comparison of the dislocation density in martensitic

ISIJ International, Vol. 50 (2010), No. 6, pp. 875–882

1. Introduction

1.1. Background

In investigating the mechanical properties of martensiticsteels,1) dislocation density is one of the critical key factorsthat should be evaluated.2) Morito et al.3) observed disloca-tions and evaluated the dislocation density by TEM in Fe–Cand Fe–Ni martensitic steels. It was also evaluated byKehoe et al.4) and Noström5) in steels with 0.01 to0.1 mass% carbon. In addition to the TEM method, X-raydiffraction (XRD) peak broadening caused by strain haslong been investigated in connection with dislocation den-sity.6,7) The XRD peak width, d , is related to the strainthrough the following Williamson–Hall (WH) equation.8)

.......................(1)

...................................(2)

Here, q , l , e , and D are the diffraction angle, X-ray wave-length, strain, and average particle size, respectively. And dis measured on a scale of 2q . Then, the dislocation density,r , is derived as follows9):

................................(3)

Here, b is the magnitude of the Burgers vector. Thebroadened XRD peaks then give dislocation densitythrough Eqs. (1)–(3), which is referred to as the WHmethod. However, in many cases, strong anisotropy in linebroadening was indicated to occur and the left hand side ofEq. (1) cannot be a monotonous function of sin q . Disloca-tions in distorted crystals induce not only the peak broaden-ing but also such anisotropy in peak width.6) In order to su-press the anisotropy, Ungár et al.10,11) modified the WHequation exploiting the kinematical diffraction theory ofdistorted crystals.6) Meanwhile, Warren and Averbach12,13)

had investigated the effect of distortion on the Fourier com-ponents of the XRD peaks, where each component can berepresented as the product of a size-effected component anda distortion-effected component. Ungár et al. related theirmodified WH equation to the analysis of Warren and Aver-bach to obtain the dislocation density, which is referred toas the modified Warren–Averbach (MWA) method.10,11)

Since then, it has often been used to study steels. Yin etal.14) applied it to study the dislocation density in ultrafine-grain steels. Kunieda et al.15) employed it to obtain the dis-

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Comparison of the Dislocation Density in Martensitic SteelsEvaluated by Some X-ray Diffraction Methods

Shigeto TAKEBAYASHI,1,3) Tomonori KUNIEDA,1) Naoki YOSHINAGA,1) Kohsaku USHIODA2) and Shigenobu OGATA3)

1) Steel Research Labs., Nippon Steel Corp., 20-1 Shintomi, Futtsu, Chiba 293-8511 Japan. E-mail: [email protected] 2) Technical Development Bureau, Nippon Steel Corp., 20-1 Shintomi, Futtsu,Chiba 293-8511 Japan. 3) Graduate school of engineering science, Osaka University, 1-3 Machikaneyama, Toyonaka,Osaka 560-8531 Japan.

(Received on January 20, 2010; accepted on March 19, 2010 )

X-ray diffraction (XRD)-based modified Warren–Averbach (MWA) analysis, in comparison with theWilliamson–Hall (WH) analysis, was applied to 0.3 mass% carbon martensitic steels, as-quenched and sub-sequently tempered at various temperatures, to give their dislocation densities. For the as-quenchedmartensite, the WH method gives a value of around 2.0�1016 m�2, which could be overestimated. Mean-while, the MWA method gives a value of around 6.3�1015 m�2, which is below the possible upper limit ofdislocation density, 1016 m�2. The MWA-derived value for the as-quenched steel seems to be 1.6–4.8 timeshigher than those expected from the precedent results derived by transmission electron microscope (TEM)observations. However, considering that the TEM-derived value gives the microscopically local averagewhile the XRD-derived value gives the macroscopic average, such discrepancy between the TEM-derivedvalue and MWA-derived value is tolerable. For the steels tempered at 723 K and 923 K, the MWA and WHmethods give comparable values ranging in 1014 m�2, where the rearrangement of dislocation structure isobserved by TEM. However, in these steels where the XRD peaks are narrower and the instrumental widthof the present XRD system could be significant, care should be taken over the peak width correction.

KEY WORDS: martensitic steels; line broadening; dislocation densities; X-ray diffraction; transmission elec-tron microscopy.

