Anomaly in the nuclear curvature energy

By use of semiclassical mean-field methods, we study the dependence of the curvature-energy coefficient and other surface properties of nuclear matter upon the energy-density functional. This is done both by solving the Euler-Lagrange equation with a simplified phenomenological functional and by obtaining a variational solution with a fourth-order extended Thomas-Fermi functional. The calculated curvature-energy coefficient a_c decreases with increasing value of the bulk nuclear incompressibility coefficient K for physically relevant values of K, but always remains larger than 3 MeV in either approach when the volume-energy coefficient, saturation density, surface-energy coefficient and surface diffuseness are constrained to their experimental values. The calculated values of the curvature-energy coefficient a_c are significantly larger than experimental values obtained from analyses of fission-barrier heights and ground-state masses of nuclei throughout the periodic table. Among possible resolutions of this anomaly, we suggest that relativistic effects or correlations may play a significant role in the nuclear surface, or that the leptodermous expansion may break down in regions of large curvature, such as occurs for highly deformed shapes and for light nuclei.     

Finite nuclei to nuclear matter: a leptodermous approach

The liquid drop model (LDM) expansions of energy and incompressibility of finite nuclei are studied in an analytical model using Skyrme-like effective interactions to examine, whether such expansions provide an unambiguous way to go from finite nuclei to nuclear matter, and thereby can yield the saturation properties of the latter, from nuclear masses. We show that the energy expansion is not unique in the sense that, its coefficients do not necessarily correspond to the ground state of nuclear matter and hence, the mass formulas based on it are not equipped to yield saturation properties. The defect is attributed to its use of liquid drop without any reference to particles as its basis, which is classical in nature. It does not possess an essential property of an interacting many-fermion system namely, the single particle property, in particular the Fermi state. It is shown that, the defect is repaired in the infinite nuclear matter model by the use of generalized Hugenholtz–Van Hove theorem of many-body theory. So this model uses infinite nuclear matter with well defined quantum mechanical attributes for its basis. The resulting expansion has the coefficients which are at the ground state of nuclear matter. Thus a well defined path from finite nuclei to nuclear matter is found out. Then using this model, the saturation density 0.1620 fm⁻³ and binding energy per nucleon of nuclear matter 16.108 MeV are determined from the masses of all known nuclei. The corresponding radius constant r₀ equal to 1.138 fm thus determined, agrees quite well with that obtained from electron scattering data, leading to the resolution of the so-called ‘r₀-paradox’. Finally a well defined and stable value of 288±20 MeV for the incompressibility of nuclear matter K_∞ is extracted from the same set of masses and a nuclear equation of state is thus obtained.     

Do analogs of the Hoyle state exist in <Superscript>16</Superscript>O?

The root-mean-square radii of short-lived 0⁺-states in ¹⁶O were obtained for the first time from an analysis of α+¹⁶O diffraction scattering. Data on elastic and inelastic α+¹⁶O scattering were analyzed on the basis of the modified diffraction model in the range of projectile energies between a few tens of MeV units to 400 MeV. No case of a significant enhancement of the radius with respect to its ground-state value was observed. In particular, this concerns the 15.1-MeV 0 ₆ ⁺ state, which lies near the threshold for breakup to four alpha particles and for which we did not confirm a giant radius predicted by the alpha-particle-condensate model. This result disproves the hypothesis that the ¹⁶O nucleus in the 0 ₆ ⁺ state has a rarefied structure and appears to be an analog of the known Hoyle state at 7.65 MeV (0 ₂ ⁺ ) in ¹²C.     

Binding energies and radii of α-cluster-type nuclei from calculations based on the strictly restricted dynamics model

The properties of α-cluster-type nuclei with 4≤A≤80 are studied by employing the microscopic strictly restricted dynamics model (SRDM). The SRDM parameter set found via a fit to the experimental and theoretical values of nuclear binding and excited-level energies, classified according to the ground-state SU ₃ configurations, includes a nuclear-radius parameter r ₀ entering into the expression R=r ₀ A ^1/3 [fm], as well as the parameters of the effective central NN-interaction potential. The use of the microscopic SRDM allows one to obtain additional information about nuclei along the Z=N line, including binding energies, radii, and spectra of low-lying levels. The calculated nuclear binding energies and nuclear root-mean-square radii 〈r ²〉^1/2 are in reasonable agreement with their experimental counterparts.     

