Reaction dynamics in clustered nuclear systems

Dottorato di Ricerca in Fisica Doctorado en Ciencias y Tecnologías Físicas Salvatore Simone Perrotta Reaction dynamics in clustered nuclear systems PhD thesis Supervisors: Maria Colonna José Antonio Lay Vincenzo Greco XXXIV ciclo – 2018 - 2022 ii Contents Introduction 1 1 Phenomenology of nuclear reactions induced by light charged particles below the Coulomb barrier 7 1.1 Bare-nucleus non-resonant cross-section . . . . . . . . . . . . . 7 1.1.1 Transmission through a potential barrier . . . . . . . . . 8 1.1.2 Barrier penetrability in WKB approximation . . . . . . 15 1.1.3 Application the Li 6+p→He 3+αreaction . . . . . . 20 1.2 Screeningeffects .......................... 24 1.2.1 The screening potential approach . . . . . . . . . . . . . 24 1.2.2 Screening by atomic electrons . . . . . . . . . . . . . . . 26 1.2.3 The atomic electron screening problem: application to the Li 6+p→He 3+αreaction ............. 29 2 Ground-state properties of clustered systems 41 2.1 Anti-symmetrisation in first-quantisation formalism . . . . . . . 41 2.2 Overlap functions and spectroscopic factors . . . . . . . . . . . 43 2.2.1 General treatment . . . . . . . . . . . . . . . . . . . . . 43 2.2.2 Spectroscopic factors within the independent-particle shellmodel ......................... 51 2.3 Expectation value of one-body observables . . . . . . . . . . . . 56 2.3.1 Introduction ........................ 57 2.3.2 General treatment of observables quadratic in the spatialposition......................... 59 2.3.3 Root-mean-square radius . . . . . . . . . . . . . . . . . 71 2.3.4 Electric quadrupole moment . . . . . . . . . . . . . . . . 73 3 Direct transfer nuclear reactions 77 3.1 Formal exact expressions for the reaction cross-section . . . . . 77 3.1.1 Introduction ........................ 78 3.1.2 Plane-wave formulation . . . . . . . . . . . . . . . . . . 80 3.1.3 Distorted-wave formulation . . . . . . . . . . . . . . . . 85 3.1.4 Generalized distorted-wave formulation . . . . . . . . . 86 3.2 One-particle transfer . . . . . . . . . . . . . . . . . . . . . . . . 89 3.2.1 Distorted-wave Born approximation (DWBA) . . . . . . 90 3.2.2 Coupled-channels approaches . . . . . . . . . . . . . . . 91 3.3 Two-particle transfer . . . . . . . . . . . . . . . . . . . . . . . . 94 3.3.1 First-order (“simultaneous”) contribution . . . . . . . . 95 3.3.2 Second-order (“sequential”) contribution . . . . . . . . . 101 iii Contents 4 The Li 6+p→He 3+αtransfer reaction 105 4.1 DWBA deuteron transfer . . . . . . . . . . . . . . . . . . . . . 106 4.1.1 Optical potentials . . . . . . . . . . . . . . . . . . . . . 106 4.1.2 Overlap functions . . . . . . . . . . . . . . . . . . . . . . 110 4.1.3 Transfer cross-section . . . . . . . . . . . . . . . . . . . 111 4.2 DWBA p +ntransfer.......................117 4.2.1 Single-nucleon states . . . . . . . . . . . . . . . . . . . . 117 4.2.2 Three-particle wave-functions . . . . . . . . . . . . . . . 121 4.2.3 Transfer cross-section . . . . . . . . . . . . . . . . . . . 128 4.3 GGW deuteron transfer . . . . . . . . . . . . . . . . . . . . . . 141 5 Impact of Li 6ground-state deformation on barrier penetrability 145 5.1 Classical cluster model . . . . . . . . . . . . . . . . . . . . . . . 145 5.2 Quantum-mechanical cluster model . . . . . . . . . . . . . . . . 149 5.2.1 Construction of the Li 6di-cluster model deformed state 150 5.2.2 Construction of the projectile-target potential . . . . . . 156 5.2.3 The Li 6+p barrier penetrability . . . . . . . . . . . . . 160 Conclusions 165 A Angular momentum and spherical harmonics 171 B Computational aspects 177 B.1 Experimental data sources and treatment . . . . . . . . . . . . 177 B.2 Adopted potentials . . . . . . . . . . . . . . . . . . . . . . . . . 178 Bibliography 181 iv List of Figures 1.1 Li 6+p→He 3+αbare-nucleus experimental astrophysical factor ................................ 22 1.2 Li 6+p→He 3+αbare-nucleus penetrability fit . . . . . . . . 23 1.3 Li 6+p→He 3+αlow-energy experimental astrophysical factor 30 1.4 Li 6+p→He 3+αscreening enhancement factor . . . . . . . 31 1.5 Li 6+p→He 3+αscreening potential . . . . . . . . . . . . . 32 1.6 Fit of Li 6+p→He 3+αdirect data measurements . . . . . . 34 1.7 Li 6+p→He 3+αdata rescaled by adiabatic screening . . . . 37 1.8 Non-standard phenomenological models for Li 6+p→He 3+α39 4.1 Li 6+p elastic-scattering phase-shifts . . . . . . . . . . . . . . 107 4.2 He 3+αelastic-scattering cross-section sample . . . . . . . . . 109 4.3 Li 6+p→He 3+αdeuteron-transfer calculation . . . . . . . . 112 4.4 Li 6+p→He 3+αdeuteron-transfer partial-wave expansion . 114 4.5 Deuteron-transfer calculation with different He 3–αpotentials . 115 4.6 Li 6single-nucleon radial wave-functions . . . . . . . . . . . . . 119 4.7 p+n+p radial probability density function . . . . . . . . . . . 124 4.8 α+p+n radial probability density function . . . . . . . . . . . 127 4.9 Role of Li 62sshell on Li 6+p→He 3+αtwo-nucleon transfer 129 4.10 Simultaneous and sequential contributions to Li 6+p→He 3+α131 4.11 Li 6+p→He 3+αtwo-nucleon-transfer partial-wave expansion132 4.12 Two-nucleon-transfer partial-wave expansion for alternative Li 6133 4.13 Prior-post comparison for Li 6+p→He 3+αtwo-nucleon transfer ...............................135 4.14 Comparison of Li 6+p→He 3+αoneand two-particle transfer136 4.15 α–d (effective) potential . . . . . . . . . . . . . . . . . . . . . . 138 4.16 Li 6+p→He 3+αcalculations rescaled on experimental data 140 4.17 Preliminary Li 6+p→He 3+αGGW deuteron-transfer calculation ...............................142 5.1 Li 6– Li 6potential within classical cluster model . . . . . . . . . 147 5.2 Li 6– Li 6average barrier penetrability within classical cluster model ................................148 5.3 α–d radial wave-functions . . . . . . . . . . . . . . . . . . . . . 152 5.4 Li 6–p form factor for a deformed Li 6wave-function . . . . . . 160 5.5 Li 6–p barrier penetrability for a deformed Li 6wave-function . 161 v List of Tables 5.6 Li 6–p average barrier penetrability for a spherical or deformed Li 6wave-function..........................162 List of Tables 1.1 Screening potential determination for several reactions . . . . . 34 4.1 Properties of Be 7bound states . . . . . . . . . . . . . . . . . . 109 4.2 Decomposition of the He 3three-particle wave-function . . . . . 123 4.3 Decomposition of the Li 6three-particle wave-function . . . . . 126 B.1 Experimental ground-state structure data . . . . . . . . . . . . 178 B.2 Parameters of the adopted potentials . . . . . . . . . . . . . . . 179 vi Introduction Since the dawn of the field, with the first experiments related to α-particle decay, and up until the most recent advances in e.g. ab initio methods, a wide fraction of nuclear physics studies revolve around the interplay between the structure adopted to depict nuclei and/or nucleons and the models describing their interactions, both bound to the role of approximated representations due to the complex and non-perturbative nature of the underlying fundamental interactions. This also caused the developments in the two directions to be correlated to a certain extent. Very heavy nuclear systems more easily lend themselves to a description in terms of spatial density and other properties that may be commonly encountered in macroscopic physics (or at least their quantum counterpart), as they show a behaviour comparable to that of a fluid, including the possibility of undergoing phase transitions, and often respond well to semi-classical treatments. In the “intermediate” mass regime, shellmodel descriptions of the nuclear structure proved to be a great success, as the mutual interaction between a set of nucleons, thanks to the effects of anti-symmetrisation, can in fact be reasonably accounted for as the result of a self-generated mean-field, which provides an accurate description of the system when appropriately corrected by the residual correlations. Nuclear reactions are then routinely employed to deduce relevant properties regarding such structure, and vice-versa. In very light nuclei, which are the subject of study of the present work, the available degrees of freedom are sufficiently small in number that more microscopic techniques can be attempted, both for the structure of isolated systems (as the No-Core Shell Model [1] or Quantum Monte Carlo [2] methods) and even for nuclear reactions (as the Resonating Group Method [3]). These approaches bear considerable predictive power, being based on the elementary interaction at the nucleon level with little reliance on phenomenological constraints. At the same time, such lack of flexibility complicates the physical interpretation of disagreements with expected or measured properties, and makes it harder to identify which ingredients or features of the model are responsible of a given result. More macroscopic models, in contrast, find their roots in general features observed in the regime of interest, and need to be supplied with several phenomenological ingredients, mainly in the form of model Hamiltonians and general structure properties, adapted to the specific situation at hand. This also implies that the impact of each ingredient on the model predictions can be explored with relative ease, opening for the possibility of a qualitative (and not merely numerical) understanding of the observations. In this sense, macroscopic approaches can complement and 1 Introduction complete ab-initio studies, and their application thus maintains significant scientific interest. Specifically, in regard to the description of scattering and reaction phenomena between light particles, the same sort of techniques often adopted for somewhat heavier systems can be applied with profit: optical models, distorted-wave Born approximation calculations and coupled-channels schemes (an in-depth treatment of such concepts may be found for instance in [4]) are routinely employed in this regime, for instance in the description of reactions involving exotic nuclei (see for instance [5]). As for the representation of the structure of light nuclear systems, an excellent framework is provided by cluster models. In view of the general expectation that the characteristics of the nuclear interaction and anti-symmetrisation can favour the formation of correlated clusters of particles within a nucleus (see for instance [6, sec. 3.2]), a system might be described as a bound state of a very small number of subsystems (rarely more than three, in practice), each with properties resembling those of the corresponding nuclide in isolation. Most or all the non-essential degrees of freedom regarding the relative motion of particles forming each cluster are then neglected, simplifying the problem to a tractable level. Such description is especially fruitful at relatively small excitation energies, where resonances are sparse and there is often sufficient experimental information to pinpoint the main characteristics of each excited state and the clustered structure which best describes it. This work is in particular concerned with the study of nuclear reactions between light charged ions at incident energies around and below the reactants Coulomb barrier, with a focus on the energy range of astrophysical interest for Big Bang and quiescent stellar processes. Cross-sections of reactions pertaining to this regime are of great interest in the modelling of associated astrophysical scenarios, in particular for nucleosynthesis and stellar evolution, where they appear as key physical ingredients. For instance, reactions destroying Li 6are interesting with regard to the lithium-depletion problem, and in the study of pre-main-sequence stars [7]. Other examples include several reactions involving lithium, beryllium and boron isotopes, which are all nuclei known to exhibit a pronounced cluster structure. Some of these processes can also be important in the field of controlled nuclear energy production. The investigation of this class of collision processes is a challenging pursuit from both the theoretical and experimental point of view. Regarding the former, despite ongoing efforts current microscopic calculations are often not able to provide a fully satisfactory account of available experimental data regarding cross-sections (see e.g. [8] and references therein for the case of the Li 6+p→He 3+αreaction, and [9] and references therein for the Li 6+p→Be 7+γ), and in some cases even ground-state structure properties (see [10] for a review, although not so recent, in particular regarding Li 6). In this respect, the possible influence of clustering effects on the lowenergy reaction dynamics has received some attention (see e.g. [11] for a Faddeev Li 6+p→He 3+αcalculation including dynamic excitations in the 2 inter-cluster motion) and still represents an interesting field of study. From the experimental point of view, the reactants electrostatic repulsion causes the cross-section to depend exponentially on energy, being reduced to values of the order of the nanobarn or less within the energy range of interest. Direct measurements of these reactions are consequently very difficult and time-consuming, and normally require heavy shielding from all background sources (see e.g. [12] for a review). A valuable contribution in the field is given by indirect measurement methods, in which the theoretical understanding on nuclear reaction processes is employed to devise techniques yielding the desired information through experiments which are easier to perform than the corresponding direct measurement (see e.g. [13] for a review). Another complication affecting the processes of interest in this work is that, when approaching the stellar energy range, the structure of matter above the nuclear level (atoms, plasmas, solids,…) starts playing a relevant role. For instance, it is well known that the cross-section of a reaction between charged particles directly measured in a fixed target experiment at low energies shows an enhancement due to the screening of the nuclear charge operated by atomic electrons. In order to correctly describe the process of astrophysical interest, a sufficiently accurate prediction of the screening effects acting both in laboratory and in the relevant astrophysical site is thus required. Extensive theoretical work exist in this direction (see [14, 15], just to mention two relevant examples). From the experimental perspective, a direct measurement of screening effects in plasma environments with conditions comparable with the astrophysical ones is challenging. Concerning standard laboratory measurements employing atomic targets, the situation is rather involved, as the cross-section enhancement determined experimentally for several reactions is larger than the upper limit predicted by atomic theory (see table 1.1 or [16] for some examples). This anomalous observation is known as the atomic electron screening problem. It was proposed in [16] that these deviations might be connected to clustering phenomena. The main goal of this work is to investigate the sensitivity of nuclear reaction dynamics, and more specifically the cross-section predictions, to the description of the structure of each reactant, and in particular to clustering phenomena. Emphasis is given to results concerning the astrophysical energy range, given the interest of its applications, and the reactions affected by the electron screening problem, with the aim of advancing its understanding. The practical applications are focused on the Li 6nucleus, and precisely the Li 6+p→He 3+αtransfer reaction and the barrier penetrability of the Li 6+p system. This thesis is structured as follows. After this introduction, chapter 1 treats some phenomenological aspects of the class of nuclear reactions of interest in this work, both taking place in vacuum (“bare-nuclei”) or immersed in an external medium. The tools developed in this chapter, in spite of (and thanks to) their simplicity, allow to understand the main features of the non-resonant sub-Coulomb reaction cross-section, providing analytical expressions that can 3 Chapter 1. Phenomenology of sub-Coulomb nuclear reactions spherical coordinates, see [20, sec. VIII, eq. (B-20)]) is5 Jr(r) = ¯h mIm(ψ∗(r)∂rψ(r)) (1.1.4) For the systems of interest, V(r)is always the sum of the nuclei electrostatic repulsion and of a “nuclear potential” which is short-ranged. Hence, in the limit of high distances (r→+∞), the potential always goes asymptotically as 1/r. The potential is expected to not diverge negatively anywhere, i.e. infr V(r)>−∞. A positive divergence can be accepted only toward r→0: this is found in some descriptions of the Coulomb or nuclear core repulsion6. Regarding the definition of Rn, the nuclear radius should arguably depend only on V(r)itself (and not, for instance, on the collision energy). If Vis not isotropic, an angle-dependent (or angular-momentum dependent) radius may in principle be considered. For simplicity, in the following qualitative discussion let V0(r)be the central part of V(r), and restrict to only V0to define Rn. If V0has a very sharp profile, Rnmay be defined as the radius where the nuclear part of the potential becomes appreciably different from zero. In general, the effective nuclear radius could be simply set independently of the specific form of Vthrough some convention. A common choice in this direction (see e.g. [21, eq. (4-105)]), in view of the general systematics regarding nuclear sizes, is Rn=r0·(A1/3 1+A1/3 2)(1.1.5) where Aiis reactant imass number, and the reduced radius r0is taken as a phenomenological constant, usually of the order of 1fm. The approach favoured in this work is instead to set Rnto the outermost local maximum of the V(r): at radii smaller than this threshold, the nuclear potential must be attractive, and growing fast enough to counter the trend given by the Coulomb repulsion. The choice is also advantageous in case a WKB solution is sought (see section 1.1.2), because it helps meeting the validity conditions of the approximation. Partial-wave expansion If V(r)is central, or more generally includes only terms which conserve both the projectile-target orbital angular momentum modulus and the total angular momentum (as spin-orbit and spin-spin couplings), σ(E)may be decomposed in partial waves (see for instance [4, sec. 4.3], [22, eq. (3.2.29)]) with definite projectile-target relative orbital angular momentum modulus quantum number l, total reactants spin modulus quantum number S, total angular momentum modulus Jand total projection M(a different coupling order for 5Im zdenotes the imaginary part of z. 6When decomposing in partial waves (see below), such a divergence always appears in the effective radial potential for any orbital angular momentum greater than 0, thus a diverging V(r)does not complicate the model significantly. 10 1.1 Bare-nucleus non-resonant cross-section the spins may be more appropriate, depending on the potential), yielding σ(E) = π k2∑︂ l,S,J 2J+ 1 (2s1+ 1)(s2+ 1)Tl,S,J (E)(1.1.6) where Tl,S,J (E)is the radial transmission coefficient for the given partial wave, independent of M, which can be found in terms of the reduced radial wavefunction, ul,S,J (r), for the partial wave of interest. The general complete solution of the system, |Ψ⟩, can be expanded in partial waves (see appendix A), |Ψ⟩=∑︂ l,S,J,M cl,S,J,M ul,S,J (r) r|l, S, (J, M)⟩(1.1.7) where the coefficients care found from the boundary conditions, and the angular-momentum state |l, S, (J, M)⟩can be expanded using equation (A.6) if useful. Each reduced radial wave-function uis in turn an eigenstate of the effective radial projectile-target Hamiltonian of interest, Hl,S,J , with the desired boundary conditions and eigenvalue E. Taking into account equation (A.5), it is Hl,S,J =−d2 dr2+Vl,S,J (r),Vl,S,J (r) = Vl,S,J (r) + ¯h2 2m l(l+ 1) r2(1.1.8) where Vl,S,J (r)is the relevant component of the “actual” potential of the system, and Vthe effective radial potential for partial wave l. Note that s1+s2 ∑︂ S=|s1−s2| l+S ∑︂ J=|l−S| (2J+ 1) = (2s1+ 1)(s2+ 1)(2l+ 1) (1.1.9) which confirms that, if Vl,S,J (r)actually does not depend on Sand J, the spins can be ignored in equation (1.1.6). Finally, perform the integral in equation (1.1.3) using the complete solution Ψwritten in equation (1.1.7) as wave-function, employing equations (A.2) and (A.7), and noting that some terms cancel out as they are purely real, finding: −r2∫︂4πJr(r)dΩ = −¯h m∑︂ l,S,J,M |cl,S,J,M |2Im(︁u∗ l,S,J (r)∂rul,S,J (r))︁(1.1.10) As a result, it can be convenient to define a “reduced” radial current density for each partial wave as J[ul,S,J ](r) = ¯h mIm(︁u∗ l,S,J (r)∂rul,S,J (r))︁(1.1.11) absorbing the r2in equation (1.1.3) through the use of reduced functions, so that the contribution from each partial wave appears as a purely one11 Chapter 1. Phenomenology of sub-Coulomb nuclear reactions dimensional problem. More explicitly, equation (1.1.3) is rewritten as ∂t∫︂4 3πr3|ψ(r)|2d3r=−∑︂ l,S,J |cl,S,J |2J[ul,S,J ](r)(1.1.12) where |cl,S,J |2=∑︁M|cl,S,J,M |2. Since the nuclear potential is short-ranged, at high distances the effective radial potential is central (does not depend on Sand J) and includes only the Coulomb and centrifugal repulsion: Vl(r≫Rn) = ¯hcαe Z1Z2 r+¯h2 2m l(l+ 1) r2(1.1.13) where αeis the fine-structure constant and cthe speed of light. In the same region, the reduced radial solution is a combination of the spherical Coulomb wave-functions, H+ l(kr)and H− l(kr), defined in [23, sec. 33]7. Note that these functions are dimensionless, while the square modulus of a physical reduced radial wave-function bears the dimensions of a one-dimensional spatial density: such dimension is included in the normalisation factor. In practice, results are independent of the chosen normalisation. The reduced radial probability current density, defined in equation (1.1.11), associated to the wave-function AH− l(kr) + BH+ l(kr), with Aand Barbitrary constants, is found to be, using [23, eq. 33.2.13, 33.4.4], J[AH− l+BH+ l](r) = (︁B2−A2)︁¯hk m(1.1.14) Impose decay boundary conditions. In the region where equation (1.1.13) holds, the solution is proportional just to the spherical outgoing Coulomb wave-function: ul,S,J (r≫1) →B H+ l(kr)(1.1.15) with arbitrary normalisation B. Asymptotically, in the limit of r→+∞ (and fixed energy), H+ lbecomes an outgoing plane-wave (as the potential vanishes). Transmission for sharp-edge nuclear potentials The outgoing radial probability current at r=Rndepends on the details of the nuclear potential VlSJ (r). Consider a “sharp-edge” nuclear potential, namely the limit where VlSJ (r)is exactly the point-like Coulomb potential for all r > Rn. This implies that, in the exterior region, the electrostatic repulsion can be treated as the Coulomb potential between two point sources, and the nuclear interaction is zero. Then, the problem is reduced to properly 7These functions depend also on the Sommerfeld parameter η, defined in equation (1.1.22), but the associated index is omitted for brevity. 12 1.1 Bare-nucleus non-resonant cross-section matching the exterior solution with the interior one. For the choice of Rn to be sensible, it is expected that at the same time the nuclear potential is significantly different from zero for r < Rn. In the spirit of the picture where the system source is formally an outgoing wave coming from negative radii (see text commenting equation (1.1.2)), a simple possibility to model the internal region, at least near r=Rn, is to consider a combination of an outgoing plane wave with wave-number k ˜, generated by the source, plus an ingoing plane wave produced from the scattering at the interface in r=Rn: u(r) = Deik ˜r+Ae−ik ˜r(1.1.16) where Dand Aare constants to be determined by matching with the external wave-function. k ˜in general depends on kand on the features of the potential (including the quantum numbers l, S, J). For brevity, k ˜and uare written with no index. The reduced radial current at Rnconnected to the outgoing wave, J[Deik ˜r], is |D|2¯hk ˜/m. It is stressed that scattered waves (either in the internal or external solution, depending on the boundary conditions) appear whenever the transmission is not complete (TlSJ <1if and only if A= 0). The wave-function in equation (1.1.16) can be seen as the solution for a real and negative constant total potential, VlSJ (r < Rn) = V<, with k ˜=1 ¯h√︁2m(E−V<)(1.1.17) If instead a real constant Vnis assumed just for the nuclear component of the potential (“spherical-well potential”), the internal solution is a combination of spherical Coulomb functions, with k ˜=√︁2m(E−V)/¯h, which however reduces again to equation (1.1.16) in the limit of k ˜Rn→+∞, which at the low energies of interest corresponds to V→ −∞. It is also possible to model the nuclear potential adding a positive imaginary part, in which case the wave-function in the interior region, near the interface, acquires an expression analogous to that shown in [17, eq. (3.73)]. This approach is in principle more consistent, but the final qualitative result shown in the following would be the same, provided that k ˜is appropriately redefined. Imposing continuity of the wave-function and of its first-derivative between equations (1.1.15) and (1.1.16) at the interface r=Rn, one finds B D=2eik ˜Rn H+ l(kRn)−i∂rH+ l(kRn)/k ˜(1.1.18) The transmission coefficient is the ratio between the outgoing radial current densities associated to the wave-functions in equations (1.1.15) and (1.1.16): Tl,S,J (E) = J[BH+ l] J[Deik ˜r]=k k ˜/4 1 H+ l(kRn)−i∂rH+ l(kRn)/k ˜2(1.1.19) 13 Chapter 1. Phenomenology of sub-Coulomb nuclear reactions In the limit of k→ ∞, it is k ˜→kand H+ l(kr)→exp(ikr), thus as expected Tl,S,J (E→ ∞) = 1. In the more interesting limit of strong nuclear potential more precisely of k ˜→+∞, the term involving the derivative of the Coulomb function may be neglected. Furthermore, within the rather simple model employed here, it is often reasonable to choose V<to be independent of the reactants spins, so that k ˜depends only on kand l, and the transmission coefficient only on l,Eand Rn. Finally, if k≪k ˜, the momentum in the internal region can be expected to not depend strongly on k(see the previous expressions given for k ˜), and 4/k ˜ may thus be approximated as ClRn, where Clis a dimensionless constant. When using the formula to fit experimental data, Clcan be absorbed into another constant appearing in the expression (see section 1.1.3), reducing the number of free parameters. In literature (see e.g. [22, eq. (7.4.17)]), the result is normally presented as: Tl=kRn H+ l(kRn)2(1.1.20) Note that this Tlis just the penetrability factor appearing in R-Matrix theory (see e.g. [22, eq. (10.2.5)]). It is emphasised again that, as far as a transmission coefficient is of interest, the approximation in equation (1.1.20) is accurate only for small k(for instance, it admits Tl>1for klarge enough). If an approximated expression was required for high values of k, setting k ˜=Clk would be more appropriate. In fact, sometimes equation (1.1.20) is quoted without the kRncoefficient (see e.g. [21, eq. (4-124)]). However, it also pointed out that a model describing only barrier penetrability, as the present one, is useful only at energies below the Coulomb barrier height, which can be defined as the Coulomb potential at r=Rn. This in turn is, typically, much smaller than the depth of the total potential Vin the interior region near r=Rn, with the exception of partial waves with lso high that the penetrability is negligible anyway at the energies of interest. As a result, equation (1.1.20) can be expected to be accurate in the relevant regime. Finally, consider the limit of vanishing nuclear radius. Using [23, eq. 5.4.3, 5.5.1, 33.2.5, 33.5.1, 33.5.2], lim Rn→0Tl,S,J (E) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 0if l > 0 8πkη k ˜ 1 1 + 2πkη k ˜1 e2πη−12 1 e2πη −1if l= 0 (1.1.21) where ηis the Sommerfeld parameter, defined as in [23, eq. 33.22.4], η=αeZ1Z2√︃mc2 2E=αeZ1Z2 mc ¯hk (1.1.22) In particular, note that kη depends only on the reactants mass and charge. If kis very small, so that k≪k ˜, with k ˜approximately independent of k, and 14 1.1 Bare-nucleus non-resonant cross-section exp(2πη)≫1, the result can be approximated as Tl=0,SJ (E)≈CSJ e−2πη, where CSJ is an appropriate constant. Assuming a nuclear potential independent of the reactants spin, the corresponding cross-section for barrier penetration, σ(E)in equation (1.1.6), is π k2Ce−2πη. This result motivates the definition of the astrophysical S-factor 8for any given nuclear reaction between charged particles from an initial state ito a final state f,Si→f(E), in terms of the corresponding non-polarised angle-integrated cross-section σi→f(E): σi→f(E) = 1 Ee−2πη(E)Si→f(E)(1.1.23) At energies above the Coulomb barrier, the the astrophysical factor and the integrated cross-section essentially differ only by the 1/Ecoefficient. Below the barrier, S(E)is a much more slowly varying function of energy than σ(E), because the main contribution connected to the barrier penetrability has been factored out. For this reason, in this work all graphics of angle-integrated (non-elastic) cross-sections are shown as astrophysical factor plots. 