Self-Consistent Determination of the Effective Radii of Heavy Nuclei for Alpha-Particle Emission

Abstract

In attempts to use the absolute rate of natural alpha-particle decay to determine the "density of alpha particles on the nuclear surface" the penetrability of the effective barrier surrounding the nucleus plays a dominating role. A new approach to the barrier problem is proposed based on the hypothesis that for transitions of definite type—to ground states of spherical even-even nuclei beyond the double closed shells at ${\mathrm{Pb}}^{208}$—we might expect the intrinsic emission probability measured in single particle units, whatever its absolute magnitude, about which no assumption is made, to be proportional to the surface area of the nucleus only. The dispersion in the reduced widths inferred from the emission rates depends on the cut-off radius $R=r_0{A}^{\frac{1}{3}}$ that is chosen and so we can define a self-consistent potential whose radius constant $r_0$ minimizes this dispersion. The "spherical" nuclei show a well-defined minimum to the dispersion at $r_0=1.57\mathrm{}\pm 0.06$ fermis. The deformed nuclei have a different behavior as is expected. This self-consistent potential is very close to that derived by Igo from an optical model analysis of alpha-particle scattering; the penetrabilities for the natural alpha-particle emitters calculated with the two potentials agree to within a factor of about 2. It is shown that if the hypothesis is modified to allow a smooth dependence of the intrinsic emission probability on $A$ of the form $1+\epsilon A$ then the resulting minimum dispersions computed as a function of $\epsilon$ themselves show a minimum with $\epsilon$ very close to zero, thereby justifying the hypothesis in its simple form.