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Prediction of the cross-sections of isotopes produced in deuteron-induced spallation of long-lived fission products

  • The spallation cross-section data for the long-lived fission products (LLFPs) are scarce but required for the design of accelerator driven systems. In this paper, the isospin dependent quantum molecular dynamics model and the statistical code GEMINI are applied to simulate deuteron-induced spallation in the energy region of GeV/nucleon. By comparing the calculations with the experimental data, the applicability of the model is verified. The model is then applied to simulate the spallation of 90Sr, 93Zr, 107Pd, and 137Cs induced by deuterons at 200, 500 and 1000 MeV/nucleon. The cross-sections of isotopes, the cross-sections of long-lived nuclei, and the reaction energy are presented. Using the above observables, the feasibility of LLFP transmutation by spallation is discussed.
  • The study on exotic hadron states elucidates the property of QCD and also explores a new form of matter. However, the existence of exotic states remains an open question. Most of the exotic states can be classified in the conventional quark model, and the properties of the exotic states can be explained in the framework of the quark model with a few improvements. For example, the well-known exotic state [1-4], X(3872), can be explained as the traditional cˉc state with a large component D¯D+DˉD in our unquenched quark model [5].

    In fact, several experimental collaborations have been searching for exotic states for the past two decades. In 2016, the D0 Collaboration observed a narrow structure, which is denoted as X(5568), in the B0sπ± invariant mass spectrum with a 5.1σ significance [6]. Owing to the B0sπ± decay mode, X(5568) was interpreted as the sˉduˉb(sˉudˉb) tetraquark state. However, it is difficult to determine the candidate for X(5568) in various approaches, if the requirements for ordinary hadrons can be described well in the approaches [7]. In our chiral quark model calculation, all the possible candidates for X(5568) are scattering states [8], while we predicted one shallow bound state [9, 10], BˉK with 6.2 GeV, in the IJP=00+bsˉqˉq system. Indeed, other experimental collaborations did not find the existing evidence of X(5568) [11]. Recently, the LHCb Collaboration coincidentally reported their observation of the first two fully open-flavor tetraquark states named X0(2900) and X1(2900) in the csˉqˉq system, whose statistical significance is more than 5σ [12]. If these two states are confirmed by other collaborations in the future, the X(2900) could be the first exotic state with four different flavors that cannot be quark-antiquark systems.

    \begin{array}{ll}  M_{X_0(2900)} \hspace*{-3mm} &=  2866\pm 7 \;{\rm MeV}, \nonumber \\ \Gamma_{X_0(2900)} \hspace*{-3mm} &=  57\pm 3\; {\rm MeV}, \nonumber \\ M_{X_1(2900)} \hspace*{-3mm} &=  2904\pm 5\; {\rm MeV}, \nonumber \\ \Gamma_{X_1(2900)} \hspace*{-3mm} &=  110\pm 12 \;{\rm MeV}.  \end{array}

    Owing to the report on X(2900), several possible candidates have emerged to elucidate X(2900) in different frameworks [13-30], and most of them can be divided into two categories: dimeson and diquark structures. Xue et al. obtained a 0+ D¯K resonance that can elucidate the X0(2900) in the quark delocalization color screening model [13], and via the qBSE approach, He et al. also arrived at the same conclusion [14]. Karliner et al. approximately estimated the diquark structure of csˉqˉq, and obtained a resonance that can be assigned as the candidate for X0(2900) [22]. In addition, a resonance with JP=0+ of bsˉqˉq system with a mass of 6.2 GeV was also proposed. In the framework of the QCD sum rule, Chen et al. assigned X0(2900) as a 0+ D¯K molecular state, while X1(2900) was assigned as a 1 csˉqˉq diquark state [23]. However, using a similar method, Zhang regarded both X0(2900) and X1(2900) as diquark states [24]. In addition, before the report on X(2900), Agaev et al. [25] obtained a resonance with 2878±128 MeV in the 0+ csˉqˉq system. A few studies have also disfavored these findings. Liu et al. hypothesized that the two rescattering peaks may simulate the X(2900) without introducing genuine exotic states [28]. Burns et al. interpreted the X(2900) as a triangle cusp effect originating from D¯K and D1ˉK interactions [29]. Based on an extended relativized quark model, the study reported in [30] determined four resonances, 2765, 3055, 3152, and 3396MeV, and none of them could be the candidate for X0(2900) in the 0+ csˉqˉq system.

    In fact, both molecular D¯K and diquark csˉqˉq configurations have energies approximate to the mass of X(2900). System dynamics should determine the preferred structure. Hence, the structure mixing calculation is required. Owing to the high energy of X(2900), the combinations of the excited states of cˉq and sˉq are possible. More importantly, these states will couple with the decay channels, DˉK, D¯K, and DˉK. Do these states survive after the coupling? Owing to the finite space used in the calculation, a stability method has to be employed to identify the genuine resonance. In this study, a structure mixing calculation of meson-meson and diquark-antidiquark structures is performed in the framework of the chiral quark model via the Gaussian expansion method (GEM), and the excited states of subclusters are included. Therefore, four kinds of states with quantum numbers, IJP=00± and 01±, are investigated. Owing to the lack of orbital-spin interactions in our calculation, we adopt the symbol 2S+1LJ to denote P-wave excited states. Accordingly, 0 and 1 may be expressed as 1P1, 3PJ, and 5PJ. To determine the genuine resonance, the real-scaling method [31] is adopted.

    This paper is organized as follows. In Sec. II, the chiral quark model, real-scaling method, and the wave-function of csˉqˉq systems are presented. The numerical results are provided in Sec. III, and the last section summarizes the study.

    The constituent chiral quark model (ChQM) has been successful both in describing the hadron spectra and hadron-hadron interactions. Details on the model can be found in Refs. [32, 33]. The Hamiltonian of ChQM for the four-quark system is written as

    H=4i=1mi+p2122μ12+p2342μ34+p212342μ1234+4i<j=1(VGij+VCij+χ=π,K,ηVχij+Vσij),

    (1)

    where mi is the constituent mass of the i-th quark (antiquark), and μ is the reduced mass of two interacting quarks or quark-clusters.

    μij=mimjmi+mj,μ1234=(m1+m2)(m3+m4)m1+m2+m3+m4,pij=mjpimipjmi+mj,p1234=(m3+m4)p12(m1+m2)p34m1+m2+m3+m4.

    (2)

    The quadratic form of color confinement is used here:

    VCij=(acr2ijΔ)λciλcj.

    (3)

    The effective smeared one-gluon exchange interaction takes the form

    VGij=αs4λciλcj[1rij2π3mimjσiσjδ(rij)],δ(rij)=erij/r0(μij)4πrijr20(μij),r0(μij)=r0μij.

    (4)

    The last piece of the potential is the Goldstone boson exchange, which originates from the effects of the chiral symmetry spontaneous breaking of QCD in the low-energy region,

    Vπij=g2ch4πm2π12mimjΛ2πΛ2πm2πmπvπij3a=1λaiλaj,VKij=g2ch4πm2K12mimjΛ2KΛ2Km2KmKvKij7a=4λaiλaj,Vηij=g2ch4πm2η12mimjΛ2ηΛ2ηm2ηmηvηij×[λ8iλ8jcosθPλ0iλ0jsinθP],Vσij=g2ch4πΛ2σΛ2σm2σmσ[Y(mσrij)ΛσmσY(Λσrij)],vχij=[Y(mχrij)Λ3χm3χY(Λχrij)]σiσj,Y(x)=ex/x.

