Quantification of Changes in Lattice Defect Density in BCC Iron during Plastic Deformation Using Electrical Resistivity Measurements

Abstract

Change in lattice defects density in bcc pure iron due to tensile deformation was quantified by using both electrical resistivity measurements and X-ray diffraction (XRD). As bcc pure irons, ultra-low carbon steel (ULCS) and interstitial free (IF) steel are used as the model specimen. Dislocation density evaluated using Williamson Hall method with XRD shows the saturation with the value of around 3.7×10¹⁵ m⁻² for ULCS and around 1.4×10¹⁵ m⁻² for IF steel after plastic strain after ~5%. Increase in electrical resistivity was observed with increasing plastic strain. Consequently, increase in vacancy concentration occurs with increasing plastic strain of around 0.3, such as, 2.6×10⁻⁵ for ULCS and 3.4×10⁻⁵ for IF steel. Additionally, the migration of carbon atoms from grain interior to grain boundary via dislocation might occur at the initial stage of plastic deformation in ULCS.

1. Introduction

Practical metallic materials contain lattice defects, which cannot completely be eliminated. There are typical lattice defects in metals, such as, vacancies (point defects), dislocations (line defects), and grain boundaries (plane defects). It is important to determine lattice defect density, i.e., dislocation density and density of grain boundary, since they affect the strength of metallic materials via work hardening and grain refinement strengthening. Oppositely, the mechanical properties of materials can be influenced by controlling the type and density of lattice defects.

Various methods have been used to evaluate lattice defect density. For instance, positron annihilation and electrical resistivity are used to evaluate vacancy concentration; transmission electron microscopy (TEM)¹⁾ and diffraction method²⁾ are used to measure dislocation density; and, optical microscopy (OM), scanning electron microscopy (SEM), electron back-scattering diffraction (EBSD), and TEM are used to measure density of grain boundary.

When electric current flows in a metallic material, free electrons move in the opposite direction to the electric current as an average. However, they do not move straightly since they are scattered by solid-soluble atoms and lattice defects in the material.³⁾ The electrical resistivity ρ increases when a mean free path which is the distance that the free electrons travel without being scattered decreases. Therefore, the larger the density of the scattering centers, the more frequently the free electrons are scattered, and ρ becomes larger. The higher the temperature of typical metals, the higher ρ, since lattice vibrations can also be scattering centers of electrons.

A procedure for quantifying changes using electrical resistivity measurements in the density of lattice defects during severe plastic deformation which is one of plastic deformations has already been reported, e.g., pure aluminum, aluminum alloys, pure nickel, and pure copper.^4,5,6,7,8) However, there is no similar report on steel, which is the most widely-used-structural-metallic material.

In this study, ultra-low-carbon steel (ULCS) with a concentration of solid-solute carbon of less than 0.01% and interstitial free (IF) steel, in which titanium (Ti) is added to ULCS to further reduce the concentration of solid-solution carbon as carbide precipitates, were used as samples. ρ values of these samples were investigated to reveal how ρ changes with progression plastic deformation.

2. Experimental Procedure

2.1. Sample Preparation

ULCS and IF steels having compositions as shown in Table 1 were used as samples. The concentration of carbon for ULCS is suppressed to less than 10 ppm, while the IF steel has Ti addition to UCLS to reduce the concentration of solid-solute carbon as much as possible. Each molten ingot was held at 1473 K for 3.6 ks, and then, hot rolled down to a thickness of 12 mm. After hot rolling, the thickness of plates become 10 mm, since about 1 mm from the top and bottom surfaces of plates were removed by machining. Subsequently, the plates were cold rolled down to 1 mm thick sheet with the rolling reduction of 90% at room temperature. The ULCS and IF steels were applied annealing for recrystallization at 943 K for 60 s and 973 K for 120 s, respectively. Both the heating rate to the holding temperature and the cooling rate to room temperature were set to 10 K/s.

Table 1. Composition of ULCS and IF steel.

