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

with the reduced matrix element M connecting both levels. For this case, the A matrices are explicitlygiven as:Ã!0 −q a(λ0) MζA 1 =q a (λ0)(3.8)Mζ 0Ã!0 −q s(λ0) MζA 2 =q s (λ0) Mζ 0and, according to 3.5:a(0 + )=cos(q (λO)sMζ)+i sin(q s(λO) Mζ) · sin(2q a (λO) Mζ) (3.9)The above yields:a(I π )=−i sin(q (λO)sMζ) · cos(2q (λO) Mζ)aq (λO)aq s (λO) = Arccos(Re a (0+ ))Mζ= −arctgµ Im a(0 + )Im a(I π /2Mζ)(3.10)Using 3.10 it is possible to extract q parameters substituting the excitation amplitudes resulting fromthe exact calculation, i.e. the solution of 2.17. A similar procedure can be applied to find the q parameterscorresponding to ∆m = ± 1 coupling.The appropriate formula 3.5 works best in cases where the range of ξ−parameters is not excessively wide.This practically assures good performance in all cases of multiple Coulomb excitation, since, as discussedin Chapter 2, small values of ξ are required to produce significant multiple excitation. To demonstrate thereliability of the semianalytic approximation, let us consider a test case in which we simulate the Coulombexcitation of a Z =46, A = 110 nucleus by a 200 MeV 58 Ni beam. The target nucleus is described bythe level and coupling scheme (of course having nothing to do with the real 110 Pd) shown in Figure 6. Thelevel energy differences all are assumed to be equal to 0.5 MeV. All reduced E2 matrix elements are givenin units of e.b.Comparison between the exact solution of Eq. 2.17 and the approximate solution using 3.5 is as follows:Excitation amplitude (population)Levels Equation 2.17 Equation 3.50 + .494 + .018i (.244) .499 - .145i (.265)2 + .134 + .139i (.032) .161 + .096i (.035)4 + -.205 + .391i (.194) -.186 + .407i (.200)6 + -.176 + .388i (.145) -.192 + .312i (.134)2 + .276 + .266i (.119) .284 + .157i (.105)4 + -.482 - .165i (.260) -.445 - .236i (.254)The A matrix approximation is generally more than adequate to calculate derivatives of level populationswith respect to the matrix elements, using internal correction factors (see chapter 4) to account for differencesbetween the approximate and the exact approach. It also provides a useful tool to investigate (at leastqualitatively) the Coulomb excitation process. As an example, let us consider the influence of the quadrupolemoment for the two-level system shown in Figure 7 and discussed extensively in the Alder-Winthermonograph [ALD75].To simplify the notation, let us denote:31