Deuteron Inelastic Scattering on the 6Li and 7Li nuclei within the Three-Body Cluster Model

  • The problem of the deuteron interaction with lithium nuclei which treated as the systems of two coupled pointlike clusters is formulated to calculate d+Li reaction's cross sections. The d+Li reaction mechanism is described using the Faddeev theory for the three-body problem of deuteron-nucleus interaction. This theory is slightly extended for calculation of as stripping processes 6Li(d,p)7Li, 7Li(d,p)8Li, 6Li(d,n)7Be and 7Li(d,n)8Be well as fragmentation reactions yielding tritium, $\alpha$ -particles, and continuous neutrons and protons in the initial deuteron kinetic-energy region $E_d=0.5-20$ MeV. The phase shifts found for $d+^6$ Li and $d+^7$ Li elastic scattering, as part of the simple optic model with a complex central potential, were used to find the cross sections for the 6Li $(d,\gamma_{M1})^8{\rm{Be}}$ and 7Li $(d,\gamma_{E1})^9{\rm{Be}}$ radiation captures. The three-body dynamics role is also summarized to demonstrate its significant influence within $d+^7$ Li system.
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M. V. EGOROV and V. I. POSTNIKOV. Deuteron Inelastic Scattering on the 6Li and 7Li nuclei within the Three-Body Cluster Model[J]. Chinese Physics C.
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Deuteron Inelastic Scattering on the 6Li and 7Li nuclei within the Three-Body Cluster Model

  • 1. Federal State Unitary Enterprise ”Russian Federal Nuclear Center“ Academician E.I. Zababakhin All-Russian Research Institute of Technical Physics, Snezhinsk, Chelyabinsk Region, Russia and Physics Faculty, Tomsk State University, Lenina ave.36, Tomsk 634050, Russia
  • *. National Research Nuclear University MEPhI, Moscow, Russia

Abstract: The problem of the deuteron interaction with lithium nuclei which treated as the systems of two coupled pointlike clusters is formulated to calculate d+Li reaction's cross sections. The d+Li reaction mechanism is described using the Faddeev theory for the three-body problem of deuteron-nucleus interaction. This theory is slightly extended for calculation of as stripping processes 6Li(d,p)7Li, 7Li(d,p)8Li, 6Li(d,n)7Be and 7Li(d,n)8Be well as fragmentation reactions yielding tritium, $\alpha$ -particles, and continuous neutrons and protons in the initial deuteron kinetic-energy region $E_d=0.5-20$ MeV. The phase shifts found for $d+^6$ Li and $d+^7$ Li elastic scattering, as part of the simple optic model with a complex central potential, were used to find the cross sections for the 6Li $(d,\gamma_{M1})^8{\rm{Be}}$ and 7Li $(d,\gamma_{E1})^9{\rm{Be}}$ radiation captures. The three-body dynamics role is also summarized to demonstrate its significant influence within $d+^7$ Li system.

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I.   INTRODUCTION
  • The fusion burn of Li nuclei in the $ ^7{\rm{Li}}(n,n'\alpha)t $ and $ ^6{\rm{Li}}(n,\alpha)t $ reactions under the neutrons generated at the perspective facilities as a results of $ dt\to\alpha n $ controlled reaction, is of importance in tritium self-reproduction. Decreased efficiency of the thermonuclear facility and mal-synchronization of the tritium-reproduction cycle may be attributed to inadequate understanding of how the fast ions in preheated $ dt $ plasma interact with Li nuclei within the blanket material. Of the ion induced nuclear reactions in the thermonuclear reactor walls, the most interesting are the reactions to produce additional tritium recovery. These include $ (d,t) $ reaction in the 6Li(d,t)p $ \alpha $ proceeding through a compound 5Li nucleus and the 7Li(d,t)d $ \alpha $ , 7Li(d,t)6Li processes. An important feature of the deuteron induced reactions on Li nuclei is that they are mostly exothermal reactions ranging from ( $ d,\gamma $ ) radiation capture to the partial target nucleus fragmentation into three or more clusters. The energy thus taken out by the deuterons from the $ dt $ reaction site may be partially compensated by additional tritium produced in the blanket, as well as by virtue of the charged particles formed through several exothermic reactions possible for Li nuclei living up to the deuteron kinetic-energy level $ E_d = 20 $ MeV.

    Most of the experimental studies on the deuteron-Li reactions were carried out last century. Tritium formation on the 6Li and 7Li isotopes in the energy region $ E_d\le4 $ MeV was investigated in [1]. The angular distributions were not measured. More details were given in relation to formation of charged particles – helium and hydrogen isotopes in the course of the deuteron-Li reactions. Formation of the two $ \alpha $ -particles was an active area of research in [2, 3, 4, 5, 6]. The total cross sections of the (d,p or n) stripping reactions accompanied by formation of both unstable 7Be [2, 7, 8], 8Li [2, 9, 10] and stable 7Li [2, 3] nuclei were measured. Currently, there is no data on the 7Li(d,n)8Be exothermal reaction. No evidence of the three-body break up reactions with the three particles being formed in the final state was given either. Such reactions refer to the above-mentioned [7, 8], focusing on the reactions to convert an initial scattering state into a two-neutron one. This also applies to [11] where total cross sections of the 6Li(d,n3He)4He and 6Li(d,p3H)4He three-body break up reactions were measured in the energy region $ E_d\leqslant 0.8 $ MeV. Although the d+Li system has always attracted much interest there are no data on some of the d+Li reactions like as 6Li $ (d,2d')\alpha $ , 6Li $ (d,np)^6 {\rm{Li}}$ , 7Li $ (d,2\alpha)n $ , 7Li $ (d,d')t\alpha $ , 6Li $ (d,np)^7 {\rm{Li}}$ . There are also no data on (d, $ \gamma $ ) radiation capture for both 6Li, 7Li nuclei, though there is some evidence of the resonance structure in the 7Li $ (d,\gamma)^9 {\rm{Be}}$ process in the energy region $ E_{d} $ = 0.35-0.4 MeV [9].

