A new convention for the epithermal neutron spectrum for improving accuracy of resonance integrals

A new convention for the epithermal neutron spectrum component is formulated in this work, aimed at improving the accuracy of resonance integrals determination. The (1 + β)/(βE + E1+α) form here proposed, is an approximating function of the epithermal neutron spectrum based upon calculations performed by the state-of-art Monte Carlo code MVP-3. Bias effects on determination of resonance integrals, due to the application of the well-known and used so far approximating functions, such as 1/E, 1/E1+α as well as the new form (1 + β)/(βE + E1+α) are compared, where E is the neutron energy, α a shape factor, and β a new shape factor introduced in this work. The other bias effect is also investigated, which is caused by neglecting the position dependence of a neutron spectrum inside an irradiation capsule. To get a demonstration of the bias effects due to these assumptions, upon the determination of a neutron spectrum from a quantitative point of view in a practical case, the thermal neutron-capture cross section and resonance integral of 135Cs measured at a research reactor JRR-3 are re-evaluated. A superior property of the proposed new mathematical expression is discussed. The experimental method is proposed to determine the new shape factor β by a combinational use of triple flux monitors (197Au, 59Co, and 94Zr), and its analytical methodology is formulated.


Introduction
Neutron cross sections are the basic physical data for neutronics calculations in many apparatuses for fundamental nuclear physics research as well as applications in many fields. Since significant advancements on neutronics calculations have been achieved [1][2][3] by utilizing a Monte Carlo method [4] coupled to high performance computing systems, the accuracy of neutron cross sections is thus becoming a dominant factor determining the final results and their uncertainties of calculations. That being said, needs for improvements in the accuracy on neutron cross sections are expanding as also summarized by the OECD/NEA's working group WPEC/SG-C [5]. To bridge the gap between the required and current accuracy in nuclear data, national or international research projects have been conducted worldwide [6][7][8]. As an example, WPEC/SG-41 tackled an accuracy improvement on neutron-capture cross section of 241 Am by integrating the knowledge so far available on both differential and integral measurements, and pointed importance of identifying bias effects and correcting them before evaluating a recommended value [9].
The neutron activation method is one of the integral measurement approaches commonly available, which has been historically and widely utilized to get measurements on the capture cross sections for thermal neutrons as well as determination for resonance integrals [10][11][12][13]. Therefore, these cross sections in evaluated nuclear data sets such as BNL-325 [14] and Atlas [15] are basically influenced by the specific neutron activation measurement method of choice, as well as the analysis methods adopted. Recently, an identified bias effect has been examined in details in the case of 241 Am [16,17] about the thermal neutron-capture cross section measured by means of activation method, where a huge resonance does exist at energy of 0.3 eV, which is lower than the Cd This approach has been investigated and utilized in application fields of neutron activation analysis (NAA) [20,21]. However, such a 1/E 1+α form has not become popular in deducing resonance integrals as pointed out by Yücel et al [22].
In the neutron activation analysis, not only the spectrum shape but also its position dependence needs to be considered, as pointed in [11]. However, the difference of neutron flux levels measured between flux monitors and an irradiation sample has been assumed to be negligible in most cases or fully neglected.
In this paper these bias effects are evaluated by deducing neutron-capture cross sections, especially resonance integrals, from the quantitative point of view. In this regard, the neutron spectrum calculated by a continuous-energy Monte Carlo code MVP-3 [1] is used as a reference. Better approximating functions have thus been extracted for the joining energy range and the epithermal energy region, by using the reference spectrum. A new formulation for reaction rate calculations has thus been drawn by using the derived approximating functions: the new formula enables to analyze experimental data with reduced bias. In order to get demonstration on how the bias effects, originating in the assumptions on a neutron spectrum, have an impact in a practical case, the experimental data concerning the 135 Cs nuclide irradiated at the JRR-3 have been re-evaluated. A method determining shape factors in the approximating function is here proposed and examined.

