A transfer to the continuum formalism for the study of (p, pn) and (p, 2p) reactions on unstable nuclei.

A Transfer to the Continuum formalism for the study of (p, pn)and (p, 2p)reactions on unstable nuclei PhD thesis Mario Gómez Ramos Departamento de Física Atómica, Molecular y Nuclear Universidad de Sevilla October 2018 A Transfer to the Continuum formalism for the study of (p, pn) and (p, 2p)reactions on unstable nuclei (Un formalismo de Transferencia al Continuo para el estudio de reacciones (p, pn)y(p, 2p)sobre núcleos inestables) Memoria para optar al título de Doctor en Física Nuclear Mario Gómez Ramos Dirigida por el Doctor D. Antonio Matías Moro Muñoz Departamento de Física Atómica, Molecular y Nuclear Universidad de Sevilla Octubre 2018 A mi familia, en todas sus acepciones Agradecimientos Este documento es el resultado de un proyecto desarrollado durante cuatro años, o incluso cinco, contando con el primer año de máster, que me sirvió como introducción al mundo de la investigación. Que sólo mi nombre aparezca en la portada de esta memoria de tesis no hace justicia al gran número de personas que lo han hecho posible, por su apoyo académico, profesional y/o personal. Espero que los siguientes párrafos sirvan como una muestra de reconocimento a este apoyo, si bien incompleta en el número de personas que aparecen y en la contribución que se les reconoce. Mis agradecimientos no pueden sino comenzar con mi director de tesis, Dr. Antonio M. Moro Muñoz, sin el cual este trabajo no habría sido posible. Su contribución va más allá de la meramente académica, esencial para el desarrollo de este trabajo, extendiéndose a una paciente labor como mentor en el extraño mundo de la investigación, con una dedicación y disponibilidad más allá de lo razonable, que a la pregunta “¿Tienes tiempo?” siempre respondía con un “Bueno dime”, aun cuando la respuesta objetiva fuera un rotundo “No”. Este agradecimiento debe extenderse a todo el departamento, en el que he encontrado un excelente ambiente en el que afrontar esta nueva etapa. Aunque objetivamente son los miembros más “senior” del departamento aquellos que más me han aportado, en términos de experiencia y consejos, debo hacer una mención especial a todos los compañeros doctorandos con los que he compartido la experiencia predoctoral. He tenido la fortuna de coincidir durante estos cuatro años con multitud de colegas, en los que he visto reflejadas mis dudas y esperanzas, alegrías y frustraciones. Sin duda, en esta guerra que es la obtención del título doctoral, una de las facetas más positivas es la camaradería que se forma en las trincheras, extendida también a aquellos que aun habiendo terminado su batalla aún recuerdan el olor de vii viii Agradecimientos la pólvora. Durante estos cuatro años, he tenido la oportunidad de realizar dos estancias en el extranjero, una en el centro RCNP de la Universidad de Osaka, Japón, supervisada por el Dr. K. Ogata, y otra en la Universidad de Surrey, supervisada por la Dra. N. Timofeyuk. Ambas estancias me han permitido explorar regiones nuevas, tanto físicamente como en el mundo de la física nuclear, y probablemente hayan sido de los periodos más fructíferos en el ámbito formativo. Tanto al Dr. Ogata como a la Dra. Timofeyuk les estoy inmensamente agradecido por su acogida y su dedicación durante estas estancias. También aprovecho esta oportunidad para agradecer a los doctores A. Deltuva, K. Yoshida, K. Ogata, L. Atar y P. Díaz Fernández, que han proporcionado cálculos y resultados que se presentan en esta memoria de tesis, por permitirme usar sus resultados y por las fructíferas discusiones asociadas a ellos. En el ámbito económico, este proyecto ha estado financiado por la beca de Formación de Profesorado Universitario, otorgada por el pasado Ministerio de Educación, Cultura y Deporte. Este programa de becas es una de las fuentes de financiación para investigación predoctoral en ciencia fundamental más fiables y establecidas sin el cual la carrera de cientos de investigadores cada año se vería arrancada de raíz. Dados los probados beneficios de la investigación para la sociedad, programas como este deben verse apoyados, manteniendo si no incrementando el número de becas, siendo la responsabilidad de los beneficiarios el honrar la confianza inherente a esta inversión con dedicación y trabajo duro. Descendiendo a un plano más personal, tengo que agradecer a mi familia por su apoyo en todos los planos: económico, personal, anímico... Es gracias a ellos, en particular a mis padres, que he podido llegar a presentar este documento superando dificultades y dudas que sin su ayuda me habrían parecido insuperables. A lo largo de estos años ha habido muchas situaciones en las que no les he demostrado el agradecimiento y reconocimiento que se merecían. Espero que estas líneas compensen un poco esta falta. Unas últimas palabras las tengo que dedicar a los amigos que me han acompañado estos años (los que haya mencionado antes ya tenéis vuestro párrafo, no seáis avariciosos), soportando las ausencias y desplantes que resultan de la dedicación a la labor investigadora, y a la paciencia que han Agradecimientos ix demostrado ante las quejas y plañidos propios del estudiante predoctoral. En especial, tengo que agradecer a Silvia y José, que han estado conmigo desde mucho antes de empezar esta aventura, dándome muchos momentos inolvidables que han acompañado los éxitos y paliado los desencantos de esta etapa. Su confianza ciega en mí y mis capacidades ha supuesto un apoyo férreo cuyo inmenso valor sólo he sido capaz de valorar en retrospectiva. Gracias de verdad. Es probable que haya muchos que no se hayan visto reflejados en estas líneas y que aun así merezcan un reconocimiento por su aporte en el proceso que ha llevado a producir esta memoria. También a ellos debo darles las gracias y pedirles perdón por no encontrar la destreza para mostrar un agradecimiento más explícito. Mario Octubre 2018 List of Figures 1.1 Sequence of single-particle orbitals. Figure from [2]. . . . . . . 3 1.2 Description of the (p, pN)process in the quasi-free picture. Note that the momentum of the core Cis not altered during thereaction............................. 4 1.3 Single-particle transversal (top) and longitudinal (bottom) momentum distributions for 23O(p, pn)22O computed for the removal of a neutron from different nuclear levels. Figure from[3]............................... 5 1.4 Nucleon-nucleon cross section as a function of projectile energy. Notice the minimum at hundreds of MeV. Figure taken from[4]............................... 7 1.5 “Quenching factors” compared to IPM as a function of the nucleus mass, obtained from (e, e0p)reactions [8]. Figure taken from[9]............................... 8 1.6 Occupation probability (n()) and quasi-hole strength (Z()) for nuclear matter and 208Pb. The transfer of quasi-hole strength from levels below the Fermi energy Fto levels above it due to nucleon-nucleon correlations is apparent. Figure takenfrom[13]........................... 9 1.7 Spectrum of single-particle levels as a function of the deformation length δ. Levels are labelled as [N, nk,Λ,Ω]. Figure takenfrom[27]........................... 13 1.8 “Quenching factors” as a function of the difference in separation energies ∆S=Sp(n)−Sn(p)for nucleon knockout reactions with heavy targets at intermediate energies. Figure takenfrom[32]........................... 17 xvii xviii List of figures 1.9 “Quenching factors” as a function of the difference in separation energies ∆S=Sp(n)−Sn(p)for (d,3H) and (d,3He) reactions. Figure taken from [37]. The top panel corresponds to calculations where the transferred nucleon wavefunction has been computed from a Woods-Saxon potential and normalized to the spectroscopic factor from SM. The bottom panel corresponds to nucleon wavefunctions from SCGF. . . . . . . . . 19 1.10 “Quenching factors” as a function of the difference in separation energies ∆S=Sp(n)−Sn(p)for exclusive (p, 2p)reactions. Red and blue triangles denote calculations with different optical potentials, while the green circle and black asterisk are results from heavy-ion knockout and (e, e0p)shown for comparison. Figure taken from [44]. . . . . . . . . . . . . . . . . 21 2.1 Different Jacobi coordinates for a 3-body system. . . . . . . . 31 2.2 Schematics of the Transfer to the Continuum framework. Note that the initial partition p+Aserves as the source term for the final partition (pn) + (A−1) as denoted by the singledirection arrow, while all channels in the final partition are coupled to all orders, as denoted by the two-directional arrows. 39 2.3 Cross section leading to states with a defined jπ, for the 16O(p, pn)15O and 16O(p, 2p)15N reactions on the top panel, and for the elastic pn and pp cross sections, ignoring Coulomb, in the bottom panel. . . . . . . . . . . . . . . . . . . . . . . . 51 2.4 Longitudinal momentum distribution for 15C(p, pn)14C at 420 MeV/A. Calculations including and excluding distorting potentials are presented. The calculation including distortion is also shown rescaled to give the peak of the calculation without distortion.............................. 52 List of figures xix 2.5 Longitudinal and transversal momentum distribution for 22O(p, pn)21O at 415 MeV/A. Longitudinal momentum distributions are presented in the top row, while transversal momentum distributions are presented in the bottom. The left column corresponds to a neutron removed from the 2s1/2 wave while the right one corresponds to the 1d5/2wave. Calculations including and excluding distorting potentials are presented. The calculation including distortion is also shown rescaled to give the peak of the calculation without distortion. 54 2.6 Free p−nand p−pcross section (ignoring Coulomb) as a function of the laboratory energy, computed with the Reid93 potential assuming J= 5 maximum angular momentum between proton and nucleon. . . . . . . . . . . . . . . . . . . . . 55 2.7 Longitudinal momentum distribution for the 15C(p, pn)14C reaction at 100 MeV/A and 420 MeV/A with and without the relativistic corrections described in Sec. 2.3. For the energy 420 MeV/A the calculation without relativistic corrections is shown rescaled to reproduce the cross section of the calculation with relativistic corrections for the sake of comparison. . 57 3.1 Benchmark calculation between TC and DWIA formalisms for 15C(p, pn)14C at a proton energy of 420 MeV. The left panel corresponds to the full calculation where the distorting potentials have been considered as described in Sec. 3.1.1. The right panel corresponds to a “plane-wave” calculation, where the distorting potentials are set to 0. . . . . . . . . . . 63 3.2 Benchmark calculation for the same reaction as Fig. 3.1, where the separation energy has been artificially increased to Sn= 5 and 18 MeV. ....................... 65 3.3 Dependence of the DWIA momentum distribution on the selection of Epn, the energy of evaluation of the free p−ncross section. .............................. 66 3.4 Distorted wave results from Figs. 3.1, 3.2 compared to DWIA results where outgoing potentials are calculated for each of the outgoing configurations. . . . . . . . . . . . . . . . . . . . 67 xx List of figures 3.5 Transversal momentum distribution for 15C(p, pn)14C at 420 MeV proton energy computed using the Faddeev/AGS formalism through non-relativistic equations and using the Transfer to the Continuum formalism with (green dashed line) and without (red dot-dashed line) relativistic corrections. The left panel shows the full calculation while the right one shows the “plane-wave” calculation where all distorting potentials are set to0. ................................ 69 3.6 Benchmark calculation for removal of a 2sneutron. The blue dashed line corresponds to the FaddeeV/AGS calculation, the black solid line corresponds to a TC calculation with Vp11Be obtained through folding as described in Sec. 3.2.1 and the red dotted line corresponds to TC with Vp11Be from KD parametrization. The oservables presented all refer to the outgoing core 10Be in the c.m. frame and are as follows: a) transversal momentum distribution, b) longitudinal momentum distribution, c) energy distribution, d) angular distribution. 73 3.7 As Fig. 3.6 but for removal of a 1pneutron. . . . . . . . . . . 75 3.8 As Fig. 3.6 taking the Reid93 as NN interaction, assuming a neutron removed from the 2s1/2wave. ............. 77 3.9 As Fig. 3.6 taking the Reid93 as NN interaction, assuming a neutron removed from the 1p1/2wave. ............. 78 3.10 Plane wave calculation for the 11Be(p, pn)10Be reaction at 200 MeV/A, assuming the removal of the neutron from the 1p1/2 wave. Red dots correspond to the TC calculation, while the blue line corresponds to the Faddeev/AGS one. . . . . . . . . 79 4.1 Cross section for the 18C(p, pn)17C∗(0.33 MeV) reaction assuming removal from the d5/2state as a function of the final p−nangular momentum and parity jπ............. 83 List of figures xxi 4.2 Transversal pxmomentum distributions for the 18C(p, pn)17C∗ reaction. Experimental data are from [79]. The left panel corresponds to (p, pn)leading to the Ex= 0.21 MeV 17C excited state, while the right panel corresponds to the Ex= 0.33 MeV excited state. The blue solid line corresponds to the full calculation rescaled to give the experimental total cross section. The blue dashed line corresponds to the calculation removing the contribution from the (p, d)transfer reaction, rescaled by the same factor as the full calculation. All theoretical calculations have been convoluted with the experimental resolution.............................. 85 4.3 Experimental and theoretical levels for 19C. The level scheme from a) has been constructed from the results in [85, 86] while that in b) is based on the results from [87, 89, 91]. The theoretical results have been obtained from a shell model calculation using the WBP interaction [15] and from a P-AMD model [92]. Note that the first 5/2+state is either bound or unbound for different experiments and models. Figure taken from [90]. . . 87 4.4 Experimental data and Transfer to the Continuum calculations for the 20C(p, pn)19C reaction. The left panel shows the single-particle cross sections for the states 1/2+,3/2+and 5/2+in the blue, red and green dash-dotted lines, respectively, while the orange and dark green solid lines denote the total contribution for states 1/2+and 3/2+and 1/2+,3/2+and 5/2+respectively. In the right panel the orange and dark green lines have been rescaled to reproduce the experimental cross section, with the result shown in the dashed lines and the rescaling factor appearing in the legend . . . . . . . . . . 89 4.5 Proton-16O potentials at three different energies using PH and Dirac parametrizations (See text). The shaded area marks the distances up to the rms radius of 16O. ............. 92 4.6 Angular differential cross sections for the elastic scattering between pand 12C at 200, 300 and 398 MeV incident proton energy compared to theoretical calculations using PH and Dirac potentials. Experimental data are from [97–99] for the energies of 200, 300 and 398 MeV respectively. . . . . . . . . . 93 xxii List of figures 4.7 Momentum distributions for 12C(p, 2p)at 398 MeV/A. Experimental data are from [42]. Calculations are presented using PH and Dirac calculations (black and red lines respectively). The black solid and red dashed lines have been convoluted with an experimental resolution of 20 MeV for the transversal momentum pxand 50 MeV for the longitudinal one pz. Black dot-dashed and red dotted lines correspond to calculations without the experimental resolution. . . . . . . . . . . . . . . 98 4.8 Single-particle transversal (pt) momentum distributions for 12C(p, 2p)at 398 MeV/A computed for the p3/2and p1/2 orbitals using Dirac and PH potentials. It can be seen that shapes are quite similar in all cases. . . . . . . . . . . . . . . . 99 4.9 Transversal pymomentum distribution for 16O(p, 2p). Experimental data are from [23]. Calculations are presented using PH and Dirac calculations (black solid and red dashed lines respectively) rescaled to reproduce the experimental cross section. ................................100 4.10 Transversal pymomentum distribution for 22O(p, pn). Experimental data are from [43]. Calculations are presented using PH and Dirac calculations (black solid and red dashed lines respectively) rescaled to reproduce the integral of experimental data (given in arbitrary units). The dotted and dot-dashed lines correspond to the contributions of s1/2and d5/2orbitals, having all been scaled by the same arbitrary factor, and as such conserving their relative weight. . . . . . . . . . . . . . . 101 4.11 “Quenching factors” Rsas a function of ∆S, the difference between proton and neutron separation energy. Results from this work are presented in black circles using PH potentials and in red squares using Dirac potentials. The brown band corresponds to the systematics found in knockout experiments [31, 32]. Rsfrom [43], [23] and [42] are presented in green triangles, blue squares and pink diamonds respectively. . . . . 104 List of figures xxiii 4.12 To the left, “notch test” performed on the 23O(p, pn)and 23O(p, 2p)reactions (see text). The thin black line corresponds to the relative variation of the cross section with the position of the notch. The density of 23O is shown in arbitrary units for comparison. To the right the Dirac and PH potentials for 23O are presented, the yellow area denoting the “area of sensitivity” which starts roughly at the position where the notch changes the cross section by ∼5%. ...........106 4.13 Rsfrom Faddeev/AGS calculations [43] (open green triangles) and from a TC calculation (in violet triangles) where KD potentials have been used, relativistic corrections omitted and the geometry of the binding potentials modified to r0= 1.25 fm, a= 0.65 fm, following the prescription from [43]. . . . . . 109 4.14 Comparison of Dirac, PH and KD potentials for p+22O at 200MeV. .............................110 5.1 Scheme of a (p, pN)reaction on a three-body nucleus, with the relevant coordinates involved in the TC formalism. . . . . 116 5.2 Energy distribution for 11Li(p, pn)10Li using potential P3 (see Table 5.3). The left panel shows the contribution of the s and pwaves along with their sum. The right panel shows the full distribution before and after convoluting with the experimental resolution. Experimental data are from [118]. . . 126 5.3 As Fig. 5.2, but showing the results for potential P4 (see Table 5.3). ................................128 5.4 As Fig. 5.2, but showing the results for potential P5 (see Table 5.3). ................................129 5.5 As Fig. 5.2, but showing the results for potential P1I (see Table 5.3). Due to splitting, the swave gives 1−and 2− states for 10Li, while the pwave gives 1+and 2+states. . . . 130 5.6 Energy distributions computed using Eq. 5.25 (solid lines) and Eq. 5.30 (dashed lines). The factors Cαare computed for each configuration of 10Li independently. The top panel corresponds to calculations with potential P3 and the bottom one to potential P1I (See Table 5.3). . . . . . . . . . . . . . . 135 xxiv List of figures 5.7 Resonance energy spectrum from [112]. The dotted red line denotes the calculated 9Li+n+dthree-body non-resonant phasespace and the solid red line denotes the extracted background for the determination of the cross section. . . . . . . . 137 5.8 Energy distributions for the 11Li(p, d)10Li reaction obtained with models P1I, P2I, P3 and P4. The contribution of each 10Li configuration is shown, as well as its sum. The sum is also shown convolved with the experimental resolution. . . . . 139 5.9 Angular differential cross section computed for 11Li(p, d)10Li. Experimental data are from [112]. Dashed lines denote calculations with models without coupling to the 9Li spin while solid lines denote models including it. The contribution of the swave components for P1I is shown in the blue dotted line. . 141 5.10 As Fig. 5.9 including the calculation for P5. . . . . . . . . . . 142 C.1 Phase-shifts for the p−nsystem for various configurations as a function of laboratory energy. In black solid, the results of our calculations with fresco are compared to the results from [125] in red dashed. . . . . . . . . . . . . . . . . . . . . . 164 C.2 Phase-shifts for the p−psystem for various configurations as a function of laboratory energy. In black solid, the results of our calculations with fresco are compared to the results from [125] in red dashed. . . . . . . . . . . . . . . . . . . . . . 