Ahmed Osman

Faddeev equations for the6Li nucleus

SummaryThe6Li nucleus is considered as composed of an alphaparticle, a proton and a neutron. With this construction the6Li nucleus is studied as a three-body problem. The Faddeev equations are given, by taking separable potentials for the interaction between the alpha, the proton and the neutron particles in pairs. The nucleon-nucleon potential is taken as a central term of the Yamaguchi type and also including a tensor force. To investigate the short-range correlations in this three-body system, the nucleon-nucleon potential is taken to include a repulsive soft-core part in addition to the attractive part considered. The nucleon-alpha potential is taken as central and spin-orbit terms which fit the corresponding experimental nucleon-alpha phase shifts. The effect of Coulomb forces between the proton and the alpha-particle is taken into account. By solving the coupled Faddeev equations numerically, the binding energies for the ground and excited states of6Li are calculated. Inclusion of repulsive-core effects is found to change the results by 19%.RiassuntoIl nucleo6Li è ritenuto composto di una particella α, un protone ed un neutrone. Con questa struttura il nucleo di6Li è studiato come un problema di tre corpi. Si danno le equazioni di Faddeev prendendo potenziali separabili per l’interazione tra la particella α il neutrone e il protone a coppie. Il potenziale d’interazione nucleone-nucleone è dato da un termine centrale del tipo di Yamaguchi e comprendente anche una forza tensoriale. Per analizzare le correlazioni a breve raggio in questo sistema a tre corpi, si considera un potenziale nucleone-nucleone che comprenda una parte repulsiva del nocciolo molle accanto alla parte attrattiva considerata. Il potenziale nucleone-particella α è dato da termini centrali e spin-orbita che approssimano i corrispondenti spostamenti di fase sperimentali nucleone-α. Si prende in considerazione l’effetto delle forze di Coulomb tra protone e particella α. Risolvendo numericamente le equazioni di Faddeev accoppiate, si calcolano le energie di legame per gli stati fondamentale ed eccitato del6Li. Si trova che se si considerano gli effetti del nocciolo repulsivo i risultati cambiano del 19%.

Perturbation of the faddeev equations with tensor forces

SummaryPerturbation of the Faddeev equations for the three-nucleon system is considered by using nucleon-nucleon interactions with tensor forces. By applying the perturbation techniques, integral equations are obtained. The contribution of the perturbing potential to the three-nucleon binding energy makes a difference of 0.3689 MeV.RiassuntoSi considera la perturbazione delle equazioni di Faddeev per il sistema a 3 nucleoni per mezzo delle interazioni nucleone-nucleone con forze tensoriali. Applicando tecniche di perturbazione si ottengono equazioni integrali. Il contributo del potenziale perturbante all’energia di legame a 3 nucleoni dà una differenza di 0.3689 MeV.РеэюмеРассматривается воэмушение уравнений Фаддеева для трехнук-лонной системы, испольэуя нуклон-нуклонные вэаимодействия с тенэорными силами. Применяя технику теории воэмушений, получаются интегральные уравнения. Вклад воэмушаюшего потенциала в знергию свяэи трех нуклонов приводит к раэнице 0.3689 МзВ.

Two-nucleon transfer reactions with form factor models

SummaryThe theory of two-nucleon transfer reactions is considered. Nuclear reactions are considered with triton or3He particles which are used as projectiles in stripping reactions and as detected particles in pick-up reactions. In each channel we have a four-particle problem, three of them are nucleons and the fourth is a heavy particle. These transfer reactions are studied on the basis of the generalizedR-matrix method. Different channel functions of the subclusters in the triton and3He particles are included. Model form factors are obtained and are used in two-nucleon transfer reactions. Differential cross-sections of different two-nucleon transfer reactions are calculated and are found in good agreement with the experimental data. The correct normalization and spectroscopic factors are obtained.RiassuntoSi considera la teoria delle reazioni di trasferimento di due nucleoni. Si considerano le reazioni con il tritone o particelle di3He che sono usate come proiettili in reazioni di strappo e come particelle rivelate in reazioni di pick-up. In ogni canale si ha un problema a quattro particelle, tre delle quali sono nucleoni e la quarta è una particella pesante. Si studiano queste reazioni di trasferimento sulla base del metodo generalizzato della matriceR. Si includono diverse funzioni di canale dei sottocluster nel tritone e nelle particelle di3He. Si ottengono i fattori di forma del modello e si usano nelle reazioni di trasferimento a due nucleoni. Si calcolano le sezioni d’urto differenziali di diverse reazioni di trasferimento a due nucleoni e si trova che sono in buon accordo con i dati sperimentali. Si ottengono la normalizzazione corretta e i fattori spettroscopici.