1.1.2 Barrier penetrability in WKB approximation A convenient expression for the transmission coefficient is obtained employing the Wentzel-Kramers-Brillouin-Jeffreys (“WKB”) approximation (see e.g. [27, sec. VI.II], [19, sec. 2.4] or [21, sec. 4-5] and references therein, or e.g. [26, sec. 2.4.3] for the analogous barrier-slicing approach). For any given partialwave, let k(r) = √︂2m[E−Vl,S,J (r)]/¯h(1.1.24) where, as before, Eis the system collision energy, mthe reduced mass, and VlSJ (r)the effective radial potential, defined in equation (1.1.8). k(r)can be either positive or purely imaginary. In a region where the WKB approximation is valid, the solution of a one-dimensional problem (as the reduced radial wave-function for a given partial wave) can be written as a combination of two functions, u+ WKB and u− WKB, as in [19, eq. (2.4.35), (2.4.38)]. Here, these are defined to be dimensionless, and the trivial time-dependence (the state is stationary) was dropped: u± WKB(r) = √︄ k∞ k(r)exp(︃±i∫︂r r0 k(x)dx)︃(1.1.25) where k∞=limr→+∞k(r)(in section 1.1.1 this was simply denoted by k), and r0is an arbitrary point within the region under study. The reduced radial current, defined in equation (1.1.11), associated to a generic combination 8In some older references, see e.g. [24, 25], the same name is employed for a slightly different quantity, which depends on the effective nuclear radius (see also [21, eq. (4-157)]). The definition given here is the most common, especially in recent literature, see e.g. [21, eq. (4-36)] or [26, eq. (3.71)], and involves only model-independent parameters known with good accuracy. 15 Chapter 1. Phenomenology of sub-Coulomb nuclear reactions Au− WKB(r) + Bu+ WKB(r), with arbitrary constants Aand B, is (the “WKB” subscript is dropped for brevity) J[Au−+Bu+](r) = ⎧ ⎪ ⎨ ⎪ ⎩ ¯h mk∞(︂|B|2−|A|2)︂if k(r)∈R ¯h mk∞2Im(AB∗)if ik(r)∈R (1.1.26) Following [27, eq. (VI.47)], the WKB approximation may be considered to be valid when |∂rk(r)| ≪ k2(r)(1.1.27) For instance, equation (1.1.25) is exact in a region where VlSJ is constant. On the contrary, the approximation is not accurate around classical turning points, i.e. points where E=VlSJ (r). In that case, it is still possible to write WKB solutions in the regions where the approximation is valid, and then match them through specific connection formulas, discussed for instance in [27, sec. VI.9] and [19, sec. 2.4] and references therein. Such situation is in fact expected for the problem at hand at sufficiently low energies. In particular, suppose that, for r > Rn, there is only one classical turning point, at position Rc, and that the WKB approximation is valid around Rn: for this to hold, it is necessary (but not sufficient) that Rn≪Rcand E≪ VlSJ (Rn). At r→+∞, if the potential is given by equation (1.1.13) the WKB is certainly valid as well. The appropriate connection formulas are then given by [27, eq. (VI.49), (VI.50)] (see also [27, sec. VI.10]). Using decay boundary conditions, and conveniently setting the normalisation factor, in the region r≫Rcthe WKB wave-function can be written as Be−iπ/4u+ WKB(r), setting r0=Rcin equation (1.1.25). The matching solution for r≪Rcis u(r≪Rc) = B[︃1 2u− WKB(r)−iu+ WKB(r)]︃(1.1.28) The real part of this function, compared to the imaginary one, is expected to be very small in module for r≪Rc, but it is in general necessary to properly match the wave-function at r=Rn. If the system wave-function for r < Rnis equation (1.1.16), with steps analogous to those followed to obtain equation (1.1.19), the transmission coefficient, Tl,S,J (E), is 4exp(︂−2∫︁Rc Rn|k(x)|dx)︂|k(Rn)|/k ˜ [︂1 2+i∂r|k(Rn)| 4|k(Rn)|k ˜−i|k(Rn)| 2k ˜]︂e−2∫︁Rc Rn|k(x)|dx+[︂∂r|k(Rn)| 2|k(Rn)|k ˜+|k(Rn)| k ˜−i]︂ 2 (1.1.29) If the WKB approximation is valid at r=Rn, from the conditions given above it can be expected that the term involving the exponential in the denominator can be neglected. Furthermore, unless k ˜≪k, the terms involving ∂r|k(r)| can be neglected by virtue of equation (1.1.27). The expression then simplifies 16 1.1 Bare-nucleus non-resonant cross-section to Tl,S,J (E) = 4|k(Rn)|k ˜ |k(Rn)|2+k ˜2exp(︃−2∫︂Rc Rn|k(x)|dx)︃= = 4√︁(Vl(Rn)−E)(E−V<) Vl(Rn)−V< exp(︃−2∫︂Rc Rn|k(x)|dx)︃(1.1.30) as in [27, sec. VI.10]. In the last equality, k ˜was defined through equation (1.1.17). For small collision energies, the factor before the exponential may be approximated as independent of E, thus writing TlSJ =ClSJ exp(. . . ). Finally, consider the case where E≫maxr>RnVlSJ (r)(the process is taking place “above the barrier”), and equation (1.1.27) holds for all r > Rn. Then, the exterior WKB solution Bu+ WKB(r)can be applied in the same region, and the reasoning yielding equation (1.1.19) can be repeated identically changing H+ lwith u+ WKB, regardless of the precise form of the potential. The transmission coefficient is consequently Tl,S,J (E) = k(Rn) k ˜/4 1 1 + k(Rn)/k ˜+i∂rk(Rn) 2k(Rn)k ˜2(1.1.31) Again, unless k ˜≪k, the term involving ∂r|k(r)|can be neglected using equation (1.1.27), so that Tl,S,J =1 4k(Rn)k ˜/[︂k(Rn) + k ˜]︂2, which coincides with equation (1.1.30) setting Rc=Rn, and also matches equation (1.1.19) in the limit of high collision energy, where k(Rn)≈k∞and H+ l≈exp(ik∞r). Transmission for central sharp-edge nuclear potentials Equation (1.1.30) is especially useful to numerically estimate the penetrability connected to a generic potential. However, it is also instructive to apply it to the case of a central sharp-edge nuclear potential of the same kind treated exactly in section 1.1.1: for each partial wave, the effective radial potential is equation (1.1.13) for r > Rn, and a constant V<for r < Rn(remind that the region at r≪Rnis irrelevant). The depth of the inner well is allowed to depend on lto account for the contribution due to the centrifugal barrier. This configuration admits analytical results, which can be manipulated formally with greater ease than equation (1.1.19), and are thus better suited to analyse some qualitative features. There is a single classical turning point Rc(l), which is the point where Vl(Rc) = E. Let R0 c=Rc(l= 0) = ¯hcαeZ1Z2/E. Then Rc(l) = 1 2R0 c+√︃1 4(R0 c)2+l(l+ 1) k2(1.1.32) where k=√2mE/¯has before. As required to apply equation (1.1.30), restrict to the case where Rc(l= 0) > Rn, so that the classically forbidden region 17 Chapter 1. Phenomenology of sub-Coulomb nuclear reactions overlaps with the domain of interest. The integral in equation (1.1.30) can be simplified applying the variable change r=R0 cx. Let γand βbe the new lower and upper bounds of the integration domain. It is γ=Rn/R0 c, which is also the ratio between the collision energy and the Coulomb potential at r=Rn(see equation (1.1.13)), hence 0≤γ≤1(by hypothesis). Note that γη =kRn/2, where ηis the Sommerfeld parameter defined in equation (1.1.22). Similarly, β=Rc/R0 c≥ 1. Furthermore, let B=l(l+1)/(4η2), which also equals β2−β. The argument of the exponential in equation (1.1.30), labelled −Gl(as in [22, eq. (7.4.22)]) for brevity, can be rewritten as −Gl=−4η∫︂β γ√︃1 x+B x2−1dx(1.1.33) It can be verified explicitly that a primitive of the given integrand is −∫︂√︃1 x+B x2−1dx=−√︁B+x−x2+ +√Bln(︄2B+x+ 2√B√B+x−x2 x√1+4B)︄+1 2arctan(︄1 2−x √B+x−x2)︄ (1.1.34) which is always real within the integration domain. Therefore, −Gl=−πη −4η[︄√Bln(︄2B+γ+ 2√B√︁B+γ−γ2 γ√1+4B)︄+ +1 2arctan(︄1−2γ 2√︁B+γ−γ2)︄−√︁B+γ−γ2]︄(1.1.35) Finally, it is convenient to define b=γ/B=4γη2 l(l+ 1) =2ηkRn l(l+ 1) =2αeZ1Z2mc2Rn ¯hc l(l+ 1) (1.1.36) which is the ratio between the Coulomb and centrifugal potential at r=Rn (see equation (1.1.13)). bis thus dimensionless and does not depend on the collision energy. Only for 1−2γ > 0(i.e. Rn< R0 c/2), which is satisfied at sufficiently low energies, the inverse tangent can be rewritten using arctan(1/x) = π/2 − arctan(x). Restrict to this case in the following. Furthermore, for simplicity, in equation (1.1.30) approximate the scaling factor before the exponential to be independent of E(see text commenting the equation). The corresponding astrophysical factor S(E), defined as in equation (1.1.23) with respect to the cross-section in equation (1.1.6), for WKB barrier penetration with a sharp18 1.1 Bare-nucleus non-resonant cross-section edge central nuclear potential, is then S(E) = ∑︂ l Cle2πη−Gl=∑︂ l Cl(︄b√︁1+4γ/b 2 + b+ 2√1 + b−bγ )︄2√︁l(l+1) · ·exp(︄2ηarctan(︄2√γ√︁1/b+ 1 −γ 1−2γ)︄+ 2√︁l(l+ 1)√︁1 + b−bγ)︄(1.1.37) where Cl=2π¯h2 m(2l+ 1) √︁Vl(Rn)V< Vl(Rn)−V< (1.1.38) For brevity, let Sl(E) = Cle2πη−Gl, which is the contribution to S(E)from l-th partial-wave9. Strictly speaking, equation (1.1.37) is ill-defined for l= 0, even though it converges to the correct result by continuity. More explicitly, setting B= 0 in equation (1.1.34), it is S0(E) = C0exp(︃2η[︃arctan(︃2√γ√1−γ 1−2γ)︃+ 2√γ√︁1−γ]︃)︃ (1.1.39) It is now of interest to qualitatively study the WKB astrophysical factor for sharp-edge barrier penetration, in the limit of very small collision energies, in particular with regard to the role of the effective nuclear radius. As computed exactly earlier, see equation (1.1.21), for Rn→0only S0(E)is non-zero, and it is approximately a constant for small collision energies10. Both these features disappear for finite values of Rn. At zero collision energy equation (1.1.37) reduces to S(E= 0) = ∑︂ l Sl(0) = ∑︂ l Cl(︄e2√1+b 2/b+1+2√1 + b/b)︄2√︁l(l+1) (1.1.40) where S0(0) = C0exp(︂4√︁2αeZ1Z2mcRn/¯h)︂. It can be seen that all partial waves contribute. However, Sl(0) monotonically decreases for increasing l(and reasonable choices of V<as a function of l). In fact, for typical values of the parameters, only the smallest orbital angular momenta contribute significantly11. In the following, consider only the l= 0 term. The astrophysical factor dependence on the collision energy can be explored evaluating its derivate. First note from equation (1.1.35) that Gl/(4η)−π/4 is just the primitive in equation (1.1.34) evaluated at x=γ. Additionally, by direct inspection it is 9This Slha no relation with the scattering S-matrix. 10Note that the WKB validity condition in equation (1.1.27) is not satisfied for l= 0 and Rn→0, if the point-like Coulomb potential is adopted. Nonetheless, the WKB formula yields the correct qualitative result, apart from a divergence in the value of C0. 11For instance, for the Li 6+p system and Rn≤10 fm, l(l+ 1)bis ≤1.8. Taking Vland V<as constants for simplicity, l= 2 contributes by less than 0.5% on S(0). 19 Chapter 1. Phenomenology of sub-Coulomb nuclear reactions a bare proton impinging on a neutral Li 6in the ground state, see section 1.2.2), at centre-of-mass energies above 75 keV the net enhancement, fe(E)−1, is approximately 1% or smaller, which is well below typical experimental errors. At smaller collision energies the correction becomes relevant, for instance at E= 10 keV (approximately the smallest energy at which direct measurements are currently available) and Uas before it is fe(E)−1≈26 %. 1.2.2 Screening by atomic electrons Consider a standard fixed-target experiment with a single projectile nucleus (which may carry some electrons) originated far away and impinging at a given collision energy on an isolated target atom. The atomic electrons screen the nuclear charges and enhance the penetrability. Following [15], the problem can be approached assuming that the system total wave-function can be decoupled in the nuclear and electronic parts (as in the Born–Oppenheimer approximation), and treating quantum-mechanically the electronic degrees of freedom. For a sufficiently small U/Eratio, the relative change in the reactants velocity may be neglected, and the motion of the nuclear degrees of freedom approximated as free ([15, eq. (3.24)] may be adopted in a numerical computation to lift the hypothesis). Furthermore, the reaction impact parameter is very small with respect to atomic distances, so it can be approximated with 0 just for the purpose of screening potential evaluation. Under these assumptions, the nuclei follow a simple linear trajectory. Once the reactants reach distances sufficiently smaller than typical atomic lengths, the electrons Hamiltonian can be approximated to the time-independent Hamiltonian generated by the compound of the two initial nuclei. If this happens at sufficiently high distances with respect to those relevant for the nuclear process (including the barrier penetration), adopting the screening potential approach is well justified. In [15, eq. (2.17)] it is estimated that the relative error on the screened cross-section induced by the adopted approximations is of the order of U/E. Since the system is isolated, the screening potential Uis just the opposite of the change in the total electron energy (namely, the sum of electrons kinetic energy, electrons-ions potential and electrons-electrons potential) between the initial state with two isolated atoms, and the corresponding final state involving the compound system. In the initial states of interest here (standard fixed-target experiments), the electrons are always bound to the nuclei, hence, when computing their energy, it is of interest to separate the contribution involving the internal state within each isolated nucleus from the one due to the reactants relative motion. The latter term does not include any potential energy, as the reactants are initially at infinite distance, and is thus denoted here as Ke. Let indexes 1 and 2 represent target and projectile in the initial state. Each electron bound to nucleus iis initially moving with the same velocity as the nucleus itself, which, neglecting the role of electrons in fixing the system centre-of-mass, in the centre-of-mass frame has modulus ¯hk/mi, where miis 26 1.2 Screening effects nucleus imass and kis the wave-number connected to the reactants relative motion, as in section 1.1.1. Then, if there are Nielectrons bound to nucleus i, it is Ke=me m1+m2(︃N1 m2 m1 +N2 m1 m2)︃E(1.2.4) where meis the electron mass. The importance of this term depends on the specific configuration under study. For instance, for the scattering of a neutral Li 6on a neutral H 1, it is Ke/E≈5×10−4, while if the hydrogen is a bare proton, it is Ke/E≈4×10−5. Finally, let index 1+2 represent the compound atom, and let Eibe the total energy in irest frame for the electrons system in the isolated atom ifound in the required state for the reaction under study. The screening potential can then be written as U=Ke+E1+E2−E1+2 (1.2.5) Hence, the problem is reduced to correctly model the electrons evolution (using [15, eq. (3.15)]). For any given initial state, a theoretical upper limit for the screening potential can be given fixing the final state to the compound atom ground state (where electrons possess the minimum possible energy), as in [15, eq. (4.19)]. For the problems of interest here, the initial state of each atom is almost invariably the ground state as well, and the maximum screening potential can be computed in terms of atomic ionization energies, which are generally very well known (tabulated values can be found e.g. in [46]). Let Xnbe the ground state of an ion with charge state n(which can be positive, zero or even negative), and B(Xn)the corresponding binding energy for ionization of all bound electrons. Then, the maximum screening potential for the scattering of Xn and Ymwith compound Zn+mis Umax =Ke+B(Zn+m)−B(Xn)−B(Ym)(1.2.6) If some reactant is in molecular form, one may in the same spirit take into account the atomization energies (namely, the energy to break the molecule into isolated atoms) of the ground states of the initial and final partitions. Since molecular bindings generally involve energies of smaller order of magnitude with respect to atomic ones, the corrections are expected to be relatively small. [47] reports experimental values for atomization enthalpies, which may be sufficient for the present purpose given that only differences between values for different molecules are of interest. In the general case, the evaluation of the enhancement factor is complicated by dynamical effects in the electrons response to the reactants motion; [48–50] are just three examples of several works discussing the problem. However, a simple result is obtained in the limiting cases of very low and high relative velocity between reactants, the adiabatic and sudden limits14. These are covered 14The system is considered isolated in both cases: this excludes, for instance, molecular 27 Chapter 1. Phenomenology of sub-Coulomb nuclear reactions in general e.g. in [27, sec. XVII.II], and are applied to the specific problem at hand in [15], which can be consulted for a more detailed treatment. In the following, the adiabatic limit formalism is briefly reviewed, as its results are useful for the purposes of the present work. Adiabatic limit In the adiabatic approximation, valid in the limit of small collision energies, the distance between the nuclei is changing slowly, and so does the Hamiltonian for the electronic degrees of freedom, which evolves in quasi-equilibrium. If the electrons initially are in a non-degenerate eigenstate of the isolated atoms, at any later time they will be found (apart for at most a phase) in the corresponding eigenstate of the new Hamiltonian, where the correspondence is set by continuity with respect to time. Note that the notion of non-degenerate state here regards the whole system (not each ion separately). For instance, if the reactants are isotopes and each occupies a distinct atomic state, then the whole system state is approximately degenerate with the configuration obtained swapping the two reactants. The set of corresponding states in the final system will in general be a superposition of all degenerate levels in the initial system, and will not be degenerate themselves. [15, sec. 5] discusses a specific example regarding this issue. Eigenstates (even excited ones) of the electrons Hamiltonian for isolated atoms are normally rather well known, thus, in general, the main difficulty in the calculation lies in the correct identification of the electrons final state. If the initial electron state is a superposition of several eigenstates i, with coefficients αi, the final state will be a superposition of the corresponding new eigenstates, with all coefficients bearing the same modulus. For each eigenstate the above treatment can be applied, obtaining from equation (1.2.5) a different screening potential Ui. It is then found [15, eq. (4.15)] that the screened cross-section is σe(E) = ∑︂ i|αi|2σb(E+Ui)(1.2.7) A particularly simple but widely applicable case is the one where both the initial and final states are the ground-state, meaning that there is no degeneracy issue. Then, the adiabatic-limit screening potential is just the upper limit given in equation (1.2.6) with Ke= 0 (since here E→0). For instance, this case can be applied to the scattering of a system initially found in the ground state of a neutral Li and a neutral H, which maps into the ground state of neutral Be, yielding an adiabatic-limit screening potential of 182 eV [15, tab. 4]. Similarly, if the proton is bare, the screening potential in the same limit is 186 eV [15, tab. 4]. A Li+ion impinging on a neutral H yields U= 178 eV, which is still very similar to the value for neutral screening effects. The term adiabatic here refers instead to quasi-static processes. 28 1.2 Screening effects atoms, but if the lithium is initially missing two electrons (which leaves its inner electronic shell open), then the adiabatic-limit screening potential is 236 eV, as the ground-state-to-ground-state transition is energetically more convenient in this case. This shows that an accurate assessment of the beam particles charge state immediately before they trigger a nuclear reaction can have an impact to correctly evaluate screening effects. Given the very low collision energies of interest, it may be expected that beam particles will pick up electrons from the target with relative ease (and that the initial charge state was not very high to begin with), hence assuming a neutral-on-neutral collision may be reasonable. This is the working hypothesis adopted in this study. If Li is neutral and there is a neutral H2molecule in place of the hydrogen atom, following the reasoning in the text commenting equation (1.2.6), and considering a BeH molecule in the final state 15, one would deduce a screening potential of 180 eV, which is only marginally different from the purely atomic case (the correction due to the electrons kinetic energy, computed as in equation (1.2.4), is more important even at E= 10 keV). For simplicity, corrections due to the molecular state are consequently ignored in the following. Finally, it is stressed that, while the adiabatic limit sometimes coincides with the theoretical upper limit at zero collision energies, it is not only an upper limit: if the screening potential approach and the approximations employed to treat the atomic case are applicable at all, then Uis expected to tend precisely to the adiabatic limit for sufficiently small E(which is also the limit where screening corrections are most important). In particular, the theoretical expectation would be violated both by an exceedingly small and an exceedingly high screening enhancement at small energies. 1.2.3 The atomic electron screening problem: application to the Li 6+p→He 3+αreaction Figure 1.3 shows the experimental astrophysical factor for the Li 6+p→ He 3+αreaction from [7, 30–32] (the same sources of data appearing in figure 1.2) and [38] (precisely, the updated version shown in [32] is employed), at collision energies smaller than 120 keV. More data at higher energies was shown in figure 1.1. Since the present study is concerned with screening effects due only to atomic electrons, the data shown includes only reactions where reactants can reasonably be considered as atoms (or at most as molecules). In particular, the data in [38] regarding metallic targets are not included. Apart from the Trojan Horse Method data from [7], the cross-section for all measurements reported in figure 1.3 are expected to be enhanced by the screening due to atomic electrons bound to reactants. For comparison, the 15Remind that this is the “final state” only with regard to the electronic configuration, in the adiabatic limit, immediately before the quantum tunnelling between the nuclei starts. 29 Chapter 1. Phenomenology of sub-Coulomb nuclear reactions Bare penetrability fit Elwyn et al. 1979 Engstler et al. 1992 Lamia et al. 2013 Cruz et al. 2007 Cruz et al. 2005 20 40 60 80 100 120 3.0 3.5 4.0 4.5 5.0 5.5 Centre-of-mass collision energy [keV] Astrophysical S-factor [MeV b] 6Li +p→3He +α Figure 1.3: Experimentally measured astrophysical factor (defined in equation (1.1.23)) for the Li 6+p→He 3+αreaction, at collision energies below 120 keV, from [7, 30, 31] (same symbols as in figure 1.2) and [32] (violet downward and red upward triangles for data originally published in 2005 and 2007 respectively). The black solid line, shown for comparison, is the same line found in figure 1.2. figure includes the bare-nucleus penetrability fit which reproduces data at higher energies, as shown in figure 1.2, but underestimates directly measured data below approximately 50 keV. As expected, a clear disagreement is also found at the lowest energies between the direct measurement from [31] and the indirect determination from [7]. The screening effects seen in the data can be studied in several ways, briefly discussed in the following using the Li 6+p→He 3+αreaction as a concrete example. Some results reported in literature regarding other reactions are also quoted at the end of this section. Energy trend of the screening potential If the penetrability fit in figure 1.2 can be regarded as a faithful estimation of the bare-nucleus cross-section, it can be combined with screened data in figure 1.3 to extract an experimental determination of the screening enhancement factor feand the corresponding screening potential U, using equation (1.2.3). The results are shown in figures 1.4 and 1.5. The determination of the screening potential is meaningful only at relatively small collision energies, once the deviation between screened and bare cross-section becomes clearly dis30 1.2 Screening effects Engstler et al. 1992 Cruz et al. 2007 Cruz et al. 2005 20 40 60 80 100 120 1.0 1.2 1.4 1.6 Centre-of-mass collision energy [keV] Screening enhancement factor 6Li +p→3He +α Figure 1.4: Screening enhancement factor obtained taking the ratio of data points in figure 1.3 to the black solid line in the same figure. The error bars include the propagated uncertainty from the fitted bare-nucleus cross-section. tinguishable within the experimental accuracy. This is partly reflected in the uncertainties obtained for U: at higher energies, a given interval of enhancement factors is covered by wider ranges of screening potentials. In the following analysis, only data points at E < 52 keV are taken into account. In such region, all data points from [31, 32] suggest a screening potential greater that the adiabatic limit discussed in section 1.2.2 (green solid line in figure 1.5). The datasets are compatible with a constant value for U, both separately and combined16, even though data from [31] favour a higher value. The blue dashed line in figure 1.5 is the weighted average of values extracted from both [31] and [32], (399±25) eV, which is well beyond the adiabatic limit value. This suggests that the bare-nucleus cross-section shown in figure 1.2 is incompatible with a screening effect due only to atomic electrons, especially if screened data from [31] are taken into account. The analysis just discussed has the advantage of not requiring to assume a priori any specific functional form for the screening potential with respect to the energy. Its weakness lies in the necessity of first obtaining a trusted bare-nucleus cross-section, which must be derived ignoring all direct data at energies where screening effects are relevant. 16Data from [31] seem to suggest a screening potential that increases with energy. However, the uncertainties are too large to draw a conclusive statement. 31 Chapter 1. Phenomenology of sub-Coulomb nuclear reactions Data average Adiabatic limit Engstler et al. 1992 Cruz et al. 2005 10 20 30 40 50 60 70 100 200 500 1000 2000 Centre-of-mass collision energy [keV] Screening Potential [eV] 6Li +p→3He +α Figure 1.5: Screening potential extracted from the enhancement factor in figure 1.4 using equation (1.2.3). The green solid line marks U= 182 eV (adiabatic limit for neutral reactants, see section 1.2.2). The blue solid line is a fit of data at energies below 52 keV on a constant (25 degrees of freedom, χ2of 20), whose best fitting value (with its standard error) is (399 ±25) eV. Fit of direct data on a constant screening potential If the screening potential can be approximated to a constant, it is possible to directly fit equation (1.2.1) with Uas free parameter, as commonly done in literature. Sometimes, the bare-nucleus cross-section is fitted first, and the enhancement factor over the obtained trend is fitted afterwards using lowenergy direct data. This approach is essentially equivalent to computing the weighted average of the observed values of Uas in figure 1.517. It is adopted especially in older works (e.g. [31]) and when indirect data are taken into account (e.g. [7]), since these are unaffected by screening and thus cannot be fitted together with direct data18. Another possibility, discussed in [51] and references therein, is to fit both the bare-nucleus cross-section and the screening potential at the same time: the fitted function is the product of a model for the bare-nucleus cross-section (with some free parameters) and the enhancement factor in equation (1.2.2) with Uas additional free parameter. This was found to yield better fits to 17Since equation (1.2.3) is not linear, in principle a difference between the fit results could be observed. In practice, the deviations are expected to be small. 