Determination of the default curve for the unfolding procedure in the measurement of threshold neutron excitation functions

Nuclear Instruments and Methods in Physics Research A 739 (2014) 68–74

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Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima

Determination of the default curve for the unfolding procedure in the measurement of threshold neutron excitation functions Nikola Jovancevic, Laura Daraban, Stephan Oberstedt n European Commission, Joint Research Centre, Institute for Reference Materials and Measurements (IRMM), Retieseweg 111, B-2440 Geel, Belgium

art ic l e i nf o

a b s t r a c t

Article history: Received 12 September 2013 Received in revised form 4 December 2013 Accepted 5 December 2013 Available online 17 December 2013

In this study we have improved the technique for measuring the neutron activation cross-section using wide energy neutron beams (NAXSUN). We propose a method for the determination of the default function for the unfolding procedure, which is an important and critical part for extracting reaction cross-sections from this type of measurements. The new method was tested on the measurement of the excitation function from the threshold energy up to 5.6 MeV for the 113In(n,n0 )113mIn and 115In(n,n0 )115mIn reactions. & 2013 Elsevier B.V. All rights reserved.

Keywords: Spectrum unfolding Default function Neutron fluence spectrometry Neutron-induced reaction Excitation function

1. Introduction The measurement of accurate neutron activation cross-section data is very important for various research activities, such as astrophysics, geophysics, environmental protection, for fission and fusion reactor installations, for the production of medical isotopes, for the modeling of nuclear reaction cross-sections and for benchmarking the predictive power of those models etc. Up to now, many neutron activation cross-section measurements were carried out at research centres worldwide and several international data bases were established [1]. In general those data were measured pointwise at well-defined incident neutron energies. An overview of the data libraries shows that for many isotopes exist wide disagreements between the data obtained in different experiments [2]. For a considerable number of isotopes, none or only a few experimental data exist on neutron activation cross-sections. Moreover, different model calculations may show different trends to the same set of experimental data. These were some of the reasons to come up with a new experimental technique for the determination of activation cross-section data in neutron-induced reactions. At IRMM (Institute for References Materials and Measurements), a new method was recently developed for obtaining neutron excitation functions in a wide energy range. By this method, as described in Ref. [3], it is possible to determine a neutron excitation function by irradiating identical disks, containing the studied isotope, in energy overlapping neutron beams and using the unfolding spectrum

n

Corresponding author. E-mail address: [email protected] (S. Oberstedt).

0168-9002/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nima.2013.12.017

procedure to extract the reaction cross-section [4]. In the study reported in Ref. [3], the few channel spectrum-unfolding technique, which starts with an initial guess excitation function was used. A-priori information, necessary for the construction of the initial guess excitation function, can be obtained from the available experimental or calculated data. However, the problem is that the final results can be dependent on the initial guess function. Moreover, it is questionable which data can be taken in consideration for the initial guess curve if there are large discrepancies between the existing results or if there is no data at all. In the demonstrator work reported in Ref. [3], the initial guess function was practically obtained from the large and concise data set existing in literature. Then, of course, the default function is already close to the real excitation function and convergence of the unfolding routine may be expected. This approach is not possible when no or discrepant data exist. Therefore, in this study we analyzed this problem and propose a new method for the determination of the guess function which enters the unfolding procedure. By this method, it is possible to obtain the new neutron excitation function that will be practically independent form other measurements or calculations. This new approach was tested on the previous experimental data for the 113In(n,n0 ) 113mIn and 115In(n,n0 )115mIn reactions reported in Ref. [3].

2. Method The method developed at IRMM for the determination of the neutron excitation function [3] and used in this study, is based on the irradiation of some number of identical disks containing the

N. Jovancevic et al. / Nuclear Instruments and Methods in Physics Research A 739 (2014) 68–74

studied isotopes in a series of well known, but different and energy overlapping neutron fields. The gamma-activity in the disks induced by the different neutron fields were measured by means of high-resolution gamma-ray spectroscopy systems. The induced saturation gamma activity of the disk is proportional to the product of the cross-section for a certain reaction and the corresponding neutron flux ð1Þ

