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

where :d 2 σ= σ R (θ p ) X R kχ (I,I f )Y kχ (θ γ ,φdΩ p dΩ γ ) (2.41)γkχR kχ (I,I f )=12γ(I) √ π G Xkρ kχλλ 0 δ λ δ ∗ λ 0,F k (λλ, I f I) (2.42)is the decay statistical tensor describing the mixed electric and magnetic transition from a state I to a stateI f . The Coulomb excitation statistical tensors are purely real for even values of k in the frame of coordinatesintroduced above, thus ρ kχ = ρ ∗ kχ. Moreover, taking into account the selection rules for electromagnetictransitions, it is easily seen that the products δ λ δ ∗ λ, also are real. Consequently, the decay statistical tensorsare purely real.The above formulas describe γ-decay of a level fed directly and exclusively by Coulomb excitation. Withmultiple excitation, however, significant feeding from the decay of higher-lying levels must be taken intoaccount. The related modification of the statistical tensors can be expressed as:R kχ (I,I f ) −→ R kχ (I,I f )+ X nR kχ (I n ,I)H k (I,I n ) (2.43)where the summation extends over all levels I n directly feeding level I. The explicit formula for the H kcoefficients is:H k (I,I n )= [(2I +1)(2I n +1)] 1/2γ(I)X½(−1) I+In+λ+k |δ λ | 2 I(1 + c(λ))λI nII nkλ¾(2.44)where c(λ) is the internal conversion coefficient for the I n → I transition. Note that for an electric monopoletransition c(0) = 0. Formula 2.44 is used to sequentially modify the deexcitation statistical tensors, startingfrom the highest levels having non-negligible population. This operation transforms the deexcitation statisticaltensors,whichareinsertedinto2.41to give the unperturbed angular distributions following Coulombexcitation.In addition, it is necessary to take into account experiment-related perturbations, namely the effects ofthe detection methods and relativistic corrections due to in-flight decay. These effects can be significantwhen using thin targets and heavy ion beams. An overview of the methods used in GOSIA to account forexperimental perturbations is presented in the following subsections.2.2.1 Nuclear Deorientation EffectIn typical Coulomb excitation experiments, both the projectile and target particles recoil into vacuum inhighly excited ionic states which subsequently decay to a ground state. The fluctuating atomic hyperfinefields cause the depolarization of the nuclear states; this effect is known as the nuclear deorientation effect.This in turn causes an attenuation of the angular distribution of the γ-rays which can be taken into accountedby introducing the spin and lifetime dependent attenuation coefficients G k , multiplying the decay statisticaltensors. The Abragam and Pound theory [ABR53] has been used extensively to describe nuclear deorientationand has proven to work well in cases where the particles recoil into high-pressure gas. However, significantdiscrepancies from the Abragam and Pound model were detected for recoil into vacuum. Therefore GOSIAuses a modified version of the two-state deorientation model ( [BOS77], [BRE77]), which seems to correlatewell with existing data despite the far-reaching simplification enforced by the complexity of the problem.[KAV89]Within the framework of the two-state model, the electrons may either belong to a “fluctuating“ statewhile the excited electronic structure decays to the atomic ground state or to a “static“ state, correspondingto an equilibrium configuration. Since the excitation and decay of stripped electron shells is too complicatedto be described exactly, it is assumed that all atomic processes taking place in the fluctuating state arepurely random. The rate of transition from the fluctating to static state, Λ ∗ , is an adjustable parameter ofthe model. The time-dependent deorientation coefficients G k (t) then are given by :21