Effect of Grain Boundary Segregation of Interstitial Elements on Hall Petch Coefficient in Steels

Effect of Grain Boundary Segregation of Interstitial Elements

on Hall

­

Petch Coef

fi

cient in Steels

Setsuo Takaki

, Daichi Akama

, Nobuo Nakada

and Toshihiro Tsuchiyama

1_{Department of Materials Science and Engineering, Kyushu University, Fukuoka 819-0395, Japan}

2_{International Institute for Carbon Neutral Energy Research (WPI-I2CNER), Kyushu University, Fukuoka 819-0395, Japan}

The yielding behavior of interstitial-free steels and low-carbon steels with varying amounts of C and N were investigated in connection with the Hall­Petch relation. The Hall­Petch coefficient is as small as 150 MPa·µm1/2_{in interstitial-free steels but it increases to 600 MPa·µm}1/2 by adding solute carbon up to 60 ppm. Nitrogen does not have a significant effect on the Hall­Petch coefficient. The results of three-dimensional (3D) atom probe analysis indicated that carbon has 3­4 times greater segregation potential in comparison with nitrogen. The small effect of nitrogen on the Hall­Petch coefficient in steel is probably due to the small segregation potential of nitrogen. It was also confirmed that discontinuous yielding occurs when the difference between the yield stress and friction stress is increased by grain-refinement strengthening and that yielding occurs by dislocation emission from grain boundaries where primary dislocations have piled up. Carbon atoms segregated at grain boundaries seem to play a role in stabilizing dislocation emission sites at the grain boundaries, which enhances the Hall­Petch coefficient of iron. These results support the dislocation pile-up model of explaining yielding in polycrystalline metals. [doi:10.2320/matertrans.MA201314]

(Received August 21, 2013; Accepted September 26, 2013; Published November 9, 2013)

Keywords: grain refinement strengthening, Hall­Petch coefficient, interstitial elements, grain boundary segregation, dislocation pile-up, interstitial free steel, discontinuous yielding

1. Introduction

It is well known that the yield strength of polycrystalline metals,·y, increases in inverse proportion to the square root

of the grain size,d, and this relation is called the Hall­Petch relation (·y=·0+kyd¹1/2).1,2)In this equation, the constant, ·0, and the slope,ky, are referred to as the friction stress and

the Hall­Petch coefficient (HP coefficient), respectively. In

general, the HP coefficient increases with an increase in the

shear modulus of metals but the other effects have not been

clarified yet. In the case of steel, elements such as C and

N are contained interstitially and these elements have a

significant effect on yielding behavior. For example, a clear

yield point appears in mild steel containing a small amount of carbon and the yield point is reduced by removing carbon

and nitrogen using a wet hydrogen treatment.3) Therefore,

for a long time, a Cottrell locking mechanism has been accepted to explain the appearance of yield points in steels.

However, Tomimura et al. found that a yield point appears

even in an austenitic stainless steel with ultrafine grains less

than 1 µm.4) _{Recently, discontinuous yielding has been}

reported in aluminum with an ultrafine-grained structure.5)

This suggests that the appearance of a yield point is a common phenomenon in polycrystalline metals strengthened by grain refinement.

The reduction of the yield point by purification, as

mentioned above, suggests that interstitial elements such as

C and N have some influence on the yielding behavior

of steel. In this paper, the effect of carbon and nitrogen was reviewed in terms of the yielding behavior of iron. The yielding mechanism of polycrystalline metals is also discussed in connection with the dislocation pile-up model

proposed by Hall and Petch.1,2)

2. Friction Stress and the Hall­Petch Relation in Interstitial Free (IF) Steels

Before discussing the influence of interstitial elements on

the yielding behavior of iron, the yielding behavior of pure

iron should be clarified. Figure 1 shows the change in yield

strength of single-crystal pure iron as a function of testing temperature. Since the iron used in this study was single crystal, the yield strength corresponds to the friction stress

in this figure. The measurements were taken in the tensile

direction between [100] and [110] at each testing

temper-ature.6) Therefore, in this figure, the crystal orientation

dependence of the yield strength is displayed by error bars. It was found that the crystal orientation dependence of the yield strength was minimal above room temperature and the yield

strength at 25°C was evaluated to be 50­60 MPa. On the

other hand, the effect of (C+N) on the friction stress of iron

Temperature, T / °C

0 100 200

−100

Lower yield stress,

L

/ MPa

0 100 200 300 400

Strain rate: 4x10−3_/s

25 °C

50~60MPa

σ

Fig. 1 Change in lower yield stress of single-crystal iron as a function of testing temperature. Each plot shows the average of specimens tensile deformed in the direction between [100] and [110].6)

