Charge- and spin-excitation gaps for a magnetic Anderson impurity embedded in a nanoscale metallic sphere

A magnetic Anderson impurity $(S=1/2\mathrm{}$ and $\overrightarrow{U}\infty )$ placed at the center of a nanosized metallic sphere is considered. The localized f electrons are hybridized with the metallic states via a contact potential, such that only s states interact with the impurity. In nanoscale particles the conduction states have discrete energy levels, and for equally spaced energy levels for the s waves, the problem is reduced to the Bethe Ansatz solution of the Anderson impurity model in a finite box. The Bethe Ansatz equations are solved numerically for the ground state and the lowest energy charge and spin excitations. The energies of the states increase monotonically with the f-level energy. For an even number of electrons in the system (in s states and localized at the impurity), the impurity in the ground state is spin compensated into a spin singlet via the Kondo effect. The specific heat and the susceptibility are exponentially activated at low T due to the discreteness of the energy spectrum, with the gaps given by the lowest-energy charge and spin excitations. The model also represents a quantum dot as a side branch to a short quantum wire.