Element (mass%)	C	Si	Mn	P	S	Al	Ti	N
ULCS	<0.001	<0.003	<0.003	<0.002	<0.0003	<0.031	<0.002	<0.0007
IF steel	<0.001	<0.003	<0.003	<0.002	<0.0003	<0.030	0.05	<0.0007

2.2. Optical Microscopy

The specimens for optical microscopy were mechanically polished, and then, corroded by immersing in an etchant at 313 K for 60–90 s. The etchant was a mixture of 100 mL water, 5 g of dodecylbenzenesulfonic acid, 0.5 g of oxalic acid, and 3 g of picric acid, in which 0.5 g iron was dissolved, and finally 2 mL hydrochloric acid with 6 N concentration was added.

2.3. Electrical Resistivity Measurements

The electrical resistance of the specimens was measured using the four-terminal method. The current flowing through the voltmeter becomes almost zero due to the high internal resistance of the voltmeter in the case of four-terminal method. Thus, the voltage drop appearing at the contact resistance between the specimen and the wires, and the wires between the voltmeter and the specimen, can be neglected. Therefore, the electrical resistance R of the sample itself can be determined with high accuracy by four-terminal method compared with the two-terminal method, and making it suitable for the measurement of metallic samples with small ρ.

First, four-pure iron wires (0.3 mm diameter, 99.5% purity, Nilaco) were spot-welded to a specimen using a NRW-100W spot welding machine and a NA-60A welding head (Nippon Avionics). The two outer wires were used for introducing current and the two inner wires were used for voltage measurement. Then, measurements were performed at room temperature (293 K) and in liquid nitrogen (77 K) with passing current of 100 mA using a nanovoltmeter (Keithley 2182A) and a high-precision current source (Keithley 6220). In order to eliminate measurement error due to thermal electromotive force generated at the contacts of dissimilar metals in the circuit, delta mode was used, in which the thermal electromotive force was subtracted by switching the polarity of the current. ρ was derived with the distance l between the wires for voltage measurement, the cross-sectional area S of the sample, the applied current A between the terminals, and the voltage drop V between the terminals by using the following equation.

  

$$\rho =\frac{V}{A}\cdot \frac{S}{l}$$(1)

l and S before tensile deformation were expressed as l₀ and S₀, respectively, and were measured using a digital caliper and a digital micrometer. The distance between terminals and the cross-sectional area of the specimen after unloading were evaluated from the strain obtained from the tensile test with using the constant volume condition. Spot-welded wires tend to fall off due to handling until the specimens are mounted. Thus, a frame made of ABS resin was fabricated using a 3D printer to prevent peeling off the spot-welded wire due to handling. ρ at a certain temperature T is denoted as ρ_T. The details of the electrical resistivity measurement can be found in the literature.^4,5,6,7,8,9)

2.4. Simple Tensile and Cyclic Tensile Tests

Tensile tests were conducted using an Instron-type tensile testing machine (Shimadzu AG-X plus, 10 kN) with an initial strain rate of 1.0×10⁻⁴ s⁻¹. As shown in Fig. 1, the shape of the tensile specimens was dog-born shape with a gauge length of 20 mm, a gauge width of 2.6 mm, and a thickness of 1 mm. Initially, strain was measured using a strain gauge (KFGS-1-120-C1-11L1M2R, Kyowa Dengyo). The total strain after the strain gauge broken at a few percent was derived using the plastic strain obtained from the crosshead displacement.

Fig. 1. Schematic illustration of tensile specimen.

Two types of tensile tests were conducted; a simple tensile test with a constant initial strain rate, which is commonly used, and a cyclic tensile test in which the material is unloaded and loaded at approximately every 5% strain after passing the yield point. In the case of cyclic tensile tests, electrical resistance at room temperature and liquid nitrogen temperature and XRD measurements at room temperature were performed after unloading. The plastic strain of the specimens for XRD measurement was derived from the crosshead displacement.