    (5)

    In the above expressions, σ indicates the SU(2) Pauli matrices; λ and λc are the SU(3) flavor and color Gell-Mann matrices, respectively; andαs is an effective scale-dependent running coupling,

    αs(μij)=α0ln[(μ2ij+μ20)/Λ20].

    (6)

    The model parameters are determined by the requirement that the model can accommodate all the ordinary mesons, from light to heavy, considering only a quark-antiquark component. Details on the meson spectrum fitting process can be found in the work by Vijande et al. [32]. Here, we provide a brief introduction of the process. First, the mass parameters, ms,s=σ,η,κ,π take their experimental values, while the cut-off parameters, Λs,s=σ,η,κ,π, are fixed at typically used values [32]. Second, the chiral coupling constant gch can be obtained from the experimental value of the πNN coupling constant

    g2ch4π=925g2πNN4πm2udm2N.

    Finally, our confinement potential takes the ordinary quadratic form, which differs from the expression used in Ref. [32], where the effect of sea quark excitation is considered. Here, we leave the effect of sea quark excitation to the unquenched quark model. All of the parameters are presented in Table 1, and the masses of mesons obtained are presented in Table 2.

    Table 1

    Table 1.  Quark Model Parameters (mπ=0.7 fm−1, mσ=3.42 fm−1, mη=2.77 fm−1, and mK=2.51 fm−1).
    Quark masses mu=md/MeV 313
    ms/MeV 536
    mc/MeV 1728
    mb/MeV 5112
    Goldstone bosons Λπ=Λσ/fm1 4.2
    Λη=ΛK/fm1 5.2
    g2ch/(4π) 0.54
    θp/() −15
    Confinement ac/MeV 101
    Δ/MeV −78.3
    μc/MeV 0.7
    OGE α0 3.67
    Λ0/fm1 0.033
    μ0/MeV 36.976
    ˆr0/MeV 28.17
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    Table 2

    Table 2.  Meson spectrum (unit: MeV).
    D D DJ D1
    QM 1862.6 1980.5 2454.7 2448.1
    exp 1867.7 2008.9 2420.0 2420.0
    K K KJ K1
    QM 493.9 913.6 1423.0 1400.0
    exp 495.0 892.0 1430.0 1427.0
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    The csˉqˉq system has two structures, meson-meson and diquark-antidiquark, and the wave function of each structure comprises four parts: orbital, spin, flavor, and color wave functions. In addition, the wave function of each part is constructed by coupling two sub-clusters wave functions. Therefore, the wave function for each channel will be the tensor product of the orbital (|Ri), spin (|Sj), color (|Ck) and flavor (|Fl) components,

    |ijkl=A|Ri|Sj|Ck|Fl,

    (7)

    A is the antisymmetrization operator.

    1   orbital wave function

    The orbital wave function comprises two sub-clusters orbital wave functions and the relative motion wave function between two sub-clusters,

    Φ(r)=CLl1,l2,l3Ψl1(r12)Ψl2(r34)Ψl3(r1234).

    (8)

    The negative parity requires the P-wave angular momentum, and only one orbital angular momentum is set to 1. Accordingly, the following combinations are obtained: l1=1, l2=0,l3=0 as "|R1," l1=0,l2=1,l3=0 as "|R2," and l1=0,l2=0,l3=1 as "|R3." However, for the positive parity state, we set all orbital angular momentum to 0, l1=0,l2=0,l3=0 as "|R0."

    In the GEM, the radial part of the spatial wave function is expanded by Gaussians:

    R(r)=nmaxn=1cnψGnlm(r),

    (9)

    ψGnlm(r)=Nnlrleνnr2Ylm(ˆr),

    (10)

    where Nnl, which is the normalization constant, is expressed as

    Nnl=[2l+2(2νn)l+32π(2l+1)]12.

    (11)

    cn are the variational parameters, which are determined dynamically. The Gaussian size parameters are selected according to the following geometric progression

    νn=1r2n,rn=r1an1,a=(rnmaxr1)1nmax1.

    (12)

    The advantage of the geometric progression is that it enables the optimization of the ranges using just a small number of Gaussians. The GEM has been successfully used in the calculation of few-body systems [34].

    2   spin wave function

    Because there is no difference between the spin of quark and antiquark, the wave functions of the meson-meson structure has the same form as that of the diquark-antidiquark structure.

    |S1=χσ10=χσ00χσ00,|S2=χσ20=13(χσ11χσ11χσ10χσ10+χσ11χσ11),|S3=χσ11=χσ00χσ11,|S4=χσ21=χσ11χσ00,|S5=χσ31=12(χσ11χσ10χσ10χσ11),|S6=χσ12=χσ11χσ11.

    (13)

    Where the subscript of "χσiS" denotes the total spin of the tretraquark, and the superscript is the index of the spin function with a fixed S.

    3   flavor wave function

    The total flavor wave functions can be written as,

    |F1=12(cˉusˉdcˉdsˉu),|F2=12(csˉuˉdcsˉdˉu).

    (14)

    Here, |F1 (|F2) denotes the flavor wave function for the molecular (diquark-antidiquark) structure.

    4   color wave function

    The colorless tetraquark system has four color structures, including 11, 88, 3ˉ3, and 6ˉ6,

    |C1=χm111=19(ˉrrˉrr+ˉrrˉgg+ˉrrˉbb+ˉggˉrr+ˉggˉgg+ˉggˉbb+ˉbbˉrr+ˉbbˉgg+ˉbbˉbb),

    |C2=χm288=212(3ˉbrˉrb+3ˉgrˉrg+3ˉbgˉgb+3ˉgbˉbg+3ˉrgˉgr+3ˉrbˉbr+2ˉrrˉrr+2ˉggˉgg+2ˉbbˉbbˉrrˉggˉggˉrrˉbbˉggˉbbˉrrˉggˉbbˉrrˉbb).|C3=χd1ˉ33=36(rgˉrˉgrgˉgˉr+grˉgˉrgrˉrˉg+rbˉrˉb,rbˉbˉr+brˉbˉrbrˉrˉb+gbˉgˉbgbˉbˉg+bgˉbˉgbgˉgˉb),|C4=χd26ˉ6=612(2rrˉrˉr+2ggˉgˉg+2bbˉbˉb+rgˉrˉg+rgˉgˉr+grˉgˉr+grˉrˉg+rbˉrˉb+rbˉbˉr+brˉbˉr+brˉrˉb+gbˉgˉb+gbˉbˉg+bgˉbˉg+bgˉgˉb).

    (15)

    To write down the wave functions easily for each structure, the different orders of particles are adopted. However, when coupling the different structure, the same order of particles should be used.

    5   total wave function

    In this study, we investigated all possible candidates for X(2900) in the csˉqˉq system. The antisymmetrization operators are different for different structures. For the csˉqˉq system, the antisymmetrization operator becomes

    A=1(34)

    (16)

    for diquark-antidiquark, and

    A=1(24)

    (17)

    for the meson-meson structure. After applying the antisymmetrization operator, some wave function will vanish, which means that the states are forbidden. All of the allowed channels are presented in Table 3. The subscript "8" denotes the color octet subcluster, the superscript of the diquark/antidiquark is the spin of the subcluster, and the subscript is the color representation of subcluster, 3, ˉ3, 6 and ˉ6 , which denote the color triplet, anti-triplet, sextet, and anti-sextet, respectively.