164 List of Tables 3.1 Parameters for the König-Delaroche potentials used in the calculation. ............................ 68 3.2 Parameters for the potentials used in the calculation. All potentials are taken from [76] but Vpn, which is based on the parametrization from [61]. . . . . . . . . . . . . . . . . . . . . 72 4.1 Cross section and “quenching factors” Rsfor 18C(p, pn)17C∗ obtained from CDCC [79] and TC calculations. The first and second columns indicate the energy and the angular momentum and parity of the state, respectively, while the third column shows the experimental cross section. The fourth column shows the spectroscopic factor obtained with the WBP interaction for each state. The fifth, sixth and seventh columns show the CDCC single particle cross section, theoretical cross section and “quenching factor”, respectively. The eight, ninth and tenth columns do the same for TC values. . . . . . . . . . 84 4.2 Results for the considered levels of 19C. The first, second and third column denote the angular momentum and parity, excitation energy and spectroscopic factors for the states obtained using the WBP interaction respectively. The fourth column indicates the reduced radius used for the computation of the bound state wavefunction. The last column shows the single particle cross section to each state. . . . . . . . . . . . . . . . 88 4.3 Detail of the reactions considered, with the incident energy of the beam, the separation energy of the removed nucleon (proton for (p, 2p)and neutron for (p, pn)) and the reference from which experimental data have been taken. . . . . . . . . 103 xxv 4Chapter 1. Introduction removed nucleon, with both projectile and nucleon separating from the core without modifying its state of motion. With this assumption, we can envision the target nucleus Aas being composed by the struck nucleon Nand the core C,A=N+C. In the laboratory system, where Ais at rest, if N has a momentum pN“inside” A, core Cmust have the opposite momentum pC=−pNto have Abe at rest. Therefore, after the reaction, since Cdoes not change its state of motion, by measuring pC, we can recover pN, the momentum of the nucleon inside A, so the momentum distribution of the outgoing core can be related to the wavefunction of the removed nucleon in momentum space. Fig. 1.2 shows a schematics of the process. Figure 1.2: Description of the (p, pN)process in the quasi-free picture. Note that the momentum of the core Cis not altered during the reaction. From this momentum distribution (or any observable that can be related to it through conservation of energy and momentum), information can be extracted from experimental data that can be compared to nuclear structure models through a single-particle reaction calculation. The single-particle prediction is computed assuming that exactly one nucleon is removed from a definite nuclear level. The shape of the prediction as a function of angle, energy or momentum is usually very sensitive to the quantum numbers defining the level, in particular its orbital angular momentum, as can be seen in Fig. 1.3 for the core momentum distributions of a (p, pn)reaction. Within the simple assumption that the reaction takes place by removing the nucleon from the corresponding level without changing the rest of the nucleus, the cross section for the removal will be proportional to the occupation number of the level (loosely speaking, the spectroscopic factor SF ), which can then be extracted as the ratio between the total cross section and the single-particle one: SF =σexp σsp .(1.1) 1.1. Background 5 -400 -200 0 200 400 px (MeV/c) 0 0.05 0.1 dσ/dpx (mb/MeV/c) s1/2 SF=1 d5/2 SF=1 p1/2 SF=1 -400 -200 0 200 400 pz (MeV/c) 0 0.05 0.1 dσ/dpz (mb/MeV/c) 23O(p,pn)22O Figure 1.3: Single-particle transversal (top) and longitudinal (bottom) momentum distributions for 23O(p, pn)22O computed for the removal of a neutron from different nuclear levels. Figure from [3]. Let us highlight here the multiple assumptions involved in this formula: the reaction is assumed to remove a nucleon only from the least bound level, without affecting the rest of the nucleus, whose structure does not influence the reaction, except by setting the number of nucleons in the least-bound level, from which they may be removed. Of course, the QFS picture described above for nucleon removal reactions is too simplistic and flat-out inconsistent, since it assumes no interaction between the outgoing Nand C, although this interaction has to exist to bind them into Ain the first place. However, it indicates which features an experiment must show in order for information on the removed nucleon to be clearly extracted. The main of this features is the minimization of the interaction between the projectile and the core, while keeping a strong 6Chapter 1. Introduction interaction between projectile and nucleon. This criterion suggests the use of light projectiles, whose interactions with the core are expected to be less intense than those of heavier projectiles. Another feature that should be required is the minimization of the interaction between the outgoing nucleon and the core. This favours experiments with a beam energy of hundreds of MeV, for which the nucleon-nucleon interaction presents a minimum, as can be seen in Fig. 1.4. At these energies, the mean free path of the outgoing nucleon in nuclear matter is maximal, so the effects of its interaction with the core should be minimized. This condition also imposes a degree of peripherality to the reaction, since nucleons removed from the interior of the nucleus will have to cover a larger distance on their “way out” of the nucleus, which make nucleon-core interactions more likely. These features make (e, e0p)and (p, pN)(Nbeing either a proton or a neutron) reactions at hundreds of MeV an excellent probe to study singleparticle of nuclei. (e, e0p)reactions present a smaller interaction between electron and core (smaller distortion), since only the electromagnetic interaction will take place between them, but at the same time the interaction between electron and proton is small, so the overall cross section will be reduced. As such, (e, e0p)reactions require beams with high luminosity which can only be achieved in reactions with stable targets. On the other hand, (p, pN)reactions have stronger distortion effects, but the strong nuclear interaction between the incoming proton and the removed nucleon provides a larger cross section. In cases with high enough statistics, distortion effects can be further reduced by performing exclusive measurements, keeping fixed the angles and/or energies of the outgoing projectile and nucleon, selecting a geometry for which the QFS condition is best fulfilled. We note that, if both projectile and removed nucleon are detected, it is not necessary to measure the core, since its momentum and energy are conditioned by conservation laws. Of course, the detection of the core serves as a validation of the reaction mechanism. From as early as the 1960s, exclusive measurements of (e, e0p)[5] and (p, 2p)[6] reactions have been performed to extract information on singleparticle properties of stable nuclei, with experiments still being carried out to this day, with a special focus on nucleon removal from deep levels [7]. In the comparison of the results from these experiments to the IPM, it 1.1. Background 7 Figure 1.4: Nucleon-nucleon cross section as a function of projectile energy. Notice the minimum at hundreds of MeV. Figure taken from [4]. was found that the model was successful in predicting the ordering of nuclear levels. However, the occupation of the levels was found to be smaller than the (2j+1) value predicted by the IPM by 30-40% in a systematic study for (e, e0p)reactions [8] (see Fig. 1.5), leading to a “quenching factor” (the ratio between experimental and theoretical cross sections, Rs=σexp σth ) of 0.6-0.7. 8Chapter 1. Introduction Figure 1.5: “Quenching factors” compared to IPM as a function of the nucleus mass, obtained from (e, e0p)reactions [8]. Figure taken from [9]. 1.1.2 Nucleon-nucleon correlations and nuclear structure models This “quenching” of the occupation numbers when compared to the IPM has been consistently reproduced in other “single-nucleon” reactions, such as transfer [10], where the nucleon is transferred from the projectile to a bound state of the target or vice versa, and (p, 2p)reactions [7, 11], although the amount of this quenching varies depending on the reaction and the nucleus studied. Given the rather naïf picture of the IPM, these discrepancies with experiments should come as no surprise and are a clear indication of beyondmean-field effects in nuclei. From these effects, those behind the depletion of the spectroscopic factors are believed to be mostly related to nucleonnucleon correlations, appearing through the interaction between a pair of 1.1. Background 9 nucleons. Nucleon-nucleon correlations are usually separated in short-range correlations (SRC) and long-range correlations (LRC). These correlations couple different single-particle levels, reducing the occupancy of the levels below the Fermi energy (the energy of the least-bound nucleon in the nucleus), which is transferred to energies over the Fermi energy. This effect is illustrated in Fig. 1.6 for nuclear matter and for 208Pb. It must be noted that in this figure are presented both level occupancy and the related quasihole strength, which is better related to nucleon removal reactions in systems with correlations. Further information can be found in [12]. Figure 1.6: Occupation probability (n()) and quasi-hole strength (Z()) for nuclear matter and 208Pb. The transfer of quasi-hole strength from levels below the Fermi energy Fto levels above it due to nucleon-nucleon correlations is apparent. Figure taken from [13]. SRC are believed to originate in the strong tensor term and repulsive core of the nucleon-nucleon interaction, coupling states with low energy to states with high energy, and are believed to account for ∼15% of the “quenching’ of the spectroscopic factors. LRC couple single-particle motion to low-lying states of the nucleus, which appear due to collective motion of the nucleons in the nucleus resulting in effects such as nuclear deformation or superfluidity, and are assumed to provide another ∼20% reduction in the single-particle spectroscopic factors, resulting in a total reduction of ∼35%, as found in 10 Chapter 1. Introduction (e, e0p)experiments [9]. Nuclear structure models including at least some of these effects have been developed to better describe the properties of nuclei. A list of various nuclear structure models, which is not meant to be exhaustive nor comprehensive, is presented in the following: •Interacting Shell Model (ISM): An extension of the IPM, keeping the shell model scheme, is the inclusion of a residual interaction which couples different IPM levels, resulting in a Hamiltonian whose diagonalization gives the energy of the ground and excited states, whose associated eigenvectors are a combination of different single-particle states, such that the coefficient of the single-particle state can be related to its spectroscopic factor. The diagonalization of the Hamiltonian is usually performed in a truncated basis space (the valence space) from which the nucleons from deep shells are excluded, forming an inert core. The empty levels with energies far above the Fermi energy are also excluded, assuming that they couple weakly to occupied levels. As such, the valence space usually only involves occupied and empty levels around the Fermi energy (F). This truncation of the space requires the use of an effective nucleon-nucleon interaction to take into account the effect of the excluded channels, which is adjusted to reproduce the properties of nuclei in the region of interest. Due to the exclusion of the high-energy levels, this prescription is expected to treat worse SRC (which can couple these excluded states) than LRC. Many effective interactions have been generated to produce predictions for energy levels and spectroscopic factors for a wide variety of nuclei [14–16]. This model has become the standard tool for calculating spectroscopic factors, thanks to its early development [17] and the accesibility of computer codes which implement it, such as oxbash [18] or antoine [19], so much that it is now usually referred to simply as shell model (SM). •Ab initio No-Core Shell Model (NCSM): The use of a truncated valence space in SM introduces a rather strong approximation in the description of nuclei and leads to serious problems when states outside the valence space interfere strongly with those included in the calculation. As such, an improvement on SM which has been made 1.1. Background 11 available thanks to the increased computing capabilities of modern computers is the inclusion of all nucleons in the valence space, as well as extending the valence space in energy to include the relevant highenergy states, in the No-Core Shell Model (NCSM). This model is based on realistic nucleon-nucleon which reproduce two-nucleon phase-shifts up to an energy of ∼350 MeV. These interactions usually present strong repulsive cores, so they have to be renormalized so that the phase space does not become too large [20, 21]. Three-body forces are also included in the calculation. The Hamiltonian generated by the two- and threebody forces among the Anucleons in the system is diagonalized in a harmonic oscillator basis, to preserve translational invariance, though this basis yields incorrect behaviour in the tails of the wavefunction. This method is able to produce nuclear properties from nucleon-nucleon interactions without introducing extra approximations. However it is extremely computationally intensive, so results can only be produced for nuclei with A.16, without introducing some approximation, such as an inert core [22]. •Self-Consistent Green Function (SCGF): In the Self-Consistent Green Function method (SCGF), one aims to obtain the single-particle propagator gαβ(ω), from which the occupancy of a certain level can be obtained: nα=ZEA 0−EA−1 0 −∞ dω1 πIm(gαα).(1.2) The propagator can be obtained from an equation such as gαβ(ω) = g(0) αβ (ω) + X γδ g(0) αγ (ω)Σ∗ γδ(ω)gδβ(ω),(1.3) where g(0) αβ (ω)is obtained from a Hartree-Fock IPM calculation and Σ∗ γδ(ω), usually called the self-energy, is produced from g(0) αβ (ω)and includes coupling between single-particle states and high-energy states through SRC and between single-particle states and low-lying collective states through LRC. An iterative procedure can be employed where the resulting gαβ(ω)is taken as an input to produce Σ∗ γδ(ω)and solve Eq. 1.3 until convergence. The results can be proven to be independent from the starting propagator g(0) αβ (ω). Calculations have been produced 12 Chapter 1. Introduction to describe (e, e0p)experiments [9] and to produce spectroscopic factors for various oxygen isotopes [23]. •Gamow Shell Model: For nuclei near the evaporation lines, the effect of the continuum on nuclear structure becomes increasingly important. However, the harmonic oscillator basis commonly used in shell model calculations is not able to describe the continuum properly. For these cases, a better description can be obtained by working with Berggren states, which include bound states and resonances as well as the nonresonant continuum, discretized along a contour surrounding the resonant states in the complex energy plane. As in the Interacting Shell Model case, in the Gamow Shell Model, a valence space must be defined and an effective nucleon-nucleon interaction used through which to generate the matrix elements which will couple different Berggren states. Gamow Shell Model has been used to compute the structure of nuclei with different levels of exoticity, such as 18O or 6He [24]. •Nilsson model: The description of collective effects by the shell model, whose main degrees of freedom are single-particle, requires a large number of states, which is computationally heavy and sometimes infeasible. However, the collective effect of nuclear deformation (nuclei presenting non-spherical shapes) is commonplace in medium-heavy nuclei and found even for some lighter nuclei such as 10Be. For these nuclei, the spherical harmonic oscillator basis is not the best one to describe the nucleus. Instead, the eigenstates of a Hamiltonian such as H=T+1 2ω2 kz2+ω2 ⊥(x2+y2) + vs.o.~ω0(l·s) + vll(l2−1 2N(N+ 3)) (1.4) are more suitable for such a description. For these states the relevant quantum numbers are [N, nk,Λ,Ω], with Nthe principal quantum number, as in the harmonic oscillator, nkthe number of quanta corresponding to the parallel direction zand Λand Ωthe projection of the orbital and total angular momenta along z. These states separate from the spherical values with an increasing deformation length δ: δ= 3 ω⊥−ωk 2ω⊥+ωk ,(1.5) 1.1. Background 13 as can be seen in Fig. 1.7. We note that these states are not rotationally invariant. The restoration of this symmetry results in the different states developing a rotational band. The Nilsson model has shown remarkable success in reproducing the spectra of medium-heavy deformed nuclei, and it has been applied to more exotic nuclei, such as halo nuclei with a deformed core, like 11Be [25] or in the so-called island of inversion [26]. Figure 1.7: Spectrum of single-particle levels as a function of the deformation length δ. Levels are labelled as [N, nk,Λ,Ω]. Figure taken from [27]. These structure models have been used extensively to compare to experimental data, in particular the phenomenological shell model, due to being developed earlier as well as being computationally less demanding, and in general it was found that, although SM produced spectroscopic factors 20 Chapter 1. Introduction question for the knockout of deeply-bound nucleons. These nucleons have to be extracted from deeper regions inside the nucleus, so it is difficult for the target to interact with these nucleons without altering the rest of the projectile (the core) [23, 37]. As for transfer reactions, the marked dependence of the results on the ingredients of the calculation, in particular in the rather involved CRC formalism, may introduce large uncertainties in the final Rswhich may cloud the extraction of overall tendencies [41]. 1.1.5 (p, pn)and (p, 2p)reactions on exotic nuclei The asymmetry dependence of the “quenching factors” Rsremains an open problem in the nuclear physics community up to date. Further experimental data studying nuclei with varying values for ∆Sare desirable in order to build a reliable systematics. The experiment should also involve a reaction mechanism that is well understood in order to minimize the uncertainties of the reaction calculation, so that the asymmetry dependence can be solidly related to the properties of the studied nuclei. In order to clarify the situation, very recently experimental campaigns on (p, 2p)and (p, pn)reactions have been performed both by the R3B collaboration in the GSI facility in Germany as well as by the group at RIBF/RIKEN in Japan. (p, pN)reactions are particularly attractive because the nucleonnucleon interaction is well-known, so it is expected that a better description of the reaction mechanism can be obtained than for knockout with heavier targets. The QFS condition described in Sec. 1.1.1 is also pursued in order to have a “cleaner” reaction. For this reason, reactions have been performed at beam energies of 200-450 MeV/A, for which the nucleon mean free path is maximal. Experimental analyses of these reactions must deal with the problem of low luminosity inherent to the use of exotic beams. In order to increase statistics, the R3B collaboration has focused on inclusive observables, in particular inclusive momentum distributions of the emitted core, in which the angle and energy of the outgoing nucleons is not restricted. The two outgoing nucleons are detected in order to ensure that the residual core is in fact the product of a (p, pN)reaction, but are only taken as “gating” of the relevant events. Results from this experimental campaign have very 1.1. Background 21 recently been published [23, 42, 43], and are subject of analysis in Sec. 4.2 so a comment on them is left for the corresponding section. The measurements at RIBF have instead produced exclusive cross sections, where the quantities of interest were the energy and angle distributions of the emitted nucleons, using the detection of a proper residual core as “gating” for the events of interest. Results for different isotopes of oxygen (14O, 16O, 18O, 22O, 24O) have been recently presented [44]. The obtained Rsas a function of ∆Sare presented in Fig. 1.10. Two results employing different optical potential parametrizations in the theoretical analysis are presented: The red triangles correspond to potentials from the phenomenological Dirac parametrization [45, 46] while the blue ones use potentials obtained from the folding of the nuclear densities with the Melbourne g-matrix effective interaction [47]. As can be seen in Fig. 1.10, both results agree rather reasonably between themselves and with the results from (e, e0p)and heavyion knockout reactions presented in the figure. Despite a somewhat large dispersion of the Rs, no clear Rsasymmetry dependence can be appreciated, so these results seem to favour those from transfer as opposed to those from heavy-ion knockout. Figure 1.10: “Quenching factors” as a function of the difference in separation energies ∆S=Sp(n)−Sn(p)for exclusive (p, 2p)reactions. Red and blue triangles denote calculations with different optical potentials, while the green circle and black asterisk are results from heavy-ion knockout and (e, e0p) shown for comparison. Figure taken from [44]. The extraction of the Rsrelies heavily on the reaction formalism used 22 Chapter 1. Introduction to describe the reaction from which spectroscopic information is extracted, since the distortion and absorption effects taking place during the reaction influence heavily the experimental observables. As such, a proper description of these effects is paramount to obtain a good description of the reaction and in particular, of the magnitude of the cross section, which is essential for the evaluation of “quenching”. In the following section, different formalisms used to analyze (p, pn)and (p, 2p)reactions in recent works are presented. 