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

4 NUMERICAL METHODSThe code GOSIA is designed to perform various functions defined by user-specified sequence of options. Inthe simplest mode, GOSIA can be used to calculate the excitation probabilities, population of the levels andγ-decay statistical tensors, equation 2.30, thus providing an equivalent of the code COULEX. Activatingthe γ-decay module of the code extends this type of calculation to obtain the γ yields for a single valueof bombarding energy and scattering angle, as well as performing integration over specified ranges of thebombarding energy (due to the projectile energy loss in a target ) and projectile scattering angle to reproducereal experimental conditions. These calculations require an input set of matrix elements treated as fixed data.The main purpose of GOSIA is, however, to fit the electromagnetic matrix elements to reproduce availableexperimental data. GOSIA can handle simultaneously experimental γ yields (up to 48000) observedin50independent experiments. Additional data, i.e. branching ratios (max. 50), lifetimes of the nuclear levels(max. 10), E2/M 1 mixing ratios (max. 20) and previously measured Eλ matrix elements (max. 30) alsomay be included. All these data, and their experimental uncertainties, are used to construct a least-squaresstatistic, usually called χ 2 or penalty function. The minimum of this statistic, treated as a function of matrixelements, defines the solution, while its distribution in the vicinity of the minimum determines the errors offitted matrix elements. In the present version of the code, the investigated nucleus is described by maximumof 75 energy levels, with the number of magnetic substates not exceeding 600. The levels may be coupledwith up to 500 matrix elements (E1 through E6 and M1, M2), any number of them allowed to be declaredas the variables to be fitted.As mentioned before, direct use of the full Coulomb excitation formalism to perform the minimizationis out of the question due to the computer time necessary for repeated calculations. The mimimization canbe accelerated using the approximation presented in Chapter 3. A significant amount of time also can besaved if the recoil-velocity correction ( 2.2.2) is neglected. The effect of both replacing the full excitationformalism by the matrix approximation and neglecting the relativistic correction is only weakly dependenton the matrix elements, therefore it is feasible to introduce the “correction factors“, that account for thedifferences between full and approximate calculations which are assumed to be independent of the fittedmatrix elements. The minimization can be performed using only the fast, approximate formalism, withcorrection factors refreshed by running the full calculation periodically. In addition, the Coulomb excitationapproximation is applied only to ∆m =0and ∆m =±1 couplings, thus an effect of truncation of the numberof magnetic substates being taken into account is also included in the correction factors, since this is notstrongly dependent on the actual set of matrix elements.Further acceleration of the fitting procedure is made possible by replacing the integration over experimentdependentscattering angle and bombarding energy ranges with a single calculation of the Coulomb excitationinduced γ yields assuming the mean values of the scattering angle and bombarding energy. This approximationalso is not explicitly dependent on the fitted matrix elements, thus the difference between the integrationprocedure and the result of using the mean values of bombarding energy and scattering angle can be accountedfor by introducing another set of correction factors, treated as constants. Actually, it is convenientto apply this correction to the experimental yields, i.e. to rescale the experimental data according to thecomparison of integrated and “mean“ yields. This is done initially using a starting set of matrix elementsand the resulting “corrected experimental yields“ are subsequently used as the experimental values for fitting(this procedure is presented in detail in Section 4.4). Thus, the fitting of matrix elements is performed usingtwo levels of iteration - first, external, is to replace the experimental ranges of bombarding energy and scatteringangle with the average values of these parameters, while second, internal, is the actual minimizationof the least-squares statistic. After the convergence at the internal level is achieved, one should recalculatethe correction to the experimental yields with the current set of matrix elements and repeat the minimization.Usually, for thin targets and a not excessively wide range of scattering angles, the second repetition ofexternal iteration already yields negligible changes in the matrix elements found by the minimization. It hasbeen checked, that even if no particle coincidences were required, only 2-3 recalculactions on the externallevel were necessary despite the integration over the full solid angle.Estimation of the errors of fitted matrix elements is a final step for the Coulomb excitation data analysis.This rather complicated procedure is discussed in Section 4.6. A separate program, SELECT, has beenwritten to reduce the considerable computational effort required for this task using the information obtainedduring minimization. This information, preprocessed by SELECT, is fed back to GOSIA. Optionally, the34