Bias Effects on g- and s-Factors in Westcott Convention

Featured Application: This paper provides a size of bias on g- and s-factors in Westcott convention, induced by a joining function shape, neutron temperature, and sample temperature, for both 1/v isotopes and non-1/v isotopes. Its impact on neutron activation reaction rate is also discussed. Quantitative results are used for accuracy improvement of neutron activation analysis and neutron capture cross sections. Abstract: For accuracy improvement of neutron activation analysis and neutron capture cross sections, bias effects are investigated on g- and s-factors in the Westcott convention. As origins of biases, a joining function shape, neutron temperature, and sample temperature have been investigated. Biases are quantitatively deduced for two 1/v isotopes ( 197 Au, 59 Co) and six non-1/v isotopes ( 241 Am, 151 Eu, 103 Rh, 115 In, 177 Hf, 226 Ra). The s-factor calculated with a joining function deduced recently by a detailed Monte Carlo simulation is compared to s-factors calculated with traditional joining functions by Westcott. The results show the bias induced by the sample temperature is small, in the order of 0.1% for the g-factor and in the order of 1% for the s-factor. On the other hand, the bias size induced by a joining function shape for the s-factor depends signiﬁcantly on both isotopes and neutron temperature. As a result, the reaction rates are also affected signiﬁcantly. The bias size for the reaction rate is given in the case of an epithermal neutron index r = 0.1, for the eight isotopes.


Introduction
Neutron activation analysis (NAA) using well-thermalized neutrons is a powerful tool utilized in various application fields. For NAA, information of neutron capture cross sections for isotopes to be analyzed and a neutron energy spectrum at an irradiation position are required. The energy spectrum of well-thermalized neutrons is approximated to be the sum of two components, a Maxwellian distribution corresponding to neutron temperature T n and an epithermal dE/E flux distribution cut off at a suitable lower limit of energy. This flux notation is known as the Westcott convention [1] and is widely utilized in NAA. The neutron activation reaction rate R x for the isotope x is expressed as R x = N x ·φ 0 ·σ x0 ·(g x + r·s x ) (1) where N x is the number of the isotope x, φ 0 is the neutron flux, r is the epithermal index that represents the relative strength of the epithermal component, σ x0 is the cross section at 0.0253 eV neutrons, and the g-and s-factors are functions of T n . The g-and s-factors depend on the departure of the cross-section law from the 1/v form (for a 1/v law, g = 1 and s = 0). NAA is also one of the powerful tools determining neutron capture cross sections [2][3][4][5]. In order to improve the accuracy of NAA and/or neutron capture cross sections, a number of improvements have been made. The epithermal dE/E flux distribution was refined as dE/E 1+α [6], introducing an adjusting parameter α, and the s-factor was revised to 1 1 + ( 4.95 ) 7 (3)  5.88 (5) the joining functions 2 , 4 , 4 for neutron temperature = 300 K are plotted in Figure 1a by solid, dotted, and dashed line, respectively. There are bumps at around 0.23 and 0.27 eV for 4 and 4 , respectively. The cut-off energy for both 2 and 4 is about 0.13 eV, which is a little higher than 0.12 eV for 4 . The difference of a high-energy side tail of the bump should be noted; there exists a larger component in 4 compared to 4 in a neutron energy region from about 0.5 eV to a few eV. The same for neutron temperature 1000 K are plotted in Figure 1b. The cut-off energy and bump peak energy are shifted by a factor 1000/300. It is anticipated that the reaction rates for isotopes with a resonance at around the cut-off energy or bump peak energy will be affected significantly by using different joining functions. The g factor in Equation (1), defined as the ratio of pure Maxwellian flux-weighted cross section (σ m ) to the cross section at 0.0253 eV (σ 0 ), is expressed as (6) where sample temperatures (T s ) are explicitly indicated in the cross section σ(E, T s ). The s factor in Equation (1), representing the ratio of resonance integral after subtraction of the 1/v component to σ 0 , is expressed as Hereafter, the s-factor calculated using the joining functions ∆ W 2 , ∆ W 4 , ∆ H 4 as ∆(E) in Equation (7) are expressed as s W 2 , s W 4 , s H 4 . Fine-mesh numerical data of the cross section are available in a website of nuclear data center [14] for T s = 0 and 300 K, corresponding to JENDL-4.0 nuclear data library [15]. The Doppler-broadened cross section at any sample temperature can be deduced from the unbroadened cross section σ(E, 0) at T s = 0 K. The Doppler-broadened cross section for T s is expressed by the next equation based on a free-gas model [16]: the Doppler width W D in Equation (8) is given by where A is the ratio of the sample mass to the neutron mass. The capture cross sections of 241 Am deduced using Equation (8) for T s = 300 and 1000 K are shown by a black solid line and a red solid line in Figure 2a, and those of 151 Eu in Figure 2b. The number of mesh points used for integration of Equation (8) is about 65 k for 241 Am and 22 k for 151 Eu.
The g factor in Equation (1), defined as the ratio of pure Maxwellian flux-w cross section (σ ) to the cross section at 0.0253 eV (σ 0 ), is expressed as where sample temperatures ( ) are explicitly indicated in the cross section σ( , The s factor in Equation (1), representing the ratio of resonance integral after tion of the 1/v component to σ 0 , is expressed as Hereafter, the s-factor calculated using the joining functions 2 , 4 , 4 as Equation (7) are expressed as s 2 , s 4 , s 4 . Fine-mesh numerical data of the cross section are available in a website o data center [14] for = 0 and 300 K, corresponding to JENDL-4.0 nuclear dat [15]. The Doppler-broadened cross section at any sample temperature can be from the unbroadened cross section σ( , 0) at = 0 K. The Doppler-broaden section for is expressed by the next equation based on a free-gas model [16]: the Doppler width W in Equation (8) is given by where A is the ratio of the sample mass to the neutron mass. The capture cross sections of 241 Am deduced using Equation (8)

