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

It can be shown that as long as the quadratic approximation 4.40 is valid:¯D = J ¯5 0 (4.47)Using the identity resulting from the symmetry of J:¯5 0 J 2 ¯50 =(J ¯5 0 ) 2 (4.48)gives:where all terms, except R 22 , are known.expressed as:¯5 0 ¯D ¯D2R =¯ ¯D 2 ¯DJ ¯D(4.49)¯By sampling the minimized function this missing term can be¯DJ ¯D = 2 h 2 £f¡¯x0 + h ¯D ¢ − f (¯x 0 ) − h ¯5 0 · ¯D ¤ (4.50)The coefficients α 1 and α 2 , resulting from the solution of 4.45, can be rescaled arbitrarily, since we areonly interested in the direction of search, the stepsize being found by the one-dimensional minimizationprocedure outlined in a previous section. A most convenient representation of a direction of search ¯5,renormalized to avoid numerical overflows, is given by:à ¯DJ ¯D¯5 =¯ ¯D¯¯3 − ¯5 0 · ¯D!¯ ¯50 | 2 | ¯D¯¯¯5 0 + 1 ¯ ¯D¯¯Ã( ¯5!20 ¯D) ¯ ¯50¯¯2 ¯¯ ¯D¯¯2 − 1 ¯D (4.51)Although primarily designed to accommodate the narrow valleys superimposed on the least-squares statisticby the spectroscopic data, the gradient+derivative method usually gives much better results than thesimple steepest descent, providing faster fitting despite the necessity of calculating two sets of derivativesper step of minimization. Since two linearly independent vectors define a two-dimensional space, the gradient+derivativemethod is very suitable to deal with the decoupled correlated pairs of matrix elements (notethat, as could be expected, the solution of the model casediscussedabovethenisfoundinasinglestepofminimization).4.5.6 Quadratization of the S Statistic by Redefinition of the VariablesThe minimization methods outlined in previous subsections are built on the assumption that the minimizedfunction can be described locally by the first or second order approximation, therefore their efficiency isstrongly dependent on the extent to which this assumption is justified. Generally, the Coulomb excitationdata analysis problem cannot be parametrized in such a way that the least-squares statistic S (4.24) becomesstrictly a quadratic function. Some hints as how to improve the efficiency of minimization come, however,from considering extreme cases, most important of which is a perturbation-type process, characterized bydirect dependence of the excitation probabilities on the product of squares of the matrix elements directlyconnecting a given level to the ground state. Assuming a cascade-like decay with negligible feeding from aboveand negligible branching, the γ-decay intensities are expected to be proportional to the product of squaresof the level-dependent subsets of transitional matrix elements, thus the γ-yields part of the S statistic canbe expressed as a quadratic function of the logarithms of γ-ray yields versus the logarithms of the matrixelements (note that the same holds for the spectroscopic data, although the signs of the E2/M 1 mixingratios and known E2 matrix elements are to be disregarded). While expressing the dependent variables in alogarithmic scale is straightforward, the same operation for the matrix elements would mean enforcing a signidentical to that of the initial guess. To avoid this problem, the logarithmic transformation of the matrixelements is only taken to first order of Taylor expansion, resulting in the actual minimization still performedin the matrix elements space with the direction of search being modified. Toderivesuchanapproach,letusconsider the modification of a single matrix element, M, having an initial value of M 0 , during a single stepof minimization. A transformation to the logarithmic scale yields the tranformed derivative of a minimizedfunction f with respect to M:49