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

where, as a worst case, the backscattering geometry (= 1) is taken into account, thus ∆m 6= 0couplingsvanish for electric excitations. There is no magnetic excitation for backscattering, as discussed in Chapter 2,moreover, the magnetic excitation is weak enough to be neglected at the excitation stage for any scatteringangle, therefore only electric excitation is considered in this criterion. The factor 1/4 is introduced in equation4.1 to account for the further decrease of the importance of the excitation taking place at large |ω| due to thehigh-frequency oscillation introduced by the exponential term of equation 2.17a. Using the normalizationproperty of the collision functions:Z ∞−∞Q λμ ( =1,ω)dω =1 (4.2)and the asymptotic, pure exponential form of collision functions for large |ω| one finally gets:ω max ≥ α λ − 1 λ ln a c (4.3)where the values of α λ can be found from the asymptotic form of the collision functions and are given in thetable below, together with the resulting ω max assuming a c =10 −5 which is the default value in GOSIATable 4.1multipolarity α λ ω max for a c =10 −5E1 -.693 10.82E2 .203 5.96E3 .536 4.37E4 .716 3.59E5 .829 3.13E6 .962 2.88The range of integration over ω corresponding to a given accuracy level decreases with multipolarity, ascan be seen in Table 4.1. Following this observation, the coupling between energy levels corresponding to agiven multipolarity is included in GOSIA only within the integration range assigned to this multipolarity,which is a time-saving feature in cases many multipolarities have to be included. For Mλ couplings, neglectedso far, the integration ranges have been set according to:ω max (Mλ)=ω max (E(λ +1)) (4.4)The actual integration of the coupled-channel system of differential equations 2.17a is performed inGOSIA using the Adams-Moulton predictor-corrector method. According to this algorithm, first the predictedsolution a at ω +4∆ω, based on a knowledge of the solution at ω +3∆ω and the derivatives a 0 at fourpoints, ω, ω + ∆ω, ω +2∆ω and ω +3∆ω is found using:ā(ω +4∆ω) =ā(ω +3∆ω)+ ∆ω24 {55ā0 (ω +3∆ω) − 59ā 0 (ω + ∆ω)+37ā 0 (ω + ∆ω) − 9ā 0 (ω)} (4.5)andthencorrectedby:ā(ω +4∆ω) =ā(ω +6∆ω)+ ∆ω24 {9ā0 (ω +4∆ω)+19ā 0 (ω +3∆ω) − ¯5a 0 (ω +2∆ω)+a 0 (ω + ∆ω)} (4.6)where the excitation amplitudes are once again treated as a vector and the prime symbolically denotes thedifferentiation with respect to ω. The predicted solution is used to obtain the derivatives defined by the righthandside of 2.17a at ω+4∆ω, employed to evaluate the corrector 4.6. Stepsize, ∆ω,iscontrolledonabasisofthe comparison of the predicted and corrected solutions. The accuracy test parameter, d, isdefined as 1/14 ofthe absolute value of the maximum difference between the predicted and corrected excitation amplitudes atacurrentvalueofω. The stepsize is then halved if d>a c or doubled if d