Results and Discussions
The sample temperature effect on the g-and s-factors is discussed based on tions for the non 1/v isotopes 241 Am and 151 Eu, in which this effect is expected to b more clearly than in other isotopes, since they have prominent resonances near ab eV and 0.46 eV, respectively. Next, a joining function shape effect as a function of temperature is discussed for eight isotopes, 241 Am, 151 Eu, 197 Au, 59 Co, 103 Rh, 115 In, 17 226 Ra. The sample temperature is fixed at = 300 K in the discussion of a joining shape effect. 197 Au and 59 Co are well known as 1/v isotopes and used for neutron termination as monitor samples. 103 Rh, 115 In, 177 Hf, and 226 Ra are non 1/v isotopes a prominent resonances between 0.5 eV and 1.5 eV. The capture cross sections of 197 A 103 Rh, 115 In, 177 Hf, and 226 Ra obtained from JENDL-4.0 at = 300 K [14] are shown i 3a-f and used in the calculations. At last, biases originated in a joining function s the s-factor and the reaction rate are discussed using the calculation results obta all eight isotopes.

Results and Discussions
The sample temperature effect on the g-and s-factors is discussed based on calculations for the non 1/v isotopes 241 Am and 151 Eu, in which this effect is expected to be shown more clearly than in other isotopes, since they have prominent resonances near about 0.31 eV and 0.46 eV, respectively. Next, a joining function shape effect as a function of neutron temperature is discussed for eight isotopes, 241 Am, 151 Eu, 197 Au, 59 Co, 103 Rh, 115 In, 177 Hf, and 226 Ra. The sample temperature is fixed at T s = 300 K in the discussion of a joining function shape effect. 197 Au and 59 Co are well known as 1/v isotopes and used for neutron flux determination as monitor samples. 103 Rh, 115 In, 177 Hf, and 226 Ra are non 1/v isotopes and have prominent resonances between 0.5 eV and 1.5 eV. The capture cross sections of 197 Au, 59 Co, 103 Rh, 115 In, 177 Hf, and 226 Ra obtained from JENDL-4.0 at T s = 300 K [14] are shown in Figure 3a-f and used in the calculations. At last, biases originated in a joining function shape for the s-factor and the reaction rate are discussed using the calculation results obtained for all eight isotopes.

Results and Discussions
The sample temperature effect on the g-and s-factors is discussed based on tions for the non 1/v isotopes 241 Am and 151 Eu, in which this effect is expected to b more clearly than in other isotopes, since they have prominent resonances near a eV and 0.46 eV, respectively. Next, a joining function shape effect as a function of temperature is discussed for eight isotopes, 241 Am, 151 Eu, 197 Au, 59 Co, 103 Rh, 115 In, 1 226 Ra. The sample temperature is fixed at = 300 K in the discussion of a joining shape effect. 197 Au and 59 Co are well known as 1/v isotopes and used for neutron termination as monitor samples. 103 Rh, 115 In, 177 Hf, and 226 Ra are non 1/v isotopes a prominent resonances between 0.5 eV and 1.5 eV. The capture cross sections of 197 103 Rh, 115 In, 177 Hf, and 226 Ra obtained from JENDL-4.0 at = 300 K [14] are shown 3a-f and used in the calculations. At last, biases originated in a joining function s the s-factor and the reaction rate are discussed using the calculation results obt all eight isotopes.

