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Structure of finite nuclei and the free nucleon-nucleon interaction: General discussion of the effective force
Abstract
It is shown that the main features of the observed effective forces in nuclei can be calculated directly from the free nucleon-nucleon interaction. Although the unrenormalized interaction acts as a short-range force, the renormalized one contains P2, P4 and other multipole forces. The calculated strength of the P2 force agrees well with that obtained empirically.
... It begins with the G-matrix or low-momentum nucleon-nucleon (NN) potential V low-k to treat the strong short-range correlations induced by the realistic NN force [20,21]. Thus, the Q-box corrections and the folded terms are evaluated to account for the core polarization and other important correlations [22,23]. ...
... The core-polarization corrections were initially calculated by Bertsch for 18 O and 42 Sc [90], and subsequently their effects were quantitatively evaluated in shell model calculations by Kuo and Brown [24]. It has been shown that the dominant contributions of the core-polarization diagrams originate from the quadruple component, providing a microscopic interpretation for the empirical long-range quadruple force [22]. Additionally, Shurpin and the collaborators found that the core-polarization corrections are significantly influenced by the single-particle basis [87]. ...
... As illustrated in Fig. 5, it is clear that the calculated low-lying spectra are in good agreement with the experimental data for both the yrast and non-yrast bands. Specifically, the excited energies for 20,22 Ne are slightly overestimated by the present calculation, while it yields a relatively compressed spectrum for 18 Ne and even-even isotopes 24∼28 Ne. The trend of the reduced transition probabilities B(E2; 2 + 1 → 0 + 1 ) is consistent with the experimental results, despite the value for 28 Ne being overestimated. ...
The configuration interaction relativistic Hartree-Fock (CI-RHF) model is developed in this work. Compared to the conventional configuration interaction shell model calculations, the CI-RHF model can be applied to study the structural properties of a wide range of nuclei without readjusting any parameters, because the effective Hamiltonian for different model space can be deduced consistently from a universal density-dependent Lagrangian based on the Hartree-Fock single-particle basis. The convergence of intermediate-state excitations has been examined in evaluating the effective interactions, and the core-polarization effects are illustrated, by using O as an example. Employing the CI-RHF model, both the bulk properties and low-lying spectra of neon isotopes in the sd shell have been well reproduced without introducing additional parameters besides those well-defined in the phenomenological Lagrangian. Moreover, the study of the isotopic evolution of charge radii and low-lying spectra highlights the shell closure at N=14 for neon isotopes.
... Pioneered by Gerry, a large number of shell-model studies of nuclei using realistic nucleon-nucleon interactions have been carried out. [1][2][3][4][5][6][7][8] The importance of intruder states in model-space effective interactions has been investigated by Schucan and Weidenmueller and others . 9,10 In Fig.1 we display the results of a typical calculation of 18 O. 7 In this calculation, a low-momentum interaction V lowk is employed, and the shell-model effective interaction has been calculated using three effective interaction methods denoted by LS, KK and EKKO 7 in the figure. ...
... 11 The second-order core polarization diagram for nuclear effective interaction is another example, and has played an essential role in microscopic nuclear structure studies. [1][2][3][4][5][6][7][8] How would the properties of hadrons change with their surrounding medium is a quite complicated problem. But the Brown-Rho scaling 12,13 provides a remarkably simple answer, namely m * /m = 1 − Cn/n 0 where m * and m are respectively the in-medium and in-vacuum hadron mass. ...
We solve the Dirac radial equation for a nucleon in a scalar Woods–Saxon potential well of depth (Formula presented.) and radius (Formula presented.). A sequence of values for the depth and radius are considered. For shallow potentials with (Formula presented.) the wave functions for the positive-energy states (Formula presented.) are dominated by their nucleon component (Formula presented.). But for deeper potentials with (Formula presented.) the (Formula presented.)s begin to have dominant anti-nucleon component (Formula presented.). In particular, a special intruder state enters with wave function (Formula presented.) and energy (Formula presented.). We have considered several (Formula presented.) values between 2 and 8(Formula presented.)fm. For (Formula presented.) and the above (Formula presented.) values, (Formula presented.) is the only bound positive-energy state and has its (Formula presented.) closely equal to (Formula presented.), both having a narrow wave packet shape centered around (Formula presented.). The (Formula presented.) of this state is practically independent of (Formula presented.) for the above (Formula presented.) range and obeys closely the relation (Formula presented.).
... MeV with χ 2 /datum ≃ 1. In order to make such interactions applicable in the nuclear medium some kind of regularization must be applied (for example Brueckner's G matrix formalism by Kuo, Brown and collaborators [160]). This type of interactions are categorized as realistic interactions from meson-exchange potentials. ...
In this work, a physics process known since quite long ago, double-gamma decay (), has been revisited from a new perspective: providing valuable insights into neutrinoless double-beta decay () nuclear matrix elements. At the same time that an eager experimental search of decay is underway, the nuclear and particle physics communities have made huge progress during the last years. The main goal of this thesis has been to investigate one of those approaches which is the computation of nuclear observables related to decay as a way to help in determining and reducing the theoretical uncertainties in -decay NMEs. This way, we have proposed that the measurement of the double magnetic dipole decay from the double isobaric analog state in the -decay final nucleus could establish the value and reduce the uncertainty in -decay NMEs, because the NMEs of the two processes are very well correlated. We have explored the validity of this approach to predict and quantify NMEs uncertainties in the case where data is available for the nuclear observable related to decay, that is for the Standard Model allowed decay. A further objective has been to start with the theoretical characterization of the first steps toward the measurement of the proposed double magnetic dipole decay from the double isobaric analogue state. In particular, we have studied the main decaying modes that can compete with this process: single gamma decay and proton emission, vi and we have calculated the corresponding branching ratios. In addition, we have also given the first steps in the study of the relation between and -decay NMEs in the ab initio valence space in-medium similarity renormalization group.
... It is challenging to study short-range and tensor correlation impacts, specifically for the theories of nuclear structure. Multiple methods have been developed for dealing with the tensor forces and explaining their impact on the nuclear ground state [3,4]. Dellagiacoma et al. [5] provided a straightforward phenomenological technique for creating dynamical short-range and tensor correlations Da Proveidencia and Shakin [6] developed a similar correlation operator for explaining short-range correlation effects, as did Malecki and Picchi [7]. ...