+_{Corresponding author, E-mail: takaki@zaiko.kyushu-u.ac.jp}

Special Issue on Strength of Fine Grained Materials®60 Years of Hall­Petch®

at 18°C was investigated by Heslop7) _{and Cracknell}8) _and

their results are shown in Fig. 2. Friction stress of metals is composed of two components; thermal component which increases with lowering temperature and non-thermal com-ponent which is changeable depending on the amount of

solute alloying elements. In thisfigure, the value at 18°C is

represented for the thermal component. The friction stress in

this figure was determined by the respective Hall­Petch

relation. The friction stress is 60 MPa without C and N, which agrees well with the results shown in Fig. 1. The friction stress of iron is changeable depending on temperature and strain rate, but it can be concluded that the friction stress

of pure iron is 40­60 MPa under conventional tensile testing

conditions, i.e., a temperature of 10­30°C and a strain rate of 10¹4_­10¹3_/s.

As for the Hall­Petch relation in interstitial-free (IF) steel, a few experimental results have been reported and are

displayed in Fig. 3.9,10) _{A small difference is found in the}

friction stress, but it is probably due to the solid-solution strengthening of excess Ti added to remove solute carbon and nitrogen as Ti(C,N), as well as the testing temperature and

strain rate. As a result, at room temperature, the following

Hall­Petch equation was obtained for IF steel:

·y½MPa ¼ ð5010Þ þ150d½m1=2 ð1Þ

The stress­strain curve of a specimen with a grain size of

30 µm is also displayed on the right side. The yielding of IF steel is characterized by continuous yielding. In the case of IF steel, the HP coefficient is as small as 150 MPa·µm1/2,

making it difficult to see the effect of grain-refinement

strengthening.

In terms of the effect of substitutional elements on the HP

coefficient, it is already known that chromium does not affect

it,11)_{that phosphorus decreases it slightly,}12)_{and that nickel}

increases it.9) _{It has not been clarified yet why nickel}

increases the HP coefficient, but the addition of Ni is

effective for enhancing grain-refinement strengthening in

iron. For example, Fig. 4 shows the effect of 3%Ni addition

on the HP relation and the yielding behavior in IF steel. Nickel increases the friction stress of iron through

solid-solution strengthening and also enhances the HP coefficient.

It should be noted in the stress­strain curve that

discon-tinuous yielding appears and yield-point elongation occurs after yielding, even in IF steel. In comparison with Fig. 3, it was found that the difference between the yield stress

and friction stress is increased by the addition of 3%Ni.

This suggests that discontinuous yielding occurs when the difference between the yield stress and friction stress has

been increased by grain-refinement strengthening. This

confirms the claim that the presence of solute C or N is not

an essential condition for the occurrence of discontinuous yielding.

3. Hall­Petch Relation and Yielding Behavior in Steels Containing Interstitial Elements

In low-carbon steels represented by mild steel, many studies have been performed to discuss the HP relation. Figure 5 summarizes the data obtained in low-carbon

steels.13­15) In 1966, Morrison13) systematically examined

HP relations in steels with various carbon concentrations

(0.005­0.2%) and reported the identical HP coefficient of

600 MPa·µm1/2_{for every steel. This result indicates that there}

0.03 0.02

0.01 0

(C+N), (mass%) 0

100 200

Thermal component at 18 °C (J.Heslop et al.)

[MPa]=4500x(C+N) (A.Cracknell et al.)