g-Factors for 241 Am and 151 Eu for Sample Temperatures of 300 and 1000 K
The g-factor for 241 Am is calculated by inserting σ( , ) into Equation (6) a tion of neutron temperature , for two sample temperatures, i.e., = 300 K (bl 1000 K (red), and plotted in Figure 4a. Two curves are almost degenerated. The g almost equal to the unity in the neutron energy region below 400 K and grad creases as the neutron temperature rises, since the first neutron resonance at 0.31 to contribute to this increase. Figure 4b is an expanded version of Figure 4a, sho neutron energy region between 250 and 450 K. In this neutron temperature ra deviation of the g-factor from unity is from 1% to 9%. If neutron temperature mined with an uncertainty of 25 K, the g-factor can be estimated with an uncer about 1% using the data shown in this figure.  The g-factor for 241 Am is calculated by inserting σ(E, T s ) into Equation (6) as a function of neutron temperature T n , for two sample temperatures, i.e., T s = 300 K (black) and 1000 K (red), and plotted in Figure 4a. Two curves are almost degenerated. The gfactor is almost equal to the unity in the neutron energy region below 400 K and gradually increases as the neutron temperature rises, since the first neutron resonance at 0.31 eV starts to contribute to this increase. Figure 4b is an expanded version of Figure 4a, showing the neutron energy region between 250 and 450 K. In this neutron temperature range, the deviation of the g-factor from unity is from 1% to 9%. If neutron temperature is determined with an uncertainty of 25 K, the g-factor can be estimated with an uncertainty of about 1% using the data shown in this figure. Figure 4c shows the ratio of g-factor at T s = 1000 K to that at 300 K for 241 Am. Figure 4d is the expanded version of Figure 4c. The maximum deviation from the unity is about 0.5% at a neutron temperature near 600 K. It is only 0.3% for a limited region from 250 K to 450 K, as shown in Figure 4d. Therefore, for most applications not requiring accuracy below 1%, the sample temperature effect on the g-factor can be neglected. It should be remembered that this effect needs to be considered for special applications such as neutron standard cross section studies aiming at accuracy in the order of 0.1%. creases as the neutron temperature rises, since the first neutron resonance at 0.31 eV star to contribute to this increase. Figure 4b is an expanded version of Figure 4a, showing th neutron energy region between 250 and 450 K. In this neutron temperature range, th deviation of the g-factor from unity is from 1% to 9%. If neutron temperature is dete mined with an uncertainty of 25 K, the g-factor can be estimated with an uncertainty o about 1% using the data shown in this figure.   Figure 4c shows the ratio of g-factor at = 1000 K to that at 300 K for 241 Am. Figur 4d is the expanded version of Figure 4c. The maximum deviation from the unity is abou 0.5% at a neutron temperature near 600 K. It is only 0.3% for a limited region from 250 K to 450 K, as shown in Figure 4d. Therefore, for most applications not requiring accurac below 1%, the sample temperature effect on the g-factor can be neglected. It should b remembered that this effect needs to be considered for special applications such as neu tron standard cross section studies aiming at accuracy in the order of 0.1%. Figure 5 shows the g-factor calculated for 151 Eu. The change of the g-factor as a func tion of neutron temperature for 151 Eu is more moderate compared to the case of 241 Am, a shown in Figure 5a. This tendency can be explained by the fact that the first prominen resonance energy of 151 Eu is about 1.5 times higher than that of 241 Am. By comparing th g-factors of 241 Am and 151 Eu in a limited region from 250 K to 450 K, it is noticed that th difference of the slopes is significant, and the ratio of these g-factors is sensitive to neutro temperature. Although the effectiveness using a Zr-Au-Lu alloy is known to determin neutron temperature [12], the combinational use of these isotopes would provide addi tional and independent information on neutron temperature. The sample temperature ef fect for 151 Eu shown in Figure 5c,d is in the same order of that for 241 Am, as discussed above.  241 Am as a function of neutron temperature T n , for two sample temperatures, T s = 300 K (black) and 1000 K (red); (b) Expanded figure of (a); (c) Ratio of Westcott g-factor for T s = 1000 K to that for 300 K; (d) Expanded figure of (c). Figure 5 shows the g-factor calculated for 151 Eu. The change of the g-factor as a function of neutron temperature for 151 Eu is more moderate compared to the case of 241 Am, as shown in Figure 5a. This tendency can be explained by the fact that the first prominent resonance energy of 151 Eu is about 1.5 times higher than that of 241 Am. By comparing the g-factors of 241 Am and 151 Eu in a limited region from 250 K to 450 K, it is noticed that the difference of the slopes is significant, and the ratio of these g-factors is sensitive to neutron temperature. Although the effectiveness using a Zr-Au-Lu alloy is known to determine neutron temperature [12], the combinational use of these isotopes would provide additional and independent information on neutron temperature. The sample temperature effect for 151 Eu shown in Figure 5c,d is in the same order of that for 241 Am, as discussed above. = 1000 K to that for 300 K; (d) Expanded figure of (c). Figure 4c shows the ratio of g-factor at = 1000 K to that at 300 K for 241 Am. Figur 4d is the expanded version of Figure 4c. The maximum deviation from the unity is abou 0.5% at a neutron temperature near 600 K. It is only 0.3% for a limited region from 250 K to 450 K, as shown in Figure 4d. Therefore, for most applications not requiring accurac below 1%, the sample temperature effect on the g-factor can be neglected. It should b remembered that this effect needs to be considered for special applications such as neu tron standard cross section studies aiming at accuracy in the order of 0.1%. Figure 5 shows the g-factor calculated for 151 Eu. The change of the g-factor as a func tion of neutron temperature for 151 Eu is more moderate compared to the case of 241 Am, a shown in Figure 5a. This tendency can be explained by the fact that the first prominen resonance energy of 151 Eu is about 1.5 times higher than that of 241 Am. By comparing th g-factors of 241 Am and 151 Eu in a limited region from 250 K to 450 K, it is noticed that th difference of the slopes is significant, and the ratio of these g-factors is sensitive to neutro temperature. Although the effectiveness using a Zr-Au-Lu alloy is known to determin neutron temperature [12], the combinational use of these isotopes would provide add tional and independent information on neutron temperature. The sample temperature e fect for 151 Eu shown in Figure 5c,d is in the same order of that for 241 Am, as discusse above.

