Measurements of differential and angle-integrated cross sections for the 10B(n, α)7Li reaction in the neutron energy range from 1.0 eV to 2.5 MeV

  • Differential and angle-integrated cross sections for the 10B(n, α)7Li, 10B(n, α0) 7Li and 10B(n, α1) 7Li* reactions have been measured at CSNS Back-n white neutron source. Two enriched (90%) 10B samples 5.0 cm in diameter and ~85.0 μg/cm2 in thickness each with an aluminum backing were prepared, and back-to-back mounted at the sample holder. The charged particles were detected using the silicon-detector array of the Light-charged Particle Detector Array (LPDA) system. The neutron energy En was determined by TOF (time-of-flight) method, and the valid α events were extracted from the En-Amplitude two-dimensional spectrum. With 15 silicon detectors, the differential cross sections of α-particles were measured from 19.2° to 160.8°. Fitted with the Legendre polynomial series, the (n, α) cross sections were obtained through integration. The absolute cross sections were normalized using the standard cross sections of the 10B(n, α)7Li reaction in the 0.3 − 0.5 MeV neutron energy region. The measurement neutron energy range for the 10B(n, α)7Li reaction is 1.0 eV≤En < 2.5 MeV (67 energy points), and that for the 10B(n, α0) 7Li and 10B(n, α1) 7Li* reactions is 1.0 eV ≤ En < 1.0 MeV (59 energy points). The present results have been analyzed by the resonance reaction mechanism and the level structure of the 11B compound system, and compared with existing measurements and evaluations.
  • The static and dynamic properties of a nucleus predominantly dictate its shape or structure, and these properties depend on the interactions among its constituents, i.e, protons and neutrons. Most nuclei exhibit an axially symmetric, dominantly quadrupole, deformed shape in the ground state. However, there are several regions, known as transitional regions, in the nuclear chart where axial symmetry is not conserved; the triaxial mean-field approximation may be used to characterize the properties of these nuclei. Nuclei in the transitional region Z50 and N82 are characterized by softness to the γ-deformation [1], which accounts for shape coexistence in these nuclei [2]. In these nuclei, protons in the valence shell begin to fill the h11/2 shell, while the neutrons remain in the middle of that shell. Rotational alignment of these valence protons and neutrons trend to drive the nucleus towards prolate and oblate shapes, respectively [3]. Therefore, the nuclei lying in the above referred transitional region are good candidates for the study of various interesting phenomena, such as the effect of orbitals on deformation, investigation of shape changes, new excitations with regard to triaxial deformations, etc.

    In the past few decades, the use of improved and sophisticated experimental techniques has made it possible to provide sufficient data to describe the structure of nuclei in various mass regions of the nuclear chart. In particular, the transitional nuclei around mass A130 have numerous interesting features, such as odd-even staggering (OES) in the gamma band at low spins, triaxial deformation, etc. [46]. The even-even nuclei close to the proton shell closure at Z = 50 have been the subject of numerous theoretical [715] and experimental studies [16] in the past. The Xenon (Z = 54) nucleus with four protons more outside the 50-proton closed shell is positioned in the transitional region of nuclear chart, where nuclear structure varies from a spherical nature to a deformed one. Thus, this region is observed to be rich in the nuclear structure, as it provides a testing ground to probe the interplay between triaxial and axially deformed nuclear shapes. This region is an exciting field in nuclear-structure studies because of the reported presence of low-lying intruder states, as well as the occurrence of a triaxial shape. Hence, the present study attempts a theoretical investigation of a chain of even-even Xe nuclei within the application of the TPSM approach. In recent times, as well as in the past few decades, various theoretical studies had been used to investigate various nuclear structure properties, such as energy levels, electric quadrupole moments, and B(E2) of the Xe nuclei [14, 1721]. These studies suggest that these nuclei are soft with regard to the γ-deformation with a almost maximum effective triaxiality of γ = 30 [7]. Nuclei above the closed shell at Z = 50 and with a neutron number close to the N = 66 midshell exhibit large ground-state deformations, with β2 0.2–0.3 [22, 23]. For the Z = 54 Xenon isotopes, the ground state deformation is at its maximum for 120Xe66, which is evident from the minimum value of excitation energy of the first 2+ state [E(2+1)]. Below 120Xe, the values of [E(2+1)] increase down to the lightest known isotope 114Xe, indicating a decrease in the deformation as the N = 50 closed shell is approached. This trend is quantitatively supported by calculations presented in Ref. [24], which predict that the tendency continues all the way to the closed shell. Furthermore, 120Xe is a well-studied nucleus [25, 26] with a ground state rotational band (E4+/E2+2.5) and a quasi-band with a band head at 876.0 keV. Furthermore, 120Xe lies in a complex nuclear transition region, which evolves from a vibration-like structure near N = 82, through a γ-soft region near N = 74, towards rotational configurations for lighter neutron-deficient nuclei. In the N = 66 region, 120Xe is the heaviest even-even nucleus showing a signature of shape coexistence [27]. Moreover, quasi-ground-state bands and quasi-rotational γ-vibrational bands with unstable γ-deformations have also been studied for Xe in this nuclear region [28].

    The Xenon nucleus thus belongs to the region of the nuclear segre chart where the effect of triaxiality on the nuclear structure is evident. Therefore, the purpose of the present study is to investigate the observed band structures and, in particular, the observed triaxiality in 118128Xe isotopes using the triaxial projected shell model (TPSM) approach [29, 30] in a simple and comprehensive manner. The formalism of the TPSM that we are using in the present study has been successfully employed in the past [3133] to study high-spin structures of triaxially deformed nuclei. Further, the present chain of isotopes 118128Xe has not been studied to date with the application of TPSM; therefore the present work was taken up as a challenge to describe their structure within TPSM.

    The paper is organized as follows. Section 2 provides a brief overview of the theoretical approach (TPSM) used in the present work. The results of the measurement for the Xenon isotopes, followed by a discussion and their comparison with the available experimental data, are presented in Section 3. Finally, the study is summarized in Section 4.

    In the present study, the three-dimensional angular momentum projection technique was successfully adopted to study the γ-vibrational band structures in triaxially deformed nuclei. The TPSM approach has also provided theoretical support for observation of the γγ-band in those nuclei [33]. Because the studied 118128Xe isotopes are of the even-even type with even protons and neutrons, the TPSM basis chosen for carrying out the present study is composed of the projected 0-qp state (or qp-vacuum), two-proton qp, two-neutron qp, and 4-qp configurations, namely,

    ˆPIMK|Φ>,ˆPIMKap1ap2|Φ>,ˆPIMKan1an2|Φ>,ˆPIMKap1ap2an1an2|Φ>,

    (1)

    where |Φ> represents the triaxial vacuum state. The three-dimensional angular-momentum projection operator [34] is given by

    ˆPIMK=2I+18π2dΩDIMK(Ω)ˆR(Ω),

    (2)

    with the rotational operator

    ˆR(Ω)=eıαˆJzeıβˆJyeıγˆJz,

    (3)

    Ω denotes a set of Euler angles (α,γ = [0, 2π], β = [0, π]) and J denotes angular-momentum operators. In this present approach, triaxiality is included in the deformed basis, which helps perform an exact three-dimensional angular momentum projection. In this manner, the deformed vacuum state obtained is significantly improved, as it permits all possible components of K. The projected shell model version employed for the description of axially deformed nuclei uses the pairing plus quadrupole-quadrupole Hamiltonian [35] with a quadrupole-pairing term also included, as follows:

    ˆH=ˆH012χμˆQμˆQμGMˆPˆPGQμˆPμˆPμ,

    (4)

    and the equivalent triaxial Nilsson mean-field Hamiltonian is given by

    ˆHN=ˆH023ω{ϵˆQ0+ϵˆQ+2+ˆQ22},

    (5)

    where the first term in Eq. (4) is the single-particle spherical Hamiltonian, which has a proper spin-orbit force represented by Nilsson parameters [36]. The force strength quadrupole-quadrupole (QQ) χ is determined in such a way that it holds a self-consistent relation with the quadrupole deformation. In Eq. (5), ϵ and ϵ specify the deformation parameters corresponding to the axial and triaxial deformation, respectively, and they are related to the triaxiality parameter by γ = tan1(ϵϵ). The calculations were performed with deformation parameters displayed in Table 1. The pairing strength, i.e, monopole pairing strength (GM), is adjusted to obtain the known energy gaps:

    Table 1

    Table 1.  Axial deformation parameter ϵ and triaxial deformation parameter ϵ, employed in calculation for 118-128Xe. The γ deformation is related to the two parameters through γ = tan1(ϵϵ).
    Xeϵϵγ
    1180.2150.16237°
    1200.2280.15835°
    1220.2170.125030°
    1240.2100.13533°
    1260.1580.11335°
    1280.1450.11338°
    DownLoad: CSV
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    GM=(G1G2NZA)1A(MeV),

    (6)

    with the '–' sign is for neutrons and the '+' sign for protons. The present study on 118128Xe isotopes was performed with both neutron and proton major shells numbers N = 3, 4, and 5 and with pairing strengths G1 = 21.24 and G2 = 13.86. The quadrupole pairing strength GQ is assumed to be proportional to GM, with the proportionality constant set to 0.18 for all isotopes considered in this study. These calculations were conducted with the same set of input parameters for all isotopes under consideration.

    Calculations were carried out for 118128Xe isotopes within the quasiparticle basis space, which is formed by the use of deformation parameters ϵ and ϵ (as shown in Table 1). The axial quadrupole deformation parameter ϵ has been selected in the present calculations with the aim to reproduce the correct shell filling of the nucleus. The nonaxial deformation parameter, ϵ, is assumed in such a way that the behaviour of the γ-band is properly explained, and its value is in accordance with those obtained from the minimum of the potential energy surface (PES), as shown in Fig. 1. In this figure, the projected ground-state energy is plotted as a function of the triaxial parameter ϵ with the axial deformation parameter ϵ held fixed. These deformation parameters are required to solve the triaxial potential that is used to construct the deformed basis states in present calculations. The discussion on calculated TPSM results for 118128Xe isotopes and the comparison to their experimental counterparts on various nuclear structure properties is presented in the following sub-sections.

    Figure 1

    Figure 1.  (color online) Variation of potential energy surfaces (PES) of ground state as a function of triaxiality parameter ϵ for 118128Xe.

    The collection of projected energies for various intrinsic configurations leads to the formation of a diagram known as the band diagram. In the present TPSM calculations, these projected energies arise from the diagonal matrix elements used before mixing of the configurations. These band diagrams are known to be quite useful and result oriented, as they play a major role in explaining the underlying intrinsic structures of various bands. Zero-quasiparticle, two-quasiparticle, and four-quasiparticle configurations have been employed for calculating the angular-momentum projected energies with the use of deformation parameters (given in Table 1), and these projected energies are plotted in Figs. 2(a)(f) for the Xenon isotopes of this study. The projection from the 0-qp configuration provides band structures corresponding to K = 0, 2, 4, denoted as (0, 0), (2, 0), and (4, 0) bands, which relate to the ground, γ, and γγ band, respectively. These bands are considered to be the main components of the band diagram, as they are responsible for obtaining the lowest energy bands after configuration mixing. It is abundantly clear from the present set of calculations that the projection from the 0-qp state gives rise to the formation of band with only even values of K, i.e, K = 0, 2, 4,... . No odd values of K are obtained here because of the symmetry requirement for the vaccum configuration.

    Figure 2

    Figure 2.  (color online) Band diagrams for 118-128Xe isotopes. Labels (0, 0), (2, 0), (4, 0), (1, 2n), (3, 2n), (1, 2p), (3, 2p), (2, 4), and (4, 4) correspond to ground, γ, 2γ, and 2n-aligned γ band on this 2n-aligned state, 2p-aligned γ band on this proton-aligned state, (2n+2p)-aligned band, and γ band built on this four-quasiparticle state.

    For 118Xe (Fig. 2(a)), the band head energy of the γ-band was obtained at an excitation energy of 0.93 MeV from the ground state band. Here, the two-quasineutron configuration states with K = 1 and 3 cross the ground-state band at I = 10. These bands represent the γ-band built on the two-quasineutron-aligned configurations. The two quasiproton states (1, 2p) and (3, 2p) are closer in energy compared to the two-neutron states, and therefore, cross the ground state band at I = 14. Finally, at I = 16 and above, the 4-qp structures (two-quasineutron plus two-quasiproton) with K = 2 and 4 cross the ground-state band, thereby giving rise to the yrast states that originate from these quasiparticle configurations for at least up to the last calculated spin values shown in the figure. In the case of 120Xe in Fig. 2(b), the band head energy of the γ-band is at an energy of 0.85 MeV above the ground state from the present calculation. Notably, the ground state band (0, 0) is crossed by two quasiparticle bands with (1, 2n) and (3, 2n) configurations at I = 12, and is yrast up to this spin value. Further, the two-aligned proton bands with configurations (1, 2p) and (3, 2p) and a 4-qp state (2, 2n2p) band cross the ground state band at I = 18 and decrease in energy until the last calculated spin values.

    For Fig. 2(c) in 122Xe, the calculated band head energy of the (2, 0) band is about 0.86 MeV from the ground state band (0,0). The ground-state band is crossed by two quasiparticle bands with (1, 2n) and (3, 2n) configurations at I = 10. The figure shows that the two-proton aligned band with (1, 2p) configuration also crosses the ground-state band at I = 14, and from I = 16 the two-quasiparticle proton band (3, 2p) and four-quasiparticle state (2n2p) attain the lowest energy, leading to the formation of yrast. In the band diagram for 124Xe, the obtained energy of the band head of the γ-band is at about 0.86 MeV above the the ground state band. The ground-state band with the (0, 0) configuration is crossed at I = 10 by two two-neutron aligned configurations, (1, 2n) and (3, 2n). The two-proton (1, 2p) and (3, 2p) bands decrease in energy and cross the ground band at I = 14. Beyond this spin, it is expected that four-quasiparticle configurations dominate the yrast band along with all the above-mentioned bands.

    Similarly for 126Xe, the theoretically calculated band head energy of the γ-band is 0.93 MeV above the ground state. The two two-neutron bands (1, 2n) and (3, 2n) cross the ground state band (0, 0) at spin I = 10, and both qp-neutron bands become degenerate in energy. At spin values I = 14 and I = 16, the two 2-qp proton bands and two four-quasiparticle bands, respectively, cross the ground band and decrease in energy, thereby, joining the two 2-qp neutron band and becoming yrast states. Finally, the band diagram for 128Xe is also presented in Fig. 2(f),where the calculated band head of the γ-band is at an excitation energy of 1.2 MeV above the ground band. The crossing of the (0, 0) and (2, 0) bands by the two-neutron aligned bands (1, 2n) and (3, 2n), respectively, occurs at I = 8. After spin I = 12 and I = 16, the two-quasiparticle proton bands and four-quasiparticle states (2n2p) attain the lowest energy and become yrast at larger values of angular momentum.