2.5. XRD Measurements

An X-ray analyzer X’Pert PRO MPD (PANalytical) was used for XRD measurements. The conditions were as follows; wavelength λ = 0.15418 nm (Cu-Kα line), tube voltage 45 kV, tube current 40 mA, and step angle 0.008356°. Samples with the same geometry as in the electrical resistivity measurement were prepared for XRD measurements, and measurements were performed after unloading during cyclic tensile tests. The measured XRD peaks were used to analyze dislocation density using the Williamson-Hall method (W. H. method). First, Williamson-Hall plots were prepared according to the following equations.^10,11)

  

$$\frac{\text{(}\mathrm{\Delta }\text{2}\theta \text{)cos}\theta }{\lambda }=\frac{k_{\text{s}}}{D}+\frac{\text{2}e\text{sin}\theta }{\lambda }$$(2)

where λ is the X-ray wavelength, D is the crystallite size, e is the lattice strain, and k_s is the Scherrer constant which is usually 0.9.¹²⁾ e and crystallite size D can be evaluated from the slope and intercept of the linear-fitted to a Williamson-Hall plot, respectively. Then, dislocation density L_v can be derived from e using the following equation proposed by Williamson and Smallman.¹³⁾

  

$$L_{\text{v}}=K{e}^2/b^2$$(3)

Here, K is a constant depending on the crystal structure, and 14.4 was used for the BCC sample used in this study. b is the magnitude of the Burgers vector. The standard error was also obtained when determining the slope of the Williamson-Hall plot.

3. Experimental Results and Discussion

3.1. Optical Microscopy

Figure 2 shows optical microscopy images. The average grain size d was evaluated using Eq. (4), by applying the sectioning method¹⁴⁾ for the photographs,.

  

$$d=L/n_L$$(4)

here, L is the total length of the lines drawn on the image, and n_L is the number of grains crossed by the line. d was 21.1 μm for ULCS and 14.4 μm for IF steel.

Fig. 2. Optical micrograph of (a) ULCS and (b) IF steel.

3.2. Tensile Tests

Figure 3 shows the nominal stress-nominal plastic strain (σ_n−ε_n,p) curves obtained from cyclic tensile tests. The Ultimate Tensile Strength (UTS) was 212 MPa for ULCS and 230 MPa for IF steel at a strain of 0.3. The 0.2% proof stress was 125 MPa for ULCS and 130 MPa for IF steel.

Fig. 3. Nominal stress-nominal strain curve of (a) ULCS and (b) IF steel.

3.3. Electrical Resistivity Measurments

ε_n,p dependence of ρ₂₉₃ and ρ₇₇ was measured after unloading of cyclic tensile tests (Fig. 4). ρ₂₉₃ is higher than ρ₇₇, due to the effect of lattice vibration. In this study, the value of ρ is derived from R, assuming that the bar specimen is uniformly deformed. Therefore, ρ cannot be derived once reaching to UTS, since the necking occurs after the UTS due to non-uniform deformation. Therefore, ρ is evaluated before UTS in this study, since the bar-shaped specimen is considered to be deformed uniformly. It implies that there is no location dependence of cross-sectional area along tensile direction.

Fig. 4. Nominal plastic strain dependence on electrical resistivity of ULCS and IF 1steel under 293 K and 77 K during intermittent tensile tests.

In the case of ULCS, ρ₂₉₃ linearly increased with increasing ε_n,p from 104 nΩm to 116 nΩm. In contrast, ρ₇₇ decreased from 12.8 nΩm to 11.1 nΩm when ε_n,p initially increased from 0 to 0.075. Then, ρ₇₇ monotonically increased up to 12.5 nΩm with increasing ε_n,p. In the case of IF steel, with increasing ε_n,p, ρ₂₉₃ and ρ₇₇ monotonically increased from 107 nΩm up to 132 nΩm and from 14.0 nΩm up to 17.7 nΩm, respectively. When discussing the change in electrical resistivity of an alloy, ρ₇₇ is often used instead of ρ₂₉₃. This is because it is relatively difficult to maintain the room temperature, exactly at 293 K, while the liquid nitrogen temperature can be regarded as a constant. Therefore, changes in ρ₇₇ will be used in the discussion in this study.

