Summary
The numerical calculations of separation energies and quasi-particle excitations for spherical nuclei (A>20) are given in some detail. The method (described in part I) consists of a reduced Hartree-Bogoliubov scheme with a realistic (effective) two-body interaction and a phenomenological nuclear potential. The relevant numerical values which occur in the computations (single-particle energies, reduced matrix elements and the gap matrices) are tabulated (Tables I–IX, Appendix A). In addition, the characteristic occupation probabilitiesv /2 j of pairs over the various subshells are plotted as functions of the nucleon numbers (Fig. 2–4). The low-lying quasi-particle states of odd nuclei were determined throughout the periodic table for all «isotopes» on an «ideal» middle line. Within the lower spherical region the complete set of levels is plotted in a graphical survey (Fig. 5–6). There is a fairly good agreement with the corresponding empirical data within a relatively large interval (28<A<90).
Riassunto
Si forniscono dettagliatamente i calcoli numerici delle energie di separazione e delle eccitazioni delle quasi-particelle per nuclei sferici. Il metodo (descritto in un precedente articolo) consiste di schemi ridotti di Hartree-Bogoliubov con una interazione reale (effettiva) a due corpi ed un potenziale nucleare fenomenologico. I relativi valori numerici dei calcoli (energia di una singola particella, elementi di matrice ridotti e matrici di gap) sottoposti in Tabella (Tabelle I–IX, Appendice A). In aggiunta le probabilità caratteristiche di occupazionev /2 j di coppie sui vari sottolivelli sono considerate in funzione dei numeri nucleari (Fig. 2–4). Gli stati energetici inferiori delle quasi-particelle di nucleo dispari sono stati determinati, per tutti gli «isotopi», tramite la tabella periodica su di una linea media ideale. All’interno della più interna regione sferica, si traccia graficamente l’insieme completo dei livelli (Fig. 5–6). Vi è un accordo abbastanza buono con i corrispondenti dati sperimentali in un intervallo relativamente largo (28<A<90).
Реэюме
Подробно приводятся численные вычисления знергий раэделения и кваэи-частичных воэбуждений для сферических ядер (A<20). Метод (описанный в Части 1) представляет приведённую схему Хартри-Боголюбова с реальным (зффективным) двух-частичным вэаимодействием и феноменологическим ядерным потенциалом. Соответствуюшие численные величины, которые встречаются при вычислениях (одночастичных знергий, приведённых матричных злементов и матриц раэрыва) табулированы. (Таблицы 1–9, Приложение А). Кроме зтого вычерчиваются характеристические вероятности эаполненияV /2 j для пар по раэличным подоболочкам, как функции атомного номера. Определяются ниэколежашие кваэи-частичные состояния нечётных ядер по всей периодической таблице для всех «иэотопов» на «идеальной» средней линии. Внутри ниэщей сферической области графически иэображена полная система уровней. Сушествует довольно хорощее согласие с соответствуюшими зкспериментальными данными внутри относительно больщого интервала (28<A<90).
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References
K. Bleuler M. Beiner, andR. de Tourreil:Nuovo Cimento,52 B, 45 (1967).
M. Beiner, K. Bleuler andK. Erkelenz:Effective nuclear forces, to be published inNuovo Cimento. The introduction of effective (nonsingular) nuclear forces is required by the fact that our model is based on an HB approximation which would be meaningless if forces with infinite (or strong) repulsive cores were used.
See paper I andM. Beiner:Separationsenergien und mittleres phänomenologisches Potential der Atomkerne, Forschugsberichte des Landes Nordrhein-Westfalen (1964), No. 1407.
R. de Tourreil:Thesis, Une application de la théorie de Bogoliubov-Valatin à la structure nucléaire, Bonn, 1966. It is, however, obvious that the adapted oscillator does not reproduce truly the energy spectrum of a Woods-Saxon potential.
T. A. Brody andM. Moshinsky:Tables of Transformation Brackets, Monografias del Instituto de Fisica (Mexico, 1960).
M. Moshinsky:Nucl. Phys.,13, 104 (1959).
Our calculations showed that the pairing effects due to the «short-range» terms {V es LL (r)·L 12 andV ot LS (r)L·S} of our effective forces are not very important. (TheL 12 force has negligible effects (1÷2)%, whereas theL·S force reduces the pairing energies by (7÷9)%.) In addition, we neglected the Coulomb interaction in the proton pairing. A recent check showed, however, that its influence reduces the gap values by about 20%. On the other hand, improved calculations which, apart from the complete force, also contain all contributing levels in the gap equation match the experimental data fairly well too. Detailed results will be given in ref. (2).
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Bleuler, K., Beiner, M. & de Tourreil, R. Pairing approximation in spherical nuclei — II. Nuovo Cimento B (1965-1970) 52, 149–186 (1967). https://doi.org/10.1007/BF02710660
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DOI: https://doi.org/10.1007/BF02710660