150

50 Friction stress _{of pure iron at 18 °C}

Δ σ

Friction stress,

0

/MPa

σ

Fig. 2 Effect of solute (C+N) on friction stress of iron at 18°C.7,8)

Elongation, (%)

1 2 3 4

0

Grain refinement strengthening (28MPa)

Friction stress

0 20 40 60 80 100 120

0 0.1 0.2 0.3

d−1/2_/μm−1/2 :this study :R.Matoba et al. :W.B.Morrison et al. Grain size, d /μm

25

100 11

IF steel (Fe−0.02%Ti)

ky=150

Y

ield strength,

0.2

/ MPa

σ

is no carbon-concentration dependence regarding

grain-refinement strengthening in steel, at least in the

carbon-concentration range above 0.005%. After that, the current

authors examined the HP relation in ultrafine-grained iron

with a grain size up to 0.2 µm and proposed the following equation for polycrystalline iron:16­19)

·y¼100þ600d1=2 ð2Þ

In the grain size region up to 1 µm, much data were added as the result of national projects in Japan, and it was

recon-firmed that the above equation holds in 0.15%C steel up to

a grain size of 1 µm.14,15) _{As a result, it can be concluded}

that the HP coefficient is approximately 600 MPa·µm1/2

in steels with more than 0.005% carbon. In Fig. 5, the HP

relation in IF iron is shown by a broken line in order to clarify the difference with low-carbon steels. This result

suggests some influence of a small amount of carbon on the

mechanism of grain-refinement strengthening.

On the other hand, Low et al. reported the effect of

purification on yielding behavior in mild steel and the results

are shown in Fig. 6.3)_{Purification was performed by a wet}

hydrogen treatment at 725°C. As carbon and nitrogen are

removed from specimens as CO and NH3 by this treatment,

specimens are purified more with increasing holding time.

The yielding of the original material is characterized by discontinuous yielding, and the yield point is reduced by

the purification and finally disappears. Because the Cottrell

locking model provides a convenient explanation for this discontinuous yielding in steel, it has been accepted for a long time. However, the current authors recently found

that the HP coefficient is changeable in iron with less than

0.005%carbon.20)_{Figure 7 shows the effect of solute carbon}

and nitrogen on the HP coefficient of iron. It was found that

the HP coefficient is large enough in iron with 41 ppm solute

carbon, but the coefficient decreases with decreasing amounts

of solute carbon. In contrast, nitrogen does not have a

significant influence on the HP coefficient. The difference

between carbon and nitrogen will be discussed later in relation to grain-boundary segregation behavior. For now, the behavior shown in Fig. 6 can be explained as shown in Fig. 8. Solute carbon enhances the friction stress slightly

but it has a greater effect on the HP coefficient. Thus, the

difference between yield stress and friction stress is increased by increasing amounts of solute carbon. In other words, the difference between yield stress and friction stress decreases

as a result of purification, which changes the yielding

behavior from discontinuous yielding to continuous yielding. Here, it should be noted that very small amounts of carbon 0.8 1.0

0.6 0.4 0.2 0

0 100 200 300 400 500 600 700

800 100 20 10 5 3 2 1

Grain size, d /μm

d−1/2_/μm−1/2

ky=600

Matsukura et al. (1999)

Morrison (1966)

0.05C−1Si−1.5Mn−0.02Nb 0.05C−1Si−1.5Mn M. Etou et al. (2008)

0.09C 0.20C 0.005C

0.13C 0.05C0.15C 0.15C−0.74Mn

Low carbon steels (0.005 ~ 0.2%C)

IF steel

Y

ield strength,

y

/ MPa

σ

Fig. 5 Hall­Petch plots in low-carbon steels.13­15)

Elongation, (%) 1

Grain refinement strengthening (75MPa)

Friction stress

0 2 3 4

100 50 20 15

Grain size, d / μm

0 0.1 0.2 0.3

d- −1/2_{/ μm}−1/2 0

50 100 150 200

Y

ield stress,

0.2

/ MPa

ky=290

Fe−3%Ni−0.008%Ti

IF steel

Solid solution strengthening

σ

Fig. 4 Hall­Petch plot and stress­strain curve in IF­3%Ni steel.

