold rolling. These results demonstrate that the neutron transmission analysis is useful for microstructural characterization of steels and the sequential change of the microstructures.

1. Introduction

Neutron scattering techniques are one of the fundamental experimental means to characterize microstructures in steels because of a large gauge volume and high penetration power. Neutron diffraction is widely used to analyze crystallographic features, e.g., phase fractions, lattice strain, crystallographic textures, and dislocation density.^1,2,3,4,5) In contrast, small-angle neutron scattering (SANS) can provide nanostructural information about precipitates and inclusions.^{6,7,8,9,10,11)} The combination of neutron diffraction and SANS is effective to understand the total behaviors of the microstructures in steels, for example, simultaneous progress of the texture evolution and precipitation. In addition, neutron transmission measurements are also useful for the microstructural characterization.^{1,12,13,14,15,16,17,18,19)} The scattered neutrons by diffraction and SANS can be observed as the decrease in the neutron transmission. The wavelength dependence (spectrum) of the decrease in the neutron transmission by neutron diffraction is well-known as Bragg-edge transmission.^{1,12,13,14,15,16,17)} Although the Bragg-edge transmission gives limited information for crystallographic orientations, it can still characterize various crystallographic parameters even from neutron imaging experiments. A contribution from SANS can be also observed in the neutron transmission and effective for the characterization of the precipitates as well as the application to the neutron imaging experiments.^18,19) While these neutron transmission techniques are frequently used with dedicated analysis codes such as RITS (Rietveld Imaging of Transmission Spectra) in the neutron imaging experiments, they can also be applied to various experiments without such software. The characteristic shapes of the neutron transmission spectra allow direct understanding of structural information similar to the relation between Bragg peaks and crystal structures.

These neutron transmission measurements are particularly suitable for in-situ experiments where samples are surrounded by large sample environment equipment such as tensile testing machine and furnace. In such case, there is often a trade-off between the performance of these devices and the accessible angular range of neutron detectors. The neutron transmission measurements can solve this problem because the neutron transmission can be measured using only a transmission monitor and requires only small windows for incoming and transmitted neutron beams.

For further application to the in-situ experiments, continuous change in the neutron transmission spectra must be traced during the entire measurements. Therefore, this study focused on the neutron transmission spectra of Fe-2 mass% Cu alloys (Cu steel). Adding a trace amount of Cu is known to improve mechanical properties of steels as seen in tensile strength of Cu-bearing high strength steels^20,21) because of large age hardenability of Cu particles precipitated in steel matrix. Therefore, the characterization of the dispersed Cu particles is crucial for the mechanical properties in the Cu steels. Previous studies have already confirmed the precipitation and deformation of Cu particles with aging heat treatment and subsequent cold rolling in the Cu steel.^22,23) Moreover, the neutron transmission spectra of the undeformed Cu steel were examined as a model sample for the analyses of the Bragg edge transmission and SANS contribution.^18,24) In this study, the sequential change of the neutron transmission spectra was investigated in the Cu steel with the cold rolling by an ex-situ experiment to obtain such basic knowledge as shapes of the neutron attenuation coefficient spectra and their changes with plastic deformation.

2. Experimental Procedure

The Fe-2 mass% Cu alloy was solution-treated at 1173 K followed by water quenching, and then aged at 873 K to disperse Cu particles in ferrite matrix. The alloy was then cold-rolled with the thickness reductions of 50 and 90%, which are equal to equivalent strains of 0.8 and 2.7, respectively. For the neutron experiments, the alloy was cut into plates with the thicknesses of about 1 mm as the rolling plane was parallel to the surfaces of the samples. Details of the sample preparation have been described in the previous studies.^22,23)

