coulomb excitation data analysis codes; gosia 2007 - Physics and ...

esults of minimization and error runs can be used to evaluate quadrupole sum rules by a separate codeSIGMA ( Chapter 6 ).The extraction of the matrix elements from experimental data requires many runs of GOSIA. Duringthese runs, GOSIA creates and updates a number of disk files, containing the data needed to resume theanalysis or to execute SELECT or SIGMA codes. The details of permanent file manipulation are presentedin Chapter 7.Relatively modest central memory requirements of GOSIA (about 1.5MB) are due to the sharing ofthe same memory locations by different variables when various options are executed and to replacing thestraightforward multidimensional arrays (such as e.g. matrix elements) with catalogued vectors and associatedlogical modules. The description of the code given in this chapter therefore will not attempt to accountfor its internal organization, which is heavily dependent on the sequence of options executed and, in general,of no interest to the user. Instead, it will concentrate on the algorithms used and the logic employed inGOSIA. The basic knowledge of the algorithms is essential since the best methods of using the code arestrongly case-dependent, so much freedom is left to the user to choose the most efficient configurationsaccording to the current needs.All three codes- GOSIA, SIGMA and SELECT- are written in the standard FORTRAN77 to make theirimplementation on various machines as easy as possible. The necessary modifications should only involvethe output FORMAT statements, which are subject to some restrictions on different systems. Full 64 bitaccuracy is strongly recommended since the results can be untrustworthy when run using 32 bit accuracy.4.1 Coulomb Excitation Amplitudes and Statistical TensorsThe state of a Coulomb excited nucleus is fully described by the set of excitation amplitudes, a IM (M 0 ),defined by the solution of Eq. 2.17a at ω = ∞, or, approximately, by the matrix expansion 3.5., used forminimization and error estimation. To set up the system of coupled-channel differential equations 2.17ait is necessary first to define the level scheme of an excited nucleus. Certainly, from a practical point ofview, the level scheme should be truncated according to the experimental conditions in such a way thatreasonable accuracy of the excitation amplitudes of the observed states is obtained with a minimum of thelevels included in the calculation. As a rule of thumb, two levels above the highest observed state in eachcollective band should be taken into account to reproduce a given experiment reliably. Truncation of thelevel scheme at the last observed level leads to an overestimation of the excitation probability of this leveldue to the structure of the coupled-channels system 2.17a, while including additional levels above, even iftheir position is only approximately known, eliminates this effect.The solution to the coupled-channels system 2.17a should, in principle, involve all magnetic substates ofagivenstate|I >, treated as independent states within a framework of the Coulomb excitation formalism.However, due to the approximate conservation of the magnetic quantum number in the coordinate systemused to evaluate the Coulomb excitation amplitudes (as discussed in Chapter 2) it is practical to limit thenumber of the magnetic substates taken into account for each polarization of the ground state, M 0 . Inany case, the excitation process follows the “main excitation path“, defined as a set of magnetic substateshaving the magnetic quantum number equal to M 0 , the remaining magnetic substates being of less andless importance as the difference between their magnetic quantum number and M 0 increases. The relativeinfluence of the excitation of magnetic substates outside the main excitation path is experiment-dependent,therefore GOSIA allows the user to define the number of magnetic substates to be taken into accountseparately for each experiment. This choice should be based on the requested accuracy related to the qualityof the experimental data, keeping in mind that reasonable truncation of the number of the magnetic substatesinvolved in Coulomb excitation calculations directly reduces the size of the coupled channels problem to besolved.The integration of the coupled differential equations 2.17a should be in theory carried over the infiniterange of ω, which, practically, must be replaced with a finite range wide enough to assure the desired accuracyof the numerical solution. To relate the effect of truncating the ω-range to the maximum relative error ofthe absolute values of the excitation amplitudes, a c , the following criterion is used:R ∞1 −∞ Q λ0( =1,ω)dω − R ω maxQ−ω max λ0 ( =1,ω)dωR4∞−∞ Q ≤ a c (4.1)λ0( =1,ω)dω35