LONG RANGE DEVIATIONS FROM THE COULOMB POTENTIAL

For various reasons the potential energy between the two

colliding nuclei is not given strictly by eq. 2.5. In this section other sources of potential energy are considered, but the nuclear

force is specifically ignored — it is considered separately in the following section.

2.4.1 Method of Treatment

Following the philosophy of the semiclassical theory, one can identify two ways in which additional potentials will affect the excitation probability.' They will cause the orbit to be different

from that given by eq. 2.18, thus altering the values of the orbit integrals, and they may have non-zero matrix elements between nuclear states, thus modifying the strength functions (eqs. 2.22 and 2.23). It is conventional to ignore this second effect, the justification being that the additional potential energies are very small compared with the Coulomb energy, even at closest approach in backscattering. In addition, the modifications to the orbit are treated approximately with the same justification.

The correction consists of determining the value of the

additional potential energy at closest approach and subtracting that value from the beam energy [A175, p.292]. This retains the hyperbolic shape of the orbit, but gives the correct distance of closest approach which is where the Coulomb interaction is the strongest. Thus the bombarding energy E is replaced in all of the above equations by an effective bombarding energy E given by [A175, p.293]

E = E -h(l +A /A ) (1 + sin 0/2)

Av(d)

where AV(d) is the increase in the potential energy over the Coulomb value at the distance of closest approach d, which is given by

d = a(l+cosec 0/2) . (2.94)

2.4.2 Electron Screening

From a consideration of the Hartree model of the atom and

experimental atomic binding energies, Foldy [Fo51] concluded that the electrostatic potential at the nucleus due to the orbital electrons is given to a good approximation by

he

ev] "

Zt

P

screening potential. If, on the other hand, the collision is so slow that the electrons are able to adjust to the presence of the

projectile, then some of the projectile's energy goes into tighter binding of the orbital electrons. It has been estimated that the

2 2/5 energy so lost, the "adiabatic correction", is equal to 22.85 Z Z eV

P t [Se52]. For 24Mg on 208Pb this is ~ 20 keV. Thomas [Th54] has suggested a procedure for deciding what fraction of the adiabatic correction should be applied in the case of alpha decay. A suitable adaptation of his approach for heavy ion scattering might be to

include a fraction f of the adiabatic correction which is equal to the fraction of orbital electrons having velocities higher than the

projectile's asymptotic velocity. The idea is that those electrons which are moving faster than the projectile are able to adjust to its presence. It is reasonable to use the projectile's asymptotic

velocity since the radius of maximum charge density of the Is

electrons in Pb is ~ 1 pm and the potential energy of an 0 and a Pb nucleus separated by this distance is ~ 1 MeV or ~ 2% of the

projectile speeds ~ 0.1 c. The speed of orbital electrons is often taken to be [Th54] V e[c] = 0.00730 p- ,_{* '} n (2.96)

where n* is the effective principal quantum number, s the shielding constant [S130] and the fine structure constant. A calculation of s and n* shows that all those electrons in Pb outside of the 4p shell have velocities less than 0.1 c, that is, f = 0.44. Similarly, f = 0.45

for Hg.

From the above discussion, the additional potential due to

t

electronic screening is taken to be

AV = -32.65 Z zl/5 + 22.85 f Z 2Z 2/5 . (2.97)

es p t p t

For 24Mg on 208Pb, AV = - 1 8 0 keV or ~ 0.2% of the bombarding energy. es

Several authors [e.g. Ra72, Ra76, Mu761 have noted the existence of a "molecular potential" due to the nonlinearity of the binding energy of the orbital electrons as a function of the nuclear charge. This potential energy is defined as the total electron binding energy when the target and projectile are in close proximity minus the total binding energies of the target and projectile atoms when separated.

It has been graphed for the system Br + Br and U + U as a function of the internuclear separation by Miiller and Greiner [Mu76] . The U + U molecular potential energy is 6 MeV or ~ 0.8% of the Coulomb energy when the surfaces of the two nuclei are touching. An extrapolation from the graphs of ref. Mu76 suggests that the corresponding figures

j*

It may be noted that eq. 2.97 differs slightly from the generally adopted expression [Sa69]

AV = -32.65 Z z!/5 + 40 Z Z . (2.98)

es p t p t

It is not clear to me how eq. 2.98 was obtained, but I suspect that it was from an incorrect generalization of a formula correctly quoted by Perlman and Rasmussen [Pe57] in the context of alpha decay.

for B r + B r are ~ 300 keV and ~ 0.2%. These calculations assume that the collision is adiabatic and that both the colliding atoms have their full complement of electrons [Ra72]. In addition, the mutual shielding of the electrons is neglected [Mu76, Ra76].

In the- analysis below, molecular potentials are ignored. The reasons for this are:

(a) the ratio of the molecular to Coulomb potentials at closest approach is less for B r + B r than for U + U . Although none of the references above give the dependence of the molecular potential on the two atomic numbers involved, they leave the

impression that the molecular potential is negligible for collisions involving light ions.

(b) each of the three approximations mentioned above contributes to an overestimation of the molecular potential present in the collision of a fast, highly stripped ion such as 06+ or Mg7+ with an atom. Thus, even if there were potentials available for e.g. O + P b , they may not be useful in our experimental situation.

In support of this neglect of the molecular potential, it may be noted that Rafelski [Ra76], in his calculation of the effect of various small potentials on elastic scattering cross sections, ignores the molecular potential when considering the O + Pb system.

2.4.3 Vacuum Polarization

It is well known that the Coulomb potential is only an

approximate description of the potential betwen two charges. For distances of separation less than the reduced Compton wavelength X^ of the electron

(X

^ t

approximated by [Li74, p.427]

Eq. 2.99 is different from that given by Alder and Winther [A175] whose source is unclear. It differs also from that given by Blomqvist

[B172], but is the same as is quoted by Huang [Hu76]. Eq. 2.99 is adopted since it is explicitly derived by Lifshitz and Pitaevskii

Av