i

where Φki are the values of neutron fluxes for different energy Ei in the case of irradiation of disk k, si are the values of the neutron excitation function, which correspond to the energy Ei, m is the number of irradiated identical disks and c is the number of bins in the neutron spectra and the excitation function curve. In this study, the following values were used m ¼ 10 and c ¼ 140. The system of Eq. (1) is undetermined with an infinite number of solutions (m⪡c). However, it is possible to obtain the si values when the values Φki are well known, by the measurement of Ck using a technique for the few channel spectrum unfolding. In this work, the maximum entropy code MAXED [5] is applied for this purpose. The MAXED code starts with a guess excitation function and, from all the curves that fit the measured saturation gamma activity (1), the curve that maximizes the relative entropy is chosen    Z  sðEÞ S¼  sðEÞln def ð2Þ þ sdef ðEÞ  sðEÞ dE s ðEÞ where sdef ðEÞ is the guess excitation function or usually called default excitation function curve. The standard procedure is that the default curve can be chosen from the corresponding evaluated activation cross-section data file [6,7] or from other existing experimental data. The problem apparently is what can be chosen as default excitation function, if the measurement data do not exist or the disagreement between different experimental and calculation data is large. In order to solve this problem, we made several simple approximations by which we transform the undetermined system of Eq. (1) into a determined system (c ¼ m). By solving this determined system of equations, we obtained the default guess function for the start of the unfolding procedure. For each irradiation and different neutron fields, the average value of the cross-section is defined by 〈s〉k ¼

∑i Φki sk ∑i Φki

ð3Þ

Combining the above equation with (1) gives 〈s〉k 

Ck

Φk

∑i si Φki Ei ∑i si Φki

ð5Þ

∑i Φki Ei

Φk

3.1. Materials The applicability of the method and its advantages are demonstrated on the determination of the excitation functions for the reactions 113In(n,n0 )113mIn and 115In(n,n0 )115mIn. A sufficient number of measurements has been done up to now on the crosssections for those two reactions [7]. The IRMM method for obtaining the neutron excitation function of the 115In(n,n0 )115mIn and 113In(n,n0 )113mIn reactions was used Ref. [3]. In that case the default excitation functions were determined by fitting a 9th-order polynomial to the existing experimental data [7]. That means that guess functions were already well defined and close to real values at the beginning of the unfolding procedure. The obtained final unfolded excitation function is in very good agreement with the existing experimental data and it is presented in Figs. 1 and 2. In this work, we used the spectroscopic data from Ref. [3] for obtaining the cross-section function of the 115In(n,n0 )115mIn and 113 In(n,n0 )113mIn reactions. However, instead of fitting the existing experimental data as it was done in Ref. [3], we defined the default neutron excitation function by the new method described above. In this way, we can prove the applicability of our new method by comparing the results of the two approaches to the unfolding procedure on the same data set. 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05

where Ei is the energy of the excitation function or neutron fluence bin. Since here the values si are unknown, we make the following simplification: 〈E〉k 

3. Measurement

ð4Þ

where Φk ¼ ∑i Φki is the integral neutron flux during the irradiation of disk k. By this approximation, the undetermined system (1) is transformed into a determined system (4). This means that for each irradiation it is possible to calculate one value of hsik , if the total neutron fluence and the detected specific gamma activities (Ck) are well known. The standard way of presentation in the neutron dosimetry files is that for the average cross-section value hsik , the corresponding average neutron energy is given as 〈E〉k ¼

Based on the above expression, we can calculate the average neutron energy hEik , which corresponds to the average values of the cross-sections 〈s〉k . The performed analysis shows that the approximation made in Eq. (6) gives valuable results due to the specific and energy well defined neutron fluence. The details about the neutron fluence used for the disks irradiation are presented in the next chapter. The k values of hsik from Eq. (4), which correspond to the k energy values hEik from Eq. (6), are used as a starting point for obtaining the default guess function. The next step in this procedure is the linear interpolation of the energy dependence of the obtained cross-section values hsik on energy Ek. This interpolated function was used as the default excitation function sdef ðEÞ for the unfolding procedure. In this way, the obtained shape of the default neutron excitation function is obtained from the experiment itself and therefore, independent from any other experimental or calculated data set.

Cross section [b]

C k ¼ ∑ Φki si ; k ¼ 1; :::; m; i ¼ 1; :::; c

69

ð6Þ

0.00 0

1

2

3

4

5

Neutron Energy [MeV] Fig. 1. The excitation function for the 113In(n,n0 )113mIn reaction. Open dots: EXFOR experimental data file for the [7]; black dots: unfolded data from [3].

