Effective Screened Coulomb Potential from e−- e− to 208Pb -208Pb Systems

We confirmed that the Coulomb phase shifts can not be given with a screened Coulomb potential by a monotonic increase of the range R. However, several screening “discrete range-bands” reproduce the Coulomb phase shift in a wide energy region. If the range-band of a proton-proton system for example is obtained, then the range-bands are generalized by three universal arguments: the universal range R(k)(≡ kR), the Sommerfeld parameter η(k)(≡ Z1Z2e2ν/k), and the universal asymptotic phase F(k)(≡ 2krc). Finally, the phase shifts of any other systems from e−−e− to heavy ions such as 208Pb−208Pb can be reproduced automatically by an individual range R = R(k)/k on the universal range-bands R(k)