The neutron experiments were conducted at the small- and wide-angle neutron scattering instrument BL15 TAIKAN installed at the Materials and Life Science Experimental Facility (MLF) of J-PARC.²⁵⁾ The incident neutron beam was normal to the sample surface. The neutron transmission spectra in the wavelength range between 0.2 and 0.76 nm were measured using an incident neutron monitor and a transmission monitor filled with nitrogen gas placed before and after the sample, respectively. Based on previous studies, the accuracy of the data was estimated using Poisson statistics, where the error of the neutron count is represented as the square root of the neutron count.²⁶⁾ Each neutron transmission spectrum T(λ) was calculated as,²⁵⁾   

$$\begin{array}{c}T(\lambda )=[I_{\text{m}2}^{\text{S}}(\lambda )/I_{\text{m}1}^{\text{S}}(\lambda )]/[I_{\text{m}2}^{\text{D}}(\lambda )/I_{\text{m}1}^{\text{D}}(\lambda )],\end{array}$$(1)

where $I_{\text{mx}}^{\text{i}}(\lambda )$ is the intensity spectrum of the neutron beam measured by the incident monitor (mx = m1) and transmission monitor (mx = m2). The parameter i = S, D denotes the measurements with and without the sample, respectively. The accuracies of T(λ) were estimated based on Eq. (1) with the propagation of error. Each intensity spectrum was measured for 2.5 hours with the beam power of 300 kW at MLF of J-PARC. Details of the experimental geometry have been provided elsewhere.¹⁸⁾ A magnetic field of 1 T was applied to the samples perpendicular to the incident neutron beam for magnetic saturation. In the following chapters, the neutron attenuation coefficient μ(λ) is discussed instead of the neutron transmission T(λ) because μ(λ) has an advantage that the difference in the sample volume is normalized in μ(λ) based on the Beer-Lambert law of optics, T(λ) = exp[−μ(λ)t].²⁷⁾ Here, λ denotes the neutron wavelength.

3. Results and Discussion

Figure 1 shows the neutron attenuation coefficient spectra μ(λ) of the solution-treated Cu steel before the cold rolling as well as of the as-aged Cu steel reproduced from the reference.¹⁸⁾ A clear jump is observed at λ = 0.4 nm in both the solution-treated and as-aged samples (shown by a vertical dotted line). This feature is the Bragg edge originated from 110 lattice planes of ferrite and typically seen in ferritic steels with weak crystallographic textures.^{12,13,14,15,16,17,18)} Small jumps also appear at 0.29 nm and 0.24 nm corresponding to the 200 and 211 Bragg edges, respectively. The shapes of the Bragg edges, which reflect the crystallographic texture of the ferrite grains,^{12,13,14,15,16,17)} are almost same between the solution-treated and as-aged samples. This indicates that the texture is not affected by the thermal aging at 873 K.

Fig. 1.

Neutron attenuation coefficient spectra of as-aged and as-solution-treated Cu steels before cold rolling. Filled circles and cross marks denote the as-aged and as-solution-treated Cu steels, respectively. Solid line is neutron attenuation coefficient calculated from incoherent scattering and absorption contributions of Fe-2 mass% Cu. Vertical dotted lines indicate the wavelengths of Bragg edges calculated for ferritic steel. The spectrum of the as-aged Cu steel was reproduced from ref. [18] with permission of the International Union of Crystallography. (Online version in color.)

The wavelength of 0.4 nm is so-called Bragg cutoff of the ferritic steels and determined by the largest lattice spacing as derived from the Bragg’s law λ = 2d₁₁₀sin(θ = 90°), where d₁₁₀ and θ denote the 110 lattice spacing of the ferritic steels (= 0.2 nm) and the half scattering angle, respectively. This means that the Bragg edge is no longer generated in the longer wavelength region. Figure 1 also shows μ(λ) calculated from only incoherent scattering and absorption contributions of Fe-2 mass% Cu as a solid line.^13,27) In the range of λ longer than the Bragg cutoff, μ(λ) of the as-aged Cu steel is obviously larger than the calculated spectrum. This larger μ(λ) is explained by the SANS contribution μ_SANS(λ) of the Cu particles and confirms the precipitation of the Cu particles by the thermal aging.¹⁸⁾