1.2 Reaction formalisms for (p, pn)and (p, 2p)reactions In this section some reaction formalisms used in the literature for the analysis of (p, pn)and (p, 2p)reactions are briefly described: 1.2.1 Plane-Wave Impulse Approximation (PWIA) Although it is a rather unrealistic approximation, the plane-wave impulse approximation is a useful model to shed some light on the dynamics of the (p, pN)process. It assumes a similar portrayal of the reaction as that taken in the QFS model from Sec. 1.1.1, in which both the incoming proton and the outgoing proton, nucleon and core can be described by plane waves, thus ignoring the interactions among them. In this approximation, and assuming an infinitely massive core, whose internal degrees of freedom are not altered by the reaction, the matrix element for the reaction can be expressed as: Tif =De−ikp·rpe−ikN·rN|TpN |eik0·rpϕnljm(rN)E,(1.8) where we ignore the spins of the nucleons for simplicity and where the p, N and 0 subscripts correspond to the outgoing proton and nucleon and the incoming proton respectively and ϕnljm denotes the wave function of the bound nucleon. TpN is the transition matrix between the initial and final states corresponding only to the VpN interaction. Next, the Impulse Approximation approximates TpN by the free nucleonnucleon t-matrix between proton and nucleon. This neglects the interaction of the proton with the rest of the target nucleus, as well as ignoring the effect 1.2. Reaction formalisms for (p, pn)and (p, 2p)reactions 23 of the binding energy during the collision: TpN =κ0 pN |tpN |κpN ,(1.9) with κpN the relative momentum between proton and neutron. Assuming a slow variation of tpN with κpN , it is possible to perform a factorization of TpN which results in a zero-range approximation: Tif =De−ikp·rpe−ikN·rN|δ(rp−rN)|eik0·rpϕnljm(rN)Eκ0 pN |tpN |κpN , (1.10) which can be related to the Fourier transform of ϕnljm: Tif =ZdrNe−iQ·rNϕnljm(rN)κ0 pN |tpN |κpN ,(1.11) with Q=kp+kN−k0. Since we have neglected the effect of binding energy and distorting potentials, the elements Dκ0 pN |tpN |κpN Emust be onshell due to conservation of energy. This allows to relate them to the free nucleon-nucleon cross section. |κ0 pN |tpN |κpN |2=(2π~2)2 µ2 pN dσpN dΩpN (EpN , θpN ),(1.12) where EpN and ΩpN are the proton-nucleon relative energy and scattering angle and are computed from κpN and κ0 pN . From here the cross section for the (p, pN)is found to be proportional to the product of the free nucleonnucleon cross section and the square of the Fourier transform of ϕnljm(rN): dσ dkpdkN∝dσpN dΩpN (EpN , θpN )1 2j+ 1 X mZdrNe−iQ·rNϕnljm(rN) 2 . (1.13) If, as usual, we normalize ϕnljm(rN)to 1, and extract its norm as a spectroscopic factor, we find: dσ dkpdkN∝SFnlj dσpN dΩpN (EpN , θpN )1 2j+ 1 X mZdrNe−iQ·rNϕnljm(rN) 2 , (1.14) which shows that, in the PWIA limit, the cross section is sensitive in its magnitude to the spectroscopic factor of the level from which the nucleon 24 Chapter 1. Introduction is removed, and in its shape, to its Fourier transform, and therefore to its angular momentum. This marks the remarkable spectroscopic power of the reaction. We note that the plane-wave approximation is too extreme to give a good description of the reaction, in particular of the magnitude of the cross section. However, it has seen an extensive use in (e, e0p)reactions [48]. 1.2.2 Distorted-Wave Impulse Approximation (DWIA) The DWIA formalism faces the description of the A(p, pN)Creaction through the so-called Impulse Approximation, as PWIA. It however includes the effects of distortion originating in the interaction between proton and target, proton and core and nucleon and core, which are included in the T matrix by substituting the plane waves by distorted waves: Tif =Dχ(−) p,kpχ(−) N,kN|tpN |χ(+) p,k0ϕnljmE.(1.15) Here χ(±)denote distorted waves corresponding to an incoming proton with momentum k0, and outgoing proton and nucleon with momenta k1and k2 respectively. In order to present more realistic formulae, in this section the formalism for a core with finite mass is presented. The neglect of off-shell effects is still maintained. We will be following the formulation in [49]. With these approximations, the core momentum distribution in the system where nucleus Ais at rest is given by the expression: dσ dpA C =C0SFnlj Zdp1dp2ηA Mφ1δ(EA f−EA i)δ(pA f−pA i)dσpN dΩpN ×X m (2π)2ZdRχ(−)∗ p,kpχ(−)∗ N,kNχ(+) p,k0ϕnljme−ik0·R/A 2 , (1.16) where the δimpose conservation of energy and momentum, the superscript Adenotes that the quantities are computed in the rest frame of Aand where C0=EA 0 (~c)2pA 0 1 2l+ 1 ~5 (2π)3µ2 pN (1.17) and ηA Mφ1=E1E2EB EA 1EA 2EA B ,(1.18) 1.2. Reaction formalisms for (p, pn)and (p, 2p)reactions 25 ηA Mφ1being the Möller factor, which stems from the transformation from the N−Nframe to the N−Aframe. Here the quantities without superscript are computed in the center of mass system. More information can be found in [50]. We note that this expression is similar to the PWIA one,but with a distorted Fourier transform of the nucleon wavefunction. The DWIA formalism has been used in the analysis of [44]. 1.2.2.1 Eikonal DWIA Due to the large energies involved in the (p, pn)and (p, 2p)reactions measured in GSI and RIBF, it is possible to simplify somewhat the calculations by using the eikonal approximation [51], which assumes the particles involved to describe linear trajectories during the reaction. Following [52], in the eikonal approximation the distorted waves for the incoming proton and outgoing proton and nucleon can be expressed as plane waves multiplied by “survival amplitudes” Sin(out), which are functions of the impact parameter b: χin(out) p(N)(r) = Sin(out)(b)eikin(out) p(N)·r,(1.19) Sin(out)(b) = exp "−i ~vZbin(out) ain(out) dz0Uin(out) p(N)(r0)#,(1.20) where, for the incoming proton the integral is performed between the point where it starts to feel the interaction with the nucleus (formally −∞) and the collision point, and for the outgoing proton and nucleon the integral starts at the collision point until they stop feeling the interaction with the core (formally ∞). It must be remarked that the incoming proton and outgoing nucleons describe straight-line trajectories. However, they are not parallel, as in standard eikonal theory, since the collision between two particles of similar mass (the proton and the nucleon) can lead to a strong change in direction. As such z0and bhave a different value for each distorted wave. The cross section as a function of Q=kp+kN−A−1 Ak0, the transferred momentum (which corresponds to the definition in Sec. 1.2.1 in the infinite- 26 Chapter 1. Introduction mass limit), can be expressed as: dσ dQ=1 (2π)2 SFnlj 2j+ 1 X mdσpN dΩpN QZdre−iQ·rhS(b)iQϕnljm(r) 2 .(1.21) In this expression the brakets denote an average over all the configurations that lead to a transferred momentum Q, and S(b)is the product of the three survival amplitudes corresponding to the incoming proton, the outgoing proton and the outgoing nucleon. Since the transferred momentum can be easily identified with the momentum of the core in the frame of reference where A is at rest, this cross section can be immediately translated the momentum distribution of the core. This formalism has been used in the analysis of [23, 42] and further details can be found in [52]. 1.2.3 Faddeev/AGS formalism The Faddeev/AGS [53, 54] formalism allows to obtain the exact solution of a non-relativistic three-body problem, through the evaluation of the operators Uβα, whose on-shell matrix elements correspond to the transition amplitudes and which are obtained solving the three-body AGS integral equations. As opposed to the previous formalisms, it does not consider any approximation beyond the assumption that the nucleus Acan be treated as a two-body system, and thus the p+Areaction can be treated as a three-body problem. The operators Uβα fulfil: Uβα =¯ δβαG−1 0+X γ ¯ δβγtγG0Uγα,(1.22) where α, β, γ denote states corresponding to one of the three possible partitions of the problem: p+ (n+C),C+ (p+n),n+ (p+C)or to the breakup state p+n+C,¯ δβα = 1 −δβα and tγcorresponds to the two-body t-matrix in the 3-body medium: tγ=vγ+vγG0tγ,(1.23) where the odd-one out notation is used so for example, assigning 1 to the p+ (n+C)partition v1=Vpn +VpC. Finally G0is the free resolvent G0= (E+i −H0)−1in the limit →0. Breakup observables are obtained from the on-shell matrix elements of U0α:T0α=qp|U0α|ψα, where α corresponds to the initial state: p+(n+C), and p,qare chosen as the Jacobi 1.2. Reaction formalisms for (p, pn)and (p, 2p)reactions 27 momenta for the C+ (p+n)partition. From here the double differential cross section for the energy and angle of the core can be computed as: dσ dECdΩC = (2π)4mn+mC Klab mCkC mnmp mn+mpZdˆpF|T0α|2,(1.24) with kCthe momentum of the core, Klab that of the projectile in the laboratory frame and F=s2mnmp mn+mp E−mnmpM mC(mn+mp)2k2 C+m2 C M2KT−2mC Mkc·kT, (1.25) where M=mp+mn+mCand KTis the total momentum, which is 0 in the center of mass frame and Klab in the laboratory frame. More information can be found in [55–57]. The Faddeev/AGS formalism has been used in the analysis of [43]. It is the objective of this work to develop a new reaction formalism able to compute (p, 2p)and (p, pn)reactions at low, intermediate and high energies. For this, it will not be based on the impulse approximation, thus allowing for its use at lower energies than PWIA and DWIA, and it includes relativistic kinematics, which are not present in the Faddeev/AGS formalism presented in the previous section and which are found to introduce important modifications in the cross section at high energies. This formalism is called Transfer to the Continuum and is presented and developed in Chapter 2. Then, in Chapter 3, it is benchmarked to other reaction formalisms, DWIA and Faddeev, in the regions and with the approximations to make the comparison meaningful. In Chapter 4, calculations using this new formalism are compared with experimental data and special interest is devoted to the analysis of the “quenching factors” Rsfor the experiments carried out by the R3Bcollaboration [23, 42, 43]. In Chapter 5, the formalism is extended to study the energy distributions of the fragments obtained in (p, pn)reactions on Borromean nuclei and results are presented for 11Li. Finally, an overall summary of the results of the present project is presented in Chapter 6, establishing an outlook for future developments. Chapter 2 Formalism: Transfer to the continuum Someone told me that each equation I included in the book would halve the sales Stephen Hawking The formalism used throughout this work is referred to as Transfer to the Continuum (TC) and has been used previously to study breakup reactions at low and intermediate energies [58, 59]. In short, it can be described as a CCBA-like (Coupled-Channel Born Calculation) calculation akin to a transfer reaction leading to continuum states of the projectile. In the next section, the formalism is detailed, highlighting the modifications required for its extension to high-energy reactions, where relativistic effects are of importance. 2.1 Coupled equations A reaction of the type A(p, pN)Ccan be studied within a three-body model where the relevant bodies are the incoming proton p, the removed nucleon Nand the residual nucleus or core C. Of these, only the core Cmay have internal degrees of freedom ξ, which will be assumed to be collective, in nature. The excitation of the nucleons to a ∆resonance requires too much energy to be of relevance in the range of energies considered. As such, the 29 36 Chapter 2. Formalism: Transfer to the continuum to the collective degrees of freedom of the core. From this the dependence of the potentials on the internal degrees of freedom ξcan be ignored, assuming a similar interaction between the nucleon por Nand all relevant states of the core Cand the third term of Eq. (2.24) can be eliminated, since no coupling will take place between different states of C. DΦC α˜ φjπ n|H−E|˜ φjπ nXjπnαΦC αE+X j0π0n06=jπnΦC α˜ φjπ n|H−E|˜ φ0j0π0 n0Xj0π0n0αΦC α +   X α06=α j0π0n0 ΦC α˜ φjπ n|H−E|˜ φ0j0π0 n0Xj0π0n0α0ΦC α0+DΦC α˜ φjπ n|H−E|XAφAE= 0. (2.25) Let us now focus on the last term. DΦC α˜ φjπ n|H−E|XAφAE=DΦC α˜ φjπ n|TR+Tr+HC+VpN +UpC +UNC −E|XAφAE= =DΦC α˜ φjπ n|TR+VpN +UpC −Ecm,i|XAφAE,(2.26) where we have performed the same transformation as in Eq. (2.12). Now we can add and subtract UpA to the equation, noting that we can cancel TR+UpA −Ecm,i since XAverifies Eq. (2.14): DΦC α˜ φjπ n|H−E|XAφAE=DΦC α˜ φjπ n|VpN +UpC −UpA|XAφAE,(2.27) forming the so-called source term of Eq. (2.25). We note that the only terms dependent on the internal variables of the core ξare ΦC α(ξ)and φA(r, ξ). Therefore we can perform the integral over them: ZdξΦ∗ Cα(ξ)φA(r, ξ) = X lj ˜ϕAC ljα (r),(2.28) obtaining the overlap function ˜ϕAC ljα (r)that only depends on coordinate r. l, j correspond to the orbital and total angular momentum of the removed nucleon that couple Ain its ground state and Cin state α. The integral from Eq. (2.28) involves in principle the position of all nucleons in nucleus Cso it is in general infeasible in most calculations. As such, it is usual to approximate it by a single-particle wavefunction obtained from a simple potential and normalized to 1, ϕAC ljα (r). The norm is obtained from a nuclear 2.1. Coupled equations 37 structure calculation which gives the spectroscopic amplitude, AAC ljα , whose square is the spectroscopic factor, SAC ljα . ˜ϕAC ljα (r)∼AAC ljα ϕAC ljα (r)(2.29) SAC ljα =|AAC ljα |2(2.30) Let us remark that in this derivation no attention has been paid to the antisymmetrization of identical protons and neutrons. This will gain relevance in Sec. 2.4, but for the present discussion we believe a disgression to treat it would be counterproductive. Furthermore, we will assume that only one l, j configuration will be relevant for the reaction, in order to simplify the notation. With this approximation, Eq. (2.25) results in DΦC α˜ φjπ n|H−E|˜ φjπ nXjπnαΦC αE+X j0π0n06=jπnΦC α˜ φjπ n|H−E|˜ φ0j0π0 n0Xj0π0n0αΦC α +AAC ljα D˜ φjπ n|VpN +UpC −UpA|XAϕAC ljα E= 0, (2.31) which expanding the hamiltonian Hcan be expressed as DΦC α˜ φjπ n|TR+Tr+HC+VpN +UpC +UNC −E|˜ φjπ nXjπnαΦC αE+ X j0π0n0˜ φjπ nΦC α|TR+Tr+HC+VpN +UpC +UNC −E|˜ φj0π0 n0Xj0π0n0αΦC α+ AAC ljα D˜ φjπ n|VpN +UpC −UpA|XAϕAC ljα E= 0. (2.32) Now, for the first and second terms, we note that the kinetic term can be reexpressed as a function of the coordinates in Jacobi set (2): TR+Tr=TR0+ Tr0. We can then use the fact that the bin wavefunctions are approximately eigenfunctions of the Vpn potential: ˜ φjπ n|Tr0+VpN |˜ φj0π0 n0=δnjπ,n0j0π0en pN .(2.33) We also note that the only dependence on ξlies in ΦC αand HC, and from 38 Chapter 2. Formalism: Transfer to the continuum Eq. (2.18) we easily get ΦC α|HC|ΦC α=EC α,(2.34) and now from Eq. (2.17) EC α+en pN −E=−En cm,f , which can be related to Ecm,i through En cm,f =E−EC α−en pN Ecm,i =E−EA En cm,f =Ecm,i +EA−EC α−en pN =Ecm,i −SA N−εα−en pN ,(2.35) where SA Nis the nucleon separation energy for nucleus Aand εαis the excitation energy of state αof core C. In order to simplify the notation we will also define Uββ0(R)as Uββ0(R0) = Zdr0˜ φjπ∗ n(r0) [UpC(rpC) + UNC(rNC)] ˜ φj0π0 n0(r0),(2.36) where βdenotes the numbers required to define the final bin: n, jπ. After these transformations the equations result as (TR0−Ecm,f +Uββ)Xβα +X β06=β Uββ0Xβ0α+ AAC ljα D˜ φjπ nVpN +UpC −UpA XAϕAC ljα = 0. (2.37) It can be put into question whether Ψbin bα is a good description of the original three-body wavefunction Ψ3b bα. It is long known that wavefunctions obtained through discretization and truncation of the continuum do not reproduce the correct asymptotic behaviour of breakup components [68]. However we note that in the reaction under study we are interested in the wavefunction in the region where VpN +UpC −UpA is large. Since UpA and UpC are effective interactions between the proton and nuclei Aand Crespectively, we expect them to be similar and thus the so-called remnant term UpC −UpA will be quite small. Therefore, we are only interested in the wavefunction in the region where VpN is strong, that is, for small values of r0, the relative coordinate between pand N. Since we have chosen the bins to be computed based on the eigenfunctions of VpN , we expect an expansion on them to work well for small values of VpN , which is our region of interest, so Ψbin bα should describe well the reaction of interest. In order to verify the adequacy of 2.2. Cross sections and momentum distributions 39 Figure 2.2: Schematics of the Transfer to the Continuum framework. Note that the initial partition p+Aserves as the source term for the final partition (pn)+(A−1) as denoted by the single-direction arrow, while all channels in the final partition are coupled to all orders, as denoted by the two-directional arrows. this approximation, benchmarks with the more rigorous three-body Faddeev formalism have been performed and will be presented in Chapter 3. In Fig. 2.2 the features of the formalism are shown schematically, remarking how the initial p+Aserves as a source for the (pn) + (A−1) partition, in which all channels are coupled to all orders. 2.2 Cross sections and momentum distributions Using standard reaction theory, it is possible to compute the angular differential cross section to a given bin: dσjπn dΩ =1 (2sp+ 1)(2JA+ 1) µpAµpn−C (2π~2)2 K0 n KpA X mTjπn if  2,(2.38) where spis the spin of the proton and JAthe one of nucleus Aand mdenotes the spin projections of the incoming pand Aand the outgoing Cand pN system. Ωdenotes the scattering angle of the core in the center of mass frame. From this cross section it is possible to obtain a double differential cross section on Ωand the energy of the pN system by dividing the cross section by the energy width of the bin ∆n=h−l, assigning the obtained 40 Chapter 2. Formalism: Transfer to the continuum value of the cross section to the nominal energy of the bin: dσjπ dpN dΩpN =n pN ≃1 ∆n dσjπn dΩ .(2.39) From here we can obtain the distribution on the energy of the core and its scattering angle through energy conservation: Ecm,f =E−Cα−SA N−pN EC=mpN mpN +mC Ecm,f dσjπ dECdΩ =mpN +mC mpN dσjπ dpN dΩ, (2.40) and of course, we can obtain the total double differential cross section summing over all considered angular momenta and parities of the p−nsubsystem: dσ dECdΩ =X jπ dσjπ dECdΩ (2.41) Since the cross section is now expressed as a function of the energy and scattering angle of the core, it is rather simple to obtain the momentum distribution of the core in the center of mass: EC=p2 C 2mC dECdΩ = pC mC dpCdΩ = 1 mCpC d3pC dσ d3pC =1 mCpC dσ dECdΩ. (2.42) 2.3 Relativistic prescription Since the reactions we are interested in correspond to a proton accelerated to ∼200-400 MeV, it reaches a speed of 0.57-0.71c. Therefore, relativistic effects are relevant to the reactions involved and must be taken into account in order to obtain a satisfactory description of the experimental results. Relativistic effects appear in two distinct parts of our calculations: in the calculation of the coupled Schrödinger equations and in the kinematic transformations leading to the energy and momentum distributions. 