Evolution of nuclear charge radii in copper and indium isotopes

Systematic trends in nuclear charge radii are of great interest due to universal shell effects and odd-even staggering (OES). The modified root mean square (rms) charge radius formula, which phenomenologically accounts for the formation of neutron-proton (np) correlations, is here applied for the first time to the study of odd-Z copper and indium isotopes. Theoretical results obtained by the relativistic mean field (RMF) model with NL3, PK1 and NL3^* parameter sets are compared with experimental data. Our results show that both OES and the abrupt changes across \begin{document}$ N = 50 $\end{document} and 82 shell closures are clearly reproduced in nuclear charge radii. The inverted parabolic-like behaviors of rms charge radii can also be described remarkably well between two neutron magic numbers, namely \begin{document}$ N = 28 $\end{document} to 50 for copper isotopes and \begin{document}$ N = 50 $\end{document} to 82 for indium isotopes. This implies that the np-correlations play an indispensable role in quantitatively determining the fine structures of nuclear charge radii along odd-Z isotopic chains. Also, our conclusions have almost no dependence on the effective forces.     

Symmetry dependence of rms Charge Radii

The nucleon number dependence of rms charge radii is often approximated by some simple formula containing the mass number only, R(A)=r(A)A^1/3, where r(A) is a slowly varying function of A. However, for nuclei off the stability line, the mass number A=N + Z is not enough to characterise the dependence of the R(Z, N) radius surface on the nucleon numbers Z and N. In the present work, an additional term has been included, depending on the symmetry parameter I=(N ? Z)/A. Several parametrisations were tried, using weighted least-squares procedures for the fit to a present-day data base. The best fit (with χ²/ń=17) was found for R(A, I)=r(A)A^1/3 + b_I/(I ? I_stab), where I_stab=(N_stab ? Z_stab)/A is the value of the symmetry parameter of the stable isobar with mass number A, and b_I=?0.83 fm. The formula R(A, I)=[r(A) + a_I(I ? I_stab)]A^1/3 is only slightly inferior to the previous one, moreover, it is supported by simple model calculations; here a_I=?0.20 fm (χ²/ń=20). The difficulty in determining the right parametrisation is caused by the fact that the surface R_exp(A, I) is not smooth: there are strong shell and deformation effects. To avoid the distorting effect of these deviations on the parameter values, more than half of the original data had to be omitted.     

Backbendings of superdeformed bands in ~(36,40)Ar

Experimentally observed superdeformed(SD) rotational bands in ~(36)Ar and ~(40)Ar are studied by the cranked shell model(CSM) with the pairing correlations treated by a particle-number-conserving(PNC) method.This is the first time that PNC-CSM calculations have been performed on the light nuclear mass region around A=40.The experimental kinematic moments of inertia J~((1))versus rotational frequency are reproduced well. The backbending of the SD band at frequency around ω =1.5 Me V in ~(36)Ar is attributed to the sharp rise of the simultaneous alignments of the neutron and proton 1 d_(5/2)[202]5/2 pairs and 1 f_(7/2)[321]3/2 pairs, which is a consequence of the band crossing between the 1 d_(5/2)[202]5/2 and 1 f_(7/2)[321]3/2 configuration states. The gentle upbending at low frequency of the SD band in ~(40)Ar is mainly affected by the alignments of the neutron 1 f_(7/2)[321]3/2 pairs and proton 1 d_(5/2)[202]5/2 pairs.The PNC-CSM calculations show that besides the diagonal parts, the off-diagonal parts of the alignments play an important role in the rotational behavior of the SD bands.     