Figure 2 shows μ(λ) of the cold-rolled Cu steels. While the 110 Bragg edge has a sawtooth shape in the as-aged sample as seen in steels with random orientation distributions,^13,15) it drastically changes to a peak with increasing the equivalent strain (shown by the arrow in Fig. 2). The peak shape is directly related to the probability of the grains having specific values of θ (pole density) based on the Bragg’s law.¹³⁾ As the incident neutron beam is along the normal direction (ND), the value of θ is geometrically connected to the tilt angle Ψ of <110> to ND as Ψ = 90° – θ (Fig. 3). Hence, the evolution of the peak indicates that the {110} planes become aligned to a specific direction. The sharper the peak width is, the more the {110} planes are concentrated to the specific direction. Figure 4 shows the values of Ψ and the peak width (standard deviation) evaluated from the peak of the 110 Bragg edge in Fig. 2 by curve fit to a Gaussian function and the relation θ = sin⁻¹(λ/2d₁₁₀) derived from the Bragg’s law. The values of the peak center and the standard deviation directly obtained from the Gaussian fit were 0.34 nm and 0.015 nm for the equivalent strain = 2.7, respectively. These values correspond to Ψ = 32° and the relative standard deviation of 4.3% based on the Bragg’s law. The relative standard deviation is the ratio of the standard deviation to the peak center. This indicates that the {110} planes are rotated toward Ψ = 32° with increasing the equivalent strain. In addition, the 200 Bragg edge becomes higher value at the equivalent strain = 2.7. Since the orientations of the grains just above the edge jump are equal to θ = 90° and Ψ = 0°, this indicates that the probability of the {100} planes in the rolling plane increases with the cold rolling. Contrarily, the 211 Bragg edge shows no clear change with the cold rolling. As the results, the Bragg edge transmission spectra successfully trace the rotation of the ferrite grains sequentially.

Fig. 2.

Neutron attenuation coefficient spectra of cold-rolled Cu steels. Filled circles, open diamonds, and filled triangles denote the Cu steels with the equivalent strains of 0, 0.8, and 2.7, respectively. Vertical dotted lines indicate the wavelengths of Bragg edges calculated for ferritic steel. Arrow indicates the position of the peak developed with cold rolling. (Online version in color.)

Fig. 3.

Schematic image of experimental setting. Incident neutron is normal to sample surface (= rolling plane). Neutrons diffracted to scattering angle 2θ by the lattice planes with tilted by tilt angle Ψ cause the characteristic decrease in neutron transmission as Bragg edges.

Fig. 4.

Tilt angle Ψ and relative standard deviation of gaussian vs equivalent strain estimated from curve fit of 110 Bragg edge. Circles and diamonds denote Ψ and standard deviation, respectively. (Online version in color.)

The information about such crystal rotations can be connected with the evolution of the texture in the Cu steels. In the current experimental conditions (Fig. 3), the Bragg edges reflect the preferred orientations along ND.^13,15) The degree of the preferred orientation is represented by the March-Dollase coefficient m.²⁸⁾ Here, m = 1 means completely random texture, while m approaches to 0 with increasing the grains having the corresponding preferred orientation. For the obtained information, comparison to theoretical study is essential to effectively characterize the crystallographic texture. In case of the cold rolling in polycrystalline iron, Inagaki has reported that the crystal rotations can be summarized as the following two paths:²⁹⁾