    Table 3

    Table 3.  All of the allowed channels. We adopt |ijkl to donate different states. "i,j,k,l" are indices that denote the orbital, spin, flavor, and color wave functions, respectively.
    csˉqˉq
    |ijkl 3PJ |ijkl 1P1 |ijkl 5PJ
    |1311 D1¯K |1111 D1ˉK |1611 DJ¯K
    |1312 [D1]8[¯K]8 |1112 [D1]8[ˉK]8 |1612 [DJ]8[¯K]8
    |1411 DJˉK |1211 DJ¯K |2611 D¯KJ
    |1412 [DJ]8[ˉK]8 |1212 [DJ]8[¯K]8 |2612 [D]8[¯KJ]8
    |1511 DJ¯K |2111 D¯K1 |3611 (D¯K)P
    |1512 [DJ]8[¯K]8 |2112 [D]8[¯K1]8 |3612 ([D]8[¯K]8)P
    |2311 D¯KJ |2211 D¯KJ |1624 [cs]1,P6[ˉqˉq]1ˉ6
    |2312 [D]8[¯KJ]8 |2212 [D]8[¯KJ]8 |2623 [cs]13[ˉqˉq]1,Pˉ3
    |2411 D¯K1 |3111 (DˉK)P |3624 ([cs]16[ˉqˉq]1ˉ6)P
    |2412 [D]8[¯K1]8 |3112 ([D]8[ˉK]8)P |ijkl 1+
    |2511 D¯KJ |3211 (DˉK)P |0311 D¯K
    |2512 [D]8[¯KJ]8 |3212 ([D]8[ˉK]8)P |0312 [D]8[¯K]8
    |3311 (DˉK)P |1123 [cs]0,P3[ˉqˉq]0ˉ3 |0411 DˉK
    |3312 ([D]8[ˉK]8)P |1224 [cs]1,P6[ˉqˉq]1ˉ6 |0412 [D]8[ˉK]8
    |3411 (DˉK)P |2124 [cs]06[ˉqˉq]1,Pˉ6 |0511 D¯K
    |3412 ([D]8[ˉK]8)P |2223 [cs]13[ˉqˉq]1,Pˉ3 |0512 [D]8[¯K]8
    |3511 (DˉK)P |3123 ([cs]03[ˉqˉq]0ˉ3)P |0324 [cs]06[ˉqˉq]1ˉ6
    |3512 ([D]8[ˉK]8)P |1324 [cs]1,P6[ˉqˉq]1ˉ6 |0423 [cs]13[ˉqˉq]0ˉ3
    |1324 [cs]0,P6[ˉqˉq]1ˉ6 |ijkl 0+ |0524 [cs]16[ˉqˉq]1ˉ6
    |1423 [cs]1,P3[ˉqˉq]0ˉ3 |0111 DˉK
    |1524 [cs]1,P6[ˉqˉq]1ˉ6 |0112 [D]8[ˉK]8
    |2323 [cs]03[ˉqˉq]1,Pˉ3 |0211 D¯K
    |2424 [cs]16[ˉqˉq]0,Pˉ6 |0212 [D]8[¯K]8
    |2523 [cs]13[ˉqˉq]1,Pˉ3 |0123 [cs]03[ˉqˉq]0ˉ3
    |3324 ([cs]06[ˉqˉq]1ˉ6)P |0224 [cs]16[ˉqˉq]1ˉ6
    |3423 ([cs]13[ˉqˉq]0ˉ3)P
    |3524 ([cs]16[ˉqˉq]1ˉ6)P
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    In this section, we present our numerical results. In the calculation of the csˉqˉq system, two structures, the meson-meson and diquark-antidiquark structures, and their coupling are considered. Because the mass of X(2900) is larger than the threshold of the csˉqˉq system, the possible candidates must be resonance states rather than bound states. To verify whether the states survive the coupling to the open channels, DˉK, D¯K, and DˉK, the real-scaling method (RSM) is employed to test stability of these candidates.

    In the JP=0+csˉqˉq system, there are four channels in the meson-meson structure and two channels in the diquark-antidiquark structure (see Table 3). The lowest eigen-energy of each channel is provided in the second column of Table 4. The eigen-energies of the entire channel coupling are presented in the rows that are marked "c.c," and the percentages in the table represent the percent of each channel in the eigen-states with corresponding energies (in the last row of the table). The two lowest eigen-energies and the eigen-energies of approximately 2900 MeV are given. In the channel coupling calculation, we obtain four energy levels, E1(2836), E2(2896), E3(2906) and E4(2936), which could be the candidates for X0(2900). However, the eigen-state with E2(2896) has ~89% of D¯K, and the energy is higher than its threshold, 2894 MeV, and the single-channel calculation of D¯K indicates that the state is unbound, such that it should be in a D¯K scattering state rather than a resonance; hence, E9(2906) does satisfy this. However, both E1(2837) and E4(2936) have more than 30% of the diquark structure, which indicates that the two states may be in resonance states. The stability of these states have to be verified when assigning these resonances to be the candidate for X0(2900).

    Table 4

    Table 4.  Results for IJP=0+,1+ states ("c.c." means channel coupling).
    0+csˉqˉq
    s.c. 1st 2nd 7th 8th 9th 10th
    DˉK 2357.0 90.1% 99.4% 61.9% 6.1% 24.0% 26.2%
    [D]8[ˉK]8 3098.2 0.3% 0.0% 1.5% 0.4% 0.9% 4.2%
    D¯K 2895.8 0.5% 0.0% 0.3% 88.8% 58.8% 28.9%
    [D]8[¯K]8 2863.7 1.5% 0.1% 2.6% 0.1% 1.5% 0.4%
    [cs]03[ˉqˉq]0ˉ3 2656.5 6.9% 0.1% 7.3% 0.3% 10.1% 0.9%
    [cs]16[ˉqˉq]1ˉ6 2965.7 0.7% 0.4% 26.3% 4.4% 4.9% 29.8%
    c.c. 2340.1 2358.9 2836.3 2896.7 2906.9 2935.8
    1+csˉqˉq
    s.c. 1st 10th 11th 12th 13th 14th
    D¯K 2777.6 0.3% 25.4% 0.5% 0.4% 0.2% 9.3%
    [D]8[¯K]8 3111.8 0.4% 3.4% 1.1% 1.2% 1.7% 0.4%
    DˉK 2475.3 87.4% 34.2% 30.1% 55.2% 63.7% 54.8%
    [D]8[ˉK]8 3110.7 0.4% 1.6% 0.6% 1.8% 1.2% 0.9%
    D¯K 2895.9 0.3% 0.6% 52.5% 11.1% 3.7% 0.9%
    [D]8[¯K]8 3005.0 1.1% 0.9% 1.5% 2.8% 3.4% 4.6%
    [cs]06[ˉqˉq]1ˉ6 3112.3 0.1% 0.4% 1.2% 7.3% 4.8% 19.7%
    [cs]13[ˉqˉq]0ˉ3 2690.6 9.8% 0.5% 1.2% 0.8% 4.5% 5.5%
    [cs]16[ˉqˉq]1ˉ6 3040.4 0.1% 16.1% 11.4% 19.4% 17.0% 3.9%
    c.c. 2464.3 2857.1 2896.3 2904.2 2920.3 2941.7
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    Due to three combinations of spin in the JP=1+csˉqˉq system, more channels are presented in Table 3. Consequently, five energy levels near X(2900), which include E5(2857), E6(2896), E7(2904), E8(2920), and E9(2941) emerge in the channel coupling calculation. Similar to the JP=0+ case, the E6(2896) and E7(2904) are dominated by the meson-meson scattering states. However, both E8(2920) and E9(2941) have almost 28% of the diquark structure, which is beneficial to the existence of resonance.