    The theoretical treatment in this field primarily covers the refined optical models with parameters selected to describe differential-angular distributions of the deuteron elastic scattering by Li nuclei. The most detailed study of the deuteron elastic scattering is given in [12] at $ E_d $ = 3-50 MeV and in [13] at $ E_d $ = 10-50 MeV. The last of these works reports on sets of optical model parameters found for the considered energy region. They are used in calculations within the continuum discretized coupled-channels (CDCC) formalism to describe differential cross sections of the deuteron elastic scattering. The optical models used in the context of the coupled-channel formalism can provide useful information on the total contribution of various inelastic channels in their total cross sections (so called reaction cross section), as well as on the partial cross sections but the available references do not give such predictive calculations. There are only calculated cross sections of the inclusive 7Li $ (d,Xn) $ and 7Li $ (d,Xp) $ reactions to predict the total yield of continuum neutrons and continuum protons in the dipole giant-resonance energy region $ E_d\ge $ 20 MeV [14] wherein the main contribution into cross-section comes through Glauber (diffraction) scattering. In addition, there are crude estimates for the s-wave cross section of the deuteron induced reactions on the 7Li [15] nucleus in the energy region $ E_d $ = 0.4-1 MeV. Use of statistical Hauser-Feshbach model with formation of a combined nucleus and its break-up, invariant of the triggering mechanism, appears to be dubious and of limited value in [16] for a light nuclei with the open p-shell and a small number of neutron resonances. Currently, the microscopic models of the deuteron-Li reactions are unavailable and they have not been studied systematically yet. The present work focuses on redressing the lack of understanding in this area.

    In this work we consider the target nucleus as a bound two-body system. This allows one to solve the scattering problem based on the integral Faddeev equations. These equations are further used to describe the nuclear-reaction cross sections. With such an approach, we avoid searching an efficient optical potential to describe the target nucleus. At the same time, couple of inelastic two-body channels in d+Li reaction, is equally considered while in preparation of the seed two-body interactions. Besides, the few-body microscopic approach resolves the question about how this or that cluster interaction contributes to the nuclear-reaction intensity.

    Chapter 1 briefly describes the nucleus cluster model and the three-body deuteron-Li reactions in the energy region $ E_d $ = 0.5-20 MeV. Chapter 2 gives classification of the most relevant deuteron-Li reactions, in terms of the input into their total cross section, along with the associated measured cross sections. The final part of the work provides Discussion of the Results succeeded by Conclusion.

  • II.   DEUTERON SCATTERING IN THE TARGET NUCLEUS CLUSTER MODEL
    • In the cluster model, the lowest-energy state of the nucleus-target is determined by pairwise interaction of the ( $ \alpha-d $ ) clusters forming the 6Li and ( $ \alpha-t $ ) clusters forming the 7Li. Thus, the cluster system is in a state wherein the quantum numbers are equivalent to the corresponding numbers of target nucleus. Let the $ \vec p $ momentum specifies motion in the coupled cluster and $ \vec q $ characterizes motion of the spectator-particle with respect to the coupled pair. The prime over the marks corresponds to the momenta resulting from the particle transfer. The $ \alpha,\beta,\gamma\in(1,2,3) $ indices specify the type of the particular three-body cluster state and repetitive rearrangement of Greek indices provides a set of independent values X-characterizing the probability amplitude for transition from one cluster channel to the other. Then, the integral three-body Faddeev equations, considering the adopted notations, become as follows

      $\begin{aligned}[b] X_{\beta\alpha}(\vec q',\vec q,E) =& Z_{\beta\alpha}(\vec q',\vec q,E)+\sum\limits_{\substack{\gamma = 1 \\ \gamma\not = \beta}}^3\int\frac{d^3q''}{(2\pi)^3}Z_{\beta\gamma}(\vec q',\vec q'',E)\\&\times\tau_{\gamma}(\vec q'',E)X_{\gamma\alpha}(\vec q'',\vec q,E).\end{aligned} $

      (1)

      In equation (1) $ \tau_{\gamma}(\vec q'',E) $ are two-body propagators which depend on three-body kinetic energy E. The driving terms $ Z_{\beta\alpha} $ are the matrix elements of the free three-body Green's function $ G_0(E) $ :

      $ Z_{\beta\alpha}(\vec q',\vec q,E) = (1-\delta_{\beta\alpha})\langle\vec q',\beta\mid G_0(E)\mid\vec q,\alpha\rangle $

      (2)

      Condensed index $ \gamma $ (as well as $ \alpha,\beta $ ) determines the following sets of states

      $\begin{aligned}[b] \mid \vec q,\vec p,\gamma\rangle_{^6{\rm{Li}}} =& \begin{cases} \mid d-(d\alpha),\vec q,\vec p\rangle,\phantom{111}\gamma = 3,\\ \mid d-(d\alpha),\vec q,\vec p\rangle,\phantom{111}\gamma = 2,\\ \mid \alpha-(dd),\vec q,\vec p\rangle,\phantom{111}\gamma = 1;\\ \end{cases} \\\mid \vec q,\vec p,\gamma\rangle_{^7{\rm{Li}}} =& \begin{cases} \mid d-(t\alpha),\vec q,\vec p\rangle,\phantom{111}\gamma = 3,\\ \mid t-(d\alpha),\vec q,\vec p\rangle,\phantom{111}\gamma = 2,\\ \mid \alpha-(dt),\vec q,\vec p\rangle,\phantom{111}\gamma = 1.\\ \end{cases} \end{aligned}$

      (3)

      The on-shell three-body kinetic energy is determined by the two-body kinetic energy and the energy $ E_b $ of bound state. In general, the solution technique of the equation (1) is similar to determining the solutions in [17] and will not be touched on here. It will only be noted that the first two states in (3) for 6Li can be combined. We will discuss more details on the two-body cluster subsystems.