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0.45

Table 1 Measured specific activities in 10  21 Bq atoms  1 for the different irradiations. The first column indicates the neutron producing reaction and the ion energy in keV (see caption of Fig. 3) [3].

0.40

Cross section [b]

0.35 0.30 0.25 0.20 0.15 0.10 0.05

Reaction

Activity concentration (10  21 Bq atoms  1)

Id

113m

Tp1463 Tp1912 Tp2384 Tp2620 Tp2866 Tp3647 Dd1142 Dd1480 Dd1949 Dd2400

– 4.5(2) 18.7(3) 21.5(3) 30.1(3) 44.1(4) 7.30(14) 14.90(18) 12.44(17) 21.1(2)

115m

In

In

0.11(1) 4.1(2) 24.6(11) 14.9(7) 36.4(17) 29.8(14) 4.02(19) 8.5(4) 7.5(3) 25.6(12)

0.00 0

1

2

3

4

5

Neutron energy [MeV] Fig. 2. The excitation function for the 115In(n,n0 )115mIn reaction. Open dots: EXFOR experimental data file for the [7]; black dots: unfolded data from [3].

A detailed explanation of the experimental set up and the measurement is given in Ref. [3]. Therefore, a brief description of the used neutron field, irradiation and measurement of the gamma activity is presented in the following, only. The irradiation of the indium disks was done at the IRMM in the Van de Graff neutron laboratory. All disks were made from natural indium. They had an identical shape with a diameter of 20 mm and 5 mm thick. The very important part of this method is the generating of a well-characterized neutron field for the disk irradiation. The neutron spectra should be of different shape, overlapping and covering the energy range of interest in order to obtain the neutron excitation function. Those spectra were obtained by a controlled energy distortion of a well-known quasi-monoenergetic neutron field, generated by an electrostatic ion beam accelerator, using the 2H(d,n)3He and 3H(d,n)4He nuclear reactions. In this study, we achieved a neutron field distortion by scanning the disks during irradiation over different angles relative to the ion beam. Samples were irradiated by neutrons in front of a neutron-producing target over a certain angular interval and therefore, the disk was exposed to different neutron energies and intensities. By using this technique, a total neutron spectrum over a broad energy region can be achieved. Each disk has been irradiated at one neutron energy, as listed in Table 1, and at 41 different positions in an interval from 01 to 801 relative to the direction of the beam, in steps of 21. The radiation time was increased with the angle to compensate for the decreasing flux at increasing angles. The neutron spectra were simulated by the Monte-Carlo code TARGET [8] and verified by the measurement using the IRMM Bonner-spheres neutron spectrometer. Fig. 3 depicts the obtained neutron flux distributions used for the irradiation of different indium disks. The gamma-ray measurements of the neutron-induced activities were carried out using low-background high-purity germanium detectors. From the number of net counts in the peak of interest, N, the specific activity per atom of the activated target isotopes, C, was calculated as   NM λ C¼ ð7Þ  eλtc N a mε P γ I A 1  e  λ t m where M is the atomic weight (g mol  1), NA is the Avogadro's constant (6.02214  1023 mol  1), m is the mass of the disk (g), ε is

Neutron intesity [arbitary unit]

3.2. Neutron fields, irradiation and gamma-ray measurements

10000

1000

100 Dd1142 Dd1949 Dd2400 Dd1480 Tp2866 Tp3647 Tp2620 Tp2384 Tp1912 Tp1463

10

1

0.1 0

1

2

3

4

5

6

Neutron energy [MeV]

Fig. 3. Total neutron flux spectra used for the irradiation of the iridium disks [3]. Reaction notations: Dd—2H(d,n), Tp—3H(p,n), number—energy, in keV, of the ions inducing the reaction.

the full energy peak efficiency, Pγ is the gamma-ray emission probability, IA is the isotopic abundance, λ is the decay constant (s  1), tm is the measurement time (s) and tc is the cooling time (s). The specific activity results from the gamma ray measurements are given in Table 1 [3].