Deuteron stripping reactions with the Tabakin potential

SummaryDeuteron stripping reactions are considered. Due to the strong repulsion between nucleons at very short distances, we have investigated the nuclear short-range correlations. The neutron-proton nuclear potential in the deuteron is taken as a short-range repulsive core surrounded by a long-range attractive potential. The neutron-proton potential is taken as the Tabakin separable potential to take into account the short-range correlations. The differential cross-sections for deuteron stripping reactions have been calculated in two different cases by taking Yamaguchi or Breitet al. type parameters for the Tabakin potential used. The angular distributions for different (d, p) stripping reactions on the different target nuclei28Si,32,34S,36Ar,40,48Ca,50,52,54Cr have been calculated by means of the DWBA calculations. Our present theoretical calculations for the angular distributions of the different reactions considered have been fitted to the experimental data, with which good agreement is obtained. The extracted spectroscopic factors from the present work are found to be more reliable.RiassuntoSi considerano reazioni di stripping del deuterone. A causa della forte repulsione tra nucleoni a distanze molto brevi, abbiamo studiato le correlazioni nucleari a breve raggio. Si prende il potenziale nucleare neutrone-protone nel deuterone come un nocciolo repulsivo a corto raggio circondato da un potenziale attrattivo a lungo raggio. Per il potenziale neutron-protone si prende il potenziale separabile di Tabakin per tener conto delle correlazioni a breve raggio. Si sono calcolate le sezioni d’urto differenziali per le reazioni di stripping del deuterone in due differenti casi, prendendo parametri del tipo di Yamaguchi o di Breitet al. per il potenziale di Tabakin usato. Si sono calcolate le distributioni angolari per differenti reazioni di stripping (d, p) sui differenti nuclei bersaglio28Si,32,34S,36Ar,40,48Ca,50,52,54Cr mediante calcoli DWBA. I nostri attuali calcoli teorici per le distribuzioni angolari per le diverse reazioni considerate sono state adattati ai dati sperimentali coi quali si è ottenuto un buon accordo. Si trova che i fattori spettroscopici estratti dal presente lavoro sono più attendibili.РеэюмеРассматриваются реакции стриппинга дейтрона. Вследствие сильного отталкивания между нуклонами на очень малых расстояниях, мы исследуем ядерные короткодействуюшие корреляции. Нейтрон-протонный ядерный потенциал в дейтроне выбирается в виде короткодействуюшего отталкиваюшего остова, окруженного длиннодействуюшим притягиваюшим потенциалом. Нейтрон-протонный потенциал выбирается в виде раэделяемого потенциала Табакина для того, чтобы общяснить короткодействуюшие корреляции. Вычисляются дифференциальные поперечные сечения для реакций стриппинга дейтрона в двух раэличных случаях, выбирая параметры типа Ямагучи или Брейта и др. для испольэованного потенциала Табакина. Определяются угловые распределения для раэличных реакций (d, p) стриппинга на раэличных ядрах мищени28Si,32,34S,36Аг,40,48Са,50,52,54Cr, испольэуя DWВА вычисления. Нащи теоретические вычисления угловых распределений для рассмотренных раэличных реакций хорощо согласуются с зкспериментальными данными. Получено, что иэвлеченные спектроскопические факторы иэ настояшей работы являются надежными.

Sensitivity of the Three-Body Calculations to Different Effects

The bound state of few-body systems in light nuclei is studied as a three-body problem. The three-body problem is solved following the different approaches of the Faddeev formalism as well as the unitary pole approximation. Separable approximations are introduced to reduce the three-body problem to a set of coupled integral equations. Numerical calculations are carried out for the resulting integral equations and the separable expansion. In the present work, we calculate the ground-state binding energy of the bound three-nucleon system3H. The main interest of the present work is to investigate the sensitivity of the three-body binding energy to different effects in the problem. For this reason, we study the dependence of the three-body binding energy of different forms of local and separable two-body potentials, on the effective range of the two-body potentials, and on the percent of theD state in the deuteron wave function. Also, we test the sensitivity of the three-body binding energy to the considered number of terms from the separable expansion.