Although ρ₇₇ decreases during the initial deformation stage of ULCS, ρ almost never decreases with plastic deformation. This is because the lattice defect density of general alloys increases with plastic deformation, resulting in an increase in ρ. As an exception, there is a phenomenon that ρ decreases as plastic deformation progresses, which is known as the K effect.¹⁵⁾ ρ decreases with progressing the plastic deformation, as a result of decrease in the density of domains of the ordered phase formed in a disordered matrix which work as scattering centers of free electrons. The K effect has been observed in a binary alloys, such as, Ni–Cr alloys, and has also been reported in Alloy625,¹⁶⁾ which is a practical Ni-Base superalloy, and high-entropy alloys.¹⁷⁾

The ULCS used in this study is pure iron with an extremely low carbon concentration, but, it still contains solid-solution carbon on the ppm order. For example, it has been reported that solid-solution carbon atoms in the grain interior at the initial stage of deformation migrate to the grain boundary by dislocation core diffusion during plastic deformation of ULCS with a carbon concentration of 10 ppm.¹⁸⁾ If the similar phenomenon occurs in the ULCS used in this study, the concentration of solid-solution carbon in the grain interiors after the first unloading in the cyclic tensile test would be lower than that before the tensile test. In such a case, the decrease in ρ is associated with the decrease in the concentration of solid-solution carbon in the grain interior which work as the scattering centers. It can be considered that the concentration of the solid-solution carbon atoms in grain interior does not change once the solid-solution carbon atoms in grain interior move to the grain boundary. Therefore, it is not unusual that ρ₇₇ in ULCS monotonically increases after the first unloading in the cyclic tensile test, since it is associated with the increase in lattice defect density as plastic deformation proceeds. Here, the increase in ρ compared with the ρ measured after the first unloading is denoted as δρ, and the maximum value of δρ was evaluated. Then, δρ₂₉₃ = 52 nΩm and δρ₇₇ = 4.2 nΩm for ULCS, and, δρ₂₉₃ = 76 nΩm and δρ₇₇ = 9.8 nΩm for IF steel, were obtained.

3.4. XRD Measurements

Figure 5(a) shows the XRD measurement results for 11% nominal plastic strain, and Fig. 5(b) shows a Williamson-Hall plot obtained using the Williamson-Hall method. Similar plots were drawn for increasing ε_n,p, and L_V was derived from the slope of the plots. As a result, ε_n,p dependence on L_V is shown in Fig. 5(c). The standard error of the slope of the Williamson-Hall plot was calculated, but the values were so small that it is within the symbols in the figure when it is displayed as error bars.

Fig. 5. (a) XRD patterns of ULCS for 11% nominal plastic strain (b) W. H. plot of ULCS for 11% nominal plastic strain (c) Change in dislocation density of ULCS and IF steel depending on the nominal plastic strain.

Figure 5(c) shows that L_V increases with the first deformation, reaching 3.7 × 10¹⁵ m⁻² at ε_n,p = 0.051 for ULCS, and, 1.4 × 10¹⁵ m⁻² at ε_n,p = 0.036 for IF steel. After the first cyclic of the cyclic tensile test, L_V remained almost unchanged with increasing ε_n,p, saturating to the mid 10¹⁵ m⁻² after about 0.05 of strain for ULCS, and to about 10¹⁵ m⁻² after about 0.03 strain for IF steel. It is considered that the measured values are reasonable, since it has been reported that dislocation density reaches saturation at 5–10% strain^19,20) and that the maximum dislocation density is approximately to 10¹⁶ m⁻² order.²¹⁾