s-Factors for 241 Am and 151 Eu for Sample Temperatures of 300 and 1000 K
The s-factors for 241 Am are plotted in Figure 6a as a function of neutron temperatu , for two sample temperatures, = 300 K(black) and 1000 K (red), which are calculate by inserting σ( , ) into Equation (7). The s-factors corresponding to 2 ( ), 4 ( and 4 ( ), that is, s 2 , s 4 , s 4 , are shown by solid-, dotted-, and dashed lines, respe tively. The difference for two sample temperatures is so small that black and red lines a not separated in Figure 6a. To show the difference between two sample temperatures, th ratios of s-factors are plotted in Figure 6c,d for three kinds of joining functions, indicate by solid-, dotted-, and dashed lines, respectively. The deviation of the ratio from the uni is less than 0.5% for a wide neutron temperature range from 0 K to 1000 K.

s-Factors for 241 Am and 151 Eu for Sample Temperatures of 300 and 1000 K
The s-factors for 241 Am are plotted in Figure 6a as a function of neutron temperature T n , for two sample temperatures, T s = 300 K(black) and 1000 K (red), which are calculated by inserting σ(E, T s ) into Equation (7). The s-factors corresponding to are shown by solid-, dotted-, and dashed lines, respectively. The difference for two sample temperatures is so small that black and red lines are not separated in Figure 6a. To show the difference between two sample temperatures, the ratios of s-factors are plotted in Figure 6c,d for three kinds of joining functions, indicated by solid-, dotted-, and dashed lines, respectively. The deviation of the ratio from the unity is less than 0.5% for a wide neutron temperature range from 0 K to 1000 K.

