Theory of core excitation components in the ground states of the calcium isotopes

Wave functions of the ground states of the calcium isotopes are calculated. The configurations are restricted to f _{7 2}^(n)JT and [d _{3 2}^-2 I_H T_Hf _{7 2}⁽ⁿ⁺²⁾ I_P T_P] JT where I_H, T_H, I_P, T_P are the angular momenta and isobaric spins of the two holes and (n + 2) particles respectively. Using the abreviated notation [T_HT_P] T for the core excitation components we show that the matrix elements between these components and the shell model states f _{7 2}⁽ⁿ⁾ assume a very simple form for all possible T_H, T_P and n. A relationship between the neutron and proton strengths in d, p reactions to opposite parity states and the average number of neutron holes and proton holes in the ground states of the calcium isotopes is worked out and is discussed in simple physical terms. It is found that the [T_H = 1 T_P = T + 1] T states are mainly responsible for the neutron strengths in (d, p) reactions, whereas the [T_H = 1 T_P = T - 1]T states contribute most to the proton strengths in (³He, d) reactions to opposite parity states. In order to reproduce the experimental trend in which the neutron strength decreases as one goes through the even calcium isotopes one must choose the energies of the [T_H = 1 T_P = T + 1]T states to be an increasing function of n. This is contrary to a previous prediction by this author. The work of Gerace and Green on the calcium isotopes is discussed and compared with the present work. We obtain much smaller neutron strengths for the (d, p) reactions in ⁴⁰Ca and ⁴²Ca to J = 3 2⁺ states than are obtained by some of the experimentalists. For example, the reaction ⁴⁰Ca (d, p) ⁴¹Ca J = 3 2⁺ state has 0.78 as the value of the neutron strength according to Belote et al¹. In our model this would imply that the percentage of core-excitation component in the ground state of ⁴⁰Ca is at least 78%.