0.75h

1.5h

3h

0 10 20 30 40 50

300

0 100 200

Elongation, (%)

Fe−0.05%C

Holding time at 725 °C:5h

Nominal stress,

/ MPa

σ

have a significant influence on yield strength by changing the

HP coefficient. Thus, in a discussion of the HP relation, the

true effect of substitutional elements cannot be evaluated under the coexistence with interstitial elements. In fact, completely different experimental results have been reported

for HP coefficients in Fe­Cr­C alloys21) _{and Fe­P­C}

alloys,22) _{in contrast with our results in IF Fe­Cr alloys}11)

and IF Fe­P alloys.12)

4. Mechanism of Yielding in Polycrystalline Metals

The dislocation pile-up model is a reasonable model for explaining the yielding of polycrystalline metals, and some direct evidences showing dislocation emission from grain

boundaries has been reported.23­25) In 1965, dislocation

emission from grain boundaries was found in optical

micrographs of an Fe­3.65%Si alloy.23) In 304-type

auste-nitic stainless steel, dislocation emission from grain bounda-ries was dynamically observed with transmission electron

microscope (TEM).24) _{Recently, the current authors have}

succeeded in observing dislocations emitted from a grain boundary to a neighboring grain in a high-nitrogen austenitic

steel.25) _{Figure 9 shows the TEM image of a specimen}

with 0.2% tensile deformation. It was found that primary

dislocations pile up at grain boundaries and secondary dislocations are emitted from the area where primary

dislocations have piled up. It was also confirmed that such

Grain size, d / μm

100

0 200

0 0.1 0.2

d−1/2_{/ μm}−1/2 _d−1/2_{/ μm}−1/2

100 30 20

(a) Fe−C

Solute carbon: 41ppmC

IF steel

17ppmC

150

50 250

0.3 100

0 200

0 0.1 0.2

100 30 20

(b) Fe−N

IF steel

Solute nitrogen: 48ppmN

20ppmN

0.3 250

150

50

Grain size, d / μm

Y

ield strength,

y

/ MPa

σ

Y

ield strength,

y

/ MPa

σ

Fig. 7 Effect of solute carbon (a) and nitrogen (b) on Hall­Petch coefficient. Specimens were water quenched from 700°C after holding for various lengths of time to control grain size.20)

Elongation, (%)

2 3 4

1 0 100

0 200

0 0.1 0.2

100 30 20

(a) Fe−C

Solute carbon: 41ppmC

IF steel

17ppmC 150

50 250

0.3 Grain size, d / μm

d−1/2_{/ μm}−1/2

Y

ield strength,

y

/ MPa

σ

Fig. 8 Hall­Petch plots and stress­strain curves in IF iron and iron containing small amounts of solute carbon.

500nm G.B. Dislocations emitted from G.B.

Piled up dislocations

microyielding takes place in each crystal grain and that the number of microyielded grains increases during macroscopic

yielding.23) _{The authors pointed out that macroscopic}

yielding starts when the total amount of microyielded crystal

grains has reached 70­80% in volume fraction.10)

According to dislocation pile-up models,1,2) _{the following}

equation has already been proposed for explaining grain-refinement strengthening in shear stress"¸y:

¸y¼ ðGb¸=k³Þ1=2d1=2 ð3Þ

whereG,b,k,dand ¸*denote, respectively, shear modulus, the Burgers vector, a constant depending on the character of the dislocation, the grain size, and the critical grain boundary strength, which corresponds to the shear stress required for generating dislocations at grain boundaries. Grain boundaries

form barriers against dislocations generated from a Frank­

Read source, whereas they work as dislocation sources

when the pile-up stress has exceeded ¸*. In bcc metals, the

following equation is constructed for the theoretical HP relation by including a Taylor factor of 2:

·y¼·0þ2ðGb¸=k³Þ1=2d1=2 ð4Þ

where·0is the friction stress. The HP coefficient,ky, is given

by ky=2(Gb¸*/k³)1/2 and, in steel, it has experimentally

been obtained as a function of solute carbon content, as

shown in Fig. 7. Figure 10 shows changes ofky and¸* that

were calculated from ky and eq. (4) as a function of solute

carbon content. In specimens water quenched from 700°C,ky

increased with increasing carbon content, and it became

constant at approximately 600 MPa·µm1/2 _{above 60 ppm}

carbon. The value of¸*also increased with increasing carbon

content, corresponding to the change ofky. It should be noted

that¸*is as small as 0.4 GPa in IF steel, but it is significantly enhanced by adding small amounts of carbon. For instance,

