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Helium effects on microstructural change in RAFM steel under irradiation: Reaction rate theory modeling
Nuclear Instruments and Methods in Physics Research B 352 (2015) 115–120
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Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Helium effects on microstructural change in RAFM steel under irradiation: Reaction rate theory modeling Y. Watanabe a,⇑, K. Morishita b, T. Nakasuji c, M. Ando a, H. Tanigawa a a
Japan Atomic Energy Agency, Rokkasho, Aomori 039-3212, Japan Institute of Advanced Energy, Kyoto University, Uji, Kyoto 611-0011, Japan c Graduate School of Energy Science, Kyoto University, Uji, Kyoto 611-0011, Japan b
a r t i c l e
i n f o
Article history: Received 11 July 2014 Received in revised form 11 December 2014 Accepted 15 December 2014 Available online 14 January 2015 Keywords: Ferritic alloy Helium effects Defect clusters Nucleation Reaction rate theory
a b s t r a c t Reaction rate theory analysis has been conducted to investigate helium effects on the formation kinetics of interstitial type dislocation loops (I-loops) and helium bubbles in reduced-activation-ferritic/martensitic steel during irradiation, by focusing on the nucleation and growth processes of the defect clusters. The rate theory model employs the size and chemical composition dependence of thermal dissociation of point defects from defect clusters. In the calculations, the temperature and the production rate of Frenkel pairs are fixed to be T = 723 K and PV = 106 dpa/s, respectively. And then, only the production rate of helium atoms was changed into the following three cases: PHe = 0, 107 and 105 appm He/s. The calculation results show that helium effect on I-loop formation quite differs from that on bubble formation. As to I-loops, the loop formation hardly depends on the existence of helium, where the number density of Iloops is almost the same for the three cases of PHe. This is because helium atoms trapped in vacancies are easily emitted into the matrix due to the recombination between the vacancies and SIAs, which induces no pronounced increase or decrease of vacancies and SIAs in the matrix, leading to no remarkable impact on the I-loop nucleation. On the other hand, the bubble formation depends much on the existence of helium, in which the number density of bubbles for PHe = 107 and 105 appm He/s is much higher than that for PHe = 0. This is because helium atoms trapped in a bubble increase the vacancy binding energy, and suppress the vacancy dissociation from the bubble, resulting in a promotion of the bubble nucleation. And then, the helium effect on the promotion of bubble nucleation is very strong, even the number of helium atoms in a bubble is not so large. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction Reduced-activation-ferritic/martensitic (RAFM) steel is proposed as one of the candidates for blanket structural materials in a nuclear fusion reactor. Blanket structural materials suffer from 14 MeV high-energy-neutron bombardments, in which many types of point defect such as vacancies, self-interstitial atoms (SIAs) and helium gas atoms are produced by atomic displacement and nuclear transmutation. Those produced point defects thermally migrate and form defect clusters, e.g., interstitial type dislocation loops (I-loops), voids and helium bubbles. Such athermal lattice defects induce the microstructural change of a material, leading to the performance degradation and deformation. Especially, helium is known to enhance formation of voids, and promote void swelling and high temperature intergranular embrittlement; ⇑ Corresponding author. Tel.: +81 175 71 6669; fax: +81 175 71 6602. E-mail address: [email protected] (Y. Watanabe). http://dx.doi.org/10.1016/j.nimb.2014.12.031 0168-583X/Ó 2014 Elsevier B.V. All rights reserved.
therefore, detailed investigation of the helium effects is necessary for the study of nuclear fusion materials. In the present study, helium effects on the formation kinetics of I-loops and helium bubbles in RAFM steel during irradiation was numerically investigated by means of reaction rate theory (mean field cluster dynamics modeling), with focusing on the nucleation and growth processes of the defect clusters. 2. Methods 2.1. Outline of rate theory model The model assumptions employed here are as follows: at first, the target microstructure of RAFM steel in the present study is a lath martensite which consists of Fe–8Cr based matrix containing heterogeneous dislocations and several precipitates. For simplicity, the lath martensite is assumed to be a homogeneity field with a constant of dislocation density. In irradiation, Frenkel pairs of
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vacancies (V) and SIAs (I) are homogeneously produced by atomic displacement. The production rate of Frenkel pairs is PV in the unit of dpa/s, where an effective production rate is described as PV ð1 er Þð1 C eq V Þ with the spontaneous recombination fraction er and the thermal equilibrium vacancy concentration C eq V . Helium atoms (He) are also homogeneously produced, as interstitial helium atoms, by nuclear transmutation reaction. The production rate is PHe in the unit of appm He/s. Those produced point defects freely migrate depending on temperature. Dislocations play a role of main sink for migrating point defects with the dislocation bias Bk ¼ ðZ I Z k Þ=Z I , where Zk is the capture efficiency of a point defect k (V, I and He) to a dislocation. Vacancies and SIAs interact with each other, and mutually annihilate (recombine). Helium atoms and SIAs do not interact with each other because both point defects have compressive fields around themselves. In contrast, helium atoms and vacancies interact with each other, in which vacancies occupied by helium atoms cannot move. Notice that the vacancies occupied by helium also can recombine easily with SIAs, emitting the helium atoms into the matrix. Namely, in the present study it is assumed that there is no difference between vacancies occupied by helium and vacancies with no helium in the recombination of SIAs. This is because the total formation energy becomes lower in the both reactions. As to formation of defect clusters, SIAs interact with each other, and form SIA-clusters (I-loops). When I-loops interact with vacancies, they become shrunk. On the other hand, vacancies and helium atoms form helium-vacancy-clusters (helium bubbles). When helium bubbles interact with SIAs, the bubbles become shrunk. Notice that a helium bubble and an SIA can interact with each other, even the bubble has a relatively high internal pressure. This is because the strain field around such a bubble is not isotropic where the one side is compressive and the opposite side is dilative [1]. As to thermal stability of defect clusters, point defects are assumed to be emitted from I-loops and helium bubbles into the matrix, depending on binding energy and temperature.
Matrix
(a)
J kIN,B( nHe , nV ) V I
J kOUT , B( n He , n V )
B(nHe, nV)
He
Matrix
(b)
J kIN,L( nI ) I V
J
L(nI)
OUT k , L( n I )
Fig. 1. Illustration for reactions between defect clusters and point defects: (a) for helium bubble B(nHe, nV) and (b) for I-loop L(nI).