    Our next aim is to obtain the energies of states obtained after configuration mixing. To this end, the shell model Hamiltonian of Eq. (3) is diagonalized with the obtained projected energies, giving rise to the energies of the states known as yrast states. The energies of the yrast states for 118128Xe are presented in Fig. 3 in comparison with the experimental data [3742]. For the purpose of discussion in terms of the relevant physics, only the lowest three bands from the 0-qp configuration and the lowest two bands for other configurations are shown in the band diagrams. However, during the diagonalization of the Hamiltonian, the basis states employed are more numerous, which includes, for example, those with K = 1, 3, 5, and 7 with κ = 1 and K = 0, 2, 4, 6, and 8 with κ = 0, where the index κ indicates the basis states. The lowest three bands, obtained after diagonalization for each angular momentum, that only contribute to the formation of yrast energies, are shown in Fig. 3 for all studied Xe isotopes. From the figure, the agreement between the TPSM and experimental energies for the yrast and γ-bands is excellent for all the chosen 118128Xe isotopes. The calculated spectra for these isotopes are well fitted to experimental data for low as well as high-spin states. The predicted γ-bands are calculated for the spins higher than the available experimental data, and we hope that future high-spin experimental studies are able to populate these bands.

    Figure 3

    Figure 3.  (color online) Comparison of theoretically calculated energies (TPSM) after configuration mixing with available experimental data (Expt.) for 118128Xe.

    From the experimental point of view, the evolution of nuclear structure of the Xe isotopes with A = 118 to 128 can be inferred from Fig. 4, where the variation of [E(2+1)] is shown with respect to neutron numbers. Fig. 4 shows that the energy of the first-excited state [E(2+1)] is at its minimum for N = 66 (for 120Xe) and then rises smoothly from N = 68 to N = 74. Both the experimental and calculated results follow the same trend, indicating satisfactory agreement for [E(2+1)]. Moreover, a dip in energy [E(2+1)] is clearly observable for 120Xe, and as per Grodzin's rule [22], this isotope 120Xe can be considered to be the most deformed among all other studied isotopes because of the inverse variation of [E(2+1)] with the quadrupole moment (ϵ2). Further, the experimental and calculated energy ratios R4/2 = [E(4+1)/E(2+1)] with neutron numbers are presented in Table 2. Casten et al. [43] showed that for even-even nuclei, the shape phase transition is associated with a sudden change in nuclear collective behaviour, as a result of which the ratio R4/2 = [E(4+1)/E(2+1)] suddenly increases from the spherical value of 2.0 to the deformed γ-soft nuclei value of 2.5, and the maximum value of 3.33 for an ideally symmetric rotor. Table 2 indicates that the comparison of observed energy ratios with experimental data is quite satisfactory, and the experimental R4/2 ratios for the 118128Xe isotopes lie between 2.33 and 2.5. These theoretical values suggest that the values of R4/2 increase gradually from 2.4 to 2.5 with the increase in mass number A = 118 to 122, and then decreases from 2.5 to 2.33 on moving towards 128Xe, indicating the presence of γ-soft shapes in these nuclei. This confirms the triaxial nature of the chosen Xenon isotopes in the present study. Further, the heavier Xe isotopes beyond 122Xe are less deformed than the lighter ones.The ratio attains its maximum value in 122Xe near 120Xe, indicating nuclei with a large deformation and more rotational properties.

    Table 2

    Table 2.  Comparison of observed energy ratios of energies of first two excited states R4/2 = [E(4+1)/E(2+1)]with experimental data.
    XeR4/2 = [E(4+1)/E(2+1)]
    Expt.TPSM
    1182.42.5
    1202.462.51
    1222.52.53
    1242.482.49
    1262.422.48
    1282.332.42
    DownLoad: CSV
    Show Table

    Figure 4

    Figure 4.  (color online) Experimental (Expt.) and calculated (TPSM) systematics of first excited energies (E2+1) relative to yrast 0+ for 118128Xe.

    The nature of triaxial shapes in all chosen 118128Xe isotopes can be interpreted in terms of the staggering parameter, defined as

    S(I)=[E(I)E(I2)][E(I1)E(I2)]E(2+1),

    (7)

    S(I) is plotted for the γ-bands in Fig. 5. The attempt has been made in the present study to describe the odd-even staggering in the γ-bands obtained after configuration mixing of the studied Xe isotopes. Fig. 5 shows that the experimental staggering parameter for the known energy levels is reproduced quite accurately by the TPSM calculations, except for 118,120Xe. Furthermore, the staggering S(I) is quite small at lower spins. This is attributed to the fixed deformation parameters used in the calculations, and shape fluctuations are not taken into account. Above I = 8, the staggering increases for almost all isotopes under consideration, which might be due to the crossing of the two-quasineutron-aligned band with γ-bands (2, 0) at this spin value (as noted from the earlier discussion presented in Section 3.1 for band diagrams). In particular, all studied Xe isotopes can be considered to possess the same staggering phase in S(I). However, the results obtained are particularly interesting after a full mixing of quasiparticle configurations at spin values I1418 for 118128Xe, where an inversion in the staggering pattern is observed. A study by R. Bengtsson et al. [44] proposed that the signature inversion, i.e., exchange of energetically favored and unfavored spins, might be evidence for a triaxial shape in those nuclei. Thus, the presence of inversion in the staggering pattern of the 118128Xe isotopes also provides an important indication of the triaxiality in the chosen nuclei.

    Figure 5

    Figure 5.  (color online) Comparison of experimental (Expt.) and calculated (TPSM) staggering parameter S(I) for the γ-band in 118128Xe.

    The essential process of the backbending phenomena [45] is considered as a quasiparticle level crossing between an vacant high-j intruder orbital and the most high-lying populated orbital, resulting in a sudden increase of the moment of inertia along the yrast level and decrease in the rotational frequency. K. Higashiyama et al. [45] have likewise successfully demonstrated the shell model description of the backbending phenomena in Xe isotopes in the A130 region. In Fig. 6, the results of moment of inertia J(1) and the square of the rotational frequency ω2 are plotted for nuclei 118128Xe, where J(1) and ω2 are defined as follows:

    Figure 6

    Figure 6.  (color online) Plots of kinematic moment of inertia (2J(1)/2) as a function of (ω)2 for 118128Xe.

    2J(1)/2=4I2EIEI2,

    (8)

    2ω2=(I2I+1)(EIEI2)(2I1)2.

    (9)

    The sudden drop in the value of rotational frequency of the yrast band at the band crossing is well reproduced for the even-even 118128Xe isotopes. Fig. 6 clearly shows that for the low spin region, the g-band is the dominant band in all 118128Xe isotopes, and the slope of calculated backbend at I812 is more prominent than that of the experimental backbend, where the first 2-qp band crossing takes place for all isotopes under study. For 118Xe, the first backbend is obtained at spin I = 10, which is the same spin value of the first band crossing. This occurs because the rotational frequency of the yrast band decreases suddenly at a band crossing and thus leads to the S-shaped (backbending) diagram. However, the experimental backbend occurs at spin I = 16. Similarly for 120Xe, the calculated backbend is obtained at spin I = 10; however, no experimental backbend is observed as the data at high spins is inadequate. Furthermore for 122Xe, the calculated and experimental backbends take place at spin I = 10, where the first band crossing is also reported in the band diagram for 122Xe. Further, for 124Xe, the calculated backbend is obtained at spin I = 10. However, due to the lack of experimental data at high spins, its accuracy cannot be guaranteed. For 126Xe, the successive band crossings take place in succession as the spin increases at spin values I = 8, 10. In contrast, the experimental backbend is obtained at I = 12. Finally, for 128Xe, the first calculated backbend is obtained at I = 8, whereas the experimental observed backbend is at I = 10. The backbending phenomena in these isotopes are, thus, considered to be caused by the crossing of the two-quasi-particle neutron bands and the ground-state band. At higher spins, the two quasi-proton aligned state is favourable in energy and crosses the g-band in all Xenon isotopes in the present study. One of the most critical factors for the sharpness of the backbend is the crossing angle of the g-band and other multi-quasi particle bands at the crossing spin. Finally, we conclude that the two 2-qp neutron aligned bands, which crossed the g-band at the (first) band crossing, remain as yrast bands until they are crossed by other 2-qp and 4-qp bands (usually a 2ν2π 4-qp band). The agreement between theory and experiment for backbendng is well reproduced for some, but not all of the isotopes under study. These results may improve with the inclusion of higher quasiparticle states in the present applied model. The theoretical calculations and analyses show that the backbending phenomenon may be the result of the crossing of the ground-state band and the band with two-quasi-neutrons in all Xenon isotopes.

    Another part of the present TPSM calculations involves the study of the electric quadrupole transition probabilities B(E2). In this study, B(E2) values are calculated according to the following procedure. The projected states thus obtained are employed as new basis states for diagonalization of the shell model Hamiltonian in the TPSM basis. Then, with the help of the diagonalized Hamiltonian, eigenfunctions are obtained, which are used to calculate the electric quadrupole transition probabilities as follows:

    B(E2:(Ii,Ki)(If,Kf))=12Ii+1|ΨKfIf||ˆQ2||ΨKiIi|2,

    joining initial state (Ii,Ki) and final states (If,Kf). In the present set of calculations for B(E2), the proton and neutron effective charges are denoted by eπ and eν, respectively, which are defined through the relations as eπ=1+eeff and eν=eeff. In the present set of calculations, the value of eeff is taken as 0.5. In Fig. 7, the calculated B(E2) transition probabilities along the yrast band are compared with known experimental values [46, 47] and HFB results [46]. The transition probabilities are efficiently reproduced by present TPSM calculations. In particular, the increasing trend of B(E2) for high-spin states are well described by TPSM calculations, and the observed drop in the B(E2) transitions with spin is likewise correctly reproduced, except for 118Xe. Furthermore, the B(E2) values for 120Xe are excellent, and this can be attributed to the presence of mid shell neutrons, N = 66, where the the number of active neutrons and hence the neutron-proton correlations are at their maximum. The transition probabilities in the band-crossing region are clearly reduced, as these values of B(E2) are evaluated between the predominant ground state and two-quasiparticle aligned configurations. The decrease in B(E2) values is related to the crossing of the two-quasiparticle neutron configuration with the ground-state band. Thus, the known experimental values and the calculated B(E2) values are efficiently reproduced by the present study.

    Figure 7

    Figure 7.  (color online) Comparison of calculated (TPSM) B(E2) values with experimental (Expt.) and other HFB ([46]) data in 118128Xe.

    The ongoing discussion regarding calculated results and their comparison with the existing experimental data led to various important interpretations for Xenon isotopes 118128Xe within the applied quantum mechanical framework, i.e., triaxial projected shell model. The following inferences are drawn from the present study.

    The TPSM approach with an extended basis was adopted in the present work to investigate the ground state and gamma bands of even-even 118128Xe isotopes, which were efficiently reproduced by the method. The calculated spectra for the yrast and γ-band provided an excellent fit to experimental data for low-spin states as well as high-lying states for all isotopes. Moreover, the bandhead of the yrast band for all isotopes is likewise well reproduced. The chosen qp basis states in the present study are appropriate for describing high-spin states properties of Xenon isotopes.

    To obtain information about the nuclear structure and evolution of the chosen Xe isotopic chain, the comparison of the experimental energy ratios with the calculated data was carried out, and the agreement between the two is well reproduced. Furthermore, the energy ratios clearly indicate the presence of γ-soft shapes in studied Xe isotopes. The calculations presented in the present study provide evidence for triaxiality in the chosen Xe isotopes.

    The odd-even staggering phase of the γ-band is well reproduced, and the staggering magnitude increases towards higher spins. This increase in S(I) is related to the importance of the two-neutron aligned configuration above I = 8, and it is also evident from the band diagrams.

    The relationship between backbending at a particular spin with the crossing of the band at the corresponding spin is efficiently described by the approach employed in this study. The results clearly show that the decrease in the moment of inertia immediately after the first band crossing takes place for all 118128Xe isotopes as a result of the rotational alignment of the 2qp neutron state, and the amount of spin alignment is given by the spin value at which the band energy assumes its the minimum value after crossing. Thus, this phenomenon of backbending conveys important information on the interplay between the ground band and bands with alignment of a pair of 2qp neutron quasiparticles.

    The present TPSM approach provides an accurate description of the measured B(E2) for all isotopes chosen for this study. In the high-spin region, the drop in the transitions is attributed to the rotational alignment of neutrons.

    Thus, the present study clearly demonstrates that a model based on systematic pairing and quadrupole-quadrupole interaction, with the incorporation of three-dimensional angular-momentum projection, efficiently describes near yrast band structures and various other nuclear structure properties in the studied transitional nuclei.

    The authors of the present paper express their gratitude to Prof. Y. Sun and K. Hara for their collaborations.

