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Over the past two decades high energy-resolution inelastic proton scatteringstudies were used to gain an understanding of the origin of fine structureobserved in the isoscalar giant quadrupole resonance (ISGQR) and the isovectorgiant dipole resonance (IVGDR). Recently, the isoscalar giant monopoleresonance (ISGMR) in $^{58}$Ni, $^{90}$Zr, $^{120}$Sn and $^{208}$Pb wasstudied at the iThemba Laboratory for Accelerator Based Sciences (iThemba LABS)by means of inelastic $\alpha$-particle scattering at very forward scatteringangles (including $0\circ$). The good energy resolution of the measurementrevealed significant fine structure of the ISGMR.~To extract scales by means ofwavelet analysis characterizing the observed fine structure of the ISGMR inorder to investigate the role of different mechanisms contributing to its decaywidth. Characteristic energy scales are extracted from the fine structure usingcontinuous wavelet transforms. The experimental energy scales are compared todifferent theoretical approaches performed in the framework of quasiparticlerandom phase approximation (QRPA) and beyond-QRPA including complexconfigurations using both non-relativistic and relativistic density functionaltheory. All models highlight the role of Landau fragmentation for the dampingof the ISGMR especially in the medium-mass region. Models which include thecoupling between one particle-one hole (1p-1h) and two particle-two hole(2p-2h) configurations modify the strength distributions and wavelet scalesindicating the importance of the spreading width. The effect becomes morepronounced with increasing mass number. Wavelet scales remain a sensitivemeasure of the interplay between Landau fragmentation and the spreading widthin the description of the fine structure of giant resonances.

Fine structure of the isoscalar giant monopole resonance in $^{58}$Ni, $^{90}$Zr, $^{120}$Sn and $^{208}$Pb