B(nHe, nV). Here, the binding energy is defined as energy required to remove a point defect from a defect cluster, and is corresponding to the thermal dissociation rate of point defects from a defect cluster; therefore, this energy is a very important parameter to evaluate the nucleation and growth processes of defect clusters. As for helium k;BðnHe ;nV Þ
bubbles, the binding energy can be written by Ebind
At first, the shape of defect clusters in the present study is assumed to be spherical for helium bubbles and discoidal for Iloops. The growth or shrinkage of a defect cluster is determined by a balance between the influx and the outflux of point defects into/from the defect cluster. The influx and the outflux correspond to absorption rate and emission (thermal dissociation) rate of point defects, respectively. Fig. 1(a) shows an illustration for reactions between a helium bubble B(nHe, nV) and point defects k (V, I and He). B(nHe, nV) denotes a helium bubble consisted of nHe helium atoms and nV vacancies. In the present study, the size of a helium bubble is described by nV. It is noted that B(1, 0), B(0, 1) and B(0, nV) denote an interstitial helium atom, a single vacancy and an empty void with the size of nV, respectively. Going back to B(nHe, nV), the chemical composition ratio defined as nHe/nV (the number of helium atoms per vacancy) is called the helium-to-vacancy ratio (He/V). In other words, this ratio is the helium density in a bubble and corresponds to the internal pressure of a bubble. The influx of point defects k into a bubble B(nHe, nV) is described by J IN k;BðnHe ;nV Þ
1
¼ 4pRB Dk C k =X in the unit of s
, where Dk and Ck are
the diffusion coefficient and concentration (atomic fraction) of point defects in the matrix. X is the atomic volume and RB is the bubble radius with the relation for the volume nV X ¼ 4pR3B =3. On the other hand, the outflux of point defects from a bubble is given k;BðnHe ;nV Þ =kB TÞ=X in the unit of s1, by J OUT k;BðnHe ;nV Þ ¼ 4pRB Dk expðEbind k;BðnHe ;nV Þ
where Ebind
is the binding energy of a point defect k to a bubble
l
,
where EkF is the k;BðnHe ;nV Þ
¼ EkF
formation energy of point defect k in
is the chemical potential of point defect the matrix and l k in a bubble B(nHe, nV). Morishita et al. [2] derived the binding energy as a function of He/V, based on detailed molecular dynamics simulation results and continuum level equation. In the present work, the derived binding energy was employed to estimate the thermal dissociation rate of point defects from helium bubbles. Fig. 2 represents the binding energy of a point defect (V, I and He) to a bubble B(nHe, nV) as a function of He/V. Notice that the area in shadow (He/V > 6) is beyond the application range of the continuum model description based on the linear elastic theory. As shown in the figure, with increasing the helium density in a
8
Binding energy of a point defect to a helium bubble (eV)
2.2. Reaction model for the growth and shrinkage of defect clusters
k;BðnHe ;nV Þ
7
He
V
vacancy (V) SIA (I) helium (He)
k, B( n , n )
E bind
I
6 5 4
He
3 2 1 0 0.01
V
nV =100 nV =10
0.1
nV =1
1
10
Helium-to-vacancy ratio (He/V) Fig. 2. Binding energy of a point defect k (V, I and He) to a helium bubble B(nHe, nV) as a function of He/V.
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bubble, the binding energy for SIAs and helium atoms decreases, while the binding energy for vacancies increases. Focusing on the vacancy binding energy, the energy becomes higher as the number of helium atoms in a bubble increases, resulting in a decrease of the vacancy thermal dissociation rate from the bubble. Now, the net growth rate of a helium bubble is described as dnV dt
OUT IN OUT ¼ J IN V; BðnHe ;nV Þ J V; BðnHe ;nV Þ J I; BðnHe ;nV Þ þ J I; BðnHe ;nV Þ . And then, the
net increase rate of helium atoms in a bubble is given as He
dn dt
¼
J IN He; BðnHe ;nV Þ V
J OUT He; BðnHe ;nV Þ .
1
2
3
Lgroup
J IOUT
J IIN
J VIN
J VOUT
4
5
49
50
nI
Fig. 4. Size distribution of I-loops treated in the present study.
Notice that dnV/dt > 0 means growth,
while dn /dt < 0 means shrinkage. Fig. 3 shows the chemical composition of helium bubbles treated in the preset study. Each of grid points denotes a bubble B(nHe, nV). As shown in the figure, bubbles individually calculated are limited to the rage of 1 6 nV 6 50 and 0 6 nHe 6 50. Bubbles beyond the grid edge of nV = 50 are classified into a group named as class-1, while bubbles beyond the grid edge of nHe = 50 are classified into a group named as class-2. Class-1 and class-2 individually have the average nHe and nV of the grouped bubbles. In the same way, the formation of I-loops was also investigated. Fig. 1(b) shows an illustration for reaction between an I-loop and point defect k (I and V). L(nI) denotes an I-loop consisted of nI SIAs. The size of an I-loop is described by nI. Notice that L(1) denotes a single SIA. The influx of point defects k into a loop L(nI) is described 1 , where RL is the loop by J IN k;LðnI Þ ¼ pZ k RL Dk C k =X in the unit of s 2
radius with the relation for the volume nI X ¼ pbRL . On the other hand, the outflux of point defects from a loop is given by k;LðnI Þ
1 , where J OUT k;LðnI Þ ¼ pZ k RL Dk expðEbind =kB TÞ=X in the unit of s k;LðnI Þ
Ebind is the binding energy of a point defect k to a loop L(nI). In the present study we employed the binding energy which was derived as a function of nI, by Morishita et al. [1,2], with the formation energy of I-loops and some relation expressions. It is noted that the derived binding energy for SIAs and vacancies to a loop is much higher than 2.5 eV in a wide range of nI, which leads to that the thermal dissociation rate of SIAs and vacancies from I-loops are quite low even at relatively high temperatures. I OUT The net growth rate of an I-loop is written as dn ¼ J IN I; LðnI Þ J I; LðnI Þ dt OUT I I J IN V; LðnI Þ þ J V; LðnI Þ , where dn /dt > 0 means growth, whereas dn /dt < 0
Fig. 3. Chemical composition of helium bubbles treated in the preset study.