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    [10] H. Jing, J. Tang, H. Tang et al, Nucl. Instrum. Methods A, 621: 91 (2010) doi: 10.1016/j.nima.2010.06.097
    [11] L. Zhang, H. Jing, J. Tang et al, Appl. Radiat. Isot., 132: 212 (2018) doi: 10.1016/j.apradiso.2017.11.013
    [12] B. Qi, Y. Li, D. Zhu et al, Measurement of neutron beam spot distribution using a Micromegas detector for the Back-n white neutron facility at China Spallation Neutron Source, submitted to Nucl. Phys. A
    [13] G. Luan, Q. Wang, J. Bao et al, Nuclear Techniques, 40: 110501 (2017)
    [14] Y. Chen, G. Luan, J. Bao et al, Eur. Phys. J. A, 55: 115 (2019) doi: 10.1140/epja/i2019-12808-1
    [15] B. He, P. Cao, D. Zhang et al, Chin. Phys. C, 41: 016104 (2017) doi: 10.1088/1674-1137/41/1/016104
    [16] H. Yi, T Wang, Y Li et al, Double-bunch unfolding methods for the Back-n white neutron source at CSNS, submitted to Nucl. Instrum. Methods A (unpublished)
    [17] R. Bevilacqua, F.-J. Hambsch, M. Vidali et al, Eur. Phys. J. Conf., 146: 11010 (2017) doi: 10.1051/epjconf/201714611010
    [18] S. J. Friesenhahn, V. J. Orphan, A. D. Carlson et al, Conf. on Nucl. Cross-Sect. and Techn., 1: 232 (1975)
    [19] R. L. Macklin and J. H. Gibbons, Phys. Rev., 165: 1147 (1968) doi: 10.1103/PhysRev.165.1147
    [20] R. A. Schrack, G. P. Lamaze, and O. A. Wasson, Nucl. Sci. Eng., 68: 189 (1978) doi: 10.13182/NSE78-A27289
    [21] G. Viesti and H. Liskien, Annals of Nuclear Energy, 6: 13 (1979) doi: 10.1016/0306-4549(79)90090-2
    [22] R. A. Schrack, O. A. Wasson, D. C. Larson et al, Nucl. Sci. Eng., 114: 352 (1993) doi: 10.13182/NSE93-A24044
    [23] S. L. Hausladen, R. O. Lane, and J. E. Monahan, Phys. Rev. C, 4: 380 (1971) doi: 10.1103/PhysRevC.4.380
    [24] S. L. Hausladen, R. O. Lane, and J. E. Monahan, Phys. Rev. C, 5: 277 (1972) doi: 10.1103/PhysRevC.5.277
    [25] S. L. Hausladen, C. E. Nelson, and R. O. Lane, Nucl. Phys. A, 217: 563 (1973) doi: 10.1016/0375-9474(73)90412-0
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Haoyu Jiang, Wei Jiang, Huaiyong Bai, Zengqi Cui, Guohui Zhang, Ruirui Fan, Han Yi, Changjun Ning, Liang Zhou, Jingyu Tang, Qi An, Jie Bao, Yu Bao, Ping Cao, Haolei Chen, Qiping Chen, Yonghao Chen, Yukai Chen, Zhen Chen, Changqing Feng, Keqing Gao, Minhao Gu, Changcai Han, Zijie Han, Guozhu He, Yongcheng He, Yang Hong, Hanxiong Huang, Weiling Huang, Xiru Huang, Xiaolu Ji, Xuyang Ji, Zhijie Jiang, Hantao Jing, Ling Kang, Mingtao Kang, Bo Li, Chao Li, Jiawen Li, Lun Li, Qiang Li, Xiao Li, Yang Li, Rong Liu, Shubin Liu, Xingyan Liu, Guangyuan Luan, Qili Mu, Binbin Qi, Jie Ren, Zhizhou Ren, Xichao Ruan, Zhaohui Song, Yingpeng Song, Hong Sun, Kang Sun, Xiaoyang Sun, Zhijia Sun, Zhixin Tan, Hongqing Tang, Xinyi Tang, Binbin Tian, Lijiao Wang, Pengcheng Wang, Qi Wang, Taofeng Wang, Zhaohui Wang, Jie Wen, Zhongwei Wen, Qingbiao Wu, Xiaoguang Wu, Xuan Wu, Likun Xie, Yiwei Yang, Li Yu, Tao Yu, Yongji Yu, Linhao Zhang, Qiwei Zhang, Xianpeng Zhang, Yuliang Zhang, Zhiyong Zhang, Yubin Zhao, Luping Zhou, Zuying Zhou, Danyang Zhu, Kejun Zhu and Peng Zhu. Measurements of differential and angle-integrated cross sections for the 10B(n, α)7Li reaction in the neutron energy range from 1.0 eV to 2.5 MeV[J]. Chinese Physics C. doi: 10.1088/1674-1137/43/12/124002
Haoyu Jiang, Wei Jiang, Huaiyong Bai, Zengqi Cui, Guohui Zhang, Ruirui Fan, Han Yi, Changjun Ning, Liang Zhou, Jingyu Tang, Qi An, Jie Bao, Yu Bao, Ping Cao, Haolei Chen, Qiping Chen, Yonghao Chen, Yukai Chen, Zhen Chen, Changqing Feng, Keqing Gao, Minhao Gu, Changcai Han, Zijie Han, Guozhu He, Yongcheng He, Yang Hong, Hanxiong Huang, Weiling Huang, Xiru Huang, Xiaolu Ji, Xuyang Ji, Zhijie Jiang, Hantao Jing, Ling Kang, Mingtao Kang, Bo Li, Chao Li, Jiawen Li, Lun Li, Qiang Li, Xiao Li, Yang Li, Rong Liu, Shubin Liu, Xingyan Liu, Guangyuan Luan, Qili Mu, Binbin Qi, Jie Ren, Zhizhou Ren, Xichao Ruan, Zhaohui Song, Yingpeng Song, Hong Sun, Kang Sun, Xiaoyang Sun, Zhijia Sun, Zhixin Tan, Hongqing Tang, Xinyi Tang, Binbin Tian, Lijiao Wang, Pengcheng Wang, Qi Wang, Taofeng Wang, Zhaohui Wang, Jie Wen, Zhongwei Wen, Qingbiao Wu, Xiaoguang Wu, Xuan Wu, Likun Xie, Yiwei Yang, Li Yu, Tao Yu, Yongji Yu, Linhao Zhang, Qiwei Zhang, Xianpeng Zhang, Yuliang Zhang, Zhiyong Zhang, Yubin Zhao, Luping Zhou, Zuying Zhou, Danyang Zhu, Kejun Zhu and Peng Zhu. Measurements of differential and angle-integrated cross sections for the 10B(n, α)7Li reaction in the neutron energy range from 1.0 eV to 2.5 MeV[J]. Chinese Physics C.  doi: 10.1088/1674-1137/43/12/124002 shu
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Measurements of differential and angle-integrated cross sections for the 10B(n, α)7Li reaction in the neutron energy range from 1.0 eV to 2.5 MeV

    Corresponding author: Guohui Zhang, guohuizhang@pku.edu.cn
  • 1. State Key Laboratory of Nuclear Physics and Technology, School of Physics, Peking University, Beijing 100871, China
  • 2. Institute of High Energy Physics, Chinese Academy of Sciences (CAS), Beijing 100049, China
  • 3. Spallation Neutron Source Science Center, Dongguan 523803, China
  • 4. State Key Laboratory of Particle Detection and Electronics
  • 5. Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China
  • 6. Key Laboratory of Nuclear Data, China Institute of Atomic Energy, Beijing 102413, China
  • 7. Institute of Nuclear Physics and Chemistry, China Academy of Engineering Physics, Mianyang 621900, China
  • 8. Northwest Institute of Nuclear Technology, Xi’an 710024, China
  • 9. University of Chinese Academy of Sciences, Beijing 100049, China
  • 10. Department of Engineering and Applied Physics, University of Science and Technology of China, Hefei 230026, China
  • 11. School of Physics, Beihang University, Beijing 100083, China

Abstract: Differential and angle-integrated cross sections for the 10B(n, α)7Li, 10B(n, α0) 7Li and 10B(n, α1) 7Li* reactions have been measured at CSNS Back-n white neutron source. Two enriched (90%) 10B samples 5.0 cm in diameter and ~85.0 μg/cm2 in thickness each with an aluminum backing were prepared, and back-to-back mounted at the sample holder. The charged particles were detected using the silicon-detector array of the Light-charged Particle Detector Array (LPDA) system. The neutron energy En was determined by TOF (time-of-flight) method, and the valid α events were extracted from the En-Amplitude two-dimensional spectrum. With 15 silicon detectors, the differential cross sections of α-particles were measured from 19.2° to 160.8°. Fitted with the Legendre polynomial series, the (n, α) cross sections were obtained through integration. The absolute cross sections were normalized using the standard cross sections of the 10B(n, α)7Li reaction in the 0.3 − 0.5 MeV neutron energy region. The measurement neutron energy range for the 10B(n, α)7Li reaction is 1.0 eV≤En < 2.5 MeV (67 energy points), and that for the 10B(n, α0) 7Li and 10B(n, α1) 7Li* reactions is 1.0 eV ≤ En < 1.0 MeV (59 energy points). The present results have been analyzed by the resonance reaction mechanism and the level structure of the 11B compound system, and compared with existing measurements and evaluations.

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    1.   Introduction
    • 10B, a stable isotope of boron with natural abundance of 19.9%, is a crucial material in nuclear engineering, including radiation protection, neutron detection, reactor control, boron neutron capture therapy (BNCT), etc [1]. For neutron induced nuclear reactions of 10B, the 10B(n, α)7Li reaction is the dominate reaction channel for En < 1.0 MeV. In addition to various applications, the study of this reaction can enhance the understanding of nuclear reaction mechanism for light nuclei [2]. The 10B(n, α)7Li reaction has two main reaction channels, which are the 10B(n, α0)7Li (Q = 2.79 MeV) and 10B(n, α1)7Li* (Q = 2.31 MeV) reactions. Many measurements of the 10B(n, α)7Li reaction have been conducted since 1954 [3]. The cross sections of the 10B(n, α)7Li and 10B(n, α1)7Li* reactions have been recommended as neutron cross section standard in the region from thermal to 1.0 MeV energy [4].

      In the MeV region, however, due to the small cross sections and strong interference of background, discrepancies among different measurements and evaluations are apparent [3, 5]. Furthermore, existing measurements of the differential cross sections of the 10B(n, α)7Li reaction, as well as those of the 10B(n, α0)7Li and 10B(n, α1)7Li* reactions, are scarce. Only three measurements of angular distributions and differential cross sections (Sealock [6], Stelts [7] and Hambsch [2]) can be found in EXFOR for En ≤ 1.2 MeV and there is no data in the 1.2 MeV < En < 2.5 MeV region. Taking these factors into consideration, accurate measurements of differential and angle-integrated cross sections for the 10B(n, α)7Li reaction are demanded.

      In the present work, a LPDA (Light-charged Particle Detector Array) system, which mainly consisted with a silicon detector array in a vacuum chamber, was built to study the neutron induced charged particle emission reaction at CSNS (China Spallation Neutron Source) Back-n white neutron source [8]. With 15 silicon detectors distributed from 19.2° to 160.8°, the differential and angle-integrated cross sections were obtained for the 10B(n, α)7Li reaction in the 1.0 eV ≤ En < 2.5 MeV region (67 energy points), as well as the two reaction channels, 10B(n, α0) 7Li and 10B(n, α1) 7Li*, in the 1.0 eV ≤ En < 1.0 MeV region (59 energy points). The present results have been analyzed with the resonance reaction mechanism and the level structure of the 11B compound system, and compared with existing measurements and evaluations.

    2.   Experimental details

      2.1.   Neutron source

    • The neutrons were produced by double bunched proton beam (1.6 GeV, ~ 20 kW) bombarding a tungsten target at CSNS Back-n white neutron source [9]. The repetition rate of the beam pulse was 25 Hz and the pulse width was ~ 41 ns. The interval between the two proton bunches was 410 ns [10]. The experiment was conducted at Endstation #1, where the length of the flight path was 57.99 m, and the neutron flux was ~ 3.5×106 n/(cm2·s). The beam spot size at Endstation #1 was determined by the apertures of the shutter and Collimator-1 [11]. The diameter of the collimation aperture of the shutter was 50 mm and that for Collimator-1 was also 50 mm in the present work. The full width at half maximum (FWHM) of the neutron beam spot was 54 – 58 mm [12]. The relative neutron intensity could be monitored by the number of protons in the beam and a Li-Si detector array mounted in the beamline. Using the single-bunch operation mode, the neutron energy spectrum was measured by a multi-layer 235U fission chamber at Endstation #2, where the length of flight was 75.76 m [13, 14]. The details of the neutron energy spectrum could be found in Refs. [8] and [14].

      The neutron energy spectrum and the neutron energy bins used in the present work are shown in Fig. 1. The error bar in Fig. 1 represents the uncertainty of relative neutron fluence (φE_bin), which is 0.5% - 21.4% (for 49 of the 67 energy points, this uncertainty is less than 5%). Compared with the origin neutron energy spectrum in Ref. [14], the new wider energy bins of the spectrum were defined in the present work. Sixty-seven energy points called E_bin were specified from 1.0 eV to 2.5 MeV, and each E_bin was correlated with a neutron energy bin. The energy points (E_bin) were specified as follows: 49 equally spaced points were defined in the logarithmic coordinate between 1.0 eV and 0.1 MeV, 8 points were defined with equal interval of 0.1 MeV in the linear coordinate between 0.1 MeV and 1.0 MeV. Above 1.0 MeV, the interval was 0.2 MeV up to 2.5 MeV. The neutron energy corresponding to each event was obtained from TOF, then the linearly nearest E_bin was searched, and the event was counted into the corresponding neutron energy bin.

      Figure 1.  (color online) The neutron energy spectrum with uncertainty presented by blue bars (below 2.5 MeV).

    • 2.2.   Samples

    • Two enriched (90%) 10B samples were prepared as shown in Fig. 2. Each 10B sample was evaporated on an aluminum sheet 50 μm in thickness. The two samples were 5.0 cm in diameter both, and 82.59 and 85.05 μg/cm2 in thickness, respectively. The two 10B samples were back-to-back mounted at one of the four sample positions of the sample holder as shown in Fig. 3. At other sample positions, two back-to-back 241Am α sources and two aluminum sheets 50 μm each in thickness were mounted. The 241Am α sources were used to calibrate the detectors and the DAQ (Data Acquisition) system, and the aluminum sheets were used for the background measurement. The angle between the normal of the samples and the neutron beam line was 60° as shown in Fig.4 (a) so that the energy loss of α-particles in the samples could be minimized.

      Figure 2.  (color online) The 10B samples.

      Figure 3.  (color online) The diagram of the sample holder.

      Figure 4.  (color online) (a) The sketch of the silicon detectors. (b) The photo of the detectors.

    • 2.3.   Detectors

    • The charged particles were detected by the LPDA system, which mainly consisted of a silicon detector array and a vacuum chamber as shown in Fig. 4(b). Apart from the silicon detectors, other detectors such as a gridded ionization chamber (GIC) and three ΔE-E detectors were installed and tested. Fifteen rectangular (2.0 cm × 2.5 cm) silicon detectors 500 μm in thickness could cover the emission angle of the particles from 19.2° to 160.8°, and their solid angles were (0.0123 - 0.0125) (±0.3%) sr according to Monte Carlo simulation. The distance between the center of the silicon detector and that of the 10B sample was 20.0 cm. The angle between the normal of the silicon detectors and the horizon was 16° in order to avoid shielding ΔE-E detectors.

    • 2.4.   DAQ system

    • The DAQ system was designed based on PXIe platform [15]. The sampling rate of the DAQ system was 1 GHz with the resolution of 12 bits. When the signal amplitude exceeded the predefined threshold of the corresponding channel, the full signal waveform with a time window 15 μs would be recorded. In order to obtain the starting time of the signal, the original signal was filtered and differentiated, and the starting time was determined by the position of the one-tenth maximum height of the differentiated signal. The TOF of the neutron could be calculated by

      TOF=TeventT0+Lc,

      (1)

      where Tevent is the starting time of the event signal, T0 is the generation moment of the related neutrons determined using the starting time of γ-flash events, L is the length of flight path, c is the velocity of light. The neutron energy distribution due to the double-bunched operation mode will be unfolded as described in Section 3.3.

    • 2.5.   Experimental process

    • In the experiment, the 15 silicon detectors and the DAQ system were firstly calibrated using the 241Am α sources. Then, the events from the 10B samples and the Aluminum backing sheets were measured in turns (~ 16 h for measurement foreground and ~ 8 h measurement for background for each turn). The total beam duration was ~ 357 h.

    3.   Data analysis and results
    • With the recorded signal waveforms and the corresponding TOFs, the En-Amplitude two-dimensional spectrum could be obtained, and the valid area of α events could be determined. Next, the events were counted into the corresponding neutron energy bins as described in Section 2.1, and then the background was subtracted to obtain the net events. After that, the neutron energy distribution caused by the neutron energy bin width and by the double proton bunches, and the spread of the detection angle were unfolded using the iterative method. Next, the relative differential cross sections of the 10B(n, α)7Li reaction were obtained, and then the relative angle-integrated cross sections were calculated via integration. The results were normalized using the standard cross sections of the 10B(n, α)7Li reaction in the 0.3 − 0.5 MeV region. After that, the ratios of the 10B(n, α0)7Li and 10B(n, α1)7Li* reactions was calculated by the unfolded spectrum of net α events, and the differential and angle-integrated cross sections for these two reaction channels were obtained. The process of data analysis is shown in Fig. 5.

      Figure 5.  The flow chart of data analysis.

    • 3.1.   The statistics of the events and the background subtraction

    • The measurement data have been sorted into two-dimensional distributions (En-Amplitude) at every detection angle. The amplitude of each event could be obtained from the recorded waveform, and the corresponding En could be calculated from its TOF. An En-Amplitude two-dimensional spectrum is shown in Fig. 6 as an example, in which the areas of α0 and α1 events, as well as the Li and Li* events, and recoil proton events are labeled. From the two-dimensional spectrum, the valid-event-area of α events could be decided. Then, the events in the valid-event-area were projected into their corresponding neutron energy bins which was described in Section 2.1.