The study of neutron-rich nuclei's form factors, root-mean-square radius (rms), and nuclear density distributions is the focus of this work for the nuclei (22N,23 O and 24F). With the use of a strong short-range effect and a strong tensor force, the nucleons distribution function of the two oscillating harmonic particles in a two-frequency shell model operates with two different parameters: the first (bcore), for the inner (core) orbits, and the second (bvalence) for the outer (halo) orbits. This work demonstrated the existence of neutron halo nuclei for the nuclei (22N, 23O and 24F) in the (2s1/2) shell. The computed density distribution of neutron, proton and matter for these nuclei displayed the long tail performance. Using the Borne approximation of the plane wave, the elastic form factor of the electron scattering from the alien nucleus was calculated, this form factor is independent of the neutrons that make up the halo, but rather it results from a difference in the proton density distribution of the last proton in the nuclei. Fortran 95 power station program was used to the neutron, proton and matter density, elastic electron scattering form factor, and rms radii. The calculated outcomes for these exotic nuclei were in good agreement with the experimental data.
... Nuclear density distribution is an important factor [2] it's an essential component of both experimental and theoretical research, where the experimental matter density distributions of unusual nuclei are characterized mostly by long tail behavior [3]. Numerous solutions have been put out to deal with tension forces since including short-range correlations and tension forces is a difficult challenge, particularly for the theory of nuclear structure [4,5]. Dellagiacoma et al. presented a simple phenomenological strategy for establishing dynamic short-range and tensor effects [6]. ...
The distributions of nuclear density, root mean square radii, and elastic electron scattering form factor are calculated for nuclei (9C) (core +2p) and (26F) (core +2n) with the two different nuclear potential parameters for (bc) and (bv), were correlations for both the (effects tensor force and short-range) are used, and the appearance of the long extension is observed in Nuclear density distributions for these nuclei. Fortran 95 power station was used to program nuclear properties such as nucleon density (matter, neutron, and proton), elastic electron scattering form factor, and rms radii. The computed results for these exotic nuclei are determined to correspond pretty well with the experimental data.
... En este método se comienza a partir de la interacción entre nucleones libres y se introducen las modificaciones necesarias para tener en cuenta la presencia de la materia nuclear. Los primeros trabajos realizados en esta línea fueron llevados a cabo por Brown y Kuo [BK67,KB66,KBB66b,KBB66a]. Algunos ejemplos de estos potenciales son el potencial de Hamada-Johnston [HJ62], el potencial de Yale [L + 62], el potencial de Tabakin [Tab64] o el potencial "soft-core" de Reid [Rei68]. ...
El objetivo de esta memoria es presentar extensiones de IBM que permitan abordar problemas recientes de estructura nuclear. En particular se estudia el problema de la anarmonicidad de la excitación colectiva γ en núcleos par-par deformados. Dicho problema se aborda tanto del sistema laboratorio como desde el sistema intrínseco, para buscar su conexión con el Modelo Colectivo. Esta discusión se muestra en detalle en el Capítulo II de esta memoria. Otro tema de interés actual, debido al desarrollo reciente de instalaciones que permiten el uso de haces radiactivos, es el papel que juega el isospín en núcleos con N Z. En el Capítulo III se discuten algunos límites de especial interés del IBM con isospín (IBM-3) [EW80]. En el Capítulo IV se estudia la estructura geométrica del IBM-3 proponiendo un nuevo formalismo de estado intrínseco. El Capítulo V se dedica a discutir el problema de la competencia entre pairingn isoscalar e isovectorial. El tratamiento de este problema en IBM requiere la introducción de un nuevo grado de libertad, pares isoscalares como IBM-4 [EE81]. En esta memoria se propone un acercamiento al problema desde el sistema intrínseco. Para finalizar se presentan el resumen y las conclusiones de este trabajo. Además de los capítulo citados anteriormente, que son la parte original de esta memoria de tesis, en el c Capítulo I se hace una breve revisión del Modelo de Capas del Modelo Colectivo y de los aspectos básicos de los Modelos Algebraicos.
El IBM es un modelo que destaca especialmente por su simplicidad de cálculo y por proporcionar un marco en el que los resultados pueden ser interpretados fácilmente. A partir de un cálculo IBM siempre es posible reproducir las tendencias de los resultados experimentales, aunque a veces resulta difícil logar una descripción satisfactoria de todos los detalles. A la hora de analizar resultados de IBM siempre hay que tener presente lo simple que ha sido realizar el cálculo, ya que en general siempre se conoce algún hamiltoniano esquemático aproximado, que resulta adecuado para estudiar el núcleo problemas. Por otro lado, la dimensión de las matrices a diagonalizar (del orden de 500 * 500) es insignificante si se compara con las dimensiones involucradas en otros cálculos más sofisticados.
En el caso del IBM-3, las ventajas de los cálculos en IBM se ven acentuadas, es decir, los cálculos se hacen aún más simples en comparación con los realizados con otros modelos en la misma región másica. Adicionalmente, la fácil interpretación de los resultados hace de los cálculos IBM-3 una valiosa herramienta a la hora de trabajar en regiones másicas poco estudiadas experimentalmente y, en las que cálculos más sofisticados implican un enorme esfuerzo computacional. En general, un cálculo IBM-3 resulta como guía antes de realizar un cálculo más elaborado por ejemplo de SM).
La importancia del estudio realizado en IBM-3 se debe a que proporciona un método sencillo de incorporar el isospín a un modelo de estructura nuclear, pudiendo estudiarse su influencia sobre energías, transiciones electromagnéticas, etc. Posiblemente un cálculo IBM-3 no sea comparable a uno realizado con SM, debido a que el segundo proporciona una descripción mucho más completa de las propiedades nucleares, mientras que el IBM-3 sólo es capaz de reproducir las tendencias generales de los espectros y de las transiciones electromagnéticas. La relevancia de los resultados de IBM-3 está en que dichos resultados se obtienen de una forma muy simple y con un esfuerzo computacional pequeño.
Finalmente, conviene citar alguna de las líneas de este trabajo no concluidas aún. En primer lugar, sería conveniente hace un análisis más exhaustivo de los términos a tres cuerpos del hamiltoniano IBM-1, con el fin de justificar de forma microscópica el alto grado de anormalidad de la vibración γ, observado en el 166Er y 168Er. En segundo, habría que completar el estudio realizado en el sistema intrínseco de IBM-4, realizando cálculos esquemáticos donde se pueda observar la competencia entre el pairing isoscalar e isovectorial.