18In fact, it is possible, albeit slightly more cumbersome, to make a simultaneous fit of direct and indirect data on different functional forms sharing some parameter, which would allow to implement the sort of fit suggested in [51] including indirect data. To the author’s knowledge this has never been done in literature. 32 1.2 Screening effects direct data and lower values of U. Such procedure is followed in [32] for the Li 6+p→He 3+αreaction, using only data from [30, 32] (including the updated ones from [38]) and modelling the bare-nucleus astrophysical factor with a third-order polynomial (4 free parameters), obtaining U= (273 ± 111) eV, which is higher than but compatible with the adiabatic limit. The result could be reproduced in this work under the same conditions. However, if the bare-nucleus cross-section is instead fitted using the same model employed in figure 1.2 (equations (1.1.6) and (1.1.20) including only l= 0 and adding an overall free scaling19), a much higher screening potential is found, (388 ± 100) eV, which resembles closely the result shown in figure 1.5, albeit which greater uncertainty. The two fits just discussed, using a polynomial or the sharp-edge barrier transmission coefficient, are compared in figure 1.6. The total (screened) cross-section are essentially the same in both models in the energy range where fitted data is present, but there is a visible difference in the bare cross-section slope at collision energies below about 75 keV, which causes the deviation in the predicted screening potential.20. In summary, even taking into account only data from [30] and [32] (which reports smaller low-energy cross-sections than [31]), and fitting bare crosssection and screening potential simultaneously, the favoured values for the U are significantly higher than the adiabatic limit. The difference can decrease considerably if the bare cross-section is allowed to take a generic functional form (implemented through a polynomial function in the S-factor), with respect to the case where the energy trend suggested by Coulomb barrier penetrability is imposed. Using the approach just described, excessively high values of the screening potential are similarly found for other reactions. Some examples are listed in table 1.1. As shown above for the Li 6(p, α)He 3case, the screening potential value extracted from data can vary significantly depending on what kind of analysis is performed, and the adiabatic limit value is also subject to small variations depending on the precise reactants state, thus a careful comparison of each available determination would be important to obtain a complete picture. On this regard, it can be useful to consult [7, 52–54], as they include a list of older measurements for each reaction mentioned in table 1.1. The collection of these anomalous observations forms the so-called “(atomic) electron screening problem”. Apart from already quoted papers presenting relevant experimental measurements, this long-standing problem was discussed in a number of data re-analyses and theoretical studies, some examples being [16, 41, 51, 56–58]. As seen, the analysis approach just discussed is rather advantageous, as 19Another fit allowing an l= 1 contribution was attempted, but as in section 1.1.3 the additional component does not contribute to the best-fitting function. 20It is underlined that such difference between the models tends to disappear if bare-nucleus data (from indirect measurements) is supplied for the fit, because in that case the fitted bare-nucleus component is directly constrained. For instance, if data from [7] is fitted using equation (1.1.20), the excitation function is very similar to the polynomial reported in the same paper. 33 Chapter 1. Phenomenology of sub-Coulomb nuclear reactions Polyonomial & screening fit Only polynomial Sharp-edge transmission & screening fit Only sharp-edge transmission Elwyn et al. 1979 Lamia et al. 2013 Cruz et al. 2007 Cruz et al. 2005 50 100 500 1000 1.5 2.0 2.5 3.0 3.5 4.0 Centre-of-mass collision energy [keV] Astrophysical S-factor [MeV b] 6Li +p→3He +α Figure 1.6: Points are a subset of data in figures 1.2 and 1.3, from [7, 30, 32], with identical symbols. Blue dotted line is a fit of only data from [30, 32] (not [7]) on a third-order polynomial for the astrophysical factor, ∑︁3 n=0 an(E/MeV)n, multiplied by the fein equation (1.2.2), with the coefficients anand the screening potential Uas free parameters, whose bestfitting values (with their standard errors) are: a0= (3.54 ±0.08) MeV b, a1= (−4.8±0.8) MeVb, a2= (6 ±2) MeVb, a3= (−3±2) MeVb, U= (267 ±129) keV. Red dashed line is a plot of only the bare-nucleus component of the fitted function. Green dot-dashed line is a fit on the same data but using for the bare-nucleus part the same model employed in figure 1.2, which has 35 degrees of freedom, returns a χ2of 17, and whose best-fitting values and standard errors for the parameters are: scaling A0= 0.41 ±0.03, radius Rn= (3.27 ±0.13) fm, screening potential U= (388 ±100) eV. Black solid line is the bare-nucleus component of this fitted function. Table 1.1: Examples in literature of screening potential experimental determination for several reactions. For each reaction, the table lists (in this order) the reference to a work including a determination for the screening potential, the quoted value and error, and the adiabatic-limit value for the given reaction between neutral atomic reactants. See text for details. Reaction Ref. Ueexper. [eV] Ueadiab. lim. [eV] Li 6(p, α)He 3[7] 355 ±100 182 Li 6(d, α)α[55] 320 ±50 182 Li 7(p, α)α[52] 425 ±60 182 Be 9(p, α)Li 6[54] 545 ±98 258 34 1.2 Screening effects it allows to employ all available data to fit both the bare cross-section and the screening effects, surpassing the limit of the analysis performed to obtain figure 1.5, at the affordable cost of requiring a model for the enhancement factor functional form (in particular, a constant screening potential is usually adopted). Nonetheless, it is relevant to point out that such analysis is normally employed mainly to establish whether the screening potential suggested by experimental data is compatible or not with the theoretical constraints: some care is due when applying the analysis results in this way. In particular, while the standard error associated to the fitted value of Uis definitely an useful indicator, it does not correspond to the interval of values of Uwhich keep the total χ2within a given interval for any possible choice of the other parameters. A study of the multi-dimensional confidence region of the fit parameters would be required to establish that in a general manner. In this sense, the results reported in table 1.1 can risk being misleading. However, a computationally much simpler approach can be adopted to address the problem at hand, as discussed in the following. Analysis of direct data rescaled by adiabatic-limit screening First note that if the fit in figure 1.6 is repeated constraining the maximum value for the screening potential with the adiabatic limit, the fitted value will simply equal the maximum allowed. This was predictable, given that the unconstrained fit suggests values of Uwhich are greater than the upper limit. Having acknowledged that data favour higher values of screening potential, it remains to be established whether experiments are at least compatible with the theoretical expectation. To this end, assume in the following that U is constant with respect to the collision energy, E, and exactly equals the adiabatic limit. This is though to be a sufficiently good approximation at this level, since the screening potential is in fact expected to reach the adiabatic limit for small E, while at higher energies the associated enhancement factor is in any case small. Having fixed this “null hypothesis”, it is then possible to examine its likelihood. For instance, data can be fitted on a reasonable model for the bare-nucleus cross-section: this is equivalent to repeating the fits in figure 1.6 after fixing the value of U. The quality of the fit can then be employed as an indication of the likelihood of the assumed scenario. This sort of statistical analysis may be performed in several ways; here, only one straightforward approach is shown for illustrative purposes. Let Fn(x)be the cumulative distribution function of the chi-square distribution with ndegrees of freedom, and Qn(x) = 1 −Fn(x). Given a fit with ndegrees of freedom and a chi-square21 of χ2, and assuming that data indeed come from a random distribution with expectation value equal to the best-fitting function, Qn(χ2) is the probability that a random sample extracted from the “true” distribution, 21This is the variance of data with respect to the prediction given by the fitted function, weighted on the experimental uncertainties (if ∆is the uncertainty on a given data point, the associated weight is 1/∆2). 35 2 Ground-state properties of clustered systems the state1, and let Vbe the set of values that each single-particle νkcan take (i.e. it must be νk∈V). Given a generic single-particle state |ψ⟩, it is possible to express it as a superposition of the basis states, |ψ⟩=∑︂ ν∈V ψ(ν)|ν⟩(2.1.1) where ψ(ν)is a shorthand for ⟨ν|ψ⟩, and the symbol ∑︁ν∈Vis employed to imply a sum over all states in the space V, regardless of its cardinality (i.e. regardless of whether the sum should actually be a discrete sum or an integral). Consider now a system of Aidentical fermions. It is postulated that its only physical states are those anti-symmetric with respect to particle exchange. Let |ψ1, . . . , ψA⟩be a state where particle 1 is in the single-particle state |ψ1⟩ and so on (thus ψ1, . . . , ψAis an ordered set): this clearly does not respect the required statistics. It is then useful to define an anti-symmetrisation operator ˆ︂ W, a projection which maps any generic A-particle state into another respecting the fermion statistics while involving only the single-particle states found in the original state. The application of ˆ︂ Won the generic state |ψ1, . . . , ψA⟩can be expressed through the following relation: ⟨︂ν1, . . . , νAˆ︂ Wψ1, . . . , ψA⟩︂=1 A!  ψ1(ν1)ψ1(ν2). . . ψ1(νA) ψ2(ν1).... . .ψ2(νA) . . .. . . .... . . ψA(ν1)ψA(ν2). . . ψA(νA)  (2.1.2) where !denotes a factorial and . . .is a matrix determinant. ˆ︂ W|ψ1, . . . , ψA⟩ itself is often referred to as a “Slater determinant”. Note that, if the singleparticle states |ψi⟩are distinct and form an orthonormal set, then √A!ˆ︂ W|ψ1, . . . , ψA⟩ is normalised to 1. Finally, consider the elements |ν1, . . . , νA⟩of the orthonormal basis for the complete A-particle system: in general, these states do not respect the required statistics and are thus non-physical. A completely generic state |Ψ⟩ for the A-particle system can be certainly written as a superposition such as |Ψ⟩=∑︂ ν1∈V··· ∑︂ νA∈V Ψ(ν1, . . . , νA)|ν1, . . . , νA⟩(2.1.3) If |Ψ⟩is a physical state, indistinguishability guarantees that Ψ(ν1, . . . , νA) changes only by a global sign under exchange of any single pair of particles, e.g. Ψ(ν1, ν2, . . . , νA) = −Ψ(ν2, ν1. . . , νA)(2.1.4) 1A typical example is a set of states with fixed spin and isospin projection, σand τ, and position, r, so that each “νi” stands for “ri, σi, τi”. 42 2.2 Overlap functions and spectroscopic factors This also implies that |Ψ(ν1, . . . , νA)|is conserved under any permutation of particles. Note that the distinct elements of the set {ˆ︂ W|ν1, . . . νA⟩}ν1∈V,...,νA∈V, once normalised, form a physical (i.e. respecting the statistics) basis for the A-particles system. For brevity, label |ν1, . . . νA⟩Wthe elements of the orthonormal physical basis. The physical state |Ψ⟩can then be expanded as |Ψ⟩=∑︂ {ν1,...,νA} ΨW(ν1, . . . , νA)|ν1, . . . , νA⟩W(2.1.5) where the sum now runs only over the distinct unordered sets {ν1, . . . , νA}. The coefficients moduli in the two basis are simply connected by the relation |ΨW(ν1, . . . , νA)|2=A!|Ψ(ν1, . . . , νA)|2(2.1.6) since equation (2.1.5) is constructed from equation (2.1.3) just by grouping together terms differing only by a permutation of particles. When treating anti-symmetrisation of a state describing a nucleus, it can be convenient to treat proton and neutrons as different isospin projections of an identical particle, a “nucleon”2, so that there is only a single species to account for. The cost of such approach is to neglect the mass difference between proton and neutron. Also note that, within the non-relativistic formulation here employed, the nucleus mass coincides with the sum of each nucleon mass, and it is consequently proportional to the number of nucleons, A. 2.2 Overlap functions and spectroscopic factors In a direct one-step transfer reaction (see chapter 3) the transferred system is passed from a reactant to the other one without altering its internal state [59, sec. 4], and the structure of the involved nuclei, in particular the affinity between the reactants wave-functions before and after the transfer, consequently plays a relevant role. This concept can be formally encoded through overlap functions, which constitute the subject of the present section. The topic is discussed, for instance, in [4] (more detailed references are given later) and [60, sec. 8.1]. 2.2.1 General treatment Let Bbe any nucleus (“composite”), and banother nucleus composed by an arbitrary subset of Bnucleons (“core”). Let Aibe the number of nucleons within nucleus i, and Ax=AB−Ab>0. Further let ΨB Jπ B,MB,TB,τB⟩︂and Ψb Jπ b,Mb,Tb,τb⟩︂be some states of interest for Band brespectively. Each state is anti-symmetric and is an eigenstate of parity and of the modulus and zprojection of total spin and isospin, with quantum numbers πi, Ji, Mi, Ti, τi 2It is then still possible to define interactions which depend on the isospin, and possibly do not conserve the isospin modulus. 43 2 Ground-state properties of clustered systems respectively (spin modulus and parity numbers are often compacted into “Jπ” for brevity). The optimal reference frame in which to compute each Ψdepends on the application: when the overlap function (see equation (2.2.3) below) is to be computed explicitly from the Ψ, it is usually most convenient to set a single coordinate system for both ΨBand Ψb(for instance using Bcentre-ofmass rest-frame). Here, since the overlap is manipulated only at formal level and the goal is to study the relative motion of clusters within B, it is assumed that each Ψis computed fixing the nucleus centre-of-mass in the origin, and thus depends only on Ai−1spatial coordinates. In the following, consider an orthogonal basis of states {Ψb}spanning the whole Ab-nucleons space. For simplicity, assume that each state is normalised3 to 1, and employ the symbol ∑︁bto imply a sum over all elements of the basis {Ψb}(the number of nucleons Abis fixed). Fractional parentage expansion and overlap function Any given state ΨBcan be expressed through the fractional parentage expansion [4, eq. (16.12), (16.17), (16.33)]: the full wave-function is expressed as a superposition of anti-symmetrised products of each possible wave-function for some core nucleus band a corresponding wave-function for the remaining system xof Axnucleons (“valence”). In a rather compact notation, ΨB Jπ B,MB,TB,τB⟩︂=∑︂ b,Tx,j Nb,Tx,j ˆ︂ W[︂Φx Tx,j⟩︁Ψb Jπ b,Mb,Tb,τb⟩︂]︂ (2.2.1) where ˆ︂ Wis the anti-symmetrisation operator defined in section 2.1 and each Nis an appropriate coefficient. Each Φx Tx,j is a wave-function in the space of states at Axnucleons, anti-symmetric under exchange of any pair of its nucleons, and accounting for both their internal motion and the relative motion between the centre-of-masses of band x. Such wave-function is an eigenstate of total angular momentum and isospin, with modulus quantum numbers j and Txrespectively, and z-projections MB−Mband τB−τb, and is similarly an eigenstate of parity with quantum number πBπb. It is stressed that jis the total transferred angular momentum, namely it includes both the intrinsic spin of the transferred system, Jx, and its orbital angular momentum with respect to the core, l. The wave-function Φis normalised following a convention similar to that adopted for Ψ(see equation (2.2.4) below), and is such that is satisfies equation (2.2.1) (which is thus basically its definition). The sum over Tx, j runs over all (compatible) values of these quantum numbers. It is of interest to consider the transition between a well-defined pair of states for core nucleus band composite nucleus B. To this end, from the 3Strictly speaking, this is in fact not possible for continuum states, for which some other convention is to be chosen. However, in practical applications, unbound states are hardly considered within this framework, and for the purposes of the present work it is sufficient to simply ignore the issue, which would complicate the formalism without adding physical insight. 44 2.2 Overlap functions and spectroscopic factors expansion in equation (2.2.1) only one term at a time is relevant, namely the projection of Bstate on the components including only the desired bstate. In particular, consider the overlap function ⟨︂Ψb Jπ b,Mb,Tb,τbΨB Jπ B,MB,TB,τB⟩︂= =∑︂ Tx,j Nb,Tx,j ⟨︂Ψb Jπ b,Mb,Tb,τbˆ︂ W[︂Φx Tx,j⟩︁Ψb Jπ b,Mb,Tb,τb⟩︂]︂ (2.2.2) When computing the overlap, Ψb⟩︁is assigned to an arbitrary but fixed subspace of the full ABnucleons space (for instance, the subspace comprising the degrees of freedom of the nucleons labelled from 1 to Ab), and a scalar product is performed on that subspace. This selects only Ab!components from ˆ︂ W[︁|Φx⟩Ψb⟩︁]︁, the ones where the nucleons assigned to bare the same on both sides of the braket: for each component, the result of the overlap is simply |Φx⟩(assigned to the sub-space of the remaining Axsingle-particle degrees of freedom). Defining conveniently the numerical factors, it is ⟨︂Ψb Jπ b,Mb,Tb,τbΨB Jπ B,MB,TB,τB⟩︂= =1 √︂(︁AB Ab)︁∑︂ Tx,j ⟨(j, MB−Mb),(Jb, Mb)|JB, MB⟩Ax Tx,j Φx Tx,j⟩︁(2.2.3) where ⟨(j, m),(Jb, Mb)|JB, MB⟩is a Clebsch-Gordan coefficient (see appendix A) and (︁AB Ab)︁=AB!/[Ab!(AB−Ab)!] (a binomial coefficient). A, which is the spectroscopic amplitude, is a number such that equation (2.2.3) is satisfied4. The modulus square of Ais the spectroscopic factor S. If the global phase of all Ψand Φis chosen consistently, the phase of Aitself can be significant for interference effects. If, for a given component, ΨB,Ψband Φxare all bound states, they can all be taken purely real with no loss of generality, so that Ais real as well. The coefficient (︁AB Ab)︁in equation (2.2.3) takes into account the desired normalisation of the anti-symmetrised states. From the discussion in section 2.1, it can be seen that each anti-symmetrised state of Aparticles carries a factor51/√A!. Furthermore, the scalar product ⟨︁ΨbΦxΨB⟩︁consists of Ab!Ax!identical terms (one for each possible ordering of Abnucleons within the core nucleus and Axnucleons within the valence system). This two contributions combined yield 1/√︂(︁AB Ab)︁. Also note that (︁AB Ab)︁is the number of distinct ways in which the ABnucleons can be arranged as a group of Ab indistinguishable nucleons in the core nucleus, and another distinct group of AB−Abindistinguishable nucleons. If ΨBcould be written as a single Slater 4In [4], the spectroscopic amplitude is absorbed in the wave-function Φx, whose norm consequently bears physical significance. 5Even if the state is not a simple Slater determinant, it can be written as a superposition of several of them, each carrying the same factor. 45 2 Ground-state properties of clustered systems determinant in some single-particle basis, then given a basis of core states {Ψb}constructed from the same single-particle basis, there would thus be precisely (︁AB Ab)︁core states giving non-zero overlap, each equal to 1/√︂(︁AB Ab)︁ times a normalised valence state, in agreement with the hypothesis that ΨB is normalised. If an explicit expression is provided for the A-body wave-functions ΨBand Ψb, equation (2.2.3) can be employed directly to derive all Φxand their respective spectroscopic amplitudes. For example, if the expansion of each Ψin the Slater-determinant basis is known, equation (2.2.17) may be applied (see the discussion later in this section). The wave-functions may be found numerically from ab-initio methods (for instance, Quantum Monte Carlo methods are discussed in [2, 61]), or an analytical form may be devised phenomenologically (as done for instance in [62]). Another approach consists in disregarding the detailed internal structure of the involved particles, adopting some cluster model, and directly provide a wave-function for the particles relative motion. Expansion of the valence states In the common occurrence of xbeing a single nucleon (this is the case treated explicitly in [4, sec. 16.4.1] and [60, sec. 8.1]), Φxreduces to the x-brelative motion appropriately coupled to the nucleon spin. In general, consider an orthogonal basis of anti-symmetrised states for the internal motion of Ax nucleons, with each element Ψx ν,πx,Jx,Mx,Tx,τx⟩︂having definite modulus and z-projection of total isospin (quantum numbers Tx,τx), total spin (quantum numbers Jx,Mx) and of parity (quantum number πx), and being normalised as the other Ψ. An additional quantum number νis also included to allow for several distinct states of system xwith equal spin and isospin. Then, Φ can be expanded in a basis of states where the internal motion within the valence cluster xand the relative motion between xand bare factorised [4, eq. (16.35)], Φx Tx,j⟩︁=∑︂ ν,l,Jx Aν,l,Jx,Tx,j ∑︂ ml⟨(l, ml),(Jx, MB−Mb−ml)|j, MB−Mb⟩ Ψx ν,(−1)lπbπB,Jx,MB−Mb−ml,Tx,τB−τb⟩︂|φν,Jx,l,ml⟩(2.2.4) where φis a function describing solely the relative motion between xand b centre-of-masses, with normalisation set analogously to the one adopted for the Ψ(for bound states, it is simply ⟨φ|φ⟩= 1), and Ais the appropriate weight6to satisfy equation (2.2.4), so that ∑︂ ν,l,Jx|Aν,l,Jx,Tx,j|2= 1 (2.2.5) 6In [4], as done for the spectroscopic amplitude A, the weight Ais absorbed into φ. 46 2.2 Overlap functions and spectroscopic factors Note that relative-motion states φwith different orbital angular momentum (l, ml)are certainly orthogonal (because spherical harmonics are), while internalmotion states Ψxwith different quantum numbers (ν, Jx, Tx)are orthogonal by construction. In equation (2.2.4), the sum over mlenforces the appropriate coupling of Ψxφto the desired total transferred angular momentum j. This also guarantees that the set of Φxwith different jare orthogonal. It may be useful to define a set of orthogonal functions Φ ˜x ν,πx,Jx,Tx,τ,l,j,M as Φ ˜x ν,π,Jx,Tx,τ,l,j,M ⟩︂= =∑︂ ml⟨(l, ml),(Jx, M −ml)|j, M⟩Ψx ν,π,Jx,M−ml,Tx,τ ⟩︁|φν,Jx,l,ml⟩(2.2.6) with Φx Tx,j⟩︁=∑︂ ν,l,Jx Aν,l,Jx,Tx,j Φ ˜x ν,(−1)lπbπB,Jx,Tx,τB−τb,l,j,MB−Mb⟩︂(2.2.7) where, in analogy with Φx Tx,j, the indexes π, M, τ in Φ ˜xwill be often dropped for brevity. Note that the treatment given here is independent from the precise framework employed to describe the internal motion of the valence system: Ψxis here in principle the full Ax-body wave-function, but it may as well be approximated as just the spin and isospin state of a structureless particle x(which is useful to perform a one-particle transfer calculation) or as a cluster-model wave-function (for multi-particle transfer). The overlap function defined in equation (2.2.3) includes the appropriate superposition of all states of the valence system xwhich can couple with the given core state to yield the desired composite state, possibly with several allowed values of Txand j, each bearing a separate spectroscopic amplitude. It can be interesting to project on a precise state Φx, obtaining a number directly proportional to the spectroscopic amplitude for the given configuration: ⟨︂Ψb Jπ b,Mb,Tb,τbΦx Tx,j ΨB Jπ B,MB,TB,τB⟩︂= =1 √︂(︁AB Ab)︁⟨(j, MB−Mb),(Jb, Mb)|JB, MB⟩Ax Tx,j (2.2.8) 47 2 Ground-state properties of clustered systems It is also possible to select a single component for the internal motion of x and its relative motion with b: using the definition in equation (2.2.6), ⟨︂Ψb Jπ b,Mb,Tb,τbΦ ˜x ν,Jx,Tx,l,j ΨB Jπ B,MB,TB,τB⟩︂= =1 √︂(︁AB Ab)︁⟨(j, MB−Mb),(Jb, Mb)|JB, MB⟩Ax Tx,jAν,l,Jx,Tx,j (2.2.9) where now the relevant amplitude is Ax Tx,jAν,l,Jx,Tx,j. The scalar product ⟨︂Ψb Jπ b,Mb,Tb,τbΦx Tx,j ΨB Jπ B,MB,TB,τB⟩︂(as the one in equation (2.2.9)) embodies the coupling of angular momenta jand Jbto yield JB, which produces the coefficient ⟨(j, MB−Mb),(Jb, Mb)|JB, MB⟩appearing in equation (2.2.3). More formally, Φx Tx,j can be seen as a (j, MB−Mb)spherical tensor, thus, by Wigner-Eckhart theorem, ⟨︂ΨB Jπ B,MB,TB,τBΦx Tx,τB−τb,j,MB−MbΨb Jπ b,Mb,Tb,τb⟩︂= =⟨(j, m),(Jb, Mb)|JB, MB⟩⟨︂ΨB Jπ B,TB,τBΦx Tx,τB−τb,j Ψb Jπ b,Tb,τb⟩︂(2.2.10) where ⟨︁ΨBΦxΨb⟩︁(the reduced matrix element) does not depend on the projections Mb, MB. This implies that the spectroscopic amplitude is the same for any spin z-projection of core and composite nuclei (provided they can actually couple, i.e. the Clebsch-Gordan coefficient is not zero), and can thus be computed for the most convenient pair of projections. On this regard, it is relevant to point out that the set of admissible components Φx Tx,j of a given overlap can depend on Mband MB. As a trivial example, valence states coupling to j= 0 are admissible only for Mb=MB(in all other cases, their contribution is cancelled by the Clebsch-Gordan coefficient in equation (2.2.10)). A similar discussion could be repeated for isospin. This is the reason why sometimes an extra factor ⟨(Tb, τb),(Tx, τB−τb)|TB, τB⟩appears in equation (2.2.1), changing the definition of A[4, sec. 16.4.1]. Extreme cluster model and Wildermuth connection In practice, it is impossible to consider a complete basis of states, and even an approximation including bound states and low-lying continuum is challenging. The extreme cluster model prescribes to approximate the overlap function ⟨︁ΨbΨB⟩︁in equation (2.2.3) including a single component Φxwhich in turn comprises only a single component Φ ˜x(defined in equation (2.2.6)), ⟨︂Ψb Jπ b,Mb,Tb,τbΨB Jπ B,MB,TB,τB⟩︂≈ ≈1 √︂(︁AB Ab)︁⟨(j, MB−Mb),(Jb, Mb)|JB, MB⟩AxΦ ˜x ν,Jx,Tx,l,j⟩︂(2.2.11) 48 2.2 Overlap functions and spectroscopic factors More explicitly, using equation (2.2.6), ⟨︂Ψb Jπ b,Mb,Tb,τbΨB Jπ B,MB,TB,τB⟩︂≈1 √︂(︁AB Ab)︁⟨(j, MB−Mb),(Jb, Mb)|JB, MB⟩· ·Ax∑︂ ml⟨(l, ml),(Jx, M −ml)|j, M⟩Ψx ν,Jx,M−ml,Tx⟩︁|φν,Jx,l,ml⟩(2.2.12) Note that, even if the stated approximation were exact, the spectroscopic amplitude could still take non-trivial values, depending on the microscopic structure of core and composite nuclei. In addition, a phenomenological value for Axmay partially correct the error introduced by the approximation. The internal motion of the valence system x,Ψx, is normally chosen to match some relevant state of system xwhen observed in isolation, often the most bound among its allowed states, such that the total spin and isospin modulus quantum numbers, sand t, can correctly couple the desired Ψband ΨB. As mentioned earlier, the precise model employed for Ψxdepends on what process or features are being described. The core-valence relative motion state, φ, can be modelled as an eigenfunction of a phenomenological potential for the interaction between band x. Its energy eigenvalue is most commonly set in terms of the experimental mass (and excitation energy) of the states of interest for B,band x. If φis a bound state, then its orbital angular momentum modulus quantum number, l, and the number of nodes of its radial part, labelled n, are often set from shell-model considerations, adopting the so-called Wildermuth connection, as follows. Consider an independent-particle shell-model state for the isolated systems b,Band x, where each nucleon iis assigned to a specific shell-model single-particle state with orbital angular momentum liand number of radial nodes ni(the minimum possible value for niis zero). The number of “energy quanta” Qjin system jis defined as Qj= Aj ∑︂ i=1 2ni+li(2.2.13) and, similarly, the b-xrelative motion is assigned an amount Qbx of energy quanta equal to 2n+l.Qbx is then found requiring that the total amount of quanta in Bmatches the sum of quanta in b,xand their relative motion, [4, eq. (16.32)] QB=Qb+Qx+Qbx (2.2.14) In general, there may be several pairs of n, l satisfying the relation, in which case the smallest permitted lis often adopted. It is also possible that none of the values of lallowed by the criterion matches the required one to describe a particular state of interest: this can be taken as a suggestion that the extremeshell-model configuration which has been chosen is not the favoured one to describe the system of interest. 