M1 Form Factors Research Articles

Effects of Velocity-Dependent Force on the Magnetic Form Factors of Odd-Z Nuclei

We investigate the effects of the velocity-dependent force on the magnetic form factors and magnetic moments of odd-Z nuclei. The form factors are calculated with the harmonic-oscillator wavefunctions. It is found that the contributions of the velocity-dependent force manifest themselves in the very large momentum transfer region (q  4fm-1). In the low and medium q region the contributions of the velocity-dependent force are very small compared with those without this force. However, in the high-q region the contributions of the velocity-dependent force are larger than the normal form factors. The diffraction structures beyond the existing experimental data are found after the contributions of the velocity-dependent force are included. The formula of the correction to the single particle magnetic moment due to the velocity-dependent force is reproduced exactly in the long-wavelength limit (q = 0) of the M1 form factor.

Transverse form factors of 117Sn.

Transverse elastic and inelastic form factors for low-lying levels in $^{117}\mathrm{Sn}$ have been measured by 180\ifmmode^\circ\else\textdegree\fi{} electron scattering in the momentum transfer range ${\mathit{q}}_{\mathrm{eff}}$=1.1\char21{}2.4 ${\mathrm{fm}}^{\mathrm{\ensuremath{-}}1}$. The simple independent-particle model fails to account for these data, predicting form factors that are much too large. In better quantitative agreement with the data are the results of more detailed calculations that allow for configuration mixing of valence nucleons, as well as first-order core polarization. As in the similar case of $^{205}\mathrm{Tl}$, these calculations successfully predict the presence of a deep diffraction minimum observed in the intermediate q region of the elastic M1 form factor. Nevertheless, the overall description of the data by the detailed calculations remains quantitatively unsatisfactory. A more complete understanding of these data may rely upon the consideration of multiparticle-multihole configurations outside the 0\ensuremath{\Elzxh}\ensuremath{\omega} basis space.

ON THE SEARCH FOR THE SCISSORS MODE

Results from quasiparticle RPA calculations on 1+ excitations in deformed nuclei, including M1 transition densities and (e, e′) form factors, are compared to the microscopic representation of the scissors state of the two-rotor model. Although this state is too collective, especially in heavy nuclei, it fragments mainly over the low lying orbital M1 excitations, which can be interpreted as a manifestation of a weakly collective scissors mode. The experimentally observed strongest M1 excitation has the same leading components in the wave function as the scissors state, the largest overlap with it, and similar transition density and form factor. Although the scissors mode is more pronounced in heavy deformed nuclei, the strong low lying orbital M1 excitations in lighter nuclei represent a better approximation to it, because the scissors state is less collective in light nuclei.

Elastic magnetic electron scattering from 13C at Q2=1 GeV2/c2.

Electron scattering from $^{13}\mathrm{C}$ was measured at a momentum transfer of 5.08 ${\mathrm{fm}}^{\mathrm{\ensuremath{-}}1}$. No electron events were observed in the vicinity of the elastic peak, giving an upper limit for the elastic cross section of 1.2\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}39}$ ${\mathrm{cm}}^{2}$/sr with a confidence level of 90%. At this momentum transfer, the square of the elastic M1 form factor apparently continues to fall exponentially with q. Comparison of the data with shell-model calculations indicates that admixtures of 2\ensuremath{\Elzxh}\ensuremath{\omega} configurations in the $^{13}\mathrm{C}$ ground state cannot entirely explain the high-q enhancement of the M1 form factor relative to 1p-shell calculations.

M1 strength and (e, e′) form factors of 46,48Ti within the RPA

The abundant experimental information on low- and high-energy M1 excitations in 46,48Ti, obtained recently through (e, e′), (p, p′) and (γ, γ′) experiments, is well described within a quasiparticle RPA approach, based on a deformed Woods-Saxon potential plus spin-spin and quadrupole-quadrupole interactions. It includes a procedure which restores the rotational invariance of the hamiltonian. The low-lying strong M1 states overlap up to 36% with the scissors state of the two-rotor model. These states, as well as some medium-energy 1+ excitations, represent a weakly collective scissors motion, more pronounced in 46Ti than in 48Ti. The titanium isotopes offer a better approximation to the scissors state than do the heavy nuclei because in lighter nuclei the scissors state is less collective.