875 © 2010 ISIJ

location density in 11Cr–0.1C martensitic steel in order toevaluate the strain energy stored in the steel. The XRD-based method was also used for analysis of the heat treat-ment effect on the dislocation density in 11.7Cr–0.2Cmartensitic steels by Pesicka et al.,16) where the TEMmethod was also applied and the rather compatible resultsbetween the XRD and the TEM method were obtained inderiving the dislocation density. However, it should be keptin mind that in an inhomogeneous system such as marten-sitic steel, where the dislocation density varies from placeto place within the grain on a microscopic scale, themethod using XRD gives a macroscopic average valuewhile TEM method gives a microscopic local value.16,17)

1.2. Analysis

In the present XRD profile analysis, a variable is usedthat is referred to as the dislocation contrast factor, whichdepends on the relative orientation between the Burgersvector and the dislocation line.6) Ungár et al.10,11) intro-duced the new scaling factor consisting of dislocation con-trast factor and the magnitude of diffraction vector to ac-count for the XRD line broadening in order to supress theanisotropic line broadening often observed in the conven-tional WH analysis. The width of a diffraction peak, de-fined by the full-width-half-maximum (FWHM), which isdenoted by DK, measured on a scale of the magnitude ofthe diffraction vector, is related to the dislocation contrastfactor, denoted by C, through the following modifiedWilliamson–Hall (MWH) equation10):

...................(4)

..............................(5)

Here, K, the magnitude of diffraction vector, equals2 sin q /l , with q , l , and a corresponding to those in Eq.(1), b and r corresponding to those in Eq. (2), M and Obeing constants depending on the effective cut-off radius ofthe dislocations.10,14,15) It should be noted that with theproper selection of C, Eq. (4) gives DK as a monotonousfunction of KC 1/2 to suppress the anisotropy in line broad-ening. Dislocation contrast factor C is replaced by the aver-age version, denoted by C

—, for the specific (hkl) diffraction

expressed as follows:

...........................(6)

......................(7)

Here, q is a constant to be determined experimentally andC—

h00 is the average dislocation contrast factor correspon-ding to (h00) diffraction determined by crystal elastic-ity.10,11) For C

—h00, 0.285 was employed as the value in pure

iron in the literature.14,15) Substituting Eq. (6) into Eq. (4),ignoring the third term on the right hand side of Eq. (4),gives

................(8)

With the proper selection of a , the left-hand side of Eq. (8)is given experimentally. The linear fitting against H 2 thengives q as an inverse of the intercept on the H 2 axis. Thusdetermined q gives C

—through Eq. (6) for (hkl) diffraction

and then leads to the new variable expressed by K 2C(�K 2C

—), which is used as an independent variable in the

following modified Warren–Averbach (MWA) equation thatrelates the Fourier coefficients of the diffraction peaks tothe dislocation density:

............(9)

...................(10)

Here, A(L) is the real part of the Fourier coefficient of the(hkl) diffraction peak depending on L, Fourier length, forwhich the values ranging from 40 to 90 nm were used. A(L)can be obtained simply by Fourier transformation, wherethe phase factor is 2pLK, of the Lorentzian that fits the dif-fraction peak. Constant g , and second-order coefficient P inEq. (9) are shown in the literature.10,11) Re is the effectivecut-off radius of dislocations.6,10,11) Fitting ln(A(L)), with Lfixed, against K2C

—, which is referred to as the MWA plot

hereafter, gives X(L) as the coefficient of the first-orderterm in Eq. (9) for a given value of L. Then, with variousLs, Eq. (10) leads to

..............(11)

through which another fitting procedure upon X(L)/L2

against ln(L) gives rpb2/2, where b�0.283 nm, as the first-order coefficient, and then finally leads to the value of r .

Thus far, the proper selection of a and q in Eq. (8) isnecessary to obtain the contrast factor C (�C

—), which

works through Eqs. (9)–(11) and leads to the dislocationdensity r . However, it seems to be somewhat laborious todetermine a and q through Eqs. (4)–(8). Kunieda et al.15)

set a so that the linear relationship should hold between theleft hand side of Eq. (8) and H 2. However, considering thata should not violate the consistency between Eq. (4) andEq. (8), a recursive iteration method is employed to give aand q in the present analysis. First, the initial value a0 issubstituted into Eq. (8). A linear fitting procedure thengives q0 as the inverse of the intercept on the H 2 axis. Thus-obtained q0 is then substituted into Eq. (6) to obtain C

—and

the quadratic fitting technique is applied to Eq. (4) to fit DKagainst KC1/2 to give renewed value a1 as the intercept onthe DK axis. a1 is then substituted into Eq. (8) and the lin-ear fitting procedure works again to obtain the renewedvalue q1. Looping these procedures until an and qn convergeis expected to give the proper selection of a and q.