49 2 Ground-state properties of clustered systems Finally, note that setting the binding energy for a bound state φamounts to put a constraint on the b-xpotential. Once the orbital angular momentum and number of nodes of the state are provided, the constraint is sufficiently detailed to completely fix one degree of freedom in the expression for the potential (for example a global scaling factor), which is useful to construct the potential phenomenologically. Expansion in the Slater-determinant basis Differently from what was done above, consider now a fixed coordinate system, and express all wave-functions ΨBand Ψbin such coordinates (instead of computing each of them in the respective centre-of-mass rest frame). As a consequence, Φx Tx,j is not cleanly factorised between internal motion of xand x-brelative motion any more, and it makes more sense to just refer to it as a valence system state computed in the same coordinate system as the other Ψ functions. Consider an orthonormal single-particle basis (for instance the one provided by a shell model), and let χB ibe elements of the orthonormal basis of the ABbody system constructed as normalised Slater determinants (defined as in equation (2.1.2)) of the aforementioned single-particle states (χB iwas labelled |ν1, . . . νA⟩Win section 2.1). In general, the composite nucleus state ΨBof interest can be expanded in such basis, obtaining ΨB⟩︁=∑︁icB iχB i⟩︁, where the care numerical coefficients such that ∑︁i|ci|2= 1 and one can proceed identically for the core nucleus state Ψb. The overlap function can then be computed explicitly in the shell-model basis: ⟨︂ΨbΨB⟩︂=∑︂ ik cb i∗cB k⟨︂χb iχB k⟩︂(2.2.15) the overlap ⟨︁χb iχB k⟩︁is either zero, if any single-particle state is occupied in χb ibut empty in χB k, or proportional to a single Slater determinant with Axnucleons filling the states occupied in χB kand empty in χb i: let χx ik be this Slater determinant, normalised as the other χfunctions. Using the same reasoning described in section 2.2.1 when discussing the binomial factor in equation (2.2.3), the proportionality factor is such that (for a pair i, k whose overlap is not zero) ⟨︂χb iχx ik χB k⟩︂=1 √︂(︁AB Ab)︁(2.2.16) The phase (which for bound states can always be defined to be just a sign) depends on the adopted conventions (it is necessary to fix an ordering for the single-particle basis elements) and on what states are occupied in χbor χx. For simplicity, the present discussion on phases will be limited to the few relative signs required to deduce the spectroscopic factors of interest within the independent-particle shell-model. It can be useful to note that, if the overlap of a given pair of compon50 2.2 Overlap functions and spectroscopic factors ents, ⟨︁χb iχB k⟩︁, is √︂Ab!Ax! AB!|χx ik⟩, then the corresponding overlap between the Slater determinants of composite and valence systems, ⟨︁χx ik χB k⟩︁, is similarly √︂Ax!Ab! AB!χb i⟩︁, where the coefficient is the same. To summarise, let fij be 0 if ⟨︂χb iχB j⟩︂= 0, or the phase of ⟨︂χb iχx ij χB j⟩︂ otherwise. Once the phase convention has been fixed, fij can be computed for any pair of Slater determinants by inspection of the involved single-particle states. Then, ⟨︂Ψb Jπ b,Mb,Tb,τbΨB Jπ B,MB,TB,τB⟩︂=1 √︂(︁AB Ab)︁∑︂ ij cb i∗cB jfij χx ij⟩︁(2.2.17) The last remaining step is to regroup the different components into valence states Φx Tx,τB−τb,j,MB−Mb⟩︂with the desired quantum numbers. The details of this step depend on the precise single-particle basis in use, but it essentially requires to couple the single-particle states to the desired total spin and isospin. For instance, in a shell-model basis state all nucleons in closed shells couple to angular momentum 0, simplifying the problem, and the remaining nucleons are in states with their intrinsic spin and orbital angular momentum already coupled. 2.2.2 Spectroscopic factors within the independent-particle shell model In many cases, experimental or ab-initio evaluations of spectroscopic factors (together with the corresponding overlap functions) can be found in literature. Even in these cases, evaluating the spectroscopic amplitudes phenomenologically, within a sufficiently simple model, can be very useful, as it allows to pinpoint the most salient features affecting the result, and to gauge the corrections introduced by the more sophisticated determinations. In the following, the predictions given by the independent-particle shell-model shell model will be discussed. Start from equation (2.2.17), and approximate the states for core and composite nucleus, Ψband ΨB, as independent-particle shell-model wavefunctions. This means that each nucleon occupies a well-defined shell among the elements of the orthogonal single-particle basis, but may not have defined angular momentum z-projection. In the simplest case, the state is just a single Slater determinant: for instance, an α-particle in its ground-state is described with its four nucleons in the 0s1/2 shell (0 nodes, orbital angular momentum l= 0, total angular momentum 1/2), with anti-symmetrisation requiring that precisely one proton and one neutron have positive spin zprojection. However, if there are several nucleons in partially-empty shells, whose angular momentum z-projections can be assigned in more than one way to obtain a nucleus with the desired spin projection, then it necessary to describe the state as a superposition of several Slater determinants. The 51 2 Ground-state properties of clustered systems in equation (2.1.3). One obtains <O> =⟨︂Ψˆ︁ OΨ⟩︂ ⟨Ψ|Ψ⟩=∑︁ i∑︁ ν1∈V··· ∑︁ νA∈V o(νi)|Ψ(ν1, . . . , νA)|2 ∑︁ ν1∈V··· ∑︁ νA∈V|Ψ(ν1, . . . , νA)|2(2.3.4) The expectation value <O> can in general depend on the specific reference frame and/or coordinate system employed (just as in classical physics). Most observables of interest acquire or retain the intended meaning when the composite system centre-of-mass is fixed in the origin. Here, such condition will be enforced when providing an explicit cluster-model expression for |Ψ⟩, see e.g. equation (2.3.19). Due to the invariance of |Ψ(ν1, . . . , νN)|under particle exchange (see equation (2.1.4)), all elements of the sum over i(running over the Aparticles) are in fact equal. Arbitrarily choosing a convenient value for i, e.g. i= 1, it is <O> =A ⟨Ψ|Ψ⟩∑︂ ν1∈V o(ν1)∑︂ ν2∈V··· ∑︂ νA∈V|Ψ(ν1, . . . , νA)|2(2.3.5) The function ρ(ν) = A∑︂ ν2∈V··· ∑︂ νA∈V|Ψ(ν, ν2, . . . , νA)|2(2.3.6) is the one-body “density” for the single-particle configuration ν, i.e. the probability for nucleon 1 to be found in state |ν⟩regardless of the state of all other nucleons (see [22, eq (5.1.21)] for a more general definition, not requiring Ψto be anti-symmetrised or computed in the centre-of-mass frame. [22, eq (5.1.20)] is instead the analogous of the definition given here). The normalisation of ρis here set such that ∑︁ν∈Vρ(ν) = A⟨Ψ|Ψ⟩. While ρwas defined as related specifically to nucleon 1, it is identical for any other one. In fact, since |Ψ(ν1, . . . , νA)|is invariant under particle permutation, it is ρ(ν) = A!∑︂ {ν2,...,νA}|Ψ(ν, ν2, . . . , νA)|2=∑︂ {ν2,...,νA}|ΨW(ν, ν2, . . . , νA)|2(2.3.7) where the sum runs only over the distinct unordered sets {ν2, . . . , νA}, and ΨW is the same appearing in equation (2.1.6). It is here relevant to highlight the difference between a sum ∑︁{ν1,ν2,...,νA}over the unordered set {ν1, ν2, . . . , νA}, and the double sum ∑︁µ∈V∑︁{µ2,...,µA}: for each term appearing in the former, there are Adistinct terms appearing in the latter (one for each possible choice of µ∈ {ν1, ν2, . . . , νA}). The expectation value of ˆ︁ Ocan then be rewritten in terms of the one-body density ρ: <O> =1 ⟨Ψ|Ψ⟩∑︂ ν∈V o(ν)ρ(ν) = A∑︁ ν∈V o(ν)ρ(ν) ∑︁ ν∈V ρ(ν)(2.3.8) 58 2.3 Expectation value of one-body observables Let now, in general, O(F)be a functional of the function F(ν), defined as: O(F) = ∑︂ ν∈V o(ν)F(ν)(2.3.9) for convenience, assume in the following that ⟨Ψ|Ψ⟩= 1, and thus ∑︂ ν∈V ρ(ν) = A , <O> =O(ρ)(2.3.10) <O> would of course be the same for any other choice for ⟨Ψ|Ψ⟩, but the given one simplifies the expressions in terms of ρ. The formalism involving the wave-function at the individual-nucleons level, employed here, was useful to derive equation (2.3.10). In the following paragraphs, instead, equation (2.3.10) will be taken as starting point to derive explicit expressions adopting a cluster model for |Ψ⟩. As a final note, remind that equation (2.3.10) neglects the nucleons internal structure, and may consequently be unsuited for computing an expectation value directly on a A-nucleons wave-function, unless the observable of interest is corrected appropriately. For instance, a nucleus root-mean-square radius would be greater than predicted by equation (2.3.10) because nucleons are not point-like (see e.g. [17, eq. (4.40)] [22, eq. (5.1.25)] for a correction on radii). This is not an issue within this work, as equation (2.3.10) is never employed directly, in favour of cluster-model expressions where the contribution from each cluster internal structure is explicitly accounted for. 2.3.2 General treatment of observables quadratic in the spatial position Consider an observable ˆ︁ Owhose single-particle acts only on the nucleon position and is diagonal in the position basis. For the purpose of deriving the relations of interest here, all other degrees of freedom can consequently be ignored, as they are integrated away in equation (2.3.5), and the νin section 2.3 can represents just the position r. The one-body density defined in equation (2.3.6) is then just the spatial particle number density of the nucleus (probability to find indifferently any nucleon in a given position r). Note that this does not imply that the other possible degrees of freedom (as the spin) do not play any role in determining the expectation values (see for instance the discussion in section 2.3.4 and chapter 5). The point is, rather, that these extra degrees of freedom can only contribute to fix the nucleus structure, i.e. to derive a complete expression for ρ(r), which is here taken as external input, but do not affect its connection with the observables of interest. With this assumption, equation (2.3.10) becomes ∫︂R3 ρ(r)d3r =A , <O> =∫︂R3 o(r)ρ(r)d3r (2.3.11) 59 2 Ground-state properties of clustered systems It will be convenient to define, in general, the functional O(F)≡∫︂R3 o(r)F(r)d3r (2.3.12) Suppose now that the composite system can be described as a bound state of ninert clusters, usually distinguishable, which will be typically expressed through a wave-function in some Jacobi coordinates related to the relative motion between the clusters. From such wave-function is possible to extract the probability density function for each cluster position, Φ ˜i(x), i.e. the probability of finding the centre-of-mass of cluster iin position x, regardless of the position of all other clusters. The procedure will be discussed explicitly in the last segment of section 2.3.2. It is convenient to fix the normalisation of each Φ ˜ito 1, and consider two other integrals of interest: ∫︂R3 Φ ˜i(x)d3x = 1 ∫︂R3 x Φ ˜i(x)d3x =X i ∫︂R3 o(x) Φ ˜i(x)d3x =O(Φ ˜i) (2.3.13) where X iis thus the average position of cluster icentre-of-mass in the system of coordinates of choice, and O(Φ ˜i)is the expected value of oon Φ ˜i(note that the expression follows the definition in equation (2.3.12)). Each cluster, in turn, has a defined and immutable one-body density for its internal structure. Let ρ˜i(r)be the i-th cluster one-body density, defined with respect to the cluster own centre-of-mass (that is, placing the centre-of-mass in the origin), so that ∫︂R3 ρ˜i(r)d3r =Ai ∫︂R3 r ρ˜i(r)d3r = 0  Ai∫︁R3o(r)ρ˜id3r ∫︁R3ρ˜i(r)d3r =∫︂R3 o(r)ρ˜i(r)d3r =O(ρ˜i) (2.3.14) where Aiis the i-th cluster mass number, so that ∑︁iAi=A, while O(ρ˜i)is the expectation value of ˆ︁ Oon the isolated cluster i, defined precisely as <O> in equation (2.3.11). If the cluster centre-of-mass is in position x, its one-body density at position r will be ρ˜i(r −x). Therefore, within the inert-clustermodel approximation, the one-body density Difor each cluster within the composite nucleus, taking into account the distribution for the cluster centreof-mass position, is defined in terms of the “intrinsic” one-body density ρ˜ias the folding (see e.g. [17, eq. (4.40)] [22, eq. (5.1.25)]) Di(r) = ∫︂R3 Φ ˜i(x)ρ˜i(r −x)d3x (2.3.15) 60 2.3 Expectation value of one-body observables and the one-body density of the composite nucleus is then ρ(r) = n ∑︂ i=1 Di(r)(2.3.16) Note that, even within an inert-cluster model, it may be necessary to take into account several possible states ρ˜ifor some cluster (for instance, the different possible spin projections), each of them possibly corresponding to a different O(ρ˜i)and Φ ˜i. If that is the case, each component has to be taken into account separately. The total one-body density for the composite nucleus would then be, using index νto enumerate the cluster configurations, ρ(r) = n ∑︂ i=1 ∑︂ ν cνDi,ν(r)(2.3.17) where cνis the appropriate weight for each configuration. As discussed in section 2.2, the formalism would in principle become exact when a sufficient (infinite) amount of possible states is included. Here, for brevity, it will be assumed that each cluster has only one possible ρ˜iand Φ ˜i. Fix the composite nucleus centre-of-mass into the origin, so that 0 =∫︂R3 rρ(r)d3r = n ∑︂ i=1 ∫︂R6 r Φ ˜i(x)ρ˜i(r −x)d3r d3x (2.3.18) Performing first the integral over r, applying the variable change y =r −x, and then using equations (2.3.13) and (2.3.14), equation (2.3.18) becomes 0 = n ∑︂ i=1 AiX i(2.3.19) It is worth reminding that, as mentioned in section 2.1, the formalism employed (non-relativistic treatment, nucleons share the same mass) causes each composite system (including each cluster separately, seen as a collection of nucleons) to have a mass proportional to their mass number. The expectation value O(ρ)of interest for the composite nucleus, as defined in equation (2.3.11), can then be computed adopting equation (2.3.16) for the total one-body density, with the goal of removing the explicit dependence on the specific form of each cluster internal density ρ˜i. Applying again the variable change y =r −x, it is O(ρ) = n ∑︂ i=1 ∫︂R3 d3x Φ ˜i(x)∫︂R3 d3y o(y +x)ρ˜i(y)(2.3.20) The result thus depends on the specific form of o(r). Given that, here, it is of interest to study the root-mean-square radius and the electric quadrupole moment, it is sufficient to specialise to the case of operators with quadratic 61 2 Ground-state properties of clustered systems dependence on the position, meaning that o(r) = ∑︂ k={x,y,z} okr2 k(2.3.21) where the sum runs over all spatial dimensions10,rkis the component of r along the k-th dimension, and the okare arbitrary constants11. Using this assumption, it is O(ρ) = n ∑︂ i=1 ∫︂R3 d3x Φ ˜i(x)∫︂R3 d3y ∑︂ k ok(yk+xk)2ρ˜i(y)(2.3.22) The mixed term 2ykxkappearing from (yk+xk)2cancels out by virtue of the second line in equation (2.3.14), i.e. because each ρi(y)is defined placing the cluster centre-of-mass in the origin. The two remaining terms are decoupled, and the three spatial components can be regrouped: O(ρ) = n ∑︂ i=1 ∫︂R3 d3x Φ ˜i(x)∫︂R3 d3y [o(y) + o(x)] ρ˜i(y)(2.3.23) Substituting the definitions given in equations (2.3.13) and (2.3.14), one finds O(ρ) = n ∑︂ i=1 [︂O(ρ˜i) + AiO(Φ ˜i)]︂(2.3.24) In equation (2.3.24) there is no explicit dependence on the clusters internal structure, but only on the observables O(ρ˜i), which can be obtained from an experimental measurement or a dedicated model. In this way, it is possible to obtain the desired result O(ρ)for the composite nucleus treating only the bound state between the clusters treated as structureless particles. Isospin-dependent one-body operators – “Charge” observables A relevant generalization to the hypothesis employed in section 2.3.2, of an operator acting solely on the nucleons position, is that of an operator distinguishing between protons and neutrons, or in other words, depending also on the nucleon isospin projection. The generic index νdefined in section 2.1 consequently must now include the isospin projection, so the system one-body density, defined in equation (2.3.6), can be written as ρτ(r), where ρp(r)and ρn(r)are the densities for protons and neutrons respectively. Similarly, the 10In particular, the “x” and “y” displayed here bear a completely different meaning than the vectors x and y. 11In general, the term “quadratic function” denotes a function which can also include linear and constant terms. Note that linear terms give no contribution to <O>, as is it calculated in the centre-of-mass coordinate system, see equations (2.3.14) and (2.3.19). Constant terms do contribute to <O>, but only as an overall offset which would add little to the discussion. 62 2.3 Expectation value of one-body observables single-particle operator ˆ︁oidefined in equation (2.3.2) is now ˆ︁oi=∑︂ τ∫︂R3 d3r |i:r, τ⟩oτ(r)⟨i:r, τ|(2.3.25) where op(r)and on(r)are the eigenvalues for the single-particle observable for protons and neutrons respectively. Equation (2.3.11) is thus generalised to ∫︂R3[︁ρp(r) + ρn(r)]︁d3r =A , oτ(r) = ∑︂ k oτ,k r2 k <O> =O(ρ) = A∑︁τ∫︁R3oτ(r)ρτ(r)d3r ∑︁τ∫︁R3ρτ(r)d3r =∑︂ τ∫︂R3 oτ(r)ρτ(r)d3r (2.3.26) The contribution for each τcan be treated separately precisely as seen earlier. To this end, it is convenient to define Oτ(F) = ∫︂R3 oτ(r)F(r)d3r (2.3.27) Furthermore, for each cluster, let Ziand Nibe the number of protons and neutrons in cluster i, so that Zi+Ni=Ai, and ρ˜τ,i the one-body density for the internal structure of cluster iregarding species s, normalised to the appropriate number of particles, e.g. ∫︁R3ρ˜p,i(r)d3r =Zi, so that ρτ(r) = n ∑︂ i=1 ∫︂R3 Φ ˜i(x)ρ˜τ,i(r −x)d3x (2.3.28) where nis again the total number of clusters. For observables obeying equation (2.3.21) (“quadratic”), one then finds, in analogy to equation (2.3.24), O(ρ) = n ∑︂ i=1 [︂Op(ρ˜p,i) + On(ρ˜n,i) + ZiOp(Φ ˜i) + NiOn(Φ ˜i)]︂(2.3.29) As before, the values for Oτ(ρ˜τ,i)are normally supplied independently, thus the observable definition and the cluster distributions Φ ˜iare again sufficient to determine the expectation value for the composite nucleus. A common occurrence, of interest for this work, is to distinguish between the “matter” and “charge” versions of an observable. “Matter” quantities arise by treating protons and neutrons equally, so that op=onas in the derivation leading to equation (2.3.24). “Charge” quantities are often more readily accessible experimentally, and as such also of greater practical interest. They are measured by probing nuclei electromagnetically, for instance through electron scattering (see e.g. [69] and references therein for a list of charge radius measurement techniques). To evaluate charge observables theoretically at the nucleus level, in treatments where nucleons are assumed to be elementary, the 63 2 Ground-state properties of clustered systems standard recipe (see e.g. [67, app. A], [70, eq. (6.120), (6.132)]) is to weight the nucleons densities on their global electric charge: this amounts to consider an operator acting exclusively on the protons, putting on(r) = 0 and reducing equation (2.3.29) to <O>ch =Op(ρp) = n ∑︂ i=1 [︂Op(ρ˜p,i) + ZiOp(Φ ˜i)]︂(2.3.30) Such recipe is adopted in the present work as well, and it is expected to represent an acceptable approximation. It is relevant to point out, however, that neutrons, despite being electrically globally neutral, are in fact not insensible to electromagnetic probes due to their internal structure12. It is also underlined that, when using equation (2.3.30), it would not be equivalent to consider a system of clusters composed exclusively of protons with density ρ˜p,i, completely neglecting the neutrons. As can be seen from equations (2.3.19) and (2.3.28), the centre-of-mass position of each cluster, which affects the total proton density, still depends on the total mass of each cluster13. Transformation to Jacobi coordinates As anticipated, the last step is to connect the cluster-model wave-function of the composite nucleus with the distribution of each cluster position, Φ ˜i, and in particular their expected value O(Φ ˜i)as defined in equation (2.3.12), or equivalently the isospin-dependent Oτ(Φ ˜i)defined in equation (2.3.27). Consider a system of nclusters, and let xibe the centre-of-mass position of the i-th cluster. Further let d ibe a set of Jacobi coordinates: d 1is the distance between clusters 1 and 2, d 2is the distance between the centre-ofmass of clusters 1 and 2 and cluster 3, and so on, while d nis the whole system centre-of-mass position. The orientation of each vector is set as in [6, 12For instance, the neutron has a negative mean-square charge radius [69, 71]. This can be reconnected to its positively and negatively charged internal constituents not sharing the same spatial distribution. 13The calculations could be repeated considering the protons centre-of-mass instead of the total one, but this is normally not the quantity of interest. As an extreme example, imagine a system composed of a purely neutronic core and a single bound proton. The experimental charge radius of such system would not be just the proton internal radius, as the proton motion around the core would for instance make the scattering target appear bigger. 64 2.3 Expectation value of one-body observables eq. (3.43)] 14, so that d 1({xi}) = x1−x2 d j({xi}) = ∑︁j α=1 Aαxα ∑︁j α=1 Aα−xj+1 d n({xi}) = ∑︁n α=1 Ajxj ∑︁n α=1 Aj (2.3.31) where the notation d j({xi})is a shorthand to signal that d jin the equation is expressed as a function of the set of single-particle coordinates x1, . . . , xn. The inverse transformation is xn({d i}) = d n−∑︁n−1 α=1 Aα ∑︁n α=1 Aα d n−1 xn−1({d i}) = d n+An ∑︁n i=1 Ai d n−1−∑︁n−2 i=1 Ai ∑︁n−1 i=1 Ai d n−2 xj({d i}) = d n+ n−1 ∑︂ β=j Aβ+1 ∑︁β+1 α=1 Aα d β−∑︁j−1 α=1 Aα ∑︁j α=1 Aα d j−1 x1({d i}) = d n+ n ∑︂ j=2 Aj ∑︁j i=1 Ai d j−1 (2.3.32) The cluster-model wave-function of the composite nucleus will be usually expressed in such a coordinate system, further setting the origin in the system centre-of-mass, that is, d n= 0 . Since in this section only observables diagonal in the position are of interest, it is sufficient to consider the wavefunction modulus-square, labelled Φ(d 1, . . . , d n−1), which is the probability density function to find the system in the configuration corresponding to the specified Jacobi coordinates15. Let fbe an auxiliary function defined as follows: f({d i}) = Φ(d 1, . . . , d n−1)δ(d n)(2.3.33) The one-cluster density Φ ˜1can then be written as Φ ˜j(xj) = ∫︂f({d i({xi})})d3x1. . . d3xj−1d3xj+1 . . . d3xn(2.3.34) Their expected value O(Φ ˜j)is thus O(Φ ˜j) = ∫︂R3 Φj(r)o(r)d3r =∫︂o(xj)f({d i({xi})})d3x1. . . d3xn(2.3.35) 14In [6] the equations refer to a set of particles with identical masses, thus some factors appear different. Also note that the opposite choice on the vectors orientation is often adopted, see e.g. [72]. 15Note that Φand Φ ˜ihave nothing to do with the Φin section 2.2. 65 2 Ground-state properties of clustered systems the integral will be computed by changing variables to the Jacobi coordinates. In general, given two vectors of variables y and x(y)(of arbitrary length) and a function f(x), it is ∫︂f(x)dx =∫︂f(x(y)) |J(y)|dy , J(y) =  ∂x1 ∂y1(y)∂x1 ∂y2(y). . . ∂x2 ∂y1(y).... . . . . . . . . . . .  (2.3.36) meaning that the measure |J(y)|is the absolute value of the determinant J(y). For the case of interest, it is convenient to compute the determinant for the variable change {d i} → {xi}. For brevity, consider only the variable change for one component of each spatial vector (say, the z-projection), as the result is identical for all components. The matrix is easily found from equation (2.3.31), yielding J= 1−1 0 0 . . . A1 A1+A2 A2 A1+A2−1 0 . . . . . .. . .. . ..... . . A1 ∑︁iAi A2 ∑︁iAi. . . . . . An ∑︁iAi  (2.3.37) Any matrix obtained by combining different rows, or different columns, of the original matrix has the same determinant [73]. By summing each column to the next one, starting from the first one, in order, then appropriately combining the rows, the matrix in equation (2.3.37) can be reduced to the identity matrix, implying that J= 1. As a consequence, O(Φ ˜j) = ∫︂o(xj({d i}))f({d i})d3d 1. . . d3d n= =∫︂o(xj({d i}, d n= 0 ))Φ(d 1, . . . , d n−1)d3d 1. . . d3d n−1(2.3.38) where the functions xj({d i}, d n= 0 ), given in equation (2.3.32), are to be computed in the coordinate system where the centre-of-mass position d nis the origin, due to the Dirac δin appearing in equation (2.3.33). In the following, for brevity, let xj({d i})be a shorthand for xj({d i}, d n= 0 ), where it is understood that calculations are performed in the system centre-of-mass. Specialise now to the case of a function o(x)defined as in equation (2.3.21) (“quadratic” and not depending on isospin). The expectation value of interest is then given by equation (2.3.24), where the relative motion contribution is given by ∑︁jAjO(Φ ˜j)(sum over all clusters). As shown in the following, this takes a notably simple form, for any Φ. Let Robe the relevant term within the integrand in equation (2.3.38), defined so that ∑︂ j AjO(Φ ˜j) = ∫︂Ro(d 1, . . . , d n−1) Φ(d 1, . . . , d n−1)d3d 1. . . d3d n−1(2.3.39) 66 2.3 Expectation value of one-body observables More explicitly: Ro(d 1, . . . , d n−1) = ∑︂ j Ajo(xj({d i})) = ∑︂ k={x,y,z} ok∑︂ j Ajx2 j,k({d i})(2.3.40) Further let A=∑︁n i=1 Ai. First, note that all “mixed” terms appearing in Rowhen squaring each xj,k, i.e. terms depending on two distinct Jacobi coordinates, cancel out when performing the complete sum. As an explicit example, consider all the terms in dn−1,kdn−2,k, which are easily extracted by inspection of equation (2.3.32): 2okAndn−1,kdn−2,k A(A−An)[An−1(A−An−An−1)−(A−An−An−1)An−1] (2.3.41) the same cancellation takes place for all other “mixed” terms as well. As a result, Rois a sum of only terms which depend on a single Jacobi coordinate d i: this simplifies the integral in equation (2.3.38). These terms also take a simple form. Let µibe the reduced mass number between cluster i+1 and the composition of clusters from 1 to i. Then, as can be found again by inspection of equation (2.3.32), it is Ro=∑︂ k={x,y,z} ok∑︂ i µid2 i,k = n−1 ∑︂ i=1 µio(d i), µi≡Ai+1 ∑︁i j=1 Aj ∑︁i+1 j=1 Aj (2.3.42) Finally, for brevity, let φibe the “one-coordinate” density derived by integrating Φon all Jacobi coordinates except d i, φi(d i) = ∫︂Φ(d 1, . . . , d n−1)d3d 1. . . d3d i−1d3d i+1 . . . d3d n−1(2.3.43) Using such definition, the expectation value of any one-body operator whose single-particle operator act only on the spatial position (in particular it is isospin-independent) and is diagonal in the spatial position basis with “quadratic” eigenvalues (as in equation (2.3.21)), given in equation (2.3.24), can be written as <O> =O(ρ) = n ∑︂ i=1 O(ρ˜i) + n−1 ∑︂ i=1 µiO(φi)(2.3.44) where O(F)and ρ˜iare defined in equations (2.3.12) and (2.3.14), respectively, and nis the number of clusters. For a two-cluster system there is only one relevant Jacobi coordinate, the distance between the two clusters (the centre-of-mass position plays no role, as mentioned), thus φ1is just the full probability density Φ. It is O(ρ) = ∑︂ i O(ρ˜i) + A1A2 AO(Φ) (2.3.45) 67 2 Ground-state properties of clustered systems each cluster iis in turn defined as a bound state of Ainucleons coupling to a definite total spin modulus and z-projection, denoted by the quantum numbers jiand j(z) i. The clusters orbital angular momenta (with respect to the composite nucleus centre-of-mass) can be coupled to form the relative orbital angular momenta of interest in the Jacobi coordinates of choice, so it is possible to consider a set of configurations with definite values for the relative orbital angular momentum moduli and z-projections, denoted by the quantum numbers λi, µi(with irunning from 1to n−1). The clusters total spins and their total orbital angular momentum (with respect to the composite nucleus centre-of-mass) further couple to form the total spin of the composite nucleus, with modulus Jand projection M. Even within an inert-cluster model, there normally will be several choices for the angular momenta j(z) i, λiand µiwhich couple to the desired M. Since the quadrupole moment does not couple the nucleons spin (as all observables treated in this section), it is possible to consider only a single possibility at a time for the set of j(z) i: the complete result can be obtained combining each component with the appropriate weight (usually a Clebsch-Gordan coefficient). An explicit example will be discussed in chapter 5. Even after this simplification, there can be several possible values for the set of λiwhich can be coupled by the spherical harmonic in equation (2.3.68). There may also be several choices for the set of µi, but these are not coupled by a Yλ0, and can thus be taken into account separately, as the spins. Using equation (2.3.45), it is immediately, for a two-cluster system, <Qm,comp>=∑︂ i <Qm,i>+4√︃π 5 A1A2 A∫︂R3 r2Y20(θ, φ) Φ(r)d3r (2.3.70) in agreement with [67, eq. (A.12)]16. This holds for a generic Φ, but given the coupling scheme mentioned above, it is of interest to specialise it to the case where the relative-motion wave-function is an eigenstate of the z-projection of the relative orbital angular momentum, µ, but not necessarily an eigenstate of the angular momentum modulus. As in equation (A.3), the probability density Φcan thus be written as the modulus square of a composition of several components ξλµ defined through the following equation: Φ(r) = ∑︂ λ cλξλ,µ(r) 2 =∑︂ λ cλ χλ(r) rYλ,µ(Ω) 2 (2.3.71) where the cλare the coefficients of the expansion, such that ∑︁λ|cλ|2= 1. 