Calculations of reference spectrum and its position dependence
In order to obtain the reference spectrum at the irradiation position where thermal neutron-capture cross sections and resonance integrals have been measured, the irradiation hole HR-1 (Hydraulic rabbit facility No. 1) in the JRR-3 (Japan Research Reactor No. 3) located at Tokai in Japan has been selected, and the neutron spectrum there available has been calculated by a Monte Carlo particle transport code MVP-3. Great care has been put in 3D geometry reconstruction in the input file of the reactor core, the irradiation hole selected, and the irradiation capsule as shown in figure 1. The fuel composition at the irradiation date has been calculated by using the burnup calculation code MVP-BURN [23] and the information incorporated in the input of the MVP-3 [24]. The temperature of the core region has been set to 334 K, while for the D 2 O filled spectrum shifter volume surrounding it being 303 K, which approximated the averaged values in the core and moderator regions calculated with the condition of the 20 MW (thermal) reactor power when the experiment was performed. Since an irradiation sample and flux monitors cannot be set simultaneously at the same position in actual experimental conditions (see [25] for example), a bias will emerge due to a position dependence of the neutron flux inside an irradiation capsule. In order to get an assessment about such a bias effect from the quantitative point of view, neutron spectra inside the irradiation capsule have been calculated, as a function of a position X inside the capsule.
In figure 2(a) the calculated neutron spectra, expected at the different irradiation positions inside the capsule set in the HR-1 hole is shown. The dependence of the neutron spectra is shown for position coordinates X=(−10, 0 (i.e. the center position in the capsule), +10 mm), where the direction of X axis is sketched in figure 1. The unit of vertical axis of figure 2(a) is chosen to be a neutron flux per unit lethargy. Based upon such an axis unit setting, an epithermal neutron flux trend of form 1/E in equation (1) or α=0 in equation (2) has a constant value when plotted function of neutron energy. In figure 2(a), an apparent discrepancy from the constant expectation, that is, the decrease of neutron flux per unit lethargy as a function of energy, is shown at the epithermal region. Position dependent neutron spectra from X=−10 mm to +10 mm calculated by 2 mm steps are summed up for improving the statistical uncertainty. This averaged spectrum is defined as the reference spectrum ( ) f E , ref which has been used for extracting the appropriate approximating functions. The epithermal neutron flux integrated over an energy region, ranging from 0.5 eV to 100 keV, is normalized to match the experimental value.

Effect of fast neutrons on activation
Contribution of fast neutrons (E>100 keV) to the total reaction rate is evaluated by using the reference spectrum. Total and fast reaction rates are expressed by the equations, The ratio of R fast and R tot is calculated using JENDL-4.0 nuclear data library [26] for several isotopes, and shown in table 1, along with their thermal neutron-capture cross sections s , 0 resonance integrals I , 0 capture cross sections averaged over a Maxwellian spectrum peaking at a neutron energy of 30 keV s , 30 keV and the ratios / s s . 30 keV 0 For comparison, those of ENDF/B-VIII.0 library [27] calculated by the nuclear data processing codes FRENDY [28] and PREPRO [29] are also tabulated. As shown, contributions of fast neutrons to total reaction rate are less than 0.01% except the case of 94 Zr as being about 0.3%. As these examples show, the contributions of fast neutrons can be neglected in the cases of capture reactions if the accuracy less than about 0.3% is not required. It should be noted that a careful evaluation is required in the cases treating fission and threshold reactions, since these cross sections do not decrease as a function of neutron energy at fast neutron region in contrast to the capture reaction.