4. Determination of the default neutron excitation function The integral neutron fluencies, Φk, and the average energies, 〈E〉k , calculated according to Eq. (5) for each irradiation, are presented in Table 2. Using this data and the values of the measured saturated activity (Table 1), the values of the neutron average cross-sections 〈s〉k were determined. Therefore, nine values of the neutron excitation function for the 113In(n,n0 )113mIn reaction and 10 values for the 115 In(n,n0 )115mIn reaction were obtained in the energy range from 0 MeV up to 5.6 MeV (presented by black dots in Figs. 4 and 5). The standard deviation for the average energy 〈E〉k was calculated by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

s〈E〉k ¼

∑ i

Φi ðE  〈E〉k Þ2 Φk i

ð8Þ

The obtained values for shEik are presented by error bars in Figs. 4 and 5. We note here that the neutron average cross-sections 〈s〉k data may not be directly compared with those obtained pointwise and with an energy-resolved neutron beam, as it is obvious from the indicated uncertainty on the energy axis in Figs. 4 and 5.

N. Jovancevic et al. / Nuclear Instruments and Methods in Physics Research A 739 (2014) 68–74

Table 2 Values of integral neutron fluxes and the corresponding average energies (for the meaning of the reaction Id see caption of Fig. 1). Reaction Id

Tp1463 Tp1912 Tp2384 Tp2620 Tp2866 Tp3647 Dd1142 Dd1480 Dd1949 Dd2400

In(n,n0 )113mIn

In(n,n0 )115mIn

113

115

〈E〉k [MeV]

Φk [103arbitrary unit]

〈E〉k [MeV]

Φk [103arbitrary unit]

– 0.71(21) 1.11(30) 1.32(43) 1.58(38) 2.24(50) 3.62(47) 3.99(54) 4.38(68) 4.77(82)

99(5) 80(4) 60(3) 55(3) 48.0(24) 8.38(42) 17.3(9) 13.3(7) 20(1)

0.32(14) 0.71(21) 1.11(30) 1.32(34) 1.58(38) 2.24(50) 3.62(47) 3.99(54) 4.38(68) 4.77(82)

46.9(23) 54.2(27) 76(4) 30.3(15) 50.5(25) 26.0(13) 4.14(21) 8.3(4) 6.5(3) 20(1)

Cross section [arbitary unit]

1.2

1.0

0.8

s113 ¼ 0.4

0.0 0

1

2

3

4

5

6

Neutron energy [MeV] Fig. 4. Determination of the default excitation function for the 113In(n,n0 )113mIn reaction: the black dots with error bars indicate the values of the cross-section calculated based on Eq. (4), the red line represents a linear interpolation of the experimental data and the dotted line gives the error bands for the default excitation function. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

1.4

f 113 s115 f 115

ð9Þ

where f113 and f115 are the corresponding integrated values for the corrected sequential linear fits as presented in Figs. 4 and 5. In the case when no integral cross-section is available for an isotope under investigation, a second material with well-characterized cross-section can be irradiated simultaneously and used as an internal standard. The finally obtained guess excitation functions are presented, together with the existing experimental data in Fig. 6 for 113 In(n,n0 )113mIn and in Fig. 7 for 115In(n,n0 )115mIn. The error bands are given as dashed lines. From both figures it is evident, that with the new approach it is possible to obtain the default neutron excitation function, which follows pretty well the trend of the data and present a good starting point for the cross-section unfolding procedure.

1.2

0.40

1.0

0.35

0.8

0.30

Cross section [b]

Cross section [arbitary unit]

The dependence of 〈s〉k on 〈E〉k was linearly interpolated between each two adjacent experimental values. The results of the linearly interpolation were calculated in the energy range between the minimum and the maximum values of 〈E〉k (Table 2) by the bin widths of ΔEn ¼0.04 MeV. Since the maximum neutron energy in this measurement was 5.6 MeV, our default excitation functions were approximated by a flat curve between 4.7 MeV and the maximum neutron energy in this measurement of 5.6 MeV. This approximation is based on expectation that the cross-section reaches a plateau extending more or less up to 8 MeV. The shape of the default neutron excitation functions are represented by a red line in Figs. 4 and 5 for the 113In(n,n0 )113mIn and 115In(n, n0 )115mIn, respectively. In Figs. 4 and 5 are also presented the uncertainty intervals for the determined default functions. These intervals are obtained as a sum of the square of the uncertainties for 〈s〉k and 〈E〉k . The default excitation function curve for the 115In(n,n0 )115mIn reaction was normalized to the integral cross-section calculated up to 5.6 MeV, to 1.380 b from the EXFOR data [3]. Since the values of the cross-sections for the 113In(n,n0 )113mIn reaction in the literature data are characterised to a lesser degree than for the 115In(n, n0 )115mIn reaction, the integrated cross-section for 115In was used as internal standard to calculated the integrated cross-section for 113 In. The excitation function for the 113In(n,n0 )113mIn reaction was normalized to the integral cross-section, calculated to 1.191 b from

