The effect of neutron-proton pairing correlations on the transfer of a neutron-proton pair

A theory of neutron-proton pairing interaction is developed considering bothJ=0T=1 andJ≠0T=0 correlations. The model of a singlej-shell is investigated explicitly forN=Z nuclei. Instead of solving the full HFB (Hartree Fock Bogoliubov) problem a variational method is used for determining the ground state energy and wavefunction. Our model shows that the best solutions contain either onlyT=0 or onlyT=1 correlations. A solution mixingT=0 andT=1 is energetically worse. It is estimated in PWBA (Plane Wave Born Approximation) that for ground state transitions at light nuclei in the transfer of a neutron-proton pair the cross section is enhanced up to a factor 3 by pairing correlations compared with shell model calculations.     

Second Coulomb energy differences of isospin triplets in the 2s-1d shell

Second Coulomb energy differences, which in the present case are proportional to the tensor Coulomb energy, are calculated for 0⁺, T = 1 ground states in the region 18 ≦ A ≦ 42 using a shell model that includes a pairing interaction. The calculation is done with a mathematical formalism that includes p-n pairs as well as p-p and n-n pairs. Besides an enhancement of proton-pair Coulomb energies, the pairing interaction is responsible for lowering the Coulomb energy of N = Z members of isospin triplets and also gives rise to an important term in the second energy difference. Using pairing strengths derived from fitting energy levels for mass-18 and mass-42 nuclei, results of the calculation reproduce experimental second energy differences extremely well.     

Effect of Wigner energy on the symmetry energy coefficient in nuclei

The nuclear symmetry energy coefficient(including the coefficient a_(sym)~((4)) of the I~4 term) of finite nuclei is extracted by using the differences of available experimental binding energies of isobaric nuclei.It is found that the extracted symmetry energy coefficient a_(sym)~*(A,I) decreases with increasing isospin asymmetry I,which is mainly caused by Wigner correction,since e_(sym)~* is the summation of the traditional symmetry energy e_(sym) and the Wigner energy ew.We obtain the optimal values J = 30.25±0.10 MeV,a_(ss)=56.18±1.25 MeV,a_(sym)~((4)) = 8.33±1.21 MeV and the Wigner parameter x= 2.38 ±0.12 through a polynomial fit to 2240 measured binding energies for nuclei with20 ≤ A ≤ 261 with an rms deviation of 23.42 keV.We also find that the volume symmetry coefficient J■ 30 MeV is insensitive to the value x,whereas the surface symmetry coefficient a_(ss) and the coefficient a_(sym)~((4)) are very sensitive to the value of x in the range 1≤x≤4.The contribution of the a_(sym)~((4)) term increases rapidly with increasing isospin asymmetry I.For very neutron-rich nuclei,the contribution of the a_(sym)~((4)) term will play an important role.     

Neutron-proton final state interaction in the 3H(3He,pα)n reaction at low bombarding energies

The ³H(³He, pα)n reaction has been studied in a kinematically complete experiment at ³He bombarding energies between 0.31 and 2.80 MeV in order to detect a possible resonant energy dependence of the spin singlet (T = 1) neutron-proton final state interaction (FSI). The data in the region of low n-p relative energies have been analyzed in terms of the Watson-Migdal formalism. Only a slight relative increase of the ¹S₀ n-p FSI is observed with increasing bombarding energy. This result is not easily related to the strong isospin violations found in the two-body reaction ³H(³He, d)⁴He at similar ³He energies.     

The role of the imaginary form factor in the microscopic description of (p,p′) scattering from nuclei with one closed shell

The inelastic scattering of protons from the lowest 2⁺ and 3^? levels in ⁴⁰Ca, Ni, Sn and N = 50 isotopes is analyzed for different incident proton energies. The addition of a collective imaginary term to the microscopic real form factor very much improves the agreement between the calculated and experimental cross section angular distributions. The variation with energy of the relative contributions of the ΔT = 1 and gDT = 0 isospin parts of the transitions is discussed.     

New mass relations and two- and four-nucleon correlations

Periodic terms in the nuclear binding energies, such as the pairing energy, are investigated. To this end we introduce a new, general set of mass relations (indicators) sensitive to specific types of periodicities in the binding-energy surface. Based on experimental data, the analysis leads to an improved expression for the (average) pairing term in the mass formulas. No trace of α-clustering is found and an upper limit for the corresponding energy gain can be deduced to be 0.4 MeV per α-particle or 1.5% of the total α-binding energy. The Wigner term, especially important for light nuclei, is accurately determined. An extra binding in odd N = Z nuclei is observed above the average of all other odd-odd nuclei.     