    For the P-wave excited csˉqˉq system, the states are denoted as 1P1, 3PJ(J=0,1,2), and 5PJ(J=1,2,3). The numerical results are presented in Table 5. The channels with negligible percentages are omitted. Because the present calculation only includes the central forces, the states with the same spin are degenerate. Consequently, the P-wave csˉqˉq threshold D1ˉK and (DˉK)P is close to X(2900), and X(2900) may be molecular states. Because every sub-cluster could be the P-wave excited state, the number of channels in this case is large. Therefore, the effects of the channels may play an important role in the formation of X(2900).

    Table 5

    Table 5.  Results for JP=0,1 ("c.c." represents channel coupling).
    1P1csˉqˉq
    s.c. 1st 6th 7th 8th 9th 10th
    D1ˉK 2943.3 0.0% 1.6% 0.0% 0.1% 2.1% 83.9%
    [D1]8[ˉK]8 3554.8 0.0% 0.6% 0.0% 0.0% 0.4% 0.1%
    DJˉK 3369.5 0.0% 0.1% 0.0% 0.1% 0.4% 0.1%
    [DJ]8[ˉK]8 3340.5 0.0% 1.7% 0.0% 0.0% 2.5% 0.3%
    DˉK1 3264.4 0.0% 6.2% 0.0% 1.5% 2.0% 1.3%
    [D]8[ˉK1]8 3544.4 0.0% 0.5% 0.0% 0.0% 1.1% 0.0%
    DˉKJ 3404.8 0.0% 0.1% 0.0% 0.0% 0.8% 0.0%
    [D]8[ˉKJ]8 3334.2 0.0% 0.9% 0.0% 0.0% 0.5% 0.1%
    (DˉK)P 2359.8 100.0% 48.0% 2.0% 10.0% 24.5% 6.9%
    (DˉK)P 2897.5 0.0% 0.1% 98.0% 74.0% 43.8% 0.9%
    [cs]03[ˉqˉq]0ˉ3 3030.1 0.0% 3.3% 0.0% 0.0% 8.6% 1.2%
    [cs]16[ˉqˉq]1ˉ6 3279.4 0.0% 29.9% 0.0% 9.8% 6.6% 4.3%
    [cs]06[ˉqˉq]1ˉ6 3483.4 0.0% 2.1% 0.0% 0.0% 5.2% 0.4%
    [cs]13[ˉqˉq]1ˉ3 3621.6 0.0% 4.5% 0.0% 1.5% 1.5% 0.3%
    c.c. 2359.8 2873 2897 2908 2932 2943.1
    3PJcsˉqˉq
    s.c. 1st 9th 10th 11th 12th 13th
    D1ˉK 3363.6 0.0% 0.9% 0.0% 0.0% 0.1% 0.2%
    [D1]8[ˉK]8 3551.3 0.0% 1.2% 0.0% 0.9% 0.5% 2.5%
    DJˉK 2950.0 0.0% 0.2% 0.0% 0.1% 0.9% 3.1%
    [DJ]8[ˉK]8 3556.2 0.0% 0.8% 0.0% 0.0% 2.1% 0.5%
    DJˉK 3370.2 0.0% 0.1% 0.0% 0.0% 0.1% 0.0%
    [DJ]8[ˉK]8 3448.5 0.0% 4.4% 0.0% 0.0% 0.1% 0.3%
    DˉKJ 3287.4 0.0% 5.9% 0.0% 0.1% 0.2% 0.4%
    [D]8[ˉKJ]8 3543.9 0.0% 0.1% 0.0% 0.1% 0.3% 0.5%
    DˉK1 3382.8 0.0% 0.8% 0.0% 0.4% 5.8% 5.8%
    [D]8[ˉK1]8 3539.9 0.0% 0.2% 0.0% 0.0% 0.3% 0.3%
    DˉKJ 3405.4 0.0% 0.1% 0.0% 0.5% 0.1% 0.9%
    [D]8[ˉKJ]8 3434.9 0.0% 0.8% 0.0% 0.0% 0.4% 0.5%
    (DˉK)P 2779.9 0.0% 49.8% 0.1% 0.4% 2.6% 2.9%
    (DˉK)P 2477.8 100.0% 6.7% 0.1% 3.1% 32.7% 39.3%
    (DˉK)P 2897.9 0.0% 0.3% 99.5% 89.0% 15.3% 3.6%
    [cs]0,P6[ˉqˉq]1ˉ6 3372.1 0.0% 12.7% 0.3% 0.2% 7.0% 0.9%
    [cs]1,P3[ˉqˉq]0ˉ3 3037.3 0.0% 5.5% 0.0% 0.6% 0.6% 2.0%
    [cs]1,P6[ˉqˉq]1ˉ6 3327.4 0.0% 3.1% 0.0% 2.7% 25.7% 30.6%
    [cs]03[ˉqˉq]1,Pˉ3 3625.5 0.0% 2.0% 0.0% 0.1% 0.8% 0.2%
    [cs]16[ˉqˉq]0,Pˉ6 3477.1 0.0% 3.3% 0.0% 0.4% 0.4% 1.2%
    [cs]13[ˉqˉq]1,Pˉ3 3640.1 0.0% 0.8% 0.0% 0.3% 3.4% 4.2%
    c.c. 2477.8 2867 2897 2908 2928 2944
    5PJcsˉqˉq
    s.c. 1st 2nd 3rd
    DJˉK 3370.2 0.0% 0.1% 0.1%
    [DJ]8[ˉK]8 3653.3 0.0% 0.7% 3.0%
    DˉKJ 3405.4 0.0% 0.6% 3.8%
    [D]8[ˉKJ]8 3649.8 0.0% 0.1% 0.5%
    (DˉK)P 2897.8 100.0% 96.5% 71.2%
    [cs]1,P6[ˉqˉq]1ˉ6 3413.2 0.0% 1.7% 17.6%
    [cs]13[ˉqˉq]1,Pˉ3 3675.6 0.0% 0.3% 2.9%
    c.c. 2897.8 2909 2933
    DownLoad: CSV
    Show Table

    For the 1P1 system, there are five energy levels near the X(2900), E10(2873), E11(2897), E12(2908), E13(2932), and E14(2943) that emerged in the channel coupling calculation. Obviously, owing to both D1ˉK and (D¯K)P being scattering states, the E11(2897) with 98% (D¯K)P may be a scattering state. Furthermore, E12(2908) and E14(2943) also have large scattering state percentages; hence, they are not possible candidate for X(2900). In contrast, E10(2873) with 40% diquark structure may be good candidate for X(2900). Regarding the E13(2932) state with only 20% diquark structure, it seems impossible for it to be the resonance; hence, further calculations are required.