s-Factors for 241 Am and 151 Eu for Sample Temperatures of 300 and 1000 K
The s-factors for 241 Am are plotted in Figure 6a as a function of neutron temperature , for two sample temperatures, = 300 K(black) and 1000 K (red), which are calculated by inserting σ( , ) into Equation (7). The s-factors corresponding to 2 ( ), 4 ( ), and 4 ( ), that is, s 2 , s 4 , s 4 , are shown by solid-, dotted-, and dashed lines, respectively. The difference for two sample temperatures is so small that black and red lines are not separated in Figure 6a. To show the difference between two sample temperatures, the ratios of s-factors are plotted in Figure 6c,d for three kinds of joining functions, indicated by solid-, dotted-, and dashed lines, respectively. The deviation of the ratio from the unity is less than 0.5% for a wide neutron temperature range from 0 K to 1000 K. Contrary to the sample temperature effect, there are noticeable differences between s-factors corresponding to 2 ( ), 4 ( ), and 4 ( ), as shown in Figure 6a,b. Figure  6e,f shows the ratio of s-factors calculated with 4 ( ) or 4 ( ) to that calculated with 2 ( ). The deviation of the ratio from the unity is as large as about 20% for a wide neutron temperature range. In Figure 6b, the expanded version of Figure 6a, the difference between s 4 and s 2 , and also between s 4 and s 2 , is clearly shown. A difference of a few % points is also noticed for the ratios between s 4 /s 2 and s 4 /s 2 in Figure 6e,f.
The s-factors for 151 Eu are plotted in Figure 7a-f, as in Figure 6 for 241 Am. As shown in Figure 7a, the s-factor for 151 Eu crosses the zero at about 160 K and 1160 K. To show the difference quantitatively between two sample temperatures, the ratios of s-factors are plotted in Figure 7c,d for three kinds of joining functions, indicated by solid-, dotted-, and dashed lines, respectively. The deviation of the ratio from the unity is about 0.5 to 1.5% in the neutron temperature range from 300 K to 400 K. However, for example, in the case of an epithermal index r = 0.1, this sample temperature effect on a reaction rate is limited to the level in the order of 0.1%, since the absolute value of the s-factor for 151 Eu is small.
There is a noticeable difference between s-factors, s 4 /s 2 , and s 4 /s 2 , as shown in Figure 7e,f. The deviation of the ratio from the unity seems to diverge at about 160 K and 1160 K. The reason is that the value of the s-factor crosses the zero point at these neutron temperatures. Since the s-factor is close to zero at these points, the effect induced by the deviations on the reaction rate is limited. However, for the important neutron temperature region at about 300-400 K, the deviation from the unity is about 30% in the case of 4 ( ) and strongly varies from 12% to 34% in the case of 4 ( ). This observation demonstrates the importance of a careful determination of the joining function shape.
Hereafter, results are shown corresponding to the sample temperature of 300 K only, since its effect is small, as discussed for 241 Am and 151 Eu. Contrary to the sample temperature effect, there are noticeable differences between s-factors corresponding to ∆ W 2 (E), ∆ W 4 (E), and ∆ H 4 (E), as shown in Figure 6a,b. Figure 6e,f shows the ratio of s-factors calculated with ∆ W 4 (E) or ∆ H 4 (E) to that calculated with ∆ W 2 (E). The deviation of the ratio from the unity is as large as about 20% for a wide neutron temperature range. In Figure 6b, the expanded version of Figure 6a, the difference between s W 4 and s W 2 , and also between s H 4 and s W 2 , is clearly shown. A difference of a few % points is also noticed for the ratios between s W 4 /s W 2 and s H 4 /s W 2 in Figure 6e,f. The s-factors for 151 Eu are plotted in Figure 7a-f, as in Figure 6 for 241 Am. As shown in Figure 7a, the s-factor for 151 Eu crosses the zero at about 160 K and 1160 K. To show the difference quantitatively between two sample temperatures, the ratios of s-factors are plotted in Figure 7c,d for three kinds of joining functions, indicated by solid-, dotted-, and dashed lines, respectively. The deviation of the ratio from the unity is about 0.5 to 1.5% in the neutron temperature range from 300 K to 400 K. However, for example, in the case of an epithermal index r = 0.1, this sample temperature effect on a reaction rate is limited to the level in the order of 0.1%, since the absolute value of the s-factor for 151 Eu is small.
There is a noticeable difference between s-factors, s W 4 /s W 2 , and s H 4 /s W 2 , as shown in Figure 7e,f. The deviation of the ratio from the unity seems to diverge at about 160 K and 1160 K. The reason is that the value of the s-factor crosses the zero point at these neutron temperatures. Since the s-factor is close to zero at these points, the effect induced by the deviations on the reaction rate is limited. However, for the important neutron temperature region at about 300-400 K, the deviation from the unity is about 30% in the case of ∆ H 4 (E) and strongly varies from 12% to 34% in the case of ∆ W 4 (E). This observation demonstrates the importance of a careful determination of the joining function shape.   197 Au, 59 Co, 103 Rh, 115 In, 177 Hf, and 226 Ra A joining function shape effect on g-and s-factor is discussed for other six isotopes, i.e., 197 Au, 59 Co, 103 Rh, 115 In, 177 Hf, and 226 Ra. The calculated g-and s-factors for these isotopes are plotted in Figures 8-13. The capture cross section of 197 Au is used as the neutron cross section standard at 0.0253 eV, and the 1/v law can be applied well, as shown in Figure 3a. The deviation from the unity of the g-factor is only 0.6% at 0.0253 eV. The absolute value of the s-factor of 197 Au shown in Figure 8c is relatively large compared to that of the other 1/v law isotope  197 Au, 59 Co, 103 Rh, 115 In, 177 Hf, and 226 Ra A joining function shape effect on g-and s-factor is discussed for other six isotopes, i.e., 197 Au, 59 Co, 103 Rh, 115 In, 177 Hf, and 226 Ra. The calculated g-and s-factors for these isotopes are plotted in Figures 8-13. The capture cross section of 197 Au is used as the neutron cross section standard at 0.0253 eV, and the 1/v law can be applied well, as shown in Figure 3a. The deviation from the unity of the g-factor is only 0.6% at 0.0253 eV. The absolute value of the s-factor of 197 Au shown in Figure 8c is relatively large compared to that of the other 1/v law isotope Hereafter, results are shown corresponding to the sample temperature of 300 K only, since its effect is small, as discussed for 241 Am and 151 Eu. 197 Au, 59 Co, 103 Rh, 115 In, 177 Hf, and 226 Ra A joining function shape effect on g-and s-factor is discussed for other six isotopes, i.e., 197 Au, 59 Co, 103 Rh, 115 In, 177 Hf, and 226 Ra. The calculated g-and s-factors for these isotopes are plotted in Figures 8-13. Figures 8-13 (a) show g-factors, and (b) is the enlarged view of (a) in the neutron temperature region from 250 K to 450 K. Figures 8-13 (c) show s-factors, and (d) is the enlarged view of (c). Figures 8-13 (e) show the ratio of s-factors, that is, s W 4 /s W 2 and s H 4 /s W 2 , and (f) is the enlarged view of (e). The capture cross section of 197 Au is used as the neutron cross section standard at 0.0253 eV, and the 1/v law can be applied well, as shown in Figure 3a. The deviation from the unity of the g-factor is only 0.6% at 0.0253 eV. The absolute value of the s-factor of 197 Au shown in Figure 8c is relatively large compared to that of the other 1/v law isotope 59 Co, since a prominent neutron resonance exists at 4.91 eV. The deviation of s H 4 from s W 2 increases from 0% to 10% as the neutron temperature increases from 0 K to 1600 K. This deviation observed in the case of s H 4 is prominent compared to the case of s W 4 . The deviation in the case of s W 4 is only about 0.6% at a neutron temperature of 300 K. Considering the large size of the s-factor of 197 Au, a bias induced by a joining function ambiguity exceeds 1%. Since the s-factor of 197 Au is sometimes used as the standard nuclear value to determine other isotopes s-factor or resonance integrals, the bias of about 1% cannot be neglected. It should be noted that the bias increases furthermore as neutron temperature increases. deviation observed in the case of s 4 is prominent compared to the case of s 4 . The deviation in the case of s 4 is only about 0.6% at a neutron temperature of 300 K. Considering the large size of the s-factor of 197 Au, a bias induced by a joining function ambiguity exceeds 1%. Since the s-factor of 197 Au is sometimes used as the standard nuclear value to determine other isotopes s-factor or resonance integrals, the bias of about 1% cannot be neglected. It should be noted that the bias increases furthermore as neutron temperature increases.  Figure 9a-f show results for 59 Co. The capture cross section of 59 Co is also used as the standard for thermal neutron capture cross sections and resonance integrals. The capture cross section of 59 Co obeys the 1/v raw well, as shown in Figure 3b. Since there is no resonance below 10 eV, the deviation of the g-factor from the unity is extremely small, as shown in Figure 9a,b. The s-factors shown in Figure 9c,d for three joining functions are almost degenerating. As shown in Figure 9e,f, the deviation originating in a joining function ambiguity is less than 0.1% for a wide neutron temperature range. From the results for 59 Co, it is expected that the bias effect originating in a joining function shape is negligible for isotopes obeying the 1/v law well. The order of the bias for isotopes obeying the 1/v law would be roughly estimated from the results of 197 Au and 59 Co.  Figure 9a-f show results for 59 Co. The capture cross section of 59 Co is also used as the standard for thermal neutron capture cross sections and resonance integrals. The capture cross section of 59 Co obeys the 1/v raw well, as shown in Figure 3b. Since there is no resonance below 10 eV, the deviation of the g-factor from the unity is extremely small, as shown in Figure 9a,b. The s-factors shown in Figure 9c,d for three joining functions are almost degenerating. As shown in Figure 9e,f, the deviation originating in a joining function ambiguity is less than 0.1% for a wide neutron temperature range. From the results for 59 Co, it is expected that the bias effect originating in a joining function shape is negligible for isotopes obeying the 1/v law well. The order of the bias for isotopes obeying the 1/v law would be roughly estimated from the results of 197 Au and 59 Co. Figure 10a-f show the results for 103 Rh. There is a huge resonance at 1.26 eV in 103 Rh. Therefore, the g-factor increases as neutron temperature increases, as shown in Figure  10a,b. The s-factor also monotonically increases as a function of neutron temperature, as  There is a huge resonance at 1.26 eV in 103 Rh. Therefore, the g-factor increases as neutron temperature increases, as shown in Figure 10a,b. The s-factor also monotonically increases as a function of neutron temperature, as for 197 Au. However, the differences among s W 2 , s W 4 , and s H 4 are more notable compared to those for 197 Au, as shown in Figure 10c,d. The deviation of s W 4 and s H 4 from s W 2 increases from 0% to about 30% as neutron temperature increases from 0 K to 1600 K. A prominent deviation is observed in the case of s W 4 for a neutron temperature less than about 1100 K. The deviation is about 7% at a neutron temperature of 300 K. Considering the large size of the s-factor of 103 Rh, a bias induced by a joining function ambiguity exceeds 3%, even in the case of epithermal neutron index r = 0.1.