    • A.   Two-Body Interactions

    • Interactions in each pair of particles are specified by the following separable potentials:

      $ v_{\gamma}^{l'l} = \lambda_{\gamma}^{l'l}\xi_{\gamma}^{l'}(p')\xi_{\gamma}^{l}(p), $

      (4)

      They are transformed as tensors in the orbital $ (l,l') $ momenta space of the interacting pairs of clusters. In calculations, for the interacting pairs we adopt the following states

      $ \begin{cases} dd - (l,J^P) = (0,0^+),\,(0,2^+),\,(1,1^-), \\ d\alpha - (l,J^P) = (0,1^+),\,(1,0^-),\,(1,1^-), \\ dt - (l,J^P) = (1,3/2^-), \\ \alpha t - (l,J^P) = (1,3/2^-). \end{cases} $

      (5)

      From the sets (5), it follows that for both 6Li and 7Li nuclei each matrix element $ X_{\beta\alpha} $ contains the p-wave scattering. This suggests that the three-body matrices have a complicated spin-angular structure. It is precisely for the simplicity of calculations, that we restrict ourselves to the $ l,l'\in(0,1) $ requirement, and further it will be demonstrated that it is quite sufficient for description of the typical cross-sectional properties in most of the considered reactions. In any two-body subsystem, for all the states (5), the $ \xi(\vec p) $ form-factors have similar functional form

      $ \xi(\vec p) = \frac{c_1}{\vec p^{\,2}+\beta_1^2}+\frac{c_2\vec p^{\,2}}{(\vec p^{\,2}+\beta_2^2)^2}. $

      (6)

      $ c_{1,2} $ and $ \beta_{1,2} $ parameters are found from the best-description condition based on the $ \delta_{l}^{J} $ phase-shift parameters and the inelasticity $ \eta_{l}^{J} $ (J-total spin) available in literature. Additionally, it should be noted that in this work, in all the cluster states, $ \lambda_{\gamma}^{l'l} $ interaction intensity is independent on the energy. The two-body T-matrix with regard to the inelastic channels up to the order of smallness with respect to $ \lambda $ intensity takes the following form

      $ \begin{aligned}[b] T(p,p') =& \xi(p)\tau(p,p^2/2\mu)\xi(p');\\ \tau(p,p^2/2\mu) =& (\lambda^{-1}-\Sigma_{el}(p)-\Sigma_{inel}(p))^{-1}, \end{aligned} $

      (7)

      wherein the energy shifts $ \Sigma_{el/inel} $ in the propagator are calculated using the form-factors (6)

      $ \begin{aligned}[b] \Sigma_{el}(p) =& \frac{\mu}{\pi^2}\int\limits_0^{\infty} \frac{p^{'2}dp'\xi^2(p')}{p^2-p^{'2}+i\varepsilon}, \\ \Sigma_{inel}(p) =& \frac{\mu}{\pi^2}\int\limits_0^{\infty} \frac{p^{'2}dp'\xi_{inel}^2(p')}{p^2-p^{'2}+i\varepsilon}, \\ \xi_{inel}(p') = &\frac{c_3}{\vec p^{'\,2}+\beta_3^2}. \end{aligned} $

      (8)

      Quantity $ \Sigma_{inel} $ as like as form-factors $ \xi_{inel} $ are introduced for to take into account some inelastic effects being important in elastic nucleus-nucleus scattering. There is no doubt this inclusion should be in coincidence with rearrangement matrix elements $ X_{\beta\alpha} $ . In this case eigenstate of initial nucleus is a vector strained on as elastic well as rearrangement channels. In this work for the sake of simplicity we avoid the direct inclusion in matrix $ X_{\beta\alpha} $ all of rearrangement channels holding on only energy shift in two-body propagators.

      For the $ \alpha d $ and $ \alpha t $ bound states with the $ E_b $ binding energy, the interaction intensity is found according to the formula

      $ \lambda^{-1} = \frac{\mu}{\pi^2}\int\limits_{0}^{\infty}\frac{(\xi^2(p)+\xi_{inel}^2))p^2dp}{(2\mu\mid E_b\mid +p^2)}. $

      (9)

      Fig. (1) shows results of the phase-shift parametrization, as well as the inelasticity parameters for $ dd $ and $ d\alpha $ scattering. For $ dt $ elastic scattering, the data on the phase shifts in the pertinent energy region are unavailable. Instead, we will rely on the fact that this interaction in the considered energy region is characterized by high inelasticity related to a neutron source reaction $ dt\to\alpha n $ . The $ \eta $ inelasticity parameter characterizes the reaction cross section $ \sigma_r $ , in the specified spin-orbital state (for the purpose of simplification, the corresponding indexes are omitted)

      Figure 1.  Real $ \mathfrak{Re}(\delta_l^J) $ and imaginary $ \mathfrak{Im}(\delta_l^J) $ parts of phase shifts for $ dd $ scattering, data taken from [18] on the left. Real $ \mathfrak{Re}(\delta_l^J) $ and imaginary $ \mathfrak{Im}(\delta_l^J) $ parts of phase shifts for $ d\alpha $ scattering, data taken from [19] on the right.