means shrinkage. Fig. 4 represents the size distribution of I-loops treated in the present study. Each of grid points denotes a loop L(nI) limited to the rage of 1 6 nI 6 50. I-loops beyond the grid edge of nI = 50 are classified into one group named as Lgroup ðnI Þ which has the average nI of the grouped I-loops. Based on the above assumptions, rate equations for defect interactions were derived to obtain the time evolution of concentration of defects. The derived simultaneous rate equations were then numerically solved using the Gear method [3] to evaluate the defect accumulations in a lath martensite of RAFM steel under irradiation. In Table 1, material parameters employed in the present model are listed. In the present calculations, the temperature and the production rate of Frenkel pairs were fixed to be T = 723 K and PV = 106 dpa/s respectively; and only the production rate of helium atoms was changed into the following three cases: (i) PHe = 0, (ii) PHe = 107 and (iii) PHe = 105 appm He/s. The cases (ii) and (iii) with PV = 106 dpa/s roughly correspond to HFIR and Fusion DEMO conditions, respectively [10]. Under the above irradiation conditions, helium effects on the formation kinetics of I-loops and helium bubbles were investigated with focusing on the nucleation and growth processes.
3. Results and discussion Fig. 5 shows the concentration of point defects (V, I and He) as a function of irradiation dose at 723 K for PV = 106 dpa/s and the three cases: PHe = 0, 107 and 105 appm He/s. At first, for each type of point defect, the concentration increases with increasing dose, and then becomes a steady-state with a balance between the production and the annihilation. An SIA is a rapid diffusing defect; therefore the nucleation process of I-loops takes place in a relatively early stage around 1012 dpa (106 s). In such a short time scale, a helium atom of a rapid diffusing defect also can fully migrate, while a vacancy cannot migrate enough. Here, helium atoms interact with vacancies, and constrain the vacancy migration; however the helium atoms trapped by vacancies are easily emitted, as interstitials, into the matrix via the recombination between the vacancies and SIAs. This is why the presence of helium does not prominently influence an increase or decrease of vacancies and SIAs in the matrix, leading to the no remarkable difference of the concentration of vacancies and SIAs for the three cases of PHe as shown in the figure. By the way, a vacancy is a slow diffusing defect; thus the nucleation process of helium bubbles takes place in a relatively late stage around 104 dpa (100 s). Fig. 6 shows the calculated number density and size of I-loops as a function of irradiation dose at 723 K for PV = 106 dpa/s and the three cases: PHe = 0, 107 and 105 appm He/s, where each plot is for Lgroup ðnI Þ of relatively large I-loops. The number density of Iloops increases with increasing irradiation dose, and then saturates. With the saturation of the number density, the size of I-loops gradually increases. Here, the number density does not depend on PHe, indicating no pronounced impact of helium on the nucleation of I-loops. As described in the previous section, the nucleation process of I-loops takes place in a relatively early stage, during which the concentration of SIAs and vacancies in the matrix hardly
Table 1 Material parameters employed in the present model. Material parameter
Value
Refs.
Lattice parameter, a0 Dislocation density, q
0.28655 m 1014 m2 1.70 eV
[4] [5] [4]
Formation energy of SIA, EIF
4.88 eV
[4]
Formation energy of interstitial helium atom, EHe F
5.25 eV
[4]
Migration energy for vacancies, EVm
1.3 eV
[6]
Migration energy for SIAs, EIm
0.3 eV
[6]
Migration energy for interstitial helium atoms, EHe m Capture efficiency of an SIA to a dislocation, Z I Capture efficiency of a vacancy to a dislocation, Z V Capture efficiency of an interstitial helium atom to a dislocation, Z He Recombination fraction of vacancies and SIAs in atomic displacement, er
0.078 eV
[4]
2.60 1.87 2.09
[7,8] [7,8] –
0.9
[9]
Formation energy of vacancy,
EVF
Irradiation time (s)
Concentration of point defects (V, I and He)
10-10 10-8 10-6 10-4 10-2 1 1 10-2 10-4 10-6 10-8
102 104 106 108
723 K PV =10-6 dpa/s PHe=0 appmHe/s PHe=10-7 PHe=10-5
V
10-10
I
10-12 -14
10
10-16
He
10-18 10-20 10-22 10-24 10-16 10-14 10-12 10-10 10-8 10-6 10-4 10-2 1
102
Irradiation dose (dpa) Fig. 5. Irradiation dose dependence of concentration of point defects (V, I and He) at 723 K for PV = 106 dpa/s and the three cases: PHe = 0, 107 and 105 appm He/s.
# density size 2x107
10-14
10-16 1x107
PHe=0, 10-7, 10-5 appmHe/s
-18
10-20
10-6 10 10
# density size
PHe
-10
=10-5
8.0x108
PHe=10-7, 10-5
PHe=0
10-14
0
20
1.2x109
appmHe/s
PHe=10-7
10-12
10-16
1.6x109
PHe= 0
-8
4.0x108
40
60
80
0.0 100
Irradiation dose (dpa) Fig. 7. Irradiation dose dependence of number density and size of helium bubbles at 723 K for PV = 106 dpa/s and the three cases: PHe = 0, 107 and 105 appm He/s.
depends on the presence of helium because of the easy recombination between SIAs and vacancies occupied by helium. This is the reason of the no remarkable helium effect on the I-loop nucleation. In fact, the net growth rate dnI/dt of relatively small I-loops, e.g., nI 6 20, at 1012 dpa was almost the same for the three cases of PHe. Fig. 7 shows the calculated number density and size of helium bubbles as a function of irradiation dose at 723 K for PV = 106 dpa/s and the three cases: PHe = 0, 107 and 105 appm He/s, where each plot for the number density is the total of bubbles in class-1 and class2, while each plot for the size is the average of those. Notice that the produced bubbles for PHe = 0 are in only class-1 and are empty voids (nHe = 0). In contrast to I-loops, the number density of bubbles strongly depends on the presence of helium, in which the number density for PHe = 107 and 105 appm He/s is much higher than that for PHe = 0, indicating a pronounced impact of helium on the nucleation of bubbles. For both the two cases of PHe = 107 and 105 appm He/s, the number density of bubbles in class-1 after 103 dpa was about three orders of magnitude greater than that in class-2, although there was no remarkable difference for the size. Fig. 8 represents He/V of bubbles in class-1 and class-2 as a function of irradiation dose. He/V gradually decreases with increasing dose, and approaches a constant value of (a) 4.3 104 for PHe = 105 and (b) 4.3 106 for PHe = 107 appm He/s, during which there is no remarkable difference of He/V between class-1 and class2 for each of PHe. Incidentally, the ratio, (a/b), for He/V is interestingly consistent with the relative fraction, (105/106)/(107/106), of heliumto-dpa ratio (PHe/PV). By the way, from the arguments for the size and He/V, both the bubbles in class-1 and the bubbles in class-2 grow in the same way, which indicates that the artificial way of
3x107
723 K PV =10-6 dpa/s
10-12
10
2.0x109
723 K PV =10-6 dpa/s
0
20
40
60
80
Average size of I-loops, nI
Number density of I-loops
10-10
10-4
Average size of helium bubbles, nV
Y. Watanabe et al. / Nuclear Instruments and Methods in Physics Research B 352 (2015) 115–120
Number density of helium bubbles
118
0 100
Irradiation dose (dpa) Fig. 6. Irradiation dose dependence of number density and size of I-loops at 723 K for PV = 106 dpa/s for the three cases: PHe = 0, 107 and 105 appm He/s. All plots are almost the same for the three cases of PHe.