Detailed study of the cluster structure of light nuclei in a three-body model: (III). Electromagnetic structure of 6Li

The multicluster dynamic model with the Pauli projection (MDMP) for treating light nuclei, developed earlier, is used to study in detail the 6Li electromagnetic structure including all the measured elastic and inelastic, transverse and longitudinal, isoscalar and isovector electromagnetic form factors of the nucleus and to make some predictions concerning other, not yet measured, form factors. The model is also used to calculate all the measured radiation widths Γ γO of the excited 6Li * states. A proper description has been obtained for most of the known electromagnetic form factors of the nucleus. The reasons for disagreement with experimental data in the case of large momentum transfers in the M1 form factors, both elastic and inelastic, are discussed. The elastic CO form factor of 6Li and, probably, of other light nuclei in the region of the secondary maximum is shown to be defined, to a very great extent, by the behaviour of the α-particle charge form factor in the region of its secondary maximum. The radiation widths of the 3 +0, 2 +0, and 1 +0 levels have been found to be highly sensitive to minor impurities of the wavefunction components with higher orbital moments for the excited states, as well as the ground state, of 6Li. The Siegert theorem is shown to be of great importance when studying the probabilities of transverse electromagnetic transitions. The copious results obtained by the present authors, as well as elsewhere, concerning the diverse aspects of the behaviour of six-nucleon system in the strong, weak, and electromagnetic processes are used to formulate the concept of using the given system as an extremely convenient theoretical laboratory in nuclear physics. The relevant proposals concerning future experiments are formulated.

Microscopic IBA calculations of (e, e') M1 form factors

General theory of transition charge and current densities is formulated for inelastic electron scattering. Single-particle densities are calculated. The many-fermion matrix element of a general one-body operator is derived for states of generalized seniority two. The Otsuka-Arima-Iachello mapping is performed to yield the M1 transition current of the interacting-boson approximation (IBA). Transition currents and DWBA form factors are computed for the strong 1 + excitations in 154Sm, 154–158Gd, 164Dy and 168Er. Particular emphasis is placed on 164Dy because of its rich experimental data. Boson g-factors are computed for these nuclei. All parameters are taken from the literature and not fitted to the scattering data. Particle-hole symmetry is discussed.

Orbital rotational vibrations in the A=130 mass region.

The rotational vibrations (${\mathit{K}}^{\mathrm{\ensuremath{\pi}}}$=${1}^{+}$ states) in 16 even-even Xe, Ba, and Ce nuclei are studied in the quasiparticle random-phase approximation with a mean field given by a deformed Woods-Saxon potential and residual forces: a self-consistent quadrupole-quadrupole interaction, a spin-spin interaction, and a force that restores the rotational invariance of the Hamiltonian. A shell effect is found which is typical for this mass region: a strong orbital character of almost all low-energy (2.5--5 MeV) excitations, while the higher-energy ones are predominantly spin flip. The comparison of random-phase approximation M1 transition densities and (e,e') form factors with the microscopic realization of the two-rotor model ${1}^{+}$ state allow us to conclude that the strongly orbital low-energy random-phase approximation states perform the scissor-type motion described by the two-rotor model, but only few particles are involved in this motion.

14C beta decay and the 14N M1 form factors.

The description of p-shell orbitals in the A=14 nuclear system in terms of relativistic wave functions, which are solutions to the Dirac equation with strong scalar and vector potentials, provides a good description of the M1 form factors in $^{14}\mathrm{N}$ and the greatly suppressed $^{14}\mathrm{C}$ beta decay rate.

A deformed model approach to elastic magnetic scattering from 29Si

The elastic magnetic form factor of 29Si is calculated using the deformed model. Agreement is found with the experimental data for the M1 form factor and static moment using Nilsson wave functions with a deformation parameter δ ⋍ −0.2. An analysis of the dependence on deformation of the magnetic form factor which applies to many odd-nucleon Nilsson orbitals is presented.

Magnetic response of closed-shell +/-1 nuclei in Dirac-Hartree approximation.

