Subatomic Physics

6.4. The Charge Distribution of Spherical Nuclei 143
Table 6.1: Probability Densities and Form Factors
for Some One-Parameter Charge Distributions. [After
R. Herman and R. Hofstadter, High-Energy Electron Scattering
Tables, Stanford University Press, Stanford, CA, 1960.]
Probability Density, ρ(r) Form Factor, F (q 2 )
δ(r) 1
ρ0 exp(−r/a) (1 +q 2 a 2 /� 2 ) −2
ρ0 exp[−(r/b) 2 ] exp(−q 2 b 2 /4� 2 )
ρ0,r ≤ R
0,r ≥ R
⎫
⎬
⎭
3[sin(|q|R/�)−(|q|R/�)cos(|q|R/�)]
(|q|R/�) 3
particle. F (q 2 ), for a specific value of q 2 , can therefore be determined with projectiles
of different energies. Equation (6.4) indicates that it is only necessary to
change the scattering angle correspondingly, and the same value of F (q 2 ) should
result. Incidentally, the fact that F (q 2 ) depends only on q 2 is true only in the first
Born approximation; it is not valid in higher order. It can therefore be used to test
the validity of the first Born approximation.
6.4 The Charge Distribution of Spherical Nuclei
The investigation of nuclear structure by electron scattering has been pioneered by
Hofstadter and his collaborators. (8) The basic arrangement is similar to the one
shown in Fig. 5.32: An electron accelerator produces an intense beam of electrons
with energies between 250 MeV and a few GeV. The electrons are transported to
a scattering chamber where they strike the target. The intensity of the elastically
scattered electrons is determined as a function of the scattering angle. Many improvements
have occurred since the early experiments by Hofstadter. In addition to
higher energies and higher intensity electron beams, which allow higher momentum
transfers to be studied, much higher resolution (∼ 100 keV or � 10 −3 of the beam
energy) has been achieved. The high resolution allows one to separate elastic from
inelastic scattering and to study inelastic scattering to individual levels in addition
to elastic scattering. The differential cross section for the scattering of 500 MeV
electrons from 40 Ca is shown in Fig. 6.3. The data can be seen to extend over 12
orders of magnitude; they yield values of |F (q 2 )| and from these values information
about the charge distribution is obtained. (9)
The crudest approximation to the nuclear charge distribution is a one-parameter
8 R. Hofstadter, H.R. Fechter, and J.A. McIntyre, Phys. Rev. 92, 978 (1953); for a review, see
C.J. Batty, E. Friedman, H.J. Gils, and H. Rebel, Adv. Nucl., Phys., ed. J.W. Negele and E.
Vogt, Plenum Press, New York, 19, 1 (1989).
9 A nice review with data tables can be found at H. De Vries, C.W. De Jager and C. De Vries,
Atom. Data Nucl. Data Tabl. 36, 495 (1987).