(A) {001}<100> → {001}<110> → {112}<110> → {223}<110>

(B) {110}<001> → {554}<225> → {111}<112> → {111}<110> → {223}<110>

The major texture components in the cold rolling are also interpreted as α (<110>//RD) and γ (<111>//ND) fibers.^29,30,31) Based on this insight, the effects of the texture evolution on the Bragg edges were simulated for several preferred orientations along ND (Fig. 5). In all the figures, the simulated spectrum with m = 1 (random texture) is shown for comparison. In the current wavelength range between 0.2 and 0.76 nm, 110, 200, and 211 Bragg edges contribute to the spectra. The vertical dotted lines in Fig. 5 indicate the wavelengths of the edge jumps of these Bragg edges. With decreasing m, all the cases show the decrease in jump height at the 110 Bragg edge and the formation of a unimodal peak or bimodal peaks at around λ between 0.3 and 0.4 nm. The preferred orientation in the experimental results can be discussed from the comparison with the variations in the positions, numbers, and heights of these simulated peaks. The unimodal peak of the 110 Bragg edge, similar to the experimental result shown by the arrow in Fig. 2, appears between λ = 0.3 and 0.4 nm in the spectra with the <111>//ND, <211>//ND, and <322>//ND orientations. Figure 6 shows the m dependences of the simulated peak positions and peak heights in these three possible orientations. The experimentally observed peak position of 0.342 nm with the equivalent strain = 2.7 coincides with only the <111>//ND orientation with m = 0.4. However, m should be smaller than 0.2 to explain the experimentally observed peak height of 2.8 cm⁻¹. Therefore, the current result is probably not exactly matched with these individual orientations and approximated by a combination of these several orientations. On the other hand, the jump height of the 200 Bragg edge increases with increasing the equivalent strain in the experimental results (Fig. 2). This feature can be observed only in the <100>//ND orientation. Moreover, little change in the 211 Bragg edge in the experimental results indicates that the <211>//ND orientation is not a main component. As the results, the major preferred orientation components can be represented as the combination of the <111>//ND, <322>//ND, and <100>//ND orientations. The <111>//ND and <322>//ND orientations are consistent with the previous study of the cold-rolling texture in the polycrystalline iron. Although the <100>//ND orientation is not typical in the cold rolling texture with large equivalent strain, this component has been reported by several previous researchers.^30,31,32) One possible origin of the <100>//ND orientation is the effects of the dispersed Cu particles. The <100>//ND orientation is a major texture component in cold-rolled single crystals.²⁹⁾ The difference in the crystallographic textures between single crystals and polycrystals is attributed to grain boundary constraints.²⁹⁾ In the Cu steels, the deformation of the ferrite grains is suppressed due to the dispersion strengthening by the uniformly dispersed Cu particles as seen in the increment of Vickers hardness.²³⁾ This can make the texture evolutions slow and relatively weaken the effect of the grain boundary constraints compared to the strengthened ferrite grains. As the results, single-crystal-like <100>//ND orientation possibly remains in the Cu steels up to the large equivalent strain.

Fig. 5.

Simulated Bragg edges for the preferred orientations of (a) <111>//ND, (b) <211>//ND, (c) <322>//ND, (d) <100>//ND, and (e) <554>//ND. The values of m are indicated in legend. Vertical dotted lines denote the positions of edge jumps for 110, 200, and 211 Bragg edges. (Online version in color.)

Fig. 6.

(a) peak position and (b) peak height of 110 Bragg edge in simulated neutron attenuation coefficient spectra. Filled circles, open squares, and filled triangles denote <111>//ND, <322>//ND, and <211>//ND preferred orientations, respectively. Horizontal dotted lines indicate the position and height of the experimentally observed peak. (Online version in color.)

In contrast to the complex changes in the Bragg edges, μ(λ) simply decreases with the cold rolling in the region λ > 0.4 nm. As this region reflects μ_SANS(λ) due to the Cu particles, the decrease of μ(λ) indicates that the Cu particles are affected by the cold rolling. Based on previous study, the relation between μ_SANS(λ) and nanostructural parameters depends on the geometry of the experiment.¹⁸⁾ Thus, μ_SANS(λ) was simulated using the geometrical parameters of the current experiments to understand the trends. The SANS contribution is described as,   

$$\begin{array}{c}{\mu }_{\text{SANS}}(\lambda )={\int }_{2{\theta }_{\text{mon}}}^{\pi }I(q)2\pi sin2\theta d2\theta ,\end{array}$$(2)