An investigation of the influence of the pairing correlations on the properties of the isobar analog resonances inA = 208 nuclei

Within the quasi-particle random phase approximation (QRPA), the method of the self-consistent determination of the isovector effective interaction which restores a broken isotopic symmetry for the nuclear part of the Hamiltonian is given. The effect of the pairing correlations between nucleons on the following quantities were investigated for theA = 208 nuclei: energies of the isobar analog 0⁺ states, the isospin admixtures in the ground state of the even-even nuclei, and the differential cross-section for the²⁰⁸Pb(³He,t)²⁰⁸Bi reaction atE(³He)=450 MeV. Both couplings of the excitation branches withT ^z = T⁰ ± 1, and the analog state with isovector monopole resonance (IVMR) in the quasi-particle representation were taken into account in our calculations. As a result of these calculations, it was seen that the pairing correlations between nucleons have no considerable effect on theT = 23 isospin admixture in the ground state of the²⁰⁸Pb nucleus, and they cause partially an increase in the isospin impurity of the isobar analog resonance (IAR). It was also established that these correlations have changed the isospin structure of the IAR states, and shifted the energies of the IVMR states to the higher values.     

中子晕核引起核反应中的同位旋效应

利用同位旋相关的量子分子动力学(IQMD)对中子晕核，⁸He和¹⁰He引起核反应中重要的同位旋效应和松散的中子晕结构影响的平均特征进行了研究.因为IQMD中的互作用势和介质中核子-核子碰撞截面灵敏地依赖于碰撞系统的密度分布.而扩展的中子晕密度分布包含了中子晕核的同位旋效应和松散的中子晕结构的平均特征，从而将这些信息通过动力学碰撞带入到反应机理中. 为了清楚地鉴别中子晕核带入反应机理重要的同位旋效应和松散中子晕结构的影响，通过比较中子晕核和相等质量稳定弹核在相同入射道条件下，所得物理观测量之间的差别加以确定.计算结果确实发现具有初始晕核信息的中子扩展密度分布将重要的同位旋效应和松散中子晕结构带入到各种物理观测量中.例如与相等质量稳定相比，中子晕核的晕特征引起了原子核阻止的降低；并明显地增加了核子发射中子-质子比和同位旋分馏比. 关键词： 中子晕核 原子核阻止 核子发射中子-质子比 同位旋分馏比     

Pairing vibrational and isospin rotational states in a particle number and isospin projected generator coordinate method

Pairing vibrational and isospin rotational states are described in different approximations based on particle number and isospin projected, proton-proton, neutron-neutron and proton-neutron pairing wave functions and on the generator coordinate method (GCM). The investigations are performed in models for which an exact group theoretical solution exists. It turns out that a particle number and isospin projection is essential to yield a good approximation to the ground state or isospin yrast state energies. For strong pairing correlations (pairing force constant equal to the single-particle level distance) isospin cranking (-ωT_x) yields with particle number projected pairing wave function also good agreement with the exact energies. GCM wave functions generated by particle number and isospin projected BCS functions with different amounts of pairing correlations yield for the lowest T = 0 and T = 2 states energies which are practically indistinguishable from the exact solutions. But even the second and third lowest energies of charge-symmetric states are still very reliable. Thus we conclude that also in realistic cases isospin rotational and pairing vibrational states may be described in the framework of the GCM method with isospin and particle number projected generating wave functions.     

The influence of correlations on the odd-mass Nuclei21Ne and25Mg

Excitation energies and electromagnetic properties of the low energy spectra of²¹Ne and²⁵Mg have been calculated using the Multi-Configuration-Hartree-Fock (MCHF) model. BothT=0 andT=1 pairing correlations are found to be simultaneously important in those odd mass nuclei. Furthermore, though axial symmetry was requested, quite good agreement with the experimental data is reached.     

Mass and charge distributions with correlated exchange

The dependence of the inclusive variancesσ _A ² ,σ _Z/2 andσ _N ² in nuclear collisions is investigated as a function of the correlation coefficient ρ for isospin correlated nuclon exchange on the dinuclear potential energy surface (PES). Variances for neutron-proton cluster exchange (single step process with zero net momentum between the constituents) are also evaluated. This analysis is helpful to understand the nature of the exchange mechanism. There exist a few experimental data which are at variance with either uncorrelated neutron-proton exchange or with isospin correlated exchange. An explanation of this data may possibly lie in coexisting neutron-proton cluster exchange.     

The empirical form of the effective nucleon-nucleon interaction in a model space with correlatedJ=0 pairs

Like in earlier work by Schiffer et al. the effective interaction is derived from experimental two-body multiplets. However, now the assumption is that a multiplet state is formed by two unpaired fermions relative to a core of correlatedJ=0 pairs. Then the need for two ranges, as proposed by Schiffer, disappears for the force between identical nucleons in a model space which is large enough to include pairing correlations. A form with a single attractive medium range is preferred for the identical nucleon interaction in order to reproduce collective 2⁺ states in even-even nuclei. In contrast, the proton-neutron force requires a very short range or two ranges to reproduce the empirical values of multipole coefficients, observed in odd-odd nuclei. Therefore we discuss the fact that the effective interaction is not always isospin invariant. As a typical case broken-pair calculations in theN=50 region are considered. But the conclusions drawn, will also apply to other regions of the periodic table.     