Faddeev equations for deuteron break-up on alpha-particles

SummaryFaddeev equations for deuteron break-up on alpha-particles are given. Total cross-sections for the reaction process d + α → n + p + α are calculated by means of the Faddeev equations and separable potentials. The numerical solution of the resulting equations gives the total cross-section for the deuteron break-up process up to 30 MeV deuteron energy. The present theoretical calculations are in good agreement with the experimental data.RiassuntoSi presentano equazioni di Faddeev per la frantumazione del deuterone su particelle α. Si calcolano le sezioni d’urto totali per il processo di reazione d + α → n + p + α per mezzo di equazioni di Faddeev e di potenziali separabili. La soluzione numerica dell’equazioni che risultano dà la sezione d’urto totale per il processo di frantumazione del deuterone fino ad un’energia del deuterone di 30 MeV. I presenti calcoli teorici sono in buon accordo con i dati sperimentali.РеэюмеПриводятся уравнения Фаддеева для расшепления дейтрона при вэаимодействии с альфа-частицами. Испольэуя уравнения Фаддеева и раэделяемые потенциалы, вычисляются полные поперечные сечения для процесса d + α → n + p + α. Численное рещение полученных уравнений дает полное поперечное сечение для процесса расшепления дейтрона вплоть до 30 МзВ. Предложенный теоретический расчет хорощо согласуется с зкспериментальными данными.

Coulomb (d, p) stripping reactions

SummaryThe theory of deuteron stripping reactions at incident energies below the Coulomb barrier has been considered. A closed analytical form for the differential cross-section has been obtained by considering three cases, according to the incident-deuteron and outgoing-proton energies. The agreement between our theoretical calculations and the experimental data is satisfactory. By fitting our theoretical calculations with the experimental data, more reliable values for the spectroscopic factors are extracted.RiassuntoSi considera la teoria di reazioni di smantellamen to del deuterone ad energie incidenti inferiori alla barriera di Coulomb. Si è ottenuta una forma analitica chiusa per la sezione d’urto differenziale prendendo in considerazione tre casi in base alle energie del deuterone incidente e del protone uscente. L’accordo tra i nostri calcoli teorici e i dati sperimentali è soddisfacente. Accordando i nostri calcoli teorici con i dati sperimentali si ottengono valori più attendibili per i fattori spettroscopici.РеэюмеРассматривается теория реакций стриппинга дейтрона при знергиях падаюших частиц ниже кулоновского барьера. Исследуя три случая знергий па-даюшего дейтрона и вылетаюшего протона, получается эамкнутая аналитическая форма для дифференциального поперечного сечения. Нащи теоретические вычисле-ния удовлетворительно согласуются с зкспериментальными данными. Испольэуя подгонку нащих теоретических вычислений с зкспериментальными данными, удается получить более достоверные эначения для спектроскопических факторов.

Three-body model for stripping nuclear reactions

A three-body model for the deuteron stripping nuclear reactions is presented. A set of three integral equations is obtained for the wave functions of the three-body problem by introducing a decomposition into angular momentum states into the Lippmann-Schwinger equation. Simple two-particle interactions with separable potentials are used. These separable potentials reduce the three-body problem to the solution of coupled sets of one-dimensional Fredholm integral equations. The angular distributions for28Si(d,p)29Si and40Ca(d, p)41Ca stripping reactions are calculated. From the extracted spectroscopic factors, good agreement with the experimental measurements is obtained.

Coulomb forces in the three-body problem with application to direct nuclear reactions

Faddeev equations are considered in the case of three charged particles interacting with both separable nuclear two-body interactions and also including Coulomb forces. Modified Faddeev equations with Coulomb Greens functions are introduced. The three-body amplitudes are given into pure Coulomb and distorted-Coulomb amplitudes. Introducing a decomposition in the angular momentum states, a set of three-body integral equations is obtained. The effect of pure coulomb amplitudes is studied in direct nuclear reactions and found to give a large contribution to the cross sections. The three-body integral equations obtained are applied for direct nuclear reactions. The angular distributions for12C(6Li,d)16O,16O(6Li,d)20Ne, and12C(6Li,α)14N transfer reactions are calculated as well as for the6Li elastic scattering on12C. From the good agreement between the theoretically calculated and experimental data, better spectroscopic factors are extracted. The effect of including Coulomb forces in the three-body problem is found to improve the results by about 16.26%.

Nucleon-nucleon interaction in the three-nucleon system

The three-nucleon system is reconsidered. The Faddeev equations are given leading to a set of integral equations. Solving these integral equations, suitable forms are considered for the nucleon-nucleon interaction. In the bound state of three-nucleon system, the form of the nuclear forces from the nucleon-nucleon interaction is important. In the present calculations, we consider the nuclear forces resulting from the nucleon-nucleon interaction by the exchange of a scalar meson, a pseudoscalar meson, and a massless vector meson. With this different meson exchange nucleon-nucleon interaction, the binding energy of the three-nucleon system is calculated by solving the Faddeev integral equations giving a value of 8.452 MeV.