    Similar to the 1P1 case, there are five energy levels, E15(2867), E16(2897), E17(2908), E18(2928) and E19(2944) in the 3PJcsˉqˉq system. E16(2897) and E17(2908) may be scattering states that would decay to (D¯K)P threshold, while E18(2928) and E19(2944) may be possible resonances of X(2900). Regarding the 5PJcsˉqˉq system, the lowest single energy level is the P-wave (D¯K)P, and the lowest energy level is the scattering state. In contrast, the second energy level is DJ¯K with 3370MeV is significantly larger than X(2900). Thus, all of the resonances in the 5PJcsˉqˉq system may be unsuitable for elucidating X(2900).

    In the last subsection, we obtain several possible candidates for X(2900) owing to the structures mixing. However, the LHCb only determined two resonances approximate to 2900 MeV, and the number of candidates may be too rich for X(2900). There are two reasons why our model provides so many candidates. First, we simultaneously consider two different structures in our calculation, which results in molecular and diquark energies filling our energy spectrum. Second, the calculation is performed in a finite space, the behavior of scattering states is similar to that of bound states. Calculations in finite spaces always offer discrete energy levels. Consequently, to check if these states are genuine resonances, the real-scaling method [31] is employed. In this method, the Gaussian size parameters rn for the basis functions between two sub-clusters for the color-singlet channels are scaled by multiplying a factor α, i.e. rnαrn. Then, any continuum state will fall off towards its threshold. A resonant state should not be affected by the variation of α when it stands alone, the coupling to the continuum indicates that the resonance would act as an avoid-crossing structure, as presented in the Fig. 1. The top line represents a scattering state, which would decay to the corresponding threshold. However, the down line, resonance line, would interact with the scattering line, which could result in an avoid-crossing structure. The emergence of the avoid-crossing structure is because the energy of scattering states will get close to the energy of the genuine resonance, with an increase in the scaling factor, and the coupling will become stronger. The avoid-crossing structure is a general property of interacting two-level systems. If the avoid-crossing structure can be repeated with the increase in α, the avoid-crossing structure may be a genuine resonance, and the width can be determined by the following formula [31]:

    Figure 1

    Figure 1.  Resonance shape in the real-scaling method.

    Γ=4V(α)(kr×kc)|krkc|.

    (18)

    V(α) is the minimal energy difference, while kr indicates the slope of the resonance state, and kc represents slope of scattering state.

    The real-scaling results for the positive (Table 4) and negative parity states (Table 5) are illustrated in Figs. 2-6. Here, we only focus on the energy range from 2800 to 3000 MeV because we are interested in the candidates for X(2900); only the D¯K threshold is relevant in our calculation, which is marked with a red line. In Fig. 2, the resonance E1(2836) rapidly falls to the lowest threshold, DˉK with the spaces increases, and both E3(2906) and E4(2936) would decay to the D¯K channel. However, the E4(2936) may combine with a higher energy level into one avoid-crossing structure, and the structure will be repeated at α=2.2, which indicates the existence of a resonance R0(2914), a possible candidate for X0(2900). In the 0+ case, the E4(2936) serves as a resonance level, because it has 30% percent of the diquark structure. According to Eq. (18), we estimate its width to be approximately 42 MeV. In the 1+ case (Fig. 3), there are three possible resonances, E7(2904), E8(2920), and E9(2941), and they all have approximately 30% of the diquark structure. Consequently, the shape of the figure may be very complex. Based on the requirement of the repetitivenes of the avoid-crossing structure, we select one possible resonance, R1(2906). However, the width of R1(2906), which is 29 MeV, may be unsuitable for X1(2900), which has a width of 110 MeV.

    Figure 2

    Figure 2.  (color online) Energy spectrum of 1S0csˉqˉq system.

    Figure 4

    Figure 4.  (color online) Energy spectrum of 1P1csˉqˉq system.

    Figure 5

    Figure 5.  (color online) Energy spectrum of 3PJcsˉqˉq system.

    Figure 6

    Figure 6.  (color online) Energy spectrum of 5PJ.

    Figure 3

    Figure 3.  (color online) Energy spectrum of 3S1csˉqˉq system.

    Now, we consider the negative parity states presented in Table 5. Owing to the threshold of the p-wave D meson and K meson close to X(2900), the resonance states may couple to the scattering states strongly, and the pattern may be more complicate. Similar to the 00+csˉqˉq system, the lines with E12(2908), E13(2932) fall to the threshold of D¯K, and E14(2943) falls within the threshold of D1ˉK (see Fig. 4). However, the state E13(2932) state with 22% of the diquark structure indicates an avoid-crossing structure, which may be a possible resonance, R1(2912) with Γ=10 MeV. Regarding the E10(2873) state, it rapidly decays to the lower threshold and is not a possible candidate for X(2900) for the lower energy. Because the 3PJcsˉqˉq system has 21 channels including two thresholds near X(2900), the pattern of the 3PJcsˉqˉq system is very complex. We determine two possible candidates, RJ(2920) with Γ=9 MeV and RJ(2842) with Γ=24 MeV, for X(2900) (Fig. 5). Finally, the channels in the 5PJcsˉqˉq system have higher energies than X(2900) , and DˉK is a p-wave excited scattering state. From the figure, it can be observed that no resonance survives the coupling to the p-wave scattering state (see Fig. 6). Although the energies of the above resonance states with negative parity are close to the mass of X1(2900), the larger width of X1(2900) prevents the formation of a conclusion.

    In the framework of the chiral constituent quark model, we systematically studied csˉqˉq states to determine the candidates for X(2900), which were reported by the LHCb Collaboration recently. Both the molecular structure, as well as the diquark-antidiquark, with all the possible color, flavor, and spin configurations are considered in the present calculation. The obtained results indicate that there are several states with energies of approximately 2900 MeV in the csˉqˉq system after structure-mixing. These superabundant resonances may be triggered by the structure mixing and finite calculation space. Therefore, the real-scaling method, a stablization method, is adopted to identify the genuine resonances. We obtained five possible resonances, R0(2914) with Γ=42 MeV, R1(2906) with Γ=29 MeV, R1(2912) with Γ=10 MeV, RJ(2920) with Γ=9 MeV, and RJ(2842) with Γ=24 MeV. All the resonances obtained are diquark-antidiquark states. For X0(2900), the resonance, R0(2904) with Γ=42 MeV, may be a good candidate. In this case, X0(2900) would be the positive parity state. However, it is possible to assign candidates for X1(2900) based on energy, but the decay width prevent us from making a definite conclusion. Hence, more information on X(2900) and further studies are required.