In spite of the development of the line broadening analy-sis, comparison between the two methods described her, hasnot been made. Therefore, it is the purpose of the presentstudy to make a quantative comparison between the two

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ISIJ International, Vol. 50 (2010), No. 6

876© 2010 ISIJ

methods, WH and MWA methods. For the as-quenchedsteel, the present results were compared with those esti-mated from the precedent TEM analysis. Furthermore, a se-ries of tempered martensitic steels were investigated to de-termine their mechanical properties. TEM observation wasalso performed on some of the specimens to observe thedislocations therein.

2. Experiment

A steel with 0.3 mass% carbon, the chemical composi-tion of which is listed in Table 1, was used. The ingot wasprepared by vacuum induction melting. It was then homog-enized at 1 473 K for 2 h and forged down to a thickness of15 mm. Bars of 15 mm in width and 180 mm in length werethen sampled from the plate. The bars were austenitized at1 153 K for 5 min, and were then water-quenched. Each ofthe specimens was tempered for 90 min at 453 K, 623 K,723 K and 923 K, respectively. The test pieces were thenmachined out for tensile testing with a cross-head speed of10�3 m/min and a gauge length of 0.025 m and for Vickershardness measurement with a diamond pyramid under theforce of 9.8 N. A Rigaku-Denki RINT 1500 system withCuka radiation was used for XRD measurement. A mono-chrometer employing graphite diffraction was used to elim-inate the Cukb radiation. The X-ray tube was operated at40 kV and 150 mA. The diffraction lines were recordedfrom 2q�35 to 145° with a step of 0.01° to cover the dif-fractions of (110), (200), (211), (220), (310) and (222). Theobserved diffraction peaks were corrected by splitting intothe peaks diffracted by Cuka1 and Cuka2 radiations,18,19) fol-lowed by the Lorentz Polarization correction.20) The peakscorresponding to Cuka1 radiation were fitted with Lorentz-ian functions and used for the analysis. Then, the corre-sponding instrumental width was subtracted from eachpeak width. The instrumental line broadening was evalu-ated by using the standard reference material of LaB6

(SRM660a supplied by NIST).7,21) The instrumental widthdepends upon the peak position and it is approximated by apolynomial function. The average instrumental width corre-sponding to the measured 6 peaks was found to be about6�10�3 nm�1.

The structure of the dislocations in the as-quenched steel,the 623-K tempered steel and the 923-K tempered steelwere observed by TEM.

3. Results and Discussion

3.1. Analysis

The tempered Fe–C martensitic steels have been investi-gated extensively.22–28) The hardness, tensile stress and totalelongation of the present steels are plotted against the tem-pering temperature in Figs. 1(a), 1(b) and 1(c), respectively.The Vickers hardness of the as-quenched steel shows about560 Hv, which is consistent with the result by Grange etal.29) for as-quenched martensitic steel with 0.3 mass% car-

bon, and it decreases to around 230 Hv as the temperingtemperature rises to 923 K. Optical micrographs of the as-quenched and the 923-K tempered specimens are shown inFigs. 2(a) and 2(b), respectively. The figures show thatplain and simple tempered martensitic steels are the objec-tives of the present analysis.

The variation of the diffraction peaks, (110), (200) and(310) are shown in Figs. 3(a), 3(b) and 3(c), respectively,where the peaks are corrected by Cuka1 splitting and theLorentz-polarization scheme, but are yet to be corrected bythe instrumental width. Each peak is clearly shown to be-come narrower and more intense in the tempered specimenthan in the as-quenched type. The precise analysis of the


877 © 2010 ISIJ

Table 1. Chemical compositions of the steel in mass%.

Fig. 1. Mechanical properties plotted against tempering temper-ature; (a), (b) and (c) show hardness, tensile stress andtotal elongation, respectively.

