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

een exceeded; third, the user-given convergence limit has been achieved, i.e. the difference between twosubsequent set of matrix elements (taken as a length of the difference vector) is less than this limit. Thefirst two conditions fulfilled terminate a current run, while, after the convergence limit has been hit, theminimization is resumed if it was specified to lock a given number of the matrix elements influencing theS function to the greatest extent to allow the weaker dependences to be fitted (the details of the usage ofthe user-defined steering parameters are given in Chapter V). To further reduce the numerical deficiency,GOSIA monitors the directions of two subsequent search vectors and fixes the matrix element having thelargest gradient component if two subsequent directions are almost parallel, which usually signifies that aspurious result of the numerical differentiation of S with respect to this matrix element inhibits the fittingof the less pronounced dependences. A warning message is issued if this action was taken.An additional reduction of the free variables can be done by coupling of the matrix elements, i.e.fixing the ratios of a number of them relative to one “master“ matrix element. This feature is useful ifthe experimental data available do not allow for a fully model-independent analysis, therefore some modelassumptions have to be introduced to overdetermine the problem being investigated. The “coupled“ matrixelements will retain their ratios, as defined by the initial setup during the fitting, a whole “coupled“ set istreated as a single variable.Finally, GOSIA requires that the matrix elements are varied only within user-specified limits, reflectingthe physically acceptable ranges. The matrix elements are not allowed to exceed those limits, neither duringthe minimization nor the error calculation.4.5.8 Sensitivity MapsAs a byproduct of the minimization, GOSIA provides optional information concerning the influence of thematrix elements on both the γ-ray yields and excitation probabilities. The compilation of those maps is,however, time-consuming and should be requested only when necessary. The yield sensitivity parameters,α y ki,define the sensitivity of the calculated γ-ray yield k with respect to the matrix element i as:α y ki =δ ln Y kδ ln |M i | = M iδY kY k δM i(4.55)The excitation probability sensitivity parameters, α p ki, are expressed similarly, the yields being replacedby the excitation probabilities. The code allows the six most pronounced dependences to be selected for eachexperimentally observed yield for the printout, while a number of the matrix elements for which the mostimportant probability sensitivity parameters are printed out can be selected by the user.By definition, the sensitivity parameters provide (to the first order) a relationship between the relativechange of the excitation probabilities, p, (or the calculated γ-ray yields) and the relative change of the matrixelements according to:∆p kp k= α p ∆M iki(4.56)M ithus supplying locally valid information on the dependence of experimental data on matrix elements. Anidentical relationship holds for the yield sensitivity parameters, although the code calculates them only forthe γ detector labeled as #1 for each experiment, thus the angular distribution of the γ rays, affected forexample by the mixing ratios, is not accounted for fully. It is suggested that a γ-ray detector yielding themost complete and accurate yields should be defined as the first one in the description of an experiment toensure that the yield sensitivity maps are calculated for all the yields observed.The sensitivity parameters are not strongly dependent on the actual matrix elements, therefore thesensitivity maps can be treated as an indication of the features of the excitation/deexcitation process at anystage of minimization. As an extreme case, it should be noted that if the lowest-order perturbation theoryapplies, i.e. the excitation probability of a given state is proportional to the product of the squares of matrixelements connecting this state with the ground state, then the probability sensitivity parameter will equal 2for all such matrix elements and will vanish for all others, no matter what the actual values are. The sameholds for the yield sensitivity parameters, provided that the γ-ray decay follows a simple cascade with nobranching (the feeding from above can be neglected as a consequence of assuming the applicability of thelowest-order perturbation approach).51