2.3. Relativistic prescription 41 2.3.1 Relativistic corrections to the coupled equations In Section 2.1, the derivations considered to arrive to the set of coupled equations from Eq. (2.37) were performed assuming non-relativistic kinematics. This is specially apparent in the fact that the equation to be solved is Schrödinger’s equation and not Dirac’s. In this respect, analyses of various observables from proton-nucleus elastic scattering using Schrödinger and Dirac formalisms were performed in the 90s [69] using the “best” ingredients available at the time and it was found that the description of the observables was equally good for both formalisms, with proton energies ranging from 20 to 800 MeV. We believe these results validate the use of non-relativistic equations to describe (p, pN)reactions at the energies of interests, or paraphrasing [69], relativistic dynamics are not necessary to describe these. reactions. On the contrary, relativistic kinematics, with the nomenclature from [69], must be taken into account when studying these reactions. In order to have a clearer view on how this affects our coupled equations, let us take the still non-relativistic Eq. (2.37) and multiply it all by 2µpN−C/~2keeping in mind that TR0=−~2∇2 R0/2µpN−C: X j0π0n0−∇2 R0−K02 nδββ0+2µpN−C ~2Uββ0Xj0π0n0Cα+ qSAC ljα 2µpN−C ~2D˜ φjπ n|VpN +UpC −UpA|XAϕACαE= 0. (2.43) Unfortunately, there is not a unique prescription to include relativistic kinematics in Schrödinger’s equation. Following the relativistic prescription from [69], we will compute K0 n relativistically from the laboratory energy of the projectile, through standard relativistic formulas which are given in Appendix B. Energy conservation in the initial and final partitions must also be imposed relativistically. The next step in the relativistic prescription is to substitute the reduced mass µpN−C by the reduced energy pN−C, defined as pN−C=CpN C+pN ,(2.44) where Ccorresponds to its relativistic energy in the center of mass system C=qm2 Cc4+~2c2K0 n2.(2.45) 42 Chapter 2. Formalism: Transfer to the continuum The same is true for the pN system. However, since we are considering bins which can have a relatively high relative energy between pand Nthis energy must be included in the nominal mass of the pN system: pN =qm∗2 pN c4+~2c2K0 n2=rmpc2+mNc2+en pN 2+~2c2K0 n2. (2.46) Let us now introduce a factor γβ, defined as γβ=pN−C µpN−C ,(2.47) so the substitution of the reduced mass by the reduced energy can be expressed by a multiplication of the potentials by the factor γβ: X j0π0n0−∇2 R0−K02 nδββ0+γβ 2µpN−C ~2Uββ0Xj0π0n0Cα+ qSAC ljα γ2µpN−C ~2D˜ φjπ n|VpN +UpC −UpA|XAϕACαE= 0, (2.48) which is a specially convenient way to write the equation since it shows it is possible to modify non-relativistic equations to include relativistic equations by multiplying the involved potentials by the factor γβ. Note that factor γ depends on the channel considered βand thus changes for different equations. Let us also remember that XAwas obtained from a Schrödinger’s equation which also requires a relativistic modification: −∇2 R+2µpA ~2γAUpA −K2 pAXA= 0.(2.49) where KpA is computed relativistically and now γAis computed from the reduced energies of pand A. Another prescription to include relativistic kinematics in Schrödinger’s equation can be found in [70] and is very similar to the one described before. The equations are also solved considering the relativistic momentum, but the factor γβbecomes instead: γβ=2(E−mA) E−mA+mp or 2(E−mC) E−mC+m∗ pN ,(2.50) where Eis the total relativistic center of mass energy. 2.3. Relativistic prescription 43 Different relativistic prescriptions may yield different cross sections and it is not easy to determine which one is most suitable for the different cases. In the following a consistency criterion has been taken: the nucleon-nucleus potentials used in the calculations have been taken from parametrizations which generate the potentials assuming certain relativistic prescriptions for elastic scattering. When using these potentials, the relativistic prescription assumed for their construction has been followed, which is in general the one where γβ=pN−C µpN−C 2.3.2 Relativistic corrections to cross sections and momentum distributions After including the modifications indicated above in the coupled-channel equations, care must be taken when computing the cross sections and momentum distributions, since, once again, the relativistic relation between energy and momentum must be applied. The relativistic density of states for a certain energy is of special relevance in the relation between the transition matrix and the cross section. Fortunately, there is a rather simple method to correct it in the standard non-relativistic formulas [71], which consists in replacing the reduced mass by the reduced energy defined in Eq. (2.44). As such, the differential cross section can be computed as: dσjπn dΩ =1 (2sp+ 1)(2JA+ 1) pApn−C (2π~2)2 K0 n KpA X mTjπn if  2,(2.51) where K0 nand KpA are computed relativistically. The relation between en pN and ECmust also be revised, leading to a change in the formula for the cross section Eq. (2.40), where we now define ECas a the kinetic energy of core Cas opposed to its full relativistic energy C, and use Mandelstam variable s. More details are found in Appendix B. dσjπ dECdΩ =√s m∗ pN dσjπ dpN dΩ,(2.52) Likewise, the modification for the momentum distribution is rather straightforward: dσ d3pC =1 CpC dσ dECdΩ.(2.53) 44 Chapter 2. Formalism: Transfer to the continuum Here it must be noted that the energy and momentum distributions have been computed for their center of mass values, while there are other reference frames which also appear in the literature, namely, the laboratory frame, where the target nucleus is at rest and the projectile reference frame (rfp), where the projectile is at rest. The laboratory frame is the one where the actual measurements take place so it allows for a more direct comparison with experimental data. In experiments in inverse kinematics, where the nucleon is removed from the projectile, the rfp frame is of special interest because the effects of the kinematics of the reaction are somewhat removed from the momentum distribution so comparisons of reactions at different energies are most significant in this reference frame. Given the importance of these reference frames, we indicate in the following the relativistic transformations from the center of mass momentum distribution to the ones in laboratory and rfp frames. For this let us define pz, the longitudinal momentum as the component of the momentum parallel to the direction of the projectile beam and pxand pyas the perpendicular momentum, that perpendicular to it. Let us note that, as long as neither the projectile nor the target are polarised, the direction of pxand pyare irrelevant (given they are perpendicular to the beam) and thus, the distribution on pxis the same as in py. With these definitions we obtain dσ dpCx =ZdpCydpCz dσ d3pC (2.54) dσ dpCz =ZdpCxdpCy dσ d3pC (2.55) dσ dpCx c.m. =dσ dpCx lab =dσ dpCx rfp (2.56) dσ dpCz lab =C(c.m.) C(lab) dσ dpCz c.m. (2.57) dσ dpCz rfp =C(c.m.) C(rfp) dσ dpCz c.m. ,(2.58) (2.59) where the momentum and energy of the core Cin the different systems are 2.4. Antisymmetrization: the (p, 2p)case 45 related to those in the center of mass through standard relativistic formulae: C(rfp/lab)=γrfp/lab(C(c.m.)−βrfp/labpCz(c.m.))(2.60) pCz(rfp/lab)=γrfp/lab(pCz(c.m.)−βrfp/labC(c.m.))(2.61) pCx(rfp/lab)=pCx(c.m.),(2.62) where βrfp =pTp(2mp+Tp) mp+Tp+mA βlab =−pTA(2mA+TA) mA+TA+mp (2.63) γrfp/lab =1 q1−β2 rfp/lab ,(2.64) where TAis the kinetic energy of nucleus Ain the laboratory system (assuming inverse kinematics), while Tpis the kinetic energy of the proton in the rfp system, which would give the equivalent direct kinematics, and can be shown to follow the very simple relation: Tp=mp mA TA.(2.65) 2.4 Antisymmetrization: the (p, 2p)case In this section, we pay special attention to the antisymmetrization of the wavefunction for identical particles in the Transfer to the Continuum reaction formalism, since it has a very important effect in (p, 2p)reactions. We will follow Section 2.11.3 from Satchler’s [72]. From Eq. (2.178) in [72] we get that the antisymmetrized transition matrix can be expressed as T(as) βα =X Πβ ΠβDΠβΨ(−) β|Vα|ΦαE,(2.66) where for our case Vα=Vpn +UpC −UpA and. Πβcorresponds to a permutation between identical particles and the sum is performed over all possible permutations. Πβcorresponds to (−1)sΠβwhere sΠβdenotes the number of identical-fermion pairs that have been swapped in permutation Πβ. Let us now focus on the (p, 2p)on a nucleus Awith Zprotons and Nneutrons. 52 Chapter 2. Formalism: Transfer to the continuum Up15C,Up14Cand Up14Chave been removed, only leaving Vpn in Eq. (2.37) (labelled “Plane wave”). This is equivalent to a calculation in which a free proton collides with a neutron bound in a wave corresponding to that of the valence neutron of 15C, with neither of them interacting further with the 14C, thus neglecting all distortion and absorption effects. Since no comparison to experimental data will be considered, the removed neutron is assumed to populate a 2sstate with SF = 1. -400 -200 0 200 400 pz (MeV/c) 0 0.1 0.2 0.3 0.4 0.5 dσ/dpz (mb/(MeV/c)) Distorted wave Plane wave Distorted wave (rescaled) Figure 2.4: Longitudinal momentum distribution for 15C(p, pn)14C at 420 MeV/A. Calculations including and excluding distorting potentials are presented. The calculation including distortion is also shown rescaled to give the peak of the calculation without distortion. As can be seen in Fig. 2.4, the inclusion of distortion effects strongly reduces the cross section, which is of 40.6 mb for the plane-wave calculation and falls to 21.9 mb when considering distortion and absorption. It also modifies the shape of the momentum distribution, which can be more clearly seen comparing the plane-wave curve to the rescaled distorted-wave one. The most remarkable modification is the severe reduction of the shoulders of the distribution in the distorted-wave calculation. This reduction can be understood from the PWIA description of the reaction, where the momentum distributions are found to be proportional to the Fourier transform of the wave function of the bound nucleon times the free nucleon-nucleon cross section (times a phase-space factor that should not be too relevant at these high energies [50]). In the Fourier transform, the shoulders seen in the planewave calculation appear because the wavefunction presents a node (being a 2.5. Features of the calculation 53 2s1/2state) at small distances (∼2 fm). At these small distances, the effects of absorption are intense so, when they are considered, the reaction loses sensitivity to this part of the wavefunction and the effect of the node (the appearance of shoulders) is reduced in the final momentum distribution. In 15C the valence neutron has a very low binding energy, Sn= 1.22 MeV, and it is an established halo nucleus. As such, its wave function extends to large distances when compared to other nuclei in its vicinity. This makes it likely that the effects of the distorting potentials are somehow reduced, since the “collision” between the proton and the neutron may happen at large distances of the core, where the effects of the potential are not that strong. It would be also desirable to check the effects of distortion on waves which do not include a node, to explore how the shapes of their momentum distributions are modified by distortion. In order to answer these questions, in Fig. 2.5, the reaction 22O(p, pn)21O is explored in a similar way to that of Fig. 2.4. For this nucleus the binding energy is larger, Sn= 6.85 MeV, and the nucleus is no longer a halo. The valence neutrons for 22O populate both the 2s1/2and the 1d5/2waves, so that realistic calculations can be performed for both waves. As before, the removed neutron is assumed to populate either the 2s1/2or the 1d5/2 waves with SF = 1. In this figure, transversal momentum distributions are presented along longitudinal momentum distributions. Looking at the longitudinal momentum distribution for the 2s1/2wave, it is apparent that the reduction of the cross section is stronger than in the 15C case, the cross section dropping from 44.2 mb without distortion to 8.9 mb with it. As for the shape the same overall features than for 15C are found, with a reduction of the relative importance of the shoulders due to absorption. This can be understood due to the larger binding energy, which restricts the neutron wave function to smaller distances to the core, at which the effects of the potentials, and in particular the absorption, are more intense. There is also a slight shift in the distribution which could also be noticed in the 15C case, being somewhat larger for 22O. This shift can be understood as being due to distortion effects shifting the momentum distribution, as discussed in [50]. However, the shift is so small that it could be a numerical artifact. The reduction in cross section for the 1d5/2is similar, going from 44.5 mb to 8.7 mb. The width of the distribution is reduced by the distorting 54 Chapter 2. Formalism: Transfer to the continuum -400 -200 0 200 400 pz_rfp (MeV/c) 0 0.05 0.1 0.15 0.2 0.25 dσ/dpz_rfp (mb/(MeV/c)) -200 0 200 400 pz_rfp (MeV/c) 0 0.05 0.1 0.15 0.2 -400 -200 0 200 400 px (MeV/c) 0 0.05 0.1 0.15 0.2 0.25 dσ/dpx (mb/(MeV/c)) -200 0 200 400 px (MeV/c) 0 0.05 0.1 0.15 0.2 Distorted Wave Plane Wave Distorted Wave rescaled 22O(p,pn) s1/2 22O(p,pn) d5/2 Figure 2.5: Longitudinal and transversal momentum distribution for 22O(p, pn)21O at 415 MeV/A. Longitudinal momentum distributions are presented in the top row, while transversal momentum distributions are presented in the bottom. The left column corresponds to a neutron removed from the 2s1/2wave while the right one corresponds to the 1d5/2wave. Calculations including and excluding distorting potentials are presented. The calculation including distortion is also shown rescaled to give the peak of the calculation without distortion. potentials. This can once again be understood noting that the potential is highly absorptive at small distances. This makes the reaction sensitive only to the value of the wavefunction at large distances, effectively resulting in a “more extended” wave function, whose Fourier transform is narrower. Similar effects have been described for nucleon knockout reactions with heavier targets [73]. The distorted longitudinal momentum distribution for 1d5/2shows a smaller asymmetry than the “plane-wave” one. There is no simple picture to describe this behaviour, which results from two competing effects: one is the distortion of the momentum distribution by the real part of the distorting potential (described in [50]) and the other is the absorption of configurations 2.5. Features of the calculation 55 with uneven energies between proton and neutron. These configurations lead to asymmetry in the “plane-wave” momentum distribution, because they are related to different free nucleon-nucleon cross sections, as shown in Fig. 2.6, but they are strongly absorbed in “distorted-wave” calculations because they result in one of the particles having a small energy with respect to the core, which make it more likely to be absorbed. As such, these asymmetry-generating configurations are reduced in the “distorted-wave” calculation, leading to some restoration of the asymmetry. The asymmetry of the distorted momentum distribution thus depends on a complicated interplay between effects, which does not permit a simple explanation. Since the momentum distribution for the s-wave is more strongly peaked, it is more accurately described by the quasi-free mechanism, and the effect described above is less likely to become apparent. 0 100 200 300 Elab (MeV) 0 200 400 600 800 1000 σ (mb) p-n p-p Figure 2.6: Free p−nand p−pcross section (ignoring Coulomb) as a function of the laboratory energy, computed with the Reid93 potential assuming J= 5maximum angular momentum between proton and nucleon. The transversal momentum distributions show similar features to the longitudinal ones, with the exception of the shift and asymmetry, since transversal momentum distributions are symmetric thanks to rotational invariance. 56 Chapter 2. Formalism: Transfer to the continuum 2.5.3 Relativistic effects Given the large energies (∼200-400 MeV/A) at which (p, pn)and (p, 2p) reactions have been measured, relativistic effects, englobed in the relativistic prescriptions described in Sec. 2.3, are bound to influence strongly reaction observables. In order to quantify their importance, in Fig. 2.7 the longitudinal momentum distribution for 15C(p, pn)14C is presented for two incident energies, 100 and 420 MeV/A. The longitudinal momentum distribution is computed with and without the relativistic corrections from Sec. 2.3. As can be seen in the figure the effects of the relativistic prescription is much more important for an incident energy of 420 MeV/A, where their inclusion increases the cross section by 45%, than for an incident energy of 100 MeV/A, where the cross section is modified only 3%. If compared to the results from Fig. 3.5, which correspond to the same reaction, the effects of relativistic corrections are somewhat larger in this comparison. This arises from the choice of optical potentials, which are taken from Dirac parametrization here, while in Fig. 3.5 were taken from Köning-Delaroche (KD) parametrization. Since Dirac potentials are deeper than KD ones, the effect of relativistic corrections (multiplying by γfrom Eq 2.47) over them is more important when using Dirac potentials. It is remarkable that the shape of the distributions are not modified much by the relativistic corrections giving virtually coinciding shapes. This can be understood because the factor γdoes not change much for the different final states considered. Therefore, to a first approximation it can be seen as a global factor which affects the overall cross section, not modifying much the shape of the momentum distributions. It is also of note how different the two momentum distributions are, despite corresponding to a neutron removed from the same orbital. The reason is twofold: first, at 100 MeV, the impulse approximation, which allows to establish a clear relation between the momentum distribution and the Fourier transform of the bound nucleon wave function, breaks down. Secondly, even if we assume that the momentum distribution can still be described as the wave function Fourier transform times the nucleon-nucleon cross section, at lower incident energies the nucleon-nucleon cross section is evaluated at lower energies, which, as can be seen in Fig. 2.6, correspond to a much steeper variation of the cross section, which alters heavily the 2.5. Features of the calculation 57 -300 -200 -100 0 100 200 300 pz_rfp (MeV/c) 0 0.1 0.2 0.3 0.4 0.5 0.6 dσ/dpz_rfp (mb/(MeV/c)) 15C(p,pn)14C @ 100 MeV/A -300 -200 -100 0 100 200 300 pz_rfp (MeV/c) 0 0.1 0.2 0.3 0.4 Relativistic Non Relativistic Non Relativistic rescaled 15C(p,pn)14C @ 420 MeV/A Figure 2.7: Longitudinal momentum distribution for the 15C(p, pn)14C reaction at 100 MeV/A and 420 MeV/A with and without the relativistic corrections described in Sec. 2.3. For the energy 420 MeV/A the calculation without relativistic corrections is shown rescaled to reproduce the cross section of the calculation with relativistic corrections for the sake of comparison. momentum distribution. In particular, this explains the increase of the cross section at high momenta for the incident energy of 100 MeV/A, since these high momenta correspond roughly to a high energy of the core in the center of mass system, which in turn, through conservation of energy, relates to a small proton-nucleon energy, which leads to a high cross section. Chapter 3 Benchmark with other reaction formalisms Science is really about individual experts reaching a consensus Alan Stern Before comparing with experimental data, we assess the reliability and accuracy of the proposed reaction formalism against other methods used in the literature for the analysis of (p, pN)data. As such, benchmark calculations are presented in this chapter for (p, pn) reactions where the Transfer to the Continuum (TC) formalism is compared to other reaction formalisms. Two reactions are analysed, 15C(p, pn)14C for a proton energy of 420 MeV/c, which is compared to DWIA (Distorted- Wave Impulse Approximation) and to published results employing the Faddeev/AGS formalism [57]; and 11Be(p, pn)10Be at 200 MeV per nucleon, which is compared to the Faddeev/AGS formalism. For each reaction a description of the different inputs of the calculation is presented followed by the results of the calculations, which will be focused on the momentum distributions of the outgoing core. 3.1 15C(p, pn)14C at 420 MeV/A The DWIA calculations shown in this section were performed by Dr. K. Yoshida and Dr. K. Ogata, to whom the author of this thesis is greatly 59 60 Chapter 3. Benchmark with other reaction formalisms thankful. 