g-and s-Factors for
to those for 197 Au, as shown in Figure 10c,d. The deviation of s 4 and s 4 from s 2 in creases from 0% to about 30% as neutron temperature increases from 0 K to 1600 K. A prominent deviation is observed in the case of s 4 for a neutron temperature less than about 1100 K. The deviation is about 7% at a neutron temperature of 300 K. Considering the large size of the s-factor of 103 Rh, a bias induced by a joining function ambiguity ex ceeds 3%, even in the case of epithermal neutron index r = 0.1.  In. There is a huge resonance at 1.46 eV in 115 In The neutron temperature dependence of the g-and s-factor is close to that observed fo 103 Rh. At the neutron temperature of 300 K, the bias induced by a joining function ambi guity reaches also about 3% in the case of epithermal neutron index r = 0.1.     Figure 13a-f shows the results for 226 Ra. There is a huge resonance at 0.54 eV in 226 Ra. The neutron temperature dependence of the g-factor is similar to that for 241 Am, although the absolute value for 226 Ra increases more rapidly as neutron temperature increases. On the other hand, the neutron temperature dependence of the s-factor is similar to that for 151 Eu. The differences among s 2 , s 4 , and s 4 are more notable compared to those for 103 Rh, 115 In, and 177 Hf at the neutron temperature of 300 K, as shown in Figure 13e,f. Therefore, a huge bias effect originating in a joining function ambiguity is expected at the neutron temperature at 300 K. Figure 13a-f shows the results for 226 Ra. There is a huge resonance at 0.54 eV in 226 Ra. The neutron temperature dependence of the g-factor is similar to that for 241 Am, although the absolute value for 226 Ra increases more rapidly as neutron temperature increases. On the other hand, the neutron temperature dependence of the s-factor is similar to that for 151 Eu. The differences among s W 2 , s W 4 , and s H 4 are more notable compared to those for 103 Rh, 115 In, and 177 Hf at the neutron temperature of 300 K, as shown in Figure 13e,f. Therefore, a huge bias effect originating in a joining function ambiguity is expected at the neutron temperature at 300 K.

