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The nuclear current is divided into the sum of an irrotational and a vortical current. This vortical current is a transverse current whose source is the vorticity. We investigate how to determine experimentally the transition vorticity and vortical current densities, using data on inelastic electron scattering from 0954-3899/25/3/006/img5, leading to the 0954-3899/25/3/006/img6 state at 16.11 MeV. We also discuss how to calculate these quantities theoretically.

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We review a recent progress in investigation of the isoscalar toroidal dipole resonance (TDR). A possible relation of the TDR and low-energy dipole excitations (also called a pygmy resonance) is analyzed. It is shown that the dipole strength in the pygmy region can be understood as a local manifestation of the collective vortical toroidalmotion at the nuclear surface. Application of the TDR as a measure of the nuclear dipole vorticity is discussed. An anomalous splitting of the TDR in deformed nuclei is inspected.

Two basic concepts of nuclear vorticity, hydrodynamical (HD) and Rawenthall-Wambach (RW), are critically inspected. As a test case, we consider the interplay of irrotational and vortical motion in isoscalar electric dipole E1(T =0) modes in 208Pb, namely the toroidal and compression modes. The modes are described in a self-consistent random-phase approximation (RPA) with the Skyrme force SLy6. They are examined in terms of strength functions, transition densities, current fields, and form factors. It is shown that the RW conception (suggesting the upper component of the nuclear current as the vorticity indicator) is not robust. The HD vorticity is not easily applicable either because the definition of a velocity field is too involved in nuclear systems. Instead, the vorticity is better characterized by the toroidal strength which closely corresponds to HD treatment and is approximately decoupled from the continuity equation.

Two basic concepts of nuclear vorticity, hydrodynamical (HD) and
Rawenthall-Wambach (RW), are critically inspected. As a test case, we consider
the interplay of irrotational and vortical motion in isoscalar electric dipole
E1(T=0) modes in $^{208}$Pb, namely the toroidal and compression modes. The
modes are described in a self-consistent random-phase-approximation (RPA) with
the Skyrme force SLy6. They are examined in terms of strength functions,
transition densities, current fields, and formfactors. It is shown that the RW
conception (suggesting the upper component of the nuclear current as the
vorticity indicator) is not robust. The HD vorticity is not easily applicable
either because the definition of a velocity field is too involved in nuclear
systems. Instead, the vorticity is better characterized by the toroidal
strength which closely corresponds to HD treatment and is approximately
decoupled from the continuity equation.

The last years developments in experimental technique improved the experimental resolution at large angles of the registration of elasticaly scattered complex particles. This enables the observation of fragmented momentum states as a signature of Bose- Einstein condensation in light nuclei. Given the exchange reaction mechanism, anomalous backward scattering may indicate the presence of vortex states in nuclei. In the framework of Cranking model we considered vortex states in light nuclei. These vortex states can be detected at backward angles where the angular momentum operator applied to the wave function gives eigenvalue proportional to the number of particles involved into vortex movement (maximal eigenvalue). The possible candidates for the vortex states could be 3− state in12C and 4+ state in16O.

The toroidal, compression and vortical dipole strength functions in
semi-magic $^{124}$Sn (and partly in doubly-magic $^{100,132}$Sn) are analyzed
within the random-phase-approximation method with the SkT6, SkI3, SLy6, SV-bas,
and SkM* Skyrme forces. The isoscalar (T=0), isovector (T=1), and
electromagnetic ('elm') channels are considered. Both convection $j_c$ and
magnetization $j_m$ nuclear currents are taken into account. The calculations
basically confirm the previous results obtained for $^{208}$Pb with the force
SLy6. In particular, it is shown that the vortical and toroidal strengths are
dominated by $j_c$ in T=0 channel and by $j_m$ in T=1 and 'elm' channels. The
compression strength is always determined by $j_c$. It is also shown that the
'elm' strength (relevant for (e,e') reaction) is very similar to T=1 one. The
toroidal mode resides in the region of the pygmy resonance. So, perhaps, this
region embraces both irrotational (pygmy) and vortical (toroidal) flows.

The transition density and current provide valuable insight into the nature of nuclear vibrations. Nuclear vorticity is a quantity related to the transverse transition current. In this work, we study the evolution of the strength distribution, related to density fluctuations, and the vorticity strength distribution, as the neutron drip line is approached. Our results on the isoscalar, natural-parity multipole response of Ni isotopes, obtained by using a self-consistent Skyrme–Hartree–Fock + Continuum RPA model, indicate that, close to the drip line, the low-energy response is dominated by L>1 vortical transitions.