Fig. 2. Microstructures of (a) as-quenched and (b) 723-K tem-pered steel.

diffraction peak profile requires detailed discussion.30,31)

Though some of the peaks are irregularly shaped, whichmight be better fitted with the pseudo Voigt function, aweighted linear combination of Lorentzian and Gaussianfunction,30) simple Lorentzian fitting is employed in thepresent analysis.14,15) The instrumental width, which is not aconstant value, is then subtracted from the peak width. Thevariation of the corresponding corrected peak width, cor-rected additionally with instrumental peak width, is shownin Fig. 4. As the tempering temperature rises, especially at723 K, the peak width steeply decreases to be rangingbelow 4�10�2 nm�1, where the instrumental width,6�10�3 nm�1 on average, could be rather significant. Thus,the peaks of the specimens tempered at 723 K and 923 Kare so narrow that they may be keen to the errors includedduring measurement.

In showing the WH plots, Eq. (1) was reformulated in

terms of magnitude of diffraction vector K. Using the rela-tionship K=2 sin q /l , Eq. (1) is equivalently written as:

.............................(12)

Here, the left hand side is FWHM measured on a scale ofK. WH plots are shown in Fig. 5 for the as-quenched andthe 723-K tempered specimens.

In order to obtain the MWH plot expressed by Eq. (4),the previously-mentioned recursive iteration method men-tioned before was applied. The final plots to give the con-verged q in Eq. (8) are shown in Figs. 6(a) and 6(b) for theas-quenched and the 723-K tempered specimens, respec-

ΔK K� �α ε


878© 2010 ISIJ

Fig. 3. XRD peaks of as-quenched steel (dotted line) and 723-Ktempered steel (solid line). (a), (b) and (c) correspond to(110), (200) and (310) diffraction peaks, respectively.

Fig. 4. Variations of diffraction peak width plotted against tem-pering temperature. Circles, rectangles and trianglesshow (110), (200) and (310) peak width, respectively.

Fig. 5. WH plots of (a) as-quenched and (b) 723-K temperedsteel. Plots are not monotonous because of the anisotropyin peak widths. Equation (12), which is equivalent to Eq.(1), is used for plotting.

tively. The reciprocal of the intercept on the H 2 axis in eachfigure gives q. Then the dislocation contrast factors are ob-tained through Eqs. (6) and (7), leading to MWH plots,which are shown in Fig. 7. In these figures, the value of ab-scissa, KC1/2, of each diffraction in one specimen differsfrom that in the other mainly due to the difference in the Cvalue. Although neglecting the highly deviating points inthe WH plot is reported to work in evaluating the disloca-tion density,18) in the present analysis, the literal WH analy-sis with all points is compared with the MWA analysis. Theintercept on the DK axis in Fig. 7, representing the particlesize broadening through Eq. (2), is in the order of10�4 nm�1 for the as-quenched specimen shown in Fig.7(a). The observed peak widths for the as-quenched speci-men are in the order of 10�2–10�1 nm�1. Thus, it is possibleto consider the particle size broadening to be negligiblysmall. Pesicka et al. also found it negligible in investigatingthe tempered martensite ferritic steels.16) However, asshown in Fig. 7(b), for the 723-K tempered specimen, theobserved widths decrease to the order of 10�3–10�2 nm�1,which is comparable with the instrumental broadening.Then the intercept on the DK axis increases to the order of10�3 nm�1, which is not small enough to be ignored. Dis-cussion on the particle size requires a detailed analysis ofthe peak profile.30,31) However, in the present analysis, asthe particle size broadening term does not directly affectthe dislocation density, it is possible to ignore the term alsoin the 723-K tempered specimen. Now that the dislocationcontrast factor C has been obtained, it is possible to build


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Fig. 6. Plots of Eq. (8) after some iterations and q have con-verged. The reciprocal of the intercept on the H2 axisgives converged q, where (a) and (b) correspond to as-quenched and 723-K tempered steel, respectively.

Fig. 7. MWH plots of (a) as-quenched and (b) 723-K temperedsteel. The plots are monotonous and the anisotropyshown in the WH plots has been supressed.