      $ \sigma_r = \frac{\pi}{k^2}\sum\limits_{l = 0}^{\infty}(2l+1)(1-\eta^2). $

      (10)

      Here, k-is a wave-number of the relative motion vector for the d and t nuclei. Then using the parametrized differential cross section $ \sigma(\theta) $ of the neutron-source reaction reported in [20] we numerically integrate it for each l over the solid angle. The obtained partial cross sections of the reaction are substituted into the formula (10) to express the inelasticity parameter $ \eta $ in the specified spin-orbital state. Fig. (2) shows the inelasticity parameter $ \eta $ obtained by this manner for $ dt $ interaction. Comparison to the data in [21] on the real part of the phase shift $ \mathfrak{Re}\delta_{1}^{3/2} $ for $ \alpha t $ elastic scattering is also presented here. Table (1) summarizes the two-body cluster interaction parameters.

      Sys. (l, $ J^P $ ) $ \lambda^{-1} $ [MeV $ ^{-1} $ ] $ c_1 $ [MeV $ ^{2} $ ] $ c_2 $ [MeV $ ^{2} $ ] $ c_3 $ [MeV $ ^{2} $ ] $ \beta_1 $ [MeV] $ \beta_2 $ [MeV] $ \beta_3 $ [MeV]
      $ dd $ (0, $ 0^+ $ ) -0.3 $ \cdot 2\pi^2 $ -430 345 49/ $ \lambda $ 106 57 1
      (0, $ 2^+ $ ) -7.75 $ \cdot 2\pi^2 $ -991 319 14/ $ \lambda $ 300 101 5
      (1, $ 1^- $ ) -1 $ \cdot 2\pi^2 $ -989 588 75/ $ \lambda $ 350 156 3.6
      $ d\alpha $ (0, $ 1^+ $ ) (12) -310 320 82/ $ \lambda $ 140 64 24
      (1, $ 0^- $ ) -40 $ \cdot 2\pi^2 $ -500 0 245/ $ \lambda $ 225 - 10
      (1, $ 1^- $ ) -25 $ \cdot 2\pi^2 $ -989 2 560/ $ \lambda $ 490 100 17
      $ dt $ (1, $ \tfrac{3}{2}^- $ ) -3 $ \cdot 2\pi^2 $ -745 0 112/ $ \lambda $ 674 - 30
      $ \alpha t $ (1, $ \tfrac{3}{2}^- $ ) (12) -435 100 - 169 53 -

      Table 1.  Intensity $ \lambda $ of the two-body interactions and $ c, \beta $ parameters which appear in the form-factors (6) and (8). In the 3d column, the equation number is given in parenthesis.

      Figure 2.  Inelasticity parameter $ \eta_1^{3/2} $ for $ dt $ scattering, the results are based on [20] (see in the text) on the left. Real part of $ \delta_1^{3/2} $ phase shift for $ \alpha t $ scattering, data taken from [21] on the right.

    • B.   d+Li Elastic Scattering

    • In the adopted (3) notations, T-matrix of the elastic scattering corresponds to $ X_{33} $ , matrix associated with the partial amplitude $ F^L(q) $ of the elastic scattering

      $ F^L(q) = -\frac{\mu_{d{\rm{Li}}}}{2\pi} N X_{33}^L(q,q,E). $

      (11)

      Here, the reduced mass of the deuteron-Li is denoted by $ \mu_{d{\rm{Li}}} $ , N-a normalization factor calculated from the binding energy of the cluster $ E_{b} $ [17]. Fig. (3) shows the measured cross sections of $ d^6 {\rm{Li}}$ and $ d^7 {\rm{Li}}$ elastic scattering. Presence of the resonance-like structures in the regions $ E_d\approx 4 $ MeV and $ E_d\approx5 $ MeV is to be noted. In this paper, their formation mechanism will not be discussed in detail. The only thing to be noted here, is that the full information on all the inelastic processes, possible within the adopted cluster scheme (3) within the used parametrization (6), has already entered in the $ X_{13} $ , $ X_{23} $ and $ X_{12} $ matrices, against which the matrix $ X_{33} $ is expressed. Taking into account that there is no experimental data d+Li elastic scattering, the major criterion of accurate prediction is its agreement with the other model approximation, specifically, with the optical-model predictions. We have found that the calibrated optical predictions based on the available differential distributions are in reasonable agreement with the data in Fig. (3) for the elastic-scattering cross sections on the level of 800-900 mb in the region $ E_d\geqslant $ 8 MeV.

      Figure 3.  The deuteron elastic scattering on the 6Li (on the left) and 7Li (on the right) nuclei.