... This results in an additional term , the core-polarisation shown diagramatically in Fig. (2) It was first included in calculations by George Bertsch [16] for 18 O and 42 Sc using the Kallio-Kolltveidt force. [17] This paper was followed by the seminal papers by Tom Kuo and Gerry Brown on the "Structure of Finite Nuclei and the Free Nucleon-Nucleon Interaction" [18,19,20]. Here the reaction matrix as well as the core-polarisation was calculated from the Hamada-Johnston potential [21] using the separation method. ...
This is a very short presentation regarding developments in the theory of
nuclear many-body problems, as seen and experienced by the author during the
past 60 years with particular emphasis on the contributions of Gerry Brown and
his research-group. Much of his work was based on Brueckner's formulation of
the nuclear many-body problem. It is reviewed briefly together with the
Moszkowski-Scott separation method that was an important part of his early
work. The core-polarisation and his work related to effective interactions in
general are also addressed.
... The G-matrix derived from the free NN interaction was successfully used, for instance, by Kuo and Brown 18) as the effective interaction in finite nuclear structure calculations. Here, we apply the nuclear-matter ΛN G-matrix interactions to shell model analyses for typical Λ hypernuclei. ...
The properties of the new YN * YY interaction model ESC08are studied on the basis of the G-matrix theory. G-matrix calculations in nuclear matter are performed in the cases of Lambda N-, Sigma N-, Xi N- and LambdaLambda-starting channels. In the Lambda N case, the spin-spin and spin-orbit terms are shown to be of reasonable strengths. The folding-model analyses are done extensively for observed spectra of Lambda hypernuclei. The experimental energy spectra are nicely reproduced. The shell-model analyses for typical Lambda hypernuclei are performed with the G-matrix interactions. In the Sigma N case, the repulsive values of U_Sigma are obtained owing to the effect of the quark Pauli-forbidden states which are taken into account phenomenologically in ESC08. In the Xi N case, the moderately attractive values of U_Xi are obtained mainly due to the strongly attractive contributions in (3) S_1 T=1 Xi N-LambdaSigma-Sigma\ Sigma coupled states, which leads to extensive existence of Xi hypernuclei. In the Lambda Lambda case, the B_{Lambda Lambda} values derived from ESC08 are found to be consistent with the experimental values.
The atomic nucleus is a quantum many-body system whose constituent nucleons (protons and neutrons) are subject to complex nucleon-nucleon interactions that include spin- and isospin-dependent components. For stable nuclei, several decades ago, emerging seemingly regular patterns in some observables could already be described successfully within a shell-model picture that results in particularly stable nuclei at certain magic fillings of the shells with protons and/or neutrons: N, Z=8, 20, 28, 50, 82, 126. However, in short-lived, so-called exotic nuclei or rare isotopes, characterized by a large N/Z asymmetry and located far from the valley of β stability on the nuclear chart, these magic numbers, viewed through observables, were shown to change. These changes in the regime of exotic nuclei offer an unprecedented view at the roles of the various components of the nuclear force when theoretical descriptions are confronted with experimental data on exotic nuclei where certain effects are enhanced. This article reviews the driving forces behind shell evolution from a theoretical point of view and connects this to experimental signatures.
The properties of the oxygen isotopes provide diverse examples of progress made in experiments and theory. This chain of isotopes has been studied from beyond the proton drip line in (Formula presented.)O to beyond the neutron drip line in (Formula presented.)O. This short survey starts with the microscopic G matrix approach for (Formula presented.)O of Kuo and Brown in the 1960’s and shows how theory has evolved. The nuclear structure around the doubly-magic nucleus (Formula presented.)O is particularly simple in terms of the nuclear shell model. The nuclear structure around the doubly-magic nucleus (Formula presented.)O exhibits the coexistence of single-particle and collective structure.
Here we consider the large-scale structure of the atomic nucleus. A nucleus comprises a system of A nucleons, which, when well separated from each other and outside the range of the strongly attractive nuclear force, has as its energy just the sum of the rest mass of the individual nucleons (Fig. 3.1 left) . In bringing, in a hypothetical way, the nucleons close together, at the range of the strong force (approximately 10-14–10-13m), condensation into the bound atomic nucleus will result (Fig. 3.1 right) with a release of the corresponding binding energy. This causes an observable deficit in the mass of the actual nucleus as compared to the sum of the masses of the individual nucleons. This “loss” of mass is known as a mass defect.
Model-space effective interactions derived from free-space
nucleon-nucleon interactions are reviewed. We employ a double
decimation approach: first we extract a low-momentum interaction
from using a T-matrix equivalence decimation method. Then
is obtained from by way of a folded-diagram effective interaction
method. For decimation momentum , the
interactions derived from different realistic models are nearly model
independent, and so are the resulting shell-model effective interactions. For
nucleons in a low-density nuclear medium like valence nucleons near the nuclear
surface, such effective interactions derived from free-space are
satisfactory in reproducing experimental nuclear properties. But it is not so
for nucleons in a nuclear medium with density near or beyond nuclear matter
saturation density. In this case it may be necessary to include the effects
from Brown-Rho (BR) scaling of hadrons and/or three-nucleon forces ,
effectively changing the free-space into a density-dependent one. The
density-dependent effects from BR scaling and are compared with those
from empirical Skyrme effective interactions.
Effective interactions and transition operators have a long history in nuclear physics. From the early works of Brueckner and Bethe to the present time, microscopic approaches based on realistic nucleon–nucleon interactions, fit to phase shifts, have produced many useful results and insights. Recently, an ab initio approach has been developed that successfully weds a no-core finite basis state treatment, conserving center of mass motion, with an Hermitian effective Hamiltonian derived from realistic two-nucleon and tri-nucleon interactions. We present this approach with a sample of recent results.
Core-polarization (CP) and Brown-Rho (BR) scaling were among Gerry's most favorite topics. In this contribution, we will discuss some of the early history as well as more recent work associated with these two fascinating phenomena. I (TTSK) will begin with some recollections of Princeton, where I met Gerry for the first time in 1964 and worked as his postdoc. Core polarization was in fact the first topic he assigned to me. JWH started working with Gerry at Stony Brook from 2003 and was Gerry's last student in nuclear physics. We three had teamed up well, working closely and pleasantly on both CP and BR scaling, particularly on the latter's connection to the anomalously-long beta-decay lifetime of carbon-14. We shall here briefly review these topics, including a recently developed new Brown-Rho scaling based on a Skyrmion half-Skyrmion two-phase model.
