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Results of Sensitivity Analyses

Sensitivity analyses were conducted for several selected nuclides in Zircaloy-2, SUS304 stainless steel, and INCONEL alloy. Analyses in Zircaloy-4 were skipped because the sensitivity coefficients were thought to be almost the same as that in Zircaloy-2 because calculation conditions were similar. For SUS304 stainless steel, activations using the cross-section library of void ratio 0 % were evaluated because the concentrations in the case of void ratio 0 % were larger than that of void ratio 70 %.

The sensitivity coefficients of initial compositions are shown in Table 20.7. As defined in Eq. (20.1), the value shows the relative amount of variation in concentration of the target nuclide when the initial composition of element varies by a unit amount. Therefore, the source elements leading to the generation of target nuclides was clarified from the results. For example, Table 20.7a shows that Fe-55 is generated from both iron and nickel and that the contribution from iron is dominant. The results can also be useful in the evaluation of the error propagated from the measurement uncertainty of initial composition.

As defined in Eq. (20.2), a sensitivity coefficient of a cross section shows the relative amount of variation in the concentration of the target nuclide when the

Table 20.7 Sensitivity coefficients of initial composition

Target nuclide

Zr-93

Ni-59

Ni-63

Co-60

C-14

Nb-94

Sb125

Ca-41

K-40

Fe-55

Tc-99

Mo-93

Be-10

Mn-54

Ag108 m

H-3

Zn-65

(a) Zircaloy-2

H

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0.9995

0

B

0

0

0

0

0

0

0

0

0

0

0

0

0.0334

0

0

0

0

C

0

0

0

0

5E-05

0

0

0

0

0

0

0

0.9666

0

0

0

0

N

0

0

0

0

1.0000

0

0

0

0

0

0

0

0

0

0

0.0003

0

Mg

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Al

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Si

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Ca

0

0

0

0

0

0

0

1.0000

1.0000

0

0

0

0

0

0

0

0

Ti

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Cr

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Mn

0

0

0

0

0

0

0

0

0

0

0

0

0

0.0032

0

0

0

Fe

0

0

0

0.0004

0

0

0

0

0

0.9564

0

0

0

0.9967

0

0

0

Co

0

0

0

0.9980

0

0

0

0

0

0

0

0

0

0

0

0

0

Ni

0

1.0000

0.9904

0.0015

0

0

0

0

0

0.0436

0

0

0

0

0

0

0.0001

Cu

0

0

0.0096

0.0001

0

0

0

0

0

0

0

0

0

0

0

0

0.9999

Zr

1.0000

0

0

0

0

0.0001

0

0

0

0

0.0289

0

0

0

0

0

0

Mo

0

0

0

0

0

0.0001

0

0

0

0

0.9711

1.0000

0

0

0

0

0

Nb

0

0

0

0

0

0.9998

0

0

0

0

0

0

0

0

0

0

0

Cd

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1.0000

0

0

Sn

0

0

0

0

0

0

1.0000

0

0

0

0

0

0

0

0

0

0

Hf

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

W

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