    III.   DEUTERON-LI REACTIONS
    • Among the considered deuteron-Li reactions in the energy region $ E_d $ = 0.5-20 MeV, the most significant are the ones listed below

      $\begin{aligned}[b]& d+^6{\rm{Li}}\to \begin{cases} \gamma+^8{\rm{Be}}, Q = 22.279, \\ 2\alpha, Q = 22.37, \\ p+^7{\rm{Li}}, Q = 5.026, \\ n+^7{\rm{Be}}, Q = 3.382, \\ n+p+^6{\rm{Li}}, Q = -2.224, E_{th} = 2.969, \\ 2d'+\alpha, Q = -1.473, E_{th} = 1.967,\\ t+p+\alpha, Q = 2.559, \\ ^3{\rm{He}}+n+\alpha, Q = 1.7951; \end{cases}\\& d+^7{\rm{Li}}\to \begin{cases} \gamma+^9{\rm{Be}}, Q = 8.623, \\ t+^6{\rm{Li}}, Q = -0.994, E_{th} = 1.279, \\ p+^8{\rm{Li}}, Q = -0.192, E_{th} = 0.247, \\ n+^8{\rm{Be}}, Q = 15.029, \\ n+p+^7{\rm{Li}}, Q = -2.224, E_{th} = 2.863,\\ \alpha+d+t, Q = -2.467, E_{th} = 3.176, \\ 2n+^7{\rm{Be}}, Q = -3.869, E_{th} = 4.979, \\ 2\alpha+n, Q = 15.121. \end{cases}\end{aligned}$

      (12)

      The masses of all nuclei were calculated using the recent values of their mass excess from [22]. In (12) the reaction energy Q, and the threshold energy $ E_{th} $ are given in MeV, respectively. The amount of exothermic reactions, which (as it has been mentioned in the introduction) might produce a significant effect on the energy balance of the advanced fusion facilities, is to be noted. The cross sections $ \sigma $ of the stripping reactions associated with the so called "two by two" reaction are expressed using the formula:

      $\begin{aligned}[b] \frac{d\sigma}{d\Omega_c} =& \frac{E_f E_{{\rm{Li}}} \omega_d \omega_c}{(2\pi W)^2}\frac{p_f}{p_i}\frac{1}{3(2J_{{\rm{Li}}}+1)}\\&\times \sum\limits_{M_iM_f} \Big| \sum\limits_{\gamma}\xi_{\gamma}(p) X_{\gamma 3}(p_i,p_f,E) N\Big|^2. \end{aligned}$

      (13)

      Kinematic notations:

      W - total system energy;

      $ \omega_d $ , $ \omega_c $ - the deuteron and light particle energy–reaction product;

      $ E_{{\rm{Li}}} $ , $ E_f $ - Li-nucleus and heavy particle energy–reaction product;

      $ p_i $ , $ p_f $ , $ \Omega_c $ - relative momenta for the particles before and after reaction, as well as, the solid angle escape of the light particle, correspondingly.

      Summation in (13) is taken over the complete-spin (spin-channel) projections of the initial and final nuclear system. Index $ \gamma $ runs over the values [1,2] and for the given stripping or rearrangement reaction these values give the required final state. $ \xi_{\gamma}(p) $ is an additional vertex form-factor (where $ \vec p $ is the relative momentum of neutron or proton and removal particle see Fig.(4)) used in stripping processes calculation. Stripping processes, for example, 7Li(d,n)8Be or 6Li(d,p)7Li, seem to be two-step processes under the three-body approach where deuteron is split off on the first step and final nucleus is formatted on the second one. Moreover, it is required the integration over relative momentum of splitting particles. Instead of that avoiding the inevitable numerical complexity we neglect propagation of splitting particle (proton, neutron, deuteron and two neutron quasi-particle – 2n) in nuclear medium that takes place in stripping processes. For the stripping reactions with formation of short-lived or stable nuclei, it is typical that the interaction of incident particle with target nucleus seems to be of a pole type (see Fig.(4)) where pole takes a role of reduced width for only removal proton for 6Li(d,p)7Li and 7Li(d,p)8Li or only removal neutron for 6Li(d,n)7Be and 7Li(d,n)8Be. In this paper, for the stripping reactions, we actually use just two types of this vertices. The first one is (pn) vertex wherein the interaction is specified by the simple rank-one separable potential [25], the second is (nd) vertex achieved on the separable rank one potential with the parameters for $ 1/2^+ $ state [17]. Here we are not distinguished neither nd nor p(2n) interactions where (2n) is quasi-bound two neutron state. In this simplification the integration over relative momentum of splitting particles replaced by vertex form-factor $ \xi_{\gamma}(p) $ multiplier calculated at on mass shell relative momentum $ \vec p $ . Another simplification has made for three-body nature of residual nuclei 7Li, 8Li, 7Be and 8Be that underlies in present stripping processes calculations and should be carefully taken into account in searching for bound energies. However, we avoid the direct solution of three-body problems for n- $ \alpha $ -d, (2n)- $ \alpha $ -d, p- $ \alpha $ -d and (np)- $ \alpha $ -d interactions in final states and related with them internal structures of residual nuclei 7Li, 8Li, 7Be and 8Be. Instead of precise calculation of three-body final states in stripping processes, we have made an attempt to reproduce stripping cross sections from kinematic point of view in which we explicitly have phase space for reactions 6Li(d,p)7Li, 7Li(d,p)8Li, 6Li(d,n)7Be and 7Li(d,n)8Be with residual nuclei 7Li, 8Li, 7Be and 8Be and emitted protons or neutrons.

      Figure 4.  Diagrammatic representation of rearrangement and stripping processes. Solid circles correspond to an additional vertex form-factor $ \xi_{\gamma}(p) $ used in stripping processes. Propagators $ \tau $ are denoted by shaded boxes. Indexing for probability amplitudes X is determined by (3).

      It should be emphasize some information about stripping processes is already partially contained in $ \Sigma_{inel} $ quantities. These terms must be in coincidence with elastic nd, np and dt scattering data needed for $ \xi_{\gamma}(p) $ search. Here for the sake of simplicity we restrict calculations by independent searching of $ \Sigma_{inel} $ and nd, np and dt two-body form-factors. The amplitudes for elastic pn and nd scattering are taken into account with the use of separable rank one potential of [25] and [17], correspondingly. For the dt potential as it has previously mentioned we are fitted inelastic parameter $ \eta $ under the separable rank one approximation. Figs. (5,6,7) show calculated total cross sections of the considered rearrangement and stripping reactions.