      Figure 6.  (color online) The En-Amplitude two-dimensional spectrum at the detection angle near 26.9°.

      The net events in each energy bin at each detection angle could be obtained after the background subtraction shown in Fig. 7 as an example. The normalization factor was decided by the ratio of the number of the protons in the beam during the foreground measurement over that during the background measurement. Although the background from the sample itself, such as the charge particles from the 11B(n, p) (Q = 0.23 MeV) and 10B(n, t2α) (Q = 0.32 MeV) reactions, could not be subtracted, these interferences can be ignored in En < 1 MeV region because of their fairly small cross sections [2]. In the 1.0 MeV ≤ En < 2.5 MeV region, the valid-event-area could be separated from background area because the charged particles from the background reactions with small Q-values have quite low energies. For En ≥ 2.5 MeV, the energies of the emitted background particles are high enough to interfere with the valid-event-area. Besides, the recoil protons from hydrogen adsorbed in the samples would be another notable source of the background. Therefore only the cross sections of the 10B(n, α)7Li reaction below 2.5 MeV region were obtained in the present measurement.

      Figure 7.  (color online) The measured α events at the detection angle 26.9° and at the neutron energy 0.50 MeV.

    • 3.2.   The unfolding of the neutron energy bin width

    • The unfolding is necessary due to the influence of the width of neutron energy bin. Each event was weighted as

      wE,θ=σE_bin,θσE,θ,

      (2)

      where w is the weight of a single event, E is the energy of the neutron calculated from TOF, θ is the detection angle, E_bin is the linearly nearest neighbor energy point described in Section 2.1, and σE_bin,θ and σE,θ are the 10B(n, α)7Li differential reaction cross sections at the neutron energy of E_bin and E, respectively. The calculations of σE_bin,θ and σE,θ will be described in Section 3.6. In the first step of iteration, the weight of every event was set as 1, and the 10B(n, α)7Li reaction cross sections could be calculated. And in all the following steps, the weight of events at every neutron energy bin was calculated by the cross sections obtained from the last step, then the cross sections were recalculated. Through the unfolding, the uncertainty of the neutron energy could be decreased from 4.2% − 50.0% to 0.4% − 17.2%. The uncertainty of the differential cross section introduced from the unfolding of the neutron bin width is 0.1% − 51.1% (for 944 of the 1005 σE_bin,θ results, this uncertainty is less than 5%).

    • 3.3.   The unfolding of the neutron energy distribution caused by the double-bunched operation mode

    • The interval (410 ns) of the double bunched proton beams would lead to fairly big uncertainty of the neutron energy especially in high energy region. For 0.02 MeV ≤ En < 2.5 MeV, the uncertainty of the neutron energy is 1.5% − 17.2% which is not negligible, and the unfolding is thus needed. The details of the unfolding method could be found in Refs. [8] and [16], which will be described briefly here.

      At each detection angle, every event was split into two child events and each of them was weighted as

      {wEn1,θ=IEn1σlast_iteraEn1,θIEn1σlast_iteraEn1,θ+IEn2σlast_iteraEn2,θwEn2,θ=IEn2σlast_iteraEn2,θIEn1σlast_iteraEn1,θ+IEn2σlast_iteraEn2,θ,

      (3)

      where the subscripts of En1 and En2 are the neutron energies determined by the TOF of the event plus 205 ns and minus 205 ns, respectively. IEn1 and IEn1 are the unit neutron fluence (i.e. n/ns) (the neutron spectrum is transformed into the number of neutrons per unit time of TOF), σlast_iteraEn1,θ and σlast_iteraEn2,θ are the differential cross sections obtained from the last iteration. Using Eq. (3), the two child events would be counted into related bins, and the new differential cross sections described in Section 3.6 could be obtained. The unfolding could decrease the uncertainty of the neutron energy to 0.4% − 1.5%. However, the unfolding would lead to the uncertainty of the differential cross sections. In the present work, the correction was processed for En ≥ 0.02 MeV and the corresponding uncertainty of wE_bin,θ introduced from the unfolding of the neutron energy distribution caused by the double-bunched operation mode is 1.2% − 11.9% (for 305 of the 360 σE_bin,θ results above 0.02 MeV region, this uncertainty is less than 5%).

    • 3.4.   The calculation of the relative differential cross sections

    • The relative differential cross section σreE_bin,θ can be obtained using

      σreE_bin,θ=WE_bin,θφE_binΩθNBεE_bin,θ,

      (4)

      where

      WE_bin,θ=EE_binwE_bin,θ

      (5)

      is the total weight of the net events, φE_bin is the relative neutron fluence shown in Fig. 1, Ωθ is the detection solid angle of the corresponding silicon detector, NB is the number of the 10B atoms in the sample (the two 10B samples are not the same), εE_bin,θ is the detection efficiency for α-particles (> 97.7 % deduced from the dead time of the DAQ system). Sources of the uncertainty of WE_bin,θ include the errors of statistics, background subtraction and uncertainty of the valid-event-area determination with the magnitudes of 0.7% − 6.9%, 0.1% − 9.0% and 0.1% − 6.5%, respectively. The uncertainties of Ωθ , and NB are 0.3% and 1.0%, respectively.

    • 3.5.   The deconvolution of the spread of the detection angle

    • The detection angle of each silicon detector was obtained from the Monte Carlo simulation. In the simulation, the particles were assumed to be emitted isotopically from the random position in the sample and then reached the detector. According to the simulation, the spread of the detection angle for each detector was expected to be 3.8° − 4.0° as shown in Fig. 8, which lead to the uncertainty of the detection angle. The iterative method was used to perform a correction for the spread of the detection angle.

      Figure 8.  (color online) The simulated distributions of the receiving angles of the silicon detectors.

      For each neutron energy bin, the 15 measured σreE_bin,θ were fitted using the Legendre polynomial series, then the fitting curve were deconvoluted with the simulated distribution shown in Fig. 8. The process of deconvolution and iteration would take the anisotropy into account. After that, the new corrected relative differential cross section C_σreE_bin,θ was obtained. Next, the deconvolution cross section cor_σreE_bin,θ was calculated by

      cor_σreE_bin,θ=σreE_bin,θσreE_bin,θC_σreE_bin,θ.

      (6)

      This deconvolution process was iterated until the variation of cor_σreE_bin,θ was less than 0.1% (usually 10 times). This correction could reduce the uncertainty of the detection angles to ~ 0.01°. The uncertainty of the differential cross section introduced from the angle deconvolution is 0.1% − 0.9%.

    • 3.6.   The calculation of the cross sections of the 10B(n, α)7Li reaction

    • It is commonly accepted that the relative differential cross section could be represented by the Legendre polynomial series

      freE_bin(cos(θ))=Mi=0AiPi(cos(θ)),

      (7)

      where M is the maximum Legendre polynomial order (M = 1 in the 1.0 eV ≤ En < 1.0 keV region, M = 2 in the 1.0 keV ≤ En < 0.1 MeV region, M = 3 in the 0.1 MeV ≤ En < 2.5 MeV region), Ai is the ith coefficient determined by fitting the cor_σreE_bin,θ at each E_bin, Pi(cos(θ)) is the ith-order Legendre polynomial. Then through the integration of freE_bin, the relative angle-integrated cross sections of the 10B(n, α)7Li reaction σreE_bin could be obtained, and the uncertainty of the fitting is 0.2% − 9.5%. Since the absolute neutron flux was not measured, the measured cross sections were normalized according to standard cross section σstandardE_bin from ENDF/B-VIII.0 in 0.3 − 0.5 MeV region, where the results obtained in the present work have relatively small uncertainties [5]. The uncertainty of normalization is 1.3% deduced from the information of standard library [4, 5]. Then, the cross sections were iterated using the method presented in Sections 3.1 − 3.5 until the variation was less than 1.0% (iterated for 15 times as shown in Fig. 5). Finally, the absolute differential cross section σE_bin,θ and the angle-integrated cross section σE_bin of the 10B(n, α)7Li reaction can be obtained. The total uncertainty of the differential cross section σE_bin,θ is 2.6% − 53.0% (for 868 of the 1005 σE_bin,θ results, this uncertainty is less than 10%), and those of the angle-integrated cross section σE_bin is 2.1% − 21.5% (for 43 of the 67 σE_bin results, this uncertainty is less than 5%). Sources of uncertainty and their magnitudes were shown in Table 1. The uncertainty of the 10B(n, α)7Li reaction is fairly big around En = 1.0 MeV due to few counts of α events near the valley in the excitation function. The results are presented in Tables A1-A3 of Appendix A. Selected results of differential cross sections are shown in Fig. 9, and those of angle-integrated cross sections are shown in Fig. 10. The variation trend of the differential cross sections will be discussed in Section 5.

      sources of uncertaintymagnitude (%)
      differential cross sectionsangle-integrated cross sections
      relative neutron fluence (φE_bin)0.5 − 21.4a, 0.6 − 1.9b, 0.7 − 0.8c0.5 −21.4a, 0.6 − 1.9b, 0.7 − 0.8c
      unfolding of the neutron energy bin width (wE_bin,θ)0.1 − 4.2a, 0.1 − 7.8b, 0.3 − 51.1c0.1 − 3.8a, 0.1 − 3.4b, 0.9 − 7.4c
      unfolding of the expanding of neutron energy due to the double-bunched operation mode (wE_bin,θ)1.2 − 10.9b, 2.9 − 11.9c0.4 − 1.3b, 1.0 − 1.6c
      uncertainty of neutron energy (E_bin, lateral error)0.4 − 1.4a, 0.6 − 1.0b, 1.0 − 1.5c0.4 − 1.4a, 0.6 − 1.0b, 1.0 − 1.5c
      statistical error of the valid α events (WE_bin,θ)0.7 − 3.1a, 1.0 − 5.6b, 2.6 − 6.9c0.2 − 0.8a, 0.3 − 1.0b, 0.8 − 1.5c
      background subtraction (WE_bin,θ)0.1 − 3.4a, 0.1 − 9.0b, 0.1 − 8.7c< 0.6a, < 0.7b, < 0.7c
      determination the valid-even-area (WE_bin,θ)0.1 − 3.8a, 0.3 − 6.5b, 0.4 − 4.4c< 0.4a, < 0.3b, < 0.2c
      detection solid angle (Ωθ)0.30.3
      number of the 10B atoms (NB)1.01.0
      deconvolution of the spread of the detection angle (cor_σreE_bin,θ)< 1.0< 0.1
      fitting using the Legendre polynomial series (freE_bin)0.2 − 1.6a, 0.6 − 2.0b, 1.8 − 9.5c
      normalization using the standard library (σstandardE_bin)1.31.3
      ratios of the 10B(n, α0)7Li reaction (R0E_bin,θ)2.9 − 25.8a, 3.6 − 36.4b1.7 − 4.3a, 1.9 − 4.6b
      ratios of the 10B(n, α1)7Li reaction (R1E_bin,θ)1.0 − 6.6a, 1.7 − 21.6b0.4 − 2.0a, 0.9 − 4.3b
      total uncertainty of the 10B(n, α)7Li reaction (σE_bin,θ and σE_bin)2.8 − 21.6a, 2.6 − 17.3b, 4.5 − 53.0c2.2 − 21.5a, 2.1 − 4.4b, 3.1 − 12.4c
      total uncertainty of the 10B(n, α0)7Li reaction (σ0E_bin,θ and σ0E_bin)4.7 − 26.3a, 4.7 − 36.7b3.2 − 21.6a, 2.8 − 6.3b
      total uncertainty of the 10B(n, α1)7Li reaction (σ1E_bin,θ and σ1E_bin)3.1 − 21.8a, 3.3 − 27.7b2.5 − 21.5a, 2.3 − 5.7b
      a: Uncertainties for 1.0 eV ≤ En < 0.02 MeV. b: Uncertainties for 0.02 MeV ≤ En < 1.0 MeV. c: Uncertainties for 1.0 MeV ≤ En < 2.5 MeV.

      Table 1.  Sources of uncertainty and their magnitudes.

      Figure 9.  (color online) The present differential cross sections of the 10B(n, α)7Li reaction at selected energy points as a function of the θlab compared with existing results of evaluations and measurements [3, 5].

      Figure 10.  (color online) The angle-integrated cross sections of the 10B(n, α)7Li reaction compared with existing results of evaluations and measurements since 1965 [3, 5, 17].