Fig. 8. MWA plots of (a) as-quenched and (b) 723-K temperedsteel.

up a new scaling variable K2C which scales the MWH plotexpressed by Eq. (9). The corrected diffraction peaks ap-proximated by Lorentzians underwent Fourier-transforma-tion to give the component A(L), on the left hand side ofEq. (9). The MWA plot is shown in Fig. 8 for the as-quenched and the 723-K tempered specimens. The MWAplots for the as-quenched specimen, shown in Fig. 8(a), arehighly nonlinear, on the other hand, those for the 723-Ktempered specimen, shown in Fig. 8(b), are rather straight.The MWA plot gives X(L) in Eq. (9) as the coefficient ofthe first order term of K2C. Then, according to Eq. (11),plotting X(L)/L2 against ln(L) gives dislocation densitythrough its coefficient of ln(L). Figure 9 shows theseX(L)/L2�ln(L) plots.

3.2. Dislocation Density

The dislocation density obtained from WH and MWAanalysis is shown in Fig. 10. For the specimens temperedbelow 623 K, the WH results show values around 1.2�1016–2.0�1016 m�2, while MWA results show 2.5�1015–6.3�1015 m�2. The WH values are somewhat higher thanthe upper limit value 1�1016 m�2 estimated by Nakashimaet al.32) The MAW plot in Fig. 8(a), corresponding to theas-quenched specimen with the dislocation density of6.4�1015 m�2 by the MWA analysis, shows a highly nonlin-ear curve. The high nonlinearity as such is observed also inthe 453-K tempered and the 623-K tempered specimens,whose MWA-derived dislocation densities are 5.0�1015 m�2 and 2.5�1015 m�2, respectively. On the otherhand, the 723-K tempered specimen, as in Fig. 8(b), showsa moderate nonlinearity, whose MWA-derived dislocationdensity is 6.8�1014 m�2. These observations seem to imply

that, in the present MWA analysis, the coefficient of thethird term on the right hand side of Eq. (9) seems to be sig-nificant in the case where the dislocation density exceeds1015 m�2. The ultrafine-grain ferritic steels investigated byYin et al.14) show 2�1013–1.4�1014 m�2, where the second-order term is negligible in Eq. (9). However, the results ofKunieda et al.15) show that the as-quenched state of11Cr–0.1C martensitic steel requires a quadratic fitting ofthe MWA plot to give a value of 5�1014 m�2. Hence, thelinearity of the MWA plot is not necessarily constrainedonly by dislocation density but by carbon content or otherfactors as well.

It is important to compare the present results with thoseof the TEM analyses. As noted previously, Morito et al.,3)

Kehoe et al.4) and Noström5) investigated the variation ofthe dislocation density against the carbon content in the as-quenched martensitic steels. They showed that it is linearlydependent on the carbon content. In Fig. 11, the TEM-de-rived results in the literature for the as-quenched marten-sitic steels are compared with the present results. TheMWA method gives dislocation density of about 6.3�1015

m�2 for the as-quenched specimen, which is relativelyhigher than the value of about 1.3�1015 m�2 derived by in-


880© 2010 ISIJ

Fig. 9. Plots of Eq. (11) for (a) as-quenched and (b) 723-K tem-pered steel.

Fig. 10. Dislocation density evaluated by MWA (circles) andWH analysis (triangles) plotted against tempering tem-perature.

Fig. 11. Dislocation densities of as-quenched steel evaluated bythe XRD method and the precedent TEM method. TheMWA and WH method correspond to the solid circleand solid triangle, respectively. Solid line interpolatingthe open rectangles, chained line extrapolating the opentriangles and broken line extrapolating the open circlescorrespond to Ref. 3), Ref. 4) and Ref. 5), respectively.

terpolating the results along the carbon content presentedby Morito et al.,3) the value of about 3.8�1015 m�2 and2.7�1015 m�2 given by linearly extrapolating the resultspresented by Kehoe et al.4) and Noström,5) respectively.These discrepancies might be inevitable because of the in-homogeneous distribution of dislocations in martensiticsteels.16) Breuer et al.17) evaluated the dislocation density inCo-based alloy using XRD profile analysis and the TEMmethod, where they found the tendency for the XRDmethod to give a higher value than the TEM method whenthe dislocation distribution is inhomogeneous. Meanwhile,the WH method gives a difference higher by nearly an orderof magnitude. These observations lead to the conclusionthat, as far as the as-quenched specimen is concerned,MWA analysis is reliable as compared with the WH analy-sis. The same is expected to hold for the 453-K and the623-K tempered specimens. On the other hand, for the 723-K and the 923-K tempered specimens, as shown in Fig. 10,

comparable values, 3�1014–8�1014 m�2, are given by eachmethod. The broadening anisotropy of the WH plot shownin Fig. 5(b) might become less effective because of the nar-row range of the peak width distribution. However, the nar-rower the peaks are, the more sensitive they are to the ex-trinsic errors involved during measurement or analysis, es-pecially in case where the instrumental width is comparablewith the observed width.

