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

adjustment of the stepsize according to the strength of the interaction and is performed every n steps, n beingan adjustable parameter defined by the user ( n =1is used as a default in GOSIA ). The Adams-Moultonintegration algorithm, with stepsize control, usually is faster than Runge-Kutta type methods, thus it hasbeen employed in GOSIA despite some drawbacks. The most important drawback is when the interaction,defined by the left-hand side of 2.17a, is weak over most of the integration range, peaking only around somevalue of the independent variable. In this case, the stepsize will be subsequently doubled and can becomeexcessively large when the strong interaction region is reached, consequently, even though the loss of accuracyis detected, the overall accuracy of integration would be irretrievably lost. Note that convergence problemsare more likely to occur for high excitation energy transitions, > 2.5MeV wherethelargeadiabaticityparameter ξ kn in equation 2.17a leads to a rapid oscillation relative to the stepsize ∆ω. Fortunately forthese large adiabaticities the excitation probabilities are well below the experimental sensitivity of currentexperiments. The loss of accuracy for high adiabaticity transitions can be detected by checking the sumof excitation probabilities provided in the output of GOSIA. The recommended procedure if the sum ofprobabilities differs significantly from unity, is to switch off the stepsize control for a given experiment usingthe INT switch of CONT suboption ( see V.3 ) rather than decrease the accuracy parameter a c . This isdue to the fact that GOSIA uses the table of Q lm functions and hyperbolic functions with a tabulation step∆ω =0.03 which defines the minimum stepsize of the integration independent of the accuracy requested,allowing the highest accuracy corresponding to a c =10 −6 . Moreover, changing the requested accuracy a cmay not solve the stepsize control problem, but result only in extending the integration range, thus makingthis problem more likely to occur.Another drawback of the Adams-Moulton method, compared to the Runge-Kutta algorithms, is thatAdams-Moulton algorithm is not self-starting, requiring the initial solutions at four points. This problemcan be overcome by employing the Runge-Kutta algorithm to provide the starting values, then switchingto the more efficient Adams-Moulton method (note that the same procedure is to be applied when thestepsize is changed, since restarting the integration with different stepsize requires the knowledge of thesolution at new independent parameter intervals). The Runge-Kutta integration algorithm is used in thecode COULEX to provide the starting solution for both the initialization of the integration and the changesof stepsize. In GOSIA the starting solutions are found using a first-order perturbation approach, valid atlarge values of ω, using the asymptotic form of the collision functions. It is assumed, that the ground state isconnected to at least one excited state with E1, M1 or E2 matrix element. The initial excitation amplitudesthen can be found as combinations of trigonometric and integral-trigonometric functions, the latter beingevaluated using a rational approximation ( given for example in [ABR72] ). To fully eliminate switchingto the Runge-Kutta algorithm, the new starting values, when changing the stepsize, are evaluated usingbackward interpolation. This procedure is reliable enough to assure reasonable accuracy while considerablyspeeding up the integration.The integration of the system 2.17a in principle should be repeated for each possible polarization of theground state, M 0 . However, the reflection symmetry in the plane of the orbit in the coordinate system usedto evaluate the excitation amplitudes yields:a Im (M o )=(−1) ∆π+I o−I a I−m (−M o ) (4.7)where ∆π= 0if there is no parity change between the ground state and a given excited state |I >,and ∆π= 1otherwise. Using the symmetry relation 4.7 one only has to solve the coupled-channel system 2.17a for theground state polarizations M 0 ≥ 0, the solution of the coupled-channel system for M 0 ≤ 0 being definedby the relation 4.7. As mentioned before, due to the approximate conservation of the magnetic quantumnumber in the frame of coordinates used, one practically has to take into account only a limited subset ofmagnetic substates beyond the main excitation path. This explicitly means that if n magnetic substateshave been specified by the user to be included in the Coulomb excitation calculation, then for each excitedstate, I, and ground state polarization, M 0 , GOSIA will catalog the magnetic substates according to theinequality:min(I,−M 0 + n) ≥ m ≥ max(−I,−M 0 − n) (4.8)which, by virtue of 4.7, defines simultaneously a set of excitation amplitudes obtained with the inversepolarization of the ground state in a range given by:37