      En /MeVσE_bin,θ /(mb/sr)
      19.2°26.9°36.5°46.7°57.3°68.0°
      1.00×10−6±4.2×10−94.99×104±2.1×1035.00×104±2.2×1035.05×104±2.2×1034.96×104±2.2×1035.06×104±2.1×1035.00×104±2.0×103
      1.26×10−6±5.4×10−94.13×104±1.2×1034.18×104±1.2×1034.22×104±1.2×1034.22×104±1.2×1034.24×104±1.2×1034.16×104±1.2×103
      1.58×10−6±6.8×10−93.30×104±9.7×1023.24×104±9.4×1023.27×104±9.4×1023.27×104±9.5×1023.29×104±9.5×1023.29×104±9.5×102
      2.00×10−6±8.7×10−93.07×104±1.0×1033.10×104±1.0×1033.08×104±1.0×1033.06×104±1.0×1033.13×104±1.0×1033.13×104±1.0×103
      2.51×10−6±1.1×10−82.64×104±1.2×1032.65×104±1.2×1032.63×104±1.2×1032.67×104±1.2×1032.61×104±1.2×1032.61×104±1.2×103
      3.16×10−6±1.4×10−82.94×104±1.1×1032.96×104±1.1×1032.92×104±1.1×1033.04×104±1.1×1032.98×104±1.1×1033.01×104±1.1×103
      3.98×10−6±1.8×10−82.09×104±2.5×1032.16×104±2.5×1032.12×104±2.5×1032.13×104±2.5×1032.14×104±2.5×1032.09×104±2.5×103
      5.01×10−6±2.3×10−82.07×104±1.5×1032.07×104±1.5×1032.06×104±1.5×1032.12×104±1.6×1032.07×104±1.5×1032.13×104±1.6×103
      6.31×10−6±2.9×10−82.03×104±1.1×1032.11×104±1.1×1032.03×104±1.1×1032.03×104±1.1×1032.07×104±1.1×1032.05×104±1.1×103
      7.94×10−6±3.7×10−81.70×104±3.5×1031.69×104±3.5×1031.70×104±3.6×1031.73×104±3.6×1031.75×104±3.7×1031.75×104±3.7×103
      1.00×10−5±4.7×10−81.68×104±6.5×1021.61×104±6.3×1021.66×104±6.4×1021.64×104±6.3×1021.62×104±6.3×1021.65×104±6.4×102
      1.26×10−5±6.4×10−81.45×104±1.3×1031.50×104±1.3×1031.44×104±1.3×1031.44×104±1.3×1031.45×104±1.3×1031.45×104±1.3×103
      1.58×10−5±8.6×10−81.11×104±1.7×1031.14×104±1.7×1031.12×104±1.7×1031.13×104±1.7×1031.16×104±1.7×1031.12×104±1.7×103
      2.00×10−5±1.2×10−79.50×103±1.7×1039.39×103±1.7×1039.45×103±1.7×1039.42×103±1.7×1039.46×103±1.7×1039.19×103±1.7×103
      2.51×10−5±1.6×10−71.04×104±5.1×1021.03×104±5.1×1021.07×104±5.2×1021.05×104±5.1×1021.02×104±5.0×1021.03×104±5.1×102
      3.16×10−5±2.1×10−77.29×103±1.6×1037.35×103±1.6×1037.40×103±1.6×1037.53×103±1.6×1037.49×103±1.6×1037.19×103±1.6×103
      3.98×10−5±2.7×10−77.73×103±7.1×1027.36×103±6.7×1027.55×103±6.9×1027.45×103±6.8×1027.60×103±6.9×1027.49×103±6.8×102
      5.01×10−5±3.6×10−77.24×103±5.2×1027.22×103±5.2×1027.00×103±5.0×1026.92×103±5.0×1027.25×103±5.2×1027.07×103±5.1×102
      6.31×10−5±4.8×10−75.40×103±5.4×1025.42×103±5.4×1025.37×103±5.4×1025.34×103±5.3×1025.37×103±5.4×1025.08×103±5.1×102
      7.94×10−5±6.4×10−75.40×103±5.1×1025.24×103±5.0×1025.16×103±4.9×1025.18×103±4.9×1025.00×103±4.8×1025.10×103±4.9×102
      1.00×10−4±8.4×10−74.76×103±3.7×1024.78×103±3.7×1024.62×103±3.6×1024.80×103±3.7×1024.74×103±3.7×1024.55×103±3.5×102
      1.26×10−4±1.0×10−64.48×103±4.7×1024.00×103±4.3×1024.23×103±4.4×1024.32×103±4.5×1024.41×103±4.6×1024.13×103±4.3×102
      1.58×10−4±1.3×10−63.72×103±2.9×1023.72×103±2.9×1023.57×103±2.8×1023.62×103±2.8×1023.86×103±3.0×1023.61×103±2.8×102
      2.00×10−4±1.6×10−63.37×103±2.8×1023.07×103±2.6×1023.08×103±2.6×1023.15×103±2.6×1023.14×103±2.6×1023.04×103±2.5×102
      2.51×10−4±2.0×10−63.08×103±2.7×1022.87×103±2.5×1022.91×103±2.5×1022.93×103±2.5×1022.85×103±2.5×1022.86×103±2.5×102
      3.16×10−4±2.5×10−62.78×103±1.9×1022.41×103±1.7×1022.47×103±1.7×1022.29×103±1.6×1022.55×103±1.7×1022.36×103±1.6×102
      3.98×10−4±3.1×10−62.63×103±1.8×1022.49×103±1.7×1022.28×103±1.6×1022.40×103±1.6×1022.40×103±1.7×1022.27×103±1.6×102
      5.01×10−4±3.9×10−62.28×103±1.2×1022.04×103±1.1×1022.04×103±1.0×1022.30×103±1.1×1022.19×103±1.1×1022.10×103±1.1×102
      6.31×10−4±4.8×10−61.98×103±8.4×1011.74×103±7.7×1011.87×103±8.1×1011.81×103±7.8×1011.89×103±8.2×1011.73×103±7.9×101
      7.94×10−4±6.1×10−61.79×103±9.4×1011.62×103±8.8×1011.67×103±8.6×1011.64×103±8.2×1011.60×103±8.9×1011.62×103±8.4×101
      1.00×10−3±7.7×10−61.40×103±6.4×1011.40×103±6.2×1011.42×103±6.1×1011.49×103±6.3×1011.38×103±6.0×1011.25×103±5.7×101
      1.26×10−3±9.8×10−61.24×103±4.8×1011.19×103±4.6×1011.20×103±4.3×1011.32×103±4.4×1011.24×103±4.4×1011.13×103±4.6×101
      1.58×10−3±1.3×10−51.17×103±4.3×1011.08×103±5.0×1011.18×103±4.0×1011.11×103±4.0×1011.13×103±4.0×1011.08×103±4.2×101
      2.00×10−3±1.6×10−51.06×103±3.9×1011.04×103±3.8×1011.03×103±3.6×1011.03×103±3.6×1011.04×103±3.6×1019.41×102±4.0×101
      2.51×10−3±2.1×10−58.72×102±5.7×1018.82×102±5.5×1019.22×102±5.6×1019.55×102±5.8×1019.03×102±5.8×1017.98×102±5.5×101
      3.16×10−3±2.7×10−58.74×102±3.6×1018.29×102±3.3×1017.96×102±3.0×1018.36×102±3.1×1017.91×102±3.0×1017.39×102±3.5×101
      3.98×10−3±3.6×10−57.05×102±3.1×1017.71×102±3.3×1017.62×102±2.8×1017.54×102±2.7×1017.60×102±2.8×1016.85×102±3.0×101
      5.01×10−3±4.8×10−55.90×102±2.5×1016.07×102±2.5×1016.39×102±2.4×1016.32×102±2.3×1016.21×102±2.4×1015.44×102±2.5×101
      6.31×10−3±6.4×10−56.05×102±2.5×1015.94×102±2.4×1015.63×102±2.3×1015.91×102±2.2×1015.66×102±2.2×1015.53×102±2.3×101
      7.94×10−3±8.6×10−55.11×102±2.7×1015.40×102±2.5×1015.06×102±2.4×1015.35×102±2.3×1015.29×102±2.4×1014.81×102±2.4×101
      Continued on next page

      Table A1.  The differential cross sections of the 10B(n, α)7Li reaction in the laboratory reference system.

      En/MeVσE_bin,θ /(mb/sr)σE_bin /mb
      133.2°143.5°153.1°160.8°
      1.00×10−6±4.2×10−95.16×104±2.1×1035.16×104±2.2×1035.11×104±2.2×1035.12×104±2.1×1036.38×105±2.7×104
      1.26×10−6±5.4×10−94.36×104±1.2×1034.29×104±1.2×1034.34×104±1.2×1034.30×104±1.2×1035.37×105±1.4×104
      1.58×10−6±6.8×10−93.42×104±9.9×1023.33×104±9.7×1023.35×104±9.9×1023.39×104±1.0×1034.19×105±1.1×104
      2.00×10−6±8.7×10−93.24×104±1.1×1033.21×104±1.1×1033.18×104±1.0×1033.17×104±1.1×1033.95×105±1.2×104
      2.51×10−6±1.1×10−82.73×104±1.3×1032.68×104±1.3×1032.71×104±1.3×1032.71×104±1.3×1033.37×105±1.5×104
      3.16×10−6±1.4×10−83.10×104±1.1×1033.07×104±1.1×1033.04×104±1.1×1032.96×104±1.1×1033.80×105±1.3×104
      3.98×10−6±1.8×10−82.16×104±2.6×1032.26×104±2.7×1032.21×104±2.6×1032.21×104±2.6×1032.73×105±3.2×104
      5.01×10−6±2.3×10−82.12×104±1.6×1032.15×104±1.6×1032.21×104±1.6×1032.11×104±1.6×1032.65×105±1.9×104
      6.31×10−6±2.9×10−82.11×104±1.1×1032.11×104±1.1×1032.08×104±1.1×1032.09×104±1.1×1032.62×105±1.4×104
      7.94×10−6±3.7×10−81.73×104±3.6×1031.78×104±3.7×1031.76×104±3.7×1031.76×104±3.7×1032.19×105±4.6×104
      1.00×10−5±4.7×10−81.61×104±6.3×1021.68×104±6.6×1021.74×104±6.8×1021.72×104±6.8×1022.10×105±7.5×103
      1.26×10−5±6.4×10−81.49×104±1.3×1031.49×104±1.3×1031.48×104±1.3×1031.43×104±1.2×1031.85×105±1.6×104
      1.58×10−5±8.6×10−81.16×104±1.7×1031.18×104±1.8×1031.16×104±1.7×1031.13×104±1.7×1031.44×105±2.1×104
      2.00×10−5±1.2×10−79.35×103±1.7×1039.67×103±1.7×1039.48×103±1.7×1039.19×103±1.7×1031.19×105±2.1×104
      2.51×10−5±1.6×10−71.07×104±5.3×1021.07×104±5.3×1021.07×104±5.3×1021.06×104±5.4×1021.33×105±6.0×103
      3.16×10−5±2.1×10−77.60×103±1.6×1037.66×103±1.7×1037.39×103±1.6×1037.35×103±1.6×1039.34×104±2.0×104
      Continued on next page

      Table A3.  The differential cross sections in the laboratory reference system and angle-integrated cross sections of the 10B(n, α)7Li reaction.

    • 3.7.   Determination of the ratios of the cross sections of the 10B(n, α0)7Li and 10B(n, α1)7Li* reactions

    • After the processes presented in Sections 3.1 − 3.6, the differential and angle-integrated cross sections of the 10B(n, α)7Li reaction have been obtained. The 10B(n, α)7Li reaction has two reaction channels which are the 10B(n, α0)7Li and 10B(n, α1)7Li* reactions. Their differential cross sections, σ0E_bin,θ and σ1E_bin,θ, can be obtained by

      σ0E_bin,θ=R0E_bin,θσE_bin,θ

      (8)

      and

      σ1E_bin,θ=R1E_bin,θσE_bin,θ,

      (9)

      where R0E_bin,θ and R1E_bin,θ are the ratios of the differential cross sections of the 10B(n, α0)7Li and 10B(n, α1)7Li* reactions over that of the 10B(n, α)7Li reaction, respectively. After the processes presented in Sections 3.1 − 3.6, the weight of the events has been corrected at the corresponding neutron energy bin and the detection angle, and the ratios can be obtained from the spectrum of the net α events. The peaks of α0 and α1 could be separated directly from the spectrum in the low neutron energy region from 1.0 eV to 0.01 MeV except for several detection angles, such as 19.2° and 160.8°. However, as the neutron energy increases, overlap occurs between the two peaks. The cross sections of the two reaction channels were obtained from 0.01 MeV to 1.0 MeV using the double Gaussian functions fitting method; above 1.0 MeV, the two peaks overlap almost completely. The calculation detail for R0E_bin,θ and R1E_bin,θ is described as fellows.

      Firstly, from the spectrum at the corresponding neutron energy bin and the detection angle, the valley between the two peaks of α0 and α1 could be found. If the minimum height of the valley was less than the one-tenth maximum height of the lower peak (usually α0 peak), a threshold at the least-count position would separate the two peaks directly as an example shown in Fig. 11(a). In this condition, the area of the overlap is less than 2% of the α0 or α1 peak area, and the overlap is thus negligible. Below 0.01 MeV, the α0 and α1 peaks of most spectra could be separated directly, except for the spectra of several detection angles, such as 19.2° and 160.8°.

      Figure 11.  (color online) The separation of the α0 and α1 peaks by (a) threshold at the detection angle 46.8° and at the neutron energy 1.00×10−6 MeV and (b) the fitting method at the detection angle 46.8° and at the neutron energy 0.50 MeV.

      Secondly, if the overlap between the two peaks in the spectrum of net α events cannot be ignored, then double Gaussian functions were used to fit them. Compared with other fitting functions, such as the exponential function and the polynomial function, the double Gaussian function agrees better with the measurement spectrum. Then the area of the overlapping region would be separated by the fitting result. A typical example is shown in Fig. 11(b).

      After the calculated R0E_bin,θ and R1E_bin,θ, σ0E_bin,θ and σ1E_bin,θ would be obtained from Eqs. (6) and (7). The uncertainties of the calculations of R0E_bin,θ and R1E_bin,θ are 2.9% − 36.4% and 1.0% − 21.6% (for 503 of the 885 σ0E_bin,θ results, this uncertainty is less than 10%; for 864 of the 885 σ1E_bin,θ results, this uncertainty is less than 10%).

      The differential cross sections would be fitted by the Legendre polynomial series, then the ratios of the angle-integrated cross sections of the two reaction channel would be calculated using fitting results. Finally, the angle-integrated cross sections of the 10B(n, α0)7Li and 10B(n, α1)7Li* reactions, σ0E_bin and σ1E_bin, were obtained. The total uncertainties of the differential cross sections σ0E_bin,θ and σ1E_bin,θ are 4.7% − 36.7% and 3.1% − 27.7%, respectively (for 271 of the 885 σ0E_bin,θ results, this uncertainty is less than 10%; for 738 of the 885 σ1E_bin,θ results, this uncertainty is less than 10%), and those of the angle-integrated cross sections σ0E_bin and σ1E_bin are 2.8% − 21.6% and 2.3% − 21.5% (for 31 of the 59 σ0E_bin results, this uncertainty is less than 5%; for 37 of the 59 σ1E_bin results, this uncertainty is less than 5%). Sources of uncertainty and their magnitudes were shown in Table 1. The uncertainty of the 10B(n, α0) 7Li reaction is quite large for En < 0.1 MeV because of the small counts of α0-particles. Besides, the uncertainties of the ratios, R0E_bin,θ and R1E_bin,θ, are quite large at the detection angles of 19.2° and 160.8°, because the emitted α-particles at these two emission angles would have longer track in the sample, which lead to increased energy loss and more overlap of the two peaks. The final results are presented in Tables A4-A6 and Tables A7-A9 of Appendix A. Selected results of differential cross sections are shown in Figs. 12 and 13, and the angle-integrated cross sections are shown in Figs. 14 and 15. The differential cross sections of the 10B(n, α1)7Li* reaction are very similar to those of the 10B(n, α)7Li reaction below 1.0 MeV, because the cross section of 10B(n, α1) 7Li* reaction takes a large proportion (> 75% below 0.5 MeV region and > 55% in the 0.5 MeV ≤ En < 1.0 MeV region) of the total 10B(n, α)7Li reaction cross section. The differential cross sections will be discussed in Section 5.

      Figure 12.  (color online) The present differential cross sections of the 10B(n, α0)7Li reaction at selected energy points as a function of the θlab compared with existing results of evaluations and measurements [3, 5].

      Figure 13.  (color online) The present differential cross sections of the 10B(n, α1)7Li* reaction at selected energy points as a function of the θlab compared with existing results of evaluations and measurements [3, 5].

      Figure 14.  (color online) The angle-integrated cross sections of the 10B(n, α0)7Li reaction compared with existing results of evaluations and measurements since 1965 [3, 5, 17].

      Figure 15.  (color online) The present cross sections of the 10B(n, α1)7Li* reaction compared with existing results of evaluations and measurements since 1965 [3, 5, 17].