We have measured excitation functions of the γ rays resulting from the bombardment of N15 by polarized and unpolarized protons in the energy range Ep=2.5−9.5 MeV with emphasis on identifying dipole decays to the first (0+) and second (3−) excited states in O16. Resonances in γ12 are observed at Ex=16.21, 16.45, 16.82, 17.12, 18.03, 18.98, 19.90, and 20.41 MeV. The 16.21 and 17.12 MeV resonances in γ12 are identified as M1 decays of the 1+ T=1 states to the 6.05 MeV 0+ state in O16. The measured ratio of reduced strengths B(M1, γ1)B(M1, γ0) is 0.48±0.03 for decays from the 16.21 MeV state and 0.55±0.04 for decays from the 17.12 MeV state. The 18.03 MeV resonance is due to a 3− T=1 state in O16 with a strength ΓpΓγ2Γ=(1.96±0.27) eV and the 18.98 MeV resonance is due to the 4− T=1 stretched particle-hole state with a strength of (0.85±0.10) eV. We determine absolute particle and γ widths for these states. The M1 γ2 width of the 18.98 MeV state, (7.1±3.1) eV, is in agreement with a shell-model calculation. Resonances in γ3 are observed at 16.82 and 17.27 MeV and in γ4 at 17.88 MeV. The excitation energies and widths of these levels as well as the strengths of the γ transitions suggest a T=1 character for all of the resonances for which capture γ rays are observed. Correspondences of our resonances to levels in N16 are given. Strong α1 branches for many of these states indicate isospin impurities. We compare γ widths, including ground-state M1 decays, and allowed β transition rates in A=16 nuclei with shell model calculations and obtain rough agreement with the experimental results. Additional shell model calculations for M1 and Gamow-Teller decays in the A=14, 15, 17, and 18 nuclei are presented, which indicate that Gamow-Teller matrix elements are quenched by ∼20% relative to shell model predictions and also relative to the spin part of the M1 matrix elements.
Ground-state binding energies and excited-state energies of nuclei in
the 1 s-O d shell are analyzed in terms of a variety of two-body
interactions. In addition to simple interactions such as the delta and
surface-delta, we consider potentials of the finite-range,
one-boson-exchange type for the central, tensor, and spin-orbit
components of the interaction, with multipole terms added to the central
part. The relative importance of these components, together with that of
the antisymmetric spin-orbit component, is considered. The antisymmetric
spin-orbit component is found to be the least important. Both
density-dependent and density-indpendent interactions are considered.
Our results favor a density-dependent form for the central and
spin-orbit components. We develop a semi-empirical "best-fit"
interaction based on a 14-parameter density-dependent two-body potential
which reproduces 447 sd-shell binding-energy data to within an rms
deviation between experiment and theory of 260 keV. This semiempirical
interaction is compared to typical G-matrix interactions as well as to
the purely empirical interactions obtained by using two-body matrix
elements or Talmi integrals as parameters of the fit.
Inelastic scattering is a source of much useful information about core polarization effects in nuclei near closed shells. Although there have been many theoretical treatments of core polarization effects reported in the literature, the results of these calculations have rarely been applied to the interpretation of inelastic scattering data. In the present paper we review the microscopic models for the treatment of inelastic proton and electron scattering and the microscopic models for the treatment of core polarization. Estimates are made of core excited admixtures in the wave functions for low-lying states in 42Ca, 50Ti, 89Y, 90Zr, 207Pb, and 209Bi. The resulting wave functions are used to calculate theoretical (p,p′) cross sections and (e,e′) form factors for comparison with available experimental data. "Realistic" G matrix interactions are used as the starting point in both the structure and the (p,p′) calculations. In the structure calculations the interaction is modified by means of a "bootstrap" prescription to account for important long-range core correlations and in the (p,p′) calculations it is modified by the addition of an imaginary component. It is concluded that the overall features of the experimental data can be understood from these calculations.NUCLEAR STRUCTURE, NUCLEAR REACTIONS (p,p′) and (e,e′); nuclei near closed shells; microscopic model for core polarization; calculated σ(θ) and F(q).
The neutron spectrum following the beta-decay of 9Li is studied with a time-of-flight spectrometer. A neutron group of 0.68 MeV is seen corresponding to the decay to the ground state of 8Be of the 2.43 MeV state of 9Be; a group of energy 1.2±0.1 MeV and width 1.0±0.2 MeV corresponds to a level in 9Be at 3.0±0.1 MeV. It is suggested that this is the “missing” level predicted by the independent particle model. The log ƒt value (5.2±0.2) for the 9Li β-decay to this state and its absolute width agree with the expectation of the model. 9Be(e,e′)9Be data are consistent with such a state and provide a measure of M1 radiative width to the ground state in adequate agreement with the model.
A shell-model calculation of 18F and 18O is performed in which the Tabakin interaction is used as the residual shell-model interaction. Two kinds of second-order corrections are included by perturbation theory. In one (the second Born corrections), plane wave intermediate states are used; in the other, particle-hole core-excited states are included. The results agree quite well with experiment. A detailed comparison is made between the results of this calculation and that made by Kuo and Brown with the Hamada-Johnston hard-core potential. The matrix elements of the two potentials agree surprisingly well. The second Born corrections are important in improving the agreement between the two sets of T = 0 matrix elements.
The energy levels of 208Bi are calculated with the reaction matrix elements deduced from the Hamada-Johnston potential, and compared with the recent experimental data of Alford, Schiffer and Schwartz. The calculated wave functions for most low-energy states show very little configuration admixture. The energy levels have been calculated with and without the inclusion of the core-polarization effects, and the results of the two calculations show no significant difference. Remarkable agreement with experiment is obtained for all 36 experimentally identified low energy levels. But a large discrepancy is observed for the isobaric analogue state of 208Pb; the calculated position being about 4 MeV too low. A multipole analysis of the effective particle-hole interactions has been carried out.
In the contributions to the effective nucleon-nucleon interaction coming from core polarization involving positive parity excitations, it is found adequate to employ lowest-order perturbation theory (one “bubble”). This is checked by explicitly constructing the core polarization phonons and coupling them with the two particles. When the exchange of negative parity phonons is involved, it is important to include also the higher order bubble diagrams. Applications have been made for nuclei in the 0p, 0d and 1s shell.