3.1.1 Inputs and details for the calculation In this subsection, the inputs for the Transfer to the Continuum calculation performed for the benchmark are presented, as well as specific details of the calculation. The bound state for the neutron has been computed as the eigensate of a Woods-Saxon potential: V(r) = V0 1 + e r−r0A1/3 a0 ,(3.1) corresponding to the quantum numbers 2s1/2. The parameters for the potential are taken as r0= 1.25 fm, a0= 0.65 fm and A= 14, corresponding to the outgoing core. The depth of the potential V0is varied in order to obtain the desired separation energy. The 14C core is assumed to have angular momentum and parity 0+, so that the ground state of 15C corresponds to a 1/2+state. The distorting potentials p−15C, p−14C and n−14C are taken from the EDAD2 parameter set from Dirac phenomenological parametrization [45, 46]. These potentials are produced for a Schrödinger equation following the relativistic prescription from [69]. Therefore, this same prescription is followed in the TC calculations. This prescription gives potentials which are energy dependent. In the initial partition, p+15C, the energy is defined by the laboratory energy of the proton, which is taken as 420 MeV. For the final partition, p+n+14C, however, the energy between pand 14C and between nand 14C, is not defined for the final states considered in TC, where the energy that is defined is the one between 14C and the pn pair. The “quasifree” nature of the (p, pn)reaction at high proton energies results in the most likely outcome of the reaction being an equal distribution of energy between the incoming proton and the removed neutron. As such, we have chosen for the outgoing potential of the p+14C and n+14C pairs the one resulting from the EDAD2 Dirac parametrization at half the proton incident energy, that is, 210 MeV. In the DWIA formalism, the outgoing energies of the proton and neutron are well defined for the final states studied. This allows us to study the effect of the described choice of the outgoing potential in the reaction 3.1. 15C(p, pn)14C at 420 MeV/A 61 observables. The p−Npotential is paramount for the description of the (p, pn)reaction, since it is the main interaction that drives the reaction. For it, we have chosen the Reid93 [74] parametrization, a revision and extension of the Reid Soft Core potential [75], developed by the Nijmegen group. This potential consists of central, spin-orbit and central terms and it has been fitted to describe elastic p−nand p−pscattering observables for energies up to 350 MeV (see Appendix C). Although the energy of the incoming proton (420 MeV) is higher than the maximum for which the Reid93 potential was fitted, given the smooth energy dependence of the p−nscattering cross section, we expect this potential to describe accurately the p−ninteraction at the energies of interest. In order to have a more meaningful comparison with the DWIA formalism, the so-called remnant of the transfer vertex is neglected in the TC calculations: UpC −UpA = 0. In the TC calculations, the maximum angular momentum of the p−n subsystem was taken as Jmax = 5. In order to be consistent between the DWIA and TC formalisms, the double differential cross section in energy and angle was produced for the free p−nreaction with the restriction Jmax = 5 and read in the DWIA calculation. Given the small contribution of higher angular momenta to the free p−ncross section, it is expected that the contributions left out are small and will not modify the conclusions of the benchmark. As described in Sec. 2.5.1, we uncouple the states with different p−n angular momentum and parity. Given the uncertainties in the cross sections due to the choice of potentials and the limits of the calculations, we deem this approximation to be well justified. Different binning meshes for the outgoing channels have been taken for the calculations for different angular momenta and parities of the final p−n system. For the angular momenta and parities jπ= 0,1,2±the bins are evenly spaced in energy with a width ∆E= 10 MeV for a p−nenergy between 30 and 330 MeV. For jπ= 3±, the width was increased to ∆E= 15 MeV while for jπ= 4,5±it was taken as ∆E= 25 MeV. The reason for the increase in the width is twofold: first, the majority of the cross section was found to correspond to jπ= 1,2±, so a smaller width should be taken for these states in order to have a better description of these states, which 68 Chapter 3. Benchmark with other reaction formalisms inary part with Woods-Saxon and Woods-Saxon-derivative components: V(r) = −V0 1 + e r−r0A1/3 a0−iW0 1 + e r−r0A1/3 a0−4iWs e r−r0sA1/3 a0s 1 + e r−r0sA1/3 a0s2,(3.2) where Ais taken as 14, corresponding to the core. From the cases presented V0(MeV) W0(MeV) r0(fm) a0(fm) Ws(MeV) r0s(fm) a0s(fm) Vp−14C13.121 13.115 1.136 0.676 0.195 1.304 0.526 Vn−14C16.161 11.044 1.136 0.676 0.138 1.304 0.542 Vp−15C13.345 13.121 1.140 0.676 0.205 1.303 0.527 Table 3.1: Parameters for the König-Delaroche potentials used in the calculation. in [57] we have chosen the one explored in the previous benchmark: a neutron removed from a 2sstate with separation energy Sn= 1.22 MeV. We present the transversal momentum pxdistribution in Fig. 3.5. In the figure, in the left panel, the Faddeev/AGS calculation taken from [57] is presented in a blue solid line, while two Transfer to the Continuum calculations are presented: the red dot-dashed line corresponds to a calculation without relativistic corrections while the green dashed line includes them. It can be seen that the agreement between the Faddeev/AGS and non-relativistic Transfer to the Continuum calculation is excellent. Since the Faddev/AGS calculation was computed non-relativistically we conclude that Transfer to the Continuum and FaddeeV/AGS give fully consistent results. It must be remarked that the Vpn interaction used in both calculations is different: the Faddeev/AGS calculation employs CD-Bonn interaction [77] while the Transfer to the Continuum uses Reid93, as mentioned above. In the elaboration of this study, we were not able to implement CD-Bonn potential in our reaction code, so a comparison using the same Vpn interaction was not feasible. However, we must note that both potentials reproduce experimental p−nphaseshifts up to energies of hundreds of MeV, so they present the same on-shell behaviour and should yield very similar observables in the (p, pn)reaction. In order to isolate the effect of the Vpn interaction, in the right panel a “plane-wave” calculation is presented where all the distorting potentials have been set to 0, which can be compared to a similar calculation shown in [57]. The agreement is still excellent, from where we conclude that both interactions 3.1. 15C(p, pn)14C at 420 MeV/A 69 -200 -100 0 100 200 px (MeV/c) 0 0.1 0.2 0.3 0.4 dσ/dpx (mb/(MeV/c)) DW -200 -100 0 100 200 px (MeV/c) 0 0.1 0.2 0.3 0.4 0.5 dσ/dpx (mb/(MeV/c)) TC FAGS TC (NR) PW Figure 3.5: Transversal momentum distribution for 15C(p, pn)14C at 420 MeV proton energy computed using the Faddeev/AGS formalism through non-relativistic equations and using the Transfer to the Continuum formalism with (green dashed line) and without (red dot-dashed line) relativistic corrections. The left panel shows the full calculation while the right one shows the “plane-wave” calculation where all distorting potentials are set to 0. seem to result in the same cross sections, which should not be surprising since both reproduce the same free p−nobservables. We must mention that the inclusion of the relativistic prescription results in a marked increase in the cross section computed for the Transfer to the Continuum formalism, of around 30%, which corresponds roughly to the factor γ2(see Eq. (2.47)) when following the prescription from [69]. We conclude from here that relativistic corrections can severely affect the cross sections obtained and therefore must be taken into account when trying to obtain information from comparing theoretical calculations to experimental data. 70 Chapter 3. Benchmark with other reaction formalisms 3.2 11Be(p, pn)10Be at 200 MeV/A In this case we perform non-relativistic calculations, since the Faddeev/AGS implementation is non-relativistic. As such we have chosen a lower proton energy (200 MeV) so as not to be so influenced by relativity. This energy is also chosen because it is the upper limit for the prescription used for the distorting potentials: Köning-Delaroche’s [76]. The Faddeev/AGS calculations shown in this section were performed by Dr. A. Deltuva, to whom the author of this thesis is greatly thankful. 3.2.1 Inputs and details for the calculation In this calculation all excitations of 10Be will be ignored. In reality the ground state of 11Be has a finite overlap with the excited state 10Be(2+)[78], but for the sake of simplicity we will assume a ground state for 11Be with only one channel: 10Be(0+). As in the previous benchmark, the bound state for the neutron has been generated with a Woods-Saxon potential, which in this case has the following geometry: R0= 1.39×101/3fm, a0= 0.52 fm. The depth of the potential is adjusted to obtain the eigenstate at the neutron separation energy Sn= 0.5 MeV. For the distorting potentials p−10Be, n−10Be the KD parametrization has been used, selecting a nucleon energy of 200 MeV for its evaluation. In the Faddeev/AGS calculation, for the n−10Be partial wave supporting the initial bound state, the potential must be fixed to that generating the bound state. In that respect, there is a difference between the Faddeev/AGS calculation and the Transfer to the Continuum one. In the Faddeev/AGS calculation all partial waves but that supporting the initial bound state are assigned the KD potential, while the one supporting the bound state is assigned the binding Woods-Saxon potential. In the Transfer to the Continuum one, the final states do not have a defined n−10Be configuration, so different potentials cannot be assigned to different configurations. As such, in the coupled-channel calculation, the n−10Be potential is always taken from KD, and the binding potential is only used to generate the bound neutron wavefunction. Another difference arises in the treatment of the p+11Be potential. As 3.2. 11Be(p, pn)10Be at 200 MeV/A 71 mentioned above, the Faddeev/AGS formalism does not require this potential as an input, since it only uses binary interactions between p, n,10Be, while in TC this potential is involved in the calculation of the distorted wave for the incident channel (Eq. (2.14)). For the TC calculation, two choices for this potential are explored: the first computes this potential by folding Vpn and Vp10Be with the bound neutron wavefunction in order to produce Vp11Be Ufold p11Be(R) = Zdrn10Be |ϕ11Be(rn10Be)|2Vpn(rpn) + Up10Be(rp10Be).(3.3) The second takes Vp11Be from KD parametrization at 200 MeV. We note that, when the final state wavefunction is the exact solution of Schrödinger’s equation the choice of Vp11Be is arbitrary and does not modify the transition matrix. As such, if our approximation for the final wavefunction Ψbin bis a good representation of the exact solution for the region of space explored by the reaction, we expect the choice of Vp11Be not to modify much the reaction observables. For Vpn, a Gaussian interaction is assumed [61]. V=−V0e−r2 a2 0.(3.4) The Gaussian interaction is taken as attractive for the waves with even p−n orbital angular momentum and as repulsive for those with odd p−norbital angular momentum. This choice was taken for simplicity and because the Reid93 potential was not implemented in the Faddeev/AGS program at the moment of this benchmark. All distorting potentials as well as Vpn are presented in Table 3.2. The potentials are taken as spin-independent, which allows us to take all involved particles as spinless. Coulomb interaction is also ignored, as it is expected to influence little the reaction observables. This reduces the computational cost of the calculations, and in particular allows all partial waves for the outgoing p−nsystem to be coupled in the Transfer to the Continuum calculation, in contrast to the previous benchmark. In the calculations, pand nare assumed to have a mass of 1 average nuclear mass 1 a.n.m.= 1.00797 amu, while 10Be and 11Be masses are taken as 10 and 11 a.n.m. respectively. The maximum orbital angular momentum considered for the p−npair is taken as lmax = 3, 72 Chapter 3. Benchmark with other reaction formalisms V0(MeV) W0(MeV) r0(fm) a0(fm) Ws(MeV) r0s(fm) a0s(fm) Vp−10Be 13.445 13.095 1.116 0.676 0.204 1.308 0.524 Vn−10Be 15.970 10.997 1.116 0.676 0.128 1.308 0.543 Vp−11Be 13.740 13.100 1.122 0.676 0.217 1.307 0.524 Vpn leven 72.15 1.484 lodd -72.15 1.484 Table 3.2: Parameters for the potentials used in the calculation. All potentials are taken from [76] but Vpn, which is based on the parametrization from [61]. since most of the cross section is expected to be contained in the considered partial waves and higher lmake the calculation both significantly heavier and less stable. The maximum total angular momentum for the three-body system is taken as Jmax = 60. Test calculations with Jmax = 70 showed negligible differences. In the Transfer to the Continuum calculations the binning mesh used is the following: bins are computed evenly spaced in energy with a width of ∆E= 5 MeV. For lpn = 0,1the mesh is extended from 0 to 160 MeV, while for lpn = 2,3the mesh is taken from 20 to 160 MeV. A test calculation with ∆E= 2.5MeV shows a change in the cross section of ∼0.6%, so we consider the calculation with ∆E= 5 MeV to be sufficiently converged. Let us remark that in contrast to the previous calculation, the remnant term Vp10Be −Vp11Be is kept in the calculations, since the “recoil” effect introduced by this term is also accounted for by the Faddeev/AGS formalism. 3.2.2 Results In this section the results for the benchmark are presented. Two different cases are studied in this benchmark: in the first one, the neutron is assumed to be removed from a 2sstate, corresponding to the component with highest population probability for the halo neutron of 11Be. The second assumes a neutron extracted from a 1pstate. Both calculations are performed assuming the same neutron separation energy Sn= 0.5MeV. This allows us to explore the dependence of the agreement between both formalisms on the quantum numbers of the removed nucleon, which have a fundamental influence on the shape of the observables, in particular of the momentum distributions. The observables we have chosen to present are the following, all referring to the outgoing 10Be core: in the top left, the transversal momentum 3.2. 11Be(p, pn)10Be at 200 MeV/A 73 distribution: dσ dpx , in the top right, the longitudinal momentum distribution measured in the center of mass system: dσ dpz , in the bottom left, the energy distribution measured in the center of mass: dσ dE10Be and in the bottom right, the angular differential cross section in the center of mass: dσ dΩ. The results of the benchmark assuming a neutron in the 2sstate are presented in Fig. 3.6. For each calculation, two TC calculations are presented which differ in the choice of Vp11Be. In Fig. 3.6, the blue dashed line corresponds to the -200 -100 0 100 200 px (MeV/c) 0 0.2 0.4 0.6 dσ/dpx (mb/(MeV/c)) 200 400 600 800 pz (MeV/c) 0 0.2 0.4 0.6 0.8 dσ/dpz (mb/(MeV/c)) TC FOLD σ=38mb TC KD σ=38 mb Faddeev σ=40 mb 0 10 20 30 E10Be c.m. (MeV) 0 5 10 15 dσ/dE10Be c.m. (mb/MeV) 0510 15 20 θc.m. (deg) 0 1000 2000 3000 4000 dσ/dΩ (mb/srad) a) b) c) d) Figure 3.6: Benchmark calculation for removal of a 2sneutron. The blue dashed line corresponds to the FaddeeV/AGS calculation, the black solid line corresponds to a TC calculation with Vp11Be obtained through folding as described in Sec. 3.2.1 and the red dotted line corresponds to TC with Vp11Be from KD parametrization. The oservables presented all refer to the outgoing core 10Be in the c.m. frame and are as follows: a) transversal momentum distribution, b) longitudinal momentum distribution, c) energy distribution, d) angular distribution. Faddeev/AGS calculation while two Transfer to the Continuum calculations are presented: the black solid line corresponds to a calculation where Vp11Be 74 Chapter 3. Benchmark with other reaction formalisms is obtained from the folding procedure mentioned in Sec. 3.2.1, while the red dotted line corresponds to a calculation where Vp11Be is taken from the KD parametrization. It must be noted that the cross section leading to the deuteron, 11Be(p, d)10Be, has been included in the Transfer to the Continuum calculations but not in the Faddeev/AGS ones. The cross section for this channel only amounts to 0.2 mb, so we find this difference not to be essential for the benchmark. In the longitudinal momentum and 10Be energy distributions the deuteron contribution can be seen at the high energy and momenta and has been given an artificial width of 1 MeV in order to emulate the effects of finite experimental energy and momentum resolution and to avoid an infinite peak in the cross section. We can see that the agreement is remarkably good for all observables except for the angular differential cross section at small scattering angles, where the Faddeev/AGS result falls under both Transfer to the Continuum calculations, which agree rather well between themselves. We note that due to the solid angle factor sin(θ), these small angles contribute little to the cross section, which explains why this difference is not so apparent in the rest of the observables. The integrated cross section is larger for the Faddeev/AGS calculation, as can be seen in the legend of the figure. Closer inspection shows the Faddeev/AGS calculation to give a slightly bigger peak in the energy distribution and a somewhat greater contribution in the low momentum tail in the longitudinal momentum distribution. It is difficult to elucidate a physical reason for these differences, although it could be argued that the in the Transfer to the Continuum, the use of the KD potential for all partial waves may introduce a small extra source of absorption, when compared to the Faddeev/AGS calculation, which uses the real binding potential for the l= 0 partial wave. With these cautionary notes, we may conclude that the agreement between both formalisms, which use slightly different inputs and may not give a priori compatible results, is rather satisfactory. We also note that the Transfer to the Continuum observables is remarkably similar for both choices of Vp11Be. This can serve as an indication that our approximation for the final three-body wavefunction is reasonable in these reactions, since formally the exact solution is independent of the choice of Vp11Be. In Fig. 3.7, the same observables as in Fig. 3.6 are presented for the calculation where the neutron is removed from a 1porbital. In general, 3.2. 11Be(p, pn)10Be at 200 MeV/A 75 -400 -200 0 200 400 px (MeV/c) 0.05 0.1 0.15 0.2 0.25 dσ/dpx (mb/(MeV/c)) 200 400 600 pz (MeV/c) 0 0.05 0.1 0.15 0.2 0.25 dσ/dpz (mb/(MeV/c)) 0 10 20 30 E10Be c.m. (MeV) 0 1 2 3 4 5 dσ/dE10Be c.m. (mb/MeV) 0510 15 20 25 30 θc.m.