3.5. Lattice Defects Density

In this study, the amount of change in lattice defect density was calculated from the change in electrical resistivity, as a procedure to calculate the lattice defect density during tensile tests. The electrical resistivity ρ of a metal at a certain temperature can be expressed by the following equation.^4,5,6,7,8,9)

  

$$\rho ={\rho }^{\text{pure}}+{\displaystyle \sum }_i^n{C}^i\mathrm{\Delta }{\rho }^i+N_{\text{v}}\mathrm{\Delta }{\rho }^{\text{vac}}+L_{\text{v}}\mathrm{\Delta }{\rho }^{\text{dis}}+S_{\text{v}}\mathrm{\Delta }{\rho }^{\text{GB}}$$(5)

ρ^pure is the magnitude of electrical resistivity of a single crystal without impurities, Cⁱ (at%) is the concentration of the i-th solid solution element, Δρⁱ (nΩm/at%) is the contribution of the i-th solid solution element to the electrical resistivity. N_v (at%), L_v (m⁻²), S_v (m⁻¹) are the vacancy concentration, dislocation density and density of grain boundary, respectively. Δρ^vac (nΩm/at%), Δρ^dis (nΩm³), Δρ^GB (nΩm²) are the contributions to electrical resistivity of vacancy concentration, dislocation density, and density of grain boundary, respectively.

The first term ρ^pure in Eq. (5) represents the contribution of lattice vibration, which is highly dependent on temperature. It can be neglected when considering changes between before and after the measurements since the temperature is constant during, and before and after tensile deformation. The sample used in this study has a purity of more than 99.9%, and the highest concentration of impurities is only 0.03% aluminum (Table 1). Therefore, the second term can be neglected if the concentration of solid-solution elements can be regarded as unchanged during tensile deformation. In the present study, it is considered that the concentration of solid-solution elements in the grain interior decreases in the early stage of plastic deformation of ULCS, and then remains constant. Therefore, it can be neglected after the first unloading of cyclic tensile deformation of ULCS. In the case of IF steel, the concentration of solid-solution carbon in the grain interior is considered to be extremely low due to the effect of added Ti. Furthermore, Ti is considered to form carbides, which are not destroyed during tensile deformation and do not re-solidify. Thus, the concentration of solid-solution Ti is considered to be constant. Therefore, this term can be neglected.

The fifth term of Eq. (5), which is about the contribution of the change in density of grain boundary to the electrical resistivity, can be neglected, because it is much smaller than the change in electrical resistivity caused by other factors. Thus, the amount of change in dislocation density can be expressed as follows.

  

$$\delta \rho =\delta N_{\text{v}}\mathrm{\Delta }{\rho }^{\text{vac}}+\delta L_{\text{v}}\mathrm{\Delta }{\rho }^{\text{dis}}$$(6)

Here, δN_v and δL_v are the change in vacancy concentration and dislocation density, respectively, and, Δρ^vac for iron is 240 (nΩm/at%)²²⁾ and Δρ^dis is 12.6×10⁻¹⁶ (nΩm³).²³⁾ Whereas, the Δρ^GB is 15.8×10⁻⁷ (nΩm²).²³⁾ The density of grain boundary of pure iron with a grain size of 10 μm is 3 × 10⁵ m⁻¹, when a rectangular grain shape is assumed. Although grain refinement is unlikely to occur during a tensile test with a strain of about 0.5, assuming a change in density of grain boundary of only 10%. In such a case, the change in grain boundary density would be 3 × 10⁴ m⁻¹, resulting in a change in electrical resistivity of 0.05 nΩm due to the change in the density of grain boundary. This is less than 1% of the smallest value of δρ₇₇ of 4.2 nΩm. Even if grain refinement did occur, it occurred at the later stage in the tensile deformation, and not at the early stage in the process. Therefore, the change in grain boundary density did not have a significant effect on the change in electrical resistivity, and can be ignored in this analysis.