(b) SUS304 stainless steel

Target nuclide

Ni-59

Ni-63

Fe-55

Co-60

Mn-54

Be-10

C-14

Cl-36

C

0

0

0

0

0

1.0000

1.0000

0

Si

0

0

0

0

0

0

0

0

P

0

0

0

0

0

0

0

0

S

0

0

0

0

0

0

0

1.0000

Cr

0

0

0

0

0

0

0

0

Mn

0

0

0

0

0.0027

0

0

0

Fe

0

0

0.9890

0.0819

0.9974

0

0

0

Ni

1.0000

1.0000

0.0110

0.9181

0

0

0

0

(c) INCONEL alloy 718

Target nuclide

Ni-59

Ni-63

Co-60

Nb-94

Mo-93

Tc-99

Fe-55

Zr-93

Mn-54

Be-10

Cl-36

C-14

Zn-65

Sr-90

Si-32

H-3

B

0

0

0

0

0

0

0

0

0

0.5830

0

0

0

0

0

0

C

0

0

0

0

0

0

0

0

0

0.4170

0

1.0000

0

0

0

0

Al

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0.0012

Si

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0.7059

0.0011

P

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0.2926

0.0002

S

0

0

0

0

0

0

0

0

0

0

1.0000

0

0

0

0.0015

0.0005

Ti

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0.0007

Cr

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0.0121

Mn

0

0

0

0

0

0

0

0

0.0019

0

0

0

0

0

0

0.0001

Fe

0

0

0.0001

0

0

0

0.7411

0

0.9982

0

0

0

0

0

0

0.0258

Co

0

0

0.9970

0

0

0

0

0

0

0

0

0

0

0

0

0.0004

Ni

1.0000

0.9994

0.0028

0

0

0

0.2589

0

0

0

0

0

0.0014

0

0

0.9557

Cu

0

0.0006

0

0

0

0

0

0

0

0

0

0

0.9985

0

0

0.0012

Nb

0

0

0

0.9999

0

0

0

0.9839

0

0

0

0

0

0.9841

0

0.0003

Mo

0

0

0

0.0001

1.0000

1.0000

0

0.0161

0

0

0

0

0

0.0159

0

0.0008

Table 20.8 Sensitivity coefficients of cross sections

Target nuclide

Sensitivity coefficient of cross section

First largest

Second largest

Others

(a) Zircaloy-2

Zr-93

Zr-92

(n, γ)

0.98

Zr-94

(n, 2n)

0.02

Ni-59

Ni-58

(n, γ)

0.99

Ni-63

Ni-62

(n, γ)

0.97

Co-60

Co-59

(n, γ)m

0.46

Co-59

(n, γ)

0.42

C-14

N-14

(n, p)

1.00

Nb-94

Nb-93

(n, γ)

1.00

Sb-125

Sn-124

(n, γ)

0.56

Sn-124

(n, γ)m

0.48

Ca-41

Ca-40

(n, γ)

1.00

K-40

Ca-40

(n, p)

1.00

Fe-55

Fe-54

(n, γ)

0.95

Ni-58

(n, α)

0.04

Tc-99

Mo-98

(n, γ)

1.00

Mo-97

(n, γ)

0.03

Zr-96

(n, γ)

0.03

Mo-93

Mo-92

(n, γ)

0.99

Be-10

C-13

(n, α)

0.97

B-10

(n, p)

0.03

Mn-54

Fe-54

(n, p)

1.00

Ag-108 m

Cd-106

(n, γ)

1.00

Ag-107

(n, γ)m

0.97

H-3

H-2

(n, γ)

1.00

H-1

(n, γ)

0.78

He-3

(n, p)

0.01

Zn-65

Zn-64

(n, γ)

1.00

Cu-63

(n, γ)

1.00

(b) SUS304 stainless steel

Ni-59

Ni-58

(n, γ)

1.00

Ni-63

Ni-62

(n, γ)

1.00

Fe-55

Fe-54

(n, γ)

0.99

Ni-58

(n, α)

0.01

Co-60

Ni-60

(n, p)

0.92

Fe-58

(n, γ)

0.08

Co-59

(n, γ)

0.04

Co-59

(n, γ)m

0.04

Mn-54

Fe-54

(n, p)

1.00

Be-10

C-13

(n, α)

1.00

C-14

C-13

(n, γ)

1.00

Cl-36

S-34

(n, γ)

1.00

Cl-35

(n, γ)

1.00

(c) INCONEL alloy 718

Ni-59

Ni-58

(n, γ)

0.99

Ni-63

Ni-62

(n, γ)

0.98

Co-60

Co-59

(n, γ)m

0.46

Co-59

(n, γ)

0.42

Nb-94

Nb-93

(n, γ)

1.00

Mo-93

Mo-92

(n, γ)

0.99

Tc-99

Mo-98

(n, γ)

