The authors review the present status of self-consistent mean-field (SCMF) models for describing nuclear structure and low-energy dynamics. These models are presented as effective energy-density functionals. The three most widely used variants of SCMF's based on a Skyrme energy functional, a Gogny force, and a relativistic mean-field Lagrangian are considered side by side. The crucial role of the treatment of pairing correlations is pointed out in each case. The authors discuss other related nuclear structure models and present several extensions beyond the mean-field model which are currently used. Phenomenological adjustment of the model parameters is discussed in detail. The performance quality of the SCMF model is demonstrated for a broad range of typical applications.

Self-consistent mean-field models for nuclear structure

http://cds.cern.ch/record/298578/files/SCAN-9603159.pdf

Review of mathematics, numerical factors, and corrections for dark matter experiments based on elastic nuclear recoil

Adopted values for the reduced electric quadrupole transition probability, B(E2)↑, from the ground state to the first-excited 2+ state of even–even nuclides are given in Table I. Values of τ, the mean life of the 2+ state; E, the energy; and β, the quadrupole deformation parameter, are also listed there. The ratio of β to the value expected from the single-particle model is presented. The intrinsic quadrupole moment, Q0, is deduced from the B(E2)↑ value. The product E×B(E2)↑ is expressed as a percentage of the energy-weighted total and isoscalar E2 sum-rule strengths.

Table II presents the data on which Table I is based, namely the experimental results for B(E2)↑ values with quoted uncertainties. Information is also given on the quantity measured and the method used. The literature has been covered to November 2000.

The adopted B(E2)↑ values are compared in Table III with the values given by systematics and by various theoretical models. Predictions of unmeasured B(E2)↑ values are also given in Table III.

Transition probability from the ground to the first-excited 2^+ state of even-even nuclides

Nuclear charge and magnetization density distribution parameters from elastic electron scattering

(1966). Electron scattering and nuclear structure. Advances in Physics: Vol. 15, No. 57, pp. 1-109.

https://www.annualreviews.org/doi/pdf/10.1146/annurev.ns.25.120175.001553

Electron Scattering and Nuclear Structure

A survey has been made of the differential scattering cross sections for 187-Mev electrons on the even-even nuclei $_{12}\mathrm{Mg}^{24}$, $_{14}\mathrm{Si}^{28}$, $_{16}\mathrm{S}^{32}$, $_{18}\mathrm{A}^{40}$, and $_{28}\mathrm{Sr}^{88}$. It has been possible to separate the elastic scattering from the inelastic in all cases and to resolve the inelastic groups from specific nuclear levels for at least one level in all cases. A simple Born-approximation analysis of the elastic data yields values of the effective radii and surface thicknesses of the nuclear charge densities which (if suitably corrected for failure of the Born approximation) are in substantial agreement with the results of Hahn, Ravenhall, and Hofstadter; i.e., a radius parameter of $c\ensuremath{\cong}1.08 {A}^{\frac{1}{3}}\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}$ cm (radius to half-maximum of the charge distribution) and a surface thickness of $t\ensuremath{\cong}2.5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}$ cm (thickness from 10% to 90% of the maximum of the charge distribution). Phenomenological analysis of the inelastic scattering along the lines laid down by Schiff yields some tentative multipolarity assignments, and application of some results of Ravenhall yields estimates of (radiative) partial level widths; for the $E2$ transitions these correspond to lifetimes of \ensuremath{\sim}19\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}13}$ sec (Mg 1.37 Mev) to \ensuremath{\sim}1.4\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}13}$ sec (Sr 1.85 Mev). The observed strengths of the transitions are compared to those predicted by Weisskopf theory.

Inelastic and Elastic Scattering of 187-Mev Electrons from Selected Even-Even Nuclei

We have observed inelastic scattering of 183-Mev electrons through angles of 40\ifmmode^\circ\else\textdegree\fi{}-90\ifmmode^\circ\else\textdegree\fi{} in the laboratory, leading to excitation of discrete nuclear excited states in ${\mathrm{Ni}}^{58}$, ${\mathrm{Co}}^{59}$, ${\mathrm{Pb}}^{208}$, and ${\mathrm{Bi}}^{209}$. The excitation energies were below 8 Mev. Born-approximation analysis of the measured inelastic form factors was used to deduce the multipolarities $\ensuremath{\lambda}$ (when not previously known), and, by extrapolation, the transition rates for 15 corresponding gamma transitions. A number of groups of electric transitions for $\ensuremath{\lambda}=2, 3, \mathrm{and} 4$ were observed, each group having strikingly similar form factors. In all but one of these groups the ratios $G$ of the observed gamma transition rates to the single-particle predictions were greater than 15, and for some transitions from 30 to over 100. One of the groups, in cobalt and the nickels, contains the 1.33-Mev $E2$ transition to the first excited state of ${\mathrm{Ni}}^{60}$. Another group consists of fast $E3$ transitions, seen in all five nuclei, from states known as the "anomalous levels." They included the transition to the first excited state in ${\mathrm{Pb}}^{208}$ ($G=31$) and a transition in ${\mathrm{Bi}}^{209}$ identical in energy and form factor. Among three slow $E4$ transitions in cobalt and the nickels was the 2.50-Mev ${4}^{+}$\ensuremath{\rightarrow}${0}^{+}$ transition in ${\mathrm{Ni}}^{60}$. The $E4$, $E3$, and an $E2$ transition in ${\mathrm{Co}}^{59}$ identify states analogous to the ${4}^{+}$, ${3}^{\ensuremath{-}}$, and ${2}^{+}$ seen in the neighboring even-even nuclei. The last two transitions are strongly enhanced. A pair of fast 4.30-Mev $E4$ transitions was observed in ${\mathrm{Pb}}^{208}$ and ${\mathrm{Bi}}^{209}$; their speed ($G=37$) indicates they may constitute the lowest-energy configuration of 16-pole mode of excitation of the nuclear surface. Values of the collective vibrational parameters ${C}_{\ensuremath{\lambda}}$ and ${B}_{\ensuremath{\lambda}}$ and the degree to which some of the transitions exhaust ordinary sum rules support the conclusion that the inelastic scattering process is strongly exciting nuclear collective excitations. Some of the observed results are expected on the basis of the theory of collective vibrational excited states; some are the consequence of unidentified configurations.