Heavy ion beam polarization produced by the multi-tilted-foil interaction

Nuclear polarization of the 7/2^? ground-state of⁵¹V was produced via the Multi-Tilted-Foil (MTF) interaction with a V beam. The induced polarization was determined by measuring the left-right asymmetry of Coulomb excited⁵¹V nuclei and, for a⁵¹V beam at E=100 MeV, was measured to beP _I =0.012(2). The nuclear polarization was also induced atE=50 MeV and, after further acceleration, determined at E=195 MeV to be P_I=0.010(1). These experiments demonstrate the feasibility of polarizing a great variety of heavy-ion beams at arbitrary velocities with subsequent acceleration and passage through magnetic beam-optics elements. Such polarization, albeit small, can be utilized for the determination of electromagnetic moments of exotic beams and separated reaction products.     

Polarization in quasielastic (p,2p) scattering on a 4He nucleus at 1 GeV

The polarization of secondary protons from the (p, 2p) reaction induced by 1-GeV protons incident to a ⁴He nucleus was measured in a kinematically complete experiment. By using a two-arm magnetic spectrometer, two secondary protons from this reaction were recorded in coincidence at unequal scattering angles of Θ₁ = 18°?24.21° and Θ₂ = 53.22° over a broad range of the recoil-nucleus momentum K _B between 0 and 150 MeV/c. It was found that the measured polarization of either secondary proton is less than that observed in free elastic proton-proton scattering. The magnitude of this difference is determined by the mean binding energy of s-shell protons rather than by the effective nuclear density. The polarizations measured in inclusive quasielastic and elastic scattering of 1-GeV protons on ⁴He nuclei are presented for scattering angles in the range Θ₁ = 18° ? 24.21°.     

Global dynamical correlation energies in covariant density functional theory: Cranking approximation

The global dynamical correlation energies for 575 even-even nuclei with proton numbers ranging from Z = 8 to Z = 108 calculated with the covariant density functional theory using the PC-PK1 parametrization are presented. The dynamical correlation energies include the rotational correction energies obtained with the cranking approximation and the quadrupole vibrational correction energies. The systematic behavior of the present correlation energies is in good agreement with that obtained from the projected generator coordinate method using the SLy4 Skyrme force although our values are systematically smaller. After including the dynamical correlation energies, the root-mean-square deviation predicted by the PC-PK1 for the 575 even-even nuclei masses is reduced from 2.58 MeV to 1.24 MeV.     

Nuclear radii of 17,19B and 14Be

The interaction cross sections (σ_I) of light radioactive nuclei close to the neutron drip line (^17,19B, ¹⁴Be) have been measured at around 800A MeV. The effective root-mean-square (r.m.s.) matter radii of these nuclei have been deduced from σ_I by two different methods, a Glauber-type calculation based on the optical limit approximation and a few-body reaction model. The deduced radii from both approaches agree with each other within experimental uncertainty. The r.m.s. radii of ¹⁷B (2.99±0.09 fm) and of ¹⁴Be (3.10±0.15 fm) in this work are consistent with the previously determined values, and have a higher accuracy. The r.m.s. radius of ¹⁹B (3.11±0.13 fm) was newly determined. Assuming a “core plus 2n” structure in ¹⁷B and ¹⁴Be, the mixing of ν(2s_1/2) and ν(1d_5/2) was studied and the s-wave spectroscopic factor is found to be 36±19% and 47±25%, respectively. A valence radius analysis suggests a “core plus 4n” structure in ¹⁹B.     