Bias on the s-Factor by the Joining Function
Biases on s-factor originating in a joining function are compared among 241 Am, 151 Eu, 197 Au, 59 Co, 103 Rh, 115 In, 177 Hf, and 226 Ra in Figure 14. The ratios s 4 /s 2 and s 4 /s 2 are shown by black bars and red-hatched bars. The results for neutron temperature of 300 K, 600 K, and 1000 K are shown in Figure 14a-c, respectively.
As shown in Figure 14a, large biases exceeding 10% occur in the cases of 241 Am, 151 Eu, and 226 Ra at the neutron temperature of 300 K. The largest deviation from the unity is observed in the case of 151 Eu for s 4 /s 2 , corresponding to about 29%. In the case of the neutron temperature of 600 K, the largest deviation from the unity is also observed for 151 Eu, though for s 4 /s 2 , corresponding to about 37%. In the case of the neutron

Bias on the s-Factor by the Joining Function
Biases on s-factor originating in a joining function are compared among 241 Am, 151 Eu, 197 Au, 59 Co, 103 Rh, 115 In, 177 Hf, and 226 Ra in Figure 14. The ratios s W 4 /s W 2 and s H 4 /s W 2 are shown by black bars and red-hatched bars. The results for neutron temperature of 300 K, 600 K, and 1000 K are shown in Figure 14a-c, respectively.   As shown in Figure 14a, large biases exceeding 10% occur in the cases of 241 Am, 151 Eu, and 226 Ra at the neutron temperature of 300 K. The largest deviation from the unity is observed in the case of 151 Eu for s H 4 /s W 2 , corresponding to about 29%. In the case of the neutron temperature of 600 K, the largest deviation from the unity is also observed for 151 Eu, though for s W 4 /s W 2 , corresponding to about 37%. In the case of the neutron temperature of 1000 K, the largest deviation from the unity is again observed for 151 Eu, for s W 4 /s W 2 , exceeding 50%.

Bias on the Reaction Rate by the Joining Function
In order to estimate the bias on the reaction rate originating in a joining function, Equation (1) is used. The size of the bias on the reaction rate depends on the value of the epithermal index r. Figure 15 shows biases on the reaction rate originating in a joining function in the case of r = 0.1; the value of r depends on the irradiation facility and irradiation position and ranges typically from 0 to 0.