16More precisely, in [67, eq. (A.12)] the factor 4√︁π 5does not appear, as this is included only later in the expectation value “Qmo” in [67, eq. (A.13)]. The factor is not present in [67, eq. (A.14), (A.15)] as well. 74 2.3 Expectation value of one-body observables Using bra-ket notation, the integral of each possible term is thus ⟨︁ξλ2,µ r2Y20 ξλ1,µ⟩︁≡∫︂R3 ξ∗ λ2,µ(r)r2Y20(θ, φ)ξλ1,µ(r)d3r = =∫︂4π Y∗ λ2,µ(Ω)Y20(Ω)Yλ1,µ(Ω) dΩ∫︂+∞ 0 χ∗ λ2(r)r2χλ1(r)dr≡ ≡ ⟨Yλ2,µ |Y20 |Yλ1,µ⟩⟨︁χλ2r2χλ1⟩︁(2.3.72) where the radial wave-functions are normalised so that ⟨χλ|χλ⟩= 1. The result of the angular integration is reported in equation (A.13), and is non-zero only when either λ1=λ2= 0 or λ1=λ2±2, which may reduce the number of components which have to be taken into account together in equation (2.3.71). In summary, the relative-motion contribution to <Qm,comp>in equation (2.3.70) is 4√︃π 5 A1A2 A∑︂ λ1,λ2 c∗ λ2cλ1⟨Yλ2,µ |Y20 |Yλ1,µ⟩⟨︁χλ2r2χλ1⟩︁(2.3.73) For a three-cluster system, using equation (2.3.46), O(ρ) = ∑︂ i <Qm,i>+ + 4√︃π 5[︃A1A2 A1+A2 <r2Y20>φ1+(A1+A2)A3 A<r2Y20>φ2]︃(2.3.74) where each <r2Y20>φican be treated identically to the analogous term in equation (2.3.70). Charge quadrupole moment In analogy with its matter counterpart, the charge electric quadrupole moment can be defined through equation (2.3.26), adopting as single-particle eigenvalues Qτ(r) = Zτ4√︃π 5r2Y20(θ, φ)(2.3.75) (see [17, ex. 1.10] or [67, app. A]) where Zτis the charge number of species τ(that is, 1for protons and 0for neutrons)17. Apart from the charge dependence, all considerations made for the matter quadrupole moment hold 17Consequently, the charge quadrupole moments, in the present work, bear the dimensions of an area. It is also possible to use the particle charge (Zτtimes the proton charge) instead of the charge number: this would be equivalent to replace the system number density with the charge density. 75 2 Ground-state properties of clustered systems unaltered. Using equation (2.3.50), for a two-cluster system it is <Qch,comp>=∑︂ i <Qch,i>+ + 4√︃π 5 Z1A2 2+Z2A2 1 A2∫︂R3 r2Y20(θ, φ) Φ(r)d3r (2.3.76) (again similarly to [67, eq. (A.12)]), where the integral involving the intercluster relative motion is identical to the one discussed for the matter quadrupole moment. For a three-cluster system, combining equations (2.3.49), (2.3.52) and (2.3.56), <Qch,comp>=∑︂ i <Qch,i>+2A3 A Z1A2−Z2A1 A1+A2 <3z1z2−r1·r2>Φ+ + 4√︃π 5 Z1A2 2+Z2A2 1 A2<r2Y20>φ1+ + 4√︃π 5 (Z1+Z2)A2 3+Z3(A1+A2)2 A2<r2Y20>φ2(2.3.77) 76 3 Direct transfer nuclear reactions Consider once more the scattering of two nuclei, Aand b, in vacuum. The system final state, in general, can comprise a different number of particles, and of different species. In a transfer nuclear reaction, the final state is composed again by two particles aand B(thus the process is labelled as A+b→a+B), and it is possible to identify a specific collection of nucleons, ν, such that A comprises the nucleons in aand νtogether, and similarly Bis the aggregate of systems band ν(this does not necessarily imply any cluster model). In general, any possible rearrangement of the nucleons comprising the whole system is called a partition. This chapter is concerned in particular with the description of processes that can be modelled as direct reactions, where the system initial and final states can be connected explicitly through the properties of the interaction (see e.g. [17, sec. 2.18]). More specifically, the goal of this chapter is to review some approximate approaches which allow to explicitly evaluate, in a fully quantum framework, the cross-section of a direct transfer reaction as a function of the aforementioned physical ingredients. First of all, different formal expressions of the cross-section in terms of the scattering problem exact solution are presented in section 3.1. These are then employed to draw explicit formulas in the following sections. Section 3.2 presents two different frameworks for single-particle transfers: the well-known first-order DistortedWave Born Approximation (DWBA), and a selection of continuum-discretised coupled-channels schemes. Finally, section 3.3 discusses the application of second-order DWBA to the transfer of two particles whose relative motion is explicitly taken into account. 3.1 Formal exact expressions for the reaction cross-section For definiteness, the specific case of transfer reactions is considered in the following, but note that the very same formalism can be applied to any direct process. Furthermore, some details of the derivations are skipped in this section for simplicity. Sources as [4, sec. 2], [60, ch. 3], [74, ch. 3, 5, 6] and [75, sec. 2, 9] may be consulted for a more complete treatment. Furthermore, bra-ket notation is widely adopted here for compactness: the quoted sources often provide more explicit expressions in coordinate representation. 77 3 Direct transfer nuclear reactions 3.1.1 Introduction The system Hamiltonian, H, can be expressed as Ha+Hb+Hν+Vab + Vνa +Vνb, where each Hi(with i=a, b, ν) includes all kinetic and potentials terms for the isolated isystem, and each Vij includes all interactions between systems iand j(i.e. between their internal components, if iand jare not elementary). Note that splitting the interaction in isolated pieces essentially amounts to assume that, for instance, the interaction between the nucleons in aand νis independent of the state of nucleons in b: this is strictly true only if the total interaction involves just two-body forces. It is possible to lift the aforementioned assumption in the general treatment, but this is not done for simplicity, given that all practical calculations adopt this framework. Let Kibe the kinetic energy of particle icentre-of-mass. As is customary, the coordinates are chosen so to decouple the irrelevant motion of the centreof-mass of the whole system (a+b+ν). For instance, in the initial partition, consider the three Jacobi coordinates (defined as in equation (2.3.31)) constructed from the position of particles ν, a, b (in this order): ν–adistance, rνa, then A–bdistance, R Ab, and centre-of-mass position, R cm. Let KAbe the kinetic energy of the centre-of-mass of νand a, similarly Kcm the kinetic energy of the whole system centre-of-mass, then Kνa =Kν+Ka−KAand similarly KAb =KA+Kb−Kcm. Since all Jacobi coordinates except the last one (here Rcm) are frame-invariant, each Kij is invariant as well, and in particular coincides with the total kinetic energy of iand jcomputed in i+jcentre-of-mass frame (this is essentially König’s decomposition theorem). From the standard treatment of the classical two-body problem, or explicitly for the quantum case by applying the coordinates transformation to the Laplacian operator, it can be seen that Kij is the kinetic energy of a fictitious particle having the i–j distance (i.e. the associated Jacobi coordinate) as position and the reduced mass of iand jas mass. rνb and R Ba are defined analogously in the final partition. Let Hα=Ha+Hb+Hν+Vaν −KAb −Kcm, which is is just the sum of the Hamiltonians for the internal motion of Aand b(each isolated and in its own centre-of-mass frame). Analogous consideration can be made regarding the final partition defining Hβ=Ha+Hb+Hν+Vbν −KaB −Kcm. Symbols αand βwill be thoroughly employed to denote the initial and final partition, respectively. The total Hamiltonian can be written as H=Kcm +KAb +Hα+Vab +Vbν =Kcm +KaB +Hβ+Vab +Vaν (3.1.1) In the following, set the reference frame to the system centre-of-mass rest frame, where Kcm gives no contribution. Such term is thus omitted in the following. Also note that Vab +Vbν is just the total interaction between b and the composite system A, thus the expression could be made slightly more general by writing such term as a generic VAb, which in principle could even include the distortions in the a–νinteraction caused by the presence of b. 78 3.1 Formal exact expressions for the reaction cross-section The symbols |ψj α⟩and ⟨ψj β|will instead denote eigenvectors of Hαand Hβ, i.e. the internal motion states for, respectively, projectile and target, and ejectile and residue. The index jlabels the eigenbasis elements. The eigenvalue of ψj α⟩︂for Hαis denoted by E−Ej α: thus, Ej α−Eis the total binding energy of Aand bin ψj α⟩︂, while Ej α=E−(E−Eα)is just the kinetic energy at infinite distance for A–brelative motion when the nuclei are in state ψj α⟩︂. Let Ej βbe defined similarly, in regard to the final states. In summary: (︂E−Ej α−ˆ︁ Hα)︂ψj α⟩︁= 0 (3.1.2) and similarly for partition β. If the index is omitted, as in “|ψα⟩”, the precise initial (Aand b) or final (aand B) state involved in the reaction of interest is being referenced, unless otherwise noted. Note that no requirement is made on the structure of such states (in particular with regard to the division in core, aor b, and valence system ν). In addition, let |k j α⟩and |k j β⟩be plane-wave states with momentum k j αand k j β, pertaining, respectively, to the initial-partition projectile-target motion, and to the final-partition ejectile-residue motion. Unless otherwise stated, the module kj γwill be such that ¯h2kj γ 2/2mγ=Ej γ, where mγis the reduced mass of reactants in the γpartition (where γ=αor β). Further let “rγ” be a multi-dimensional coordinate representing all internal coordinates for nuclei in γpartition 1. Finally, let Rαbe a shorthand for (R Ab, rα), and similarly Rβ= (R aB, rβ). Both Rcan describe any configuration in the full space under consideration, but either one can be more practical depending on the specific situation. In some occasions it will also be useful to consider the position eigenvectors for a given coordinate, as R Ab⟩︂etc. Finally, let |Ψ+⟩be an exact eigenfunction of the operator associated to the complete system Hamiltonian, ˆ︁ H, for a given eigenvalue E(this is the total energy, including the binding energy of all subsystems). According to the physical problem of interest, the boundary conditions are chosen to fix the desired initial state for the system as the product of a wave-function |ψα⟩, describing the reactants internal state, and a plane-wave |k α⟩directed as the beam (and oriented toward the target) for the reactants relative motion. |Ψ+⟩can thus be written as the sum of |ψαk α⟩and, asymptotically at large distances, of a spherical scattered outgoing wave for each open exit channel. Reminding that Ψ+(R)depends on several coordinates, each scattered wave can be observed when approaching |R|→∞from a specific “direction”. For instance, a scattered wave involving a bound state of ejectile and residue can be observed if all internal coordinates of partition β, “rβ”, are kept finite (otherwise the reactants internal-motion wave-function vanishes), and only RaB is sent to infinity. This statement can be written in formulas as in [74, 1In a simple cluster model where a, b, ν are inert and structureless, and their spins are not coupled, rαwould be just the three-dimensional vector rνa defined previously. 79 3 Direct transfer nuclear reactions eq. (3.34)]. The normalisation adopted for |ψαk α⟩fixes proportionally the norm of all other terms. It is also possible to consider the time-reversed version of the same problem (see [4, sec. (2.7)] and referenced sections for a more complete discussion), drawing an eigenfunction |Ψ−⟩of ˆ︁ Hwith boundary conditions that fix the final state: |Ψ−⟩is written as |ψβk β⟩, where the planewave is now directed from the target toward the detector, plus (asymptotically at infinity) ingoing waves in all initial channels compatible with the chosen final state. If the distinction is relevant (in particular if complex potentials are admitted), |Ψ+⟩and |Ψ−⟩are taken to be, respectively, a right and a left eigenfunction, meaning that 0 = (︂ˆ︁ H−E)︂|Ψ+⟩=⟨Ψ−|(︂ˆ︁ H−E)︂(3.1.3) The reaction cross-section for a given process can be found, in post form, through the projection of the complete solution Ψ+on the desired final state ψβfor the internal motion, precisely the asymptotic limit of such projection for big ejectile-residue distance in the detector direction. Analogously, in prior form, Ψ−can be projected on the initial state of interest. 3.1.2 Plane-wave formulation For definiteness, consider here the post form (similar considerations may be employed in prior form). A formal solution for the Ψ+eigenvalue problem can be expressed through the Green’s functions formalism. Start from equation (3.1.3), splitting Hin (R aB, rβ)coordinates: (E−ˆ︁ KaB −ˆ︁ Hβ)|Ψ+⟩= (ˆ︁ Vab +ˆ︁ Vaν)|Ψ+⟩(3.1.4) Seek now a Green’s function ˆ︁ Gfor the operator (E−ˆ︁ KaB −ˆ︁ Hβ), namely a distribution such that, for any two coordinates Rand X, it is ⟨︂X[︂E−ˆ︁ KaB(R aB)−ˆ︁ Hβ(rβ)]︂ˆ︁ G(R,X)R⟩︂=⟨X|R⟩(3.1.5) where the symbols “ ˆ︁ KaB(R aB)” etc. were employed here just to underline that each operator acts only on the space specified by the coordinate. Given a solution ˆ︁ G, it can be seen by direct substitution that ˆ︁ G[ˆ︁ Vab+ˆ︁ Vaν]|Ψ+⟩(i.e. ˆ︁ G times the source term in eq. (3.1.4)) is a solution for |Ψ+⟩in equation (3.1.4). In explicit notation, this is Ψ+(R) = ∫︂G(R,X)[Vab(X) + Vaν(X)]Ψ+(X)dX(3.1.6) There are in general several solutions for ˆ︁ G, each generating a different Ψ which solves equation (3.1.4)2. The set of solutions depends on the structure 2Furthermore, as for any eigenvalue problem, a linear combination of solutions Ψis still a solution. 80 3.1 Formal exact expressions for the reaction cross-section of the operator under study, and the correct one must thus be chosen to satisfy the required boundary conditions. For the problem under study, some freedom is generated because the operator E−ˆ︁ KaB −ˆ︁ Hβis not invertible. Consider any function v(R)belonging to the operator eigenspace with eigenvalue 0: given a solution G, clearly G+v(R)g(X), with gan arbitrary function, still satisfies equation (3.1.5). By substitution in equation (3.1.6), it is seen that the additional term in the Green’s function generates an additional term v(R) in the wave-function (the scaling constant is irrelevant, since vhas arbitrary norm anyway). For the present purposes, it is then convenient to add v explicitly to the expression for Ψ+, and remove the discussed degree of freedom from Gin a way that satisfies the required boundary conditions. In particular, it is customary to change the problem into ⟨︂X[︂E−ˆ︁ KaB −ˆ︁ Hβ+iϵ]︂ˆ︁ G+R⟩︂=⟨X|R⟩(3.1.7) where ϵis an arbitrary positive number, and later take the limit of ϵ→0+ (see e.g. [4, sec. 2.8]), drawing the solution |Ψ+⟩=|v⟩+ˆ︁ G+(ˆ︁ Vab +ˆ︁ Vaν)|Ψ+⟩(3.1.8) Regarding the explicit form of |v⟩, note that ˆ︁ KaB(R aB)and ˆ︁ Hβ(rβ)act on different subspaces and are not coupled. Consequently, |v⟩is any linear combination of decoupled states, each written as ψj βk j β⟩︂, where ψj β⟩︂is any eigenvector of Hβ, and k j β⟩︂is a plane wave in R aB whose momentum is such that the total energy of ψj βk j β⟩︂is E. As mentioned earlier, Ψ+is requested to be composed by ψαk α⟩︂plus scattered outgoing waves (not plane waves) in all channels. ψαk α⟩︂does not belong to the space spanned by the set of ψj βk j β⟩︂unless αand βare actually the same partition (i.e. this is not a transfer reaction). When α=β, the boundary conditions are such that the |v⟩ term in equation (3.1.8) is just the null vector (see e.g. [4, eq. (2.29)] or [60, eq. (3.39)]): |Ψ+⟩=ˆ︁ G+(ˆ︁ Vab +ˆ︁ Vaν)|Ψ+⟩(3.1.9) Explicit expression of the Green’s function Since, as already mentioned, ˆ︁ KaB(R aB)and ˆ︁ Hβ(rβ)are decoupled, it is possible to split equation (3.1.7) into two independent equations: ⟨︂X aB [︂Ej β−ˆ︁ KaB +iϵ]︂Gj PW R aB⟩︂=⟨︂X aB R aB⟩︂ ⟨︂xβ[︂(E−Ej β)−ˆ︁ Hβ]︂ˆ︁ Gj int rβ⟩︂=⟨xβ|rβ⟩ (3.1.10) 81 3 Direct transfer nuclear reactions These generate a set of solutions {Gj PW(X aB, R aB)Gj int(xβ, rβ)}j, one for each Ej βsuch that E−Ej βis an eigenvalue of Hβ. Any linear combination of the associated wave-functions is then a solution for Ψ+, and the correct one is chosen to satisfy the boundary conditions. Provided that the eigenvalue problem of Hβcan be solved (as it is normally assumed), ˆ︁ G+can thus be found as well, because the Green’s function associated to the free-particle problem can be calculated explicitly and analytically. For definiteness, set the normalisation of the position eigenvectors so that ⟨︂X aB R aB⟩︂=δ3(︂R aB −X aB)︂. For brevity, let ˆ︁ L=Ej β−ˆ︁ KaB +iϵ, and drop the index jin the following. First note that ˆ︁ Lis translational invariant, meaning that if ˆ︁ L(r)f(r) = h(r)then ˆ︁ L(r)f(r −r′) = h(r −r′). In particular, if GPW(R , X )is a solution, then GPW(R −X ,0 )must be a solution of the same equation. Let g(R , X ) = GPW(R , X )−GPW(R −X ,0 ). By construction, gbelongs to the eigenspace of ˆ︁ Lwith eigenvalue 0, but ˆ︁ Lis invertible, thus g= 0. In summary, the solution must obey GPW(R , X ) = GPW(R −X ,0 ). For brevity, let G+(R −X ) = GPW(R −X ,0 ). The equation to be solved is reduced to ¯h2 2mβ[︁∇2+k2 β+iϵ˜]︁G+(r) = δ3(r)(3.1.11) where the explicit expressions of ˆ︁ KaB and kj βwere employed, and similarly ϵ˜was defined as 2mβϵ/¯h2. Define the three-dimensional Fourier transform, F(f(x), q), as ∫︁R3f(x)exp (−iq ·x)d3x, which is linear in f. It is found that F(︁δ3(x), q)︁= 1, while, in general, F(︁∇2f(x), q)︁=−q2F(f(x), q). Transforming both sides of equation (3.1.11), an expression for F(G+(r), q)is deduced. The sought solution can the be computed applying the inverse transform, G+(r) = ∫︁R3F(G+(r), q)exp (+iq ·x)d3q. Given that F(G+(r), q)actually depends only on the modulus of q, the integral is conveniently computed in spherical coordinates, and reduces to a one-dimensional Fourier transform (see e.g. [74, eq. (6.16)]): G+(r) = −2mβ ¯h2 1 (2π)2∫︂R q ir e−iqr k2 β+iϵ˜−q2dq=−mβ 2π¯h2 1 re−√︂−(k2 β+iϵ˜) r (3.1.12) where it was taken into account that r > 0for the physical problem of interest. The sign of the exponential depends on the adopted convention on the value to adopt as square root of a complex number, √z. If this is defined as √︁|z|exp(i 2argz), with argz∈[−π, π], in the limit of ϵ˜→0+the desired sign for the spherical waves is found, and Gj PW,+(R aB, X aB) = −mβ 2π¯h2 eikj β  R aB−X aB   R aB −X aB (3.1.13) 82 3.1 Formal exact expressions for the reaction cross-section Reaction amplitude As mentioned, in order to evaluate (in post form) the reaction cross-section for a specific process, it is sufficient to find ⟨ψβ|Ψ+⟩, the projection of Ψ+ on the desired final state for the internal motion of ejectile and residue3, and not the full solution. As a consequence, in this case it is more convenient to not proceed through equation (3.1.10). Instead, start from equation (3.1.4) and project on ⟨ψβ|, using equation (3.1.2) and the separation of variables between ˆ︁ KaB and ˆ︁ Hβ: (Eβ−ˆ︁ KaB)⟨ψβ|Ψ+⟩=⟨︂ψβˆ︁ Vab +ˆ︁ Vaν Ψ+⟩︂(3.1.14) Both ⟨ψβ|Ψ+⟩and ⟨︂ψβˆ︁ Vab +ˆ︁ Vaν Ψ+⟩︂are functions of the R aB coordinate only, and the equation can be solved directly for ⟨ψβ|Ψ+⟩. This is helpful because the associated Green’s function can then be restricted to the R aB coordinate. will depend only on that degree of freedom as well. Note how the simplification of the internal-motion Hamiltonian was made possible by the choice, in equation (3.1.4), to split the complete Hamiltonian in the finalpartition expression, even though the boundary conditions of Ψ+would have suggested the opposite choice. Incidentally, the analogous equation in prior form would be ⟨Ψ−|ψα⟩(︂Eα−ˆ︁ KAb)︂=⟨︂Ψ−ˆ︁ Vab +ˆ︁ Vbν ψα⟩︂(3.1.15) Resuming with the post form, the analogous of equation (3.1.7) is now just the first line of equation (3.1.10), thus the solution is the same GPW,+shown in equation (3.1.13) with jcorresponding to the specific state |ψβ⟩of interest. Note that Gis the identity operator in the rβspace, thus, in operator notation, one could just write ˆ︁ G=ˆ︁ GPW,+. The solution for ⟨ψβ|Ψ+⟩, in analogy with equation (3.1.9), is then ⟨ψβ|Ψ+⟩=⟨︂ψβˆ︁ GPW,+(︂ˆ︁ Vab +ˆ︁ Vaν)︂Ψ+⟩︂(3.1.16) or, in explicit notation, defining ξβ(R aB) = ⟨︂ψβR aB Ψ+⟩︂, ξβ(R aB) = ∫︂ψ∗ β(xβ)GPW,+(R aB, X aB)[Vab(X)+Vaν(X)]Ψ+(X)dX(3.1.17) To evaluate the reaction cross-section, it is sufficient to study the form of ξβin the limit of RaB →+∞and R aB directed toward the detector. If the potentials appearing in equation (3.1.17) are short-ranged, it is possible to simply consider the asymptotic form of GPW,+. Note that such assumption 3When writing such projections, it is understood that the integration is performed only on the degrees of freedom appearing in both Bra and Ket, so that e.g. ⟨ψβ|Ψ+⟩is still a Ket in the a–Brelative motion space. 83 3 Direct transfer nuclear reactions sion in section 2.2.1) between the desired microscopic states for a,ν, and the composite system A. The anti-symmetrised wave-functions for the composite systems consist of several components essentially differing just for the assignment of labelled coordinates to each nucleon. At the same time, the precise identity of the nucleons removed from A(and added to B) during the transfer process is irrelevant for the result. The reaction cross-section is then the sum of several contributions, which counter the binomial coefficients appearing in the overlap functions (which are defined projecting the composite-system wave-function on a precise repartition of nucleons between core and valence), see for instance equation (2.2.3). Similarly, the Clebsh-Gordan coefficient in the overlap expression is employed when coupling the angular momenta to form a composite-system state with defined total spin. The spectroscopic amplitudes appearing in the overlaps instead do not cancel out, and the firstorder reaction amplitude is proportional to these. In summary, it is possible to compute the cross-section as a single term, adopting as φj α⟩︂the spectroscopic amplitude for the overlap times a one-body bound state normalised to 1, constructed from the functions |φν,l,s,ml⟩in equation (2.2.4) by appropriately coupling the angular momenta. In the expressions shown in the following, the symbol ψγfor the internal-motion states is retained for conciseness. Having specified a model for the reactants structure, the last missing ingredient is to choose an approximation for the full scattering solution Ψ+to obtain an explicit expression for the transition amplitude. Several approaches, parallel to the schemes employed for generating the distorted waves in section 3.1, are applicable, and some of them are discussed in the following. Many other schemes have been devised in literature, which are not covered in this work. See for instance [22] and references therein for the adiabatic and eikonal approximation, and [82] for semi-classical methods. 3.2.1 Distorted-wave Born approximation (DWBA) Consider the prior-form standard-distorted-wave transition amplitude in equation (3.1.27). Let UaB be a potential depending only on the a–Brelative motion coordinates, and find a solution ⟨︁χβ−for that potential, in complete analogy with equation (3.1.23). The first-order distorted-wave Born approximation (DWBA) prescribes8to approximate ⟨Ψ−| ≈ ⟨︁ψβχβ−. The same approach is followed in post form for Ψ+. In summary: Tpost DWBA =⟨︂ψβχβ−ˆ︁ Vab +ˆ︁ Vaν −ˆ︁ UaB ψαχα+⟩︂ Tprior DWBA =⟨︂ψβχβ−ˆ︁ Vab +ˆ︁ Vbν −ˆ︁ UAb ψαχα+⟩︂(3.2.1) In prior form, if UaB properly describes the elastic scattering between two 8See [4, sec. 2.8.7] and referenced section for a more in-depth discussion on the interpretation of the approximation. The simple recipe reported here is sufficient within the scope of this text. 90 3.2 One-particle transfer inert particles aand B, the approximation basically amounts to claiming that the exact solution is dominated by the elastic component, and all other channels are a small perturbation. The potential can also phenomenologically contain some effects related to non-elastic channels, for instance through imaginary components. The approximation could be expected to be more accurate for a smaller residual interaction (in prior, Vab +Vaν −UaB), so that χβ−resembles more the exact solution. Incidentally, if both UaB and UAb are set to zero, the exact transition amplitude reduces to the plane-wave expression in equation (3.1.22), and the exact wave-function is approximated to just its non-scattered component, |Ψ+⟩ ≈ ψαk α⟩︂etc. The result is know as “plane-wave Born approximation” (PWBA). As long as a numerical solution to the problem as posed is sought, the PWBA scheme is rarely employed in recent works, as the practical computation of the distorted waves is not particularly expensive or difficult, compared to current computational capabilities. DWBA prior and post forms are equivalent, meaning that, if the same pair of potentials is adopted in both forms, and no further approximation is made, it is Tpost DWBA =Tprior DWBA, as shown in [4, sec. 2.8.8]. Such result is not trivial because, while the exact expressions in equations (3.1.26) and (3.1.27) necessarily coincide, they are supplied with two different approximations in order to obtain equation (3.2.1). The prior-post equivalence has two relevant consequences in practical calculations. First, for any given calculation it is possible to freely choose the form giving rise to the simplest, and thus most accurate, calculation from the numerical point of view. Second, the difference between prior and post transition amplitudes is a measure of numerical inaccuracies, disentangled from the impact of physical approximations. In particular, if the computation is accurate in both forms, the results must coincide. Note that if the results do not coincide, it is still possible that the calculation in one form is accurate. There is an apparent asymmetry between the role of UaB and UAb in each (prior or post) DWBA scheme. One of the potentials is an auxiliary, in principle arbitrary, operator employed to reformulate an exact expression: note that this does not exclude that the precise choice of the auxiliary potential can have an impact on the computed amplitude, since an approximation is then being performed. The other U, instead, is introduced while attempting to generate a physically sound approximation for the complete scattering wave-function. However, the aforementioned property of equivalence between prior and post forms (in which the role of the two auxiliary potentials is opposite) actually suggest that, once the approximation of the full solution is introduced, both UaB and UAb equally contribute to determine the final result. 