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

Figure 1: Coordinate system used to evaluate the Coulomb excitation amplitudes. The origin is chosen in thecenter of mass of the target nucleus. The x-axis is perpendicular to the plane of the orbit, the z-axisisalongsymmetry axis of the incoming projectile orbit pointing towards the projectile, while the y-axisischosensuch that the y-component of the projectile velocity is positive. The scattering angle θ of the projectile isshown.ρ( −→ r ) and −→ j ( −→ r ) being spatial charge and current distributions of a free nucleus, respectively. Inserting 2.7to 2.6 one obtains the parametrization of electromagnetic excitation by the matrix elements of multipolemoment operators:da k (t)dt= −i 4πZ 1,2e~Xa n (t)exp it ~ (E k − E n ) X (−1) μ · S λμ (t) < Φ k |M(λ, −μ)|Φ n > (2.10)nλμTo express the time-dependent functions S λμ (t) it is convenient to introduce a frame of coordinates withthe z-axis along the symmetry axis of the incoming particle trajectory and y-axis in the trajectory planedefined in such a way that the incoming particle velocity component v y is positive. The x-axis then is definedto form a right-handed cartesian system of coordinates (Fig. 1).In this system of coordinates, one can describe the relative two-body motion, treated as the classical hyperbolicKepler solution, introducing two parameters ε and ω. Theparameterε, called the orbit eccentricity,is expressed by the center-of-mass scattering angle θ cm :1 =sin θ (2.11)cm2The dimensionless parameter ω, which replaces time, is given by:t = a v I( sinh ω + ω) (2.12)where a is the distance of closest approach in a head-on collision. This parametrization yields the followingexpression for the length of the radius-vector ¯r:Explicitly, cartesian coordinates are expressed by ε and ω as:r = a( cosh ω +1) (2.13)x =013