Approximating functions for joining and epithermal energy regions
In order to get an approximation of the epithermal neutron spectrum component, the equations (1) and (2) have been widely utilized. Several types of Δ(E) function forms in equations (1) and (2) have been proposed [18], which is reported as a 'cut-off' function in some [10] or a 'joining' function in other [17] and in this work, as well. In order to extract approximating functions for both the joining part and epithermal energy regions from the reference spectrum, a main Maxwellian distribution is subtracted by the reference spectrum. A thermal energy region below 0.125 eV (∼5 kT) has been considered to get a fitting by a Maxwellian function for the ( )  figure 4(a). The latter spectrum is defined as 'the reference epithermal spectrum' ( ) f E epi ref hereafter. As shown, the reference spectrum is well fitted at thermal energy region with a Maxwellian function; a fitting gives a value of 27.28 meV for kT with an uncertainty as small as 0.02 meV, which corresponds to 316.5 (3) K. The digit in parentheses is the onestandard-deviation uncertainty in the last digits of the given value.
In figure 4(b), the fitting by a 1/E 1+α form (blue solid line) can reproduce the spectrum much better than that by a 1/E form (dashed black line). The fitting function is expressed by: where D 2 is expressed as below: The fitted value of a is 0.0959 (15) and that of f 2 is 1.636 (12)×10 12 [1/sec/cm 2 ]. (blue circles). Some noticeable deviations are present in the short energy range at about 0.2-0.3 eV, and in a wide energy region, ranging from a few eV to 100 keV even if equation (5) is used. The deviation at around 0.2-0.3 eV is expected to be reduced by allowing adjustment of the parameters in the joining function. However, the deviation observed in the wide energy range from a few eV to 100 keV cannot be reduced, if any new free parameter other than a is not allowed.
Trkov et al [30] proposed some years ago the same formula / a + 1 E 1 in which the parameter a is allowed to be energy-dependent as expressed by the mathematical relation α(E)=α 0 +α 1 ×ln(E)+α 2 ×ln 2 (E). However, it is anticipated here that is difficult to determine three parameters from the limited experimental values of reaction rates or reaction rate ratios. In order to get an as practical as possible fitting operation for an energy region from a few eV to 100 keV, trial functional forms allowing only one additional free parameter have been investigated instead. The additional parameter is added to the denominator in order to allow for a function shape degree of freedom. This is expressed by the 1/(βE+E 1+α ) form function. For normalization purpose, the  factor (1+β) is multiplied in the numerator in order to get the condition (1+β)/(βE+E 1+α )=1 when E=1 [eV]. The β dependence of the new (1+β)/(βE+E 1+α ) function is plotted in figure 4(e) and, for comparison, the α 1 dependence of the form function is also plotted in figure 4(f). The parameter α 2 has been set to a fixed value of zero in order to limit the number of free fitting parameters. As shown in figure 4(e) the parameter β affects the neutron spectrum shape nearly equally in a wide energy range, while the parameter α 2 is more sensitive to high energy region as being the logarithm coefficient for the neutron energy E. The resulted best fitted spectra have at last been calculated by using the set of adjusted parameters (α, β)=(0.221, 4.21), in the case of the (1+β)/(βE+E 1+α ) form function and the (α 0 , α 1 )=(0.0373, 0.005 64) in the case of the form function under the restriction of α 2 =0. This paper therefore focuses on the utilization of the new 1/(βE+E 1+α ) form function, and comparison with the alternative function is discussed as well. In figure 4(b), the fitting by the (1+β)/(βE+E 1+α ) form epithermal spectrum multiplied by the D 4 Free is also shown by solid line; the resulting fitting function is expressed as below.
Free where the function D 4 Free is expressed by introducing four additional free parameters m 1 , m 2 , m , 1 m 2 instead of D 4 Orig given in [18], as reported below The new shape factor b is introduced in this work with the aim to achieve a superior fitting.    The reference spectrum available within a Cd capsule has been calculated by the MVP code in the same manner followed for the calculation of ( ) f E ,  [25] are reproduced in the geometry input of the MVP code.

Reaction rate expression using the generalized approximating functions
Based upon the aforementioned considerations, in the following the most general expression for reaction rate determination is formulated, by taking into account the neutron flux approximating functions discussed in the previous section.
By inserting equation (7) as f epi into equation (1), the reaction rate R in equation (3) is expressed as where ( ) g T w is the Westcott factor depending only on neutron temperature T. If we define a quantity H join by the equation The quantity H join reflects the shape of an epithermal neutron spectrum. As shown in figure 4(c), it corresponds not to a resonance integral with a Cd cut-off energy of about 0.55 eV, but to a resonance integral with a cut-off energy of about 5kT, which is determined by the joining function This formula is also different from the Høgdahl convention [31], where an epithermal neutron component in an energy region below 0.55 eV is assumed to be neglected. A reaction rate above 0.55 eV may be experimentally measured by means of the activation technique by irradiating a sample within a Cd capsule. The reaction rate R′ within a Cd capsule is expressed by where H , Cd which accounts for the resonance integral, is given by the equation where ( ) Tr E is a transmission function by the Cd capsule. The quantity H Cd corresponds to a resonance integral with a Cd cut-off energy. In the case shown in figure 4(d), it is about 0.55 eV. In the case of a b = = 0 and 0, the H Cd is the same with the normal resonance integral I . 0 The term · ( ) · f s g T 1 w 0 in equation (12) is omitted in deriving equation (13)  Am, respectively. A huge bias effect is anticipated if an analysis of resonance integrals determination is carried out by using the form 1 function, which is larger than uncertainty of the resonance integrals for almost all nuclei. Hereafter, the bias effects using not the reaction rate relation R″ but R′ is studied.