Fig. 8. Irradiation dose dependence of He/V of bubbles in class-1 and class-2 at 723 K for PV = 106 dpa/s and the two cases: PHe = 107 and 105 appm He/s.
Y. Watanabe et al. / Nuclear Instruments and Methods in Physics Research B 352 (2015) 115–120
classification into the two categories may not be necessary for the present calculation conditions. To make a detailed analysis of the helium effect on the bubble nucleation, the following investigation was conducted. Fig. 9 is a mechanism map for relatively small bubbles at 104 dpa (100 s) for the case of PHe = 107 appm He/s. Each of grid points denotes a bubble B(nHe, nV) in the rage of 1 6 nV 6 20 and 0 6 nHe 6 20. Notice that this mechanism map [2] shows which bubble grows (dnV/dt > 0) or shrinks (dnV/dt < 0). At first, as to bubbles B(0, nV) that are empty voids, dnV/dt < 0 for nV 6 14, while dnV/dt > 0 for nV > 14. This means that an empty void larger than 14 is well thermally stable, so that the nucleation takes place easily, in which the critical nucleus size nV⁄ = 15. nV⁄ is defined as the smallest size for dnV/dt > 0. It is important to mention here that nV⁄ becomes smaller when the number of helium atoms in a bubble increases. Namely, nV⁄ = 14 for B(1, nV), 13 for B(2, nV), and 2 for B(nHe, nV) of nHe P 4. This tendency has a close relationship with the outflux of vacancies from a bubble. Fig. 10 shows the influx and outflux of point defects (V and He) to a helium bubble of nV = 10 as a function of nHe at 104 dpa (100 s) for the case of PHe = 107 appm He/s. Notice that the plots for the influx and outflux of SIAs are neglected for simplicity. With focusing on vacancies, J IN V does not depend on nHe, whereas J OUT strongly depends on nHe and drastically decreases V with increasing nHe. This is because the vacancy binding energy becomes higher as the helium density in a bubble, i.e. He/V, increases. It makes the bubble nucleation easier, resulting in a drastic increase of the number density of bubbles as shown in Fig. 7. Besides, from the argument for nV⁄, it is indicated that the helium effect on the promotion of bubble nucleation is very strong, even the number of helium in a bubble is not so large. By the way, Fig. 10 says that J IN V is about six orders of magnitude greater than J IN . This means that much more vacancies than helium atoms flow He into a bubble per second. From another viewpoint, a bubble grows through a nucleation path in which the rightward pathway than the upward one is more dominantly in Fig. 3. This is the reason of: (1) the number density of bubbles in class-1 is much higher than that in class-2; and (2) the constant value of He/V is relatively low, as described in the previous section. Now, let us mention the bubble formation for the case of PHe = 105 appm He/s. As shown in Fig. 5, the concentration of helium atoms in the matrix for PHe = 105 appm He/s is about two orders of magnitude higher than that for PHe = 107 appm He/s, which leads to about a double-digit increase IN OUT of J IN and J OUT do not He in Fig. 10. Notice that the others J V , J V He change because T and PV are fixed in the present study. Under
119
Fig. 10. Influx and outflux of point defects (V and He) to a helium bubble of nV = 10 as a function of nHe at 104 dpa and 723 K for PV = 106 dpa/s and PHe = 107 appm He/s.
the above conditions, more helium atoms inflow into a bubble; IN nevertheless, J IN V remains much larger than J He by four orders of magnitude, resulting in that a bubble grows through a nucleation path similar to that for the case of PHe = 107 appm He/s. This is the reason of the no remarkable difference between PHe = 105 and PHe = 107 appm He/s for the number density of bubbles as shown in Fig. 7.
4. Summary Helium effects on the formation kinetics of I-loops and helium bubbles in a lath martensite of RAFM steel during irradiation were numerically investigated by means of reaction rate theory, with focusing on the nucleation and growth processes at 723 K for PV = 10–6 dpa/s and the three cases: PHe = 0, 10–7 and 10–5 appm He/s. In the rate theory model, the thermal dissociation rate of point defects from a defect cluster was employed as functions of size and chemical composition ratio of the defect cluster. The calculation results say that helium effect on I-loop formation is quite different from that on bubble formation. As to I-loops, the loop formation hardly depends on the presence of helium, in which the number density of I-loops is almost the same for the three cases of PHe. This is because helium atoms trapped in vacancies are easily emitted into the matrix due to the recombination between the vacancies and SIAs, which causes no pronounced change of the concentration of vacancies and SIAs in the matrix for the three cases of PHe, leading to no remarkable impact on the I-loop nucleation. As to helium bubbles, the bubble formation strongly depends on the presence of helium, in which the number density of bubbles for PHe = 10–7 and 10–5 appm He/s is much higher than that for PHe = 0. This is because helium atoms in a bubble raise the vacancy binding energy, and constrain the vacancy dissociation rate from the bubble, resulting in a promotion of the bubble nucleation. And then, the helium effect on the promotion of bubble nucleation is very strong, even the number of helium atoms in a bubble is not so large. This kind of information will be a basic of the theoretical understanding of helium effects on the defect cluster formation in RAFM steel under a wider range of irradiation conditions. Acknowledgment
Fig. 9. Mechanism map for relatively small bubbles at 104 dpa and 723 K for PV = 106 dpa/s and PHe = 107 appm He/s.
