Introduction

In the next century the world will face the need for new energy sources. Nuclear fusion is seen as a clean source of energy. Fusion will not produce CO2 or SO2 and thus will not contribute to global warming or acid rain [1]. There are not radioactive nuclear waste problems in the fusion reactors. The stress on fusion safety has stimulated worldwide research for fuel cycles other than deuterium–tritium (D–T) driven fusion reactors. The high-current proton accelerators are being researched at Los Alamos National Laboratory (LANL) and other laboratories for accelerator production of tritium (APT), transmuting long-lived radioactive waste into shorter-lived products [accelerator transmutation of waste (ATW)], converting excess plutonium, and producing energy [2]. These accelerators make use of spallation neutrons produced in (p,xn) and (n,xn) nuclear reactions on high-Z targets [3–5]. Through (p,xn) and (n,xn) nuclear reactions, neutrons are produced and are moderated by heavy water in the target region and light water in the blanket region. These moderated neutrons are subsequently captured on 3He, which flows throughout the blanket system, to produce tritium via the (n,p) reaction. The tritium self-sufficiency must be maintained for a commercial fusion power plant. For a self-sustaining fusion reactor, the tritium breeding ratio (TBR) should be greater than 1.05 [6–9].

The nuclear reaction models are frequently needed to provide the estimation of the particle-induced reaction cross-sections, especially if the experimental data are not obtained or on which they are hopeless to measure the cross-sections; due to the experimental difficulty [10–16]. Such predictions can guide the design of the target/blanket configurations and can reduce engineering over design costs. In this study, by using ultra fast neutrons (incident neutron energy >50 MeV), the (n,xt) reactions for target nuclei as 16O, 27Al, 56Fe, 59Co, 208Pb and 209Bi have been investigated. The pre-equilibrium calculations have been done by using the new evaluated hybrid model and geometry dependent hybrid (GDH) model [17]. Equilibrium effects have been calculated according to the Weisskopf–Ewing model [18]. The calculated results have been compared with the experimental data taken from the literature.

Pre-Equilibrium Calculation Models

Pre-equilibrium processes play an important role in nuclear reactions induced by light projectiles with incident energies above about 8–10 MeV. For pre-equilibrium calculations, the hybrid model was formulated by Blann [19, 20],

$$ {\frac{{d\sigma_{\upsilon } (\varepsilon )}}{d\varepsilon }} = \sigma_{R} P_{\upsilon } (\varepsilon ) $$
(1)
$$ P_{\upsilon } (\varepsilon )d\varepsilon = \sum\limits_{\begin{subarray}{l} n = n_{0} \\ \Updelta n = + 2 \end{subarray} }^{{\bar{n}}} {\left[ {{}_{n}\chi_{\upsilon } N_{n} (\varepsilon ,U)/N_{n} (E)} \right]} g_{\nu } d\varepsilon \left[ {\lambda_{c} (\varepsilon )/(\lambda_{c} (\varepsilon ) + \lambda_{ + } (\varepsilon ))} \right]D_{n} $$
(2)

where \( \sigma_{R} \)is the reaction cross section, \( g_{\nu } \) is the single particle level density for particle type ν, \( {}_{n}\chi {}_{\nu } \) is the number of particle type νν (proton or neutron) in n exciton hierarchy, \( P_{\upsilon } (\varepsilon )d\varepsilon \) represents number of particles of the type ν emitted into the unbound continuum with channel energy between ε and ε + dε. The quantity in the first set of square brackets of Eq. (2) represents the number of particles to be found (per MeV) at a given energy ε for all scattering processes leading to an “n” exciton configuration. \( \lambda_{c} (\varepsilon ) \) is emission rate of a particle into the continuum with channel energy ε and \( \lambda_{ + } (\varepsilon ) \) is the intranuclear transition rate of a particle. The second set of square brackets in (2) represents the fraction of the ν type particles at a energy which should undergo emission into the continuum, rather than making an intranuclear transition. The D n represents the average fraction of the initial population surviving to the exciton number being treated. U is the residual nucleus excitation energy, E is the composite system excitation energy (U = E−B ν−ε, where the B ν is the particle binding energy and N n (ε, U) is the number of ways).