where 2θ_mon is the acceptance angle of the transmission monitor and 5 mrad in this experiment.¹⁸⁾ The function I(q) denotes the q-dependent scattering intensity, which is measured by conventional SANS experiments and reflects the shape, size, number density, and scattering contrast of nanoscale inhomogeneities.^{6,7,8,9,10,11)} Here, q is the magnitude of the scattering vector and written as q = 4πsin(θ)/λ. In the Cu steels, transmission electron microscopy (TEM) has confirmed that the Cu particles are only slightly elongated before the cold rolling and then markedly transformed to elongated spheroids.²³⁾ For spheroidal particles with the equatorial and polar radii of R and εR, I(q) is written as,³⁰⁾   

$$\begin{array}{c}I(q)=\mathrm{\Delta }{\rho }^2{d}_{\text{N}}{V}^2{\int }_0^{\frac{\pi }{2}}{F}^2(q,R,\epsilon ,\alpha )sin\alpha d\alpha ,\end{array}$$(3)

  

$$\begin{array}{c}\begin{array}{l}F(q,R,\epsilon ,\alpha )=\\ {\displaystyle \frac{3\left\{sin[q{R}_e(R,\epsilon ,\alpha )]-[q{R}_e(R,\epsilon ,\alpha )]cos[q{R}_e(R,\epsilon ,\alpha )]\right\}}{{[q{R}_e(R,\epsilon ,\alpha )]}^3}},\end{array}\end{array}$$(4)

  

$$\begin{array}{c}{R}_e(R,\epsilon ,\alpha )=R\sqrt{{\text{sin}}^2(\alpha )+{\epsilon }^2{\text{cos}}^2(\alpha )},\end{array}$$(5)

where ε, Δρ, d_N, V, and α denote the aspect ratio, scattering contrast, particle number density, particle volume, and azimuthal angle around the incident beam axis.³³⁾ While the values of Δρ, d_N, V are constant prefactors, the λ dependence of μ_SANS(λ) is determined by R and ε. Figures 7(a) and 7(b) show the R and ε dependences of the simulated μ_SANS(λ) using Eqs. (2), (3), (4), (5) and typical values based on the previous results of the Cu steels.^18,23) The values of μ_SANS(λ) are normalized at λ = 0.8 nm for comparison. All the spectra are convex downward and increase with longer wavelength. With increasing R and ε, the curvature tends to be steep. However, since the changes by R and ε are small compared to the current experimental error, higher experimental accuracy is required to perform curve fitting analysis and determine R and ε. To obtain nanostructural insight, Fig. 7(c) shows the simulated μ_SANS(λ) in absolute units for several spheroids having constant values of Δρ, d_N, and V estimated from the previous results of the undeformed Cu particles (Δρ²d_N = 1.6 × 10³⁶ cm⁻⁷, V = 2 × 10⁴ nm³).^18,23) The simulated μ_SANS(λ) decreases with increasing ε. This is consistent with the experimental results and supports the elongation of the Cu particles with the cold rolling. This demonstrates that not only the shape but also the magnitude of μ_SANS(λ) are beneficial to the nanostructural characterization.

Fig. 7.

(a) Aspect ratio ε dependence of simulated μ_SANS(λ) for spheroidal particles with equatorial radius R = 17 nm. (b) R dependence with ε = 7. The spectra are normalized at λ = 0.8 nm. (c) Simulated μ_SANS(λ) in absolute units for several spheroidal particles with same scattering contrast Δρ, particle number density d_N, and particle volume V. (Online version in color.)

4. Summary

The neutron attenuation spectra of the cold-rolled Cu steels were measured for the crystallographic and nanostructural characterizations. The 110 Bragg edge drastically changes from the sawteeth pattern to the peak with increasing the equivalent strain. This indicates that the {110} planes are rotated to Ψ = 32° at the equivalent strain of 2.7. The sharping of the 200 Bragg edge with the cold rolling is attributed to the increase of the probability of {200} planes in the rolling plane. These changes can be described as the evolution of the <111>//ND, <322>//ND, and <100>//ND preferred orientations. While the <111>//ND and <322>//ND orientations are consistent with the typical cold-rolling texture in ferritic steel, the <100>//ND orientation is probably formed via the dispersion strengthening by the Cu particles. The decrease of μ_SANS(λ) indicates the elongation of the Cu particles with the cold rolling. These results support that the analysis of the neutron attenuation spectra is useful to characterize the microstructures and their sequential change. This feature of the neutron attenuation measurement will provide a powerful means to in-situ characterization of the microstructures during heat treatment and mechanical processing. The evolution of the microstructures can be observed as the change in the neutron attenuation spectra even through the small windows opened at the sample environment devices.