This work was supported by the ‘‘Joint Usage/Research Program on Zero-Emission Energy Research, Institute of Advanced Energy, Kyoto University (ZE26B-25)’’.
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Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Helium effects on microstructural change in RAFM steel under irradiation: Reaction rate theory modeling Y. Watanabe a,⇑, K. Morishita b, T. Nakasuji c, M. Ando a, H. Tanigawa a a
Japan Atomic Energy Agency, Rokkasho, Aomori 039-3212, Japan Institute of Advanced Energy, Kyoto University, Uji, Kyoto 611-0011, Japan c Graduate School of Energy Science, Kyoto University, Uji, Kyoto 611-0011, Japan b
a r t i c l e
i n f o
Article history: Received 11 July 2014 Received in revised form 11 December 2014 Accepted 15 December 2014 Available online 14 January 2015 Keywords: Ferritic alloy Helium effects Defect clusters Nucleation Reaction rate theory
a b s t r a c t Reaction rate theory analysis has been conducted to investigate helium effects on the formation kinetics of interstitial type dislocation loops (I-loops) and helium bubbles in reduced-activation-ferritic/martensitic steel during irradiation, by focusing on the nucleation and growth processes of the defect clusters. The rate theory model employs the size and chemical composition dependence of thermal dissociation of point defects from defect clusters. In the calculations, the temperature and the production rate of Frenkel pairs are fixed to be T = 723 K and PV = 106 dpa/s, respectively. And then, only the production rate of helium atoms was changed into the following three cases: PHe = 0, 107 and 105 appm He/s. The calculation results show that helium effect on I-loop formation quite differs from that on bubble formation. As to I-loops, the loop formation hardly depends on the existence of helium, where the number density of Iloops is almost the same for the three cases of PHe. This is because helium atoms trapped in vacancies are easily emitted into the matrix due to the recombination between the vacancies and SIAs, which induces no pronounced increase or decrease of vacancies and SIAs in the matrix, leading to no remarkable impact on the I-loop nucleation. On the other hand, the bubble formation depends much on the existence of helium, in which the number density of bubbles for PHe = 107 and 105 appm He/s is much higher than that for PHe = 0. This is because helium atoms trapped in a bubble increase the vacancy binding energy, and suppress the vacancy dissociation from the bubble, resulting in a promotion of the bubble nucleation. And then, the helium effect on the promotion of bubble nucleation is very strong, even the number of helium atoms in a bubble is not so large. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction Reduced-activation-ferritic/martensitic (RAFM) steel is proposed as one of the candidates for blanket structural materials in a nuclear fusion reactor. Blanket structural materials suffer from 14 MeV high-energy-neutron bombardments, in which many types of point defect such as vacancies, self-interstitial atoms (SIAs) and helium gas atoms are produced by atomic displacement and nuclear transmutation. Those produced point defects thermally migrate and form defect clusters, e.g., interstitial type dislocation loops (I-loops), voids and helium bubbles. Such athermal lattice defects induce the microstructural change of a material, leading to the performance degradation and deformation. Especially, helium is known to enhance formation of voids, and promote void swelling and high temperature intergranular embrittlement; ⇑ Corresponding author. Tel.: +81 175 71 6669; fax: +81 175 71 6602. E-mail address: [email protected] (Y. Watanabe). http://dx.doi.org/10.1016/j.nimb.2014.12.031 0168-583X/Ó 2014 Elsevier B.V. All rights reserved.
therefore, detailed investigation of the helium effects is necessary for the study of nuclear fusion materials. In the present study, helium effects on the formation kinetics of I-loops and helium bubbles in RAFM steel during irradiation was numerically investigated by means of reaction rate theory (mean field cluster dynamics modeling), with focusing on the nucleation and growth processes of the defect clusters. 2. Methods 2.1. Outline of rate theory model The model assumptions employed here are as follows: at first, the target microstructure of RAFM steel in the present study is a lath martensite which consists of Fe–8Cr based matrix containing heterogeneous dislocations and several precipitates. For simplicity, the lath martensite is assumed to be a homogeneity field with a constant of dislocation density. In irradiation, Frenkel pairs of
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vacancies (V) and SIAs (I) are homogeneously produced by atomic displacement. The production rate of Frenkel pairs is PV in the unit of dpa/s, where an effective production rate is described as PV ð1 er Þð1 C eq V Þ with the spontaneous recombination fraction er and the thermal equilibrium vacancy concentration C eq V . Helium atoms (He) are also homogeneously produced, as interstitial helium atoms, by nuclear transmutation reaction. The production rate is PHe in the unit of appm He/s. Those produced point defects freely migrate depending on temperature. Dislocations play a role of main sink for migrating point defects with the dislocation bias Bk ¼ ðZ I Z k Þ=Z I , where Zk is the capture efficiency of a point defect k (V, I and He) to a dislocation. Vacancies and SIAs interact with each other, and mutually annihilate (recombine). Helium atoms and SIAs do not interact with each other because both point defects have compressive fields around themselves. In contrast, helium atoms and vacancies interact with each other, in which vacancies occupied by helium atoms cannot move. Notice that the vacancies occupied by helium also can recombine easily with SIAs, emitting the helium atoms into the matrix. Namely, in the present study it is assumed that there is no difference between vacancies occupied by helium and vacancies with no helium in the recombination of SIAs. This is because the total formation energy becomes lower in the both reactions. As to formation of defect clusters, SIAs interact with each other, and form SIA-clusters (I-loops). When I-loops interact with vacancies, they become shrunk. On the other hand, vacancies and helium atoms form helium-vacancy-clusters (helium bubbles). When helium bubbles interact with SIAs, the bubbles become shrunk. Notice that a helium bubble and an SIA can interact with each other, even the bubble has a relatively high internal pressure. This is because the strain field around such a bubble is not isotropic where the one side is compressive and the opposite side is dilative [1]. As to thermal stability of defect clusters, point defects are assumed to be emitted from I-loops and helium bubbles into the matrix, depending on binding energy and temperature.
Matrix
(a)
J kIN,B( nHe , nV ) V I
J kOUT , B( n He , n V )
B(nHe, nV)
He
Matrix
(b)
J kIN,L( nI ) I V
J
L(nI)
OUT k , L( n I )
Fig. 1. Illustration for reactions between defect clusters and point defects: (a) for helium bubble B(nHe, nV) and (b) for I-loop L(nI).