13C NMR Chemical Shift of β-Alkoxyvinylketones: II. Empirical Substituent Effects in β-Aryl-β-Methoxyvinyltrihalomethylketones

Evaluation by empirically derived equations for the substituent effect (α,β,γ,δ) on the ¹³C NMR chemical shifts for C-1, C-2, C-3 and C-4 in β-aryl-β-methoxyvinylhalomethylketones 1a-g to 2a-g [R³C(O)-CH=C(Ar)-OMe, where R³ = CCl₃, CF₃ and Ar = p-YC₆H₄ (Y = H, Me, MeO, F, Cl, Br, NO₂)], taking as reference the β-ethoxyvinyltrichloromethylketone (3), is reported. From the calculated values for the α,β,γ,δ effects for each substituent it was possible to estimate the chemical shift of each carbon of the compounds 1,2. The ¹³C chemical shifts of the C-1, C-2, C-3, C-4 of these compounds, can be estimated with good to rasoable precision: 84% of the calculated chemical shifts are found to be within ±1.0ppm, and 100% are found to be within ±1.5ppm. The Y-Effects on C-3 and C-4 are compared with carbon charge densities (qr).     

Nuclear charge radii and electromagnetic moments of scandium isotopes and isomers in the f7/2 shell

Collinear laser spectroscopy experiments on the ScII transition 3d4s ³D₂→3d4p ³F₃ at λ ≈ 363.1 nm were performed on the ^42–46Sc isotopic chain using an ion guide isotope separator with a cooler–buncher. Isotope and isomer shifts and hyperfine structures of five ground states and two isomers were measured. Preliminary results on the nuclear moments and charge radii changes deduced from these measurements are reported.     

Shell structure fingerprints of tensor interaction

We address the consequences of strong tensor terms in the local energy density functional, resulting from fits to the f _5/2 -f _7/2 splittings in ⁴⁰Ca , ⁴⁸Ca , and ⁵⁶Ni . In this study, we focus on the tensor contribution to the nuclear binding energy. In particular, we show that it exhibits an interesting topological feature closely resembling that of the shell correction. We demonstrate that in the extreme single-particle scenario at spherical shape, the tensor contribution shows tensorial magic numbers equal to N(Z) = 14 , 32, 56, and 90, and that this structure is smeared out due to configuration mixing caused by pairing correlations and migration of proton/neutron sub-shells with neutron/proton shell filling. Based on a specific Skyrme-type functional SLy4_T, we show that the proton tensorial magic numbers shift with increasing neutron excess to Z = 14 , 28, and 50.     

Rotational bands in 169Re

Based on the systematic investigation of the data available for nuclei with A≥ 40, a Z ^1/3-dependence for the nuclear charge radii is shown to be superior to the generally accepted A ^1/3 law. A delicate scattering of data around R _c/Z ^1/3 is inferred as owing to the isospin effect and a linear dependence of R _c/Z ^1/3 on N/Z (or (N - Z)/2) is found. This inference is well supported by the microscopic Relativistic Continuum Hartree-Bogoliubov (RCHB) calculation conducted for the proton magic Ca, Ni, Zr, Sn and Pb isotopes including the exotic nuclei close to the neutron drip line. With the linear isospin dependence provided by the data and RCHB theory, a new isospin-dependent Z ^1/3 formula for the nuclear charge radii is proposed. Received: 23 September 2001 / Accepted: 21 January 2002     

Role of cross-shell excitations in the reaction 54Fe($$ \vec d $$, p)55Fe

The reaction ⁵⁴Fe(, p)⁵⁵Fe was studied at the Munich Q3D spectrograph with a 14MeV polarized deuteron beam. Excitation energies, angular distributions and analyzing powers were measured for 39 states up to 4.5MeV excitation energy. Spin and parity assignments were made and spectroscopic factors deduced by comparison to DWBA calculations. The results were compared to predictions by large-scale shell model calculations in the full pf -shell and it was found that reasonable agreement for energies and spectroscopic factors below 2.5MeV could only be obtained if up to 6 particles were allowed to be excited from the f _7/2 orbital into p _3/2 , f _5/2 , and p _1/2 orbitals across the N = 28 gap. For levels above 2.5MeV the experimental strength distribution was found to be significantly more fragmented than predicted by the shell model calculations.     

First Measurement of the Nuclear Charge Radii of Short-Lived Lithium Isotopes

A novel method for the determination of nuclear charge radii of lithium isotopes is presented. Precise laser spectroscopic measurements of the isotope shift in the lithium 2s → 3s transition are combined with highly accurate atomic physics calculation of the mass dependent isotope shift to extract the charge-distribution-sensitive information. This approach has been used to determine the charge radii of ^8,9Li for the first time.     