Electron-scattering study of nuclear levels in cobalt, nickel, lead, and bismuth

Many of the important properties of the stored beam in an electron storage ring are determined by integrals, ’ taken around the whole ring, of various characteristic functions of the guide fielci. Some of the integrals are handled easily, but a few are usually estimated graphically - particularly for alternating-gradient guide fields. This report describes a convenient method for evaluating numerically these recalcitrant integrals. In the usual linear approximation, the integrals we wish to consider are most conveniently expressed in terms of four (somewhat redundant) functions of the azimuthal coordinates: p(s) the radius of curvature of the design orbit, n the field index, F(s) the radial betatro; function andT(s) the off-energy (or “dispersion”) function. The Integrals We restrict our attention to guide fields made up of a number of magnetic segments - magnets or straight sections The functions p and n are assumed to have constant values within a given magnet, but vary abruptly at the entrance and erit boundaries. The integrals of interest are given by:

/pdf/evaluation-of-synchrotron-radiation-integrals-43be5f08o9.pdf

Evaluation of Synchrotron Radiation Integrals

The absolute elastic electron-scattering cross sections of Ca/sup 40/, V/ sup 51/, Co/sup 59/ In/sup 115/ Sb/sup 121//sup / ,>s/sup 123/,and Bi/sup 209/ were measured at a number of angles at a primary electron energy of 183 Mev. The cross sections were obtained by comparison with scattering from the proton. The data were compared with the previous relative angular distributions of Hahn, Ravenhall, and Hofstadter. The present data are in closer agreement with the charge distributions found from fitting a Fermi two-parameter model to the older data than with those found from a fit of the Ford and Hill charge distributions. The absolute cross sections for Bi/sup 209/ show the least agreement: they are 35% larger than the predictions of the Fermi model and about 70% larger than the Ford and Hill model. (auth)

SCATTERING OF HIGH-ENERGY ELECTRONS FROM Ca$sup 40$, V$sup 51$, Co$sup 59$, In$sup 115$, Sb$sub 121$$sup 123$, AND Bi$sup 20$$sup 9$

The Stanford Linear Collider (SLC) requires a positron source with a yield large enough to give equal number of positrons and electrons at the interaction point. In addition, the colliding positrons must have an emittance and bunch length similar to the electron beam. This report describes the design of a high yield positron source to give these characteristics. Reference 1 contains more details of the systems of the SLC and positron source. The overall positron production system in relation to the SLC is shown in Fig. 1. A single bunch of electrons of 33 GeV is extracted from the Linac at the two-thirds point by means of a fast kicker magnet and a pair of septum magnets. The extracted electron beam is transported into the positron vault where it is focused to a spot size of + or -0.6 mm at the target. Spoiler foils are placed in the extracted beam path to increase the emittance so that the spot size cannot be made so small that target damage would occur. Positrons are collected in a focusing solenoid system using an iron shaped solenoidal field of 1 Tesla at the target and a pulsed solenoidal field from a flux concentratormore » with a peak field of 5 Tesla. The positron beam emerging from the focusing solenoid system is accelerated to 200 MeV in a 1.5 meter high-gradient-accelerator of 50 MeV/meter and three 3.05 m standard accelerator sections. The positrons are contained in the accelerator aperture using a 0.5 Tesla solenoidal field in the beginning and quadrupole focusing in the final two standard accelerator sections. The positrons are then transported back to the beginning of the Linac for further acceleration to 1.2 GeV, whereupon they are sent to the positron damping ring. Later they are removed from the damping ring, accelerated to the end of the Linac, and transported around the e/sup +/ arc to the interaction point.« less

/pdf/design-of-a-high-yield-position-source-2vv4x4rtks.pdf

R. Helm

Papers

Inelastic and Elastic Scattering of 187-Mev Electrons from Selected Even-Even Nuclei

Electron-scattering study of nuclear levels in cobalt, nickel, lead, and bismuth

Evaluation of Synchrotron Radiation Integrals

SCATTERING OF HIGH-ENERGY ELECTRONS FROM Ca$sup 40$, V$sup 51$, Co$sup 59$, In$sup 115$, Sb$sub 121$$sup 123$, AND Bi$sup 20$$sup 9$

Design of a High Yield Position Source