1. Probing Nuclear Skins through Density Form Factors
Witold Nazarewicz
UWS, June 10, 2010
(2000)

2. Introduction: Electron Scattering from nuclei
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 is
KE=Kinetic energy of electron

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. j0 – spherical Bessel function of zero order
In plane wave Born approximation (PWBA) the link between the charge density distribution and the cross section is straightforward:
homogenous charge
distribution
finite diffuseness
Form factor
j0 – spherical Bessel function of zero order
q – 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 density
and the Gaussian folding function
– folding width (surface thickness)
R0 – box-equivalent radius
c – volume conservation factor
bL – set of shape deformations
In this way, the sharp density is normalized to unity

8. The form factor of the sharp density distribution
The monopole form factor becomes
At spherical shape, one obtains
The 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 radius
The surface thickness parameter can be computed from the first maximum of F0 at qm

10. Helm model, summary How to characterize nucleonic density? form factor
R.H. Helm, Phys. Rev. 104, 1466 (1956)
How to characterize nucleonic density?
form factor
spherical shape
In the Helm model, nucleonic density is approximated by a convolution of
a sharp-surface density with radius R0 with the Gaussian profile:
diffraction radius
surface thickness

11. first zero of F(q) first maximum of F(q)
The first zero of FH(q) is uniquely related to the radius parameter R0
first zero of F(q)
first maximum of F(q)

13. Calculated and experimental densities

15. Shape of a charge distribution in 154Gd
theory: Hartree-Fock
experiment: (e,e’) Bates
Shape of a charge distribution in 154Gd

16. Neutron & proton density distributions
Skin
Diffuseness
(p)
(n)
Halo
150Sn

24. Summary
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.