PHYSICAL REVIEW C 77, 024302 (2008) Analysis of fine structure in the nuclear continuum A. Shevchenko, 1 J. Carter, 2 G. R. J. Cooper, 3 R. W. Fearick, 4 Y. Kalmykov, 1 P. von Neumann-Cosel, 1,* V. Yu. Ponomarev, 1,† A. Richter, 1 I. Usman, 2 and J. Wambach 1 1 Institut f ¨ ur Kernphysik, Technische Universit¨ at Darmstadt, D-64289, Darmstadt, Germany 2 School of Physics, University of the Witwatersrand, P. O. Wits, Johannesburg 2050, South Africa 3 School of Earth Sciences, University of the Witwatersrand, P. O. Wits, Johannesburg 2050, South Africa 4 Department of Physics, University of Cape Town, Rondebosch 7700, South Africa (Received 10 September 2007; published 19 February 2008) Fine structure has been shown to be a general phenomenon of nuclear giant resonances of different multipolarities over a wide mass range. In this article we assess various techniques that have been proposed to extract quantitative information from the fine structure in terms of characteristic scales. These include the so-called local scaling dimension, the entropy index method, Fourier analysis, and continuous and discrete wavelet transforms. As an example, results on the isoscalar giant quadrupole resonance in 208 Pb from high-energy-resolution inelastic proton scattering and calculations with the quasiparticle-phonon model are analyzed. Wavelet analysis, both continuous and discrete, of the spectra is shown to be a powerful tool to extract the magnitude and localization of characteristic scales. DOI: 10.1103/PhysRevC.77.024302 PACS number(s): 24.30.Cz, 21.60.Jz, 25.40.Ep, 27.80.+w I. INTRODUCTION Electric and magnetic nuclear giant resonances are well- known examples of the striking behavior of an interacting system of fermions to form collective modes [1]. Over the years, much experimental work has gone into establishing an understanding of the global behavior of the gross features, such as centroid energies and widths, of these resonances. It is generally accepted that the width Ŵ of the resonances mainly results from two mechanisms: direct particle emission from one-particle one-hole (1p-1h) configurations giving rise to an escape width Ŵ ↑ and the evolution of these 1p-1h configurations into more complicated two-particle two-hole (2p-2h) and finally to np-nh configurations giving rise to a spreading width Ŵ ↓ . This latter scheme has implicit in it a hierarchy of widths and time scales resulting in a fragmentation of the giant resonance strength in a hierarchical manner [2]. An important theoretical problem is to explain the nature of couplings between the levels in this hierarchy and to predict the scales of the fragmentation of the strength which thus arise from it. Already about 30 years ago it became apparent from high- energy-resolution inelastic electron-scattering experiments [3,4] that there was considerable fine structure superimposed on the broad bump of the isoscalar giant quadrupole resonance (ISGQR) in 208 Pb. Further studies [5] have shown that such fine structure is physical in nature and also appears in other reaction channels. Recent high-energy-resolution (p,p ′ ) measurements demonstrated the fine structure in a wide range of nuclei for the ISGQR [6]. It has also been observed in other types of resonances like the isovector giant dipole resonance [7,8], the magnetic quadrupole resonance [9], or * vnc@ikp.tu-darmstadt.de † Permanent address: Bogoliubov Laboratory for Theoretical Physics, JINR, Dubna, Russia. the spin-isospinflip Gamow-Teller mode [10], establishing it as a generic phenomenon of nuclei. Nevertheless, a serious experimental problem has been the quantitative extraction of the scales of this fragmentation. A lower limit on observable scales is placed by the experimental resolution. The recent experiments have been made possi- ble by the exploitation of high-energy-resolution magnetic spectrometers and particle beams with energies of several hundred MeV allowing for energy resolutions of a few tens of keV. The problem then is to determine scales that occur in the range between the experimental resolution and the broad envelope of the resonances (typically several MeV). Early on, an attempt was made to analyze the data on the fine structure of the ISGQR in 208 Pb observed in Refs. [3,4] in terms of a doorway-state model [11]. It could be shown that in this case the spreading width dominates over the escape width but the deduced scales depended strongly on the assumptions about the (unknown) number of doorway states. In this work, we concentrate on the evaluation of several new methods proposed for the extraction of such scales, viz. the local scaling dimension approach [12], the entropy index method [13], and the use of wavelet techniques [6], and compare the latter to older techniques such as Fourier analysis. As a test case, we investigate data on the ISGQR in 208 Pb from high-energy-resolution (p,p ′ ) experiments and a calculation of the corresponding isoscalar E2 strength function within the quasiparticle-phonon model (QPM). Although a more extensive data set and still other calculations are available, we restrict ourselves to these examples because the focus of the article is to evaluate the advantages and limitations of the different techniques for an extraction of characteristic scales. Possible conclusions on the nature of these scales and their implications for the decay of giant resonances are subject of a subsequent article. The article is organized as follows: in Sec. II we briefly present the data sets used during the analysis. As pointed out above, these are an experimental spectrum and a theoretical 0556-2813/2008/77(2)/024302(12) 024302-1 ©2008 The American Physical Society