      En /MeVσ0E_bin,θ /(mb/sr)
      19.2°26.9°36.5°46.7°57.3°68.0°
      1.00×10−6±4.2×10−93.21×103±1.8×1023.14×103±1.7×1023.09×103±1.6×1023.02×103±1.6×1023.15×103±1.6×1023.07×103±1.5×102
      1.26×10−6±5.4×10−92.68×103±1.4×1022.51×103±1.2×1022.54×103±1.2×1022.46×103±1.2×1022.80×103±1.3×1022.72×103±1.3×102
      1.58×10−6±6.8×10−91.93×103±1.0×1022.08×103±1.0×1022.07×103±1.0×1022.13×103±1.1×1022.07×103±1.0×1022.37×103±1.1×102
      2.00×10−6±8.7×10−91.88×103±1.1×1021.92×103±1.1×1021.91×103±1.0×1021.87×103±1.0×1021.85×103±1.0×1021.92×103±1.0×102
      2.51×10−6±1.1×10−81.69×103±1.2×1021.58×103±1.1×1021.66×103±1.1×1021.75×103±1.1×1021.74×103±1.1×1021.68×103±1.1×102
      3.16×10−6±1.4×10−81.89×103±1.3×1021.87×103±1.2×1021.76×103±1.2×1021.88×103±1.2×1021.94×103±1.3×1021.89×103±1.2×102
      3.98×10−6±1.8×10−81.54×103±2.2×1021.23×103±1.8×1021.33×103±1.9×1021.41×103±2.0×1021.31×103±1.9×1021.26×103±1.8×102
      5.01×10−6±2.3×10−81.44×103±1.3×1021.25×103±1.2×1021.28×103±1.2×1021.29×103±1.2×1021.21×103±1.1×1021.46×103±1.3×102
      6.31×10−6±2.9×10−81.30×103±1.0×1021.23×103±9.6×1011.19×103±9.3×1011.22×103±9.5×1011.28×103±9.9×1011.34×103±1.0×102
      7.94×10−6±3.7×10−81.02×103±2.2×1021.12×103±2.4×1029.69×102±2.1×1029.98×102±2.2×1021.08×103±2.4×1021.25×103±2.7×102
      1.00×10−5±4.7×10−81.15×103±8.4×1019.53×102±7.3×1019.30×102±7.2×1011.03×103±7.7×1011.06×103±7.9×1011.22×103±8.5×101
      1.26×10−5±6.4×10−89.95×102±1.1×1021.04×103±1.1×1029.82×102±1.0×1028.62×102±9.2×1019.12×102±9.7×1019.31×102±9.8×101
      1.58×10−5±8.6×10−86.91×102±1.1×1026.65×102±1.1×1026.02×102±9.8×1016.94×102±1.1×1026.79×102±1.1×1027.28×102±1.2×102
      2.00×10−5±1.2×10−77.02×102±1.4×1026.60×102±1.3×1025.88×102±1.1×1025.38×102±1.1×1026.39×102±1.2×1025.92×102±1.2×102
      2.51×10−5±1.6×10−77.07×102±6.5×1017.09×102±6.5×1016.89×102±6.3×1016.97×102±6.3×1015.97×102±5.8×1017.75×102±6.8×101
      3.16×10−5±2.1×10−74.97×102±1.1×1024.14×102±9.5×1015.62×102±1.3×1024.39×102±1.0×1024.91×102±1.1×1024.43×102±1.0×102
      3.98×10−5±2.7×10−75.22×102±6.4×1014.59×102±5.7×1014.70×102±5.7×1014.97×102±6.0×1014.34×102±5.4×1014.11×102±5.2×101
      5.01×10−5±3.6×10−75.03×102±5.0×1014.81×102±4.8×1014.31×102±4.3×1015.11×102±5.0×1014.68×102±4.6×1014.40×102±4.4×101
      6.31×10−5±4.8×10−73.49×102±4.2×1013.21×102±3.9×1013.36×102±4.1×1013.67×102±4.3×1013.24×102±3.9×1013.27×102±4.0×101
      7.94×10−5±6.4×10−73.54×102±4.3×1012.89×102±3.7×1013.37×102±4.1×1013.16×102±3.9×1013.29×102±4.0×1012.66×102±3.4×101
      1.00×10−4±8.4×10−74.25×102±4.6×1013.10×102±3.5×1013.01×102±3.4×1012.89×102±3.2×1012.51×102±2.9×1012.70×102±3.1×101
      1.26×10−4±1.0×10−63.88×102±5.3×1012.56×102±4.0×1013.14×102±4.4×1012.62×102±3.8×1012.75×102±4.0×1012.58×102±3.9×101
      1.58×10−4±1.3×10−63.26×102±3.7×1011.96×102±2.6×1012.22×102±2.8×1012.40×102±2.9×1012.09×102±2.6×1012.49×102±3.0×101
      Continued on next page

      Table A4.  The differential cross sections of the 10B(n, α0)7Li reaction in the laboratory reference system.

      En/MeVσ0E_bin,θ /(mb/sr)σ0E_bin /mb
      133.2°143.5°153.1°160.8°
      1.00×10−6±4.2×10−93.21×103±1.6×1023.04×103±1.6×1023.17×103±1.7×1023.12×103±1.6×1023.93×104±1.8×103
      1.26×10−6±5.4×10−92.84×103±1.4×1022.44×103±1.2×1022.63×103±1.3×1022.66×103±1.4×1023.34×104±1.1×103
      1.58×10−6±6.8×10−92.15×103±1.1×1021.98×103±1.0×1021.90×103±1.0×1022.33×103±1.2×1022.63×104±8.6×102
      2.00×10−6±8.7×10−91.97×103±1.1×1022.02×103±1.1×1021.80×103±1.0×1021.78×103±1.1×1022.38×104±8.7×102
      2.51×10−6±1.1×10−81.70×103±1.1×1021.61×103±1.1×1021.50×103±1.0×1021.54×103±1.1×1022.08×104±1.0×103
      3.16×10−6±1.4×10−81.63×103±1.2×1021.89×103±1.3×1021.96×103±1.3×1021.66×103±1.2×1022.34×104±9.3×102
      3.98×10−6±1.8×10−81.31×103±1.9×1021.26×103±1.9×1021.44×103±2.1×1021.21×103±1.8×1021.69×104±2.0×103
      5.01×10−6±2.3×10−81.35×103±1.3×1021.37×103±1.3×1021.25×103±1.2×1021.38×103±1.3×1021.64×104±1.2×103
      6.31×10−6±2.9×10−81.32×103±1.0×1021.26×103±9.9×1011.21×103±9.6×1011.11×103±9.1×1011.58×104±8.9×102
      7.94×10−6±3.7×10−81.12×103±2.4×1021.20×103±2.6×1021.11×103±2.4×1029.62×102±2.1×1021.35×104±2.8×103
      1.00×10−5±4.7×10−81.07×103±8.3×1019.91×102±7.7×1011.01×103±7.7×1011.00×103±7.9×1011.31×104±5.6×102
      1.26×10−5±6.4×10−88.93×102±9.7×1019.37×102±1.0×1028.79×102±9.6×1018.71×102±9.6×1011.18×104±1.0×103
      1.58×10−5±8.6×10−87.94×102±1.3×1027.44×102±1.2×1026.86×102±1.1×1027.59×102±1.2×1028.82×103±1.3×103
      2.00×10−5±1.2×10−76.26×102±1.2×1026.30×102±1.2×1026.32×102±1.2×1024.67×102±9.4×1017.48×103±1.4×103
      2.51×10−5±1.6×10−75.93×102±5.8×1016.70×102±6.2×1015.73×102±5.7×1016.22×102±6.2×1018.32×103±4.4×102
      Continued on next page

      Table A6.  The differential cross sections in the laboratory reference system and angle-integrated cross sections of the 10B(n, α0)7Li reaction.

      En /MeVσ1E_bin,θ /(mb/sr)
      19.2°26.9°36.5°46.7°57.3°68.0°
      1.00×10−6±4.2×10−94.67×104±2.4×1034.69×104±2.1×1034.74×104±2.1×1034.65×104±2.1×1034.75×104±2.0×1034.70×104±2.0×103
      1.26×10−6±5.4×10−93.86×104±1.2×1033.92×104±1.2×1033.97×104±1.2×1033.97×104±1.2×1033.96×104±1.2×1033.89×104±1.2×103
      1.58×10−6±6.8×10−93.10×104±1.2×1033.03×104±9.8×1023.07×104±9.9×1023.06×104±9.9×1023.09×104±9.9×1023.05×104±9.8×102
      2.00×10−6±8.7×10−92.88×104±1.2×1032.91×104±1.1×1032.89×104±1.0×1032.87×104±1.1×1032.94×104±1.1×1032.94×104±1.1×103
      2.51×10−6±1.1×10−82.47×104±1.2×1032.49×104±1.2×1032.46×104±1.2×1032.50×104±1.2×1032.43×104±1.2×1032.44×104±1.2×103
      3.16×10−6±1.4×10−82.75×104±1.1×1032.77×104±1.1×1032.75×104±1.1×1032.86×104±1.2×1032.78×104±1.1×1032.82×104±1.1×103
      3.98×10−6±1.8×10−81.94×104±2.4×1032.04×104±2.5×1031.99×104±2.4×1031.99×104±2.4×1032.00×104±2.4×1031.97×104±2.4×103
      5.01×10−6±2.3×10−81.93×104±1.5×1031.95×104±1.5×1031.94×104±1.5×1031.99×104±1.5×1031.95×104±1.5×1031.99×104±1.5×103
      6.31×10−6±2.9×10−81.90×104±1.1×1031.99×104±1.1×1031.91×104±1.1×1031.91×104±1.1×1031.94×104±1.1×1031.92×104±1.1×103
      7.94×10−6±3.7×10−81.59×104±3.3×1031.58×104±3.3×1031.61×104±3.4×1031.63×104±3.4×1031.64×104±3.4×1031.63×104±3.4×103
      1.00×10−5±4.7×10−81.56×104±7.5×1021.52×104±6.8×1021.57×104±7.0×1021.54×104±6.9×1021.52×104±6.8×1021.52×104±6.8×102
      1.26×10−5±6.4×10−81.35×104±1.2×1031.40×104±1.2×1031.34×104±1.2×1031.36×104±1.2×1031.36×104±1.2×1031.36×104±1.2×103
      1.58×10−5±8.6×10−81.05×104±1.6×1031.08×104±1.6×1031.06×104±1.6×1031.06×104±1.6×1031.09×104±1.6×1031.05×104±1.6×103
      2.00×10−5±1.2×10−78.80×103±1.6×1038.73×103±1.6×1038.87×103±1.6×1038.88×103±1.6×1038.82×103±1.6×1038.59×103±1.6×103
      2.51×10−5±1.6×10−79.68×103±5.5×1029.58×103±5.5×1029.98×103±5.6×1029.78×103±5.5×1029.63×103±5.5×1029.52×103±5.4×102
      3.16×10−5±2.1×10−76.80×103±1.5×1036.93×103±1.5×1036.84×103±1.5×1037.09×103±1.5×1037.00×103±1.5×1036.75×103±1.5×103
      3.98×10−5±2.7×10−77.21×103±6.9×1026.90×103±6.6×1027.08×103±6.8×1026.95×103±6.6×1027.16×103±6.8×1027.07×103±6.8×102
      5.01×10−5±3.6×10−76.73×103±5.3×1026.74×103±5.1×1026.57×103±5.0×1026.41×103±4.9×1026.78×103±5.1×1026.63×103±5.0×102
      6.31×10−5±4.8×10−75.05×103±5.2×1025.10×103±5.2×1025.04×103±5.2×1024.97×103±5.1×1025.05×103±5.2×1024.76×103±4.9×102
      7.94×10−5±6.4×10−75.04×103±5.0×1024.95×103±4.9×1024.82×103±4.8×1024.86×103±4.8×1024.67×103±4.6×1024.84×103±4.8×102
      1.00×10−4±8.4×10−74.33×103±3.7×1024.48×103±3.7×1024.32×103±3.6×1024.51×103±3.7×1024.49×103±3.7×1024.28×103±3.5×102
      1.26×10−4±1.0×10−64.09×103±4.6×1023.74×103±4.3×1023.91×103±4.3×1024.06×103±4.5×1024.13×103±4.6×1023.87×103±4.3×102
      1.58×10−4±1.3×10−63.40×103±3.0×1023.52×103±3.0×1023.35×103±2.8×1023.38×103±2.9×1023.65×103±3.1×1023.36×103±2.8×102
      2.00×10−4±1.6×10−63.06×103±2.9×1022.89×103±2.8×1022.87×103±2.7×1022.93×103±2.7×1022.97×103±2.7×1022.87×103±2.7×102
      2.51×10−4±2.0×10−62.79×103±2.6×1022.70×103±2.6×1022.70×103±2.6×1022.70×103±2.5×1022.64×103±2.5×1022.64×103±2.5×102
      3.16×10−4±2.5×10−62.54×103±2.0×1022.30×103±2.0×1022.33×103±1.9×1022.15×103±1.8×1022.37×103±1.9×1022.25×103±1.9×102
      3.98×10−4±3.1×10−62.36×103±1.9×1022.34×103±1.9×1022.14×103±1.8×1022.26×103±1.8×1022.25×103±1.8×1022.10×103±1.7×102
      5.01×10−4±3.9×10−62.06×103±1.3×1021.89×103±1.3×1021.91×103±1.2×1022.16×103±1.3×1022.05×103±1.3×1021.95×103±1.2×102
      6.31×10−4±4.8×10−61.80×103±1.0×1021.65×103±1.0×1021.75×103±1.0×1021.70×103±9.7×1011.79×103±1.0×1021.60×103±9.9×101
      7.94×10−4±6.1×10−61.62×103±1.2×1021.54×103±1.1×1021.58×103±1.1×1021.51×103±9.9×1011.50×103±1.1×1021.51×103±1.0×102
      1.00×10−3±7.7×10−61.31×103±8.3×1011.33×103±8.1×1011.33×103±7.8×1011.38×103±7.9×1011.31×103±7.8×1011.15×103±7.5×101
      1.26×10−3±9.8×10−61.20×103±7.2×1011.11×103±6.7×1011.12×103±6.3×1011.23×103±6.4×1011.17×103±6.5×1011.04×103±6.5×101
      1.58×10−3±1.3×10−51.09×103±6.2×1011.02×103±6.6×1011.10×103±5.7×1011.03×103±5.6×1011.06×103±5.7×1019.97×102±5.9×101
      2.00×10−3±1.6×10−59.89×102±5.7×1019.65×102±5.5×1019.43×102±5.2×1019.59×102±5.2×1019.42×102±5.1×1018.70×102±5.6×101
      2.51×10−3±2.1×10−58.05×102±6.9×1018.27×102±6.8×1018.51×102±6.7×1018.87×102±6.8×1018.32×102±6.8×1017.62×102±6.8×101
      3.16×10−3±2.7×10−58.28×102±5.2×1017.68×102±4.9×1017.37×102±4.4×1017.80×102±4.5×1017.37×102±4.5×1016.95×102±4.9×101
      3.98×10−3±3.6×10−56.56×102±4.5×1017.24×102±4.6×1017.17×102±4.2×1017.02×102±4.0×1016.99×102±4.1×1016.39×102±4.3×101
      5.01×10−3±4.8×10−55.57×102±3.6×1015.53×102±3.5×1015.87×102±3.4×1015.90×102±3.3×1015.79×102±3.4×1014.98×102±3.5×101
      6.31×10−3±6.4×10−55.65×102±3.7×1015.54×102±3.5×1015.28×102±3.3×1015.50×102±3.3×1015.27×102±3.3×1015.10×102±3.3×101
      Continued on next page

      Table A7.  The differential cross sections of the 10B(n, α1)7Li reaction in the laboratory reference system.

