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

Z EmaxY i (I → I f )= dE 1 Z θp,maxE min( dEdx ) Y (I → I f )dθ p (4.22)θ p,minNote that both the Rutherford cross section and the solid angle factor, sinθ p , are already included in thedefinition of the “point“ yields, as well as integration over the detected particle φ angle.The electronic stopping powers, dE/dx, in units of MeV/(mg/cm 2 ),aredefined by a user-specifiedtable assuming common energy meshpoints for all experiments. The actual values of the stopping powersare obtained using Lagrange interpolation. The double integral is then evaluated numerically using thediscrete Simpson method. GOSIA performs the integration in two separate steps - first, the full Coulombexcitation coupled-channel calculation is done at each of the user-specified (θ p,E) meshpoints to evaluate the“point“ γ-yields, next, the actual numerical integration is performed according to the user-defined stepsizesin both dimensions. The “point“ yields at the (θ p ,E) points as required by the fixed stepsizes are evaluatedfrom the meshpoint values using the logarithmic scale Lagrangian interpolation. The θ p −E mesh is limitedto a maximum of 11x20 points, while up to 50 steps in angle and 100 steps in energy can be defined forthe integration. Subdivision of the calculated mesh improves the accuracy of integration, since Lagrangianinterpolation provides the information of the order dependent on the number of meshpoints, while theSimpson method is a fixed second-order algorithm. In addition, in cases for which the φ p (θ p ) dependence iscomplicated, such as for the large area particle detectors where kinematic and mechanical constraints maycreate such shapes, the user may optionally choose to input this dependence at the subdivision meshpoints.The interpolation is then performed between the values divided by φ p ranges to assure continuity, then theuser-given dependence is used to estimate the yields at the subdivision meshpoints. The calculation of yieldsat the meshpoints requires the full coupled-channel Coulomb excitation calculation which is time-consuming.Consequently, one should balance the number of meshpoints needed with the required accuracy.The integrated yields are calculated in units of mb/sterad times the target thickness (mg/cm 2 ), whichfor thick targets, should be assumed to be the projectile range in the target.The integration module of GOSIA is almost exclusively used in conjunction with the correction module,invoked by the OP,CORR command (V.4), used to transform the actual experimentally-observed yields tothe ones to which the subsequent fit of the matrix elements will be made. This operation is done to avoid thetime-consuming integration while fitting the matrix elements and is treated as an external level of iteration.An effect of the finite scattering angle and bombarding energy ranges as compared to the “point“ values ofthe yields is not explicitly dependent on the matrix elements, thus the fit can be done to the “point“ valuesand then the integration/correction procedure can be repeated and the fit refined until the convergence isachieved. Usually no more than two integration/correction steps are necessary to obtain the final solution,even in case of the experiments performed without the particle-γ coincidences, covering the full particle solidangle. The correction module of GOSIA uses both the integrated yields and the “point“ yields calculatedat the mean scattering angle and bombarding energy, as defined in the EXPT (V.8) input, to transform theactual experimental yields according to:Yexp(I c → I f )=Y exp (I → I f ) Y point(I → I f )(4.23)Y int (I → I f )where the superscript “c“ stands for the “corrected“ value. To offset the numerical factor resulting from theenergy-loss integration the lowermost yield observed in a γ-detector labeled as #1 for the first experimentdefined in the EXPT input is renormalized in such a way that the corrected and actually observed yieldare equal. This can be done because the knowledge of the absolute cross-section is not required by GOSIA,therefore, no matter how the relative cross-sections for the various experiments are defined, there is always atleast one arbitrary normalization factor for the whole set of experiments. This normalization factor is fittedby GOSIA together with the matrix elements, as discussed in the following section (4.5). The renormalizationprocedure results in the “corrected“ yields being as close as possible to the original values if the same targethas been used for the whole set of the experiments analyzed, thus the energy-loss factor in the integrationprocedure is similar for the whole set of the experiments. However, one should be aware of the fact, that thecorrection factors may differ significantly for different experiments, thus the corrected yields, normalized toauser-specified transition, always given in the GOSIA output, should be used to confirm that the result isreasonable rather than absolute values.42