We formulate a systematic approximation to the Dirac-Hartree description of closed-shell \ifmmode\pm\else\textpm\fi{}1 nuclei using the closed-shell basis. Expectation values of one-body operators include random-phase-approximation-like loop corrections to the single particle (or hole) results. We use a nonspectral form of the Hartree propagator which allows inclusion of positive- and negative-energy continuum contributions exactly. The loop corrections quench the single-particle isoscalar magnetic moments back to the Schmidt values, resulting in good agreement with experiment. Elastic M1 electron scattering form factors are also quenched at low q. However, the loop corrections vanish at q\ensuremath{\simeq}1.2 ${\mathrm{fm}}^{\mathrm{\ensuremath{-}}1}$ and change sign so as to enhance the single-particle result at still higher momentum transfers. This raises the possibility that experimental signatures of the strong relativistic nuclear dynamics characteristic of Dirac-Hartree models may be present in elastic M1 form factors for closed-shell \ifmmode\pm\else\textpm\fi{}1 nuclei.

Simultaneous GCSM description of the M1 state and the major collective bands

The generalized coherent state model is used to simultaneously describe the magnetic state 1 + and the collective bands: ground, beta and gamma. Effective operators for the M1 and the E2 transitions are derived. Also the M1 form factor for the (e, e′) scattering is treated in the plane wave Born approximation. The symmetry of the lowest gamma band is discussed. The application is made for 154Gd, 154Sm, 164Dy, 168Er, 174Yb, 232Th, 238U. The agreement with the experiment is very good for both the 1 + state and the considered collective bands.

On the scissors type mode in 46Ti and lighter nuclei

A microscopic formulation of the scissors mode, based on the angular-momentum projected Hartree-Fock Bogoljubov approximation, is applied to the Ti and Ne isotopes. Results for M1 transition strengths and form factors are presented and compared to available experimental data and shell model calculations. For 46Ti the B( M1)↑ value obtained is in excellent agreement with the experimental data for the 1 + state at 4.3 MeV.

On magnetic dipole form factors in deformed nuclei

Properties of magnetic dipole form factors for deformed nuclei are discussed in terms of the angular momentum projected Hartree-Fock-Bogoljubov approximation and the neutron-proton interacting boson model. It is pointed out that there exists a relation between the M1 form factor for the excitation of the orbital K π =1 + band, the M1 form factor for the ground-state band, and the collective M1 form factor in odd- A nuclei.

Phenomenological wave functions for the ground and 2.313 MeV states in sup14N.

Transverse electron scattering form factors have been measured for elastic scattering and for the transition to the 2.313 MeV state in $^{14}\mathrm{N}$. Existing structure models could not simultaneously describe the two M1 form factors. Consequently, new phenomenological wave functions were determined by fitting the electron scattering results and other observables. For the 2.313 MeV transition, the new wave functions give smaller L=2 transition amplitudes than the earlier models. This reduction and the deduced value for the L=0 transition amplitude appear to be supported by the measurements of the $^{14}\mathrm{N}$(p,p'${)\mathrm{}}^{14}$N, $^{14}\mathrm{C}$(p,n${)}^{14}$N, and $^{14}(\ensuremath{\gamma}$,${\ensuremath{\pi}}^{+}$${)\mathrm{}}^{14}$C reactions. Although the L=0 transition amplitude is found to be relatively small, it is not small enough to give quantitative agreement with the severely retarded $^{14}\mathrm{C}$ \ensuremath{\beta}-decay rate. The possibility that the vanishingly small \ensuremath{\beta}-decay matrix element is the result of destructive interference between the one-body matrix element and other terms is discussed.

Are the low-lying isovector 1 + states scissor vibrations?