(deg) 0 100 200 300 400 500 600 dσ/dΩ (mb/srad) TC FOLD σ=29 mb TC KD σ=28mb Faddeev σ=30 mb Figure 3.7: As Fig. 3.6 but for removal of a 1pneutron. the same features are observed as in the case with removal from the 2s orbital. All calculations give a rather good agreement, with the Transfer to the Continuum calculations giving a cross section which is somewhat smaller than that of Faddeev/AGS ones. As before the deuteron contribution is included in the Transfer to the Continuum calculations but not in the Faddeev/AGS ones. In this case the contribution is even smaller, of 0.04 mb, so this difference should not be too relevant for the comparison. The difference in the angular distribution is reduced in this case, although this is likely due to the reduction of the contribution of the lower angles, which tend to be more unstable and sensitive to the intricacies of the calculation. Although the differences are small, the Transfer to the Continuum calculations seem to differ somewhat more in this case when compared to the 2sone. The differences are however too small to be given physical meaning, since they may well be simply a result of the numerical calculation. Let us also remark that once the final wavefunction is not the exact solution of Schrödinger equation, the choice of Vp11Be may influence the cross section, and it is difficult to relate the dependence of the observables on it with the adequacy of the approximate wavefunction, beyond a merely qualitative 76 Chapter 3. Benchmark with other reaction formalisms discussion. Both Transfer to the Continuum calculations seem to give slightly narrower momentum distributions than the Faddeev/AGS calculation. We must remark that, in contrast to what was found in the benchmark with DWIA presented in Sec. 3.1, this is not related to the energy dependence of the potential, which in this case is fixed at a certain energy in both Faddeev/AGS and Transfer to the Continuum calculations. Again the difference is too small to try and give physical meaning to it. Calculations with Reid93 We note that the Gaussian interaction used up until now in the comparison with the Faddeev/AGS formalism is adjusted to describe well the binding energy and mean square radius of the deuteron and the low-energy phase shifts for the p−nsubsystem in the 3S1wave. However, it does not produce the well known dwave component of the deuteron, nor does it reproduce the phase shifts for the other waves of the p−nsubsystem. For these reasons we find it suitable to perform some other benchmark calculations using the Reid93 nucleon-nucleon potential, in order to check whether the consistency found in the previous calculations is maintained when introducing the new NN interaction. Given that the Reid93 interaction is spin-dependent we no longer can remove the spins of proton and nucleon. Instead of performing the calculation for final states with l= 0 −3relative angular momentum between proton and neutron, we have opted to consider the states with total angular momentum between them of Jπ= 0 −3±. Since the number of final states increases by a factor of 4 due to the spins of both proton and neutron, it is no longer feasible to perform the calculation coupling all final states. Following the method taken for the other calculations in this work, we uncouple states with different total angular momentum and parity jπ, computing independently the cross section leading to each set of states. The results are presented in Fig. 3.8 assuming a neutron removed from as1/2state and in Fig. 3.10 from a p1/2state. As can be seen from the figures both TC and Faddeev/AGS calculations keep showing very similar shapes for all the distributions presented, again the TC ones being slightly narrower than the ones from Faddeev/AGS. However, the TC distribution 3.2. 11Be(p, pn)10Be at 200 MeV/A 77 -400 -200 0 200 400 px (MeV/c) 0 0.1 0.2 0.3 0.4 0.5 dσ/dpx (mb/(MeV/c)) 0 200 400 600 800 pz (MeV/c) 0 0.1 0.2 0.3 0.4 0.5 dσ/dpz (mb/(MeV/c)) Faddeev σ=29.6 mb TC Fold σ=26.9 mb TC KD σ=26.8 mb 0 10 20 30 40 E10Be c.m. (MeV) 0 5 10 dσ/dE10Be c.m. (mb/MeV) 0510 15 20 θc.m. (deg) 0 1000 2000 3000 dσ/dΩ (mb/srad) Figure 3.8: As Fig. 3.6 taking the Reid93 as NN interaction, assuming a neutron removed from the 2s1/2wave. is noticeably smaller than Faddeev’s, with a reduction of ∼8%. Given the excellent agreement found in the study employing the Gaussian potential, this difference is quite intriguing. In order to try to clarify the origin of this difference and to check the implementation of the Reid93 interaction, we have performed a comparison of our plane-wave calculations for the case with the neutron removed from the pwave, whose results are shown in Fig. 3.10. For the plane-wave calculation we find that the agreement between Faddeev/AGS and TC is once more excellent. Since in this calculation the only relevant potential is the NN interaction, it should be most sensitive to the intricacies of the Reid93 interaction. The agreement of the plane-wave calculation thus indicates a compatible behaviour of the Reid93 interaction in the Faddeev/AGS and TC formalisms. The origin of the difference between both formalisms in Figs. 3.8 and 3.10 is therefore not found (only) in the Reid93 interaction and has not been clarified by the time of composition of this work. One possible cause is the effect of uncoupling the final states with different jπfor the final p−n subsystem, which could, in principle, affect the total cross section. The 84 Chapter 4. Comparison with experimental data Ex(MeV) Jπσexp C2S σCDCC sp σCDCC th RCDCC sσTC sp σTC th RTC s 0.21 1/2+11(2) 0.65 28 21 0.52(10) 21.8 15.9 0.69(13) 0.33 5/2+43(5) 2.80 18 57 0.75(9) 17.4 54.6 0.79(9) Table 4.1: Cross section and “quenching factors” Rsfor 18C(p, pn)17C∗obtained from CDCC [79] and TC calculations. The first and second columns indicate the energy and the angular momentum and parity of the state, respectively, while the third column shows the experimental cross section. The fourth column shows the spectroscopic factor obtained with the WBP interaction for each state. The fifth, sixth and seventh columns show the CDCC single particle cross section, theoretical cross section and “quenching factor”, respectively. The eight, ninth and tenth columns do the same for TC values. factor with the necessary isospin factor, Adenotes as usual the mass number of the parent nucleus and Nis the shell main quantum number for the orbital from which the nucleon is removed. We must note here that the A A−1Nfactor has been strictly derived for the Independent Particle Model in a harmonic oscillator basis [82] and its use can be questionable in the present case, where we use shell-model SF. Noting this caveat, we include this factor nonetheless, following previous authors [32] with whose results we intend to compare. We have computed the cross section for the 1/2+state at Ex=0.21 MeV assuming removal from the 2s1/2level and for the 5/2+state at Ex=0.33 MeV assuming removal from the 1d5/2level. The results for the total cross section, as well as for the “quenching factors” Rs=σexp/σth are presented in Table 4.1. Results from the CDCC calculation in [79] are presented as well for comparison. Transversal momentum distributions are presented in Fig. 4.2. As can be seen in Fig. 4.2, the agreement in shape between the TC calculation and experimental data, after convolution with the experimental resolution (FWHM=71 MeV), is very good for both excited states of 17C: 1/2+in the left panel and 5/2+in the right one. In the table we can see that the “quenching factors” Rsobtained in the TC calculation are larger than those obtained with the CDCC calculation. They are also more consistent between themselves, both agreeing within error (which was computed propagating the error from the experimental cross section) while this is not the case for the CDCC calculation. However, a very interesting feature appearing in our calculation is the 4.1. Experimental results for low- and medium-energy experiments 85 -300 -200 -100 0 100 200 300 px (MeV/c) 0 0.02 0.04 0.06 0.08 dσ/dpx (mb/(MeV/c)) s1/2 Ex=0.21 MeV -300 -200 -100 0 100 200 300 px (MeV/c) 0 0.05 0.1 0.15 0.2 dσ/dpx (mb/(MeV/c)) Exp. data (Kondo et al) TC TC no (p,d) d5/2 Ex=0.33 MeV Figure 4.2: Transversal pxmomentum distributions for the 18C(p, pn)17C∗ reaction. Experimental data are from [79]. The left panel corresponds to (p, pn)leading to the Ex= 0.21 MeV 17C excited state, while the right panel corresponds to the Ex= 0.33 MeV excited state. The blue solid line corresponds to the full calculation rescaled to give the experimental total cross section. The blue dashed line corresponds to the calculation removing the contribution from the (p, d)transfer reaction, rescaled by the same factor as the full calculation. All theoretical calculations have been convoluted with the experimental resolution. contribution of the (p, d)reaction channel. We must note that experimentally, only the 17C core was detected, so the contribution of the (p, d)channel, if significant would be incorporated to the experimental observables. For the 1/2+single-particle cross section, (p, d)contributes with 0.83 mb, which amounts only to 3% of the cross section, being rather small. However, for the 5/2+its contribution rises to 2.61 mb, 15% of the single particle cross section. In order to check whether this strong contribution of the (p, d)channel is an artifact of the calculation we have performed simple DWBA calculations for the 18C(p, pn)17C∗(Ex= 0.33)MeV reaction. A calculation using An-Cai [83] potential between dand 17C, which is valid for this range of masses and 86 Chapter 4. Comparison with experimental data energy, and the same JLM prescription of the TC calculation for the p−18C potential yields a transfer cross section of 2.18 mb, while a calculation using An-Cai for d−17C and Köning-Delaroche for p−18C yields 2.48 mb. The variation of the values with the potential is expected and known for transfer calculations [40]. However, overall value of the cross section is coherent and consistent with that found in TC. From this, we conclude that the contribution of (p, d)transfer reaction to the total cross section is indeed important at these energies. Therefore, we must discourage the use of formalisms which do not include this channel, such as CDCC, for reactions such as this, in which this channel contributes strongly, in favour of those which include it, such as TC. 4.1.2 20C(p, pn)19C at 40 MeV/A In this section, we analyze the reaction 20C(p, pn)19C using the Transfer to the Continuum formalism. This reaction was measured at the RI Beam Factory (Japan) and the experimental data published in [84]. This reaction is of particular interest because it populates the bound states of 19C, whose lowenergy spectrum is still under debate. As can be seen in Fig. 4.3, experimental evidence is conflicting on the existence of a bound 5/2+state, with results from [85, 86] favouring a bound 5/2+state while measurements from [87–89] suggest the state is rather an unbound resonance. Also, a reanalysis of the data from [85] presented in [90] shows them to be compatible with the 5/2+ state being unbound. The 20C(p, pn)19C reaction can be used to shed some light on the existence of a 5/2+bound state, since if it does exist, it should be populated in the reaction, and rather strongly, as will be shown below. In a similar way to the analysis in [84], we base our calculations in the level scheme resulting from a shell-model calculation using the effective nucleon-nucleon WBP interaction [15], taking from it the spectroscopic factors and excitation energies of the corresponding levels. The latter are used to compute the effective separation energy S∗ n=Sn+Exwith Sn= 2.9MeV. The bound states for the different states are computed using Woods-Saxon potentials with a depth adjusted to give the effective separation energy and a reduced radius adjusted to give the mean square radius of a Hartree-Fock calculation using the SkX interaction [93]. In this calculation the 1d3/2configuration was found to be unbound, so for it the value of the reduced radius from the 1d5/2 4.1. Experimental results for low- and medium-energy experiments 87 Figure 4.3: Experimental and theoretical levels for 19C. The level scheme from a) has been constructed from the results in [85, 86] while that in b) is based on the results from [87, 89, 91]. The theoretical results have been obtained from a shell model calculation using the WBP interaction [15] and from a P-AMD model [92]. Note that the first 5/2+state is either bound or unbound for different experiments and models. Figure taken from [90]. calculation has been considered. Details on the calculation are presented in Table 4.2. The optical potentials between p, n and 19,20C have been computed as in the previous section, through a folding procedure between the JLM effective interaction [80] and Hartree-Fock densities. Due to the low beam energy, the coupling between p−nstates with different angular momentum and parity becomes relevant. From the results of the previous section, which show that states with Jπ≥3do not contribute much to the cross section, we have restricted the calculations to states with Jπ<3, which are all coupled. The results of the calculations are shown in Fig. 4.4. The single-particle cross sections are shown in the dot-dashed lines, blue, red and green for the states 1/2+,3/2+and 5/2+, respectively. The orange solid line corresponds to the sum of the contributions for the 1/2+and 3/2+states, following Eq. 4.1, while the dark green solid line corresponds to the sum of the contributions 88 Chapter 4. Comparison with experimental data to the three considered states. As can be seen in Fig. 4.4, both results overestimate experimental data, though it is evident that the inclusion of the 5/2+increases greatly the cross section separating it much more from experimental data. In the right panel, the calculations excluding and including the 5/2+state have been rescaled to give the experimental cross section. The rescaling factor for the calculation excluding the 5/2+state is rescaled by a factor 0.59(22) while the one including it is rescaled by a factor 0.20(7), which we consider to be too small to be credible. It is notable that despite the different shape shown by both calculations, after rescaling both reproduce relatively well experimental data, due to the large error bars for these data. We take these results as support for the models where the 5/2+is unbound, since its inclusion worsens the agreement with experimental data, leading to a too small rescaling factor. The method employed here, which takes the spectroscopic factors from a shell model calculation, but allows for a change in the energies of the corresponding states may be put into question. However, we note that in shell model calculations spectroscopic factors tend to be more stable than the energies of the levels, which are more sensitive to the interactions and model spaces used in the calculation. We have repeated the nuclear structure calculation using the WBT [15] interaction. This has switched the positions of the 1/2+and 5/2+states, leading to a 5/2+ground state for 19C. However, the spectroscopic factors have shown changes of less than 15%, being 1.053, 3.946 and 0.202 for 1/2+,5/2+and 3/2+states respectively, which is in agreement with the previous interpretation. JπEx(MeV) C2S r0(fm) σsp (mb) 1/2+0.00 1.099 0.905 27 5/2+0.19 3.649 1.145 19 5/2+0.62 0.247 1.145 18 Table 4.2: Results for the considered levels of 19C. The first, second and third column denote the angular momentum and parity, excitation energy and spectroscopic factors for the states obtained using the WBP interaction respectively. The fourth column indicates the reduced radius used for the computation of the bound state wavefunction. The last column shows the single particle cross section to each state. 4.2. Comparison with R3B data 89 -200 0 200 pz(rfp) (MeV) 0 0.1 0.2 0.3 0.4 dσ/dpz (mb/MeV) -200 0 200 pz(rfp) (MeV) 0 0.1 0.2 0.3 0.4 Ozawa et al (2011) 1/2++3/2+ 1/2++3/2++5/2+ 1/2++3/2+ Rs=0.59 1/2++3/2++5/2+ Rs=0.20 1/2+ SF=1 3/2+ SF=1 5/2+ SF=1 Figure 4.4: Experimental data and Transfer to the Continuum calculations for the 20C(p, pn)19C reaction. The left panel shows the single-particle cross sections for the states 1/2+,3/2+and 5/2+in the blue, red and green dashdotted lines, respectively, while the orange and dark green solid lines denote the total contribution for states 1/2+and 3/2+and 1/2+,3/2+and 5/2+ respectively. In the right panel the orange and dark green lines have been rescaled to reproduce the experimental cross section, with the result shown in the dashed lines and the rescaling factor appearing in the legend 4.2 Comparison with R3B data As mentioned in Chapter 1, the R3Bcollaboration has very recently performed a campaign of (p, pn)and (p, 2p)reactions at energies of 300-500 MeV, whose fist results have only very recently been published [23, 42, 43]. The results published correspond to carbon, nitrogen and oxygen isotopes and their analysis with the Transfer to the Continuum formalism is presented in the following, with special interest in the dependence of the found “quenching factors” on the isospin asymmetry of the nuclei. 90 Chapter 4. Comparison with experimental data 4.2.1 Choice of optical potentials Fundamental ingredients in the calculations presented in this document are the optical potentials used to describe the distortion and absorption suffered by the incoming proton and the outgoing nucleons due to their interaction with the rest of the nucleus. In general we have chosen two parametrizations to produce the optical potentials, one that can be considered more “microscopic” and one more “phenomenological”. 1. Paris-Hamburg (PH) The first interaction we consider is the “microscopic” one. It is built through the folding of an effective nucleon-nucleon interaction with the nuclear densities of the involved nuclei. For the effective nucleonnucleon interaction we have chosen the Paris-Hamburg g-matrix effective interaction [94, 95], which includes nuclear medium effects (in particular, Pauli blocking) and has shown to give a good description of proton-nucleus elastic scattering for energies from 100 to 400 MeV, on various nuclei such as 12C and 208Pb. The nuclear densities are obtained from Hartree-Fock calculations based on the SkX Skyrme interaction [93], using the subroutine dens as before. The folding is performed with the code lea [96]. From now on this potential will be referred to as PH potential. 2. Dirac The other “phenomenological” potential has already been briefly introduced in Sec. 3.1[45, 46]. It is based on the application of the Dirac equation to describe the dynamics of the proton and has been fitted to a vast number of experimental data on elastic proton-nucleus scattering ranging from 12C to 208Pb and for energies from 20 to 1040 MeV. From the multiple parametrizations presented, we have chosen the EDAD2 one (Energy-Dependent A-Dependent), which provides potentials for nuclei beyond those used for the fitting and is energy-dependent, thus giving a better fit to experimental data. Even though the potentials are originally produced for the Dirac equation, they can be modified to be suitable for the Schrödinger equation we use in our calculations. From now on, we will refer to this potentials as Dirac potentials. 