Neutron-induced reaction cross-sections of ^natPd with the fast neutron based on the ⁹Be(p,n) reaction

3.1 Calculation of the threshold energy and average neutron energy

Natural palladium has six stable isotopes ^{35 , 36} with the composition of ¹⁰²Pd (1.02 %), ¹⁰⁴Pd (11.14 %), ¹⁰⁵Pd (22.33 %), ¹⁰⁶Pd (27.33 %), ¹⁰⁸Pd (26.46 %), and ¹¹⁰Pd (11.72 %), respectively. In the present work, we measured the ^natPd(n,xn)^109,101Pd and ^natPd(n,pxn)^105,106mRh reaction cross-sections. The neutron fluxes used for these reactions have been measured by using the ²⁷Al(n,α)²⁴Na monitor reaction. In the neutron-induced reactions of ^natPd, the same reaction product is produced from different Pd-isotopes, which have different threshold energies. For the products of present interest, the Q-value and threshold energies for possible contributing (n,xn) and (n,pxn) reactions of Pd-isotopes and the (n,α) reaction of ²⁷Al were calculated by using ref ⁴³ and are listed in Table 1.

The neutron spectra from the ⁹Be(p,n) reaction with the proton energies of 35 and 45 MeV were generated by using a computer code MCNPX 2.6.0, ⁴² which have been shown in Figure 2. The simulation by using MCNPX code ⁴² is justified based on the similar calculation for the ⁷Li(p,xn), ^{44 , 45 9}Be(p,xn) ^{46 , 47} and ⁹Be(d,xn) ^{48 , 49} reactions. The spectral neutron yield from the ⁷Li+p (carbon backing) reaction with the proton energies of 20, 25, 30, 35 and 45 was measured by Uwamino et al. ⁴⁴ Simakov et al. ⁴⁵ calculated the neutron spectrum from the ⁷Li+p reaction by using MCNPX simulation and reproduced the experimentally measured quasi-mono energetic neutron spectrum of Uwamino et al. ⁴⁴ The quasi-mono-energetic neutrons produced from the ⁹Be+p reaction were experimentally measured by Uwamino et al. ⁴⁶ Rakopoulos et al. ⁴⁷ at the Svedberg Laboratory (TSL) used the MCNPX code to estimate the neutron source from the ⁹Be(p,xn) reaction converters of different thicknesses with a 30 MeV proton beam. On the other hand, Wölfle et al. ^{48 , 49} used the ⁹Be+d reaction at various deuteron energies for neutron production. They characterized the neutron spectrum via multiple-foil activation and iterative unfolding technique using the SAND-II code as well as by generalized least-squares unfolding code, and found similar results. In the present work, we have not characterized the neutron spectrum from the ⁹Be+p reaction at different proton energies by using multiple-foil activation and iterative unfolding technique but generated by using the computer code MCNPX as done by Rakopoulos et al. ⁴⁷ In order to examine this, first we generated the neutron spectrum at the proton energy of 40 MeV and compared it with the experimental spectrum of Uwamino et al. ⁴⁶ and found it to be in good agreement except for the low neutron energy tail part. The aforementioned facts indicate that the MCNPX simulation can be used to generate the neutron spectrum at sample position for the ⁷Li+p, ⁹Be+p and ⁹Be+d reactions with proper geometry as suggested by Novak et al. ⁵⁰

Figure 2:

Neutron spectra from the ⁹Be(p,n)⁹B reaction for the proton energies of 35 and of 45 MeV calculated by using the MCNPX code. ⁴²

It can be seen from Figure 2 that the neutron spectra have broad energy distributions for the proton energies of 35 and 45 MeV. Based on the neutron spectra, the flux-weighted average neutron energies ( E n I , J M ) for each reaction within the range of threshold values to the maximum energy were calculated by using the following equation.

where ϕ(E _n ) is the neutron flux as a function of neutron energy estimated with the MCNPX 2.6.0 code ⁴² as shown in Figure 2 and E th I , J M is the threshold energy for the ^I Pd(n,xn) ^J Pd, ^I Pd(n,pxn) ^J Rh and the ²⁷Al(n,α)²⁴Na (monitor) reaction. The flux-weighted average neutron energies for different (n,xn) and (n,pxn) reactions of various Pd isotopes and for the ²⁷Al(n,α)²⁴Na monitor reaction on two different neutron spectra are given in Table 2.