Isospin Effect on the Emission Mechanism of Hot Nuclei in 35MeV/u 40Ar +112Sn/124Sn Reactions

As an extension of radioactive ion beam physics, the research on isospin dependent properties of hot nuclei has increasingly attracted considerable interest. The isospin effect on the decay of hot nuclei in reactions 35MeV/u ⁴⁰Ar+¹¹²Sn/¹²⁴Sn has been investigared. It is concluded that due to Coulomb repulsion and instability, proton-rich hot nuclei probably emit prides such as a with high energy to increase their neutron-proton ratio. Moreover, the decay chain for those particles is seemingly long and the emission Probability is high. Thus, the conventional observations, for instance, the ‘slope temperature’ extracted from energy spectrum, may very appreciably with the measured particles.     

Pairing and quartetting in exotic nuclei

A review is given of pair correlations in nuclei with an emphasis on the symmetry character of the superfluid solution which depends on (i) the isospin of the nucleus and (ii) the relative strength of the T=0 and T=1 pairing forces. The most general SO(8) model which accommodates neutrons and protons as well as T=0 and T=1 pairing, is solvable in three limits: only T=0 pairing, only T=1 pairing and equal strengths in the two channels. In these limits, the superfluid ground-state solution of N=Z nuclei exhibits a quartet structure. The competition between superfluidity and magicity is discussed with reference to integrable models. To cite this article: P. Van Isacker, C. R. Physique 4 (2003).     

Diquark correlations in the nucleon

We investigate the structure of the nucleon and the Δ resonance employing a constituent chiral quark model. We propose a variational approach which allows the nucleon to have a [21] flavor-spin symmetry as well as the usual [3] symmetry. This means one gives also up the requirement that the spatial wave function must be symmetric [3]. One then has also to admix the [21] spatial symmetry for a total antisymmetrization and thus one allows quark-diquark correlations, where two quarks are more closely bound than the other two possible pairs. It is found that a quark pair with isospin T = 0 and spin S = 0 in the nucleon becomes spatially very close compared with other pairs in order to gain a strong attraction due to the pion exchange in the chiral quark model, which implies a diquark correlation in the nucleon. We also calculate the proton and neutron charge square radii.     

Measurements of the neutron-proton analyzing power in the energy range from 17 to 50 MeV

Measurements of the analyzing power A_y for neutron-proton scattering in the energy range from 17 to 50 MeV are reported. These data improve considerably the precision of the np data base in this energy range. Preliminary phase-shift analyses indicate reduced uncertainties in the np ³P(T = 1) phases and in the ³D(T = 0) phase shifts.     

Isospin dynamics in heavy ion collisions: From Coulomb barrier to quark gluon plasma

Heavy Ion Collisions (HIC) represent a unique tool to probe the in-medium nuclear interaction in regions away from saturation. In this report, we present a selection of new reaction observables in dissipative collisions particularly sensitive to the symmetry term of the nuclear Equation of State (Iso-EoS). We will first discuss the Isospin Equilibration Dynamics. At low energies, this manifests via the recently observed Dynamical Dipole Radiation, due to a collective neutron-proton oscillation, with the symmetry term acting as a restoring force. At higher beam energies, Iso-EoS effects will be seen in Imbalance Ratio Measurements, in particular from the correlations with the total kinetic energy loss. For fragmentation reactions in central events, we suggest to look at the coupling between isospin distillation and radial flow. In Neck Fragmentation reactions, important Iso-EoS information can be obtained from the correlation between isospin content and alignment. The high density symmetry term can be probed from isospin effects on heavy ion reactions at relativistic energies (few AGeV range). Rather, isospin sensitive observables are proposed from nucleon/cluster emissions, collective flows and meson production. The possibility to shed light on the controversial neutron/proton effective mass splitting in asymmetric matter is also suggested. A large symmetry repulsion at high baryon density will also lead to an “earlier” hadron-deconfinement transition in n-rich matter. A suitable treatment of the isovector interaction in the partonic EoS appears very relevant.     

The T = Tz + 2 GDR and the dipole states of the A = 6 and 14 nuclei

The isospin properties of the giant dipole resonance (GDR) built on states with isospin T = T_z + 1 are discussed. Results of a shell-model calculation with realistic interactions for dipole transitions to various low-lying states in ⁶Li, ⁶He, ¹⁴N and ¹⁴C are presented. The T = T_z + 2 component of the GDR in ¹⁴N decaying to the 0⁺T = 1 state is predicted at the surprisingly low energy E_x = 26.0 MeV.     

Isospin dependence of the nuclear level width in the compound nucleus 30P

The excitation functions of ²⁹Si(p, α₀)²⁶Al, ²⁹Si(p, α₁)²⁶Al and ²⁹Si(p, α₂)²⁶Al were measured with high beam energy resolution in order to determine the isospin dependence of the nuclear level width of the composite nucleus, ³⁰P, at an average excitation energy of 19.6 MeV. From an auto-correlation analysis of these excitation functions, the level widths of the T_< = 0 and T_> = 1 isospin states are determined as 81 ± 17 keV and 104 ± 35 keV, respectively. An analysis including isospin mixing is also performed. With the aid of the statistical theory of nuclear reactions, the coherence energies are used to deduce the relative densities of two isospin states. The predictions of the Fermi gas model of level densities including isospin are in good agreement with the obtained results.