      Figure 5.  Total cross sections of the 6Li $ (d,\alpha)\alpha $ (on the left) and 7Li $ (d,t)^6 $ Li (on the right) reactions. The experimental points are from [2],[3],[4],[5],[6].

      Figure 6.  Total cross section for the 6Li(d,p)7Li (on the left) and 7Li(d,p)8Li (on the right). reactions. The experimental points are from [2],[3],[23].

      Figure 7.  Total cross sections for the 6Li(d,n)7Be (on the left) and 7Li(d,n)8Be (on the right) reactions. The experimental points are taken from [3].

    • A.   Three-body Break Up Reactions

    • Within the considered energy region, the deuteron-induced reactions may have up to 4 different channels with the three particles in the final state on both stable Li isotopes. In this regard, the measured cross sections, which are not provided with the experimental data are of interest. First of all, this applies to the reactions yielding the following sets of particles $ n+p+^6 {\rm{Li}}$ , $ 2d'+\alpha $ , $ 2\alpha+n $ , $ n+p+^7 {\rm{Li}}$ and $ \alpha+d+t $ .

      The total cross section of the three-particle break up reaction is calculated via integration over the following expression

      $ \begin{aligned}[b] \frac{d\sigma}{d\Omega_cd\omega_{12}d\Omega_{12}} =& (2\pi)^{-5}\frac{\omega_dE_{{\rm{Li}}}\omega_c p_c E_{1} E_{2} p_{12}}{p_i W^2} \frac{1}{3(2J_{{\rm{Li}}}+1)} \\&\times \sum\limits_{M_iM_f} \Big|\sum\limits_{\gamma = 1,2,3}\tau_{\gamma} (\vec p_{\gamma},E-\vec p^{\,2}_{\gamma}/2M_{\gamma}) \\&\times\xi_{\gamma}(p_{\gamma}) X_{\gamma 3}(p_i,p_c,E)N\Big|^2 \end{aligned}$

      (14)

      New notations are introduced in (14) as compared to (13):

      $ \omega_c $ , $ p_c $ , $ \Omega_c $ - energy, momentum, and solid angle for one of the three escaping particles;

      $ p_{12} $ , $ \Omega_{12} $ - relative momentum for the other pair of particles and its solid angle calculated in the center-of-mass system of the pair;

      $ E_1 $ , $ E_2 $ - energies of the other two particles in the common center-of-mass system.

      Fig. (8) shows calculated cross sections for the three-body break up reaction based on the formula (14).

      Figure 8.  Cross section of the three-body break up reaction on the 6Li (on the left) and the 7Li nucleus (on the right). Experiment on the 6Li(d,pt)4He and 6Li(d,n3He)4He reactions is in [11], data for 7Li(d,2n)7Be process taken from [7],[8]

    • B.   Radiation Capture

    • The radiation-capture processes in the form

      $ d(\omega_i,\vec q_i)+{\rm{Li}}(E_i,-\vec q_i)\to \gamma(\omega_{\gamma},\vec k_{\gamma}) + {\rm{Be}}(E_f,\vec p_f) $

      (15)

      reveal information on the internal structure of Be nucleus and relate with electromagnetic transitions between the excited discrete levels of Be nucleus and low-lying or ground state. Cross section of the process (15) is expressed by the formula

      $ \frac{d\sigma}{d\Omega_{\gamma}} = \frac{E_i E_f \omega_i \omega_{\gamma}^2}{q_i (2\pi W)^2}\frac{1}{3(2J_{{\rm{Li}}}+1)}\sum\limits_{l\mu,{\rm{EM}}} \big|T_{{\rm{Be}}\,{\rm{Li}}}(l\mu,{\rm{EM}})\big|^2. $

      (16)

      In longwave approximation, when the wave number of radiation $ k_{\gamma} $ is much smaller than the spatial dimensions of the system R, i.e. $ k_{\gamma}\cdot R\ll 1 $ , that for the special case $ \omega_{\gamma} = 10 $ MeV and $ R = 2.5 $ Fm yields $ 0.13\ll 1 $ , probability $ T_{{\rm{Be}}\,{\rm{Li}}}(l\mu,{\rm{EM}}) $ of the electromagnetic transition of the $ {\rm{EM}}\in({\rm{E}},{\rm{M}}) $ type with multipolarity $ l\mu $ may be represented [24] by the following

      $ T_{{\rm{Be}}\,{\rm{Li}}}(l\mu,{\rm{EM}}) = \frac{8\pi (l+1)k_{\gamma}^{2l+1}}{l\big((2l+1)!! \big)^2} \Big|\langle {\rm{Be}} \mid \Omega_{l\mu}({\rm{EM}}) \mid {\rm{Li}}\rangle \Big|^2 $

      (17)

      where

      $ \begin{aligned}[b] \Omega_{l\mu}({\rm{E}}) =& \sum\limits_{j = 1}^{A}\Big[e_j r_j^l Y_{l\mu}(\hat r_j) - i \frac{e \hbar c}{M_N c^2} K_j \frac{k_{\gamma}}{l+1}(\vec\sigma_{j}\times\vec r_j)\\&\times \vec\nabla_j\big(r_j^l Y_{l\mu}(\hat r_j)\big)\Big], \\ \Omega_{l\mu}({\rm{M}}) =& i\sum\limits_{j = 1}^{A}\Big[\Big(\frac{e_j\hbar c}{2M_N c^2} \frac{2}{l+1}\vec L_j +\frac{e\hbar c}{2M_N c^2} K_j \vec\sigma_j \Big)\\&\times\vec\nabla_j\big(r_j^l Y_{l\mu}(\hat r_j)\big)\Big]. \end{aligned} $

      (18)

      Notations in (18): e, $ e_j $ -elementary charge and $ e_j = e $ for protons and $ e_j = 0 $ for neutrons;

      $ r_j $ , $ K_j $ - coordinate of j-th nucleon and its magnetic moment;

      $ \vec\sigma_j $ , $ \vec L_j $ , $ \vec\nabla $ - spin operators of the orbital moment and gradient acting upon the j-th nucleon.