A method is developed for the treatment of the Bethe-Faddeev three body cluster equations in finite nuclei. A matrix method is employed to sum the three hole line graphs in 4He. For each value of a constant shift C in the intermediate state oscillator spectrum we have calculated: (i) the two body bound state binding energy self-consistently in the Brueckner approximation, (ii) the energy contribution from three hole line graphs, and (iii) the effect of single particle potential insertion in particle lines. The most important dependence on C comes from those graphs containing single particle insertions and their effect is to make the sum of (i) + (ii) + (iii) much less dependent upon C than (i) alone. The three hole line contribution for 4He comes mainly from third order graphs. Effects of truncation of the matrix are severe and calculations with a larger matrix could alter the quantitative but probably not the qualitative results. Present address: University of Sussex, Brighton, England.
The low-lying energy levels of O18 are calculated in the harmonic-oscillator shell model using as residual interaction the exponentially velocity-dependent singlet-even potential of Tabakin and Davies in combination with the triplet-odd potential of Green. The main result is that the energy levels so calculated agree closely with the energy levels of Kuo and Brown calculated without core polarization from the hard-core Hamada-Johnston nucleon-nucleon potential.
Results are reported for angular distributions of the reaction Zr90(He3, t)Nb90 at 21 MeV. It is found that for targets in the f72 shell this reaction favors specific final states in which a neutron is replaced by a proton in the same shell-model orbit, coupled to any allowed angular momentum. The angular distributions differ in a systematic way depending on the spin of the final state. The nine states of the T< (g92)2 multiplet are tentatively identified. A multipole decomposition leads to coefficients very similar to those of the (f72)2 multiplet, showing an enhanced quadrupole coefficient.
Energy gap calculations are made with the realistic Yale-Shakin interaction and the standard delta -function interaction. States below the valence shell are taken into account. It is found that the results are very close, showing that they are practically insensitive to the details of the interaction. The constant- Delta approximation is found to be very well justified.
In this chapter, we shall summarize the fundamentals of time-dependent perturbation theory. Although we shall be only concerned
with stationary problems in this book, it will be advantageous to apply time-dependent methods. We restrict ourselves in this
chapter to the nonrelativistic formalism and return to the relativistic one in later chapters.
High-spin three-particle levels populated in the decay of a 182+/-6-musec isomeric state in 207Bi have been studied using the pulsed external beam of the Stockholm 225-cm cyclotron. The isomeric state was produced in the 205T1(28-MeV alpha, 2n)207Bi reaction. The intensities and energies of the 13 delayed gamma transitions observed fit consistently into a level scheme of six excited levels having energies (keV) and spins: 2101.5 (212), 1645.4 (152), 1357.9 (152), 1240.5 (132), 931.8 (132), and 669.5 (112-) Shell-model calculations have been performed showing that the observed levels can be interpreted as resulting from the coupling of the two-neutron-hole states in 206Pb with the odd proton. From the known level structure of 206Pb and the investigation of the n-p interaction in 208Bi, the theoretically expected level structure of 207Bi has been calculated. The average energy difference between theoretically and experimentally determined levels is as low as 18 keV, and the agreement of experimental and theoretical transition probabilities is also satisfactory. The spin assignments are confirmed, and all levels except the 212 are found to have odd parity.
The low-lying energy levels of the odd-odd nucleus Bi210 and the even-even nuclei Po210 and Pb210 are calculated within the framework of the harmonic-oscillator shell model taking Green's velocity-dependent N-N potential as the residual interaction. It is found that Green's parameters do not give satisfactory agreement with experiment. Sets of parameters are determined that give good fits to the experimental spectra.
Current theories of direct two-nucleon transfer reactions are tested, by comparing (p,t) and (p,3He) transitions on odd-mass nuclei leading to mirror final states. Proton-induced reactions on 15N at 43.7 MeV and on 13C at 49.6 MeV are discussed in detail. Many mirror transitions are analyzed with DWBA calculations in an attempt to fit both angular distributions and cross-section ratios; good results for the shapes of the angular distributions are obtained. The agreement between theory and experiment for the cross-section ratios of mirror (p,t) to (p,3He) transitions improves in every case with the inclusion of a strongly spin-dependent force in the nucleon-nucleon interaction, but over-all satisfactory agreement is not obtained. The (p,t) transitions are found to be generally stronger than expected, relative to their mirror (p,3He) transitions, and three cases are discussed where the experimental ratios of these cross sections exceed the theoretical upper limit. Two possibilities, both of which introduce coherent effects, are discussed to account for this result: (1) interference terms arising through a spin-orbit interaction in the optical potential or (2) interference terms between a direct-reaction contribution and a core-excitation contribution to the cross section.
In order to obtain a numerical basis for the shell structure we solve the HartreePock (HF) equations with effective realistic 2-body forces (L~) for a complete sequence of (closed-shell) heavier spherical nuclei (A = 16 + 208). Although--in a first stagethe oscillator approximation has been used, the numerical results are very satisfactory in various respects. Our effective forces were determined--in the spirit of the Scott-Moszkowsky (s) method--by fitting the scattering data up to 150 MeV with the help of a HamadaJohnston type potential in which the inner part was set equal to zero. (Spin and paritydependent cut-off radii of the order of 1 fro.) For odd states the fit is good even up to 400 MeV, which indicates that the hard core of the original Hamada potential is not needed in this case. For even states, however, our procedure is ultimately confirmed by the numerical coincidence of the (~ separation radii ,> obtained by BROWN and Kuo (4) (1.05 fm compared to 1.07 fm for our force). This fact also allows us to use the results of the more detailed investigations of higher-order effects discussed by these authors: we may thus introduce into our calculations their extremely simple and useful formula for the second-order contribution, namely
The mass of the nucleus, through its binding energy, continues to be of capital importance not only for various aspects of nuclear physics, but also for other branches of physics, notably weak-interaction studies and astrophysics. The authors first describe the modern experimental techniques dedicated to the particularly challenging task of measuring the mass of exotic nuclides and make detailed comparisons. Though tremendous progress in these and the associated production techniques has been made, allowing access to nuclides very far from stability, it is still not yet possible to produce many nuclides involved in stellar nucleosynthesis, especially the r process, leaving no choice but to resort to theory. The review thus goes on to describe and critically compare the various modern mass formulas that may be used to extrapolate from the data towards the neutron drip line. Special attention is devoted to the crucial interplay between theory and experiment, showing how new measurements far from stability can considerably reduce the ambiguity in extrapolations to nuclides even further away.