B(nHe, nV). Here, the binding energy is defined as energy required to remove a point defect from a defect cluster, and is corresponding to the thermal dissociation rate of point defects from a defect cluster; therefore, this energy is a very important parameter to evaluate the nucleation and growth processes of defect clusters. As for helium k;BðnHe ;nV Þ
bubbles, the binding energy can be written by Ebind
At first, the shape of defect clusters in the present study is assumed to be spherical for helium bubbles and discoidal for Iloops. The growth or shrinkage of a defect cluster is determined by a balance between the influx and the outflux of point defects into/from the defect cluster. The influx and the outflux correspond to absorption rate and emission (thermal dissociation) rate of point defects, respectively. Fig. 1(a) shows an illustration for reactions between a helium bubble B(nHe, nV) and point defects k (V, I and He). B(nHe, nV) denotes a helium bubble consisted of nHe helium atoms and nV vacancies. In the present study, the size of a helium bubble is described by nV. It is noted that B(1, 0), B(0, 1) and B(0, nV) denote an interstitial helium atom, a single vacancy and an empty void with the size of nV, respectively. Going back to B(nHe, nV), the chemical composition ratio defined as nHe/nV (the number of helium atoms per vacancy) is called the helium-to-vacancy ratio (He/V). In other words, this ratio is the helium density in a bubble and corresponds to the internal pressure of a bubble. The influx of point defects k into a bubble B(nHe, nV) is described by J IN k;BðnHe ;nV Þ
1
¼ 4pRB Dk C k =X in the unit of s
, where Dk and Ck are
the diffusion coefficient and concentration (atomic fraction) of point defects in the matrix. X is the atomic volume and RB is the bubble radius with the relation for the volume nV X ¼ 4pR3B =3. On the other hand, the outflux of point defects from a bubble is given k;BðnHe ;nV Þ =kB TÞ=X in the unit of s1, by J OUT k;BðnHe ;nV Þ ¼ 4pRB Dk expðEbind k;BðnHe ;nV Þ
where Ebind
is the binding energy of a point defect k to a bubble
l
,
where EkF is the k;BðnHe ;nV Þ
¼ EkF
formation energy of point defect k in
is the chemical potential of point defect the matrix and l k in a bubble B(nHe, nV). Morishita et al. [2] derived the binding energy as a function of He/V, based on detailed molecular dynamics simulation results and continuum level equation. In the present work, the derived binding energy was employed to estimate the thermal dissociation rate of point defects from helium bubbles. Fig. 2 represents the binding energy of a point defect (V, I and He) to a bubble B(nHe, nV) as a function of He/V. Notice that the area in shadow (He/V > 6) is beyond the application range of the continuum model description based on the linear elastic theory. As shown in the figure, with increasing the helium density in a
8
Binding energy of a point defect to a helium bubble (eV)
2.2. Reaction model for the growth and shrinkage of defect clusters
k;BðnHe ;nV Þ
7
He
V
vacancy (V) SIA (I) helium (He)
k, B( n , n )
E bind
I
6 5 4
He
3 2 1 0 0.01
V
nV =100 nV =10
0.1
nV =1
1
10
Helium-to-vacancy ratio (He/V) Fig. 2. Binding energy of a point defect k (V, I and He) to a helium bubble B(nHe, nV) as a function of He/V.
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bubble, the binding energy for SIAs and helium atoms decreases, while the binding energy for vacancies increases. Focusing on the vacancy binding energy, the energy becomes higher as the number of helium atoms in a bubble increases, resulting in a decrease of the vacancy thermal dissociation rate from the bubble. Now, the net growth rate of a helium bubble is described as dnV dt
OUT IN OUT ¼ J IN V; BðnHe ;nV Þ J V; BðnHe ;nV Þ J I; BðnHe ;nV Þ þ J I; BðnHe ;nV Þ . And then, the
net increase rate of helium atoms in a bubble is given as He
dn dt
¼
J IN He; BðnHe ;nV Þ V
J OUT He; BðnHe ;nV Þ .
1
2
3
Lgroup
J IOUT
J IIN
J VIN
J VOUT
4
5
49
50
nI
Fig. 4. Size distribution of I-loops treated in the present study.
Notice that dnV/dt > 0 means growth,
while dn /dt < 0 means shrinkage. Fig. 3 shows the chemical composition of helium bubbles treated in the preset study. Each of grid points denotes a bubble B(nHe, nV). As shown in the figure, bubbles individually calculated are limited to the rage of 1 6 nV 6 50 and 0 6 nHe 6 50. Bubbles beyond the grid edge of nV = 50 are classified into a group named as class-1, while bubbles beyond the grid edge of nHe = 50 are classified into a group named as class-2. Class-1 and class-2 individually have the average nHe and nV of the grouped bubbles. In the same way, the formation of I-loops was also investigated. Fig. 1(b) shows an illustration for reaction between an I-loop and point defect k (I and V). L(nI) denotes an I-loop consisted of nI SIAs. The size of an I-loop is described by nI. Notice that L(1) denotes a single SIA. The influx of point defects k into a loop L(nI) is described 1 , where RL is the loop by J IN k;LðnI Þ ¼ pZ k RL Dk C k =X in the unit of s 2
radius with the relation for the volume nI X ¼ pbRL . On the other hand, the outflux of point defects from a loop is given by k;LðnI Þ
1 , where J OUT k;LðnI Þ ¼ pZ k RL Dk expðEbind =kB TÞ=X in the unit of s k;LðnI Þ
Ebind is the binding energy of a point defect k to a loop L(nI). In the present study we employed the binding energy which was derived as a function of nI, by Morishita et al. [1,2], with the formation energy of I-loops and some relation expressions. It is noted that the derived binding energy for SIAs and vacancies to a loop is much higher than 2.5 eV in a wide range of nI, which leads to that the thermal dissociation rate of SIAs and vacancies from I-loops are quite low even at relatively high temperatures. I OUT The net growth rate of an I-loop is written as dn ¼ J IN I; LðnI Þ J I; LðnI Þ dt OUT I I J IN V; LðnI Þ þ J V; LðnI Þ , where dn /dt > 0 means growth, whereas dn /dt < 0
Fig. 3. Chemical composition of helium bubbles treated in the preset study.
means shrinkage. Fig. 4 represents the size distribution of I-loops treated in the present study. Each of grid points denotes a loop L(nI) limited to the rage of 1 6 nI 6 50. I-loops beyond the grid edge of nI = 50 are classified into one group named as Lgroup ðnI Þ which has the average nI of the grouped I-loops. Based on the above assumptions, rate equations for defect interactions were derived to obtain the time evolution of concentration of defects. The derived simultaneous rate equations were then numerically solved using the Gear method [3] to evaluate the defect accumulations in a lath martensite of RAFM steel under irradiation. In Table 1, material parameters employed in the present model are listed. In the present calculations, the temperature and the production rate of Frenkel pairs were fixed to be T = 723 K and PV = 106 dpa/s respectively; and only the production rate of helium atoms was changed into the following three cases: (i) PHe = 0, (ii) PHe = 107 and (iii) PHe = 105 appm He/s. The cases (ii) and (iii) with PV = 106 dpa/s roughly correspond to HFIR and Fusion DEMO conditions, respectively [10]. Under the above irradiation conditions, helium effects on the formation kinetics of I-loops and helium bubbles were investigated with focusing on the nucleation and growth processes.