      Typically, radiation captures are calculated with complicated cluster systems. Each of them consists of several protons and neutrons. Therefore, it is advantageous to limit summation over j entirely by the initial nuclei with the effective charge $ \varepsilon_l $ . Then for the process (15) the effective charge equals

      $ \varepsilon_l = \Big(\frac{A}{A+2}\Big)^l+3\Big(-\frac{2}{2+A}\Big)^l $

      (19)

      Here A is the atomic number of Li nucleus. Further summation (18) over magnetic quantum numbers with the Wigner-Eckart formula and integration over $ \hat r_j $ angles with the $ d- $ Wigner function are rather time-consuming and can be found in elsewhere. We will dwell on calculations of the $ \langle{\rm{Be}}\mid r^{l} \mid {\rm{Li}}\rangle $ , $ \langle{\rm{Be}}\mid r^{l-1} \mid {\rm{Li}}\rangle $ , radial integrals included in (17). The d+Li scattering states are roughly approximated by the s-wave only. Then numerical solving of the Schrödinger equation with central complex potential taking into consideration the Coulomb potential screening at r = 1.5 Fm yields the appropriate radial wave functions. For the central complex potential given below

      $ \frac{V+iW}{1+\exp{\Big(\dfrac{r-r_0}{a}\Big)}}, $

      (20)

      the free parameters – the depths of the real and imaginary parts V [MeV] and W [MeV] of the potential, as well as, its diffuseness a [Fm] and the cutoff parameter $ r_0 $ [Fm], correspondingly, were adjusted based on reproduction of the previously-found phase shifts and inelasticity parameter for d+Li elastic scattering using the formula (11). Fig. (9) demonstrates the phase shifts as compared to the inelasticity parameters for the s-wave scattering calculated using the Faddeev equations and the optical model with the central potential (20). The potential parameters were taken to be equal to V = 70(6Li),30(7Li), W = 30(6Li),5(7Li), a = 0.15(6Li),0.1(7Li), $ r_0 $ = 1.5(6Li,7Li). Fig. (10) shows calculated cross sections of the radiation captures for the transition $ 1^+\to 0^+ $ within the process 6Li $ (d,\gamma_{M1})^8 {\rm{Be}}$ , and $ \tfrac12^+\to\tfrac32^- $ within the process 7Li $ (d,\gamma_{E1})^7 {\rm{Be}}$ .

      Figure 9.  Results of calculations based on the simple optical model (black lines) and exact calculation (red lines) of the inelasticity parameters (on the left) and the phase shifts (on the right) in degrees. Data on d+6Li elastic scattering of $ \eta_0^{1} $ , $ \delta_0^1 $ and on the d+7Li elastic scattering of the $ \eta_0^{1/2} $ , $ \delta_0^{1/2} $ are plotted.

      Figure 10.  Cross sections of the 6Li $ (d,\gamma_{M1})^8 {\rm{Be}}$ (on the left) and the 7Li $ (d,\gamma_{E1})^9 {\rm{Be}}$ radiation captures (on the right).

    IV.   DISCUSSION OF THE RESULTS
    • Structurally, the Faddeev equations (1) allow one to consider the multiple scattering within the three-body systems. The system of equations (1) solved with respect to the $ X_{33} $ elastic scattering matrix can be written in the following form

      $\begin{aligned}[b] X_{33}(\vec q',\vec q, E) =& K_{31}(\vec q',\vec q'',E)X_{13}(\vec q'',\vec q,E)\\&+K_{32}(\vec q',\vec q'',E)X_{23}(\vec q'',\vec q,E) \end{aligned}$

      (21)

      where $ K_{31} $ and $ K_{32} $ are kernels of the integral equations (1). Information on inelastic interactions accompanied by transfer of the clusters from one interacting pair of particles to the other one has already been added to the structure (21). Due to this fact, the Faddeev equations offer an advantage in describing the processes accompanied by transformation because the amplitude obtained while in constructing the elastic cross section, already contains the required information on the potential inelastic processes within the system. The thing is, that the Faddeev equations cannot be simply used until in each partial wave the inter-particle interaction have to be successfully parametrized and the potentials of pairwise interactions have to reveal all the dynamic properties typical for the processes under consideration. If parametrization of the two-particle interactions is a success and the number of these interactions approaches an actual value, the chance to reach the correctly described three-particle dynamics in the considered system becomes higher. For the same reason we belief if the parametrization of two-body interactions of charged particles is a success without Coulomb penetration factor introduction nevertheless the Coulomb forces indirectly have been taken into account.

