In practice the single-particle space has to be severely truncated. This truncation is typically based upon the single-particle energies, which is the effective energy from a mean-field potential.

Sometimes we freeze the core and only consider a valence space. For example, one may assume a frozen \( {}^{4}\mbox{He} \) core, with two protons and two neutrons in the \( 0s_{1/2} \) shell, and then only allow active particles in the \( 0p_{1/2} \) and \( 0p_{3/2} \) orbits.

Another example is a frozen \( {}^{16}\mbox{O} \) core, with eight protons and eight neutrons filling the \( 0s_{1/2} \), \( 0p_{1/2} \) and \( 0p_{3/2} \) orbits, with valence particles in the \( 0d_{5/2}, 1s_{1/2} \) and \( 0d_{3/2} \) orbits.

Sometimes we refer to nuclei by the valence space where their last nucleons go. So, for example, we call \( {}^{12}\mbox{C} \) a \( p \)-shell nucleus, while \( {}^{26}\mbox{Al} \) is an \( sd \)-shell nucleus and \( {}^{56}\mbox{Fe} \) is a \( pf \)-shell nucleus.