In the density dependent version, the geometry dependent hybrid model (GDH) takes into account the density distribution of the nucleus [20]. This means a longer mean free path at the surface of the nucleus because of a lower density, and a limit to the depth of the holes below the Fermi energy. The differential emission spectrum is given in the GDH as

$$ {\frac{{d\sigma_{\upsilon } (\varepsilon )}}{d\varepsilon }} = \pi \mathchar'26\mkern-10mu\lambda^{2} \sum\limits_{\ell = 0}^{\infty } {(2\ell + 1)} T_{\ell \, } P_{\upsilon } (\ell ,\varepsilon ) $$
(3)

where \( \mathchar'26\mkern-10mu\lambda \) is the reduced de Broglie wavelength of the projectile and \( T_{\ell } \) represents transmission coefficient for \( \ell \)th partial wave. The GDH model is made according to incoming orbital angular momentum \( \ell \) in order to account for the effects of the nuclear-density distribution. This leads to increased emission from the surface region of the nucleus, and thus to increased emission of high-energetic particles.

The nuclear density distribution used in the hybrid model is a Fermi density distribution function,

$$ \rho (R_{\ell } ) = \rho_{0} \left[ {\exp (R_{\ell } - C)/0.55 + 1} \right]^{ - 1} $$
(4)

where \( \rho_{0} \) is the density at the center of nucleus, and

$$ C = 1.18A^{1/3} \left[ {1 - 1/(1.18A^{1/3} )^{2} } \right]^{{}} + \mathchar'26\mkern-10mu\lambda $$
(5)

The radius for the \( \ell \)th entrance channel partial was defined by \( R_{\ell } \, = \, \mathchar'26\mkern-10mu\lambda \, ( \, \ell \, + 1/2) \). In the GDH model, the Fermi energies and nuclear densities are defined to impact parameter \( R_{\ell } \) [20].

Results and Discussions

In this study, the triton-emission spectra produced by (n,xt) reactions for target nuclei as 16O, 27Al, 56Fe, 59Co, 208Pb and 209Bi have been investigated between 53.5 and 96 MeV incident neutron energy. The pre-equilibrium calculations have been made using the new evaluated hybrid model and the GDH model [17]. Equilibrium effects have been calculated according to the Weisskopf–Ewing model [18]. In the equilibrium (evaporation) model, the basic parameters are binding energies, inverse reaction cross-section, the pairing energy and the level-density parameters. The equilibrium calculations with Weisskopf–Ewing model don’t include angular momentum effects. In the all equilibrium and pre-equilibrium calculations, the code as ALICE/ASH was used. The ALICE/ASH code is an advanced and modified version of the ALICE codes [17]. In the calculations of the hybrid and GDH model, we have used the initial exciton number as n o  = 3. The initial exciton numbers in the ALICE/ASH code calculations for neutron induced reactions as,

$$ X_{n} = 2{\frac{{\left( {\sigma_{np} /\sigma_{nn} } \right)Z + 2N}}{{2(\sigma_{np} /\sigma_{nn} )Z + 2N}}},\quad X_{p} = 2 - X_{n} $$
(6)

where σxy is the nucleon–nucleon interaction cross-section in the nucleus. The ratio of nucleon–nucleon cross-sections calculated taking into account to Pauli principle and the nucleon motion is parameterized

$$ \sigma_{pn} /\sigma_{pp} = \sigma_{np} /\sigma_{nn} = 1.375 \times 10^{ - 5} T^{2} - 8.734 \times 10^{ - 3} T^{{}} + 2.776 $$
(7)

where T is the kinetic energy of the projectile outside the nucleus. The generalized superfluid [21] has been applied for nuclear level density calculations in the ALICE/ASH code. In details, the other code model parameters can be found in Ref. [17].

The triton-emission spectra produced by (n,xt) reactions for 16O, 27Al, 56Fe, 59Co, 208Pb and 209Bi have been calculated with the equilibrium and pre-equilibrium reaction models in Figs. 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10. Generally for all reactions, the calculated triton-emission spectra by using ALICE/ASH codes for the equilibrium with Weisskopf–Ewing model calculations are only in agreement with the experimental data at low energy region up to 20–25 MeV. Above 20–25 MeV, the Weisskopf–Ewing model can not calculate emission spectra (even if all the calculation parameters are changed). The equilibrium spectra calculations of the emitted triton particles are approximately Maxwellian.