In order to observe the structure of dislocations, the as-quenched specimen, the 623-K tempered specimen, and the923-K tempered specimen were observed by TEM. Al-though the present analysis is not precise enough for aquantitative investigation of the dislocation structure,33) theresults are shown in Fig. 12. In the as-quenched specimen,where the dislocations originate in the plastic accommoda-tion of transformation strain,34) almost all of the views showthat they aggregate over the background of the lath struc-ture as depicted in Fig. 12(a). These may be recognized asthe transformation-induced statistically stored dislocations.Pesicka et al.16) observed aggregated dislocations similar toFig. 12(a) in their 12Cr–0.2C martensitic steel, where theyfound the dislocation density too high to be evaluated.However, they tried to evaluate at different locations to givethe density of 8�1014–1.2�1015 m�2. Therefore, consider-ing that such aggregation is observed in most of the viewsin the present specimen, it is probable that the dislocationdensity of the present as-quenched specimen is higher than1.2�1015 m�2 and the MAW-derived value of 6.3�1015 m�2

could be tolerable. In the 623-K tempered specimen, wherecarbide precipitation is expected,24) most of the views showsuch structures as shown in Fig. 12(a), but some show, as inFig. 12(b), recovery and carbide precipitation possibly giv-ing rise to a decrease of the dislocation density and a re-arranged dislocation structure, which implies a moderatedecrease in the dislocation density as compared with the as-quenched specimen, Fig. 12(a). Therefore, it is possible toestimate the dislocation density to be still in the order of1015 m�2, where the MWA-derived value of 2.5�1015 m�2

could be tolerable. In the 923-K tempered specimen, mostof the views show a totally different rearranged structure ofdislocations as shown in Fig. 12(c). The rearrangement ofdislocations suggests that the dislocation density decreasedby more than an order of magnitude. As compared againwith the observation by Pesicka et al.16) of the specimenwith low dislocation density, it is possible to estimate thedensity from Fig. 12(c) to be in the order of 1014 m�2,which is consistent with the MWA-derived value. Althoughthe 723-K tempered specimen did not undergo TEM obser-vation, it is possible that the collective rearrangement of thedislocation structure synchronously occurs with the steepdecrease in the dislocation density at the tempering temper-ature of 723 K.

4. Conclusions

In summary, in order to evaluate the dislocation densitiesin 0.3 mass%-C as-quenched and tempered martensiticsteels, the MWA method and the conventional WH methodwere compared. TEM observation was also performed onsome of the steels.

(1) The MWA method gave values ranging from 6.3�


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Fig. 12. TEM micrographs of (a) as-quenched, (b) 623-K tem-pered and (c) 923-K tempered steel.

1015 for as quenched steel down to 3�1014 m�2 for the 923-K tempered type. Meanwhile, for these steels, the WHmethod gave values ranging from 2.0�1016 for the as-quenched steel down to 3.5�1014 m�2 for the 923-K tem-pered type.

(2) For the steels tempered below 623 K, including theas-quenched type, the WH method gives values exceeding1016 m�2, which can be overestimated. Meanwhile, theMWA method gives values below 1016 m�2, which are ac-ceptable.

(3) For the specimens tempered at 723 K and 923 K,both methods give comparable values in the range of1014 m�2.

(4) Comparison with the expected values from theprecedent TEM results for the as-quenched specimen showsthat the MWA method gives a value 1.6–4.8 times higherthan the TEM method. However, they are still within thesame order of magnitude. Considering the inhomogeneityof the dislocation distribution, the observed discrepanciesare tolerable.

Acknowledgements

The present authors owe much to Dr. Manabu Takahashi,Dr. Yasushi Hasegawa and Dr. Yoichi Ikematsu of NipponSteel Corp. for the encouraging discussions. Dr. MuneyukiImafuku of Nippon Steel Technoresearch is acknowledgedfor his helpful suggestions on the instrumental width analy-sis. Also acknowledged is the support by the Ministry ofEducation, Culture, Sports, Science and Technologies inJapan, Grant-in-Aid for Scientific Research B_Grant No.20360055.

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