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    [15] C. Zhang, S. H. Zhang, Y. He et al, Chinese Physics C, 39: 117002 (2015) doi: 10.1088/1674-1137/39/11/117002
    [16] Q. L. Peng, B. Wang, Y. Chen et al, Chinese Physics C, 38: 037002 (2014) doi: 10.1088/1674-1137/38/3/037002
    [17] L. J. Wen, S. H. Zhang, Y. M. Li et al, Chinese Physics C, 40: 027004 (2016) doi: 10.1088/1674-1137/40/2/027004
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    [19] C. Meng, J. Y. Tang, S. L. Pei et al, Chinese Physics C, 39: 097002 (2015) doi: 10.1088/1674-1137/39/9/097002
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    [28] J. C. David, The European Physical Journal A, 51: 68 (2015) doi: 10.1140/epja/i2015-15068-1
    [29] K. Abdel-Waged, N. Felemban, T. Gaitanos et al, Phys. Rev. C, 81: 014605 (2010) doi: 10.1103/PhysRevC.81.014605
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2. Wang, J.-Z., Lin, Z.-Y., Wang, B. et al. Double pole structures of X1 (2900) as the P -wave D ¯ ∗k∗ resonances[J]. Physical Review D, 2024, 110(11): 114003. doi: 10.1103/PhysRevD.110.114003
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Guanming Yang, Suyang Xu, Mengting Jin and Jun Su. Prediction of the cross-sections of isotopes produced in deuteron-induced spallation of long-lived fission products[J]. Chinese Physics C, 2019, 43(10): 104101. doi: 10.1088/1674-1137/43/10/104101
Guanming Yang, Suyang Xu, Mengting Jin and Jun Su. Prediction of the cross-sections of isotopes produced in deuteron-induced spallation of long-lived fission products[J]. Chinese Physics C, 2019, 43(10): 104101.  doi: 10.1088/1674-1137/43/10/104101 shu
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Prediction of the cross-sections of isotopes produced in deuteron-induced spallation of long-lived fission products

  • Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-sen University, Zhuhai 519082, China

Abstract: The spallation cross-section data for the long-lived fission products (LLFPs) are scarce but required for the design of accelerator driven systems. In this paper, the isospin dependent quantum molecular dynamics model and the statistical code GEMINI are applied to simulate deuteron-induced spallation in the energy region of GeV/nucleon. By comparing the calculations with the experimental data, the applicability of the model is verified. The model is then applied to simulate the spallation of 90Sr, 93Zr, 107Pd, and 137Cs induced by deuterons at 200, 500 and 1000 MeV/nucleon. The cross-sections of isotopes, the cross-sections of long-lived nuclei, and the reaction energy are presented. Using the above observables, the feasibility of LLFP transmutation by spallation is discussed.

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    1.   Introduction
    • As it does not release greenhouse gases and chemical pollutants, nuclear energy is one of the low-carbon power sources that can overcome the shortage of energy and solve the environmental problems in the world [1, 2]. However, nuclear energy still faces some challenges. One of them is the highly radioactive long-lived nuclear waste generated during power production [3]. At present, most of the nuclear waste is first kept under surveillance in storage facilities, and subsequently stored permanently in deep underground storage. Due to the risk of leaks and proliferation in biosphere of nuclear waste [4], a new concept of transmutation based on the accelerator driven systems (ADS) has been introduced for disposal of the long-lived nuclear waste. The idea of ADS was first mentioned in the early 1950s [5]. Since 1980s, several countries have started research projects in ADS, including France, Japan, USA, and some other European countries [6], but there has been no successfully constructed ADS in any country yet [7]. In China, the research in ADS began in 1990s [8, 9]. In 2010, the ADS research project was started by the Chinese Academy of Sciences, and the project China initiative Accelerator Driven System (CiADS) was approved in 2015. Its goal is to build a megawatt grade ADS demonstration facility in Huizhou, Guangdong [1018].

      ADS consists of three sub-systems: proton accelerator, spallation target and sub-critical reactor [1922]. As the accelerator and the reactor are coupled by the spallation target, the spallation reaction plays an important role in ADS [2325]. The cross-section data for different target nuclei in a spallation reaction are required for the construction of ADS. However, each set of experimental data is limited and can only be used to verify part of the spallation reaction mechanism owing to the large projectile energy range, the broad target nucleus spectrum and the quantity of output results. It is impossible to cover every type of spallation reaction using only one code. Furthermore, proton and deuteron-induced spallation reactions are considered as promising mechanisms for transmutation of the long-lived fission products (LLFPs) [26]. However, the experimental cross-section data for deuteron-induced spallation of LLFPs are quite rare. The cross-sections of the spallation reactions of 90Sr and 137Cs induced by deuterons at 185 MeV/nucleon were measured in Japan [27]. For projectile energies greater than 200 MeV/nucleon, there are no experimental data. Hence, the development of spallation models is necessary [28].

      Several dynamical models based on a microscopic approach have been developed to simulate the out-of-equilibrium process of spallation. Two examples are the Boltzmann-Uehling-Uhlenbeck models [29, 30], and the quantum molecular dynamics models [3133]. To describe the decay process of spallation, principally evaporation or fission, several statistical models have been developed based on the Weisskopf [34] or Hauser-Feshbach [35] formalisms, such as ABLA [36], GEM [37], GEMINI [38], and SMM [39]. A decade ago, the International Atomic Energy Agency (IAEA) has organized an international benchmark of different spallation models in the world [40]. An effort was made to collect the data and estimate the prediction capacities of these models [28]. The codes in the IAEA benchmark are based on a two-step process, where the excitation stage of the spallation reaction uses dynamical models, and the de-excitation stage is based on the statistical models [40]. The two-step process has been successful in modeling nuclear spallation. However, there are still some observables, such as the production of the intermediate mass fragments (IMFs), where the discrepancy between the results of the models and the experimental data highlight the necessity for further improvement of the existing models. In our previous work, by increasing the dynamics evolution time, a two-step model was developed [41, 42]. In order to improve the description of IMFs, the phase space density constraint was considered, and the dynamical simulation is performed until the excitation energy of hot fragments is below the threshold energy for multi-fragmentation.

      In this paper, we predict the cross-sections of isotopes produced in deuteron-induced spallation of LLFPs in the GeV/nucleon energy region in the framework of isospin dependent quantum molecular dynamics (IQMD) model [43, 44] followed by the GEMINI code [45]. The version of the IQMD model is IQMD-BNU, which was introduced and compared with the other versions in the transport code comparison project [46]. The paper is organized as follows. In Sec. 2, the method is introduced. In Sec. 3, the results are presented. In Sec. 4, a summary is given.

    2.   Theoretical framework

      2.1.   Isospin dependent quantum molecular dynamics model

    • In the IQMD model, each nucleon is described by the wave function in the form of a Gaussian wave packet. To describe the N-body system, the total wave function is the direct product of these Gaussian wave packets. Applying the Wigner transformation of the quantum wave function, the N-body phase-space density is given by the Hamiltonian, which consists of three terms,

      H=T+UCoul+V[ρ(r)]dr,

      (1)

      where T is the kinetic energy and UCoul is the Coulomb potential energy. The third term is the nuclear potential energy density of the asymmetric nuclear matter with density ρ and asymmetry δ

      V(ρ,δ)=α2ρ2ρ0+βγ+1ργ+1ργ0+Csp2(ρρ0)γiρδ2+gs2ρ0[ρ(r)]2,

      (2)

      where ρ0 is the normal density. The parameters used in this paper are α=356.00MeV, β=303.00MeV, γ=7/6, Csp = 38.06 MeV, and γi = 0.75, gs = 120.00 MeV/fm2.

      The time evolution of nucleons in the generated mean-field is governed by Hamilton's equations of motion,

      ˙ri=piH,˙pi=riH.