0.6

0.2

71

0.6 0.4 0.2 0.0

0.25 0.20 0.15 0.10

0

1

2

3

4

5

6

0.05

Neutron energy [MeV] Fig. 5. Determination of the default excitation function for the reaction (for different symbols see caption of Fig. 4).

In(n,n0 )115mIn

115

Those uncertainties of neutron beam energy indicate the width of each distorted neutron beams from which the corresponding average energy was calculated.

0.00 0

2

4

6

Neutron energy [MeV] Fig. 6. The default excitation function for the 113In(n,n0 )113mIn reaction obtained in this work (red line) with error bands (dotted red lines). Open dots: EXFOR experimental data [7]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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0.40

0.4

0.35

Cross section [b]

Cross section [b]

0.30

0.3

0.2

0.25 0.20 0.15 0.10

0.1

0.05

0.0

0.00

0

1

2

3

4

5

6

0

1

Neutron energy [MeV] Fig. 7. The default excitation function for the 115In(n,n0 )115mIn reaction obtained in this work (red line) with error bars (dotted red lines). Open dots: EXFOR experimental data [7]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

115m

In

4

5

0.4

In

(C-M)/M (default)

Final

(C-M)/M (default)

Final

– 0.53931 0.08460 0.05954  0.01775  0.14164  0.00130 0.04901 0.04071  0.08386 0.033

– 0.06260 0.00510 0.01161 0.00204  0.03174  0.00265 0.02014 0.02185  0.03257 0.0004

2.04059 0.01968  0.20808  0.22743  0.27765  0.33444  0.18329  0.18661  0.20883 0.27191 0.579

0.09745 0.05587 0.04056 0.04137 0.00006  0.09680 0.00770 0.00609  0.02034  0.08298 0.004

“Default” indicates that the calculated data refers to the default excitation function curve and “Final” to the unfolded result curve. Id is the irradiation identifier number, SS indicates “sums of squares” and k is the number of irradiations.

Cross section [b]

Tp1463 Tp1912 Tp2384 Tp2620 Tp2866 Tp3647 Dd1142 Dd1480 Dd1949 Dd2400 SS/(k  1)

113m

3

Fig. 8. Unfolded excitation function for the 113In(n,n0 )113mIn reaction obtained in this work, presented by the full (red) line. EXFOR data is presented by open black dots [7]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 3 Relation between the measured activity data (M) and the calculated data (C) for the 113 In(n,n0 )113mIn and 115In(n,n0 )115mIn reactions. Id

2

Neutron energy [MeV]

0.3

0.2

0.1

0.0 0

1

2

3

4

5

Neutron energy [MeV] Fig. 9. Unfolded excitation function for the 115In(n,n0 )115mIn reaction obtained in this work, presented by the full (red) line. EXFOR data is presented by open black dots [7]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

5. Results and discussion For the unfolding procedure we used the software WinDONA, which was developed at IRMM [3,9]. This software is based on the MAXED algorithm. As described in the previous work [3], the MAXED procedure also allows for the uncertainty propagation of the measured data (see Table 3) and the default curve. The unfolding was performed for 140 bins (c ¼140 as stated in Eq. (1)) covering the range from En ¼0 to 5.6 MeV. The final excitation functions are given in Figs. 8 and 9, together with the experimental data compiled in EXFOR [7]. After unfolding the excitation curves were normalized to the integral cross-section values, exactly as it was done for the default excitation function. As a validation of the obtained results, the induced saturated activation data were calculated from Eq. (1) by using the crosssection data (both the default and unfolded) and the neutron fluence field used for the unfolding. In Table 3, a comparison between the measured and calculated data for the default and the unfolding data is shown. For the 113In(n,n0 )113mIn excitation function, the variance for the relative difference of the measured and the calculated data is 0.007 in the case of the unfolded excitation