Fig. 1
figure 1

The comparison of calculated triton emission spectra of (n,xt) reactions with the values reported in literature for 16O at 53.5 MeV. The experimental values were taken from Ref. [22]

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Fig. 2
figure 2

As Fig. 1 but neutron incident energy at 62.7 MeV

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Fig. 3
figure 3

The comparison of calculated triton emission spectra of (n,xt) reactions with the values reported in literature for 27Al at 53.5 MeV. The experimental values were taken from Ref. [22]

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Fig. 4
figure 4

As Fig. 3 but neutron incident energy at 62.7 MeV

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Fig. 5
figure 5

The comparison of calculated triton emission spectra of (n,xt) reactions with the values reported in literature for 59Co at 53.5 MeV. The experimental values were taken from Ref. [22]

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Fig. 6
figure 6

As Fig. 5 but neutron incident energy at 62.7 MeV

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Fig. 7
figure 7

The comparison of calculated triton emission spectra of (n,xt) reactions with the values reported in literature for 209Bi at 53.5 MeV. The experimental values were taken from Ref. [22]

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Fig. 8
figure 8

As Fig. 7 but neutron incident energy at 62.7 MeV

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Fig. 9
figure 9

The comparison of calculated triton emission spectra of (n,xt) reactions with the values reported in literature for 56Fe at 96 MeV. The experimental values were taken from Ref. [22]

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Fig. 10
figure 10

The comparison of calculated triton emission spectra of (n,xt) reactions with the values reported in literature for 208Pb at 96 MeV. The experimental values were taken from Ref. [22]

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The pre-equilibrium model calculations with the hybrid and GDH model are in good agreement with the measurements data above the triton-emission spectra energy 10–15 MeV for all target nuclei used in this study. The pre-equilibrium reactions for the heavier target nuclei are more dominate than the light nuclei. Thus, the calculated triton-emission spectra by using the pre-equilibrium reaction models are found to be in better agreement with the experimental data for the heavier target nuclei. But above 50–60 MeV, the pre-equilibrium spectra calculations of the emitted triton particles for 56Fe and 208Pb nuclei are lower than experimental data.

The calculated triton-emission spectra with GDH by using ALICE/ASH show the best agreement with the experimental data above 10–15 MeV energy. The reason is that the new developed pre-equilibrium reaction mechanism ALICE/ASH includes angular momentum conversion and so it gives us more information for new nuclear reaction researchers. When it has been taken into consideration the pairing energy and the mass shell correction, the experimental values are in better agreement with the theoretical results. Additionally, when more experimental data for the (n,xt) cross sections become available by using new technology, more reliable results for the triton-emission spectra can be obtained.

Summary and Conclusions

In this study, the (n,xt) reactions triton-emission spectra for some target nuclei as 16O, 27Al, 56Fe, 59Co, 208Pb and 209Bi have been investigated between 53.5 and 96 MeV incident neutron energy. Calculation results have been also compared with available experimental values in literature. The results can be summarized as follows:

  1. (1)

    The equilibrium model calculations of triton-emission spectra are only in agreement with the experimental data at low energy region up to 20–25 MeV. Above 20–25 MeV, the Weisskopf–Ewing model can not calculate the triton emission spectra.

  2. (2)

    The equilibrium spectra calculations of the emitted triton particles are approximately Maxwellian.

  3. (3)

    The GDH calculations show the best agreement with the experimental data above 10–15 MeV energy.

  4. (4)

    Above 50–60 MeV, the pre-equilibrium spectra calculations of the emitted triton particles for 56Fe and 208Pb nuclei are lower than experimental data.

  5. (5)

    The pre-equilibrium reaction model calculations are found to be in better agreement with the experimental data for the heavier target nuclei.

  6. (6)

    When taking the pairing energy and the mass shell correction into consideration, the experimental values are in better agreement with the theoretical results.