Bias on the Reaction Rate by the Joining Function
In order to estimate the bias on the reaction rate originating in a joining function, Equation (1) is used. The size of the bias on the reaction rate depends on the value of the epithermal index r. Figure 15 shows biases on the reaction rate originating in a joining function in the case of r = 0.1; the value of r depends on the irradiation facility and irradiation position and ranges typically from 0 to 0. As shown in Figure 15a, large biases exceeding 10% occur only in the cases of 226 Ra at the neutron temperature of 300 K, for (s 4 )/ (s 2 ). At the neutron temperature of 600 K, the largest deviation from the unity is also observed in 226 Ra, but for (s 4 )/ (s 2 ), that exceeds 20%. In the case of the neutron temperature of 1000 K, the largest deviation from the unity is again observed in 226 Ra for s 4 /s 2 and it is about 26%.
The reason that the largest bias is not observed in 151 Eu but in 226 Ra is explained as follows: for the reaction rate, the bias is moderate due to the opposite effect exerted by the g-factor and s-factor on the reaction rate. This is demonstrated for 151 Eu, by comparing Figures 5a and 7a.
The height of a bump in a joining function at its peak position is about 26-31%, as shown in Figure 1. The peak position of the bump increases as neutron temperature increases. When an energy of a first huge resonance is close to the peak position, a large bias is anticipated in the s-factor.
In this work, a bias originating in neutron nuclear data is not considered. It should be noted here that there is sometimes an uncertainty in the order of 10% even in wellinvestigated neutron nuclear data, as demonstrated in recent works on 241 Am [10,17,18].

Conclusions
Bias effects have been investigated on the g-and s-factors in the Westcott convention. As origins of biases, a joining function shape, neutron temperature, and sample temperature, have been investigated. The quantitative results are given for 8 isotopes, i.e., 241 Am, 151 Eu, 197 Au, 59 Co, 103 Rh, 115 In, 177 Hf, and 226 Ra. The results show the bias induced by sample temperature is small, in the order of 0.1% for the g-factor and in the order of 1% for the sfactor. Biases induced by a joining function shape for the s-factor are shown to be in the order of 10% for many non-1/v isotopes. The bias size for the reaction rate in the case of an epithermal neutron index r = 0.1 is also shown. The largest bias is obtained for 226 Ra among the eight isotopes. The order of bias size for the s-factor or reaction rate can be grasped from the results given for the eight isotopes in the figures. For non-1/v isotopes, the bias effect originating in a joining function shape needs to be carefully investigated. The formulas to calculate g-and s-factors in this paper enable bias calculation without a special code. If the size of the calculated bias is close to the required accuracy of NAA, a more careful study on neutron energy spectrum at the irradiation position and energydependent nuclear data is recommended.   115 In, 177 Hf, and 226 Ra at the neutron temperature of 300 K; (b) Same as (a) at the neutron temperature of 600 K; (c) Same as (a) at the neutron temperature of 1000 K.
As shown in Figure 15a, large biases exceeding 10% occur only in the cases of 226 Ra at the neutron temperature of 300 K, for R s H 4 /R s W 2 . At the neutron temperature of 600 K, the largest deviation from the unity is also observed in 226 Ra, but for R s W 4 /R s W 2 , that exceeds 20%. In the case of the neutron temperature of 1000 K, the largest deviation from the unity is again observed in 226 Ra for s W 4 /s W 2 and it is about 26%. The reason that the largest bias is not observed in 151 Eu but in 226 Ra is explained as follows: for the reaction rate, the bias is moderate due to the opposite effect exerted by the g-factor and s-factor on the reaction rate. This is demonstrated for 151 Eu, by comparing Figures 5a and 7a.
The height of a bump in a joining function at its peak position is about 26-31%, as shown in Figure 1. The peak position of the bump increases as neutron temperature increases. When an energy of a first huge resonance is close to the peak position, a large bias is anticipated in the s-factor.
In this work, a bias originating in neutron nuclear data is not considered. It should be noted here that there is sometimes an uncertainty in the order of 10% even in wellinvestigated neutron nuclear data, as demonstrated in recent works on 241 Am [10,17,18].

Conclusions
Bias effects have been investigated on the g-and s-factors in the Westcott convention. As origins of biases, a joining function shape, neutron temperature, and sample temperature, have been investigated. The quantitative results are given for 8 isotopes, i.e., 241 Am, 151 Eu, 197 Au, 59 Co, 103 Rh, 115 In, 177 Hf, and 226 Ra. The results show the bias induced by sample temperature is small, in the order of 0.1% for the g-factor and in the order of 1% for the s-factor. Biases induced by a joining function shape for the s-factor are shown to be in the order of 10% for many non-1/v isotopes. The bias size for the reaction rate in the case of an epithermal neutron index r = 0.1 is also shown. The largest bias is obtained for 226 Ra among the eight isotopes. The order of bias size for the s-factor or reaction rate can be grasped from the results given for the eight isotopes in the figures. For non-1/v isotopes, the bias effect originating in a joining function shape needs to be carefully investigated. The formulas to calculate g-and s-factors in this paper enable bias calculation without a special code. If the size of the calculated bias is close to the required accuracy of NAA, a more careful study on neutron energy spectrum at the irradiation position and energy-dependent nuclear data is recommended.