Question

A proton in an atomic nucleus is bound to other protons and neutrons in the nucleus by the strong nuclear force when it is within approximately 3 fm of another particle. What is the approximate kinetic energy of a proton that is localized to such a region? What would be the corresponding energy of an electron localized to within such a region? Compare this to typical beta energies for beta decay. Is a release of electrons

Answer 1

Approximate kinetic energy of a proton that is localized to such a region

Neutrons and protons in atomic nuclei are confined within a region whose diameter is about
10^-15m = 1 fm

Corresponding energy of an electron localized to within such a region

where ? is the electron spin density and ? the kinetic energy density. D is expected to be small in those regions of space where localized electrons are to be found. Given the arbitrariness of the magnitude of the localization measure provided by D, it is compared to the corresponding value for a uniform electron gas with spin density equal to ?(r), which is given by

The ratio,

is a dimensionless localization index that expresses electron localization with respect to the uniform electron gas. In the final step, the ELF is defined in terms of ? by mapping its values on to the range 0 ? ELF ? 1 by defining the electron localization function as

ELF = 1 corresponding to perfect localization and ELF =