Acknowledgements

We thank Prof. Elliot Paul Gilbert, Dr. Toshiro Tomida, and Dr. Yusuke Onuki for fruitful discussion. A part of this work was supported by the 26th ISIJ Research Promotion Grant, the research group “Quantum-beam Analysis of Microstructures and Properties of Steels” of ISIJ, and KAKENHI Grant Number 19K05102. The neutron scattering experiments were performed at the BL15 TAIKAN of J-PARC with the approval of CROSS (Proposal Nos. 2012B0152 and 2014A0301).

References

• 1)   Y.  Tomota: Sci. Technol. Adv. Mater., 20 (2019), 1189.
• 2)   S.  Harjo,  Y.  Tomota,  P.  Lukáš,  D.  Neov,  M.  Vrána,  P.  Mikula and  M.  Ono: Acta Mater., 49 (2001), 2471.
• 3)   T.  Tomida,  M.  Wakita,  M.  Yasuyama,  S.  Sugaya,  Y.  Tomota and  S. C.  Vogel: Acta Mater., 61 (2013), 2828.
• 4)   Y.  Onuki,  A.  Hoshikawa,  S.  Sato,  T.  Ishigaki and  T.  Tomida: J. Mater. Sci., 52 (2017), 11643.
• 5)   M.  Hayashi,  M.  Ishiwata,  Y.  Morii,  N.  Minakawa and  J. H.  Root: Mater. Sci. Res. Int., 6 (2000), 287.
• 6)   M.  Große,  A.  Gokhman and  J.  Böhmert: Nucl. Instrum. Methods Phys. Res. B, 160 (2000), 515.
• 7)   M.  Ohnuma,  J.  Suzuki,  S.  Ohtsuka,  S.-W.  Kim,  T.  Kaito,  M.  Inoue and  H.  Kitazawa: Acta Mater., 57 (2009), 5571.
• 8)   Y.  Oba,  S.  Koppoju,  M.  Ohnuma,  T.  Murakami,  H.  Hatano,  K.  Sasakawa,  A.  Kitahara and  J.  Suzuki: ISIJ Int., 51 (2011), 1852.
• 9)   T.  Ishida,  M.  Ohnuma,  B. S.  Seong and  M.  Furusaka: ISIJ Int., 57 (2017), 1831.
• 10)   T.  Tanaka,  Y.  Toji,  P.  Kozikowski,  M.  Ohnuma and  M.  Nagano: ISIJ Int., 60 (2020), 2246.
• 11)   C.  Ioannidou,  A.  Navarro-López,  A.  Rijkenberg,  R. M.  Dalgliesh,  S.  Koelling,  C.  Pappas,  J.  Sietsma,  A. A.  van Well and  S. E.  Offerman: Acta Mater., 201 (2020), 217.
• 12)   J. R.  Santisteban,  L.  Edwards,  M. E.  Fitzpatrick,  A.  Steuwer,  P. J.  Withers,  M. R.  Daymond,  M. W.  Johnson,  N.  Rhodes and  E. M.  Schooneveld: Nucl. Instrum. Methods Phys. Res. A, 481 (2002), 765.
• 13)   H.  Sato,  T.  Kamiyama and  Y.  Kiyanagi: Mater. Trans., 52 (2011), 1294.
• 14)   K.  Iwase,  H.  Sato,  S.  Harjo,  T.  Kamiyama,  T.  Ito,  S.  Takata,  K.  Aizawa and  Y.  Kiyanagi: J. Appl. Crystallogr., 45 (2012), 113.