In 1984 Richter and coworkers detected in deformed nuclei in electron scattering low-lying isovector 1 + states. In the same year Dennis Hamilton and coworkers from the University of Sussex found in (n,γ)-reactions low-lying 2 + isovector states in spherical nuclei. Such states have been predicted in several theoretical models. The first prediction was work of the author in 1966 where he extended the Bohr-Mottelson-model to include independent collective motions of the proton and the neutron distributions in nuclei. Both motions are coupled by the symmetry energy which dislikes demixing of the protons and neutrons. Thus one obtains the usual low-lying isoscalar quadrupole vibrations and rotations, but in addition also isovector states in spherical and deformed nuclei. In these states protons and neutrons move out of phase. The 1 + states found by Richter and coworkers in deformed nuclei can then be interpreted as vibrations of the deformed proton and neutron shapes in a scissor like way. In spherical nuclei the data of Dennis Hamilton and coworkers can be explained as harmonic quadrupole vibrations of protons and neutrons out of phase around the spherical equilibrium. We show here that this collective interpretation can very nicely reproduce the energies of the isovector 1 + states in deformed nuclei and of the isovector 2 + states in spherical nuclei. If the parameters are fixed to the low-lying isoscalar quadrupole rotations or vibrations the energies of the isovector states can be predicted without adjusting an additional parameter. Although the energies are reproduced very nicely one finds in deformed nuclei that the B( M1;0 +→1 +)-transition probability is too collective by a factor 5. In this collective description the M1-strength is concentrated in one state. In reality this isovector 1 + state contains only part of the Oπω-two-quasi-particle exitations. The rest of the strength is shifted to higher energies. A microscopic description using the quasi-particle random phase approximation or MONSTER can reproduce the exitation energies and transition probabilities correctly. It turns out to be important that in deformed nuclei the spurious state connected with the rotation of the whole system has to be put to zero energy by adjusting the parameter of the scalar part of the nucleon-nucleon interaction. In addition, one has to take care that the low-lying isovector 1 +-state is orthogonal to the spurious state |S>. If we request that the Hamiltonian is rotational invariant, that means that it commutes with the angular momentum operator, we can determine the nucleon-nucleon interaction from the given one-body part. For the one-body Hamiltonian we use the kinetic energies of the single nucleons, a deformed Saxon-Woods potential with spin-orbit coupling and slightly different shapes for protons and neutrons. By the quasi-particle transformation we include also a pairing interaction into the one-body piece. With the requirement that the Hamiltonian should commute with the total angular momentum, one can determine the many-body Hamiltonian parameter free if the Saxon-Woods potential and the pairing force are given. This symmetry restoring interaction is very nicely reproducing the energies and the magnetic transition probabilities from the ground state to these low-lying isovector 1 + states in the rare earth nuclei. We furthermore apply this formalism to the Ti-isotopes. There we get also similar satisfactory results using MONSTER and VAMPIR. The theoretical results indicate that the low-lying isovector 1 + states in deformed nuclei are in average half of orbital and half of spin character. A comparison of the excitation of these states with electron scattering and with proton scattering suggest that they are of more purer orbital character. A closer analysis of these measurements shows that they are not contradicting to a spin and an orbital character of these states which is roughly half and half. For the MONSTER we compare also M1-form factors with the data and find excellent agreement.

Importance of longitudinal contributions to backward inelastic electron scattering

Coulomb contributions of multipolarity 2 are calculated for backward ( θ = 180°) inelastic electron scattering by our improved DWBA code. General features are studied such as nuclear charge ( Z) dependence, energy transfer dependence, momentum transfer dependence as well as the angle dependence near θ = 180°, with the help of a simple model for the transition charge density. We compare these features with PWBA and other simple approximations. We show, as a specific example, that the C2 contribution is comparable to the M1 and E2 multipole form factors for inelastic scattering to the rotational excited states of 181Ta.

Nucleonic versus nuclear spin-isospin polarization: A study of the 48Ca and 88Sr M1 form factors

We compare standard nuclear polarization mechanisms, Δ-hole-polarization and meson-exchange-current effects in the q-dependent quenching of isovector spin transitions. Calculations are performed for the M1-transition form factors of the 1 + states in 48Ca (10.23 MeV) and 88Sr (3.48 MeV). We obtain a satisfactory description of both form factors if the repulsive part of the residual interaction in the Δ-hole channel is of similar strength to that in the nucleon-hole channel. Meson-exchange currents lead to an enhancement of M1 transitions by an amount which is small in general, but sensitive to the particular nuclear state involved.

Comparison of isoscalar and isovector (e, e′) M1 form factors at high momentum transfer

Electron scattering M1 form factors have been measured for the ground state and for the 2.313 MeV M1 transition in 14N. Whereas the ground-state form factor is in good accord with 1p-shell models, the data for the 2.313 MeV transition show an unexplained enhancement at high momentum transfers.

Parametrization of the short-range correlation and exchange matrix elements by the Landau-Migdal parameters

The effects of the short-range correlation and the exchange part of two-body matrix elements in the particle-hole and Δ-hole excitation contributions in M1 form factors are parametrized by the Landau-Migdal parameters g′ N and g′ Δ . Th momentum transfer ( q) dependences of the parametrized values of g′ N and g′ Δ are investigated.