It is important to note that the evaluation of absolute value of lattice defect density cannot be achieved using the electrical resistivity measurement. This is because what can be obtained using Eq. (6) is only from the change in electrical resistivity caused via the change in lattice defect density. Therefore, the electrical resistivity change due to the change in vacancy concentration can be calculated, by using the results of the XRD measurement for the value of L_V as the second term of Eq. (6). However, in the case of ULCS, there is a high-possibility that the concentration of solid-solution carbon is changed in addition to the change in L_V by the first unloading of the cyclic tensile test, but these contributions cannot be separated.

Therefore, in the study, the discussion will be conducted after the first unloading of the cyclic tensile test. In the region, the increase in L_V saturates according to XRD measurements, and, the concentration of solid-solution carbon is also considered to be a constant value. Then, the only remaining contribution is considered to be that of the vacancy concentration. Therefore, the change in vacancy concentration δN_v can be calculated after the first unloading of the cyclic tensile test, in other words, after the change of δρ₇₇ based on the second point from the left in Fig. 4. Evaluated figure about change in vacancy concentration vs nominal plastic strain is shown in Fig. 6.

Fig. 6. Change in Vacancy concentration of ULCS and IF steel depending on nominal plastic strain.

For both ULCS and IF steel, δN_v increased monotonically with strain in the region where L_V is considered to saturate, and δN_v was 2.6×10⁻⁵ for ULCS and 3.4×10⁻⁵ for IF steel, both at a strain of around 0.3. This indicates that the vacancy concentration increases uniformly with deformation. To the best of the authors’ knowledge, the only report on the change in vacancy concentration of steels during plastic deformation is with positron annihilation by Sugita et al.²⁰⁾ The value of the vacancy concentration at a plastic strain of approximately 0.5 in tensile deformation is 6×10⁻⁵ (~60 at. ppm), and it increases linearly up to this point. Therefore, the value of the vacancy concentration obtained in this study is considered to be reasonable.

The maximum increase in dislocation density of low-carbon steels derived using XRD measurements due to cold working is about 10¹⁵ m⁻².^24,25) Whereas, the dislocation density in the present measurements exceeds 10¹⁵ m⁻². According to Akama et al., values similar to those observed by TEM can be obtained by XRD measurement by using K=9.3 instead of K=14.4 as we used.²⁵⁾ The dislocation density obtained by TEM may be affected by a decrease in dislocation density due to mirror image forces on the surfaces, and the presence of dislocations that satisfy invisible conditions.

Therefore, it is not always necessary to coincide the dislocation densities obtained by XRD measurement and TEM observations. However, if K=9.3 can be used to derive the dislocation density more appropriately, L_V would be 65% of the above-mentioned values. Nevertheless, changes in the absolute value of the dislocation density do not affect the analysis of changes in the vacancy concentration after the first unloading. Because, the saturation of dislocation density occurs after the first loading of the cyclic tensile test.

Here, we attempted to derive absolute values of vacancy concentration by making assumptions for the change in vacancy concentration determined by the change in electrical resistivity. Sugita et al. reported that the value of vacancy concentration which was zero before tensile deformation increases monotonically due to the cutting of dislocations.²⁰⁾ Assuming that the vacancy concentration increases in proportion to the plastic strain, a linear approximation for Fig. 6 gives the results shown in Table 2. Extrapolating the value at ε_n,p = 0, and using these values to convert the change in vacancy concentration to absolute values yields 4.1 × 10⁻⁵ for ULCS and 3.4 × 10⁻⁵ for IF steel both at a strain of approximately 0.3. Using such a procedure, if the vacancy concentration increases in proportion to the plastic strain, it may be possible to estimate the absolute value of the vacancy concentration from the electrical resistivity measurement.

Table 2. Parameters of linear fitting of vacancy concentration – nominal plastic strain curve (Fig. 6).