      En /MeVσ1E_bin,θ /(mb/sr) σ1E_bin /mb
      133.2°143.5°153.1°160.8°
      1.00×10−6±4.2×10−94.84×104±2.0×1034.86×104±2.1×1034.79×104±2.1×1034.81×104±2.2×1035.98×105±2.5×104
      1.26×10−6±5.4×10−94.08×104±1.3×1034.04×104±1.3×1034.07×104±1.3×1034.03×104±1.3×1035.03×105±1.3×104
      1.58×10−6±6.8×10−93.21×104±1.0×1033.14×104±1.0×1033.16×104±1.0×1033.16×104±1.0×1033.92×105±1.1×104
      2.00×10−6±8.7×10−93.04×104±1.1×1033.01×104±1.1×1033.00×104±1.1×1032.99×104±1.1×1033.71×105±1.2×104
      2.51×10−6±1.1×10−82.56×104±1.3×1032.52×104±1.3×1032.56×104±1.3×1032.55×104±1.3×1033.16×105±1.4×104
      3.16×10−6±1.4×10−82.93×104±1.2×1032.88×104±1.2×1032.84×104±1.2×1032.80×104±1.2×1033.56×105±1.2×104
      3.98×10−6±1.8×10−82.03×104±2.5×1032.13×104±2.6×1032.07×104±2.5×1032.09×104±2.5×1032.56×105±3.0×104
      5.01×10−6±2.3×10−81.98×104±1.5×1032.01×104±1.5×1032.09×104±1.6×1031.97×104±1.5×1032.49×105±1.8×104
      6.31×10−6±2.9×10−81.98×104±1.1×1031.98×104±1.1×1031.96×104±1.1×1031.98×104±1.1×1032.46×105±1.3×104
      7.94×10−6±3.7×10−81.62×104±3.4×1031.66×104±3.5×1031.65×104±3.5×1031.67×104±3.5×1032.06×105±4.3×104
      1.00×10−5±4.7×10−81.51×104±6.9×1021.58×104±7.2×1021.64×104±7.3×1021.62×104±7.4×1021.97×105±7.2×103
      1.26×10−5±6.4×10−81.40×104±1.3×1031.40×104±1.3×1031.39×104±1.2×1031.34×104±1.2×1031.73×105±1.5×104
      1.58×10−5±8.6×10−81.08×104±1.6×1031.10×104±1.7×1031.09×104±1.6×1031.05×104±1.6×1031.36×105±2.0×104
      2.00×10−5±1.2×10−78.73×103±1.6×1039.04×103±1.6×1038.84×103±1.6×1038.72×103±1.6×1031.11×105±2.0×104
      2.51×10−5±1.6×10−71.01×104±5.7×1021.00×104±5.7×1021.01×104±5.8×1021.00×104±5.9×1021.24×105±5.8×103
      3.16×10−5±2.1×10−77.17×103±1.6×1037.15×103±1.6×1036.96×103±1.5×1036.95×103±1.5×1038.76×104±1.9×104
      3.98×10−5±2.7×10−77.36×103±7.0×1027.33×103±7.0×1027.14×103±6.9×1026.87×103±6.7×1028.97×104±8.1×103
      5.01×10−5±3.6×10−76.95×103±5.2×1026.79×103±5.1×1026.84×103±5.2×1026.61×103±5.1×1028.46×104±5.9×103
      6.31×10−5±4.8×10−75.00×103±5.1×1025.06×103±5.2×1025.21×103±5.3×1025.16×103±5.3×1026.39×104±6.3×103
      7.94×10−5±6.4×10−75.09×103±5.0×1024.95×103±4.9×1025.06×103±5.0×1024.79×103±4.8×1026.17×104±5.8×103
      1.00×10−4±8.4×10−74.30×103±3.5×1024.55×103±3.7×1024.31×103±3.6×1024.02×103±3.5×1025.50×104±4.1×103
      1.26×10−4±1.0×10−64.01×103±4.4×1023.97×103±4.4×1023.85×103±4.3×1023.71×103±4.3×1025.01×104±5.1×103
      1.58×10−4±1.3×10−63.52×103±2.9×1023.59×103±3.0×1023.50×103±2.9×1023.54×103±3.1×1024.35×104±3.3×103
      2.00×10−4±1.6×10−63.13×103±2.8×1023.05×103±2.8×1022.90×103±2.7×1023.01×103±2.9×1023.77×104±3.0×103
      2.51×10−4±2.0×10−62.81×103±2.6×1022.71×103±2.5×1022.79×103±2.7×1022.64×103±2.6×1023.45×104±2.9×103
      3.16×10−4±2.5×10−62.37×103±1.8×1022.46×103±1.9×1022.46×103±2.0×1022.15×103±2.0×1022.97×104±1.9×103
      3.98×10−4±3.1×10−62.32×103±1.8×1022.23×103±1.7×1022.19×103±1.8×1022.23×103±2.0×1022.84×104±1.8×103
      5.01×10−4±3.9×10−62.08×103±1.3×1022.07×103±1.3×1022.09×103±1.3×1022.05×103±1.4×1022.55×104±1.2×103
      6.31×10−4±4.8×10−61.91×103±1.0×1021.82×103±9.9×1011.73×103±1.0×1021.67×103±1.1×1022.19×104±8.3×102
      7.94×10−4±6.1×10−61.60×103±1.0×1021.56×103±1.0×1021.55×103±1.0×1021.39×103±1.1×1021.94×104±8.7×102
      1.00×10−3±7.7×10−61.37×103±7.6×1011.31×103±7.6×1011.26×103±7.7×1011.31×103±8.8×1011.65×104±6.1×102
      1.26×10−3±9.8×10−61.16×103±6.0×1011.19×103±6.2×1011.16×103±6.4×1011.14×103±7.5×1011.47×104±3.9×102
      1.58×10−3±1.3×10−51.05×103±5.4×1011.10×103±5.7×1011.02×103±5.7×1019.48×102±6.5×1011.33×104±3.6×102
      2.00×10−3±1.6×10−59.48×102±5.0×1019.40×102±5.1×1019.69×102±5.2×1019.50×102±6.1×1011.19×104±2.9×102
      2.51×10−3±2.1×10−58.72×102±6.4×1017.91×102±6.1×1017.97×102±6.5×1018.51×102±7.7×1011.07×104±5.8×102
      3.16×10−3±2.7×10−58.15×102±4.3×1018.05×102±4.4×1017.12×102±4.5×1017.70×102±5.4×1019.68×103±2.8×102
      3.98×10−3±3.6×10−57.02×102±3.9×1016.56×102±3.8×1017.07×102±4.2×1016.97×102±4.7×1018.55×103±2.3×102
      5.01×10−3±4.8×10−55.76×102±3.2×1015.76×102±3.3×1015.64×102±3.5×1015.33×102±4.0×1017.17×103±2.1×102
      6.31×10−3±6.4×10−55.35×102±3.1×1015.52×102±3.3×1014.45×102±3.2×1015.04×102±3.9×1016.70×103±2.0×102
      Continued on next page

      Table A9.  The differential cross sections in the laboratory reference system and angle-integrated cross sections of the 10B(n, α1)7Li reaction.

    4.   Discussions

      4.1.   Comparison of the present results with different measurements and evaluations

    • The present differential cross sections have been compared with existing measurement data and evaluations [3, 5]:

      1) For En < 0.1 MeV, the present differential cross sections of the10B(n, α)7Li, 10B(n, α0)7Li and 10B(n, α1)7Li* reactions agree well with the measurement data of Hambsch [2] (2009, 0.40 keV − 1.20 MeV, after normalization using ENDF/B-VIII.0 data) and Stelts [7] (1979, 2 keV − 24 keV, after normalization using the ENDF/B-VIII.0 data). The present differential cross sections of the 10B(n, α)7Li, 10B(n, α0)7Li and 10B(n, α1)7Li* reactions show that the differential cross sections are almost isotropic in the center-of-mass system and slightly forward-peaked in the laboratory system, as well as those of different evaluations.

      2) In the 0.1 MeV ≤ En < 1.0 MeV region, the present differential cross sections of the 10B(n, α)7Li and 10B(n, α1)7Li* reactions agree with the measurement data of Hambsch (2009, 0.40 keV − 0.98 MeV, after normalization using ENDF/B-VIII.0 data) [2], as well as the ENDF/B-VIII.0 and ENDF/B-VII.1 library.

      For the present differential cross sections of the 10B(n, α0)7Li reaction, there are differences between the present data and the measurement data of Hambsch [2], as well as those of ENDF/B-VIII.0 and ENDF/B-VII.1 library. In this neutron energy region, compared with the measurement data of Hambsch [2] and evaluation data, the present differential cross sections are commonly higher in the forward emission angles, and lower in the backward angles. However, due to the large uncertainty of the present results of the 10B(n, α0)7Li reaction, more research is needed to clarify these differences.

      Besides, the present differential cross sections of the 10B(n, α0)7Li and 10B(n, α1)7Li* don’t agree with the results of Sealock (1976, 0.20 − 1.20 MeV) [6]. Compared with the results of Sealock [6], the uncertainty of the present differential cross sections is smaller. The average uncertainty of the results of Sealock is 22.4%, while that of the present results is 8.6% in this neutron energy region.

      3) In the 1.0 MeV ≤ En < 2.5 MeV region, the discrepancies between the present differential cross sections of the 10B(n, α)7Li reaction and evaluations exist. According to the evaluations, including ENDF/B-VIII.0, ENDF/B-VII.1 and CENDL-3.1 libraries, the differential cross sections of the 10B(n, α)7Li reaction decrease monotonously with the increasing of θLab, while our present results show non-monotone decreasing trend. There is no other data in the 1.0 MeV ≤ En < 2.5 MeV region except the results of Sealock (1976, 0.20 − 1.20 MeV), which have fairly big uncertainties (in the 1.0 MeV ≤ En < 1.20 MeV region, the average uncertainty of Sealock's results is 26.4%, while that of the present results is 8.9%) [6]. So, further researches are therefore demanded in the MeV neutron energy region.

      The present cross sections have been also compared with existing measurement data from EXFOR library since 1965 and evaluations [3, 5]:

      1) For the 10B(n, α)7Li reaction, the present cross sections agree well with the measurement data of Friesenhahn (1975, 2.35 keV − 1.72 MeV) [18] for En ≤ 0.3 MeV, Sealock (1976, 0.20 − 1.20 MeV) [6] in the 0.2 MeV ≤ En ≤ 1.2 MeV region and Bevilacqu (2017, 0.50 – 3.00 MeV) [17] in the 0.5 MeV < En < 1.0 MeV region. The present cross sections of the 10B(n, α)7Li reaction are 17.7% lower than those of Friesenhahn [18] for En > 0.3 MeV, and 12.6% higher than those of Bevilacqu [17] for En ≥ 1.0 MeV.

      2) For the 10B(n, α0)7Li reaction, the present cross sections agree well with the measurement data of Macklin (1968, 0.04 − 0.52 MeV) [19] in the 0.04 MeV ≤ En ≤ 0.1 MeV region, Sealock (1976, 0.20 − 1.20 MeV) [6] in the 0.2 MeV ≤ En < 1.0 MeV region, and Bevilacqu (2017, 0.50 – 3.00 MeV) [17] in the 0.8 MeV ≤ En < 1.0 MeV region. The present cross sections of the 10B(n, α0)7Li reaction are 19.5% higher than those of Macklin [19] for En > 0.1 MeV and 8.1% lower than those of Bevilacqu [17] for En < 0.8 MeV.

      3) For the 10B(n, α1)7Li* reaction, the present cross sections agree well with the measurement data of Schrack (1978, 3.82 keV − 0.63 MeV) [20], Viesti (1979, 0.10 − 2.20 MeV) [21], Sealock (1976, 0.20 − 1.20 MeV) [6], Schrack (1993, 0.20 − 4.06 MeV) [22] and Bevilacqu (2017, 0.50 – 3.00 MeV) [17].

      Compared with different evaluations, including ENDF/B-VIII.0, ENDF/B-VII.1, JENDL-4.0, ROSFOND-2010 and CENDL-3.1 libraries, the present cross sections of the 10B(n, α0)7Li and 10B(n, α1)7Li* reactions generally agree better with ENDF/B-VIII.0 library; those of the 10B(n, α)7Li reaction agree well with ENDF/B-VIII.0 library for En ≤ 1.8 MeV, and well with ENDF/B-VII.1 library for En > 1.8 MeV [5].

    • 4.2.   The future experimental plan

    • The present results have two major sources of uncertainty: for the low neutron energy region (En < 1.0 keV), the major source is the uncertainty of the relative neutron fluence; and for the high neutron energy region (En > 2.0 MeV), the major source is the uncertainty of the unfolding of the expanding of neutron energy due to the double-bunched operation mode. As mentioned in Section 2.1, the experiment was performed at Endstation #1 (the length of flight path was 57.99 m) while the neutron spectrum was measured at Endstation #2 (the length of flight path was 75.78 m), so the actual neutron energy spectrum should be a little different between the two positions. The absolute cross sections are affected by the uncertainty of the neutron energy spectrum, while the relative angular distributions are not. The results of simulation by Fluka code was used to determine this effect [10]. According to the simulation, the deviations between the spectrum of the two position were expected to be less than 2% in the 0.3 − 0.5 MeV region which was chosen for normalization. In the 0.1 − 2.5 MeV region, the deviations were less than 4%. Besides, one could notice that the neutron energy spectrum has fairly big uncertainty for En < 1.0 keV because of the resonance of the cross section of the 235U(n, f) reaction below 2.5 keV region and the uncertainty of the moderation length of the neutron source. In the future, precise neutron energy spectrum at Endstation #1 should be measured.

      The present measurement results are limited in En < 2.5 MeV region; more work should be done in the higher neutron energy region. Besides, the peaks of α0 and α1 could not be separated for En > 1.0 MeV in the present work; thinner sample and the detectors with higher resolution should be used in the future. The unfolding of the expanding of neutron energy due to the double-bunched operation mode would introduce fairly big uncertainty for En > 2.0 MeV, so the single bunched proton beam should be used in the coming work.

    5.   Theoretical analysis
    • As shown in Figs. 10, 14 and 15, the cross sections of the 10B(n, α) 7Li and 10B(n, α1) 7Li* reactions are smooth and obey the 1/v law for En < 0.1 MeV, as well as those of the 10B(n, α0) 7Li reaction for En < 0.01 MeV. The 10B(n, α1) 7Li* reaction dominates below 0.01 MeV region where the 10B(n, α1) / 10B(n, α) cross-section ratio is 93.77%±0.92% according to the present results. For the low neutron energy region (En < 0.1 MeV), the present differential cross sections of the 10B(n, α)7Li, 10B(n, α0)7Li and 10B(n, α1)7Li* reactions are almost isotropic in the center-of-mass system and slightly forward-peaked in the laboratory system as shown in Figs. 9 (a)(d), Figs. 12 (a)(c) and Figs. 13 (a)(c).

      The big 10B(n, α1) / 10B(n, α) cross-section ratio for the low neutron energy region could be explained by the resonance reaction mechanism and the level structure of the 11B compound system [23]. When the 10B target nucleus interacts with the low-energy neutrons (ln=0), the scattering and bound states with Jπ=7/2+ of the compound nucleus 11B would be formed, then 11B would decay to 7Li or 7Li*. The s-wave scattering state of 11B has the resonance energy of En = 0.37 MeV, and the excitation energy of 7Li* is 0.48 MeV. With the broad s-wave state (Γn(C.M.) = 0.77 MeV and Γα1(C.M.) = 0.113 MeV), the 10B(n, α1) 7Li* reaction would have large cross section in the low neutron energy region. The α0 partial width (Γα0(C.M.) = 0.001 MeV) is negligible compared to the α1 partial width, so it is very difficult for the 10B(n, α0) 7Li reaction to have the resonance with this state of 11B. Besides, the two 7/2+ s-wave states are primarily responsible for the 1/v law of the excitation function for the low neutron energy region [24].