3.2 Calculation of neutron flux and conversion factor

As mentioned before, for the estimation of (n,xn) and (n,pxn) reaction cross-sections of Pd-isotopes, the experimental neutron fluxes were determined by using the ²⁷Al(n, α)²⁴Na monitor reaction. The neutron fluxes for two different average neutron energies were experimentally measured from the photo-peak activity of the 1,368.6 keV γ-line of ²⁴Na produced from the ²⁷Al(n,α) reaction. The net photo-peak area (A _net) was obtained from the gross-counts by subtracting the linear Compton background and is related to the measured neutron flux ( Φ m E n M ) by the following relation:

where N is the atomic number of target nuclei and σ M E n M the average cross-section of the ²⁷Al(n,α)²⁴Na monitor reaction at the average neutron energy, I _γ the branching intensity of the analyzed γ-ray, ε _γ the detection efficiency of the photon and λ is the decay constant (=ln2/T_1/2) for the isotope of interest. T _irr, T_C, CL, and LT are the irradiation time, decay time, clock time, and counting time, respectively. The net photo-peak area (A _net) in Eq. (2) has been multiplied by the CL/LT factor for the dead time correction. The nuclear spectroscopic data ³⁷ such as γ-ray energies, branching intensity, and half-lives of reaction products are given in Table 1.

The cross-section (σ ^M (E _n )) of monitor reaction ²⁷Al(n,α)²⁴Na with mono-energetic neutrons was taken from TALYS-1.9 code. ^{51 , 52} This is because most of the experimental cross-section (σ ^M (E _n )) of monitor reaction ²⁷Al(n,α)²⁴Na was available in literature ^{53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64} within the neutron energy of 20 MeV, which are plotted in Figure 3. Above the neutron energy of 20 MeV, some data are available in literature, which are also plotted in Figure 3. In the same figure, calculated data based on TALYS-1.9 code ^{51 , 52} were also plotted for comparison. Figure 3 shows a good agreement between experimental data within neutron energy of 20 MeV and calculated values from TALYS-1.9. Few data above the neutron energy of 20 MeV are very much scattered. In view of this, the flux-weighted average cross-section ( σ M E n M ) based on the data of TALYS-1.9 code was obtained with the following relation: ³⁰

where E thr M is the threshold energy of the monitor reaction, and E _max is the maximum neutron energy. The neutron flux distribution (ϕ(E _n )) was taken from Figure 2. The flux-weighted average cross-section ( σ M E n M ) of the ²⁷Al(n,α)²⁴Na monitor reaction at the average neutron energies of 18.7 and 23.6 MeV corresponding to the proton energies of 35 and 45 MeV were found to be 0.0329 and 0.0284 b, which are given in Table 2.

Figure 3:

Cross-sections of ²⁷Al(n,α)²⁴Na reaction as a function of neutron energy from literature ^{53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64} and calculated values from the TALYS-1.9. ^{51 , 52}

We defined ^I Pd(n,xn) ^J Pd and ^I Pd(n,pxn) ^J Rh as a main reaction for this study. It is mentioned before that the natural Pd has six stable isotopes (^IPd, where I = 102, 104, 105, 106, 108, and 110) with different isotopic composition. Table 1 shows that the threshold energy of ²⁷Al(n,α)²⁴Na monitor reaction is 3.25 MeV, whereas the threshold energies for the different (n,xn) and (n,pxn) reactions of various Pd-isotopes are different. Thus, it is necessary to modify the experiment neutron flux obtained from the ²⁷Al(n,α)²⁴Na monitor reaction for different (n,xn) and (n,pxn) reactions of various Pd-isotopes based on their threshold energies to the maximum neutron energy. In order to do that, we defined the flux conversion factor (C ^I,J (〈E _n 〉)) for the ^I Pd(n,xn) ^J Pd and ^I Pd(n,pxn) ^J Rh reaction as follows:

where ϕ(E _n ) is neutron flux distribution as a function of neutron energy E _n calculated with the MCNPX 2.6.0 code ⁴² as shown in Figure 2. The E t h r M I , J is the threshold energy of the monitor reaction ²⁷Al(n,α)²⁴Na and the interest reaction ^I Pd(n,xn) ^J Pd and ^I Pd(n,pxn) ^J Rh. For example, in the ¹¹⁰Pd(n,2n)¹⁰⁹Pd reaction at the average neutron energies of 22.0, and 28.3 MeV, the weighted average flux obtained from the ²⁷Al(n,α)²⁴Na reaction is multiplied by factors of 0.798 and 0.793, respectively. For example, the flux conversion factor ( C 110 , 109 E n 110 , 109 = 0.798 ) is the ratio of the flux for the ¹¹⁰Pd(n,2n)¹⁰⁹Pd reaction from threshold energy ( E th 110 , 109  = 8.895 MeV) to the maximum neutron energy to the flux for the ²⁷Al(n,α)²⁴Na monitor reaction from threshold energy ( E thr M  = 3.25 MeV) to the maximum neutron energy. The conversion factors of the neutron flux for various (n,xn) or (n,pxn) reactions of different Pd isotopes to the overall flux of the ²⁷Al(n,α)²⁴Na reaction are given in Table 2.