function, and 0.33 for the default excitation function. The corresponding values for the 115In(n,n0 )115mIn excitation function are 0.004 (the unfolded) and 0.58 for the default excitation function. These results show that the unfolding procedure converged to a much better description of the measured activity data than the initial default curve. In this study, we also compared our results with those from Ref. [3]. In Figs. 10 and 11 we show for both reactions 113In(n,n0 )113m In and 115In(n,n0 )115mIn the default curves from Ref. [3] and from this work, as well as the obtained unfolded functions in both works. The results show that the unfolded excitation curves from this work and from the previous work [3] are following the same trend. A very good agreement between the unfolded curves is in the region below 2 MeV, for both tested reactions. A sharp resonance-like structure appears around 2.7 MeV for the 113In(n,n0 )113mIn reaction and around 2.8 MeV for the 115In(n, n0 )115mIn reaction, which is not present in the experimental data. This structure appeared also in the previous work [3]. Especially in the case of 115In(n,n0 )115mIn the experimental data are not well described in the energy region between 3 MeV and 4 MeV. Above

N. Jovancevic et al. / Nuclear Instruments and Methods in Physics Research A 739 (2014) 68–74

0.40 0.35

Cross section [b]

0.30 0.25 0.20 0.15

default function in this work unfolded function in this work unfolded function from [3] default function from [3]

0.10 0.05 0.00 0

1

2

3

4

5

Neutron energy [b] Fig. 10. Comparison of the neutron excitation function for the 113In(n,n0 )113mIn reaction, from this study and from the previous work [3]. Dashed black line— default function from work [3], dashed red line—default function from this work, solid red line—unfolded function form this work, black line—unfolded function from [3]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

may conclude that a careful choice of shape and energy range of the distorted neutron beams prior to a measurement campaign is essential. Possible sources of uncertainties in this method come from the calculation of the neutron fields, the measurement of the induced gamma activity, the measurements of the distance between the neutron-producing target and the irradiated disks, the weighting of the disks and the measurements of the beam current. All of these contribute to the uncertainty of the determined default function and affect the uncertainty of the finally obtained excitation function. The uncertainties of the measurement of the induced gamma activity and the calculation of the neutron field are presented in Tables 1 and 2. Other listed sources of uncertainty are estimated to sum up to 1.5%. We calculated by the unfolding procedures how an uncertainty of the default function translates to an uncertainty in the excitation function. It has been done in that way that the upper and lower error bands for the default function (see Figs. 6 and 7) were taken as input function for the

0.40 0.35 0.30

Cross section [b]

0.45 0.40 0.35

Cross section [b]

73

0.30 0.25

0.25 0.20 0.15

spline default unfolded unfolded from [3] default from[3]

0.10

0.20 0.05

default function from [3] unfolded function from [3] unfolded function in this work default function in this work

0.15 0.10

0.00 0

1

0.05

0

1

2

3

4

5

Neutron energy [MeV] Fig. 11. Comparison of the neutron excitation function for the 115In(n,n0 )115mIn and reaction, from this study and from the previous work [3]. (See caption on Fig.10 for details.).

4 MeV unfolded curves follow well the EXFOR data. This result proves that the new technique, presented here, allows obtaining a reasonable default excitation function, which at the end converges to a realistic neutron excitation function. The sharp resonance-like structure appearing around 2.7 MeV can be explained as being due to the neutron-energy distributions which were used for irradiation. The “resonance” energy corresponds to that energy region, where two adjacent neutron distributions do not overlap very well (see Fig. 3 and the uncertainty on the energy axis in Figs. 4 and 5). Therefore, the unfolded function in that energy region is highly affected by the chosen initial default function (Figs. 10 and 11). The absence of measured values 〈s〉k for energies between 2.6 MeV and 3.2 MeV does not lead to a good convergence of the unfolding routine in this region. For possible improvement we tried also a cubic spline interpolation instead of the linear one for determining the default function. The corresponding results, presented in Figs. 12 and 13, show a much better agreement with the data. Of course, this is a consequence of the fact that the spline-interpolated default function is closer to the existing experimental cross-section values. From this we

3

4

5

Fig. 12. Comparison of the neutron excitation function for the 113In(n,n0 )113mIn reaction, from this study and from the previous work [3]. Dashed black line— default function from work [3], dashed red line—default cubic spline interpolated function, solid red line—unfolded function in this work, black line—unfolded function from [3]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

0.4

Cross sectin [b]

0.00

2

Neutron energy [MeV]

0.3

0.2

unfolded function default splin function default from [3] unfolded from [3]

0.1

0.0 0

1

2

3

4

5

Neutron energy [MeV] Fig. 13. Comparison of the neutron excitation function for the 115In(n,n0 )115mIn and reaction, from this study and from the previous work [3]. (See caption on Fig. 12 for details.).

