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

Figure 3: The coordinate system used to describe the γ-decay of a Coulomb-excited nucleus. The z-axis ischosen along the incoming beam direction, the x-axis is in the plane of the orbit such that the x-componentof the velocity is positive. A γ-decay in the direction θ γ ,ϕ γ is indicated.The statistical tensors ρ kχ( π 2 , (π+θ)2,π):in the new coordinate frame are obtained by rotation with Euler anglesρ kχ → X χ 0ρ kχ 0D k χ 0 χµ π2 , π + θ2,π(2.37)the rotation functions Dχχ k 0(α, β, γ) being defined according to [BOH69]:withD k χχ 0(α, β, γ) =eiχα d k χχ 0(β)eiχ0 γ(2.38)Xµmk + χk − χ 0 − m∙ (k + χd k 0 χχ 0(β) = )!(k − χ 0 ¸1/2)!·(k + χ!(k − χ)!µ k − χm∙(−1) k−χ0 −mcos β 0 ¸2m+χ+χ ∙sin β 0¸2k−2m−χ−χ22χ 0(2.39)Inserting 2.38 and 2.39 into 2.37 gives:ρ kχ → (−1) χ · X µ π + θi χ0 d k χ 0 χ ρ2 kχ 0 (2.40)A complete description of the γ-decay also must involve correction for the depolarization of the excitedlevels due to the hyperfine interaction, i.e. the interaction between the stripped electron shells and thenucleus. This effect can be taken into account by introducing the k-dependent attenuation coefficients,G k , multiplying the statistical tensors (this approach is described in Section 2.2.1). The attenuation of theangular distribution due to the hyperfine interactions is usually significant in case of thin-target experiments,while all the G k ’s equal unity if the decaying nucleus has been stopped in the target prior to the γ decay.The basic formula 2.33 can be expressed in terms of the decay statistical tensors, R kχ , which include thehyperfine interaction effects:20