1.00

Fe-55

Fe-54

(n, γ)

0.74

Ni-58

(n, α)

0.26

Zr-93

Nb-93

(n, p)

0.98

Mo-96

(n, α)

0.02

Mn-54

Fe-54

(n, p)

1.00

Be-10

B-10

(n, p)

0.58

C-13

(n, α)

0.42

Cl-36

S-34

(n, γ)

1.00

Cl-35

(n, γ)

0.97

C-14

C-13

(n, γ)

1.00

Table 20.8 (continued)

Target nuclide

Sensitivity coefficient of cross section

First largest

Second largest

Others

Zn-65

Zn-64

(n, γ)

0.99

Cu-63

(n, γ)

0.99

Sr-90

Zr-93

(n, α)

1.00

Nb-93

(n, p)

0.98

Si-32

Si-31

(n, γ)

1.00

Si-30

(n, γ)

0.71

P-31

(n, p)

0.29

H-3

H-2

(n, γ)

1.00

H-1

(n, γ)

1.00

Ni-58

(n, p)

0.95

He-3

(n, p)

0.01

(n, γ)m means the (n, γ) reaction yielding to meta-stable state

Fig. 20.2 Examples of complicated generation pathways of activation products. The values in bold font in the figures show the sensitivity coefficients of the cross sections. a Be-10 generation in Zircaloy-2. b Cl-36 generation in SUS304 stainless steel

cross section varies by a unit amount. Therefore, a positive value of this coefficient indicates that the target activation product is generated through the nuclear reaction. Thus, if a sensitivity coefficient is positive and large, the cross section of the nuclear reaction is significant for the generation of the target activation products. In the analyses, the objectives of reaction were six reactions treated in ORLIBJ40 library; the reaction of (n, γ), (n, 2n), (n, α), and (n, p) yielding to nuclides of ground state and the reaction of (n, γ) and (n, 2n) yielding to nuclides of meta-stable state. The summary of the results of sensitivity analyses of cross sections are shown in Table 20.8, where the sensitivity coefficients that are positive and more than 0.01 are extracted from all the results and listed in descending order. The results clarified the nuclear reaction dominating the generation of target nuclides. For example, it is thought that Fe-55 in Zircaloy-2 can be generated from the (n, γ) reaction of Fe-54, the (n, α) reaction of Ni-58, and the (n, 2n) reaction of Fe-56. Table 20.8a clearly shows the (n, γ) reaction of Fe-54 is dominant in the generation of Fe-55.

It was remarkable that the dominant generation pathways were clarified even for the target nuclides generated through complicated pathways. Some of the examples are shown in Fig. 20.2.

Figure 20.2a shows an example of nuclides generated with some contributing pathways. Be-10 is generated in Zircaloy-2 mainly through two pathways, the (n, p) reaction of Be-10 and the (n, α) reaction of C-13. It is not predictable which pathway is dominant from the initial composition of the material. The sensitivity

Table 20.9 Initial composition of SUS304 stainless steel

Element

Value based on measurement data

Value based on the standard specification

C

0.08

N

0.05

Si

1.00

P

0.045

S

0.004

0.030

Cl

0.001

K

4.0E-05

Cr

19.00

Mn

2.00

Fe

72

68.60

Co

0.1

Ni

9.25

9.25

Cu

0.16

Zr

0.00032

Nb

0.02

Mo

0.13

Th

2.0E-07

U

2.0E-07

0.0001

coefficients clearly showed that the (n, α) reaction of C-13 is the dominant pathway for Be-10 generation in Zircaloy-2.

Figure 20.2b shows an example of nuclides generated through long and complicated generation chains. The source nuclide of Cl-36 generated in SUS304 stainless steel is ambiguous because the initial composition in this analysis does not contain chlorine, which could be the dominant source element of Cl-36. The sensitivity coefficients quantitatively clarified that S-34 is the source nuclide of Cl-36 even for the long and complicated chain.

 
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