      (3)

      In addition, the nucleon-nucleon (NN) collisions are included in the IQMD code to simulate the short-range residual interaction. The differential cross-section of the NN collision is the product of three parameters,

      (dσdΩ)i=σfreeifanglifmedi,

      (4)

      where σfree is the cross-section of the NN collision in free space, fangl refers to the angular distribution, and fmed is the in-medium factor. The parametrization used above of the cross-section σfree and the angular distribution fangl are taken from Ref. [47]. The in-medium factor fmed is from Ref. [48]. The subscript i represents different channels of the NN collision: i = pp represents elastic proton-proton scattering, i = nn elastic neutron-neutron scattering, i = np elastic neutron-proton scattering, and i = in inelastic NN collision.

      In order to include the fermionic nature of the N-body system, the method of phase space density constraint and the Pauli blocking, which were introduced in the constrained molecular dynamics (CoMD) model proposed by M. Papa et al. [49], are applied in the IQMD model. According to the phase space density constraint, the integration is performed on a hypercube of volume h3 in the phase space, centered around the i-th nucleon, and the phase space occupancy probability is calculated as,

      ¯fi=nδτn,τiδsn,sih31π33e(rrn)22L(ppn)2L2d3rd3p.

      (5)

      At each time step, the phase space occupancy probability of each nucleon is calculated. The parameter kfcon in the IQMD model is used to estimate the phase space occupancy probability. If fi>kfcon, the many-body elastic scattering is carried out for the i-th nucleon. Moreover, fi is calculated for each NN collision. By taking into account the Pauli blocking, only those NN collisions with fi < 1.0 in the final state are accepted. We choose kfcon = 1.15.

    • 2.2.   Gemini

    • The two-step process in the IAEA benchmark includes the dynamical code, and the statistical code. The dynamical code is used to model the excitation stage of the target nucleus impacted by the incident particle. The emission of heavy fragments and light particles is then simulated by the statistical code. In this work, the model used is different: the decay process of the excited nucleus is included in the IQMD code, so that the dynamical evolution calculated by the IQMD code is longer than in the IAEA benchmark. A parameter Estop is used in the IQMD code to extend the dynamical description of a spallation reaction. After the excitation stage, the simulation in the IQMD code continues until the excitation energy of the heaviest hot fragments of the target nucleus is below the model parameter Estop, when the statistical code GEMINI is turned on. The smaller the value of Estop , the longer is the simulation by the IQMD code. In this work, Estop = 2 MeV/nucleon. The output of the IQMD code, which is the charge, mass number and the excitation energy of each hot fragment, is the input for GEMINI. By using the Monte Carlo method [42], the GEMINI code simulates the sequential decay of hot fragments, including the light-particle evaporation and symmetric fission. The simulation continues until the excitation energy of the hot fragments reaches zero. The partial decay width from the Hauser-Feshbach formalism, is given by,

      ΓJ2(Z1,A1,Z2,A2)=2J1+12πρ0J0+J2l=|J0J2|EBErot0Tl(ε)ρ2×(EBErotε,J2)dε,

      (6)

      where Zi, Ai, Ji and ρi are the charge, mass number, spin and level density. The subscript i takes the value of 0 for the initial fragment, 1 for the emitted light particle, and 2 for the residual fragments. E*, Erot, B and ε refer to the excitation energy, rotational energy of the ground-state configuration, separation energy and kinetic energy. The separation energy B is calculated from the nuclear masses, where the tabulated masses are employed [50, 51].

    3.   Results and discussion
    • Transmutation of LLFPs in nuclear waste by spallation has attracted attention for minimizing radioactive hazard of nuclear waste. However, there is a lack of experimental cross-section data for LLFPs, especially for reactions induced by deuterons at energies greater than 200 MeV/nucleon. Therefore, the above model is applied to predict the cross-sections of LLFPs produced in deuteron-induced spallation reactions.

      Before simulating deuteron-induced spallation reactions, the reliability of the simulations needs to be verified. Figs. 1 and 2 show the cross-sections of isotopes produced in the spallation reactions 136Xe + p at 500 and 1000 MeV/nucleon. The elements are from Nb to Ba for 500 MeV/nucleon, and from As to Ba for 1000 MeV/nucleon. The solid circles in the figures refer to the experimental data and the blue curves to the simulations. The figures show that the simulations agree with the experimental data. The discrepancies mainly appear on both sides of the peaks, where the cross-sections are small. In the zone around the peaks, the cross-sections are more significant, and the simulations agree quite well with the data. Hence, in the case of calculations of transmutation, the discrepancies have only a small influence on the results, and the calculations by the model can be considered as reliable.

      Figure 1.  (color online) Isotope distribution of cross-sections for the residual nuclei (Z from 33 to 56) produced in 136Xe + p at 500 MeV/nucleon. The experimental data, shown as solid circles, are taken from Ref. [52].

      Figure 2.  (color online) Same as Fig. 1 but for the residual nuclei (Z from 43 to 56) with the projectile energy 1000 MeV/nucleon. The experimental data are taken from Ref. [53].

      After verifying the performance of the model, the deuteron-induced spallation reactions of four LLFPs, 90Sr, 93Zr, 107Pd and 137Cs , at 200, 500 and 1000 MeV/nucleon were simulated. As an example, the predictions of the cross-sections of isotopes produced in 2H + 137Cs are shown in Fig. 3. The black circles in the figure correspond to the spallation reaction at 200 MeV/nucleon, the blue squares at 500 MeV/nucleon, and the red triangles at 1000 MeV/nucleon. The cross-sections display a peak near the stable isotope, except for Z > 53. For the elements near the target nucleus, there is another peak near A = 137. For the elements from Se to In, the cross-sections increase with increasing projectile energy. However, the energy dependence becomes weak as the atomic number of the element increases. The calculations overlap for elements from Sn to Ba. During transmutation of LLFPs, large production of noble gases may be harmful for ADS. When noble gases contain long-lived nuclei, there is a risk of radioactive leaks. The cross-sections predicted for the noble gas Kr have a maximum around 10 mb. The predictions suggest that the minima of the cross-sections for spallation of 90Sr, 93Zr is at 1000 MeV/nucleon, and of 107Pd, 137Cs at 200 MeV/nucleon. Hence, in order to reduce the production of long-lived nuclei 81Kr and 85Kr, the projectile energy should be as high as possible for 90Sr + 2H and 93Zr + 2H , and as low as possible for 107Pd + 2H and 137Cs + 2H.

      Figure 3.  (color online) Isotope distribution of cross-sections for the residual nuclei (Z from 34 to 57) produced in 137Cs + 2H spallation at 200, 500 and 1000 MeV/nucleon. The data for 185 MeV/nucleon are also shown.

      The transmutation of LLFPs with ADS aims at burning the long-lived radioactive nuclei by producing short-lived nuclei using the spallation reaction. However, the spallation reaction can also produce long-lived nuclei, reducing the efficiency of transmutation. Hence, production of long-lived nuclei in a spallation reaction should also be considered. The cross-sections of the long-lived nuclei in deuteron-induced spallation are presented in Table 1. For the long-lived nuclei 41Ca to 63Ni , produced in 90Sr + 2H and 93Zr + 2H at 200 Mev/nucleon, the cross-sections increase as the projectile energy increases. The production of long-lived nuclei from 79Se to 93Mo is inversely proportional to the projectile energy in these two spallation reactions. As the projectile energy in the spallation 107Pd + 2H increases, the production of long-lived nuclei from 53Mn to 81Kr increases, and from 85Kr to 101Rh decreases. For 137Cs + 2H, the cross-sections of long-lived nuclei from 60Co to 107Pd and from 126Sn to 133Ba are respectively directly and inversely proportional to the projectile energy. Globally, when the projectile energy increases, the cross-sections of long-lived nuclei in the above spallation reactions first increase until a certain nucleus, and then decrease.

