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

2.2 Gamma Decay Following Electromagnetic ExcitationThe decay of a Coulomb-excited nucleus can be treated as completely separated in time from the excitationprocess. The initial condition for the decay can be described by a statistical tensor, expressing the state ofpolarization of the decaying level:ρ kχ (I) =(2I +1) 1/2 X MM 0 (−1) I−m0 µI−M 0kχIMa ∗ IM 0,a IM (2.30)whereweexplicitlydenoteexcitation amplitude of a substate |IM > with two-dimensional indexing.Averaging 2.30 over all possible polarizations of a ground state gives:where:ρ kχ (I) = (2I +1)1 22I o+ 1Xµ(−1) I−m0M o MM 0I−MkχIMa ∗ IM 0(M o)a IM (M o ) (2.31)ρ 00 (I) = P I (2.32)In an arbitrary coordinate frame with the origin at the decaying nucleus the angular distribution ofgamma radiation can be expressed as:d 2 σ1= σ R (θ p )dΩ p dΩ γ 2γ(I 1 ) √ πXkevenχρ ∗ kχ(I 1 ) X δ λ δ ∗ λ 0F ¡k λλ 0 ¢ ¡ ¢I 2 I 1 Ykχ θγ, φγλλ 0(2.33)Here σ R (θ, p ) denotes the scattering angle dependent Rutherford cross-section, F k (λλ, I 2 I 1 ) are the γ − γcorrelation coefficients (as defined, e.g., in FRA65), Y kχ (θ γ ,θ γ ) are normalized spherical harmonics. Thequantities δ λ are the I 1 → I 2 transition amplitudes for multipolarity λ, related to the emission probabilityγ(I 1 ) by:γ(I 1 )= X λn|δ λ (I 1 → I n )| 2 (2.34)An explicit form of F k (λλ 0 ,I 2 I 1 ) can be expressed in terms of Wigner’s 3 − j and 6 − j symbolsF k (λλ 0 ,I 2 I 1 )=(−1) I1+I2−1 [(2k +1)(2I 1 +1)(2λ +1)(2λ 0 +1)] 1/2µ λ1λ 0−1k0½ λI 1λ 0I 1kI 2¾(2.35)while δ λ is given by:whereµ 1/2 µ λ+1/2δ λ = i n(λ) 1 8π(λ +1) Eγ (2λ +1)!!~ λ+1 (2.36)λc(2I 1 +1) 1/2½ λn(λ) =λ +1for Eλ transitionfor Mλ transitionNote that the transition amplitude for an electric monopole transition, δ 0 , is defined by equation 2.71.The coordinate system used for evaluation of excitation probabilities is no longer convenient to determineangular distributions of gamma radiation, as it is not fixed with respect to the laboratory frame. Therefore,it is useful to define the z-axis along the beam direction with the x-axis in the plane of orbit in such a way,that the x-component of the impact parameter is positive. The y-axisisthendefined to form a right-handedsystem (Fig. 2.3).19