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

leads to excitation of a growing number of levels, while the number of matrix elements influencing the processincreases even faster.The product ξ relates the collision time to the natural time period for a nuclear transition. Appreciableexcitation of nuclear levels is most probable when the relative oscillation of their eigenfunctions is slow onthe scale of the effective collision time. That is, when the Fourier transform of the electromagnetic impulsehas significant probability at the transition energy. This implies that the Coulomb excitation cross sectionsgenerally increase with decreasing adiabaticity ξ. Taking into account that the excitation energy is smallcompared to the bombarding energy, expressed in MeV, and expanding 2.17b in a series, yields:ξ kn ∼ Z 1Z 2 A 1 2112.65E n− E kE ppEp(2.23)Coulomb excitation is most probable for small transition energy differences in contrast to the deexcitationprocess, that is, the adiabaticity limits Coulomb excitation to a band of states within about an MeV abovethe yrast sequence. Increasing the bombarding energy of the projectile enhances the Coulomb excitation notonly because of increased coupling parameters ζ but also because of decreased values of ξ.A convenient way to visualize the excitation process is to introduce so-called orbital integrals R(ε, ξ),defined as:R λμ (, ξ) =Z ∞−∞Q λμ (, ω)exp(iξ( sinh ω + ω))dω (2.24)The orbital integrals R directly determine the excitation amplitudes, provided that the first-order perturbationapproach is applicable. Generally, the formal solution of 2.17a can be expressed by an infiniteintegral series as:½Z ∞ā(ω = ∞) =ā o +A(ω o )dω o )−∞½Z ∞ Z ωn... + A(ω n )dω n A(ω n−1 )dω n−1 ...−∞−∞¾½Z ∞ Z ω1¾ā o + A(ω 1 )dω 1 A(ω o )dω o ) ā o + (2.25)−∞−∞¾Z ω1−∞A(ω o )dω o )ā o + ...where ā stands for the vector of the amplitudes and A(ω) is the right-handside matrix operator of 2.17a.The initial vector ā o has the ground state amplitude equal to one, all other components vanish. Each termof 2.25 corresponds to given order excitation, i.e., connects a number of matrix elements equal to the numberof integrations. Assuming a weak interaction, the first-order perturbation theory expresses the excitationamplitudes of the states k directly coupled to the groundstate as the linear term of 2.25:a k = X λζ (λμ)ko· R λμ (, ξ ko ) (2.26)μ = M o − M kwhere the index “0“ denotes the ground state. Note that 2.26 (and more general 2.17) is valid for a fixedpolarization of a ground state, thus for non-zero ground state spin one must average the excitation amplitudesover all possible magnetic substates of the ground state. This explicitly means:1 Xa k =a(2I o+ 1) 1/2 k (m o ) (2.27)m oandP k = a k a ∗ k (2.28)where a k (m 0 ) denotes the excitation probability of state k with ground state polarization m o . The formula2.26 has been used extensively to determine reduced excitation probabilities:16