Effect of Configuration Mixing and Core Excitation on Predominantly (f 7/2)n States in the Lower fp Shell Nuclei

Shell model calculations in the lower fp shell region, (for ^44,45Ca and ⁴⁷Sc nuclei) have been performed, with different model spaces, to probe the effect of configuration mixing from the p^3/2, p^1/2 and f_5/2 orbitals on the predominantly (f_7/2)ⁿ states and the contribution arising from the excitation of the N = Z = 16 core. Our calculations indicate that excitation of nucleons across the N = Z = 16 magic shell closure do contribute significantly towards the wavefunctions of the observed level structures of ^44,45Ca nuclei. However the inclusion of these configurations did not result in a better agreement for the observed level structure of ⁴⁷Sc nulcei. A plaussible explaination for this phenomena could be attributed to the two-body matrix elements used and calls for a detailed micro-scopic calculations involving fundamental interactions substantiated by additional spectroscopic data such as lifetime measurements to have an unambigious understanding of the intrinsic configurations of nuclei in this region.     

High spin states in 63Cu

Excited states of ⁶³Cu were populated via the ⁵²Cr+¹⁰O (65 MeV) reaction using the gamma detector array equipped with charged particle detector array for reaction channel separation. On the basis of γ-γ coincidence relations and angular distribution ratios, a level scheme was constructed up to E _n=7 MeV and J ^π=23/2⁽⁺⁾. The decay scheme deduced was interpreted in terms of shell model calculations, with a restricted basis of the f _5/2, p _3/2, p _1/2, g _9/2 orbitals outside a ⁵⁶ ₂₈Ni core.     

On the liquid drop model mass formulae and charge radii

An adjustment to 782 ground-state nuclear charge radii for nuclei with N, Z 3 \ge8 leads to R₀ = 1.2257 A^1/3\ensuremath R_0 = 1.2257 A^{1/3} fm and s \sigma = 0.124 fm for the charge radius. Assuming such a Coulomb energy E_c = \frac35 e²Z²/1.2257 A^\frac13\ensuremath E_c = \frac{3}{5} e^2Z^2/1.2257 A^{\frac{1}{3}} , the coefficients of different possible mass formulae derived from the liquid drop model and including the shell and pairing energies have been determined from 2027 masses verifying N, Z 3 \ge8 and a mass uncertainty ￡ \le150 keV. These formulae take into account or do not the diffuseness correction ( Z²/A\ensuremath Z^2/A term), the charge exchange correction term ( Z^4/3/A^1/3\ensuremath Z^{4/3}/A^{1/3} term), the curvature energy, the Wigner terms and different powers of I = (N - Z)/A . The Coulomb diffuseness correction or the charge exchange correction term play the main role to improve the accuracy of the mass formulae. The different fits lead to a surface energy coefficient of around 17-18MeV. A possible more precise formula for the Coulomb radius is R₀ = 1.2332A^1/3 + 2.8961/A^2/3 - 0.18688A^1/3I\ensuremath R_0 = 1.2332A^{1/3} + 2.8961/A^{2/3} - 0.18688A^{1/3}I fm with s \sigma = 0.052 fm.     

Emergence of new magic numbers,N = 16 and 32 by tensor interaction in Skyrme–Hartree–Fock theory

Melting of N = 20 shell and development of N = 16 and 32 shells for neutron-rich nuclei have been studied extensively by including tensor interaction in Skyrme–Hartree–Fock theory optimized to reproduce the splitting Δ1f shells of ^40,48Ca and ⁵⁶Ni nuclei. Evolution of gap generated by the energy difference of single-particle levels ν2s _1/2 and ν1d _3/2 has been found to be responsible for shell closure at N = 16. The splitting pattern of spin–orbit partners 2p shell model state in Ca, Ti, Cr, Fe and Ni isotopes indicates the formation of a new shell at N = 32 region.     