The multipole vortical, toroidal, and compression modes are analyzed.
Following the vorticity concept of Ravenhall and Wambach, the vortical operator
is derived and related in a simple way to the toroidal and compression
operators. The strength functions and velocity fields of the modes are analyzed
in $^{208}$Pb within the random-phase-approximation using the Skyrme force
SLy6. Both convection and magnetization nuclear currents are taken into
account. It is shown that the isoscalar (isovector) vortical and toroidal modes
are dominated by the convection (magnetization) nuclear current while the
compression mode is fully convective. The relation between the above concept of
the vorticity to the hydrodynamical vorticity is briefly discussed.

The states of C12 and O16 are computed in the framework of the particle-hole model of nuclear excitations, using a finite range force, with an exchange mixture determined by a least square fit. The particle-hole energies are obtained from the spectra of the neighbouring odd-mass nuclei, in the hypothesis of a spherical scheme and of pure j-j coupling. The results given respectively by the Elliot and Flowers procedure and the random phase approximation (RPA) are compared. The parametric behaviour of the areas of strong discrepancies is delimited. An overall good agreement with the odd parity states O16 is reached. In the case of C12, while reproducing many features of its spectrum, it is not found possible to obtain such a close agreement with forces consistent with spherical stability of the nucleus.

We examine the importance of the gauge dependence of model calculations of transverse electric form factors coming from the violation of the Schrödinger equation by the model wavefunctions. This is done by comparing Tamm - Dancoff approximation (TDA) and random phase approximation (RPA) calculations of transverse electric form factors for collective states of negative parity in . In our discussion we isolate the term responsible by the gauge dependence and clarify its physical meaning. We also point out some general properties of the RPA currents which help us to understand the behaviour of the gauge dependence in RPA model calculations.

We present experimental data on inelastic electron scattering from 12C leading to the 2+ state at 16.11 MeV. The E2 and C2 form factors are analysed in a conventional harmonic-oscillator shell model. Several sets of nuclear structure coefficients arc extracted by adjusting the theoretical form factors to reproduce the experimental data. One is restricted to the 1p shell only and can be used for other reactions involving this level, while another one allows for the admixture of higher configurations. The implications of the Siegert theorem are also discussed. Employing the nuclear structure input we performed calculations for the process using a nonlocal DWIA approach. The elementary pion production operator is assumed to be given by the Blomqvist-Lyget amplitude, while the final-state interaction of the existing pion is provided by an optical potential that is in agreement with pion scattering. We compare our computations with experimental data (γ, π) at a number of different photon lab energies.

In high-energy electron-nucleus scattering, a new quantity for characterizing the additional information about the structure of a nuclear transition contained in the current distributions is defined, the vorticity density. It is symmetrical between the states involved in the transition, and it is completely unconstrained by the transition charge distribution. It is evaluated and discussed for single-particle harmonic oscillator transitions, for hydrodynamic collective transitions, and then for 3− transitions in 208Pb as described by the random phase approximation. Its relationship to collectivity is discussed.

Form-factor measurements in the momentum-transfer range 0.6q2.7 fm-1 are presented for the 1- levels in O18 at 4.46, 6.20, 7.62, and 8.04 MeV, the 3- levels at 5.10, 6.40, and 8.29 MeV, and the 5- levels at 7.86 and 8.13 MeV. These are the first measurements of the 5- states by inelastic scattering and the first electron-scattering measurements of the higher 1- and 3- states. A Rosenbluth separation of the longitudinal and transverse form factors was performed by fitting the data with a phenomenological polynomial-times-Gaussian parametrization motivated by the form of theoretical form factors when harmonic-oscillator wave functions are used. Comparisons are made with structure models. The Coulomb form factors of several levels indicate the significance of small admixtures of 3Latin small letter h with stroke components in the negative-parity wave functions.

Comparison is made between calculations of transverse electric form factors using the standard expression for the electric multipole operator and those obtained by invoking current conservation in the long wavelength limit and for arbitrary momentum transfer. In all cases, only the free-nucleon one-body current and charge operators are explicitly used. Results are presented for select E2 transitions in 12C, 20Ne, 24Mg, and 28Si. It is found that the form factors yielded by the various operators differ significantly when the conventional 0homega shell model wave functions are used, confirming that these do violate conservation of the usual free-nucleon current. For these cases, the data are best reproduced using the operator with current conservation invoked in the long-wavelength limit. However, the variation between results is much smaller when multishell models of nuclear structure are used, all three forms of the operator yielding good agreement with existing data, suggesting that the wave functions obtained from these still practicable models are significantly closer to being current conserving.