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

q (20)s (0 + → 2 + ) · M 1 ζ (20)0 + 2 + = qq (20)a (0 + → 2 + ) · M 1 ζ (20)0 + 2 + = q 1 (3.11)q (20)s (2 + → 2 + ) · M 2 ζ (20)2 + 2 + = QNote that there is no antisymmetric component of the interaction for the quadrupole moment, sinceζ =0.The A matrices can then be written as:µ 0 −q1A 1 =(3.12)q 1 0µ 0 qA 2 =q Qyielding:µ cos q1 ¯+sinqexp(±A 1 )=1(3.13)± sin q 1 cos q 1Ãcos p −1exp(−iA 2 )=e iQ/2 2 i Q p sin p − iq p sin p!− iq p sin p cos p + 1 2 i Q p sin pwhereUsing 3.5 gives:p = 1 2 (Q2 +4q 2 ) 1/2a 2 + = 1 sin pieiQ/22 p (Q sin 2q 1 − 2q cos 2q 1 ) (3.14)which results in an excitation probability:P 2+ = 1 sin 2 p4 p 2 (Q sin 2q 1 − 2q cos 2q 1 ) 2 (3.15)The formula 3.15 is a generalization of the second-order perturbation theory result. Taking into accountonly the lowest order terms in q, q 1 ,q 1 and Q gives:P 2+ ≈ q 2 (1 − Q · q1q )2 (3.16)as predicted by second-order perturbation theory. It should be observed, that the influence of the quadrupolemoment is significant only if the ratio of antisymmetric to symmetric q parameters is high, which physicallycorresponds to a large value of ξ. As can be seen from 3.14, theprimaryeffect of the static moment is rotationof the complex excitation amplitude, due to the exp(iQ/2) factor. This allows rather accurate measurementsof static moments in cases of interfering paths of excitation, i.e., when a state is excited in a comparable wayvia two or more sequences of couplings. In this situation the phases that the partial excitation amplitudesare summed are of primary importance.In general, 3.10 is only a semianalytical formula, since for complex cases, exponents of the A matricesmust be evaluated numerically. Nevertheless, numerical determination of exp(A) operators is much fasterthan the integration of a system of differential equations. Application of the A matrix approximation forfitting of matrix elements to reproduce experimental data will be discussed in some more detail in Chapter4. It is worth observing that the truncation of Taylor series approximating the exp(A) operators providesa way to generate perturbation theory of a given order, thus being useful for investigating weak excitationprocesses.33