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0.50 0.45

Cross section [b]

0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0

1

2

3

4

5

Neutron energy[MeV] Fig. 14. Comparison of the default and the unfolded functions for 113In(n,n0 )113mIn. Solid and dotted black lines—default function with error bars, solid and dotted red lines—unfolded function. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

1.0 0.9

Cross section [b]

0.8 0.7

approach we transformed an underdetermined system of equations into a symmetric, determined system of equations, where the number of equations is determined by the number of measured disks. This is possible, when each neutron field has a broad energy range, with sufficient overlap between adjacent neutron fields and different mean energy. We tested this new method by determining the neutron excitation function for the 113In(n, n0 )113mIn and 115In(n,n0 )115mIn reactions. The obtained results show that this approach can provide reliable input data without taking into account as a-priori information other existing experimental results for constructing the default function. In this way, the IRMM method to measure neutron excitation functions can be applied when the existing experimental data and/or theoretical calculations cannot provide enough information necessary to obtain a reasonable default guess function for the unfolding procedure. The new method was applied here for the determination of neutron excitation functions in the energy range from threshold up to 5.6 MeV. With input data extracted from the NAXSUN experiment [3], directly reasonable agreement with the EXFOR data was achieved. With the presented solution to the correct choice of the default input function to the unfolding procedure the neutron activation cross-section measurements using wide energy neutron beams (NAXSUN) becomes feasible. It can be used to obtain information about the shape, threshold and plateau value of the neutron excitation function over a wide energy range. We plan to apply the method to neutron-induced reactions where present experimental data show large discrepancy, as well as for the verification of models used for the calculation of the neutron excitation function.

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Acknowledgment

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We are grateful to G. Lövestam and M. Hult for very useful advice and discussion.

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References

0.0 0

1

2

3

4

5

Neutron energy [MeV] Fig. 15. Comparison of the default and the unfolded functions for (see caption on Fig. 14 for details).

In(n,n0 )115mIn

115

unfolding procedures. The corresponding results are depicted in Fig. 14 for 113In(n,n0 )113mIn and in Fig. 15 for 115In(n,n0 )115mIn as dashed lines. The obtained confidence bands reflect well the uncertainties in the guess functions. Of course, in the region where the distorted neutron beams do not show sufficient overlap, unfolding remains most unstable and the error bands show the largest spread. 6. Conclusion In this work, we investigated a new approach for obtaining the guess default function for the unfolding procedure. Within this

[1] The IAEA Reference Neutron Activation Library, RNAL, 〈http://www.nds.iaea. org/nbspub/rnal/www/〉. [2] A. Plompen (coordinator), Neutron Activation Cross-section Measurement from Threshold to 20 MeV for the Validation of Nuclear Models and their Parameters, International Evaluation Co-operation, Report NEA/WPEC-19, OECD Nuclear Energy Agency, Le Seine Saint-Germain, Issy-les-Moulineaux, France, ISBN:92-64-01070-X, 2005. [3] Göran Lövestam, Mikael Hult, Andreas Fessler, Thierry Gamboni, Joël Gasparro, Wouter Geerts, Ricardo Jaime, Patric Lindahl, Stephan Oberstedt, Hamid Tagziria, Nucl. Instr. Meth. A 580 (2007) 1400. [4] M. Reginatto, Radiat. Meas. 40 (2010) 1323. [5] M. Reginatto, P. Goldhagen, Health Phys. 77 (1999) 579. [6] The International Reactor Dosimetry File IRDF-2002, 〈http://www-nds.iaea.or. at/irdf2002/〉. [7] The EXFOR, Experimental Nuclear Reaction Data, 〈http://www.nea.fr〉. [8] D.J. Thomas, A.V. Alevra, Nucl. Instr. Meth. A 476 (2002) 12. [9] J.S. Elisabeth Wieslander, Göran Lövestam, Mikael Hult, Andreas Fessler, Joël Gasparro, Pierre Kockerols, Radiat. Prot. Dosim. 138 (3) (2010) 205.