3. Results and discussion Fig. 5 shows the concentration of point defects (V, I and He) as a function of irradiation dose at 723 K for PV = 106 dpa/s and the three cases: PHe = 0, 107 and 105 appm He/s. At first, for each type of point defect, the concentration increases with increasing dose, and then becomes a steady-state with a balance between the production and the annihilation. An SIA is a rapid diffusing defect; therefore the nucleation process of I-loops takes place in a relatively early stage around 1012 dpa (106 s). In such a short time scale, a helium atom of a rapid diffusing defect also can fully migrate, while a vacancy cannot migrate enough. Here, helium atoms interact with vacancies, and constrain the vacancy migration; however the helium atoms trapped by vacancies are easily emitted, as interstitials, into the matrix via the recombination between the vacancies and SIAs. This is why the presence of helium does not prominently influence an increase or decrease of vacancies and SIAs in the matrix, leading to the no remarkable difference of the concentration of vacancies and SIAs for the three cases of PHe as shown in the figure. By the way, a vacancy is a slow diffusing defect; thus the nucleation process of helium bubbles takes place in a relatively late stage around 104 dpa (100 s). Fig. 6 shows the calculated number density and size of I-loops as a function of irradiation dose at 723 K for PV = 106 dpa/s and the three cases: PHe = 0, 107 and 105 appm He/s, where each plot is for Lgroup ðnI Þ of relatively large I-loops. The number density of Iloops increases with increasing irradiation dose, and then saturates. With the saturation of the number density, the size of I-loops gradually increases. Here, the number density does not depend on PHe, indicating no pronounced impact of helium on the nucleation of I-loops. As described in the previous section, the nucleation process of I-loops takes place in a relatively early stage, during which the concentration of SIAs and vacancies in the matrix hardly
Table 1 Material parameters employed in the present model. Material parameter
Value
Refs.
Lattice parameter, a0 Dislocation density, q
0.28655 m 1014 m2 1.70 eV
[4] [5] [4]
Formation energy of SIA, EIF
4.88 eV
[4]
Formation energy of interstitial helium atom, EHe F
5.25 eV
[4]
Migration energy for vacancies, EVm
1.3 eV
[6]
Migration energy for SIAs, EIm
0.3 eV
[6]
Migration energy for interstitial helium atoms, EHe m Capture efficiency of an SIA to a dislocation, Z I Capture efficiency of a vacancy to a dislocation, Z V Capture efficiency of an interstitial helium atom to a dislocation, Z He Recombination fraction of vacancies and SIAs in atomic displacement, er
0.078 eV
[4]
2.60 1.87 2.09
[7,8] [7,8] –
0.9
[9]
Formation energy of vacancy,
EVF
Irradiation time (s)
Concentration of point defects (V, I and He)
10-10 10-8 10-6 10-4 10-2 1 1 10-2 10-4 10-6 10-8
102 104 106 108
723 K PV =10-6 dpa/s PHe=0 appmHe/s PHe=10-7 PHe=10-5
V
10-10
I
10-12 -14
10
10-16
He
10-18 10-20 10-22 10-24 10-16 10-14 10-12 10-10 10-8 10-6 10-4 10-2 1
102
Irradiation dose (dpa) Fig. 5. Irradiation dose dependence of concentration of point defects (V, I and He) at 723 K for PV = 106 dpa/s and the three cases: PHe = 0, 107 and 105 appm He/s.
# density size 2x107
10-14
10-16 1x107
PHe=0, 10-7, 10-5 appmHe/s
-18
10-20
10-6 10 10
# density size
PHe
-10
=10-5
8.0x108
PHe=10-7, 10-5
PHe=0
10-14
0
20
1.2x109
appmHe/s
PHe=10-7
10-12
10-16
1.6x109
PHe= 0
-8
4.0x108
40
60
80
0.0 100
Irradiation dose (dpa) Fig. 7. Irradiation dose dependence of number density and size of helium bubbles at 723 K for PV = 106 dpa/s and the three cases: PHe = 0, 107 and 105 appm He/s.
depends on the presence of helium because of the easy recombination between SIAs and vacancies occupied by helium. This is the reason of the no remarkable helium effect on the I-loop nucleation. In fact, the net growth rate dnI/dt of relatively small I-loops, e.g., nI 6 20, at 1012 dpa was almost the same for the three cases of PHe. Fig. 7 shows the calculated number density and size of helium bubbles as a function of irradiation dose at 723 K for PV = 106 dpa/s and the three cases: PHe = 0, 107 and 105 appm He/s, where each plot for the number density is the total of bubbles in class-1 and class2, while each plot for the size is the average of those. Notice that the produced bubbles for PHe = 0 are in only class-1 and are empty voids (nHe = 0). In contrast to I-loops, the number density of bubbles strongly depends on the presence of helium, in which the number density for PHe = 107 and 105 appm He/s is much higher than that for PHe = 0, indicating a pronounced impact of helium on the nucleation of bubbles. For both the two cases of PHe = 107 and 105 appm He/s, the number density of bubbles in class-1 after 103 dpa was about three orders of magnitude greater than that in class-2, although there was no remarkable difference for the size. Fig. 8 represents He/V of bubbles in class-1 and class-2 as a function of irradiation dose. He/V gradually decreases with increasing dose, and approaches a constant value of (a) 4.3 104 for PHe = 105 and (b) 4.3 106 for PHe = 107 appm He/s, during which there is no remarkable difference of He/V between class-1 and class2 for each of PHe. Incidentally, the ratio, (a/b), for He/V is interestingly consistent with the relative fraction, (105/106)/(107/106), of heliumto-dpa ratio (PHe/PV). By the way, from the arguments for the size and He/V, both the bubbles in class-1 and the bubbles in class-2 grow in the same way, which indicates that the artificial way of
3x107
723 K PV =10-6 dpa/s
10-12
10
2.0x109
723 K PV =10-6 dpa/s
0
20
40
60
80
Average size of I-loops, nI
Number density of I-loops
10-10
10-4
Average size of helium bubbles, nV
Y. Watanabe et al. / Nuclear Instruments and Methods in Physics Research B 352 (2015) 115–120
Number density of helium bubbles
118
0 100
Irradiation dose (dpa) Fig. 6. Irradiation dose dependence of number density and size of I-loops at 723 K for PV = 106 dpa/s for the three cases: PHe = 0, 107 and 105 appm He/s. All plots are almost the same for the three cases of PHe.