      137Cs 107Pd 93Zr 90Sr
      2005001000200500100020050010002005001000
      53Mn0.090.562.060.321.864.260.402.094.80
      55Fe0.110.662.170.382.084.580.552.615.18
      60Fe0.020.190.610.100.581.020.241.011.51
      60Co0.010.150.610.100.611.800.462.183.700.933.304.62
      59Ni0.010.120.480.121.043.170.833.696.521.294.536.73
      63 Ni0.020.210.770.100.802.080.812.724.132.064.325.52
      79Se0.000.291.100.340.840.991.180.800.812.722.081.98
      81Kr0.010.682.303.625.405.4712.668.707.6517.0813.2411.73
      85Kr0.010.240.760.140.090.110.580.480.423.953.643.39
      90Sr0.000.300.660.030.020.022.472.372.17
      93Zr0.051.071.780.350.260.26
      92Nb0.091.913.666.264.133.941.501.190.63
      94Nb0.091.592.631.140.960.84
      93Mo0.142.374.1014.869.808.35
      97Tc0.203.234.606.135.164.43
      98Tc0.162.433.403.122.762.41
      99Tc0.192.303.032.522.011.93
      101Rh0.585.356.1913.5513.2211.65
      107Pd0.641.851.86
      126Sn0.120.060.03
      125Sb0.580.510.47
      129I4.194.073.89
      134Cs15.6518.7416.52
      135Cs22.3732.4428.98
      133Ba4.913.412.30

      Table 1.  Cross-sections (µb) of the long-lived nuclei in deuteron-induced spallation at 200, 500, and 1000 MeV/nucleon.

      Figure 4 shows the cross-sections of the long-lived and short-lived nuclei produced in the transmutation of 137Cs predicted by the model. The results are compared to the calculated cross-sections from TENDL-2015 [55] and the experimental data for 136Xe. The spallation reaction is induced by protons. The open circles refer to the data for 136Xe, the solid squares to the calculations of our model, and the lines to TENDL-2015. The red color corresponds to the long-lived and the black color to the short-lived nuclei. The cross-sections obtained by TENDL-2015 are only for projectile energies less than 200 MeV/nucleon. Our model gives cross-sections from 50 to 1000 MeV/nucleon. The results of our model agree much better with the experimental data than TENDL-2015. Thus, the model is suitable for predicting the cross-sections of LLFPs produced in transmutation due to the wider range of projectile energies and higher accuracy.

      Figure 4.  (color online) Cross-sections of the long-lived and short-lived nuclei produced in proton-induced spallation as a function of projectile energy. The experimental data for the target nucleus 136Xe are taken from Refs. [52-54]. The calculated cross-sections of the target nucleus 137Cs are from the IQMD+GEMINI model and TENDL-2015 [55].

      The cross-sections of the long-lived and short-lived nuclei produced in deuteron-induced spallation reactions given by the model and TENDL-2015 are shown in Fig. 5. The total number of produced nuclei are shown as function of the projectile energy for the reactions 90Sr + 2H, 93Zr + 2H, 107Pd + 2H and 137Cs + 2H . The results obtained by TENDL-2015 are for projectile energies less than 200 MeV/nucleon, while the results of our model are for the range 200 to 1000 MeV/nucleon. The cross-sections predicted by the model show that the production of long-lived nuclei is one to two orders of magnitude lower than of short-lived nuclei. These results indicate that the transmutation of LLFPs by deuterons has sufficient efficiency for applications. Moreover, the cross-sections of long-lived nuclei produced in these spallation reactions increase with projectile energy. Thus, a lower projectile energy is suggested for transmutation of LLFPs ​​​​with the purpose of reducing the production of long-lived nuclei.

      Figure 5.  (color online) Cross-sections of long-lived and short-lived nuclei as a function of projectile energy. The spallation reactions are 90Sr + 2H, 93Zr + 2H, 107Pd + 2H, and 137Cs + 2H.

      Figure 6 shows the probability distribution of the energy released in the spallation reactions of four LLFPs, 90Sr, 93Zr, 107Pd, and 137Cs , for projectile energies of 200, 500 and 1000 MeV/nucleon. The black circles correspond to the projectile energy of 200 MeV/nucleon, the blue squares to 500 MeV/nucleon, and the red triangles to 1000 MeV/nucleon. The simulations are given by the IQMD+GEMINI model, and are important for determining the temperature field inside ADS. The energies released presented in the figure are all with a negative sign, which indicates that these spallation reactions are endothermic. The probability distribution curves of the four spallation reactions have the same bow shape for the three projectile energies. They all reach their peaks at the absorbed energy of around 100 MeV. Around the peak, the probability of spallation is inversely proportional to the projectile energy. However, when the absorbed energy is greater than 180 MeV, or less than 40 MeV, the probability increases with projectile energy. In general, the area under the probability curve increases with projectile energy, meaning that the energy absorbed in the spallation reaction is higher for a higher projectile energy. Considering the generating efficiency of ADS, a lower projectile energy is recommended.

      Figure 6.  (color online) Probability distribution of energy released in the spallation reactions 90Sr + 2H, 93Zr + 2H, 107Pd + 2H, and 137Cs + 2H for projectile energies 200, 500 and 1000 MeV/nucleon.

    4.   Conclusion
    • Transmutation of long-lived fission products in nuclear waste and the development of accelerator driven systems require a large amount of data for cross-sections of spallation reactions. The isospin dependent quantum molecular dynamics model was applied to study the cross-sections of spallation reactions. The predictions of the model were verified for the spallation reactions 136Xe + p at 500 and 1000 MeV/nucleon, and 137Cs + 2H at 185 MeV/nucleon. The model was then applied to the transmutation of four LLFPs, 90Sr, 93Zr, 107Pd and 137Cs , induced by deuterons. The projectile energy was between 200 and 1000 MeV/nucleon. The predicted cross-sections of isotopes produced in spallation reactions were presented. The results allow to fill in the blank regions in the experimental data, and give some recommendations for the practical cases. For example, to reduce the production of radioactive noble gas Kr, higher projectile energy in 90Sr + d, 93Zr + d , and lower projectile energy in 107Pd + 2H, 137Cs + 2H , are suggested. A comparison of the production of long-lived and short-lived nuclei and the energy released in spallation reactions were also discussed. The production of long-lived nuclei was related to the efficiency of transmutation of LLFPs. The cross-sections of the long-lived nuclei are predicted to be one to two orders of magnitude lower than of the short-lived nuclei. Thus, transmutation of LLFPs could be considerable. The obtained probability of energy released contributes to the calculation of the temperature field inside ADS. The predictions show that the most probable energy absorbed in the four spallation reactions is around 100 MeV, and that the energy absorbed in a spallation reaction is higher for a higher projectile energy. Hence, a lower projectile energy is recommended to increase the generating efficiency of ADS.

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