Discussions on the bias effects
3.1. Biases due to assumption on the shape of neutron spectrum In order to evaluate biases due to assumptions on neutron spectrum, reaction rates within a Cd capsule, R′ (fitted approximating functions), are compared with R′ (MVP). For the calculations of the R′ (fitted approximating function), the fitted parameters reported in table 2 are used. Co to match with experimental data reported in section 4 for allowing direct comparison.
The results of R and R-R′ are also tabulated. The R-R′ is determined mainly by a Maxwellian component and partly by a component in an energy region of about 0.1−1 eV. As shown, the value of R-R′ calculated by any fitting function reproduces that by the reference spectrum within 0.5%, except for the form 1 function applied to 241 Am: this is the special case where a huge resonance does exist at 0.3 eV as studied in [16,17]. Even in this special case, the value of R-R′ for 241 Am can be reproduced within 0.2% if advanced functions other than of form 1 are utilized. On the other hand, there are noticeable fragmentations on the values of R′ as expected from table 3. Figure 5 shows the ratios of R′ (fitted functions) and R′ (MVP) for 197 Au, 59 Co, 94 Zr, 135 Cs and 241 Am isotopes. The numerical values are also given in the rows of R'/R'(MVP) in table 4. In the case of form 1 function (1/E trend in an epithermal region), the ratio of reaction rates calculated with fitted functions and the MVP calculation varies from 0.72 to 1.54. Therefore, huge bias effects of about 50% need Cs. If 197 Au is used as a flux monitor to measure the resonance integral of 135 Cs, the bias of about 20% needs to be considered. On the other hand, if a 59 Co is used as the flux monitor, the bias will be as small as 1%. As demonstrated here, the detailed study of the neutron spectrum in advance of activation experiments will help to select a suitable flux monitor.
In case of function of the form 2 (1/ a + E 1 in an epithermal region), the range of the ratios is limited within 0.99 to 1.04. It is 0.94 to 1.08 for function of the form 3, and 0.95 to 1.02 for function of the form 4. It should be noted that the difference of about 5% exists between the ratios of 197 Au and 59 Co in the case of function of the form 2, although both are commonly used as standards for neutron activation measurements. In these cases, a bias of about 5% need to be considered.
On the other hand, the function of the form 5 introduced in this paper significantly reduces the range: from 1.002 to 1.016. A bias need to be considered in such a case, which is much reduced compared to the function of the form 1 to 4. The difference between the ratios of 135 Cs and 59 Co is low as 0.1%. If a 59 Co is used as the flux monitor to measure the resonance integral for 135 Cs, the bias due to the assumption of the approximating function of the form 5 will be expected as low as 0.1%. The function of the form 6, that is, also reduces the range very well: from 1.004 to 1.011.
The difference of R′ on 241 Am is as high as 5% between the form 4 and the form 5. This difference can be explained by the huge resonance located at about 0.3 eV in the capture cross section of 241 Am. The difference of the form 4 and the form 5 is significant at the energy region of about 0.2-0.5 eV as shown in figure 4(c). If the uncertainty of resonance integrals is required to be less than 1%, it is thus to be considered the form 5 or 6 as the approximating function. In addition to 241 Am, there are several isotopes, whose resonances exist in this region, such as 239,241 Pu, 237 Np, 242 Am, 231,232 Pa, 251 Cf, 151 Eu, 151 Sm, 167 Er. For these isotopes, the bias pointed here will have a value to be investigated.

3.2.
A bias due to position dependence of neutron flux As shown in figure 3, an intensity of an epithermal neutron flux changes about 0.8% per 1 mm. For example, in a case that a position difference between a sample and a flux monitor is 4 mm, about 3% bias needs to be considered in determining a resonance integral deduced from R′. On the other hand, a bias in R-R′, which is almost proportional to a thermal neutron-capture cross section, is limited to about 0.6% even if there is the same position difference of 4 mm. By setting more than two flux monitors for each isotope placed at symmetric positions, sandwiching a sample and averaging reaction rates of these flux monitors, the bias due to the position dependence is cancelled: amounts of flux monitors are assumed enough small not to induce any effect on the neutron flux at a sample position. This position dependence needs to be considered in planning experiments including choice and setting of flux monitors. The use of a mixed alloy such as a Zr-Au-Lu [32] will be useful in planning a set of plural multiflux monitors inside an irradiation capsule. Nevertheless, a difference of fluxes at a target sample and a flux monitor positions needs to be considered. The use of a homogeneously mixed alloy including both a target sample and flux monitors may decrease the bias due to position dependence, if atomic and isotopic ratios in the mixed alloy can be determined with a required accuracy.

