Root Mean Square Radii of the Λ-Particle Orbits in Hypernuclei Using Rectangular Shape Potentials in a Relativistic Treatment

Abstract

The root mean square radii of the Λ-particle orbits in hypernuclei are calculated in the ground and first excited states using the Dirac equation with scalar and vector potentials of rectangular shape and of the same radius. An exact analytic and also approximate expressions are derived for the root mean square radius of the Λ-particle orbit in its ground state. It is shown analytically that in the ground state the r.m.s. radius varies, as in the non-relativistic case, to a good approximation, linearly with A1/3core namely: ≪rΛ2 >1/2s1/2 = cA1/3core + b for the higher mass hypernuclei, where the constants c and b are related to the potential parameters. On the basis of this treatment and the assumptions made, the upbending of the curve ≪rΛ2 >1/2s1/2 versus A1/3core observed in the region of the lower mass hypernuclei is also understood.