Thermostatic properties of symmetric nuclear matter

The thermostatic properties of symmetric nuclear matter are calculated by extension of a recent Thomas-Fermi approach to ground-state nuclei by Myers and Swiatecki [1]. We have computed the free energy per nucleon f(T, n) in Landau's quasiparticle approximation and have derived from it the relevant thermostatic properties. In view of its application to finite excited nuclei, the degenerate limit of nuclear matter is discussed in particular. As an interesting result we find at higher temperatures van-der-Waals-like isotherms in the p-n plane. Below the critical temperature T_c = 17.3 MeV two phases of nuclear matter, liquid and vapour, are defined by these. Comparing these results with the reduced phase transition data of ³He, ⁴He, and “inert gases,” we find that nuclear matter is similar to the He-isotopes, but differs considerably from the inert gases.     

Error propagation in extrapolated nuclear mass predictions

The properties of the error of the nuclear masses calculated from the transverse mass relations are analysed. The work assumes that the calculated errors of the nuclei whose masses are known experimentally behave as a sample selected randomly from a normal population having a zero mean and a standard deviationσ. It is found that the errors of the calculated masses of nuclei far from the line of beta-stability behave asc ₁ d ^3/2 wherec ₁ is a constant andd is the distance of the nucleus from the line of beta-stability. It is shown also that the errors related to the calculated mass differences behave asc ₂ d ^1/2 wherec ₂ is another constant.     

Comparison of the noncoplanar 6Li(p,pd)4He reactions at 120 and 200 MeV

The ⁶Li(p, pd)⁴He reaction was studied at 200.2 MeV, at the quasi-free angle pair (θ_p, θ_d) = (54°, ?48.9°), for noncoplanarity angles φ from 0° to 28°. ⁶Li αd spectroscopic factors of 0.84 and 0.76 are deduced from our coplanar data at this energy and 120 MeV, respectively, for ground-state 2S Woods-Saxon wave functions. A recent microscopic three-body calculation predicts spectroscopic factors from 0.70 to 0.75; using the ground-state wave functions from this study, we deduce a factor of 0.76 from the 200 MeV data. DWIA calculations fit the measured integrated cross sections versus φ for spectator momenta P_α ? 100 MeV/c at both bombarding energies, but underpredict them for larger P_α. Momentum form factors were better reproduced with 1S αd cluster wave functions for a soft-core bound-state potential than with the 2S Woods-Saxon wave functions, but the former wave functions generate unphysically large (～1.25) spectroscopic factors.     

Alpha scattering from nonclosed shell nucleus 12C and alpha-nucleus interaction

In order to understand the α-nucleus interactions for the α+¹²C～α+¹⁶O systems systematically, α scattering from ¹²C is studied in a microscopic coupled-channel model using a folding potential and realistic microscopic wave functions of ¹²C. The experimental angular distributions of elastic and inelastic scattering to the 2⁺(4.43 MeV), 3^? (9.64 MeV), and 0 ₂ ⁺ (7.66 MeV) states in the range of E _α = 41–172.5 MeV are analyzed.     

Channel coupling and exchange of an alpha-particle cluster in deuteron scattering on 6Li nuclei

Existing experimental data on elastic and inelastic deuteron scattering on ⁶Li nuclei in the energy range from 8 to 50 MeV were analyzed within the approach of coupled reaction channels. The coupling of elastic scattering and inelastic scattering accompanied by the transition to the 3⁺ state at E _x = 2.186 MeV and the mechanism involving the exchange of an alpha-particle cluster were taken into account in respective calculations. The phenomenological potentials obtained from the present analysis describe well experimental angular distributions at all energies and in full angular ranges. The depths of the real and imaginary parts of the potentials in question depend smoothly on energy at fixed values of the remaining parameters. The energy dependence of relevant volume integrals agrees well with similar data for the p + ⁶Li, ?? + ⁶Li, and ¹²C + ¹²C systems and with the predictions of a microscopic theory.     

Correlation between the optical model potential and matter distribution parameters in heavy-ion elastic scattering

Starting from the analytical expression for the folding model potential for the heavy-ion scattering, a simple relationship between the potential and the nuclear density parameters is obtained. This relationship is consistent with the analysis of the proximity forces and is confirmed around the strong absorption radius through the results of the folding model calculations for¹⁶O projectile incident on a variety of targets ranging from⁴⁰Ca to²⁰⁸Pb. These relations help to deduceR _x the distance where the folding potential has a specific valuex and the densities of the target nuclei. The predictions are found to be in good agreement with the experiment.     