Proton-neutron correlations in nuclei with N = 30

Structure of the nuclei with N = 30 and Z = 20–28 is investigated by the nuclear shell model within the proton-neutron configurations (1f_{$\text{7}\text{2}$})^z?20_p × (2p_{$\text{3}\text{2}$}, 2p_{$\text{case1}\text{2}$}, 1f_{$\text{case5}\text{2}$})²_n. Effective proton-neutron interactions determined by a least-squares fit to the observed spectra of N = 29 nuclei are adopted. Agreement of the calculated spectra with experimental spectra is satisfactory. Strong correlations between protons and neutrons break down the pairing scheme and lower the first J = 2 levels in doubly even nuclei, which is shown from the resultant wave functions. A relation between the shell model and collective rotational model is discussed concerning the calculated rotation-like spectrum of ⁵⁶Fe. Electromagnetic properties and spectroscopic factors of single-nucleon transfer reactions are calculated. They are in good agreement with experiments.     

Isotopic and Isotonic Chains of Nuclei in the 2p−1f−1gRegion in the Relativistic Mean Field Approach

A complete systematic set of relativistic mean field calculations are presented for the 2p−1f−1gshell nuclei with 28Z40; 26N50, both for isotopic and isotonic chains. Two sets of Lagrangian parameters NL1 (most successfully used in the past) and NL-SH (recently proposed) are used. Also sensitivity of the calculated results on the gaps required to account for pairing is investigated. The calculated binding energies, one (two) nucleon separation energies, the deformations and quadrupole moments, the point and the charge radii, isotopic shifts, densities, alpha staggering and the related quantities are systematically analyzed. The calculations are in good agreement with the experiment. The set Nl-SH seems to have an edge over the set NL1 especially for the neutron rich nuclei.     

Evolution of the N = 20 and N = 28 shell closures in neutron-rich nuclei

The isospin dependence of shell closure phenomena is studied for light neutron-rich nuclei within a microscopic self-consistent approach using the Gogny force. Introducing configuration mixing, ³²Mg is found to be dynamically deformed, although the N = 20 spherical shell closure persists at the mean-field level for all N = 20 isotones. In contrast, the N = 28 spherical shell closure is found to disappear for N - Z≥ 10 whereas deformed shell closures are preserved and lead to shape coexistence in ⁴⁴ S. Configuration mixing shows that the ground state of this nucleus is triaxially deformed. The first 2⁺ excitation energy E_x = 1.46 MeV and the reduced transition probability B(E2;0⁺ _gs→ 2⁺ ₁)= 420 e ² fm ⁴ obtained with our approach are in good agreement with experimental data. Received: 26 July 2000 / Accepted: 30 August 2000     

The odd-even staggering of the nuclear charge radii of Pb isotopes

Previous laser-spectroscopic studies of the hyperfine structure splittings and isotope shifts of the (6p^{2 3}P₀-6p7s ³P₁; 283.3 nm) PbI resonance line have been extended to further radioactive Pb nuclides: ^{196, 197, 197m, 211, 214}Pb Using an improved theoretical result for the specific mass shift, we recalibrate the isotope shifts of the full series (A = 196–214) of measured isotope shifts in terms of the nuclear parameter and determine the odd-even staggering parameters. The variation of the nuclear charge radii of the Pb nuclides exhibits distinct shell effects, and the odd-even staggering shows a conspicuous trend, being more pronounced for neutron-deficient Pb isotopes.     

Subshell effect in mean square charge radii of stable even cadmium isotopes

Isotope shifts of the 467.8 nm line have been measured by Doppler-free saturation spectroscopy for the even Cd isotopes 106 through 116. The differential changes in mean square charge radii derived from the field shifts show a pronounced break in their trend atN=64, giving evidence for the closure of theg _7/2 neutron subshell. The average charge radii determined from our data and the rms radius of¹¹⁴Cd measured with muonic X-rays verify the general trend predicted by the droplet model but are off by about 35 mfm on an absolute scale. A consistent interpretation of δ〈r ²〉 further supports the necessity for considering changes in the skin thickness of the nuclear charge distribution.