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

4.5 MinimizationThe minimization, i.e. fitting the matrix elements to the experimental data by finding a minimum of theleast-squares statistic is the most time-consuming stage of Coulomb excitation analysis. A simultaneous fitof a large number of unknown parameters (matrix elements) having in general very different influences onthe data, is a complex task, which, to whatever extent algorithmized, still can be slowed down or speeded updependent on the way it is performed. GOSIA allows much freedom for the user to define a preferred strategyof the minimization. A proper use of the steering parameters of the minimization procedure can significantlyimprove the efficiency of fitting dependent on the case analyzed - in short, it still takes a physicist to get theresults!. The following overview of the fitting methods used by GOSIA is intended to provide some ideasabout how to use them in the most efficient way.4.5.1 Definition of the Least-square StatisticA set of the matrix elements best reproducing the experimental data is found by requesting the minimumof the least-squares statistic, S( ¯M). The matrix elements will be treated as a vector ordered according tothe user-defined sequence. The statistic S is in fact a usual χ 2 -type function, except for the normalizationto the number of data points rather than the number of degrees of freedom which cannot be defined due toaverydifferent sensitivity of the excitation/deexcitation process to various matrix elements, as discussed inmore detail in Section 4.6. The statistic S, called CHISQ in the output from GOSIA, is explicitly given by:ÃS(M) = 1 S y + S 1 + X !w i S i (4.24)Niwhere N is the total number of data points ( including experimental yields, branching ratios, lifetimes,mixing ratios and known E2 matrix elements ) and S y , S 1 and S i are the components resulting from varioussubsets of the data, as defined below. Symbol w i stands for the weight ascribed to a given subset of data.The contribution S y to the total S function from the measured γ-yields following the Coulomb excitation,is defined as:S y = X I e I dw IeI dX1σ 2 k(I e,I d ) k(C IeI dY ck − Y ek ) 2 (4.25)where the summations extend over all experiments (I e ), γ-detectors (I d ) and experiment- as well as detectordependentobserved γ-yields, indexed by k. The weights ascribed to the various subsets of data (w IeI d)canbe chosen independently for each experiment and γ-detector, facilitating the handling of data during theminimization (particularly, some subsets can be excluded by using a zero weight without modifying theinput data). The superscript “c“ denotes calculated yields, while the superscript “e“ stands for experimentaldata, with σ k being the experimental errors. The coefficients C Ie I dare the normalization factors, connectingcalculated and experimental yields (see 4.5.2).The next term S 1 of equation 4.24 is an “observation limit“ term, intended to prevent the minimizationprocedure from finding physically unreasonable solutions producing γ-ray transitions which should be, buthave not been, observed. An experiment and detector dependent observation upper limit, u(I e,I d ) is introduced,which is the ratio of the lowest observable intensity to that of the user-specified normalizationtransition (see V.29). S 1 is defined as:S 1 = X jµ Yc 2j (I e ,I d )1Yn c − u(I e ,I) ·(I e ,I d) u 2 (I e ,I d )(4.26)the summation extending over the calculated γ transitions not defined as experimentally observed andcovering the whole set of the experiments and γ-detectors defined, provided that the upper limit has beenexceeded for a given transition. The terms added to S 1 are not counted as data points, thus the normalizationfactor N is not increased to avoid the situation in which the fit could be improved by creating unreasonabletransition intensities at the expense of the normalization factor.The remaining terms of 4.24 account for the spectroscopic data available, namely branching ratios, meanlifetimes, E2/M1 mixing ratios and known E2 matrix elements. Each S i term can be written as:43