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

y = a( 2 − 1) 1/2 sinh ω (2.14)z = a(cosh ω + )Note that the closest approach corresponds to ω =0.The functions S(t) are replaced by dimensionless “collision functions“ Q(, ω):for electric excitations, andµ 1/2Q E λ (2λ − 1)!! πλ,μ(, ω) =a r(ω)S E(λ − 1)! 2λ +1λ,μ(t(ω)) (2.15a)Q M λ,μ(, ω) = c vµ 1/2(2λ − 1)!! πaλ r(ω)S M(λ − 1)! 2λ +1λ,μ(t(ω)) (2.15b)for magnetic excitations. The explicit expressions for the collision functions Q(ε, ω) for E1 to E6 and M1,M2 are compiled in Table 2.1.It is convenient to replace the multipole operator matrix elements by reducedmatrix elements< I s ||M(λ)||I f > using the Wigner-Eckart theorem:=(−1) I s −M sµIs−M sλμI f (2.16)fIt is presumed that the phase convention for the wavefunctions |I> is such, that the reduced matrixelements are real. Insertion of 2.11 through 2.16 into 2.10 yields the final system of differential equationsfor the excitation amplitudes a k . The following formulae are given already in a numerical representation ofphysical constants, corresponding to energies in MeV, reduced electric multipole matrix elements in eb λ/2and reduced magnetic multipole matrix elements in μ n .b (λ−1)/2 ,μ n being the nuclear magneton.da kdω = −i X Q λμ (, ω)ζ (λμ)kn · · exp(iξ kn ( sinh ω + ω)) · a n (ω) (2.17a)λμnThe one-dimensional indexing of excitation amplitudes a k involves all magnetic substates of states |I>,which from a point of view of a theory of electromagnetic excitation are treated as independent states. Theadiabaticity parameters ξ kn , which represent the ratio of a measure of the collision time and the naturalnuclear period for a transition, are given by:ξ kn = Z 1Z 2√A16.34977 ((E p − sE K ) −1/2 − (E p − sE n ) −1/2 ) (2.17b)s =(1+A 1 /A 2 )E p being the bombarding energy. The above is valid for target excitation, while for projectile excitationindices 1 and 2 are to be interchanged. The adiabaticity parameters have been symmetrized with respect toinitial and final velocities in a manner that does not violate unitarity and time reversal invariance.withThe same convention is valid for the coupling parameters ζ (λn)knζ (λn)kn =(2λ +1)1/2 (−1) In−Mn µIn−M nψ kn = C E(M)λλμgiven by:I kψM knk√Z 1 A1(sZ 1 Z 2 ) λ {(E p − sE k )(E p − sE n )} (2λ−1)/4(2.17c)(2.17d)where the numerical coefficients Cλ E (M) are different for electric and magnetic excitation and are givenexplicitly as:14