• 15)   Y.  Oba,  S.  Morooka,  H.  Sato,  N.  Sato,  K.  Ohishi,  J.  Suzuki and  M.  Sugiyama: ISIJ Int., 55 (2015), 2618.
• 16)   R.  Woracek,  J.  Santisteban,  A.  Fedrigo and  M.  Strobl: Nucl. Instrum. Methods Phys. Res. A, 878 (2018), 141.
• 17)   K.  Oikawa,  Y.  Su,  R.  Kiyanagi,  T.  Kawasaki,  T.  Shinohara,  T.  Kai,  K.  Hiroi,  S.  Harjo,  J.  Parker,  Y.  Matsumoto,  H.  Hayashida,  S.  Zhang,  Y.  Tomota and  H.  Sato: Physica B, 551 (2018), 436.
• 18)   Y.  Oba,  S.  Morooka,  K.  Ohishi,  J.  Suzuki,  S.  Takata,  N.  Sato,  R.  Inoue,  T.  Tsuchiyama,  E. P.  Gilbert and  M.  Sugiyama: J. Appl. Crystallogr., 50 (2017), 334.
• 19)   Y.  Oba,  T.  Shinohara,  H.  Sato,  Y.  Onodera,  K.  Hiroi,  Y.  Su and  M.  Sugiyama: J. Phys. Soc. Jpn., 87 (2018), 094004.
• 20)   A.  Ghosh,  S.  Das and  S.  Chatterjee: Mater. Sci. Eng. A, 486 (2008), 152.
• 21)   M.  Kapoor,  D.  Isheim,  G.  Ghosh,  S.  Vaynman,  M. E.  Fine and  Y.-W.  Chung: Acta Mater., 73 (2014), 56.
• 22)   Y.  Imanami,  M.  Murakami,  N.  Nakada,  T.  Tsuchiyama and  S.  Takaki: ISIJ Int., 49 (2009), 1225.
• 23)   T.  Tsuchiyama,  S.  Yamamoto,  S.  Hata,  M.  Murayama,  S.  Morooka,  D.  Akama and  S.  Takaki: Acta Mater., 113 (2016), 48.
• 24)   Y.  Oba,  S.  Morooka,  K.  Ohishi,  N.  Sato,  R.  Inoue,  N.  Adachi,  J.  Suzuki,  T.  Tsuchiyama,  E. P.  Gilbert and  M.  Sugiyama: J. Appl. Crystallogr., 49 (2016), 1659.
• 25)   S.  Takata,  J.  Suzuki,  T.  Shinohara,  T.  Oku,  T.  Tominaga,  K.  Ohishi,  H.  Iwase,  T.  Nakatani,  Y.  Inamura,  T.  Ito,  K.  Suzuya,  K.  Aizawa,  M.  Arai,  T.  Otomo and  M.  Sugiyama: JPS Conf. Proc., 8 (2015), 036020.
• 26)   J. R.  Santisteban,  L.  Edwards,  A.  Steuwer and  P. J.  Withers: J. Appl. Crystallogr., 34 (2001), 289.
• 27)   A. W.  Hewat and  G. J.  McIntyre: Neutron Data Booklet, ed. by A.-J. Dianoux and G. Lander, Old City Publishing, Philadelphia, (2003), 2.9-1.
• 28)   W. A.  Dollase: J. Appl. Crystallogr., 19 (1986), 267.
• 29)   H.  Inagaki: ISIJ Int., 34 (1994), 313.
• 30)   D.  Raabe: Mater. Sci. Eng. A, 197 (1995), 31.
• 31)   S.  Takajo,  C. N.  Tomé,  S. C.  Vogel and  I. J.  Beyerlein: Int. J. Plast., 109 (2018), 137.
• 32)   R.  Saha and  R. K.  Ray: Scr. Mater., 57 (2007), 841.
• 33)   J. S.  Pedersen: Adv. Colloid Interface Sci., 70 (1997), 171.