3.2.2 Coupled-channels approaches Another relevant class of approximate methods includes all the cases where a wave-function is computed using a continuum-discretised coupled-channels 91 3 Direct transfer nuclear reactions scheme. This is the case when a generalised-distorted-waves formalism is adopted, computing the distorted wave as described in section 3.1.4, or when the full solution of the scattering problem is approximated by a CDCC wavefunction. The possible combinations are listed below, followed by a general discussion on the features of such methods. Standard distorted-waves scheme Start from the standard-distorted-wave reaction amplitude in equations (3.1.26) and (3.1.27), but now approximate Ψ+with a continuum-discretized coupled channels solution of the full problem equation (3.1.3). Such scheme is essentially a simplified version of the coupled-channels Born Approximation discussed in [4, sec. 3.6]. The differential equation for the solution |Ψ+⟩(required in post-form), equipped with the desired boundary condition, is expressed using the prior-form coordinates and the corresponding rearrangement of the complete Hamiltonian, H: [︂E−ˆ︁ KAb(R Ab)−ˆ︁ Hα(rα)−ˆ︁ Vab(R Ab, rα)−ˆ︁ Vbν(R Ab, rα)]︂|Ψ+⟩= 0 |Ψ+⟩=ψαk α⟩︂+outgoing scattered waves (3.2.2) analogously, in prior: ⟨Ψ−|[︂E−ˆ︁ KaB(R aB)−ˆ︁ Hβ(rβ)−ˆ︁ Vab(R aB, rβ)−ˆ︁ Vaν(R aB, rβ)]︂= 0 ⟨Ψ−|=⟨︂ψβk β+outgoing scattered waves (3.2.3) It is also possible to supply a simplified version of the full problem, including only simpler couplings within the initial partition states. The differential equation is then solved approximately trough a scheme entirely analogous to the one employed for Φin equation (3.1.33), and the result is inserted in equation (3.1.26) or (3.1.27). It is underlined that, in principle, any set of coordinates and any form for Hwould be acceptable, and the choice is dictated by practical matters. When deriving the exact transition amplitude, the goal was to analytically evaluate the projection of the unknown full solution on the desired final (or initial) state, thus the Hamiltonian was expressed so to simplify such task. Here, where an explicit solution for Ψis sought, the given choice for the coordinates removes any difficulty in properly enforcing the boundary condition, and allows to describe with good accuracy elastic and inelastic couplings from the initial (or final) state of interest, which presumably give rise to the dominant components in the full wave-function. The disadvantage is that the two wave-functions appearing in the reaction amplitude are originally computed in different coordinates, thus a coordinate change will be required to obtain the final result, with the main effect of complicating the explicit form of the angular-momentum expansions. Note that this sort of issue already arises in 92 3.2 One-particle transfer equation (3.2.1). Generalised-distorted-waves schemes Consider now the expression in equation (3.1.30) for the exact reaction amplitude, together with the CDCC solution for the generalised distorted wave, computed from equation (3.1.33). In order to approximate the full solution, Ψ, one possible approach is to apply again the CDCC method (see text commenting equation (3.2.2)). Another possibility is to employ the same approximation adopted when deriving the DWBA, illustrated in section 3.2.1: ⟨Ψ−|is set to ⟨︁ψβχβ−, where χβ−is the solution of a one-body problem with the form of equation (3.1.23). Such construction is simpler to implement and less demanding computationally, and was thus employed in the coupled-channels calculation discussed in chapter 4. Common features of coupled-channels schemes The CDCC solution for either Ψor Φis not limited to the elastic scattering component. This framework thus allows to take into account some “dynamical” effects, namely features that cannot be explained solely in terms of ground-state properties of the reactants, and which manifest quantummechanically as coupling to excited states. If the spectrum of ψj αis approximated with acceptable accuracy, Ψ+(or Φα) may provide a good account of inelastic excitations of nucleus Ainduced by the interaction with nucleus b. This approach may thus be expected to be superior to equation (3.2.1). However, as mentioned when discussing equation (3.1.33), the practical execution of the CDCC recipe can be significantly more complex with respect to the standard first-order DWBA prescription. On this regard, remind that the reactants structure assumed in this section (associated to the transfer of an inert cluster) causes ψαand ψβto reduce to just the ν–aand ν–brelative-motion wave-functions, greatly simplifying the problem. Even in the case where both the full solution and the generalised distortedwave are computed in CDCC, prior and post forms are in general not equivalent. Of course, in the limit where the approximated Ψ±approach the exact solutions, the calculated amplitudes would tend to the same, exact value. Consequently, in this framework, prior-post invariance cannot be used to test specifically the formal consistency of the calculation and the appropriate implementation of the numerical algorithm, but can only give a suggestion on the “global” accuracy of the calculation, being sensitive also (but not only) to the adopted physical approximations. At the same time, achieving satisfactory numerical accuracy can be difficult, thus, even more than in DWBA, for each specific calculation it is useful to identify the most convenient form regarding the amount of required computational resources. Some information from this point of view can be deduced checking convergence with respect to 93 3 Direct transfer nuclear reactions the numerical parameters of the calculation (see e.g. [78]). 3.3 Two-particle transfer If the transferred system is a composite nucleus, especially a loosely bound one, it can be interesting to take into account its internal structure and the associated impact on the reaction. Applications in literature almost invariably involve the transfer of two nucleons, very often two neutrons. A discussion on the associated formalism, from different points of view, can be found in [4, sec. 3.7], [59, 83–86]. Such calculations are customarily performed in second-order distorted-waves Born approximation (DWBA). In chapter 4, the formalism will be applied to the transfer of a proton and a neutron. As in section 3.2, the core particles of both reactants are treated as inert and their internal structure is neglected. Notwithstanding the formal similarities with the two-neutron-transfer case, the application of the same scheme to p+n transfer reactions is quite new, with the very first published calculation apparently dating back to 2017 [87]. In the second-order DWBA formalism, (see [4, sec. 3.7]), the transition amplitude can be decomposed into the coherent sum of a first-order term, connected to the one-step (or “simultaneous”) transfer of both particles, and a second-order contribution, associated with two-step processes, and in particular the “sequential” transfer of each particle separately. It is found that the equivalence between prior and post forms of the distorted-wave Born approximation holds order-by-order (provided that non-orthogonality corrections are taken into account) [4, sec. 3.7.2]. The considerations made in section 3.2.1 on this regard can thus be applied here to the simultaneous and sequential contribution separately. In principle, the calculation of each contribution to the two-particle-transfer transition amplitude involves addressing a four-body problem (the two core nuclei and the two transferred particles), where the reactants internal-motion wave-functions depend on two independent coordinates (e.g. the position of each transferred particle). Here, the full problem is instead approximated so that all internal-motion states can be constructed in terms of two-body problems, which can be solved with greater ease. The following parts of this section are principally dedicated to the formal development of such approximation. The discussion will be focused in particular on approaches that can be implemented in the coupled reaction channel calculations code Fresco [88], which was employed to perform the practical calculations in chapter 4. The system Hamiltonian The notation employed in this section mirrors the one found in section 3.1.1, with some modification to allow explicit discussion of the additional degree of freedom, discussed in the following. Consider the transfer reaction A+b→ a+B, while nucleus Ais a bound system of three elementary clusters, a+ν+µ, 94 3.3 Two-particle transfer and similarly B=b+µ+ν. The transferred system is N=ν+µ. Also let A=a+µ,A′=a+ν,B=b+ν, and B′=b+µ. As before, given any pair of systems i, j and their composition J=i+j, let Kij =Ki+Kj−KJ, where Kiis the kinetic energy of the centre-of-mass of i. Assume all interactions can be described by two-body forces, with the potential energy between any pair of particles i, j being Vij. Let rij be the distance between particles iand j. The system Hamiltonian, is the same in section 3.1.1: updating the notation and explicitly splitting the terms involving νor µ(rather than their composition N), it is H=∑︁i∈{a,b,µ,ν}Ki+Vab +Vaν +Vbν +Vaµ +Vbµ +Vνµ. All calculations are performed in the system centre-of-mass rest frame. Hcan be rewritten in several ways, for instance H= (Kaµ +Vaµ)+(KAν +Vaν +Vµν )+(KAb+Vab +Vbµ +Vbν)(3.3.1a) which is useful to perform a calculation in prior form, or analogously H= (Kbν +Vbν)+(KBµ +Vbµ +Vµν)+(KBa+Vab +Vaν +Vaµ)(3.3.1b) which is the post-form rearrangement. Similar expressions, involving A′and B′, can be similarly obtained (note that the role of µand ν, and thus Aand A′, is entirely symmetrical). 3.3.1 First-order (“simultaneous”) contribution From the formal point of view, the DWBA simultaneous contribution to the reaction amplitude has precisely the same form in equation (3.2.1), see [4, eq. (3.61)], with the relevant difference that now the Hamiltonian depends also on rµν, and full internal-motion states ψαand ψβreduce to three-particle wave-functions explicitly describing the a+ν+µand b+µ+νrelative motion, scaled, as in section 3.2, by the spectroscopic amplitude for the associated overlaps (see section 2.2). In the following, the symbol ψγwill still denote the wave-function for reactants internal motion in partition γ, as in section 3.1.1, but ignoring the trivial factors related to the internal state of a, b, µ, and ν. Consider for instance the “post-form” Hamiltonian in equation (3.3.1b). The grouping of terms enclosed in braces suggests a formal procedure to construct all required ingredient for the simultaneous transfer calculation. First, the spectrum for the b–νrelative-motion is constructed as the solution of a standard two-body problem under the Hamiltonian Kbν +Vbν , obtaining an orthogonal set of wave-functions φj bν indexed by j. Then, ψβis found expanding it in the b–νbasis and solving for the B–µmotion under the potential Vµb +Vµν, trough a procedure analogous to equation (3.1.33)9. The internal state for Ais constructed analogously. Finally, the first-order DWBA amp9In section 3.1.4, the a–Bmotion was studied expanding in a basis of reactant internalmotion states ψj β. Here, a single state ψβis studied expanding in a basis for the b–ν motion. 95 3 Direct transfer nuclear reactions litude is derived precisely as in equation (3.2.1), using Vab +Vaν +Vaµ −UaB as transition potential. Note that explicit three-body structure calculations are usually not performed in this manner10 (see instead e.g. [68] an actual example): the purpose of the formal construction just illustrated is rather to clarify the meaning of approximations shown later, which are aimed at decoupling the two degrees of freedom appearing in both ψαand ψβ. Regardless of how ψβis originally computed, to perform a practical calculation it is convenient to express it in the (rµν, rNb)Jacobi coordinates (sometimes known as “t” coordinates), and similarly ψαin (rµν , rNa)coordinates [59, sec. 4]: note that rµν is the same vector in both cases. In this way, taking into account that it is possible to express rAband rNa as functions of rBaand rNb, the first-order DWBA transition amplitude could be written as T=⟨︁ψβ(rµν, rNb)χβ−(rBa)Vψα(rµν, rNb, rBa)χα+(rNb, rBa)⟩︁(3.3.2) where Vis a shorthand for the appropriate transition potential appearing in the scalar product, which depends on all coordinates, rµν , rNb, rBa. For any fixed value of R Nb and rBa, perform first the integration on rµν, defining ˜︁ ψ∗ β(rNb)˜︁ V(rNb, rBa)˜︁ ψα(rNb, rBa) = =∫︂ψ∗ β(rµν, rNb)V(rµν, rNb, rBa)ψα(rµν , rNb, rBa)d3rµν (3.3.3) so that T=⟨︂˜︁ ψβ(rNb)˜︁ V(rNb, rBa)˜︁ ψα(rNb, rBa)⟩︂, which is formally identical to a standard one-particle-transfer transition amplitude, and can thus be computed using the same numerical approaches. Hence, it is relevant to point out that a function expressed in (raµ, raν)or (rbν , rbµ)coordinates (also known as “v” coordinates) can be mapped to the corresponding one in “t” coordinates using the Moshinsky transformation [89]. The change from (raµ, rAν)and (rbν, rBµ)(known as “y” coordinates) to “t” coordinates is instead governed by the Raynal-Revai transformation [90]. The “heavy-ion” approximation The Hamiltonian in equation (3.3.1b) may be rewritten, in post-form rearrangement, as H= [(Kbν +Vbν)+(Kbµ +Vbµ) + Vµν +KB+KB′−Kb−KB] + +KBa+Vab +Vaν +Vaµ (3.3.4) 10One difficulty is that a good account of b–µand B–νmotion separately as described would normally require inclusion of continuum states. 96 3.3 Two-particle transfer or H= [(KB′ν+Vbν +Vµν)+(KBµ +Vbµ +Vµν)−Vµν+ +KB+Kb−KB−KB′] + KBa+Vab +Vaν +Vaµ (3.3.5) These exact expressions by themselves bring no profit. However, suppose now that the core nuclei aand bare much heavier than the transferred system N. Then, the centre-of-mass position of each composite system, B,B′and B, approximately coincides with bposition, so that rbµ ≈rBµ and so on. This also implies that the expectation value of ˆ︁ KB+ˆ︁ Kb−ˆ︁ KB−ˆ︁ KB′on any physical state is approximately zero (and similarly for a). Furthermore, assume that the interaction between the two transferred particles, Vµν, is negligible compared to their interaction with the cores, for instance Vbµ, so that Vbµ(rbµ) + Vµν(rµν)≈Vbµ(rbµ), which, if convenient, may also be seen as a potential VBµ(rBµ)depending only on the distance between µand the centre-of-mass of B. The Hamiltonians are as a result approximated to: Hpost HI-cc = [(Kbν +Vbν)+(Kbµ +Vbµ)] + KaB+Vab +Vaν +Vaµ Hpost HI-CC = [(KB′ν+Vbν)+(KBµ +Vbµ)] + KaB+Vab +Vaν +Vaµ (3.3.6) which could be labelled, respectively, as “core-core” and “composite-composite” schemes of the “heavy-ion” approximation. In this limit, the problems for the b–νand B–µmotions are decoupled, so that any state of Bcan be obtained as a linear combination of products of solutions for each independent problem. Note that, as a result of the adopted approximations, the solutions ψαand ψβ deduced from the model will not coincide with the “exact” ones. This is not a serious concern in itself, given that the “exact” Hin equation (3.3.1b) is of course a model Hamiltonian as well, and if phenomenological forms for Vbν, Vbµ and so on are chosen appropriately, it is possible to partially correct for the neglected terms. For instance, the potentials are usually selected so that the sum of the binding energies of each product of single-particle solutions matches the experimental binding energy of the composite nucleus. In order to mimic phenomenologically an interaction between µand ν, the potential Vbν +Vbµ may be set to depend in a non-standard manner on the quantum numbers characterising the solutions11. In studies aimed at investigating correlations between the two transferred particles, a framework as the one just described may be employed to generate a starting basis of solutions, which is then coupled adding an explicit Vµν interaction, often focused on pairing effects (a lot of work exists in this direction, see e.g. [91] just for a recent example). In the “core-core” scheme, states for the B–µproblem can be approximated as solutions of Kbµ +Vbµ, thus the complete state is constructed using only bound-state wave-functions of the core nucleus bwith each valence particle 11For example, the depth of the nuclear potential may depend on the angular momentum in a way that could not be described as a spin-orbit potential. 97 3 Direct transfer nuclear reactions independently, labelled φj bν and φj bµ, which are most conveniently described using “v” coordinates (see comment to equation (3.3.2)). In the same way, the “composite-composite” scheme instead allows to describe the complete state in terms of removal of one valence particle from the given composite state B. This sort of approach is particularly convenient when µand νare the same nuclide, so that Vbµ(r) = Vbν(r)and φj bν(r) = φj bµ(r), further simplifying the practical computation. Furthermore, in the heavy-ion approximation, the transition potential Vab + Vaν +Vaµ can be obtained as the sum of Vab and of the two binding potentials required to construct ψαin the same approximate scheme, Vaν and Vaµ. Such condition, which simplifies the explicit computation of the transition amplitude, arises naturally in one-particle transfers (see equation (3.2.1)), but is not found in the full four-body treatment of the two-particle transfer (see equation (3.3.1)), where Vµν appears in the equations defining ψαand ψβbut not in the transition operator. With the same reasoning employed for equation (3.3.6), the prior form of the “core-core” scheme is found to be Hprior HI-cc = [(Kaµ +Vaµ)+(Kaν +Vaν)] + KAb+Vab +Vbν +Vbµ (3.3.7) this is different than the corresponding post form shown above. The discrepancy in the Hamiltonians causes a difference in the calculated transition amplitudes, thus the heavy-ion approximation breaks the DWBA prior-post invariance. In particular, Hpost HI-cc −Hprior HI-cc = (Kbµ −KBµ)−(Kaν −KAν)(3.3.8) thus the difference lies only in the kinetic terms and becomes negligible in the limit of heavy core nuclei. If the difference is instead significant, it is expected that the most accurate form will be the one related to the partition where the heaviest core particle is bound to the transferred system12. Regarding the Li 6+p→He 3+αreaction studied in chapter 4, the approximation appears to be unsuitable in both the initial and final partition. The “core-composite” approximation It is here noted that the shortcomings of equation (3.3.6) can be mitigated by correctly computing the kinetic terms. In the exact expression for the total Hamiltonian, in equation (3.3.1), let VAν(raν , rµν) = Vaν(raν ) + Vµν(rµν), and similarly define VBµ =Vbµ +Vµν, obtaining H= [(Kaµ +Vaµ)+(KAν +VAν )] + (KAb+Vab +Vbν +VBµ −Vµν) H= [(Kbν +Vbν)+(KBµ +VBµ)] + (KBa+Vab +Vaµ +VAν −Vµν )(3.3.9) 12More explicitly, if the target is the heaviest reactant, the smallest error related to this issue is expected in prior for a pick-up reaction and in post for a stripping one. 98 3.3 Two-particle transfer which is still exact. If now VAν is approximated by a potential depending only on the distance between Aand ν, and similarly VBµ is simplified to a function of only rBµ, the problem for A–νand B–µrelative motion is again decoupled from the equations for Aand Binternal state. However, in this case, the Hamiltonian is the same for both prior and post form, and the scheme can be applied indifferently to heavy and light systems. Regarding the choice of the approximate potentials, from the formal point of view one may for instance set VAν to the “diagonal” component of the full potential on the solution φaµ for the a–µrelative motion, ⟨φaµ(rµν)|Vaν(rAν , rµν) + Vµν(rµν )|φaµ(rµν)⟩. In practice, the potentials are usually constructed phenomenologically. The proposed approach presents two issues from the practical point of view. The first one is that, differently than in equation (3.3.6), the transition operator, e.g. Vaµ +VAν −Vµν in post form, does not coincide with the sum of the binding potentials employed to construct ψα(here Vaµ +VAν)13. The problem disappears in the limit where Vµν is negligible, as assumed when writing equation (3.3.6): then equations (3.3.9) and (3.3.10) coincide, and VAν(rAν )can be formally thought as ⟨φaµ(rµν)|Vaν(rAν, rµν )|φaµ(rµν)⟩, for instance14. In general, it is possible to overcome the limitation, at least partially, adopting a suitable phenomenological form for VAν and VBµ. Formally, define ˜︁ VAν(rAν , rµν) = Vaν(rAν , rµν) + 1 2Vµν(rµν )and ˜︁ VBµ =Vbν +1 2Vµν, and rewrite the total Hamiltonian as H=[︂(Kaµ +Vaµ) + (︂KAν +˜︁ VAν)︂]︂+(︂KAb+Vab +Vbν +˜︁ VBµ)︂ H=[︂(Kbν +Vbν) + (︂KBµ +˜︁ VBµ)︂]︂+(︂KBa+Vab +Vaµ +˜︁ VAν)︂(3.3.10) which, if the complete expression of all potentials is employed, is still exact: in principle, the calculation scheme suggested by this form could be employed to address the full four-body problem15, just as equation (3.3.1) or (3.3.9). If ˜︁ VAν and ˜︁ VBµ are approximated to potentials depending only on, respectively, rAν and rBν, the resulting calculation scheme has the same features found from equation (3.3.9) and additionally the desired simplified form for the transition potential. Once again, the corresponding solutions for ψαand ψβdo not coincide with the “exact” ones (namely, the solutions of e.g. the Hamiltonian Kaµ +Vaµ +KAν +Vaν +Vµν), but this is true for all approximate schemes discussed here, and the deviation may in fact be less severe in equation (3.3.10) than in equation (3.3.6) (without adding further corrections), depending on the choice of the approximate potentials. 13This is required, for instance, in the Fresco code. 14Such prescription is expected to be superior to simply setting VAν (rAν )to Vaν computed at the same value for the function variable, Vaν (raν ≡rAν ). 15Given a basis of eigenfunctions of the modified “internal” Hamiltonians (the terms within square brackets in equation (3.3.10)), the physical initial and final states for the system can be expressed in such basis, and subsequently the transition amplitude for these states can be computed, using Vab +Vbν +˜︁ VBµ as prior-form transition operator (and similarly in post form). Of course, there is no advantage in such approach with respect to the standard one, as long as no further approximations are introduced. 99 4 The Li 6+p→He 3+αtransfer reaction All cross-sections and phase shifts shown in this chapter were computed using the Fresco code [88]. The practical computation of the transfer crosssections is performed solving directly the relevant coupled differential equations, instead of numerically evaluating the expressions for the reaction amplitude introduced in chapter 3. It was shown that the two approaches are formally equivalent [93]. 4.1 DWBA deuteron transfer The first part of this section, up to section 4.1.2, presents and comments all the physical ingredients required to compute the cross-section of interest, together with some details regarding how they were derived. Some of these ingredients are later employed also in the other transfer calculations reported in this chapter, and in the Li 6+p barrier penetrability estimation in section 5.2. The results on the transfer channel are then shown in section 4.1.3. 4.1.1 Optical potentials The following paragraphs discuss the form of some potentials mainly aimed at describing the elastic scattering between the particles involved in the transfer reaction. The projectile-target interactions are employed to construct both the distorted waves, as discussed in section 3.1.3, and to approximate the full scattering solution, using the approach reported in section 3.2.1. The “core-core” potential is the Vab appearing in the transition amplitudes in equation (3.2.1), and refers to the interaction between reactants deprived of the transferred system. Li 6+pprojectile-target potential Most optical potentials existing in literature for Li 6+p elastic scattering include at most spin-orbit couplings. However, it is not possible to reproduce the experimental phase-shifts without a component coupling the spins of both reactants together. Furthermore, no published energy-independent potential known to the author compares acceptably with elastic scattering cross-section in the energy range of interest here. For this work, an energy-independent potential was adjusted to reproduce the most relevant partial waves in the experimental phase-shifts, namely the s-waves and the p5/2 wave (projectiletarget orbital angular momentum of 1, total angular momentum of 5 2), the latter showing a resonant trend, attributed to a Be 7level, which is visible also in the transfer channel (see the discussion later in section 4.1.3). The result is shown in figure 4.1. Note that the experimental phase-shifts for other waves are not reproduced, but the adjusted potential introduces no spurious resonances (namely, resonances with incorrect energy or quantum numbers) within the region of interest: this is a relevant advantage of this potential with respect to others tested before. An imaginary component was then added to 106 4.1 DWBA deuteron transfer 0 1000 2000 3000 4000 Centre-of-mass collision energy [keV] -60 -30 0 30 60 90 120 150 Phase-shift [deg] L = 0, S = 1/2, Jπ = 1/2+ L = 0, S = 3/2, Jπ = 3/2+ L = 1, S = 3/2, Jπ = 5/2Experimental data 6Li + p elastic scattering Figure 4.1: Points are experimental Li 6+p elastic-scattering phase-shifts, collected in [94] from earlier experiments. Lines in corresponding colours are the prediction of the optical potential discussed in section 4.1.1. In the figure legend, Lis the Li 6–p relative orbital angular momentum modulus quantum number, Sthe modulus quantum number for the sum of Li 6and p spins, J the modulus quantum number for the sum of L and S , and πthe parity of the state. 107 4 The Li 6+p→He 3+αtransfer reaction the potential to improve the agreement with elastic-scattering cross-sections at the relevant energies. The precise form and parameters of this potential are reported in appendix B.2. He 3+αprojectile-target potential Different α– He 3potentials are available in literature, either related to scattering [95] or bound state properties [67, 96]. Generic parametrizations, fitted on the elastic scattering of He 3(or α) on heavy-ion targets at energies above the Coulomb barrier, are also available. However, all potentials considered within the present project poorly reproduced the experimental elastic scattering at the energies of interest, and yielded an unsatisfactory description of the transfer cross-section as well. The projectile-target optical potential for the final partition adopted in the present work was created by fitting the experimental elastic scattering differential cross-sections found in [97, 98], using SFresco [88] (based on Minuit [99]). The potential included a spin-orbit component and an imaginary surface term. The complete set of parameters is reported in column “α– He 3” of table B.2a. For illustration, a sample of the results is shown in figure 4.2. Attempts to fit the data using a purely real potential, with or without using phase-shifts in [98] as guide, yielded inferior results. While a visual comparison of datasets in [97] and [98] suggests that some experimental errors are underestimated, it is apparent that the fitted potential does not describe perfectly the data (a similar result is found when attempting to fit phase-shifts, as only few waves at a time can be reasonably adjusted). Given the fit complexity and the high number of free parameters, more than one local minimum in χ2can be found through the fitting procedure, depending on the computation specific details. Some of the potentials obtained in this manner were discarded as they displayed extreme parameter values1. The remaining candidate potentials yielded similar results regarding the transfer cross-section, which suggests that the specific choice adopted here does not influence the conclusions significantly. The real part of the adopted potential was also employed to check the Be 7 bound states it generates, as shown in table 4.1. It is stressed that no explicit adjusting was performed on these quantities. α–pcore-core potential [68, eq. 4.3] quotes a real energy-independent α-nucleon potential, found by fitting scattering phase-shifts. Such potential was also tested here, finding good qualitative agreement with experimental elastic scattering cross-sections from [100–108] (in this regard, it should be taken into account that the different datasets do not fully agree with each other). No information about the Coulomb repulsion term is given in [68], thus, in the present work, the shape 1For instance, a vanishing diffuseness for the spin-orbit component. 