4.2. Comparison with R3B data 91 It must be remarked that in order to give meaningful results, the production of the potentials must be consistent with the equation to be solved. In particular, the relativistic prescription used in this work (the substitution of the reduced mass µby the reduced energy and the momentum by its relativistic value, see Sec. 2.3) has to be consistent with the codes used to generate the potential. The code LEA indeed generates potentials assuming this prescription and the program we have used to generate our Dirac potentials has an option to consider this prescription as well. As in Sec. 3.1, a choice must be made for the outgoing proton-core and neutron-core potentials. In our formalism, the energy between outgoing nucleon and core is not well defined for the final states considered, so it is difficult to assign an energy at which to evaluate the potentials. Using the same criteria as in Sec. 3.1 we have evaluated the potentials for the outgoing channels at half the incident energy, since the maximum of the cross section is expected at this nucleon-nucleus energy. Based on the results presented in Fig. 3.4, we expect the effects of the energy dependence of the potential (which we do not include) to be relatively minor, although we note that we have only tested them in one case, at high energies and corresponding to removal from an swave, so the extension of these results to other cases can be put into question and would deserve further study. As a summary, in Eq. (2.37), the potentials used to generate Uββ0 [Eq. (2.36)] are evaluated at half the incident energy, while the potential UpA is evaluated at the incident energy. For the potential UpC in the third term of Eq. (2.37) (the reaction vertex), we have chosen to evaluate it at the incident energy, since it is to be applied on XA, the wavefunction in the incoming channel. This choice also allows to minimize the so-called remnant term,UpC −UpA. In Fig. 4.5 both PH and Dirac potentials are presented between proton and 16O at three different energies: 400, 200 and 100 MeV. As can be seen in the figure, at higher energies potentials become highly absorptive. This results in the global (p, pN)corresponding to only a few percent of the full reaction cross section. It also implies that (p, pN)are not very sensitive to the nuclear interior, although it can be argued that they are not as peripheral as transfer or knockout reactions with heavier targets [52]. They also become repulsive at small distances. However, due to the strong absorption, the features of the real potential in the nuclear interior are not reflected much in 92 Chapter 4. Comparison with experimental data 0246 R (fm) -40 -30 -20 -10 0 10 20 V (MeV) 400 MeV 0 2 4 6 R (fm) -40 -30 -20 -10 0 10 20 200 MeV 0 2 4 6 R (fm) -40 -30 -20 -10 0 10 20 Dirac Real Dirac Imag PH Real PH Imag 100 MeV Vp16O Figure 4.5: Proton-16O potentials at three different energies using PH and Dirac parametrizations (See text). The shaded area marks the distances up to the rms radius of 16O. the reaction observables. The behaviour at small distances is rather different for both potentials, even though they have been adjusted to reproduce similar experimental data. This can be understood easily due once again to the insensitivity of the observables (in this case, elastic differential cross sections) to the nuclear interior, where the value of the potential is not relevant for the reaction. In order to test the potential sets and our implementation of them, in Fig. 4.6 the elastic differential angular cross section is computed with both potentials and compared to experimental data at energies of 200, 300 and 398 MeV [97–99]. Both calculations agree well with experimental data at small angles, but for larger angles Dirac potentials agree systematically better. This should come as no surprise, since the Dirac parametrization was obtained through fitting of very similar data, the experimental data at 200 and 398 MeV actually being used for the fitting procedure. The agreement with experimental data serves as a validation for the use of these potentials 4.2. Comparison with R3B data 93 0 20 40 60 80 θc.m.(deg) 10-3 100 dσ/dΩ (mb/srad) p+12C @ 200 MeV 10 20 30 40 θc.m. (deg) 10-2 100 102 p+12C @ 300 MeV 0 10 20 30 θc.m. (deg) 10-2 100 102 PH Dirac p+12C @ 398 MeV Figure 4.6: Angular differential cross sections for the elastic scattering between pand 12C at 200, 300 and 398 MeV incident proton energy compared to theoretical calculations using PH and Dirac potentials. Experimental data are from [97–99] for the energies of 200, 300 and 398 MeV respectively. as descriptors of the interaction between nucleon and nucleus in (p, pN), noting that elastic scattering is less sensitive to the nuclear interior than (p, pN)reactions, so that a part of the potentials that may be relevant for the latter is not being probed by this test. 4.2.2 Spectroscopic factors and binding potentials Before the comparison to experimental data of (p, pn)and (p, 2p)reactions at high energies, the parameters chosen to generate the spectroscopic factors and the binding potentials which bind the to-be-removed nucleon to the core must be described. 100 Chapter 4. Comparison with experimental data •16O(p, 2p)@ 451 MeV/A In Fig. 4.9, the transversal momentum distribution (px) is presented for 16O(p, 2p)15N at an incident energy of 451 MeV/A, for which experimental data are taken from [23]. -400 -200 0 200 400 py (MeV/c) 0 0.05 0.1 dσ/dpy (mb/(MeV/c)) Atar et al. Dirac PH Figure 4.9: Transversal pymomentum distribution for 16O(p, 2p). Experimental data are from [23]. Calculations are presented using PH and Dirac calculations (black solid and red dashed lines respectively) rescaled to reproduce the experimental cross section. The same procedure as for 12C(p, 2p)has been followed to generate the theoretical cross sections. The waves p1/2and p3/2have been considered for the removed proton, with SF 2.00 and 4.09 respectively. The Dirac and PH calculations have been multiplied by the “quenching factor” Rs 0.74 and 0.78 respectively. A similar result is found as for 12C(p, 2p). Both PH and Dirac calculations present similar shapes, both of which agree reasonably well with the experimental data, though they are somewhat narrower than experiment. In this case, it is the PH calculation the one that is slightly narrower than the Dirac one, as opposed to the previous case. As in 12C, the shape of the single-particle momentum distributions is very similar for removal from p1/2and p3/2orbitals. •22O(p, pn)@ 414 MeV/A In this section the transversal pymomentum distribution for 22O(p, pn) is presented together with the experimental data from [43] in Fig. 4.10. Theoretical calculations have been obtained through the same proce- 4.2. Comparison with R3B data 101 dure as the previous cases, though since in this case experimental data are given in arbitrary units results have been rescaled to reproduce the integral of experimental data, instead of by the Rsfactor. -400 -200 0 200 400 py (MeV/c) 0 0.1 0.2 0.3 dσ/dpy (arb. units) Díaz-Fernández et al. Dirac s1/2 Dirac d5/2 PH s1/2 PH d5/2 Dirac PH Figure 4.10: Transversal pymomentum distribution for 22O(p, pn). Experimental data are from [43]. Calculations are presented using PH and Dirac calculations (black solid and red dashed lines respectively) rescaled to reproduce the integral of experimental data (given in arbitrary units). The dotted and dot-dashed lines correspond to the contributions of s1/2and d5/2 orbitals, having all been scaled by the same arbitrary factor, and as such conserving their relative weight. For this reaction two orbitals with different orbital angular momentum intervene, s1/2and d5/2. Both waves show momentum distributions with markedly different shapes, as can be seen in Fig. 4.10, where the dotted lines correspond to s1/2and the dot-dashed ones to d5/2. Therefore, different relative weights for both orbitals lead to different shapes of the total cross section, so its comparison to the experimental shape has spectroscopic value. Instead of trying to fit the shape of the distribution, we have represented the momentum distribution resulting from the sum of both waves scaled with the SF obtained from a shellmodel calculation, as described in 4.2.2, showing the contribution of each orbital (dotted lines for s1/2and dot-dashed ones for d5/2). Let us remark that since both PH and Dirac calculations have been rescaled to give the total integral of the data, the difference in magnitude for both calculations (with the Dirac cross section being 7% larger than 102 Chapter 4. Comparison with experimental data the PH one) is not apparent in the figure. We find a good agreement in the distribution shape for the shell-model SF, which serves as a test for these values. As with the previous cases, both Dirac and PH yield distributions with very similar shapes, which are somewhat narrower than experimental data. In general, we find a reasonably good agreement between experimental momentum distributions and our theoretical calculations, although the former seem to be systematically broader than the latter. We also find that the shape of momentum distributions is rather insensitive to the prescription for optical potentials chosen, even with the magnitude of the full cross section showing a stronger dependence on these potentials, as will be shown in the following section. A thorough comparison between the distributions produced with PH and Dirac potentials seems to show that the calculation resulting in a smaller cross section tends to produce a (slightly) narrower cross section. This would relate a stronger absorption to a narrower momentum distribution, which agrees with the qualitative description in Sec. 2.5.2. Given that our distributions are narrower than experimental ones, this could indicate an overestimation of the effect of absorption in our calculations. Another explanation for the narrowness of our distributions is the effect of energy dependence in the potentials. As was shown in Sec. 3.1, the inclusion of energy dependence for the optical potentials in DWIA calculations broadens slightly the distributions without modifying much the total magnitude. Since we are not including energy-dependence, our distributions may be narrower for this exclusion. In spite of the previous caveats, we find the agreement between our calculations and the momentum distributions to be in general good for all of the considered cases, which span nuclei with high and low nucleon binding energies, and as such, take it as a validation of the method, which will in the following be applied to study the systematics of the “quenching factors” Rs. 4.2.4 Comparison with integrated cross section. “Quenching factors” In this section we present our results for the “quenching factors” Rs, the ratios between experimental and theoretical cross sections and which have 4.2. Comparison with R3B data 103 Reaction E/A (MeV) Sp(n)(MeV) Ref Reaction E/A (MeV) Sp(n)(MeV) Ref 13O(p, 2p)401 1.51 [23] 21O(p, 2p)449 20.99 [23] 14O(p, 2p)351 4.63 [23] 21N(p, pn)417 4.59 [43] 15O(p, 2p)310 7.30 [23] 21N(p, 2p)417 19.60 [43] 16O(p, 2p)451 12.13 [23] 22O(p, pn)414 6.85 [43] 17O(p, 2p)406 13.78 [23] 22O(p, 2p)414 23.26 [43] 18O(p, 2p)368 15.94 [23] 23O(p, pn)445 2.73 [43] 12C(p, 2p)398 15.96 [42] 23O(p, 2p)445 24.71 [43] Table 4.3: Detail of the reactions considered, with the incident energy of the beam, the separation energy of the removed nucleon (proton for (p, 2p)and neutron for (p, pn)) and the reference from which experimental data have been taken. already been described in the introduction. As described in Secs. 4.2.2, only (p, pN)processes leading to states of the residual core which are below the nucleon separation energy and which give a SF larger than 0.1 are considered. Single-particle cross sections (with SF=1) have been computed using both Dirac and PH potentials in order to test the dependence of our results to the choice of potential and they are used to compute total theoretical cross section through Eq. 4.1, as in Sec. 4.1.1. Single-particle cross sections have been computed for the different states assuming a certain orbital from which the nucleon is removed. These single particle cross sections are detailed in Appendix A. When two states correspond to the same orbital and lie close in energy we have opted to use the same σsp for both states, usually taking the value computed for the state which lies lower in energy. States for which the σsp has not been computed present an asterisk in the value of σsp. Theoretical cross sections have been computed for all the cases presented in Table 4.3, which are the ones that have been published by the R3B collaboration up to the date of composition of this document. For 22O(p, 2p)and 23O(p, 2p), two different but compatible analysis were performed in [43] and [23]. We have taken the experimental cross sections from [43] in both cases. For all of these reactions, the “quenching factors” Rs=σth/σexp have been computed, using both Dirac and PH potentials. The values are shown in Table 4.4. The Rsare presented as a function of ∆S(Sp−Snfor (p, 2p) reactions and Sn−Spfor (p, pn)ones) in Fig. 4.11 in order to study their dependence on ∆Sand compare it to that found in knockout experiments with heavier targets [31, 32]. In the figure, Rsfactors from [23, 42], which 104 Chapter 4. Comparison with experimental data were obtained using an eikonal DWIA formalism [52], are also shown as well as those from [43], which were obtained from the Faddeev/AGS formalism. The errors in Rshave been propagated from the errors in the experimental cross sections. -30 -20 -10 0 10 20 30 ∆S (MeV) 0.2 0.4 0.6 0.8 1 Rs 17O23O 21O 22O 14O16O 21N 21N 22O 23O18O 12C 13O 15O Faddeev/AGS (Díaz Fernández et al.) Eikonal DWIA (Panin et al) Eikonal DWIA (Atar et al.) TC: Dirac pots. TC: PH pots. Linear fit (Dirac) Linear fit (PH) Proton removal Neutron removal Figure 4.11: “Quenching factors” Rsas a function of ∆S, the difference between proton and neutron separation energy. Results from this work are presented in black circles using PH potentials and in red squares using Dirac potentials. The brown band corresponds to the systematics found in knockout experiments [31, 32]. Rsfrom [43], [23] and [42] are presented in green triangles, blue squares and pink diamonds respectively. Despite the relatively large dispersion in the values of Rsfor both potential sets, in particular for PH potentials, it can be clearly seen that both sets of Rspresent a dependence on ∆Swhich is much milder than the one found in knockout experiments (represented with the shaded area). In order to quantify their dependence on ∆S, both sets of quenching factors have been fitted with a linear function, which is presented in Fig. 4.11 in a dashed line (black for PH and red for Dirac). We obtain from this fit a dependence such as Rs= 0.697(17) −2.5(12) ·10−3∆Sfor the calculations employing 4.2. Comparison with R3B data 105 Dirac potentials and Rs= 0.766(18) + 0.4(13) ·10−3∆Sfor the calculations with Paris-Hamburg potentials. The corresponding reduced χ2values are, 1.15 and 0.76, for Dirac and PH calculations, respectively. The very small value of the slope for both sets of calculations serves as a clear indication of the small dependence of the reduction factors on ∆S. It is rather remarkable that the slope for PH potentials is positive instead of negative. However, given that the error error of the slope is larger than its value, we can only conclude from its value that the PH calculation is compatible with a flat Rs with no dependence on ∆S. When comparing our results with those from [23] and [42] we find that the agreement is reasonably good for 14O, 16O, 12C and 17O, with our Rs being overall larger than those from [23] and [42]. However, for 21O and 23O, our Rsshow a marked disagreement with those of [23]. First, we must remark that we are using SF resulting from shell model calculations while in [23] they used the independent particle model (so that the associated SF are 2 for 21O(p, 2p)and 23O(p, 2p)). As can be seen in Table 4.4, the total SF for 23O(p, 2p)is quite close to 2, so the comparison is meaningful at this respect. This is not the case for 21O(p, 2p), where the total SF is 1.88. However, if we assume a SF of 2 to compare to the Rsfrom [23], we obtain a Rsof 0.67±0.05 with Dirac potentials and one of 0.76(6) with PH potentials, which are still incompatible to the Rs= 0.58(4) from [23]. Given that both analyses are based in different formalisms which use different inputs (in particular different optical potentials) it can be argued that the comparison between results of such different analyses is not meaningful. However, the eikonal DWIA used in [23] is formally similar to the one presented in the benchmark calculation in Sec. 3.1, where we found an excellent agreement when using the same input ingredients. As well, the agreement found for the other nuclei (14O, 16O, 12C and 17O) seems to suggest that both formalisms are reasonably compatible. We note that the nuclei for which the disagreement is larger are located to the right of Fig. 4.11, so that they correspond to the knockout of the most bound species of the nucleus. We also note that, overall, the agreement between TC calculations with PH and Dirac potentials is better to the left of the figure (for removal of less bound species) than to the right (for removal of more tightly bound nucleons). This suggests that TC (p, pn)and (p, 2p) cross sections for more deeply bound nuclei are more sensitive to the optical 106 Chapter 4. Comparison with experimental data potentials describing the interaction between nucleon and nucleus. In order to test this idea, we have chosen the 23O(p, pn)and 23O(p, 2p) reactions, which are situated to the extremes of Fig. 4.11 and we have performed a “notch test”: we have added a deep and very narrow Gaussian term (V(r) = V0e−(r/a)2V0=-50 MeV, a=0.2 fm) to the potential Vp23Oand have analyzed the sensitivity of the cross section to the position of the notch. In order to speed up calculations, this test has only been performed on the cross section leading to p−Nstates with jπ= 1−, which yield the largest cross section (See Sec. 2.5.1). The results of the test are presented in the left column of Fig. 4.12, where the black line denotes the relative change in the cross section as a function of the position of the node. 0 1 2 3 4 5 0 0.05 0.1 0.15 0.2 ∆σ1-/σ1- (mb) 0 1 2 3 4 5 Rnotch (fm) 0 0.05 0.1 0.15 0.2 ∆σ1-/σ1- (mb) Notch in Vp23O 23O density (a.u.) 0 1 2 3 4 5 6-40 -20 0 20 V (MeV) 0 1 2 3 4 5 6 R (fm) -40 -20 0 20 V (MeV) Vp23O Dirac Vp23O PH 23O(p,pn)22O s1/2 Sn=2.73 23O(p,2p)22N p1/2 Sp=24.71 MeV Figure 4.12: To the left, “notch test” performed on the 23O(p, pn)and 23O(p, 2p)reactions (see text). The thin black line corresponds to the relative variation of the cross section with the position of the notch. The density of 23O is shown in arbitrary units for comparison. To the right the Dirac and PH potentials for 23O are presented, the yellow area denoting the “area of sensitivity” which starts roughly at the position where the notch changes the cross section by ∼5%. As can be expected, when the notch is situated at small distances, the 4.2. Comparison with R3B data 107 cross section is only weakly sensitive to it. This is an indication of the peripherality of the reaction and a result of the strong absorption of the potentials. As the notch moves to larger distances, the reaction becomes more sensitive to it and the cross section varies more strongly. We note that the 23O(p, pn)reaction seems to require the notch to be at larger distances to be sensitive to it than the 23O(p, 2p)reaction: for the former the cross section is modified by ∼5% when the notch is found at ∼3.5 fm, while for the latter this modification is already found at ∼2.5 fm. In Fig. 4.12 the density of 23O is also plotted in arbitrary units. From the notch test we can see that the 23O(p, pn)reaction is clearly sensitive only to the tail region of the density distribution, while the 23O(p, 2p)one is more sensitive to the interior. This can be easily understood by noting that the wave function of a more deeply bound nucleon (such as the proton in 23O) drops exponentially faster and is more concentrated near the origin. Therefore, in order to extract the nucleon the incoming proton has to probe deeper inside the nucleus, thus being more sensitive to the features of the interaction in this deeper area. This also relates naturally to the reduction of the cross section with separation energy, a larger separation energy requires the proton to probe deeper regions where absorption is stronger. Taking (rather arbitrarily) the position of the notch for which the cross section is changed 5% as the radius of sensitivity for the reaction, we can check which areas of the Vp23Opotential are explored by the reaction. This is shown in the right column of Fig. 4.12, where starting at the radius of sensitivity, the “area of sensitivity” is presented in yellow. We can see that for the 23O(p, pn)reaction, this area corresponds to the tail of the potentials, where PH and Dirac potentials are rather similar. On the other hand, the 23O(p, 2p)is sensitive to deeper regions where potentials differ more strongly, thus leading to more different cross sections. We admit this description to be rather heuristic and phenomenological but we find it gives a qualitative and intuitive explanation for the differences found between the PH and Dirac calculations and between them and the results from [23] (where the nucleon-nucleus interactions are obtained through a very different method [52]). We now compare our results to those presented in [43] (open triangles in Fig. 4.11), finding a larger disagreement. For the negative ∆S, the difference is of about 25%, while for the positive ∆S, our Rsare about twice larger 108 Chapter 4. Comparison with experimental data than those from [43]. More importantly, their results seem to show a greater dependence on ∆Sthan ours. Given the relatively good agreement found in the benchmark calculation with the Faddeev/AGS formalism (See Sec. 3.2), this disagreement cannot be fully attributed to the difference in the used formalisms. We note three main differences between our reaction inputs and those of [43]: 1. The geometry of the binding potential: Here we use, as described in Sec. 4.2.2 a diffuseness a= 0.7fm and a radius adjusted to give the Hartree-Fock rms radius. In [43] the geometry was taken as a= 0.65 fm and r0= 1.25 fm. 2. The optical potentials: Here we use Dirac and PH potentials, while in [43] they use Köning-Delaroche (KD) potentials. 