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

elements is done to find the correlation axis. The maximum correlation direction is then defined as a vectordifference of the resulting set of the matrix elements and the central point. In this way, a correlation curveis approximated at a range consistent with the order of magnitude of the finalerror.However,itshouldbenoted that performing the one-step gradient minimization to find the correlation axis is very time-consumingif all the matrix elements are taken into account. Moreover, since the gradient method will only affect thevalues of the matrix elements strongly correlated with the one investigated, the calculation of the derivativeswith respect to all others is simply redundant and does not yield any improvement in predicting the correlationaxis. To speed up the full error calculation, GOSIA offers a mechanism for reducing the number ofmatrix elements being taken into account, by predicting the subsets of mutually correlated matrix elements.This requires that at a final stage of minimization, the yield sensitivity map is compiled and the full set ofthe sensitivity parameters α y ki(4.55) isstoredinapermanentfile. A separate program, SELECT, uses themap of the sensitivity parameters to predict the correlation. Each two matrix elements are correlated if theyaffect the same data points, thus, for each matrix element M l the correlation with a matrix element M k canbe measured by:c lk = X i|α y ki αy li | (4.68)the summation extending over all observed γ-ray yields. The absolute values are taken to prevent the neglectof the correlation due to coincidental cancellation of opposite sign terms. For each matrix element, indexedby the row index l, SELECT identifies those matrix elements M k having values of c lk greater than 10% of themaximum value of c lk in this row. As a result, SELECT produces a binary matrix whose elements are either0, if no strong correlation was predicted, or 1. The matrix elements explicitly correlated by the spectroscopicdata are assumed to be correlated by definition. The correlation matrix then is used during the full errorcalculation to fix the matrix elements for which the correlation was not predicted when searching for thecorrelation axis.GOSIA issues messages if lower values of the S statistichavebeenfound. Thesetofthematrixelements producing an overall best value of S is stored on a permanent file, which could be used eventuallyto improve the minimization as a starting guess. However, it should be remembered that the uncorrectedfast approximation used during the error estimation can be a source of inaccuracies, thus the small changesin the S statistic usually can be disregarded.4.7 Identification of Erroneous Data PointsFitting a large set of parameters, as in the case of Coulomb excitation analyses, one must be aware of thepossible existence of local minima, which can cause the fitting procedure to be trapped in an erroneoussolution. While there is no general method of distinguishing a local minimum from a global one, somejudgement is still possible based on the value of the normalized least-squares statistic S and the comparisonof experimental and fitted data. A local minimum could be created artificially by including erroneous piecesof data, such as incorrectly assigned γ yields, or, even if there were no mistakes in the data, could result fromthe local dependence on the parameters. The information provided by GOSIA during the minimization anderror estimation usually is sufficient to determine whether the solution is acceptable. To further facilitatethe identification of the local solutions, GOSIA is equipped with a “troubleshooting“ module, designed todetect inconsistencies within the data set. This module uses the information obtained at a current stageof minimization concerning the dependence of the observed γ yields on matrix elements. Only the γ yieldsobserved in the detectors labeled #1 for each experiment are used, as a result of the convention that theγ detector providing the most complete and reliable set of data should be defined as the first one. Thetroubleshooting routine first calculates the normalized S statistic disregarding the other γ detectors andthe spectroscopic data, assuming the independent, code-calculated normalization, thus overriding the userspecifiednormalization constants. A large difference between a value of S calculated this way, and thevalue resulting from the minimization, points to either a wrong relative normalization or a problem withreproduction of the γ-ray angular distribution, which may be due to the errors in assigning the angularpositions of the γ detectors ( inconsistencies within the spectroscopic data are easy to notice by comparingthe calculated and experimental values). A next step is to select the experimental γ yields data, whichlocally inhibit minimization with respect to the individual matrix elements. Keeping in mind the gradient56