108 4.1 DWBA deuteron transfer 77.5 88.5 99.5 10 3He laboratory energy [MeV] 6 8 10 12 14 16 18 20 22 (dσ/dΩ) / (dσ/dΩ)Rutherford Spiger et al., 1967 Tombrello et al., 1963 Fit 6Li+p threshold 3He + α elastic scattering, θcm = 90.0° Figure 4.2: Points are the experimental He 3+αelastic-scattering differential cross-section at a centre-of-mass scattering angle of 90◦, rescaled to the Rutherford cross-section, as a function of He 3laboratory energy, from [97, 98]. Blue line is the prediction of the fitted optical potential discussed in section 4.1.1. Green vertical line is the threshold for the Li 6+p channel. Table 4.1: Experimental values for the binding energies of Be 7bound states and the ground-state root-mean-square radius are compared with the predictions of the real part of the He 3+αoptical potential discussed in section 4.1.1. Quantity Experimental He 3+αpotential Be 7g.s. 3/2−BE 1587 keV 2022 keV Be 71st 1/2−BE 1158 keV 951 keV Be 7g.s. rms radius 2.48 fm 2.64 fm 109 4 The Li 6+p→He 3+αtransfer reaction in equation (B.2) was assumed, fitting the radius of the uniformly charged sphere, RC, on experimental cross-sections, using the SFresco code. The result is reported in table B.2c. 4.1.2 Overlap functions The description of the ingredients for the DWBA transfer calculation is now completed by a discussion on the model Hamiltonians for the internal motion within each nucleus, and the overlap functions for both projectile and target systems. In fact, within the inert di-cluster model employed here for the structure, the required physical input is reduced to the potentials between the transferred system and each core nucleus, since all inter-cluster relativemotion wave-functions are constructed as appropriate eigenstates of Hamiltonians involving these potentials. ⟨︁αLi 6⟩︁overlap The Li 6nucleus is being described here as a bound state between two structureless clusters, an αparticle and a deuteron, hence the ⟨︁αLi 6⟩︁overlap function depends only on the distance between the clusters centre-of-mass: the normalised wave-function was constructed as detailed in section 5.2.1, using the binding potential given in [96, eq. (13)] and including both the 2sand 1dcomponents. Most calculations in this chapter adopt a value of −√0.05 for the amplitude of the L= 2 component, c2in equation (5.2.1): this is similar to the value which reproduces the experimental magnetic dipole moment, see the discussion in section 5.2.1. Some calculations employing c2=−0.0909 (the value reproducing the electric quadrupole moment) are reported in figure 4.5. Regarding the spectroscopic factor for the overlap, S, the independentparticle shell model (in which c2= 0) would suggest to assign a value of 1 (see section 2.2.2). The Variational Monte Carlo calculation in [2, sec. V.C] yields a value of S= 0.841 including both L= 0 and 2 components2. In this calculation, an overall spectroscopic factor of 0.85 was adopted (meaning that the spectroscopic factor for each specific component is c2 LS). ⟨︁pHe 3⟩︁overlap Using the same reasoning and notation as in section 4.1.2, the He 3ground state is described as a p+d bound state with orbital angular momentum L= 0 and zero nodes (“1s”). An L= 2 component (a 1dstate) is also admissible. Both components were constructed using the binding potentials published in 2A spectroscopic factor of 0.85 ±0.04 is also suggested in [109] through the analysis of a radiate-capture experiment. However, more recent literature suggests that the result may be coincidental. In [110] it is stated that the determination in [109] is inaccurate due to the choice of α–d wave-functions, in particular regarding the number of nodes of the radial component. In [111] it is similarly argued that, if the analysis of radiative-capture data through a di-cluster model is of interest, it is most correct to assign a spectroscopic factor of 1 to the ⟨︁αdLi 6⟩︁overlap. 110 4.1 DWBA deuteron transfer [61, tab. VIII] and reported in table B.2b, under the columns “p – d (L= 0)” and “(L= 2)” respectively. These potentials were fitted in order to reproduce the ⟨︁pHe 3⟩︁overlap functions calculated through the Green’s function Monte Carlo method, and at the same time to provide the correct binding energy for the system, and thus the correct asymptotic form for the corresponding solutions. [61, tab. IV] also lists the spectroscopic factors found within the same calculation for each component, 1.31 and 0.0221 respectively, which are adopted here3. The total spectroscopic factor is slightly smaller than the value found within the independent-particle shell model, 1.5, and is similar to the values found in past literature using different models, see for instance [112, sec. IV.A] and references therein. For comparison, very similar results are found for the Li 6+p→He 3+αcross-section employing instead the overlap function (including the interaction potential and spectroscopic factor) from [112]. It is pointed out that these overlaps functions have slightly higher rootmean-square extensions drms than would be expected within the cluster model. Restricting to the L= 0 component for simplicity, the overlap from [61] has drms = 2.79 fm, which, adopting experimental values in table B.1 for each cluster rms charge radius and using equation (2.3.66), corresponds to an He 3rms charge radius of 2.20 fm, against the experimental value in [69] of (1.966 ±0.003) fm. For comparison, an rms inter-cluster distance of drms = 2.07 fm would be required in order to reproduce the experimental He 3radius. 4.1.3 Transfer cross-section The ingredients discussed earlier in this section are here employed to estimate the non-polarised angle-integrated cross-section of the Li 6+p→He 3+αreaction described as the transfer of a structureless deuteron in DWBA. Figure 4.3 shows both the computed astrophysical factors and a selection of experimental data for comparison. The results may be contrasted, for instance, with more microscopic calculations in [3, 8, 11]. To this end, note that, in this chapter, cross-sections obtained from direct fixed-target experiments are always shown corrected by the expected adiabatic-limit atomic screening (see section 1.2). As in the quoted earlier work, the predicted cross-section overestimates barenucleus data at low energies, while (see in particular [3, fig. 8]) the resonance at about 1.5MeV above the Li 6+p threshold is not reproduced. However, the calculation predicts the correct order of magnitude for the astrophysical factor trough the whole energy range, suggesting that the physical ingredients of the calculation are at least qualitatively appropriate. 3The relative phase between the L= 0 and 2 components is discussed explicitly in [2, sec. V.A], and may be inferred visually from [61, fig. 7] (which refers to ⟨︁dH 3⟩︁). 111 4 The Li 6+p→He 3+αtransfer reaction 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Center-of-mass collision energy [keV] 0 1 2 3 4 5 Astrophysical S-factor [MeV b] Elwyn et al. 1979 Engstler et al. 1992 Lamia et al. 2013 d transfer - spherical reactants d transfer - deformed reactants (α-d: 5% L=2) 6Li + p -> 3He + α, direct data rescaled with U = 182 eV Figure 4.3: Points are experimental Li 6+p→He 3+αbare-nucleus astrophysical factors, from [30] (black circles), [31] (orange squares), [7] (green diamonds). Data from direct measurements was rescaled by the atomic adiabaticlimit screening enhancement factor as in figure 1.7. For readability, only a selection of data is shown. Brown dashed line is DWBA deuteron transfer calculation where both Li 6and He 3are spherical. Blue solid line is the same calculation including a deformed component for both nuclei, as detailed in section 4.1.2. 112 4.1 DWBA deuteron transfer The Be 75/2−resonance The peak in the data at about 1.5MeV is associated in literature (see e.g. [37]) to the Be 75/2−level at an excitation energy of about 7.2MeV, manifesting in the p-wave (i.e. projectile-target angular momentum of 1) Li 6+p scattering4. Note that, in the He 3–αchannel, a 5/2−level necessarily correspond to a projectile-target orbital angular momentum of 3. To observe such resonance in a calculation, it is thus necessary that the model includes a mechanism not conserving projectile-target orbital angular momentum. All calculations shown here are such that the total orbital angular momentum (sum of the projectile-target and inter-cluster orbital momenta) is conserved during the reaction. As a consequence, if both projectile and target have a spherical structure (inter-cluster orbital momentum zero) the sought resonance cannot be populated. Figure 4.3 shows two calculations, considering respectively spherical (brown crosses) and deformed (blue solid line) reactants. Each of these astrophysical factors is expanded in partial waves in the two panels in figure 4.4. In accordance with the considerations made above, the calculation involving spherical nuclei gives no outgoing cross-section through the p5/2 wave in the Li 6–p channel, while the other case displays such contribution, albeit small, approximately at the correct energy. Another component which is allowed when reactants are deformed is the s3/2 wave in the Li 6–p channel (which translates to an orbital angular momentum of 2 between He 3and α). However, it is interesting to note that the total s-wave cross-section, which is the dominant component at low energies, remains essentially unaltered in both calculations, thus the astrophysical region is not affected by this difference. It is also relevant to stress that the lack of a clearly visible peak in the excitation function is ultimately connected to the choice on the adopted interactions, rather than an intrinsic limitation of the structure model (once deformations are introduced). In particular, the potentials employed in this work are mainly focused on describing the sub-Coulomb energy range without introducing spurious resonances, and not on reproducing accurately the region relevant for the peak. To understand to what extent better agreement in the resonant region is possible, an alternative He 3–αpotential was generated adjusting it on the elastic scattering phase-shifts involving the resonant waves (f5/2 and f7/2), degrading at the same time the agreement with the total elastic cross-section. It was not possible to obtain agreement with all experimental phase-shifts using, for all waves, a single energy-independent potential with only spin-orbit coupling5. The results on the transfer calculation ob4There is another, much broader 5/2−level in the Be 7spectrum at about 6.7MeV of excitation energy, which is instead visible in the He 3–αelastic scattering (see e.g. figure 4.2), together with a 7/2−level which is however about 1MeV below the Li 6+p threshold. 5Adding an explicit dependence of the potential on angular momentum is an effective way to add non-central components, which pose computational difficulties in the transfer calculation, see the discussion later in this section. Limiting the potentials to standard spin-orbit components appeared as a good compromise between keeping the numerical approximations under control and providing the desired properties for the distorted 113 4 The Li 6+p→He 3+αtransfer reaction 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Center-of-mass collision energy [keV] 0 1 2 3 4 5 Astrophysical S-factor [MeV b] s1/2 s3/2 p1/2 p3/2 p5/2 6Li + p -> 3He + α, decomposition in 6Li + p partial waves d transfer, spherical reactants 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Center-of-mass collision energy [keV] 0 1 2 3 4 5 Astrophysical S-factor [MeV b] s1/2 s3/2 p1/2 p3/2 p5/2 d transfer, deformed reactants (α-d: 5% L=2) Figure 4.4: Upper panel: partial-wave decomposition for the astrophysical factor for spherical reactants in figure 4.3. Each line represents the contribution of a partial wave in the Li 6+p channel as per the legend (“s” and “p” refer to a projectile-target orbital angular momentum of 0 or 1, the subscript number is the total angular momentum). Bottom panel: same for the case of deformed reactants (brown dashed line in figure 4.3). 114 4.1 DWBA deuteron transfer 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Center-of-mass collision energy [keV] 1 2 3 4 5 6 7 8 9 10 Astrophysical S-factor [MeV b] 3He-α on cross-section, α-d: 0.8% L=2 3He-α on cross-section, no imaginary, α-d: 0.8% L=2 3He-α on phase-shifts, α-d: 0.8% L=2 3He-α on phase-shifts, no imaginary, α-d: 0.8% L=2 6Li + p -> 3He + α, direct data rescaled with U = 182 eV Figure 4.5: Points are the same in figure 4.3 (legend suppressed for readability): direct data was rescaled as in figure 1.7. Turquoise solid line is the same calculation as the blue solid line in figure 4.3 but with a smaller deformed component for Li 6. Orange dashed line is the same calculation as the turquoise solid line but with a different He 3–αpotential, see text or details. Indigo dotted and brown dot-dashed lines are the same as, respectively, turquoise solid and orange dashed lines, but removing all imaginary parts from all potentials. tained using this potential and the other one presented in section 4.1.1 (fitted on elastic cross-sections) are compared in figure 4.5, assigning the more modest Li 6deformation suggested by reproducing the electric quadrupole moment (c2≈ −0.091, see the discussion in section 5.2.1). If the same deformation adopted in figure 4.3 is instead employed, the resonant contribution is enhanced in all calculations, but the qualitative conclusions remain unaltered. From figure 4.5 (brown dot-dashed line), it can be seen that a prominent structure is found using the He 3–αinteraction fitted on phase-shifts and including only the real parts of all projectile-target potentials (also for Li 6–p). This structure peaks at about the experimental resonance energy and has the expected quantum numbers, as found by performing a partial-wave expansion (analogous to the one in figure 4.4). If the calculation is rescaled by an appropriate constant, a qualitatively reasonable energy trend is found throughout the energy range under study. However, the computed absolute cross-section waves. 115 4 The Li 6+p→He 3+αtransfer reaction a component with quantum numbers l=L= 0, another small one with l=L= 2, and in principle infinitely many others10. In the calculations shown here, all wave-functions were truncated to components with l≤2. In a test calculation where all terms with l≤4were included, the cross-sections did not change appreciably. A similar issue arises regarding the radial components in the two coordinate systems. More specifically, the Moshinsky transformation is performed expanding the complete three-particle state in an harmonic oscillator basis, truncating the expansion to a finite number of terms, and then transforming each single term. In the figures shown in this thesis, the wave-functions space in t coordinates was truncated to a maximum distance between the two transferred nucleons of 28 fm, and further discretised so that, for each angular component (definite values of l,L,S, etc.), the adopted harmonic oscillator truncated basis included about 28 elements11. Additional preliminary calculations showed that a moderate variation in the precise values of the computed cross-sections is found when including about 140 harmonic oscillator basis elements for each angular component12. However, the conclusions discussed in this work are not affected by these numerical details. He 3system Table 4.2 shows the decomposition in “t”-coordinates angular components of the Moshinsky-transformed He 3three-particle wave-function. As mentioned earlier in this section, the complete state is in fact constructed as the combination of a wide number of components, thus the table only includes the overall norm of each radial wave-function13. It can be seen that the transformed wave-function has a total norm of about 0.964: the missing norm is connected to the truncation of the space mentioned earlier in this section. The transformed wave-function is not rescaled to fix the total norm to 1: doing so would amount to merely rescale the cross-section by a factor. Rather, it is expected that including the missing components would cause only a small change, and not necessarily an increase, in the reaction cross-section. Consequently, it appears to be a better approximation to simply neglect the norm associated with the discarded components. 10Odd values of lare not allowed in this specific case. Given the simple shell-model structure adopted for He 3single-particles states (both transferred nucleons are found only in the 1s1/2 shell), as commented in section 2.2.2, if the transferred system has total isospin 0, then jmust be 1. Given that L= 0, this implies S= 1. Anti-symmetrisation of the transferred system state then imposes lto be even. 11Components whose contribution in norm falls below a given threshold are discarded, thus the number of basis elements is not exactly the same for all wave-functions. Also note that, in the Fresco code, the integration over the distance between the two transferred particles is performed using Gaussian quadrature [88, sec. 5.3.2]. 12Further increase of the number of basis elements, up to about 560 elements for each angular component, did not cause any appreciable difference in the results. 13The amplitude assigned to each single component is the one obtained applying the Moshinsky transformation [89] as implemented in the Fresco code [88]. 122 4.2 DWBA p+ntransfer Table 4.2: Decomposition of the Moshinsky-transformed p +n+p threeparticle wave-function (in “t” coordinates) employed to describe the ground state of He 3. Each row lists the total norm of all components with definite quantum numbers l,S,Jx,L(notation as in section 4.2.2). l S JxLNorm [10−2] 0 1 1 0 94.28 2 1 1 2 0.4327 2 1 2 2 0.7212 2 1 3 2 1.010 Finally, when calculating the transfer cross-section, the wave-function will be assigned an overall spectroscopic factor of 1.31, the same value employed for the spherical component of He 3in the transfer of a structureless deuteron, see section 4.1.2. Such factor is not included in section 4.2.2 or figure 4.7. Figure 4.7 shows the radial probability density function (square-modulus of the wave-function integrated over all angular components) associated to the computed He 3wave-function. Let rNN be the distance between the transferred nucleons, and R ct the distance between core and centre-of-mass of the transferred system. In general, configurations with rNN significantly greater than Rct represent “cigar-like” structures, with the two nucleons found in anti-correlation. Conversely, the region with rNN < Rct (or, even better, a structure with rNN approximately fixed and Rct running through a wide range) corresponds to correlated, or “clustered”, components. Finally, configurations in which each particle is equally distant from the other two lie on the red solid line in figure 4.7. Qualitatively, He 3dominant configurations are expected to resemble an equilateral triangle, apart from deformations due to the different electric charge and the nucleons spin projection. The result found here agrees with this expectation. The root-mean-square charge radius of the system can be evaluated through equation (2.3.67). Using for simplicity only the l=L= 0 component of the wave-function (which incorporates most of the wave-function norm), the value of 2.16 fm was found. This is slightly smaller than the di-cluster prediction given in section 4.1.2, but still 10 % bigger than the experimental value. Li 6system Since several single-particle states were defined for Li 6, the total wave-function will be a combination of all products of α+p and Li 5+n states coupling to the desired total angular momentum and parity. In particular, only singleparticle states with equal parity can be paired. For instance, there will be a component where the α+p system occupies the 1p3/2 state and Li 5+n occupies the 1p1/2 state, which for brevity will be denoted as “1p3/2 ×1p1/2”; similarly, a 1p3/2 ×1p3/2 component is allowed, and so on, but no term such as 1p1/2 ×2s1/2 is possible. 123 4 The Li 6+p→He 3+αtransfer reaction 2 4 6 8 10 [fm] NN r 1 2 3 4 5 6 7 [fm] core-NNcm R 0.02 0.04 0.06 0.08 0.1 ], 1s only -2 Radial pdf p+n+p [fm NN = r core - NNcm R NN /2 * r3 = core - NNcm R Figure 4.7: Contour plot of the reduced radial probability function for the p+n+p state constructed as described in text. The x-axis is the distance between the transferred nucleons, the y-axis is the distance between core and centre-of-mass of the transferred system (Rct in text). The lines follow the equations given in the legend. The weights to be assigned to each product of single-particle wave-functions were deduced by comparing with the results of a three-body calculation in [68], performed using the potential in [120] for the nucleon-nucleon interaction. Note that the α–nucleon interaction adopted in [68] is the same employed here to construct the single-particle states involving the Li 6(see again section 4.2.114). Hence, by additionally adopting the same weights, it is expected that the calculated three-particle state will be consistent to some extent. [68, tab. 2] (in particular under the column “dTS”) lists the norms of interest here, each associated to a component with (using the notation presented in the first part of this subsection) fixed l1,l2,Land S. These weights were then converted to the required weight for each product of specific single-particle states, in jj coupling scheme, using the analogous of equation (A.10) (see below for more details). Note that the three components involving a single-particle state occupying adshell (l1or l2equal to 2) appearing in [68, tab. 2] are discarded here, as it would not be possible to include them consistently in the sequential 14Furthermore, as mentioned at the beginning of section 4.2, the same potential is also used as core-core interaction in the simultaneous transfer. 124 4.2 DWBA p+ntransfer calculation without also adding all other possible d-shell components. Finally, since the present calculation is performed assuming that Li 6is found in a state with definite isospin modulus Tequal to 0, the T= 1 component in [68, tab. 2] (l1=l2=L=S= 1) was discarded as well15. The remaining components of the Li 6state, adopted here, comprise a total norm of 0.936. These components were not rescaled to fix the total norm, with the same rationale applying regarding the Moshinsky-transformed wave-function (see the discussion on He 3earlier in this section). In order to perform the conversion from the “LS” basis adopted in [68, tab. 2] and the “jj” one required here, it is necessary to know not only the absolute weight of each component, but also its sign. However, the data included in [68] are not sufficient for extracting all phases in a straightforward manner16. To find a set of reasonable signs, the weight of each component was compared with those given by a three-body calculation in Hyperspherical Harmonics formalism [121], which appears to be in good agreement with the one in [68], meaning that, with an appropriate choice of signs, they predict very similar norms for each component17. The signed amplitudes in jj basis adopted in this work are reported in table 4.3a. Table 4.3b lists the corresponding weights in t coordinates for the Moshinsky-transformed wave-function, which has a total norm of 0.873, to be compared with the initial value of 0.936 (the same considerations made earlier for He 3apply here). When calculating the transfer cross-section, the wave-function will be additionally assigned an overall spectroscopic factor of 0.85 (not included here), the same value employed in the transfer of a structureless deuteron, see section 4.1.2. Figure 4.8 shows the the radial probability density (pdf) function associated to the Li 6wave-function computed in this manner. In order to show the role of the s-shell component on the result, the same figure includes the probability density function obtained inverting the sign of such component. The impact of all other components is not visually distinguishable in the probability density function. The Li 6radial pdf shows two peaks. As commented earlier, one peak may be associated to clustered configurations, where the two transferred nucleons are more strongly correlated in space, while the other can on the contrary be connected to “cigar-like” configurations. The tail behaviour of both peaks is different for each Li 6state considered in figure 4.8. The choice of signs yielding the best agreement with three-body calculations, which is 15It is interesting to note that, of the remaining p-shell components, all but the L= 0 one are either small or do not contribute to the simultaneous transfer, thus a good account of the simultaneous calculation could be obtained restricting to the L= 0 p-shell component and the s-shell one. However, such approximation is not performed here. 16These could be found, for instance, by comparison of the radial components of the threeparticle wave-function computed here (for each possible choice of signs) and in the original work. 17Except for the relative importance of pand s-shells. The sign of the 2s1/2 ×2s1/2 component was thus fixed comparing the radial probability density function of the present calculation and of [121], which suggested to pick the sign generating a stronger clustered configuration. 125 4 The Li 6+p→He 3+αtransfer reaction Table 4.3: Weights of the components of the α+p+n three-particle wavefunction employed to describe the ground state of Li 6, deduced from [68, tab. 2] as detailed in text. (a) Each row refers to a pair of single-particle states in “jj” angular momentum coupling scheme, as specified in the first two columns, marking the proton and neutron state respectively. The last column is the amplitude associated to the corresponding component of the Li 6wave-function. p shell n shell Amplitude 1p3/2 1p3/2 0.7482 1p3/2 1p1/2 −0.4044 1p1/2 1p3/2 0.4044 1p1/2 1p1/2 −0.1228 2s1/2 2s1/2 −0.1843 (b) Each row lists the total norm of all components of the Moshinskytransformed wave-function (in “t” coordinates) with definite quantum numbers l,S,Jx,L(notation as in section 4.2.2). l S JxLNorm [10−2] 0 1 1 0 78.13 0 1 1 2 0.1774 1 0 1 1 5.027 2 1 1 0 0.1774 2 1 1 2 1.075 2 1 2 2 1.415 2 1 3 2 1.262 126 4.2 DWBA p+ntransfer 0 2 4 6 8 10 [fm] NN r 1 2 3 4 5 6 [fm] ct R 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 =0+1-2)-2s t ], 1p(L -2 +n+p [fmαRadial pdf nn = r ct R nn /2 * r3 = ct R 0 2 4 6 8 10 [fm] NN r 1 2 3 4 5 6 [fm] ct R 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 =0+1-2)+2s t ], 1p(L -2 +n+p [fmαRadial pdf nn = r ct R nn /2 * r3 = ct R Figure 4.8: Same as figure 4.7 for the Li 6three-particle system. Upper panel corresponds to the state constructed using the amplitudes given in table 4.3a (namely with the choice of relative signs yielding the best agreement with three-body calculations, see text for details). Lower panel shows for comparison the opposite choice on the sign of the 2s1/2 ×2s1/2 component. In the figure titles, “2s” and “1p” refer to the two single-particle shells under consideration, “Lt” is L(notation as in section 4.2.2), and the signs mark the relative phases adopted for each component of the wave-function. 