	Gradient [at%/εn,p]	Gradient [−/εn,p]	Intercept [at%]	Intercept [−]	R2
ULCS	1.26×10−2	1.26×10−4	−1.50×10−3	−1.50×10−5	0.991
IF steel	1.49×10−4	1.49×10−6	−1.06×10−5	−1.06×10−7	0.994

4. Conclusions

Values of stress and change in electrical resistivity with increasing plastic strain were obtained from tensile tests using ULCS and IF steel, in order to clarify the behavior of change in lattice defect density in pure iron during plastic deformation. Uniform deformation cannot be assumed after the ultimate tensile strength because the reduction of the localized cross-sectional area occurs due to necking. Therefore, conversion from electrical resistance to electrical resistivity can be performed in the region before ultimate tensile strength. The dislocation density increased sharply at the beginning with plastic deformation in both ULCS and IF, and then saturated, according to the XRD measurements. The values of ULCS and IF at saturation were about 3.7×10¹⁵ m⁻² after plastic strain of 0.051 and about 1.4×10¹⁵ m⁻² after plastic strain of 0.036, respectively. The calculated vacancy concentration increased monotonically with strain with respect to the first unloading point, and was 2.6 × 10⁻⁵ for ULCS and 3.4 × 10⁻⁵ for IF steel both at a strain of approximately 0.3. If the assumption that the vacancy concentration increases in proportion to the plastic strain during tensile deformation can be made, the vacancy concentration is 4.1 × 10⁻⁵ for ULCS and 3.4 × 10⁻⁵ for IF steel both at a strain of about 0.3. Using such a procedure, it could be possible to estimate not only the change in vacancy concentration but also the absolute value from the electrical resistivity measurement. In the early stage of tensile deformation of ULCS, carbon atoms in solid solution in the grain interiors migrate to the grain boundaries via diffusion through dislocations, which may affect the change in electrical resistivity.

Acknowledgments

A part of this work was supported by the 24th Grant-in-Aid for Promotion of Steel Research. Y. M. appreciate discussions with Prof. Susumu Onaka and Mr. Kazuki Fujioka at Tokyo Institute of Technology. Y. M. is grateful for discussions on steel materials with Mr. Kengo Takeda at Nippon Steel Corporation.