In this tutorial-style article we discuss the time-independent perturbation formalism of Brandow for effective interactions and operators in a nonrigorous, intuitive way. The simple example of a 2×2 matrix is used to introduce the basic concepts, for instance the notion of folded diagrams. The same example is used subsequently in a discussion of the branch points arising from "level crossings" and the associated convergence difficulties with the perturbation series. Numerical calculations of the effective interaction and the effective charge are then reviewed, focussing, for the most part, on the (1s0d) shell. Whenever possible, simple physical models are used to illustrate the results. A fairly comprehensive summary of the calculations to date is given, and we attempt to synthesize and draw conclusions.
The Landau theory for infinite Fermi systems is discussed, and it is then shown how one can apply the Landau technique of functional differentiation of the energy with respect to quasiparticle occupation numbers to obtain the effective interaction, beginning from the Brueckner-Bethe approximation to the energy. Calculations of this effective interaction by Bäckman from a nucleon-nucleon potential are discussed and criticized. Results of Bäckman's calculation are compared with the parameters in Migdal's effective force, evaluated at the center of the nucleus, the latter parameters having been derived phenomenologically by fitting various nuclear phenomena. There is a large discrepancy between the calculated value of the compressibility of nuclear matter and that obtained phenomenologically. The advantages of introducing a model space in order to improve the accuracy of the calculated results is discussed, and it is pointed out that such an introduction would also be advantageous in recent calculations by Barrett and Kirson of third-order terms in the effective interaction; also, that a model space is effectively introduced in recent calculations with the Midgal theory when it comes to deriving the quadrupole-quadrupole interaction. Calculation of the effective force between two atoms in 3He is discussed briefly, and it is pointed out that this is a much more ambitious task than in nuclear matter. Calculations carried out within the Brueckner-Bethe formalism are reviewed.
The beta+/EC decay of mass-separated A=80 nuclei was studied by means of beta- and gamma-ray spectroscopy after the bombardment of a 24Mg target with 58Ni ions at 190 MeV. A new 1- isomer at 228.5(1) keV has been identified in 80Y with a half-life of 4.7(3) s. This new isomer decays to the 4- ground state with an 81(2)% branch and by beta+/EC transitions to levels in 80Sr with a 19(2)% branch. The half-life of the 80Y ground-state decay has been remeasured to be 30.1(5) s. A level scheme for the low-lying states in 80Y is presented. Hartree-Fock-Bogolyubov calculations show a large prolate deformation for these states, and two-quasiparticle + rotor model calculation results suggest that the low-lying states can be reproduced by inclusion of an effective proton-neutron residual interaction. The dominating Nilsson configurations for the 4- ground state and the 1- isomeric state have been found to originate from the parallel and antiparallel coupling of the proton [422]5/2+ and the neutron [301]3/2- orbitals.
The core-renormalization of the effective interaction between nucleons in the pf-shell is found to be very much larger for an RPA-description of the core than for the lowest-order perturbation theory description. The difference between the two descriptions is much larger than in the sd-shell.
The contributions of effective three-body forces to the binding energies of the Ca-isotopes are calculated with G-matrix elements of the Hamada-Johnston potential, and are found to be small. Work supported in part by Norges Almenvitenskapelige Forskningsråd, Oslo, Norway.
Two- and four-quasiparticle theories with the nucleon-nucleon potential of Tabakin are applied to study the even parity states of the even tin isotopes. The core polarisation corrections defined by Kuo and Brown are found to be most important in reproducing the existing experimental data.
The effective charges of the protons outside the closed shell Z = 28 are calculated with G-matrix elements of the free nucleon-nucleon potential, including an approximate estimate of the effect of the exclusion principle.
The folded diagram expansion for the effective hamiltonian of a system of three valence nucleons in the 1s0d shell is investigated. Beside the one-body and two-body operators, which already occur in the folded diagram expansion for systems with one and two valence nucleons, also folded diagrams involving three nucleons are obtained. These terms, which can be interpreted as a contribution to an effective three-nucleon force, yield a non-negligible contribution which is repulsive for the low-lying states. Two different techniques are studied for the summation of the folded diagrams. A very good convergence is obtained using the Lee-Suzuki iteration scheme.
In the Brueckner-Hartree-Fock theory of finite nuclei, the problems of renormalization, potential energy of intermediate states and re-arrangement effects are examined. For practical calculations it is shown that satisfactory solutions to them can be obtained if the parametrization chosen for the single-particle potential in finite nuclei is linked closely to nuclear matter results. We make the link with similar problems in the density dependent Hartree-Fock theory and emphasize the possibility of such a parametrization by recalling the existence of a formal solution for the Hartree-Fock single-particle potential if the effective interaction is of the delta-function type. A method of solution of the Bethe-Goldstone equation is then presented which separates the intermediate states into those of an ``open shell'' and of a ``continuum''. Finally results of model calculations of 16O and 40Ca with harmonic oscillator functions are presented in which the parametrization chosen for the BHF single-particle potential is taken from the Skyrme-Vautherin delta-function force. A self-consistent determination of certain parameters of this form of force leads to values in close agreement with the empirical estimate made by Vautherin and Brink in 16O, with the exception of the spin-orbit splittings. Limitations and possible improvements of this type of approach are discussed for 40Ca.
We have made various calculations of two number-conserving sets of diagrams which contribute to the effective interaction in third order using a variety of forces. While there is in general a cancellation between the individual diagrams in these two sets, the degree of cancellation is not sufficiently high that the contribution to the effective interaction of third-order number-conserving sets may be neglected as small in comparison with most other third-order diagrams.
Shell-model calculations are performed for five nuclei 210Po, 210Bi, 208Bi, 208Tl and 206Tl, by using the K-matrices deduced from the Brueckner-Gammel-Thaler and Hamada-Johnston potentials. Then the effects of the core polarization are examined with the use of the P2 and P4 forces. Good results are obtained with the Hamada-Johnston potential. The effects of the P2 and P4 forces improve these results slightly.