Fig. 8. Irradiation dose dependence of He/V of bubbles in class-1 and class-2 at 723 K for PV = 106 dpa/s and the two cases: PHe = 107 and 105 appm He/s.
Y. Watanabe et al. / Nuclear Instruments and Methods in Physics Research B 352 (2015) 115–120
classification into the two categories may not be necessary for the present calculation conditions. To make a detailed analysis of the helium effect on the bubble nucleation, the following investigation was conducted. Fig. 9 is a mechanism map for relatively small bubbles at 104 dpa (100 s) for the case of PHe = 107 appm He/s. Each of grid points denotes a bubble B(nHe, nV) in the rage of 1 6 nV 6 20 and 0 6 nHe 6 20. Notice that this mechanism map [2] shows which bubble grows (dnV/dt > 0) or shrinks (dnV/dt < 0). At first, as to bubbles B(0, nV) that are empty voids, dnV/dt < 0 for nV 6 14, while dnV/dt > 0 for nV > 14. This means that an empty void larger than 14 is well thermally stable, so that the nucleation takes place easily, in which the critical nucleus size nV⁄ = 15. nV⁄ is defined as the smallest size for dnV/dt > 0. It is important to mention here that nV⁄ becomes smaller when the number of helium atoms in a bubble increases. Namely, nV⁄ = 14 for B(1, nV), 13 for B(2, nV), and 2 for B(nHe, nV) of nHe P 4. This tendency has a close relationship with the outflux of vacancies from a bubble. Fig. 10 shows the influx and outflux of point defects (V and He) to a helium bubble of nV = 10 as a function of nHe at 104 dpa (100 s) for the case of PHe = 107 appm He/s. Notice that the plots for the influx and outflux of SIAs are neglected for simplicity. With focusing on vacancies, J IN V does not depend on nHe, whereas J OUT strongly depends on nHe and drastically decreases V with increasing nHe. This is because the vacancy binding energy becomes higher as the helium density in a bubble, i.e. He/V, increases. It makes the bubble nucleation easier, resulting in a drastic increase of the number density of bubbles as shown in Fig. 7. Besides, from the argument for nV⁄, it is indicated that the helium effect on the promotion of bubble nucleation is very strong, even the number of helium in a bubble is not so large. By the way, Fig. 10 says that J IN V is about six orders of magnitude greater than J IN . This means that much more vacancies than helium atoms flow He into a bubble per second. From another viewpoint, a bubble grows through a nucleation path in which the rightward pathway than the upward one is more dominantly in Fig. 3. This is the reason of: (1) the number density of bubbles in class-1 is much higher than that in class-2; and (2) the constant value of He/V is relatively low, as described in the previous section. Now, let us mention the bubble formation for the case of PHe = 105 appm He/s. As shown in Fig. 5, the concentration of helium atoms in the matrix for PHe = 105 appm He/s is about two orders of magnitude higher than that for PHe = 107 appm He/s, which leads to about a double-digit increase IN OUT of J IN and J OUT do not He in Fig. 10. Notice that the others J V , J V He change because T and PV are fixed in the present study. Under
119
Fig. 10. Influx and outflux of point defects (V and He) to a helium bubble of nV = 10 as a function of nHe at 104 dpa and 723 K for PV = 106 dpa/s and PHe = 107 appm He/s.
the above conditions, more helium atoms inflow into a bubble; IN nevertheless, J IN V remains much larger than J He by four orders of magnitude, resulting in that a bubble grows through a nucleation path similar to that for the case of PHe = 107 appm He/s. This is the reason of the no remarkable difference between PHe = 105 and PHe = 107 appm He/s for the number density of bubbles as shown in Fig. 7.
4. Summary Helium effects on the formation kinetics of I-loops and helium bubbles in a lath martensite of RAFM steel during irradiation were numerically investigated by means of reaction rate theory, with focusing on the nucleation and growth processes at 723 K for PV = 10–6 dpa/s and the three cases: PHe = 0, 10–7 and 10–5 appm He/s. In the rate theory model, the thermal dissociation rate of point defects from a defect cluster was employed as functions of size and chemical composition ratio of the defect cluster. The calculation results say that helium effect on I-loop formation is quite different from that on bubble formation. As to I-loops, the loop formation hardly depends on the presence of helium, in which the number density of I-loops is almost the same for the three cases of PHe. This is because helium atoms trapped in vacancies are easily emitted into the matrix due to the recombination between the vacancies and SIAs, which causes no pronounced change of the concentration of vacancies and SIAs in the matrix for the three cases of PHe, leading to no remarkable impact on the I-loop nucleation. As to helium bubbles, the bubble formation strongly depends on the presence of helium, in which the number density of bubbles for PHe = 10–7 and 10–5 appm He/s is much higher than that for PHe = 0. This is because helium atoms in a bubble raise the vacancy binding energy, and constrain the vacancy dissociation rate from the bubble, resulting in a promotion of the bubble nucleation. And then, the helium effect on the promotion of bubble nucleation is very strong, even the number of helium atoms in a bubble is not so large. This kind of information will be a basic of the theoretical understanding of helium effects on the defect cluster formation in RAFM steel under a wider range of irradiation conditions. Acknowledgment
Fig. 9. Mechanism map for relatively small bubbles at 104 dpa and 723 K for PV = 106 dpa/s and PHe = 107 appm He/s.
This work was supported by the ‘‘Joint Usage/Research Program on Zero-Emission Energy Research, Institute of Advanced Energy, Kyoto University (ZE26B-25)’’.
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