127 4 The Li 6+p→He 3+αtransfer reaction thought to represent a more realistic description of the system, is the one favouring the clustered peak (top panel in fig. 4.8). The qualitative situation is similar to the one found when computing wave-functions related to twoneutron overlaps, for instance in [83]. Considering only the l=L= 0 component for simplicity, a root-meansquare charge radius of 2.35 fm was found for both wave-functions shown in figure 4.8. This is 9% smaller than the experimental value in table B.1 (compare also with the value found from the di-cluster model in section 5.2.1). In order to evaluate the electric quadrupole moment, it is instead necessary to include all angular components. Additionally, since the wave-function in use mimics the one in [68], which cannot reproduce the quadrupole moment (which is standard for three-body models of Li 6, see [10]), there is no reason to expect a better result here. 4.2.3 Transfer cross-section Figure 4.9 presents the computed astrophysical factors for the Li 6+p→He 3+ αreaction described as a two-nucleon transfer process, which are commented in detail in the following paragraphs. Similarly to what was found in the deuteron-transfer case, the predicted absolute value for the cross-sections is approximately of the correct order of magnitude throughout the energy range under study, but the experimental cross-sections are not reliably reproduced. Role of 2sshell on the cross-section Even though a reasonable assignment for all weights and phases in the components of the Li 6wave-function was found in section 4.2.2, it is interesting to consider alternative choices, as a mean to study the impact of each component on the final result. Figure 4.9 compares the calculations performed using either of the two Li 6wave-functions whose probability density function is reported in figure 4.8. While the results are qualitatively similar in both cases, it can be seen that the Li 6featuring a longer “clustered” tail (upper panel in fig. 4.8, red dot-dashed line in fig. 4.9) yields higher absolute cross-sections and enhances more the reaction at lower energies (see also figure 4.16 later). Similarly, it is found that the opposite sign choice for the 2s-shell component of Li 6wave-function (lower panel in fig. 4.8, violet dashed line in fig. 4.9), which favours “cigar-like” configurations, also hinders the transfer reaction, especially at lower energies, with respect to the case where the 2sshell is excluded from the structure. This result was found to hold in all two-particle transfer calculations performed for the present project (also those not shown in this thesis), including different choices for the form (prior or post) employed, the adopted potentials, the components included in the reactants structure. Similar conclusions are also often drawn when studying two-neutron transfer processes, where correlations between the transferred nucleons affect the reaction cross-section and are connected to pairing phenomena, see e.g. [85]. 128 4.2 DWBA p+ntransfer 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Center-of-mass collision energy [keV] 0 1 2 3 4 5 6 7 8 9 10 Astrophysical S-factor [MeV b] Elwyn et al. 1979 Engstler et al. 1992 Lamia et al. 2013 p+n transfer [6Li: 2s], total p+n transfer [6Li: 1p(Lt=0+1-2)+2s], total p+n transfer [6Li: 1p(Lt=0+1-2)-2s], total 6Li + p -> 3He + α, direct data rescaled with U = 182 eV Figure 4.9: Points are the same in figure 4.3: direct data was rescaled as in figure 1.7. Red dot-dashed and violet dashed lines are the second-order DWBA p +n transfer calculation performed as detailed in text, using the Li 6 wave-function shown respectively in the top and bottom panel of figure 4.8. Brown dotted line is the same calculation but using as Li 6wave-function only the component with transferred nucleons in the 2sshell. Symbols in the legend have the same meaning in figure 4.8. 129 4 The Li 6+p→He 3+αtransfer reaction Figure 4.9 also shows the cross-section associated to only the 2s-shell component of Li 6(brown dotted line), rescaled by the same norm this component has in the total wave-function (3.4%). The total transition amplitudes can be expressed as the coherent sum of the amplitudes connected to the 1pand 2scomponents separately. The astrophysical factors are in turn proportional to the amplitude square-modulus, through equations (1.1.23) and (3.1.21). Consequently, the cross-section for the complete wave-functions in figure 4.8, which are obtained combining the same components with different relative signs, can differ only because the phases of the amplitudes for each component show some coherence. This is particularly relevant considering that the 2s-only astrophysical factor, around 0.10 MeV b at low energies, is rather small with respect the 1pcontribution, requiring a significant degree of coherence in order to yield the observed relative difference in the total cross-sections. It can also be seen that the phases are more coherent at lower energies, where greater differences between the two complete calculations are found, notwithstanding the fact that the 2s-only absolute astrophysical factor does not change dramatically throughout the energy range under study, reaching a maximum of 0.15 MeVb at about 1MeV centre-of-mass energy. Interplay between simultaneous and sequential contributions Figure 4.10 shows the simultaneous and sequential contributions to the calculation already shown as the red dot-dashed line in figure 4.9 (more-strongly clustered Li 6wave-function). The total transition amplitude is the coherent sum of the amplitudes of each component. It can be seen that the amplitudes for simultaneous and sequential terms are in phase opposition at low collision energy, resulting in a destructive interference. Such trend partially changes at higher energies, particularly in the energy range around the resonance at 1.5MeV, where the two contributions are almost in phase quadrature (but still in slight opposition), forming a modest but visible bump. The role of interference can be analysed in greater detail by checking the expansion into the initial-channel partial-waves, shown in figure 4.11. The only wave generating a non-negligible contribution to the simultaneous term (top panel in figure) is the s1/2 ( Li 6–p with relative orbital angular momentum 0 and total angular momentum 1/2). By comparing the magnitude of this contribution between the simultaneous, sequential (middle panel) and complete (bottom panel) calculations, it can be seen that the simultaneous and sequential s1/2 transition amplitudes are in very strong phase opposition. All other waves feel no significant interference as they do not contribute appreciably to the simultaneous term. The same analysis can be carried out for the calculation involving the less-strongly-clustered Li 6wave-function (violet dashed line in figure 4.9): the partial-wave expansion of simultaneous and sequential contributions and their total is shown in figure 4.12. The situation is qualitatively the same found in figure 4.11, but the simultaneous contribution is slightly smaller and, most importantly, the s1/2 wave of the sequential term is much 130 4.2 DWBA p+ntransfer 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Center-of-mass collision energy [keV] 0 1 2 3 4 5 6 7 8 9 10 Astrophysical S-factor [MeV b] Elwyn et al. 1979 Engstler et al. 1992 Lamia et al. 2013 p+n transfer [6Li: 1p(Lt=0+1-2)-2s], simultaneous p+n transfer [6Li: 1p(Lt=0+1-2)-2s], sequential+NO p+n transfer [6Li: 1p(Lt=0+1-2)-2s], total 6Li + p -> 3He + α, direct data rescaled with U = 182 eV Figure 4.10: Points and red dot-dashed line are the same in figure 4.9: direct data was rescaled as in figure 1.7. Brown dotted and red dashed lines are the simultaneous and sequential (including non-orthogonality) contribution to the same calculation. 131 4 The Li 6+p→He 3+αtransfer reaction folded 3-particle (integrated p-n pdf) folded 3-particle (averaged p-n wf) folded free deuteron di-cluster model α-d potential 1 2 3 4 5 6 7 α-d distance [fm] -80 -60 -40 -20 α-(p+n)potential [MeV] di-cluster model α-d potential folded 3-particle (integrated p-n pdf) folded 3-particle (averaged p-n wf) folded free deuteron 6 7 8 9 10 11 α-d distance [fm] -0.4 -0.2 0.2 0.4 α-(p+n)potential [MeV] Figure 4.15: Solid line is the α–d potential employed for the di-cluster model Li 6wave-function, see sections 4.1.2 and 5.2.1. Points are a folding of the “Vαp+VLi n” potential defined in text in section 4.2.3 (central part of the α+p+n potential specified in sec. 4.2.1), over different deuteron distributions. See text for details. Upper and lower panel show different ranges of α–d distance. 138 4.2 DWBA p+ntransfer fective one found using the three-cluster model potentials and wave-functions. It is thought that this difference may account for a significant fraction of the discrepancy between the oneand two-particle transfer calculation. A similar analysis could be repeated for He 3. In the future, the form-factors computed as shown here will be employed to perform a single-particle transfer calculation (including the generation of the reactant bound states), to evaluate quantitatively the importance of the difference in the potentials. Calculations rescaled to transfer experimental data It was mentioned earlier that the choice between the moreor less-strongly clustered Li 6wave-function has a greater relative impact on the cross-sections at low energies. This may be seen visually in figure 4.9 but will now be analysed more carefully. In figure 4.16, the same calculations already shown in figure 4.14 are scaled in order to agree with experimental data at centre-ofmass collision energies slightly smaller than 1MeV. This allows to compare the predicted energy trends independently of the computed absolute values. The difference in the slopes of the astrophysical factors is more apparent, and it can additionally be seen that the impact of the choice of sign for the Li 6 2s-shell contribution is of the right order of magnitude to be of interest for the experimental discrepancy discussed in section 1.2.3. It is stressed that establishing which wave-function for Li 6is the “correct” one is not the issue at hand19. Rather, the point made here is that an accurate treatment of the reactants static structure, in particular regarding the clustering strength, can be important to obtain a good understanding of the reaction dynamics between light ions even at very low energies and far from resonances. It is also interesting to point out that, after rescaling all calculations, the transfer of a structureless deuteron (blue solid line in figure 4.16) displays a significantly steeper slope than both p+n calculations. Qualitatively, the d-transfer process may be seen as the limit of a p+n transfer in which the transferred system is forced into a fully clustered configuration. Nonetheless, it is to be reminded that, while the two shown p+n-transfer cross-sections correspond to the same calculation (apart from the commented change in the structure) and are certainly comparable, the d-transfer process is not computed through the same formalism and does not employ the same physical ingredients, thus the attribution of a given difference in the results is less clearcut. While the rescaling procedure in figure 4.16 allows an immediate and visual description, the precise energy at which the calculations are fixed to match between each other and with the data is arbitrary, and the curves can appear slightly different if this is changed. The trend suggested by each calculation at astrophysical energies may be compared in a more quantitative manner 19As mentioned in section 4.2.2, the state referenced in table 4.3 and whose probability density function is shown in the upper panel of figure 4.8 yields the best agreement with three-body calculations. 139 4 The Li 6+p→He 3+αtransfer reaction 10 100 1000 Center-of-mass collision energy [keV] 1 1.5 2 2.5 3 3.5 4 4.5 Astrophysical S-factor [MeV b] Elwyn et al. 1979 Engstler et al. 1992 Lamia et al. 2013 d transfer - deformed 6Li and 3He p+n transfer [6Li: 1p(Lt=0+1-2)+2s], total p+n transfer [6Li: 1p(Lt=0+1-2)-2s], total 6Li + p -> 3He + α, calculations rescaled on data Direct data rescaled with U = 182 eV 0100 200 300 400 500 600 700 800 900 Center-of-mass collision energy [keV] 1 1.5 2 2.5 3 3.5 4 4.5 Astrophysical S-factor [MeV b] Elwyn et al. 1979 Engstler et al. 1992 Lamia et al. 2013 d transfer - deformed reactants (α-d: 5% L=2) p+n transfer [6Li: 1p(Lt=0+1-2)+2s], total p+n transfer [6Li: 1p(Lt=0+1-2)-2s], total Figure 4.16: Same as figure 4.14, but the blue solid, violet dashed and red dot-dashed lines are rescaled respectively by factors of 0.88, 0.48 and 0.39. Upper and lower panel differ only by the x-axis scale. 140 4.3 GGW deuteron transfer by computing the ratio [∂ES(E)]/S(E)and using it in equation (1.1.43) to deduce the value of the effective nuclear radius associated to the calculated line (or similarly taking the logarithm of the astrophysical factor and using equation (1.1.42)). Such analysis is a possible future development for this work. 4.3 GGW deuteron transfer One of the calculation schemes discussed in section 3.2.2 was applied to the Li 6+p→He 3+αone-particle transfer reaction, as follows20. The generalised-distorted-wave formalism in section 3.1.4 is applied in prior form with the choice of generalised auxiliary potential yielding the “GGW” amplitude in equation (3.1.31), and the full scattering solution is approximated by the final-partition standard distorted wave as in section 3.2.1, The Li 6 structure includes a continuum of inert di-cluster model α–d states, with either parity, total angular momentum modulus quantum number up to 3, and excitation energies up to about 3.5MeV above the ground state, discretised using the binning method [122]. For He 3, only the ground state is currently included. All bound states considered here are eigenstates of the core-valence orbital angular momentum (Lin section 4.1.2), in particular the ground states include no deformed components. The adopted potentials are the same mentioned in section 4.1, with two exceptions. First, the α–d binding potential was modified by including a spin-orbit component, adjusted to reproduce the energy of the 3+first excited state of Li 6: see the “α– d (GGW)” column of table B.2c. Second, there is no Li 6–p optical potential, since a generalised auxiliary potential is employed in its place. Remind that, as discussed in section 4.1.3, non-central components of the core-core α–p are still being neglected; however, no projectile-target optical potential appears in the transition amplitude operator, thus no other term is here approximated. Figure 4.17 shows the preliminary astrophysical factor computed in this way (tick red solid line). Two very large peaks are visible in the calculation. The first one, at lower energies, is discussed later. The second peak has its maximum at an energy similar (but slightly smaller) to the experimental resonance connected to the Be 75/2−level at an excitation energy of about 7.2MeV. Decomposition in initial-channel partial-waves reveals that the peak is indeed primarily fed by p-waves ( Li 6–p orbital angular momentum 1), and thus states with negative total parity, as expected, see the discussion in section 4.1.3, but the total angular momentum is incorrectly reproduced, since the contribution of the p5/2 wave is negligible. On this regard, remind that the present formalism does not include the Li 6–p distorted wave obtained from optical potential discussed in section 4.1.1, which in the DWBA calculations was sufficient to exclude all significant spurious contributions from the transfer channel. Here, 20The study reported in this section was developed in collaboration with A. Moro. 141 4 The Li 6+p→He 3+αtransfer reaction 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 Center-of-mass collision energy [keV] 0 1 2 3 4 5 Astrophysical S-factor [MeV b] 6Li-p GGW 6Li-p GGW, π(6Li) = + 6Li-p GGW Ex<2.5 MeV 6Li-p PW 6Li-p DW only Coulomb 6Li + p -> 3He + α, preliminary d-transfer calculations Direct data rescaled with U = 182 eV Figure 4.17: Points are the same in figure 4.3 (legend suppressed for readability): direct data was rescaled as in figure 1.7. Tick red solid line is the preliminary GGW deuteron transfer calculation discussed in section 4.3. Turquoise dotted and blue thin solid lines are the same calculation including Li 6 continuum levels with, respectively, only positive parity or excitation energy below 2.5MeV above the ground state. Brown dashed line is a DWBA calculation with Li 6–p optical potential equal to zero. Maroon dot-dashed line is the same calculation with Li 6–p potential equal to the electrostatic repulsion. 142 4.3 GGW deuteron transfer the Li 6generalised distorted wave is generated through equation (3.1.29) by the α–d binding potential, which currently possesses a rather simple and phenomenological form, and the α–p core-core potential, which however must be central in the current numerical implementation, The other component of the complete Li 6–p interaction, the d–p binding potential, in this calculation is the transition amplitude operator (see again equation (3.1.31)) and thus can be always applied to the He 3state, which currently is just a spherical wave-function for the ground state, so that any non-central component of the potential would be effectively irrelevant. Additionally, the only projectiletarget optical potential employed in this calculation, the He 3–αone (discussed in section 4.1.1), was not adjusted on the elastic scattering phase-shifts. A better description of the experimental resonance might thus be obtained by including more appropriate interactions, or by computing both the initialand final-partition wave-functions through a CDCC approach. The influence of continuum states of Li 6on the complete result was studied by repeating the calculation after removing selected states from the structure. Two examples are displayed in figure 4.17 (blue thin solid and turquoise dotted lines). Note that all states are coupled in a non-trivial manner, and furthermore their contributions on the transition amplitude interfere, thus it is not possible to clearly assign fractions of the angle-integrated cross-section to any specific state. Nonetheless, it is seen that the peak at about 1.3MeV completely disappears if all Li 6negative-parity states are removed, or if the continuum is truncated to a maximum excitation energy of about 2.5MeV above the ground state. Note that 2.5MeV is approximately the collision energy at which the calculations shown in figure 4.17 were stopped, thus this is an indication that virtual excitations to closed channels can play a relevant role in the reaction. However, regardless of the way the Li 6continuum is modified, it was not possible to remove or significantly alter the strongly rising trend appearing in the computed cross-section at low energies. In fact, such trend persists even if all excited states are removed21.To get an insight into the possible origin of this trend, the following test calculations were performed. The brown dashed line in figure 4.17 is a DWBA calculation, including only a spherical ground state for each composite nucleus, in which the (central) remnant is fully taken into account, but the Li 6–p optical potential is set to zero: the astrophysical factor shows the same low-energy trend as the GGW calculations. Finally, the maroon dot-dashed line in figure 4.17 is another DWBA calculation where the Li 6–p potential is set equal to just the Coulomb potential between the two nuclei22. With this choice, the long-range Coulomb part of the transition potential, which is Vαp+Vd p −ULi p (see equation (3.2.1)), is cancelled, and 21Note that removing all excited states within the GGW formalism is equivalent to reduce the calculation to a DWBA with ˆ︁ UAb =⟨︂ψαˆ︁ Vab ψα⟩︂(compare equations (3.1.23) and (3.1.33)) in zero-remnant approximation (namely, Vab −UAb ≈0). 22For convenience in performing the numerical computation, the potential actually has the form in equation (B.2) with RC= 0.1fm. This has no influence on the discussion. 143 4 The Li 6+p→He 3+αtransfer reaction it is seen in the figure that the divergence toward low energies disappears. It is thought that in the GGW calculations, the long-range Coulomb potential appearing in the transition amplitude in equation (3.1.31) similarly causes the observed feature in the low-energy cross-section, which is thus deemed as an artefact of the calculation. It is also relevant to point out that, in general, calculations involving coupled-channels schemes at low energies are complicated by numerical instabilities arising in the inclusion of virtual couplings to states at excitation energies which are too high with respect to the collision energy. The issue, which currently prevents to draw conclusions on the energy range of astrophysical interest, is still under study. Also note that a full analysis of the convergence of the result for all numerical parameters of the calculation was not completed yet. In addition, the numerical integrations are currently performed using the Numerov method: it may turn out to be beneficial to employ instead the “calculable” R-Matrix method, see e.g. [123]. Nonetheless, it is interesting to note that, far from the two peaks, the agreement between experimental data and calculation is remarkable, given also the simple structure currently adopted for He 3. 144 5 Impact of Li 6ground-state deformation on barrier penetrability The impact of clustering on barrier penetrability has been explored in literature within semi-classical models [16]. In the present chapter, this kind of approach was investigated, expanded, and applied to some cases of interest. Section 5.1 treats the classical cluster model already discussed in [16]. By assigning a defined spatial position to the clusters forming each nucleus, it is possible to deduce an effective projectile-target potential which depends on the reactants orientation. The radial barrier penetrability associated to each possible configuration is then estimated in WKB approximation. The average result over all configurations, possibly weighted over some distribution suggested by more microscopic models, can then be compared with other calculations to explore the role of the assigned cluster structure. The practical implementation of the model was here improved, and an alternative way to analyse the results is proposed. The features were applied to the study of Li 6– Li 6barrier penetrability, to make a comparison with results found in literature. In section 5.2, the same underlying ideas are then reformulated within the framework of a quantum-mechanical cluster model. The effective projectiletarget potential for a pair of composite particles can be expressed as the form factor of the “microscopic” interaction between the elementary components of the system evaluated on the reactants internal state. In this way, the details of the reactants structure can be connected to the barrier penetrability for a reactant pair subject to the aforementioned effective potential. In particular, the formalism is employed to study the impact of ground-state quadrupole deformations in the Li 6wave-function, considering for convenience its scattering with a proton, which can be regarded as structureless. As will be seen in section 5.2.2, the deformed components in the wave-function give rise to tensor components in the form factor: the anisotropic projectile-target interaction is then treated precisely as in the classical model. 5.1 Classical cluster model Two nuclei Pand T, undergoing a collision, are assigned a classical frozen di-cluster state, meaning that Pis described as a system of two particles P1 and P2fixed (during the whole collision) at a given relative displacement d P (and similarly for T). The coordinate system may be chosen e.g. to fix the orientation of d P. The projectile-target relative orbital angular momentum, 145 Chapter 5. Impact of Li 6ground-state deformation and thus the impact parameter, is set to zero (which is expected to be the dominant component). The projectile-target potential, VT P , is defined as the sum of the potentials between each cluster, ∑︁i,j VPiTj, and thus depends on the orientation and extension of each reactant, so it may be written as VT P (R , dP, d T), where R  is the projectile-target displacement. For any fixed value of the inter-cluster displacements, and for any projectile-target orientation, the s-wave barrier penetrability is computed in WKB (using equation (1.1.30) with the factor in front of the exponential approximated to a constant), assuming a spherically symmetric potential V(R)equal to VT P (R , dP, d T). The transmission coefficient for each reactants orientation may then be weighted on some distribution phenomenologically accounting for the particles structure or dynamical effects. In [16], all orientations were averaged with equal weight to check the implications of the model in the absence of any alignment. Present implementation of the classical model In the calculation shown in [16]1, the lower bound of the integral in equation (1.1.30) was chosen as the inner classical turning point for the system. The same procedure was adopted here. It should be considered that the WKB approximation is not expected to be accurate around turning points (see equation (1.1.27)), however, the integrand in equation (1.1.30) is small around the same points, thus the errors induced by such approach are not expected to be large2. In order to practically find the classical turning points, a numerical root-finding algorithm was employed. The algorithm employed in [16] in some cases failed and returned an incorrect result. In this work, a more robust numerical algorithm was implemented, improving the practical results of the calculations. Furthermore, the potentials adopted for the calculations were slightly modified. In [16], point-like Coulomb potentials between each pair of clusters are employed, which however generate spikes in the total potential, and thus degrade the numerical computation whenever they appear in the integration domain. Here, the electrostatic repulsion potentials were parametrised as generated by an uniformly charged sphere, overcoming the issue3. Results for Li 6+Li 6barrier penetration To allow for a comparison with results in [16], the same system, Li 6+Li 6, was considered, adopting the same parameters for the potentials (with the already mentioned modification on the Coulomb repulsion). 1The source code of the algorithm employed in [16] was provided by L. Fortunato, who also contributed to the work discussed in this chapter. 2Nonetheless, in the future the calculations may be repeated using the prescription for Rn mentioned in section 1.1.1. 3It was verified that this change brought negligible differences at sufficiently high energies, where the divergences of the Coulomb potential did not appear in the integration domain. 146 5.1 Classical cluster model θp=0, ϕp=0, θt=0 θp=π 2,ϕp=0, θt=π 2 θp=0, ϕp=0, θt=π Angle average Li-Li pure Coulomb 10 15 20 0.6 0.8 1.0 1.2 1.4 Projectile - target distance [ fm ] Potential energy [MeV] Figure 5.1: Potential energy for the Li 6Li 6classical model as a function of the projectile-target distance r.θpand φpare the polar and azimuthal angle of the projectile orientation, while θtis the target polar angle. The black solid line is V= 9αe¯hc/r. Every other solid line is the projectile-target potential, defined as explained in text, at a different reactants orientation. The dashed line is the average of the same potential over all orientations. In the practical calculation performed in this work, the coordinate system was chosen by fixing the projectile-target distance on the zaxis, and fixing the xaxis so that the azimuthal angle of the target orientation is always zero. Varying the projectile orientation and the target polar angle suffices to properly account for all possible configurations. Note that this is not the coordinate system adopted in [16]. Figure 5.1 shows a sample of the radial profiles of the projectile-target potential obtained in selected configurations. The average of the penetrabilities computed on the potential for each orientation is shown in figure 5.2, and compared with the analogous calculation in [16, fig. 1]. The improvements on the computation implemented in this work are not negligible (especially in the limit of small collision energies), but do not alter the qualitative conclusion in the original paper. The same figure includes the results of the same calculation performed at lower energies (down to 10 keV). In order to show a plot including the whole investigated energy range, the transmission coefficient was converted into an astrophysical factor. Again in figure 5.2, it is also shown (red line) the penetrability under a potential obtained averaging the complete potential VT P over all directions. In this way, the non-spherical components induced by the adopted structure model are integrated away, and the averaged potential may be thought as representing a more “uniform” (less strongly clustered) system. A different 147