A Mass Formula with a Few Free Parameters and Nuclear Mass Calculations

A mass formula with deformation and shell corrections for a few free parameters was described.By means of the formula the calculations of 1440 nuclear masses and 1250 two neutron separation energies s_2n in the region of 28≤Z≤81 were performed.The root-mean-square deviations are respectively 1.01MeV for nuclear masses and 0.57MeV for s_2n.The results of the theory could also reproduce the characters of the closure shell,the subshell and the deformation of the nuclei.     

Five-dimensional fission-barrier calculations from 70Se to 252Cf

We present fission-barrier-height calculations for nuclei throughout the periodic table based on a realistic macroscopic-microscopic model. Compared to other calculations (i) we use a deformation space of a sufficiently high dimension, sampled densely enough to describe the relevant topography of the fission potential, (ii) we unambiguously find the physically relevant saddle points in this space, and (iii) we formulate our model so that we obtain continuity of the potential energy at the division point between a single system and separated fission fragments or colliding nuclei, allowing us to (iv) describe both fission-barrier heights and ground-state masses throughout the periodic table.     

Elastic scattering of polarised deuterons from 16O at 200, 400 and 700 MeV

Angular distributions of cross section, and A_y and A_yy analysing powers were measured for polarised deuteron elastic scattering from ¹⁶O at 200, 400 and 700 MeV. The data at 200 MeV bear evidence of the nuclear rainbow phenomenon while those at 400 and 700 MeV are reminiscent of the proton scattering results at equivalent energies. The data were analysed in terms of the optical model. The real central potential shape changes from an attractive Woods-Saxon form at 200 MeV to a wine-bottle-bottom form with a repulsive interior at 700 MeV. The total reaction cross sections deduced display a clear nuclear transparency effect in the present energy domain in agreement with predictions from the Glauber theory optical limit. Comparison with previous results for ⁴⁰Ca and ⁵⁸Ni targets is made.     

Antiproton-nucleus scattering with self-consistent model for medium corrections

The antiproton-nucleon t-matrix with propagation in the nuclear medium is calculated selfconsistently and applied to the construction of optical potentials for the elastic scattering of antiprotons from nuclei. We find that this treatment gives results that are sensitive to medium corrections even though the strong absorption acts to mask these corrections partially. The agreement with scattering experiments at 46.8 MeV on ¹²C is very good. We compare potentials containing medium corrections to those based on free $\text{p}\text{N}$ amplitudes for ¹²C and ⁴⁰Ca. The local approximation to the optical potential is found attractive at low energies, becoming shallower with increasing bombardment energy in the range considered here (up to about E_L = 120 MeV).     

Nuclear-matter incompressibility from fits of generalized Skyrme force to breathing-mode energies

In an attempt to extend the range of values of K_v, the incompressibility of symmetric nuclear matter, for which fits to the measured breathing-mode energies are possible, we investigate generalized Skyrme-type forces with a term that is both density- and momentum-dependent. Acceptable fits are found to be possible only for values of K_v in the range 215±15 MeV. For higher values fits are impossible, while for lower values fits are achieved only by introducing an unphysical collapse of nuclear matter. Thus our generalization of the Skyrme force does not permit a significantly wider range of values of K_v than that already given by force SkM¹. However, with a view to having a more reliable estimate of the compressional properties of the highly neutron-rich nuclear matter that comprises the core of collapsed stars, we present a new version of this latter force giving a much better fit to the masses of neutron-rich nuclei. Comparison with force SkM¹ also shows that the value of K_v extracted from the breathing-mode energies is essentially independent of the choice of effective mass. By providing a counter-example, we show that K_v cannot be extracted from masses and charge distributions alone. As for the third-order coefficient K′, we cannot be more precise than to say that it lies in the interval 700 ± 500 MeV.