3. Relativistic corrections: Here we have used the relativistic corrections described in Sec. 2.3, while in [43] the calculation was performed nonrelativistically. In order to check whether the discrepancies in the Rsvalues stem from these differences in the analysis, we have performed a calculation where we have tried to emulate all of the mentioned features of the analysis in [43]. The result is presented in Fig. 4.13. As can be seen in the figure the agreement between our new calculation and that of [43] is improved substantially. Most notably, the ∆Sdependence of the new calculation is increased, becoming similar to that of [43]. Our Rskeep being larger than those of [43], resulting from smaller single-particle cross sections when compared to Faddeev/AGS ones. This is consistent with the results found in the benchmark calculation with the Reid93 potential in Sec. 3.2. From this comparison we conclude that the stronger ∆Sdependence found in [43] is largely related to the inputs used in the calculation. In particular we find that the use of KD potentials increases heavily the cross sections, due to their relatively small absorption when compared to PH and Dirac potentials, as can be seen in Fig. 4.14. KD potentials have been fitted for proton energies up to 200 MeV, so their use in reactions of ∼400 MeV/A can be put into question, trusting better calculations using PH and Dirac 4.2. Comparison with R3B data 109 -30 -20 -10 0 10 20 30 ∆S (MeV) 0.2 0.4 0.6 0.8 1 Rs 22O23O 21N22O 21N 23O Faddeev/AGS (Díaz Fernández et al.) TC KD NR Proton removal Neutron removal Figure 4.13: Rsfrom Faddeev/AGS calculations [43] (open green triangles) and from a TC calculation (in violet triangles) where KD potentials have been used, relativistic corrections omitted and the geometry of the binding potentials modified to r0= 1.25 fm, a= 0.65 fm, following the prescription from [43]. potentials, which have been adjusted to reproduce data in the energy range of the reactions we are studying. 116 Chapter 5. Application of (p, pN)reactions to Borromean nuclei Figure 5.1: Scheme of a (p, pN)reaction on a three-body nucleus, with the relevant coordinates involved in the TC formalism. expanding Ψbin the states defined by Eq. (5.5): Ψ3b(+)(R,y,x)≃XA(R)φA(x,y)+Ψb= XA(R)φA(x,y) + ZdqX σ2,ς ϕ(−) q,σ2,ς(x)Ψ3.b. bqσ2ς(R0,r0).(5.7) We will now impose a participant-spectator approximation, analogous to the frozen core approximation performed in Chapter 2, in which we will assume that the (p, pN)reaction only modifies the motion of the removed nucleon N1, while the state of the residual nucleus Bwill not be affected by the reaction. As in Chapter 2, this results in the decoupling of the different states of B, which in the present case implies that the states with different q,σ2and ςcan be computed independently. The potentials in the reaction are assumed to be independent of the internal variables of B, which in this case corresponds to x. As such, we can reach an equation equivalent to Eq. (2.28): Zdxϕ(−)∗ q,σ2,ς(x)φA(x,y) = ϕABq,σ2,ς (y),(5.8) where now we do not impose a normalization on ϕABq,σ2,ς . In order to have expressions with a defined total angular momentum, let us define ϕAB,LJJTMT(q, y) = X L0J0 iL0ZdxfJT LJ,L0J0(q, x) x[[YL0(ˆx)⊗χs2]J0⊗χI]∗ JTMT ×φA(x,y), (5.9) 5.1. Extension of the formalism 117 where it is rather trivial to obtain: ϕABq,σ2,ς (y) = 4π qX LJJTMT YLM (ˆq)hLMs2σ2|JMJi ×hJMJIς|JTMTiϕAB,LJJTMT(q, y). (5.10) As in Chapter 2, we expand Ψ3.b. bqσ2ςin discretized states with good angular momentum and parity of the p−N1subsystem: Ψ3b bqσ2ς≃X jπn ˜ φjπ nXjπnα,(5.11) and we can compute the transition matrix to a certain state of the p−N1: Tif q,σ2,ς,n,jπ=Dϕ(−) q,σ2,ς ˜ φjπ nXq,σ2,ς jπnα |VpN +UpB −UpA|XAφAE =D˜ φjπ nXq,σ2,ς jπnα |VpN +UpB −UpA|XAϕABq,σ2,ς E, (5.12) where in the second line, we have made use of Eq. 5.8 and Eq. 5.9 under the assumption that the potentials appearing in the matrix element are independent of x, so that they can be extracted from the integration over it. We can transform the transition matrix to: Tif q,σ2,ς,n,jπ=4π qX LJJTMT YLM (ˆq)hLMs2σ2|JMJihJMJIς|JTMTi ×D˜ φjπ nXLJJTMT jπnα |VpN +UpB −UpA|XAϕAB,LJJTMTE =4π qX α YLM (ˆq)hLMs2σ2|JMJihJMJIς|JTMTiTif α, (5.13) where α={LJJTMT, njπ}. Now we note that each of the Tif αcan be computed independently as an independent (p, pN)reaction, and therefore, the TC formalism can be applied to obtain it as described in Chapter 2. Let us now get the cross section leading to a certain final state, ϕ(−) q,σ2,ς and a certain state of the p−N1subsystem ˜ φjπ n: dσ dΩ n,jπ q,σ2,ς =1 (2sp+ 1)(2JA+ 1) µpAµpn−C (2π~2)2 Kf KpA X mTif q,σ2,ς,n,jπ 2,(5.14) where as in Chapter 2, Ωcorresponds to the scattering angle of the (center of mass of) residual nucleus B. 118 Chapter 5. Application of (p, pN)reactions to Borromean nuclei Now we can apply the decomposition: dσ dΩ n,jπ q,σ2,ς =1 (2sp+ 1)(2JA+ 1) µpAµpn−C (2π~2)2 Kf KpA 16π2 q2 ×X mp,mA,md,MTX α YLM (ˆq)hLMs2σ2|JMJihJMJIς|JTMTiTif α 2 , (5.15) where we now explicitly state the spin projections of the nuclei involved in the reaction. In order to produce a differential cross section on q, we multiply by the differential element and the number of states per momentum-space element (N(q)), which for our definition corresponds to [71]: N(q) = (2π)−3,(5.16) which we note does not depend on q, so we will denote it as Nq. This expression leads to: dσ dΩdq n,jπ σ2,ς =dσ dΩq,σ2,ς Nqd3q.(5.17) Since we are only interested in the energy distribution of the residual nucleus B, we may integrate over the angles of qand sum over the spin projections of both N2and C, that is over σ2and ς. Now we can use the properties: ZdˆqY ∗ LM (ˆq)YL00 M00 (ˆq) = δLL00 δMM00 (5.18) and X M,σ2hLMs2σ2|JMJihLMs2σ2|J00 MJi=δJJ00 (5.19) to get an expression for the cross section to a certain q dσ dΩdq n,jπ =1 (2sp+ 1)(2JA+ 1) µpAµpn−C (2π~2)2 Kf KpA 16π2 ×X mp,mA,md,MTX αTif LMJTMT 2 Nq, (5.20) 5.1. Extension of the formalism 119 which can be easily recognised as dσ dΩdq n,jπ = 16π2NqX LJJTMT dσ dΩq,α .(5.21) The cross section to a state {LJJTMT}can be computed using TC by summing over the cross sections to all the bins: dσ dΩdq= 16π2X α dσ dΩq,α Nq.(5.22) Of course, integration over Ωyields the formula for the differential cross section with the modulus of q: dσ dq= 16π2X α σ|q,α Nq.(5.23) From here, we can easily obtain the distribution for the relative energy between N2and C, by multiplying by transforming from momentum to energy through: N2C=~2q2 2µN2C dq dN2C =µN2C ~2q(5.24) leading to: dσ dN2C = 16π2µN2C ~2qNqX α σ|q,α (5.25) Here the use of the non-relativistic expression for the density of states is justified by the low relative energies between N2and C, which tend to encompass only a few MeV. Finally let us note that in the previous derivation neither the integration over Ωnor the sum over the different bins n, jπis a fundamental part of the derivation. As such it is possible to obtain a differential cross section over N2Cand the center of mass momentum of the outgoing residual nucleus B(see Section 2.2) which still verifies the property that it can be constructed through the incoherent sum of the differential cross sections for the different states {LJJTMT}. 120 Chapter 5. Application of (p, pN)reactions to Borromean nuclei 5.1.1 Comment on antisymmetrization Borromean systems in which two of the bodies are nucleons (the ones we are interested in in this work) must be of the type n+n+Cor p+p+C, since a system such as p+n+Cwould not be Borromean, since the twobody subsystem p+ncan be bound forming the deuteron. Therefore, special attention must be paid to the fact that the two nucleons are identical particles so they must be properly antisymmetrized. This antisymmetrization must be taken into account in the calculation of φA, and is most easily considered by working in the T-system, where the orbital angular momentum between the two nucleons is defined. There, the components where L+S(with S= sN1+sN2) is odd can be removed, and those where it is even multiplied by a factor √2, in a similar way to Section 2.4. We must then transform the wavefunction to the Y-system as in Eq. (5.6), since there the overlap with nucleus Bis simpler. Let us remark that in the actual calculations it is common to leave the factor √2out of the calculation, so that the final cross sections obtained must be multiplied by a factor 2. 5.2 11Li(p, pn)10Li∗ 5.2.1 Brief introduction 11Li has the honour of being the first established halo nucleus, after the experiments by Tanihata et al [102] showed it to have a interaction cross section which was much larger than other nuclei of its neighbourhood. This was interpreted as 11Li having a much larger radius, which was later understood as having two valence neutrons orbiting a 9Li core at a large radius, forming the halo. Despite the numerous experiments performed on 11Li, the structure of its ground state is not clearly established yet. It is experimentally agreed upon that the ground state of 11Li is an admixture of s2 1/2and p2 1/2[103], but the weight of each component is not clear, with predictions for the weight of the (sd)2component ranging from 0 to 100% [104], with the predictions for the s2occupation based on the matter radius of 11Li being around 30-50%. The associated unbound system 10Li presents a spectrum that has yet to be clarified as well, as can be seen from Table 5.1, which has been adapted from [104]. Currently, the general consensus is that the ground state of 10Li 5.2. 11Li(p, pn)10Li∗121 Year Reaction Energy (MeV) Width (MeV) lRef 1997 10Be(12C,12 N) 0.24(4) 0.10(7) [105] 1999 9Be(9Be,8B) 0.50(6) 0.40(6) [106] 1999 fragmentation <0.05 s[107] 2001 premoval from 11Be g.s. is s[108] 2003 9Li(d, p)0.35(11) <0.32 [109] or <0.2 and 0.77(24) – and 0.62 2006 9Li(d, p)∼0s[110] ∼0.38 ∼0.2 p 2015 2premoval from 12Be 0.11(4) 0.2 p[111] 0.50(10) 0.8 p 2016 11Li(p, d)0.62(4) 0.33(7) p[112] Table 5.1: Position of the resonances and virtual states found for 10Li in various experiments, adapted from [104] corresponds to a swave with an energy very close to threshold and that it has one or more resonances corresponding to a pwave in the range of 0.2-0.7 MeV. 5.2.2 Models for 11Li and 10Li As was indicated in Sec. 5.1, the main structure ingredient required for the computation of the (p, pn)energy distribution is the overlap between the ground state of the Borromean nucleus (11Li) and the unbound residual twobody system (10Li) at different n+9Li relative energies. In order to produce this overlap, the wave functions of both systems have to be produced, in this case, employing a 3-body model for 11Li (n+n+9Li) and a two-body one for 10Li (n+9Li). The main input ingredients for these models are the potentials between the subsystems considered. Given that in the three body system two of the particles considered are identical (the two neutrons) only two potentials are required in these calculations: Vnn and Vn9Li. For Vnn, the well tested GPT interaction [113] has been chosen, which is able to reproduce the scattering length for this system. Given that the relevant energy range between the two neutrons inside 11Li will be relatively small, Enn <5MeV, we find that a good description of the scattering length is enough to describe the n−ninteraction. The n−9Li interaction is less well established since the properties of 10Li are not that well known. As such, we have produced 5 different interactions 122 Chapter 5. Application of (p, pN)reactions to Borromean nuclei which yield different spectra for 10Li. This allows us to explore the sensitivity of the reaction observables to the characteristics of the n−9Li continuum. One main property of the n−9Li interaction we explore in this work is the spin-spin coupling between nand 9Li, which leads to splitting of the structures of the continuum (resonances and virtual states) depending on the relative orientation of the spins of both systems. The shapes for the interactions including spin-spin terms is different in the literature than that of the interactions excluding this term. As such distinctions must be made between them. The interactions excluding spin-spin coupling have been based in the parametrization of Thompson and Zhukov [114]: V(L) n9Li =V(L) c(x) + V(L) so−v(x)L·sn,(5.26) where the depth of the interactions depend on the orbital angular momentum (L) between nand 9Li, with the central term following a Woods-Saxon shape and the spin-orbit term following a Woods-Saxon derivative, both with geometries with the parameters R= 2.642 fm (note it is Rand not r0) and a= 0.67 fm. Meanwhile the interactions including spin-spin coupling are based on the parametrization of Garrido et al. [115]: V(L) n9Li =V(L) c(x) + V(L) so−v(x)L·sn+V(L) so−C(x)L·sC+V(L) ss (x)sn·sC,(5.27) presenting central, spin-spin and spin orbit terms for both nand 9Li spins. The geometry for these terms is Gaussian V(r) = ve−r2/a2with a= 2.55 fm. As in the previous case the depths of the potentials are L-dependent. The depths of the different terms for the potentials considered in the following sections are presented in Table 5.2. The potentials with an Iin their names (P1I and P2I) include spin-spin terms, while those without it (P3, P4 and P5) exclude them. Looking at the depths of the central part for the swave it can be seen that they are much shallower for P1I and P2I than for P3, P4 and P5. As well, we can see that for the p-wave the potential is highly repulsive for P1I and P2I. This is done to push out the p3/2component in what is called the repulsive core approach [116]. This method results in the s-wave states produced being 1s1/2states, which do not present a node. We do not expect 5.2. 11Li(p, pn)10Li∗123 Model v(0) cv(1) cv(2) cv(0) ss v(1) ss v(1) so-c v(1) so-v P1I -5.4 260.75 – -4.50 1.00 1.00 300 P2I -5.0 260.25 – -2.00 1.00 1.00 300 P3 -50.5 -39.0 -50.5 – – – 40.0 P4 -49.6 -39.4 -50.5 – – – 35.5 P3 -50.5 -39.0 -53 – – – 40.0 Table 5.2: Potential depths for the 11Li models used in this work. Spin-orbit terms have the same depth for L= 1,2and spin-spin terms for L= 0,2. Er[p1/2](MeV) a(fm) Er[d5/2](MeV) %p1/2%s1/2%d5/2rmat (fm) rch (fm) 1+2+1−2− P1I 0.37 0.61 – -37.9 – 31 67 1 3.2 2.41 P2I 0.30 0.55 -1.1 -6.7 – 44 54 <1 3.0 2.40 P3 0.50 -29.8 4.3 30 64 3 3.6 2.48 P4 0.23 -16.2 4.3 67 27 3 3.3 2.43 P5 0.50 -29.8 1.5 35 39 23 3.2 2.42 Table 5.3: Features of the 10Li structure for the different potentials employed in this work. The second column shows the energy of the p1/2resonance while the third one shows the scattering length of the s1/2virtual state. Note that for the model with spin, both the resonance and virtual state are split. The fourth column shows the position of the d-wave resonance, only for the models without spin. The fifth, sixth and seventh columns show the weights of the p1/2,s1/2and d5/2waves in the 11Li ground state respectively, while the last two columns show its matter and charge radii. this to influence the energy distributions we intend to study, although it may change momentum distributions, which will not be explored in this work. As for P3, P4 and P5, they produce deeply-bound 1s1/2and 1p3/2com- ponents which have to be removed from the spectrum using the adiabatic projection method [117]. Apart from these potentials, the three-body calculation for 11Li includes a three-body force to reproduce the two-nucleon separation energy for 11Li, S2n=−0.37 MeV. Let us remark here that Vn9Li is used consistently for the calculations for 10Li and 11Li. The features of the 10Li continuum obtained for these potentials as well as the weight of the different waves for the n−9Li system in the 11Li ground state wave function are presented in Table 5.3. The charge and matter radii for 11Li are also presented. In our adjustment of potential depths, we have tried to reproduce the 124 Chapter 5. Application of (p, pN)reactions to Borromean nuclei overall features for the 10Li continuum, a p-wave resonance at ∼0.5 MeV and a virtual state at low energies with a (negative) scattering length of ∼20 −30 fm for the P1I and P3 potentials. Let us note that, while we have a relative freedom to displace the position of the resonances and virtual states by changing the depths of the interaction, this also modifies strongly the weights of the associated waves in the ground state of 11Li. This can be understood quite intuitively, as a resonance at lower energies is more easily populated as one at higher energies, so a model with a lower resonance in the p-wave will have a larger weight for this wave than one with the resonace at a higher energy, as can in fact be seen when comparing models P4 and P3. Actually, models P2I and P4 have been produced, not to be “realistic”, but to give a larger p-wave component. As can be seen in Table 5.3, this can only be achieved either by reducing the scattering length of the s-wave, as for P2I, effectively pushing the energy of the associated virtual state away from the threshold; or by reducing the energy of the p-wave resonance, as in P4. This shows that our method establishes a relation between the energy of the structures in the 10Li and the weight of their contribution, thus linking the height of the resonance peaks to their position. This severely restricts the distributions we can produce, thus endowing them with stronger spectroscopic capabilities. 5.2.3 11Li(p, pn)10Li∗at 280 MeV/A In this section we compare our calculations to experimental results measured at GSI, Germany, and published in [118]. In this work, the experimental momentum distribution was adjusted assuming a Breit-Wigner shape for the resonance and the shape for the virtual state from [119]. From this analysis they concluded that the spectrum of 10Li presented a p-wave resonance at ∼0.5 MeV and a virtual state with a scattering length of -20 to -30 fm. We note that in this fitting procedure the energy of the resonance, the scattering length of the virtual state and their weights are all independent parameters, as opposed to our method where the weight of each component is closely related to its energy. For this calculation, the optical potentials between pand 11Li and be- 5.2. 11Li(p, pn)10Li∗125 tween p, n and 10Li have been generated through the same method as the PH potentials in Sec. 4.2.1, through the folding of the Paris-Hamburg g-matrix effective interaction with the density of the nucleus computed from a Hartree- Fock calculation with the code oxbash. Since 10Li is unbound, we have opted to use the density of 9Li instead. We note that the use of mean-field densities for the case of such exotic and light nuclei can be questionable. However, we have not found a better prescription to build the optical potentials for these cases, so we have kept to this one. The p−ninteraction is taken as Reid93, as in the other cases presented in this document. The (p, pn)calculation has been computed for 20 to 30 p−nbin states for each configuration of the final 10Li, in order to produce the energy distribution. This means a large number of calculations has to be performed in order to produce the energy distribution. Therefore, in order to reduce computation time we have only computed the cross section leading to p−n states with angular momentum and parity jπ= 0±,1±,2±, which are the waves that contribute the most at the incident energy of 280 MeV/A. Some test calculations including higher jπsuggest that by this truncation we are underestimating the cross section by 10-15%. Since we are not interested in the magnitude of the cross section nor in “quenching factors” and noting that our choice of potentials may well induce larger uncertainties in the total cross section, we consider this precision to be enough. It can be questioned whether this approximation may alter the relative weights of different components of 10Li. Test calculations indicate that the ratio between the contributions of different jπis similar for different components of 10Li. As such, the error induced by the truncation should affect all components of 10Li similarly, not affecting their relative weights. Now we present calculations for the different models for 10Li presented in the previous section, and use the results in order to extract information on the structure of 10Li. These results have been published in [120] 5.2.3.1 P3 and P4 models: No spin splitting, no d-wave From the spectrum of [118], we try to build a 10Li spectrum with a p-wave resonance at E= 0.5MeV and a virtual state for the s-wave with a scattering length of −20 to −30 fm. Such a spectrum can be produced from potential P3 from Table 5.3. We therefore compute the energy distribution using TC