Introduction

The incompressibility of infinite nuclear matter as well as of finite nuclei has been studied in a number of theoretical papers and reviews. In the classic review by Blaizot [1] the connection between the finite-nucleus incompressibility and centroid of the Giant Monopole Resonance (GMR) was shown. This relation allows us to study incompressibility of nuclei through microscopic calculations of the monopole excitation spectra. It also brings us the possibility to directly compare theoretical results with experimental data. For examples, see the measurements presented in Refs. [2,4,3].

In Ref. [5], it was shown that the self-consistent models that succeed in reproducing the GMR energy in the doubly-magic nucleus $^{208}$Pb systematically overestimate the GMR energies in the tin isotopes. In spite of many studies related to the isospin [7,6,8], surface [9], and pairing [10,11,13,14,15,16,12] influence on the nuclear incompressibility, to date there is no theoretical explanation of the question "Why is tin so soft?" [17,5]. For an excellent recent review of the subject matter we refer the reader to Ref. [3].

Studies in Refs. [15,16] were restricted to the effect of zero-range pairing interaction. In the present paper we focus on a different kind of pairing force, namely, we implement the finite-range, fully separable, translationally invariant pairing interaction of the Gaussian form [18,19,20], together with the general phenomenological quasilocal energy density functional in the ph-channel [21]. We have performed calculations for all particle-bound semi-magic nuclei starting from $Z=8$ or $N=8$, up to $Z=82$ or $N=126$. The ground-state properties were explored within the Hartree-Fock-Bogolyubov (HFB) method, whereas the monopole excitations were calculated by using the Quasiparticle Random Phase Approximation (QRPA) within the Arnoldi iteration scheme [22]. For the numerical solutions, we used an extended version of the code HOSPHE [23].

The paper is organized as follows. In Secs. 2 and 3, we briefly outline the Arnoldi method to solve the QRPA equations and present the separable pairing interaction, respectively. In Sec. 4, we discuss the nuclear incompressibility, including its theoretical description, definitions in finite and infinite nuclear matter, and relations to monopole resonances. Then, our results are shown and discussed in Sec. 5 and conclusions are given in Sec. 6.