References

• 1)   Y.  Miyajima,  M.  Mitsuhara,  S.  Hata,  H.  Nakashima and  N.  Tsuji: Mater. Sci. Eng. A, 528 (2010), 776. https://doi.org/10.1016/j.msea.2010.09.058
• 2)   Y.  Tomota,  S.  Sato and  S.  Harjo: Tetsu-to-Hagané, 103 (2017), 73 (in Japanese). https://doi.org/10.2355/tetsutohagane.TETSU-2016-085
• 3)   C.  Kittel: Introduction to Solid State Physics, John Wiley and Sons, New York, (2004).
• 4)   Y.  Miyajima,  S. Y.  Komatsu,  M.  Mitsuhara,  S.  Hata,  H.  Nakashima and  N.  Tsuji: Philos. Mag., 90 (2010), 4475. https://doi.org/10.1080/14786435.2010.510453
• 5)   Y.  Miyajima,  T.  Shinohara,  H.  Adachi,  T.  Fujii,  S.  Onaka and  M.  Kato: Mater. Trans., 59 (2018), 393. https://doi.org/10.2320/matertrans.M2017298
• 6)   Y.  Miyajima,  K.  Fukuda,  H.  Adachi,  T.  Fujii and  M.  Kato: Mater. Trans., 61 (2020), 305. https://doi.org/10.2320/matertrans.L-M2019864
• 7)   Y.  Miyajima,  T.  Ueda,  H.  Adachi and  T.  Fujii: Mater. Trans., 62 (2021), 1247. https://doi.org/10.2320/matertrans.L-M2019864
• 8)   Y.  Miyajima,  T.  Homma,  S.  Takenaka,  C.  Watanabe,  H.  Adachi and  K.  Ishikawa: Mater. Today Commun., 33 (2022), 104587. https://doi.org/10.1016/j.mtcomm.2022.104587
• 9)   Y.  Miyajima,  S.  Komatsu,  M.  Mitsuhara,  S.  Hata,  H.  Nakashima and  N.  Tsuji: Philos. Mag., 95 (2015), 1139. https://doi.org/10.1080/14786435.2015.1021400
• 10)   W. H.  Hall: Proc. Phys. Soc. A, 62 (1949), 741. https://doi.org/10.1088/0370-1298/62/11/110
• 11)   G. K.  Williamson and  W. H.  Hall: Acta Metall., 1 (1953), 22. https://doi.org/10.1016/0001-6160(53)90006-6
• 12)   H. P.  Klug and  L. E.  Alexander: X-ray Diffraction Procedures for Polycrystalline and Amorphous Materials, 2nd ed., A Wiley-Interscience Pubication, New York, (1974).
• 13)   G. K.  Williamson and  R. E.  Smallman: Philos. Mag., 1 (1956), 34. https://doi.org/10.1080/14786435608238074
• 14)   Y.  Takayama: J. Jpn. Inst. Light Met., 44 (1994), 48 (in Japanese). https://doi.org/10.2464/jilm.44.48
• 15)   T.  Nagashima: Bull. Jpn. Inst. Met., 3 (1964), 281 (in Japanese). https://doi.org/10.2320/materia1962.3.281
• 16)   T.  Nagata,  K.  Takeda,  H.  Adachi,  K.  Ishikawa and  Y.  Miyajima: Mater. Trans., 63 (2022), 278. https://doi.org/10.2320/matertrans.MT-M2021204
• 17)   Y.  Miyajima,  T.  Nagata,  K.  Takeda,  S.  Yoshida,  S.  Yasuno,  C.  Watanabe,  I.  Kazuhiro,  H.  Adachi and  N.  Tsuji: Sci. Rep., 12 (2022), 16776. https://doi.org/10.1038/s41598-022-20932-y
• 18)   Y.  Ono,  Y.  Funakawa,  K.  Okuda,  K.  Seto,  N.  Ebisawa,  K.  Inoue and  Y.  Nagai: ISIJ Int., 57 (2017), 1273. https://doi.org/10.2355/isijinternational.ISIJINT-2016-622
• 19)   T.  Omura,  K.  Tomatsu,  Y.  Sakiyama,  K.  Sugita,  M.  Mizuno,  H.  Araki and  Y.  Shirai: Tetsu-to-Hagané, 104 (2018), 655 (in Japanese). https://doi.org/10.2355/tetsutohagane.TETSU-2018-031
• 20)   K.  Sugita,  M.  Mizuno,  H.  Araki,  Y.  Shirai,  T.  Omura,  K.  Tomatsu and  Y.  Sakiyama: Tetsu-to-Hagané, 106 (2020), 20 (in Japanese). https://doi.org/10.2355/tetsutohagane.TETSU-2019-064
• 21)   S.  Takaki: J. Jpn. Inst. Light Metals, 56 (2006), 609. https://doi.org/10.2464/jilm.56.609
• 22)   T.  Oi and  K.  Sato: Trans. Jpn. Inst. Met., 7 (1966), 32 (in Japanese). https://doi.org/10.2320/matertrans1960.7.32
• 23)   A. S.  Karolik and  A. A.  Luhvich: J. Phys. -Condens. Matter, 6 (1994), 873. https://doi.org/10.1088/0953-8984/6/4/007
• 24)   S.  Takaki,  T.  Masumura,  F.  Jiang and  T.  Tsuchiyama: Tetsu-to-Hagané, 104 (2018), 683 (in Japanese). https://doi.org/10.2355/tetsutohagane.TETSU-2018-054
• 25)   D.  Akama,  T.  Tsuchiyama and  S.  Takaki: J. Soc. Mater. Sci. Jpn, 66 (2017), 522 (in Japanese). https://doi.org/10.2472/jsms.66.522