The low-lying energy levels of 18O and 18F are calculated in the harmonic oscillator shell model taking Green's velocity-dependent N-N potential as the residual interaction. It is found that the shell-model matrix elements agree substantially with those calculated by Kuo and Brown from the unrenormalized Hamada-Johnston hard-core N-N potential. It is found furthermore that Green's parameters give rather unsatisfactory agreement with the experimental spectra. It is shown that somewhat improved agreements with experiment are possible by making appropriate parameter adjustments.
The nuclei neighbouring 88Sr and 208Pb are treated as a core plus an additional particle or hole interacting with the core vibrations. With the help of the Green function technique the core vibrations are described as collective particle-hole excitations thus depending on the single-particle propagators. An iterative calculation has been performed for 88Sr, showing that the coupling of single-particle and collective modes is very important. Because of angular momentum coupling and parity selection rules the coupling possibilities are strongly reduced in the lead region.
The low-lying non-normal parity states of 41Ca and 39K (positive-parity and negative-parity spectra respectively) are treated as a particle (hole) coupled to the low-lying negative-parity states of 40Ca (often called vibrational states). These spectra are calculated by the technique of Sherwood and Goswami1), in which the RPA is extended from a closed shell to a closed shell plus a particle or hole. The calculated spectra are compared with experimental results. It is observed that the diagonalization of the energy matrix leads to roughly twice the number of physical states. It is shown that the doubling of the number of physical states does not lead to a breakdown of the theory, and a way to distinguish the physical states from the non-physical states is indicated. A method for calculating transition rates in the phonon-particle coupling is developed and is applied to the known electromagnetic transitions in 39K. In the calculated spectra, it is found that two vibrations are usually important, namely, the lowest J = 3- and 5- states of 40Ca. It is noted that the concept of a particle coupled to a vibration is ambiguous. The non-orthogonality is handled correctly in this microscopic theory and is quite important in calculating transition rates. The phonon-particle coupling model predicts that the non-normal parity states in 41Ca and 39K lie much lower in energy and are in much better agreement with experiment than according to conventional p-h calculation (TDA). The experimental levels (in MeV) and their calculated transition rates (in Weisskopf units) are 2.82 (2.5, 9), 3.02 (15 × 10-4, 13 × 10-4), 3.60 (16, 23), 3.94 (17, 16). The 4p-3h states were not included. Free charges rather than effective charges were used in the calculation of the transition rates. Work supported in part by the National Science Foundation.
The effective interaction in T = 1 two-body states in the sd shell is parameterized in terms of radial matrix elements of the effective potential in states of relative orbital angular momenta of two nucleons. Seven of these parameters are needed to define all the T = 1 matrix elements with the assumption of a central effective potential. The values of these parameters are determined by making a least-squares fit to 16 observed level energies in the oxygen isotopes and are then compared with values calculated from various realistic free nucleon-nucleon interactions. It is shown that a major effect of core excitation and consequent renormalization of the two-body matrix elements, so forcefully pointed out by Kuo and Brown, consists of a strong modification of the contributions of the triplet-odd force. Two-body matrix elements calculated with our parameters are also found to agree well with those determined by Glaudemans et al. from a least-squares fit to observed level schemes of nuclei with A = 29-40.
This paper attempts to choose the parameters of the pairing-plus-quadrupole model as realistically as possible. A correction is introduced to take into account the different numbers of neutrons and protons. An expression is given for the deformation energy of a nucleus with an arbitrary two-body interaction. Then the pairing-plus-quadrupole matrix elements are compared with those of more realistic interactions, and it is decided to keep only matrix elements between states of two adjacent oscillator shells. An argument is given to determine a priori the strength of the quadrupole force, and various corrections to this are discussed. No similar argument is found for the pairing force. Residual effects of the remaining oscillator shells are included in an approximate fashion. Comparison with previous work along these lines is presented. Present affiliation: Niels Bohr Institute, Copenhagen, Denmark.
The diagrams of the Tamm-Dancoff and random-phase approximations have been summed to calculate the effective charge in the 1s0d and 1p0f shells. The generally large RPA result for the effective charge is reduced to roughly the TDA one when the self-screening corrections to the particle-hole and ground-state correlation vertices are included. The screening corrections serve to suppress the collectivity of the isoscalar quadrupole core phonons and are particularly important for 40Ca where the isoscalar quadrupole mode is near collapse in the RPA using the bare particle-hole interaction. The qualitative rather than the quantitative effects of these and other corrections are stressed. Permanent address.
DOI:https://doi.org/10.1103/RevModPhys.35.853
DOI:https://doi.org/10.1103/PhysRevLett.14.380
Shell-model calculations are performed for the nickel isotopes with an approximate reaction matrix, corrected in various ways for core polarization, as interaction. The resulting energy levels are compared with experiment and with the predictions of phenomenological shell-model interactions.
Using Brueckner's method for the treatment of complex nuclei, the effect of an infinite repulsive core in the interaction between nucleons is studied. The Pauli principle is taken into account from the beginning. A spatial wave function for two nucleons is defined, and an integro-differential equation for this function is derived. Owing to the Pauli principle, the wave function contains no outgoing spherical waves. A solution is given for the case when only a repulsive core potential acts. The effective-mass approximation is investigated for virtual states of very large momentum.
The odd-parity states in O16 are calculated assuming a spin-dependent two-body potential with hard core. The separation method of Moszkowsky and Scott is used, and it is shown that this method is very convenient for nuclear calculations as long as harmonic oscillator wave functions can be used.
DOI:https://doi.org/10.1103/PhysRevLett.15.758
The separation method of Moszkowski-Scott has been applied to the calculation of the properties of nuclear matter using two different nucleon-nucleon potentials, both in reasonable agreement with two-body data. Calculations with the potential of Brueckner-Gammel gave at an equilibrium densitty corresponding to kf = 1.5 f−1. The difference from the results of B and G may be caused by slow convergence of the series (especially in the triplet-even state where the tensor interaction has a large second order contribution). An important factor in obtaining nuclear saturation is shown to be the weakening of tensor interaction effects by the Fermi sea. Evidence for this may also be seen from the results obtained using a different two nucleon potential which, however, still gives good fits to two body data. The potential chosen has a much weaker tensor component and shows no sign of saturation at normal densities (at kf = 1.5 f−1, ). The difference in the two results appears to be much larger than can be accounted for either by higher order terms or by differences in the phase shift approximation to the reaction matrix.
Effective forces are derived from realistic two-body forces, and renormalizations coming from excitation of closed shells are shown to be important in achieving agreement with empirical binding energies.