      In the neutron energy range from 0.1 MeV to 1.0 MeV, the differential cross sections of the 10B(n, α1) 7Li* reaction are forward-peaked as shown in Figs. 13 (d)(f) which are also mainly contributed by the two 7/2+ s-wave states of the 11B [24]. For the 10B(n, α0) 7Li reaction, the differential cross sections are backward-peaked in 0.3 − 0.6 MeV region as shown in Fig. 12 (e), which are caused by the resonance of 5/2- p-wave state of the 11B which would occur as En ≈ 0.52 MeV. As for the excitation function, the resonance of 5/2- p-wave state would lead to a peak of the 10B(n, α0) 7Li reaction around En = 0.50 MeV. Because of the competition of the 10B(n, α0) 7Li reaction, the cross section of the 10B(n, α1) 7Li* reaction would decrease rapidly above 0.50 MeV region [2].

      For 1.0 MeV ≤ En < 2.5 MeV, the differential cross sections of the 10B(n, α)7Li reaction are approximately forward-peaked in the laboratory (lab) system, while the deviations exist between the present measurements and evaluations. For example, the anomaly was observed as shown in Fig. 9 (h) around En = 1.80 MeV. It might be due to the contribution of the resonance of the 9/2- p-wave state at En = 1.83 MeV and 5/2+ or 7/2+ d-wave state at En = 1.88 MeV of 11B [25]. These resonance states would also lead to the peak of the excitation function of the 10B(n, α) 7Li reaction near En ≈ 1.80 MeV as shown in Fig. 10.

    6.   Conclusions
    • In the present work, with 15 silicon detectors distributed from 19.2° to 160.8°, differential and angle-integrated cross sections for the 10B(n, α)7Li, 10B(n, α0) 7Li and 10B(n, α1) 7Li* reactions have been measured using the LPDA detector system at CSNS Back-n white neutron source. Compared with existing measurements, the present results have been obtained in the wider neutron energy range. The differential and angle-integrated cross sections were obtained for the 10B(n, α)7Li reaction in the 1.0 eV ≤ En < 2.5 MeV region (67 energy points), as well as the two reaction channels, 10B(n, α0) 7Li and 10B(n, α1) 7Li*, in the 1.0 eV ≤ En < 1.0 MeV region (59 energy points). There is no previous measurement datum of the differential cross section of the 10B(n, α)7Li reaction existing in EXFOR library in the 1.2 MeV < En < 2.5 MeV region; the present data are the first measurement results in this region. The measurement results have been analyzed by the resonance reaction mechanism and the level structure of the 11B compound system. Compared with most existing measurements and evaluations, the present results show a good agreement except for several energy points around 1.0 MeV and above 2.0 MeV, which need further research especially in the MeV neutron energy region.

      The authors are indebted to the operation crew of the CSNS Back-n white neutron source. Dr. Qiwen Fan from China Institute of Atomic Energy is appreciated for preparing the 10B samples. Prof. Zhenpeng Chen from Tsinghua University is appreciated for the beneficial discussions.

    Appendix A
    • Tables A1-A9 results of differential and angle-integrated cross sections of the 10B(n, α)7Li, 10B(n, α0)7Li and 10B(n, α1)7Li* reactions.

      En /MeVσE_bin,θ /(mb/sr)
      78.8°90.3°101.2°112.0°122.7°
      1.00×10−6±4.2×10−94.96×104±2.1×1035.09×104±2.1×1035.20×104±2.4×1035.09×104±2.1×1035.16×104±2.3×103
      1.26×10−6±5.4×10−94.24×104±1.2×1034.39×104±1.3×1034.34×104±1.2×1034.32×104±1.2×1034.34×104±1.2×103
      1.58×10−6±6.8×10−93.33×104±9.6×1023.38×104±9.8×1023.38×104±9.8×1023.35×104±9.6×1023.34×104±9.7×102
      2.00×10−6±8.7×10−93.08×104±1.0×1033.17×104±1.1×1033.15×104±1.0×1033.18×104±1.1×1033.18×104±1.0×103
      2.51×10−6±1.1×10−82.69×104±1.3×1032.70×104±1.3×1032.77×104±1.3×1032.69×104±1.3×1032.77×104±1.3×103
      3.16×10−6±1.4×10−82.99×104±1.1×1033.11×104±1.1×1033.07×104±1.1×1033.11×104±1.1×1033.03×104±1.1×103
      3.98×10−6±1.8×10−82.15×104±2.5×1032.25×104±2.6×1032.19×104±2.6×1032.22×104±2.6×1032.20×104±2.6×103
      5.01×10−6±2.3×10−82.05×104±1.5×1032.10×104±1.5×1032.11×104±1.5×1032.13×104±1.6×1032.12×104±1.6×103
      6.31×10−6±2.9×10−82.08×104±1.1×1032.10×104±1.1×1032.12×104±1.1×1032.09×104±1.1×1032.16×104±1.2×103
      Continued on next page

      Table A2.  The differential cross sections of the 10B(n, α)7Li reaction in the laboratory reference system.

      En /MeVσ0E_bin,θ /(mb/sr)
      78.8°90.3°101.2°112.0°122.7°
      1.00×10−6±4.2×10−93.08×103±1.6×1023.17×103±1.7×1023.02×103±1.6×1023.18×103±1.6×1023.20×103±1.8×102
      1.26×10−6±5.4×10−92.79×103±1.3×1022.74×103±1.3×1022.65×103±1.3×1022.49×103±1.2×1022.92×103±1.4×102
      1.58×10−6±6.8×10−92.15×103±1.1×1021.96×103±1.0×1022.14×103±1.1×1021.98×103±9.9×1012.16×103±1.1×102
      2.00×10−6±8.7×10−91.97×103±1.1×1021.85×103±1.0×1021.84×103±1.0×1021.98×103±1.1×1021.90×103±1.1×102
      2.51×10−6±1.1×10−81.64×103±1.1×1021.70×103±1.1×1021.57×103±1.0×1021.67×103±1.1×1021.80×103±1.2×102
      3.16×10−6±1.4×10−81.95×103±1.3×1022.00×103±1.3×1021.89×103±1.2×1021.73×103±1.1×1021.94×103±1.3×102
      3.98×10−6±1.8×10−81.35×103±2.0×1021.58×103±2.2×1021.31×103±1.9×1021.30×103±1.8×1021.35×103±2.0×102
      5.01×10−6±2.3×10−81.25×103±1.2×1021.22×103±1.1×1021.29×103±1.2×1021.29×103±1.2×1021.28×103±1.2×102
      6.31×10−6±2.9×10−81.25×103±9.7×1011.24×103±9.5×1011.37×103±1.0×1021.19×103±9.2×1011.35×103±1.1×102
      7.94×10−6±3.7×10−81.01×103±2.2×1021.03×103±2.3×1021.17×103±2.5×1029.94×102±2.2×1021.05×103±2.3×102
      1.00×10−5±4.7×10−81.04×103±7.8×1011.05×103±7.6×1011.02×103±7.6×1011.03×103±7.5×1011.10×103±8.4×101
      1.26×10−5±6.4×10−89.36×102±1.0×1029.84×102±1.0×1028.60×102±9.2×1019.22×102±9.7×1011.07×103±1.1×102
      1.58×10−5±8.6×10−87.08×102±1.1×1027.38×102±1.2×1026.15×102±1.0×1027.51×102±1.2×1026.81×102±1.1×102
      2.00×10−5±1.2×10−75.65×102±1.1×1025.38×102±1.0×1025.75×102±1.1×1025.45×102±1.1×1026.38×102±1.2×102
      2.51×10−5±1.6×10−76.21×102±6.0×1016.97×102±6.2×1016.01×102±5.6×1016.01×102±5.5×1017.75×102±7.1×101
      3.16×10−5±2.1×10−74.54×102±1.0×1024.15×102±9.4×1014.40×102±1.0×1024.32×102±9.8×1014.89×102±1.1×102
      3.98×10−5±2.7×10−74.07×102±5.3×1015.27×102±6.1×1015.02×102±5.9×1014.75×102±5.6×1014.90×102±6.1×101
      5.01×10−5±3.6×10−74.11×102±4.3×1014.81×102±4.7×1014.72×102±4.6×1014.91×102±4.7×1014.35×102±4.4×101
      6.31×10−5±4.8×10−72.79×102±3.5×1013.16×102±3.8×1013.69×102±4.3×1013.17×102±3.8×1013.49×102±4.2×101
      7.94×10−5±6.4×10−73.04×102±3.9×1012.87×102±3.5×1013.46×102±4.1×1012.66×102±3.3×1013.34×102±4.1×101
      1.00×10−4±8.4×10−72.68×102±3.2×1012.73×102±3.0×1013.08×102±3.3×1013.35×102±3.5×1013.17×102±3.5×101
      1.26×10−4±1.0×10−62.53×102±3.9×1012.63×102±3.6×1013.42×102±4.5×1012.75×102±3.8×1012.35×102±3.5×101
      1.58×10−4±1.3×10−62.30×102±3.0×1011.95×102±2.3×1012.70×102±3.0×1011.98×102±2.4×1012.26×102±2.7×101
      2.00×10−4±1.6×10−62.29×102±3.2×1011.76×102±2.3×1011.77×102±2.4×1011.91×102±2.4×1012.12×102±2.8×101
      2.51×10−4±2.0×10−61.78×102±2.7×1011.77×102±2.2×1012.06×102±2.6×1011.62×102±2.1×1011.91×102±2.5×101
      3.16×10−4±2.5×10−62.28×102±3.1×1011.24×102±1.7×1011.25×102±1.7×1011.66×102±2.0×1011.74×102±2.2×101
      3.98×10−4±3.1×10−61.82×102±2.6×1011.41×102±1.7×1011.46×102±1.8×1011.55×102±1.8×1012.03×102±2.3×101
      5.01×10−4±3.9×10−61.26×102±1.7×1011.09×102±1.2×1011.29×102±1.4×1011.32×102±1.4×1011.27×102±1.5×101
      6.31×10−4±4.8×10−67.23×101±1.2×1018.99×101±1.0×1018.78×101±1.0×1011.08×102±1.2×1011.23×102±1.3×101
      7.94×10−4±6.1×10−69.27×101±1.5×1011.21×102±1.3×1011.01×102±1.2×1011.21×102±1.3×1018.99×101±1.2×101
      1.00×10−3±7.7×10−66.06×101±1.0×1019.54×101±9.8×1009.75×101±1.0×1019.22×101±9.7×1008.81×101±1.0×101
      1.26×10−3±9.8×10−65.13×101±9.4×1008.21×101±8.7×1006.73×101±7.9×1001.05×102±9.9×1008.41×101±9.8×100
      1.58×10−3±1.3×10−55.74×101±9.2×1007.19×101±7.6×1007.16×101±7.7×1005.43×101±6.4×1008.05×101±8.9×100
      2.00×10−3±1.6×10−57.68×101±1.1×1017.01×101±7.3×1006.22×101±6.9×1006.46×101±7.0×1006.33×101±7.5×100
      2.51×10−3±2.1×10−57.30×101±1.2×1016.14×101±7.6×1004.63×101±6.7×1007.00×101±8.3×1006.19×101±8.4×100
      3.16×10−3±2.7×10−55.88×101±9.0×1005.12×101±6.2×1007.31×101±7.3×1005.61×101±6.2×1005.25×101±6.6×100
      3.98×10−3±3.6×10−55.01×101±8.1×1005.60×101±6.1×1003.76×101±4.9×1003.91×101±5.0×1004.42×101±5.9×100
      5.01×10−3±4.8×10−54.41×101±6.4×1004.04×101±4.6×1003.94×101±4.5×1004.46×101±4.8×1004.24×101±5.1×100
      6.31×10−3±6.4×10−52.28×101±5.0×1003.08×101±4.0×1003.82×101±4.6×1004.21×101±4.8×1002.98×101±4.3×100
      Continued on next page

      Table A5.  The differential cross sections of the 10B(n, α0)7Li reaction in the laboratory reference system.

      En /MeVσ1E_bin,θ /(mb/sr)
      78.8°90.3°101.2°112.0°122.7°
      1.00×10−6±4.2×10−94.65×104±2.0×1034.77×104±2.0×1034.90×104±2.3×1034.77×104±2.0×1034.84×104±2.2×103
      1.26×10−6±5.4×10−93.96×104±1.2×1034.12×104±1.3×1034.07×104±1.3×1034.07×104±1.2×1034.04×104±1.3×103
      1.58×10−6±6.8×10−93.12×104±1.0×1033.19×104±1.0×1033.17×104±1.0×1033.15×104±1.0×1033.12×104±1.0×103
      2.00×10−6±8.7×10−92.88×104±1.0×1032.98×104±1.1×1032.96×104±1.1×1032.99×104±1.1×1032.99×104±1.1×103
      2.51×10−6±1.1×10−82.53×104±1.2×1032.53×104±1.3×1032.61×104±1.3×1032.52×104±1.2×1032.59×104±1.3×103
      3.16×10−6±1.4×10−82.79×104±1.1×1032.91×104±1.2×1032.89×104±1.2×1032.93×104±1.2×1032.83×104±1.2×103
      3.98×10−6±1.8×10−82.02×104±2.5×1032.09×104±2.5×1032.06×104±2.5×1032.09×104±2.5×1032.07×104±2.5×103
      5.01×10−6±2.3×10−81.93×104±1.5×1031.98×104±1.5×1031.98×104±1.5×1032.00×104±1.5×1032.00×104±1.5×103
      6.31×10−6±2.9×10−81.96×104±1.1×1031.98×104±1.1×1031.98×104±1.1×1031.97×104±1.1×1032.03×104±1.2×103
      7.94×10−6±3.7×10−81.64×104±3.4×1031.64×104±3.4×1031.70×104±3.6×1031.66×104±3.5×1031.66×104±3.5×103
      1.00×10−5±4.7×10−81.58×104±7.0×1021.58×104±6.9×1021.59×104±7.0×1021.55×104±6.9×1021.61×104±7.3×102
      1.26×10−5±6.4×10−81.37×104±1.2×1031.36×104±1.2×1031.37×104±1.2×1031.41×104±1.3×1031.42×104±1.3×103
      1.58×10−5±8.6×10−81.06×104±1.6×1031.09×104±1.6×1031.10×104±1.6×1031.09×104±1.6×1031.14×104±1.7×103
      2.00×10−5±1.2×10−78.72×103±1.6×1039.24×103±1.7×1038.99×103±1.6×1038.68×103±1.6×1039.10×103±1.7×103
      2.51×10−5±1.6×10−71.00×104±5.7×1021.01×104±5.6×1021.01×104±5.6×1021.00×104±5.6×1029.92×103±5.7×102
      Continued on next page

      Table A8.  The differential cross sections of the 10B(n, α1)7Li reaction in the laboratory reference system.

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