3.3 Calculation of yield, yield-weighted conversion factor and the flux-weighted average cross-sections of (n,xn) and (n,pxn) reactions of Pd-isotopes

Since ^natPd has six isotopes, the production of ¹⁰¹Pd, ¹⁰⁵Rh and ^106mRh isotopes from the ^natPd(n,xn) and ^natPd(n,pxn) reactions have different yield percentages based on their threshold values and isotopic abundances. For example, the ^natPd(n,xn)¹⁰¹Pd reaction is contributed by the ¹⁰²Pd(n,2n)¹⁰¹Pd, ¹⁰⁴Pd(n,4n)¹⁰¹Pd, and ¹⁰⁵Pd(n,5n)¹⁰¹Pd reactions with different yield percentage. Similarly, the ^natPd(n,pxn)¹⁰⁵Rh reaction is contributed by the ¹⁰⁵Pd(n,p)^105gRh, ¹⁰⁶Pd(n,pn)¹⁰⁵Rh, and ¹⁰⁸Pd(n,p3n)¹⁰⁵Rh reactions with different yield percentages. The ^natPd(n,pxn)^106mRh reaction is contributed by the ¹⁰⁶Pd(n,p)^106mRh and ¹⁰⁸Pd(n,p2n)^106mRh reactions with different yield percentages. Therefore, the measured flux-weighted average cross-sections of ¹⁰¹Pd, ¹⁰⁵Rh, and ^106mRh from the ^natPd(n,xn) and ^natPd(n,pxn) reactions are the sum of the yield-weighted contributions from different isotopes of ^natPd. The normalized reaction yield contribution (Y _I,J ) for the ^I Pd(n,xn)^JPd and ^I Pd(n,xn) ^J Rh reactions were calculated by using the following relation. ^{65 , 66}

where A _I is the natural composition of ^I Pd isotopes in the ^natPd and ϕ(E _n ) is the neutron flux calculated using the MCNPX code. ⁴² The reaction cross-sections σ ^I,J (E _n ) for the ^IPd(n,xn)^JPd and ^I Pd(n,pxn) ^J Rh reactions were taken from the TENDL-2019 library. ³³

The ^natPd(n,yn)¹⁰⁹Pd reaction is contributed only by the ¹¹⁰Pd(n,2n)¹⁰⁹Pd reaction, in this case I=110 and J=109 and thus has a unique conversion factor as shown in Table 2. This is because ¹¹⁰Pd is the heaviest isotope, which can only contribute to the ¹¹⁰Pd(n,2n)¹⁰⁹Pd reaction. For the ^I Pd(n,xn)¹⁰¹Pd (I=102, 104, and 105) reactions, there are three conversion factors due to the primary contributions of ^102,104,105Pd-isotopes to the same reaction product ¹⁰¹Pd based on the average neutron energy of present work. Similarly, the ^IPd(n,pxn)¹⁰⁵Rh (I=105, 106, and 108) reactions also have three conversion factors due to the primary contributions of ^105,106,108Pd-isotopes to the same reaction product ¹⁰⁵Rh. For the ^IPd(n,pxn)^106mRh (I=106 and 108) reactions, there are only two conversion factors due to the primary contributions of ^106,108Pd-isotopes to the same reaction product ^106mRh. Thus, it is necessary to have a yield-weighted flux conversion factor C n a t , J E n n a t , J for the ^I Pd(n,xn)^JPd and ^I Pd(n,xn)^JRh reactions defined as follows:

where I and J correspond to ^I(J)Pd isotope and E n nat , J is the yield weighted average neutron energy for the ^natPd(n,xn)^JPd and ^I Pd(n,xn)^JRh reactions as defined as follows:

The yield weighted average neutron energy and the normalized yield contribution for the production of ¹⁰⁹Pd, ¹⁰¹Pd, ¹⁰⁵Rh and ^106mRh from the ^IPd(n,xn)^JPd and ^I Pd(n,xn)^JRh reactions are given in Table 3. It can be seen that for the formation cross-section, the yield percentage of ¹⁰⁹Pd is 100 % only from the ¹¹⁰Pd(n,2n)¹⁰⁹Pd reaction. Thus, the ^natPd(n,xn)¹⁰⁹Pd reaction has the contribution only from the ¹¹⁰Pd(n,2n)¹⁰⁹Pd reaction, which has a threshold value of 8.895 MeV. It can be seen from Table 1 that the radionuclide ¹⁰⁹Pd has two isomers. The m-state has a half-life of only 4.703 min, which decays by internal transition with a branching fraction of 100 %. ³⁹ Within the decay time of the irradiated samples of the present work, it decays completely, and it is not detectable. The g-state of ¹⁰⁹Pd has a half-life of 13.59 h. It has γ-ray of 88.03 keV with an intensity of 3.67 %. Table 3 also shows that at the yield-weighted average neutron energy of 22.7 MeV, the yield percentage of ¹⁰¹Pd is 99.73 % from the ¹⁰²Pd(n,2n)¹⁰¹Pd reaction and 0.27 % from the ¹⁰⁴Pd(n,4n)¹⁰¹Pd reaction. Thus, the ^natPd(n,xn)¹⁰¹Pd reaction has the contribution primarily from the ¹⁰²Pd(n,2n)¹⁰¹Pd reaction, which has a threshold value of 10.57 MeV. On the other hand, at the average neutron energy of 32.3 MeV, the normalized yield of ¹⁰¹Pd is 54.38 % from the ¹⁰²Pd(n,2n)¹⁰¹Pd reaction, 43.64 % from the ¹⁰⁴Pd(n,4n)¹⁰¹Pd reaction and 1.98 % from the ¹⁰⁵Pd(n,5n)¹⁰¹Pd reaction. The radionuclide ¹⁰¹Pd has a half-life of 8.47 h, which decays by electron capture and β⁺ with a branching fraction of 100 %. ³⁸ It has a very good γ-ray of 296.29 keV with an intensity of 19.2 %. It was detected in the γ-ray spectra of irradiated samples for both neutron energies. The radionuclides ¹⁰⁵Rh and ¹⁰⁶Rh have two isomers for each. ^{40 , 41} It can be seen from Table 1 that the radionuclide ^105mRh has a half-life of 42.8 s, which decays to the g-state by internal transition with a branching fraction of 100 %. ⁴⁰ It has a γ-ray of 129.8 keV with good intensity but is not possible to detect due to its short half-life. The radionuclide ^105gRh has a half-life of 35.341 h, which decay by β ⁻ with a branching fraction of 100 %. ⁴⁰ It has a γ-ray of 319.2 keV with good intensity and thus it is possible to detect in the γ-ray spectra. The radionuclide ^106gRh has a half-life of 30.07 s, which decays by β ⁻ with a branching fraction of 100 %. ⁴¹ It has a γ-ray of 621.9 keV with a reasonably good intensity but is not possible to detect due to its short half-life. On the other hand, the radionuclide ^106mRh has a half-life of 131 min and also decays by β ⁻ with a branching fraction of 100 %. ⁴¹ It has a number of γ-rays with very good intensities and is possible to detect in the first few γ-ray spectra.

From the normalized yield contributions of Table 3 and the flux conversion factors of Table 2, the yield-weighted flux conversion factors were obtained, which are given in Table 3. From the normalized yield contributions and the average neutron energy of each reaction in Table 2, the yield-weighted average neutron energies were obtained and are given in Table 3. The yield-weighted flux conversion factors (C ^nat,J) in Table 3 were used to calculate the modified neutron flux ( Φ J M E n nat , J ) from the measured neutron flux ( Φ m E n M ) based on the ²⁷Al(n,α)²⁴Na monitor reaction:

Once the modified neutron flux ( Φ J M E n nat , J ) was obtained, the flux-weighted average or the spectrum average cross-sections (SACS) of the ^natPd(n,xn)^109,101Pd and ^natPd(n,pxn)^105,106mRh reactions were determined from the net photo-peak area (A _net) of the γ-rays of the reaction products of present interest. The net photo-peak area corresponding to the γ-ray for the reaction products ^109,101Pd and ^105,106mRh were obtained from the gross photo-peak area by subtracting the linear Compton background. The net photo-peak area of the γ-rays for the radio-nuclides of interest and the modified neutron flux ( Φ J M E n nat , J ) were used to calculate the flux-weighted average or the spectrum average cross-section (SACS) ( σ nat , J E n nat , J ) of the ^natPd(n,xn)^109,101Pd and ^natPd(n,pxn)^105,106mRh reactions:

All terms in Eq. (9) have the similar meaning as in Eq. (2) except the modified neutron flux ( Φ J M E n nat , J ). Thus, using appropriate neutron flux, the flux-weighted average or the spectrum average cross-sections ( σ nat , J E n nat , J ) of the ^natPd(n,xn)^109,101Pd and ^natPd(n,pxn)^105,106mRh reactions were obtained at their respective average neutron energies. The nuclear spectroscopic data for the reaction products of present interest used in Eq. (9) were taken from refs. ^{38 , 39 , 40 , 41}