Witold Nazarewicz UWS June 10 2010 2000 For low energies and under conditions where the electron does not penetrate the nucleus the electron scattering can be described by the Rutherford formula The Rutherford formula is an analytic expression for the differential scattering cross secti ID: 1022453
Download Presentation The PPT/PDF document "Probing Nuclear Skins through Density Fo..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
1. Probing Nuclear Skins through Density Form FactorsWitold NazarewiczUWS, June 10, 2010(2000)
2. For low energies and under conditions where the electron does not penetrate the nucleus, the electron scattering can be described by the Rutherford formula. The Rutherford formula is an analytic expression for the differential scattering cross section, and for a projectile charge of 1, it isKE=Kinetic energy of electronhttp://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/elescat.htmlIntroduction: Electron Scattering from nuclei
3. As the energy of the electrons is raised enough to make them an effective nuclear probe, a number of other effects become significant, and the scattering behavior diverges from the Rutherford formula. The probing electrons are relativistic, they produce significant nuclear recoil, and they interact via their magnetic moment as well as by their charge. When the magnetic moment and recoil are taken into account, the expression is called the Mott cross section:
4. A major period of investigation of nuclear size and structure occurred in the 1950's with the work of Robert Hofstadter and others who compared their high energy electron scattering results with the Mott cross section. The illustration below from Hofstadter's work shows the divergence from the Mott cross section which indicates that the electrons are penetrating the nucleus - departure from point-particle scattering is evidence of the structure of the nucleus.
5. homogenous chargedistributionfinite diffusenessIn plane wave Born approximation (PWBA) the link between the charge density distribution and the cross section is straightforward:Form factorj0 – spherical Bessel function of zero orderq – three momentum transfer of electron
6. Three-dimensional form factor of nucleonic density is defined as:Multipole expansion of density:
7. In the Helm model, the nucleonic density can be written as a convolution of the sharp-surface densityand the Gaussian folding function– folding width (surface thickness)R0 – box-equivalent radiusc – volume conservation factorbL – set of shape deformationsIn this way, the sharp density is normalized to unity
8. The form factor of the sharp density distributionThe monopole form factor becomesAt spherical shape, one obtainsThe form factor of the folding function is
9. The information on the box radius and shape deformations can be extracted from measured (or calculated) density form factors.According to the convolution theorem, the Fourier transform of a convolution is the product of Fourier transforms. Consequently:In the spherical Helm model, the first zero of F0(q), defines the diffraction radiusThe surface thickness parameter can be computed from the first maximum of F0 at qm
10. Helm model, summaryR.H. Helm, Phys. Rev. 104, 1466 (1956)How to characterize nucleonic density?form factorspherical shapeIn the Helm model, nucleonic density is approximated by a convolution ofa sharp-surface density with radius R0 with the Gaussian profile:diffraction radiussurface thickness
11. The first zero of FH(q) is uniquely related to the radius parameter R0first zero of F(q)first maximum of F(q)
12.
13. Calculated and experimental densities
14.
15. Shape of a charge distribution in 154Gdtheory: Hartree-Fockexperiment: (e,e’) Bates
16. (n)(p)SkinDiffuseness150Sn(p)(n)HaloNeutron & proton density distributions
17.
18.
19.
20.
21.
22.
23.
24. Diffuseness of the density distribution is equal to the difference of radii where the density has values of 10% and 90% of the average central density.Neutron skin size is equal to the difference of radii where the neutron and proton densities have values of 50% of their respective average central densities Better quantitative measure of the skin can be formulated within the Helm model as the difference of neutron and proton diffraction radii.Neutron halo size is the difference between the neutron root-mean-squared and diffraction radii. Properties of the neutron halo are governed by the asymptotic features of tails of quantal wave functions.Summary