Eigenstates of the shell model are obtained by diagonalization of the Hamiltonian submatrix defined by a given shell model subspace. Matrix elements of the effective nuclear interaction can be determined from experiment in a consistent way. This approach was introduced in 1956 with the 38Cl-40K spectra, has been applied in many cases and its latest success is in the s,d shell. This way, general features of the effective interaction have been determined. The T = 1 interaction is diagonal in the seniority scheme as clearly demonstrated in proton configurations and in the description of semimagic nuclei by genrealized seniority. Apart from a strong and attractive pairing term, T = 1 interactions are repulsive on the average. The T = 0 interaction is attractive and is the origin of the central potential well in which nucleons are bound. It breaks seniority in a major way leading to deformed nuclei and rotational spectra. Such an interaction may be approximated by a quadrupole-quadrupole interaction which is the basis of the interacting boson model. Identical nucleons with pairing and quadrupole interactions cannot be models of actual nuclei. Symmetry properties of states with maximum T are very different from those of ground states of actual nuclei. The T = 1 interaction between identical nucleons cannot be approximated by pairing and quadrupole interactions. The rich variety of nuclear spectra is due to the competition between seniority conserving T = 1 interactions and the T = 0 quadrupole interaction between protons and neutrons.
The binding energy in nuclear matter has been calculated for two S-wave potentials (one with a hard core and the other with a Yukawa repulsive core) by the reference-spectrum method of the Brueckner-Bethe-Goldstone theory of nuclear matter. Pauli and spectral correction terms are included, and single-particle potentials are calculated self-consistently up to as high as 5 to 6 fm−1. The improved treatment of off-energy-shell effects causes a displacement of the saturation curve (i.e., total energy versus density) in the direction of decreased equilibrium density and decreased binding energy per particle. The use of a finite repulsive core increases the equilibrium binding energy per particle by 4 to 7 MeV and also significantly increases the equilibrium density. This indicates that phenomenological nucleon-nucleon potentials with finite repulsive cores should yield more satisfactory results for nuclear matter than the hard-core potentials now available.
An energy independent nucleon-nucleon potential model is described. The model represents the two-nucleon data below 315 MeV more faithfully than any other potential model known to date. The data include the effective range expansion parameters and the deuteron properties as well as the scattering data at higher energies. It is of special significance that the n-p data have been represented by the energy independent potential model in a satisfactory manner for the first time. For this purpose a quadratic LS potential was found essential in the T = 0 state. The T = 1 potential differs little from those previously proposed by several authors. The phase shifts predicted by the model are in fair agreement with the solutions YLAM (T = 1) and YLAN3M (T = 0) recently found by the Yale group.
The intention of this work is to investigate the applicability of the free nucleon-nucleon potential determined by the scattering data in the shell-model description of finite nuclei. The potential is chosen to be that of Hamada and Johnston. We have chosen 18O and 18F as our first numerical calculations. A major part of the work reported here concerns the evaluation of the shell-model reaction matrix elements. They are evaluated using the separation method for the singlet-even and triplet-even states and the reference spectrum method for the singlet-odd and triplet-odd states. The second-order Born term for the triplet-even tensor force is found to be very important. It can be calculated conveniently and with good accuracy using the closure approximation with an energy denomiator of ≈ 220 MeV. Assuming the singlet-particle wave functions to be those of a harmonic oscillator well, the single-particle energy levels of , and for 170 are calculated by letting the valence nucleon interact with 16O core. The results are very encouraring. The splitting is found to come mostly from the triplet-odd 1 · s force through the reaction matrix Hartree-Fock process. The spectra of 18O and 18F are satisfactorily reproduced by diagonalizing the model interaction GΘ3p1h in the sd shell, except for the states which are presumed to be largely deformed; G is the reaction matrix and Ω3p1h is the wave operator which takes care of the corrections arising from the one-particle-one-hole excitation of the 16O core. It is found that the conventional shell-model effective interactions are well reproduced by the model interaction deduced from the free nucleon-nucleon potential.
The linked diagram expansion of the ground state energy and wave function of a system of fermions are extended to the case where the ground state of the unperturbed system is degenerate, as, for instance, in a nucleus with an incomplete shell. The state where all the complete shells are filled is taken as vacuum state. It is then shown, to all orders in the perturbation expansion, that the energy of the first levels is given by the sum of two terms. The first term, is the energy of the core alone and is given by a linked diagram expansion of the usual form. The second term is the energy of the particles outside the core. It is obtained by solving an eigenvalue problem in the space of the degenerate unperturbed wave functions. The corresponding eigen-functions are given by linked diagram expansions very similar with the usual ones. A few generalizations of the method, and its relation to the Brillouin-Wigner perturbation method are discussed.
The two-neutron binding energy and excitation spectrum of O18 are computed using the free nucleon-nucleon potential of Brueckner-Gammel-Thaler (which contains a hard core). The problem is discussed within the framework of the Bethe-Goldstone theory using harmonic oscillator wave functions as the unperturbed solutions. The interaction is diagonalized among the degenerate states and matrix elements are computed by a transformation to relative and center-of-mass coordinates. The energies in relative s-states are obtained by a numerical integration of the relative s-wave Schrödinger equation and then corrected for the presence of the filled levels. The ordering of the first five states of O18 is given correctly and the binding energy and level spacings are quite close to the experimental values.
Using the reaction matrix theory, the effective interactions for the nickel isotopes are deduced from the Hamada-Johnston potential, and compared with the empirical values determined by Auerbach. The renormalization effects due to the core polarization are found to be very important. Impressive agreement is obtained.
The low-lying even-parity energy levels of O18 are calculated within the shell model. Core excitations from the double closed O16 system have been taken explicitly into account using the SU3 coupling scheme. The resulting energy spectrum shows nice agreement with experiments for nearly all states below 6 MeV excitation energy. The branching ratio for the E2 transitions from the first 2+ (1.98 MeV) to the ground state and from the second 0+ (3.63 MeV) to the first 2+ are well described in the calculation even if the absolute values are somewhat too small.
The correction to the interaction of two valence particles in 18O and 42Sc, due to perturbations of the closed shell wave functions, is calculated and found to be as large as 30% of the first-order interaction. The qualitative behaviour of this interaction is: attractive for T = 1, low J states; repulsive for T = 1, high J states; and small for T = 0 states.
- Bethe