Determination of an α-shape factor
In this section, the relation between an α-shape factor and reaction rate ratios between 197 Au and 59 Co within a Cd capsule is examined. Using this relation, one can estimate an α-shape factor in an experiment if R′( 197 Au) and R′( 59 Co) are given. As an example, in case of experiments by Katoh et al [25] in which the ratio R′ ( 197 Au)/ R' ( 59 Co) is given as 24.2 (7), one can roughly estimate the alpha shape factor as 0.060 (13).
Yücel et al derived an α-shape factor from the ratio of the Cd ratios (R( 197 Au)/R′( 197 Au)-1)/(R( 98 Mo)/R'( 98 Mo)-1), and claimed that the α-shape factor can be determined experimentally. Figure 6 shows that the R′ ( 197 Au)/R′ ( 59 Co) can be used to determine the α-shape factor, as well as the combinational use of 197 Au and 98 Mo. If 94 Zr is used as a second monitor in combination with 197 Au and its R′ is measured with the same uncertainty, the α-shape factor will be determined with higher accuracy since the sensitivity of the ratio on α-shape factor is much higher than those of other combinations as shown in figure 6.

Determination of both α and β-shape factors
This section describes the method determining both α and β-shape factors by a combinational use of the triple flux monitors, 197 Au, 59 Co, and 94 Zr. We assume reaction rates R′ in a Cd capsule are available experimentally for all monitors. Based on such an approach, the two ratios R′( 197  according to equation (13), which are defined here as (   is equivalent with that of c D = 1 2 in equation (16). From the contour W= / e 1 2 projects onto the α and β axes, the one standard deviations δα and δβ of α and β are deduced, respectively [33].  The ( ) · s g T w 0 can be derived by two ways: (i) a combination of R and R′ for one kind of isotope monitor, (ii) a combination of R for two kind of isotopes. In the first case, it is expressed as below In contrast to the formula for the two-flux monitor method in [12,13], the analysis method formulated here gives the thermal neutron-capture cross section in three ways as shown in the bottom of table 5. These independent values will be effectively used to check consistency of measurements. It should be noticed that the two-flux monitor method was utilized to analyze experiments utilizing a Cd filter with open-ended cylinder shape as described in the experiments reported in [34]. In the case of a fully-closed Cd filter, it is not necessary to use the two-flux monitor method.
Recently, Nakamura et al measured the resonance integral of 135 Cs at the research reactor KUR using a Gd filter [35]. Their value is 42 (3) b after correcting from their original value 45 (3) b the contribution in the energy range between 0.133 and 0.55 eV. This value is consistent with the revised value 43 (3) b in this study, as compared in table 6. However, it should be noted that their resonance integral is derived based on the traditional Westcott convention by using a combination of Au and Co flux monitors. In order to evaluate the correction factor, the neutron spectrum in the KUR irradiation position needs to be calculated using Monte Carlo simulations with detailed geometries and compositions.
The evaluated value of Atlas 6th edition [15] seems to be based on Katoh's data. Therefore, the value should be increased to be consistent with the revised value. The evaluated value of JENDL-4.0 seems to be about 23% overestimation. The data by Baerg et al [36] was deduced relative to the resonance integral of 59 Co. They used 48.6 b for that of 59 Co available at the time [37], which is about 2/3 of the current evaluated data 75.85 b. After correcting the resonance integral of 59 Co, their value increases to 96 b. There is an apparent discrepancy between the revised Kotoh's value and Baerg's value. Therefore, it would not be reasonable to calculate an averaged value from these contradicted data. According to our analysis, we recommend the revised value based on Katoh's data [25].

Conclusions
A new convention here proposed for the epithermal neutron spectrum component, that is, (1+β)/(βE+E 1+α ) form function, with two shape factors α and β, is formulated aimed at improving accuracy of resonance integrals. Bias effects on determination of resonance integrals, due to the approximating functions traditionally used, having mathematical forms 1/E, 1/E 1+α , as well as the new type (1+β)/(βE+E 1+α ) are at last compared. The possible bias due to a 1/E type assumption is shown to be up to about 50% on determination of the resonance integral. The bias is shown to be reduced to the level of about 5% by introducing a 1/ E 1+α form function. On the other hand, by utilizing the new (1+β)/(βE+E 1+α ) form function here proposed, the bias is shown to be further limited to as low as about 1%. It is also shown that the At last, a general formula for activation analysis is given, which enables to utilize the developed method. In order to get a demonstration of the discussed method and related impacts of the bias effects, the thermal neutron-capture cross section and resonance integral of 135 Cs measured at a research reactor JRR-3 have been re-evaluated. The Katoh's resonance integral data [25] is revised as 43 (3) b, which is 13% larger than the original value. The proposed method is expected to be widely applied to re-evaluate experimental data for resonance integral determination, and to plan an experiment aiming accuracy less than 1% for resonance integrals.