¸* becomes about 6 GPa by adding 60 ppm carbon. The

carbon concentration at grain boundaries increases with increasing carbon content of steel. The above results suggest that dislocation emissions from grain boundaries become more prevalent when carbon has segregated at the grain boundaries. Thus, Wilson examined the effect of 90°C aging

on the value of kyin an Fe­0.003%C alloy water quenched

from 700°C26) and Fig. 11 shows the results. In the

as-quenched specimen, ky is 310 MPa·µm1/2, and this is a

reasonable value in comparison with our results shown in Fig. 10. At 90°C, C atoms can move within a bcc Fe lattice

but it is difficult to form carbide precipitates. Here, it is

interesting thatkygradually increases with increased holding

time at 90°C, and it finally levels off at approximately

700 MPa·µm1/2_{. This result enables us to imagine that C}

atoms segregated at grain boundaries during 90°C aging, which enhanced the stability of the dislocation emission site

at the grain boundaries. The value of¸*should be enhanced

by this stabilizing dislocation emission site at the grain

boundaries and lead to an increase of ky. In polycrystalline

iron, it is probable that yielding starts by dislocation emission at grain boundaries and that carbon has a significant influence

on the mechanism of grain-refinement strengthening.

5. Grain-Boundary Segregation of Carbon and Nitrogen

As mentioned above, carbon has the ability to increase the

HP coefficient but nitrogen does not. If the dislocation

pile-up model is acceptable as the mechanism of grain-refinement

strengthening, such a difference between carbon and nitrogen may occur depending on the grain-boundary segregation behavior of each element. Figure 12 displays the results of three-dimensional (3D) atom probe analysis showing the amount of C and N segregated at the grain boundaries in

Fe­C and Fe­N alloys.27) Each specimen was prepared to

have the same atomic fraction of C or N, 0.02 at%, and it was

confirmed that the amount of solute C and N was 0.0019 at%

in both specimens. However, apparent differences can be seen between carbon and nitrogen in terms of grain-boundary segregation behavior, in that carbon has an approximately 3­4 times greater segregation potential. In the Fe­N alloys, it should be noted that almost the same amount of C and N was detected at the grain boundaries, even though the bulk carbon content was only one-tenth that of the nitrogen content. This demonstrates that carbon has a greater potential

Fe−30ppmC

ky

/ MPa·

μ

m

1/2

600 700

500

400

300

200

100

0

101 ₁₀2 ₁₀3 ₁₀4 ₁₀5

Aging time at 90 °C, t/ min

IF steel

Slowly cooled from 700°C

Water−quenched from 700 _°C

Fig. 11 Effect of aging at 90°C on Hall­Petch coefficient, ky, of Fe­ 0.003%C alloy that was water quenched from 700°C.26)

0 20 40 60 80 100

Edge (k=0.72) Screw (k=1)

Solute carbon, (ppm) IF iron (0.4GPa)

0 2 4 6 8

k=0.86

(b) 100 200 300 400 500 600 700

ky

/ MPa·

μ

m

1/2

0 (a)

(700°C WQ)

Author W.B.Morrison

/GPa

τ

∗

than nitrogen in terms of grain-boundary segregation. So far,

it is unknown whether nitrogen itself has a significant effect

on grain-refinement strengthening similar to carbon or not,

but it is clear that the effect of nitrogen on the HP coefficient of iron is difficult to determine owing to its small segregation potential.

If the HP relation is affected by the segregation of alloying elements, the grain-size dependence of the concentration at

the grain boundary, Xb, should be taken into consideration.

Xb will decrease with decreasing grain size,d, because the

grain boundary area per unit volume, Sb, increases with

decreasing grain size (Sb ;3=d). The results shown in

Fig. 12 do not give the value of Xb directly, but it can be

roughly evaluated as follows: The molar volume of bcc iron

is about 7.1 cm3/mol at room temperature, and therefore, the

number of Fe atoms per unit volume,NFe, is evaluated to be

85 atoms/nm3. The number of carbon atoms in the matrix is

calculated to be 0.016 atoms/nm3, which corresponds to a

solute carbon content of 41 ppm. The atomic density around grain boundaries must be smaller than that in the matrix but its actual value is unknown. Therefore, the current authors evaluated it by grain-boundary simulation using steel balls

and obtained a value of approximately 90% as the relative

atomic density at a grain boundary layer 1 nm thick. Thus, the number of Fe atoms around the grain boundary is estimated

to be 76 atoms/nm3_{. Since the number of excess C atoms at}

the grain boundary was 7.6 atoms/nm2in the Fe­C specimen,

Xb can be evaluated as 0.10 [=(7.6+0.016)/76] by

presuming that C atoms are replaced with Fe atoms at the grain boundary and that C atoms are distributed uniformly within the 1-nm-thick grain boundary layer. On the other

hand, the volume fraction of the grain boundary,Vb, is given

by Vb=t·(3/d) under the conditiond ºt, where tdenotes

the thickness of the grain boundary layer. From a viewpoint of mass balance, the following equation should be realized betweenXb, the concentration in the matrix,Xm, and the mean

concentration, X0:

X0¼VbXbþ ð1VbÞ Xm ð5Þ

Here, the segregation coefficient, K(=Xb/Xm), is unknown

but the enrichment factor, ¡ (=Xb/X0), is given as 522

(=0.10/0.00019) for the Fe­C specimen. The value of

K must be slightly larger than that of ¡, so K=530 was

applied for calculating the relation between d and Xb. The

results obtained from eq. (5) are shown in Fig. 13. The grain

size dependence of Xb becomes significant below 20 µm

but it is small above 20 µm. This means that the carbon concentration at the grain boundaries is almost the same as far as grain size of specimens has been controlled to be larger than 20 µm. In our study, the HP relation was examined in the grain size region above 10 µm, as shown in Fig. 7.

It is concluded that the HP coefficient was reasonably

evaluated under the same conditions as the carbon concen-tration at the grain boundaries. It is understood that the results displayed in Fig. 13 are changeable depending on the carbon content of steels, but it is true that the carbon concentration

at grain boundaries is markedly reduced by grain refinement

to a sub-micron level. Regarding the discussion of the HP

relation in metals, especially in ultrafine-grained metals, we

should pay attention to the segregation behavior of alloying elements and impurities, which is changeable depending on grain size.

6. Conclusions

In this review, the yielding behavior of interstitial-free (IF) steel and low-carbon steels containing different C and N

amounts were discussed in connection with the Hall­Petch

relation and the following conclusions were obtained:

(1) The friction stress of pure iron is 50«10 MPa at room

temperature at a strain rate of 10¹4­10¹3/s, which is usually applied to tensile testing.

(2) The Hall­Petch coefficient in IF steel is as small as

150 MPa·µm1/2, but the value increases with increasing

amounts of solute carbon. It reaches 600 MPa·µm1/2 by

adding 60 ppm carbon but tends to level off at this value in the carbon concentration range above 60 ppm. The change of

yield strength by purification in low-carbon steel was

reasonably explained by the carbon-concentration depend-ence of the Hall­Petch coefficient.

Solute N: 0.0191 at% Solute C: 0.0191 at%

0 2 4 6 8 10

(a) Fe−56ppmC−11ppmN (Grain size: 20μm)

Δ

N, (atoms/nm

)

2

(b) Fe−5ppmC−54ppmN (Grain size: 20μm)

: C

: N 7.6

2.1

1.5

Fig. 12 Results of 3D atom probe analysis showing the difference between carbon and nitrogen on grain boundary segregation.27)_{¦Ndenotes the} number of excess atoms per unit grain boundary area. Specimens were held at 700°C for 90 s and then quenched in water.

Atomic fraction of C at grain boundary; Xb

Grain size, d /μm Xb

Segregation coefficient; K=530 (=Xb / Xm)

20μm

0 0.02 0.04 0.06 0.08 0.10 0.12

0 10 20 30 40 50 60 70 80 90 100

(3) Discontinuous yielding is a common phenomenon that appears in polycrystalline metals when the difference between yield stress and friction stress has been increased

by grain-refinement strengthening. It is probable that the

yielding of polycrystalline iron is caused by dislocation emissions from grain boundaries where primary dislocations have piled up and segregated carbon atoms have stabilized the dislocation source at the grain boundaries. These results support a dislocation pile-up model to explain yielding in polycrystalline metals.

(4) In polycrystalline iron, carbon enhances the Hall­Petch

coefficient significantly but nitrogen does not.

Three-dimen-sional atom probe analysis showed significant grain boundary

segregation of carbon but little segregation of nitrogen in specimen that were water quenched from 700°C. It seems

that carbon has 3­4 times larger segregation potential in

comparison with nitrogen. The small effect of nitrogen on the

Hall­Petch coefficient in iron is probably due to the small

segregation potential of nitrogen.

Acknowledgment

This work was supported by Grant-in-Aid for Scientific

Research B Number 23360310.

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