Electron-Nucleus Non-Integrated Sum Rules and the Projection-Operator Method

Abstract

Non-integrated sum rules in terms of time-dependent correlation functions are discussed. A statistical treatment of the time-dependent correlation functions for nuclei is suggested and a projection method clue to Mori is applied to the solution of an equation-of-motion for the transition operators. The complete, nuclear response function can be obtained by assum­ ing a non-linerrr random force. The differential cross sections at fixed momentum transfer can be written as the sum of a linear and a random part. § I. Introduction Experimental research on electrodisintegration of nuclei covers both the giant resonance energy region and, at much larger energies, the quasi-elastic scattering region. In the first region the electro-excitation is treated as excitation of collective quasidiscrete states of various multipolarity. Recently the existence of \'arious multipole giant resonances other than El, postulated for a long time, has been clemon:;trated experimentally from electron and hadron scattering, at excitation en­ ergies ol 10 MeV and above.n.~) All these experiments indicate the existence at these high excitation energies of collective states, which are modes in which an appreciable fraction of the nucleons move together. In the quasi-elastic peak region a single particle approach is pursued. The electron scattering takes place as it would on a free individual nucleon. If the nucleons were free, this peak would be sharp and would occur at an energy loss of q 2/2lv1 for momentum transfer q and nucleon mass 111. The nuclear binding shifts the position of the peak. The bind­ ing also broadens the peak, due to the momentum distribution of the bound nucleons. Berthot and Isabelle 3) have found the existence of a structure in the e- 12C quasi­ elastic peak, at constant momentum transfer, which indicates that the single nucleon scattering picture of the quasi-elastic scattering is not adequate. On the other hand the dependence of the structure on the momentum transfer q does not seem to be of the collective type. Different models are used to describe the two different regions. However electroexcitation of resonances and quasi-elastic scattering are basically the same process and in principle it is impossible to separate them. At the moment a comprehensive theory of electrodisintegration of the nucleus describing