      In this work, not all of the pairwise interactions are parametrized by the phase shifts. For dt interactions, the inelasticity parameters calculated with respect to the measured cross sections for $ dt\to\alpha n $ reaction [20] were used. Our predictions have proved that the form of the $ \xi_{dt} $ form-factor determines the typical behavior of the elastic scattering in the region $ E_d<5 $ MeV. That is why the role of this interaction has been crucial for all the reactions within the $ d+^7 {\rm{Li}}$ . At the same time Fig. (5) shows that the cross section of the 7Li(d,t)6Li reaction reproduces itself quite well in the region $ E_d<5 $ MeV, and this suggests that the adopted functional relationship of the $ \xi_{dt} $ form-factor is quite comparable with the real one. Figs. (6), (7) show that although calculations of the stripping reaction cross sections have been considerably simplified, they are still reflective of the experimental cross-section features in the near-threshold region of energies.

      The three-body break up reactions are less sensitive to details of the scattering matrix as compared to the stripping reactions, as their phase space is larger and increases with the energy growth. That is why peculiarities of the pairwise interactions in the three-body reactions are only noticeable in the threshold region. Our calculations of the 6Li(d,pt)4He, 6Li(d,n3He)4He cross sections are in reasonable agreement with the experimental data in the energy region $ E_d<2 $ MeV, above this energy the theoretical cross section significantly underestimates the experimental one. The specified underestimation may be attributed to the d-wave contributions of the cluster interactions that were neglected in our calculations. For d+7Li system, only the experimental data on the 7Li(d,np)7Be reaction are available. Comparison to these data reveals agreement with the experimental value in the low-energy region $ E_d<10 $ MeV and further underestimation of the cross section with the energy increase. The remaining calculations associated with the three-body break up reaction and the radiation capture are presented in this paper for the first time.

      Neglect of multiple scattering in ( $ \alpha d $ ) and ( $ \alpha t $ ) subsystems is equivalent to substitution in (21) $ X_{13}\to Z_{13} $ and $ X_{23}\to Z_{23} $ . Fig. (11) shows differential by the invariant mass of $ (\alpha d) $ cross section calculations of 6Li(d,2d')4He and 6Li(d,pt)4He reactions at the energy $ E_d = 10 $ MeV and $ E_d = 1 $ MeV, correspondingly, versus the relative rate in $ (\alpha d) $ pair. Here, the dashed lines indicate calculations disregarding multiple $ (\alpha d) $ scattering. The obtained differential cross sections of the 7Li(d,d't)4He reactions at the energy $ E_d = 5 $ MeV and the 7Li(d,2n)7Be at the energy $ E_d = 7 $ MeV versus relative rate within the clusters $ (dt) $ and $ (nn) $ , respectively, are also presented on the Fig. (11). Calculations with neglected multiple interaction into $ (\alpha t) $ subsystem are indicated by dashed lines and taking into account multiplier 160 for the 7Li(d,d't)4He reaction and multiplier 380 for the 7Li(d,2n)7Be reaction. As it is seen, the role of multiple scattering in $ (\alpha d) $ and $ (\alpha t) $ subsystems turns out to be different. If we take real lengths of the deuteron elastic scattering on the 6Li and 7Li nuclei as a unit, then the ratio of the scattering length calculated with neglect of multiple scattering in the $ (\alpha d) $ and $ (\alpha t) $ subsystems will be equal to 1.27 (or +27%) and 2.59 (or +159%), which is also in correlation with the results in Fig. (11). Such divergence in properties in d+6Li and d+7Li systems depending on the interactions in the coupled $ (\alpha d) $ and $ (\alpha t) $ clusters within the adopted cluster scheme (3) as we believe reveals the significant role of inelastic dt interaction in the nucleus surrounding. This may be verified within the experimental investigation of dt system or in forthcoming theoretical calculations of the partial phase shifts of dt elastic scattering.

      Figure 11.  Distributions, differential by the invariant mass, in the accurate calculation and in the calculation with neglected multiple interaction in the coupled pair of clusters versus relative rate in this pair (in speed of light unity). $ \frac{d\sigma}{d\omega_{\alpha d}} $ for the 6Li(d,2d')4He at $ E_d $ =10 MeV and the 6Li(d,pt)4He at $ E_d $ =1 MeV on the left. $ \frac{d\sigma}{d\omega_{nn}} $ for the 6Li(d,2n')7Be at $ E_d $ =7 MeV and $ \frac{d\sigma}{d\omega_{dt}} $ for the 6Li $ (d,d')\alpha t $ at $ E_d $ =5 MeV.

    V.   CONCLUSION
    • The cross sections of most nuclear reactions in $ d+^6 {\rm{Li}}$ and $ d+^7 {\rm{Li}}$ systems existing in the energy range $ E_d = 0.5-20 $ MeV are presented in this work. The cross sections for radiation capture and some reactions with the (n+p+6Li, 2d'+ $ \alpha $ , $ 2\alpha $ +n, n+p+7Li, $ \alpha $ +d+t) three-body break up have been calculated for the first time. It is shown that, generally, the adopted parametrization provides an adequate characterization of the cross sections in the near-threshold energy range and in some cases, specifically in the $ 2n+^7 {\rm{Be}}$ channel, it is even possible to describe the experimental data up to 10 MeV. Despite the cluster behavior of both 6Li, 7Li nuclei, which is characterized by remarkable isolation of the cluster configurations, interaction in the coupled pair reveals itself in different ways for $ d+^6 {\rm{Li}}$ and $ d+^7 {\rm{Li}}$ . The next step in searching for d+Li stripping processes should carefully take into account the three-body nature of residual 7Li, 8Li, 7Li and 8Be nuclei. Further studies are also possible in this area with the help of the d-wave interactions in the cluster subsystems and by virtue of some other kind of parametrization for elastic $ dt $ interaction.

Reference (25)

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