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

Zd 2 σ(I → I f )Y (I → I f )=sin(θ p )dφφ pdΩ γ dΩ p (4.16)pwhere the integrand is given by equation 4.13. The integration over the φ angle is trivial since the φ-dependence is analytical, described only by a single cosine function, as seen from equation 4.13. Accordingto the input units requested by GOSIA ( Section V ) the yields will be calculated in units of mb/srad/rad (i.e. millibarns per steradian of the γ−ray solid angle per radian of the particle scattering angle ). This holdsfor the γ yields calculated using OP,POIN (V.18). The reproduction of the experimentally observed yieldsshould, however, involve the integration over the particle scattering angle including the beam energy loss inthe target, thus the fully integrated yields, obtained using OP,INTG, have a different meaning, as describedin Section 4.4.So far, we have neglected the effect of the geometric displacement of the origin of the system of coordinatesdue to in-flight decay, i.e. it is assumed that all observed decays originate at the center of a target, thusthe relativistic velocity correction is the only one needed to transform the nucleus-centered system to thelaboratory-fixed system. This approximation is adequate as long as the mean lifetimes of the decaying statesare in the subnanosecond range. For the cases involving longer-lived states, GOSIA provides an optionalfirst-order treatment to correct for the geometric displacement. To treat this effect rigorously, one has totake into account both the change of the angles of the γ detectors, as seen by the decaying nucleus, and thechange of the solid angles subtended by the detectors. For a given direction of the recoil the observed yieldof the decay of a state having the decay constant λ can be written as:Y = λZ ∞0exp(−λt) · S(t)Y (t)dt (4.17)where S(t) denotes the time dependence of the solid angle factor, while Y (t) stands for the time dependenceof the “point“ angular distribution. To the lowest order, the product S(t)Y (t) is expressed as:S(t)Y (t) ≈ Y (0) + pt (4.18)where p stands for the time derivative of this product taken at t =0(note that S(0) = 1,thusS(0)Y (0) = Y (0)). Inserting 4.18 into 4.17 and using the displacement distance, s, as an independent variable instead of timewe finally obtain for mean lifetime τ:Y = Y (0) + τp (4.19)where p is calculated numerically using a second set of yields evaluated in a point shifted by s in the recoildirection, i.e.:S(s)Y (s) − Y (0)p = (4.20)swhere S(s), is calculated assuming that the the displacement is small compared to the distance to thedetector, r 0 ;r02 S(s) =(¯r 0 − ¯s) 2 (4.21)The displacement correction requires the γ yields to be calculated twice for each evaluation, thus shouldbe requested only when necessary to avoid slowing down the execution.4.4 Integration over the Projectile scattering Angle and the Energy Loss in aTarget - Correction of Experimental γ-ray YieldsAn exact reproduction of the experimentally observed γ yields requires the integration over a finite scatteringangle range and over the range of bombarding energies resulting from the projectile energy loss in a target.The γ-decay formalism presented in Section 4.3 has so far assumed that the projectile scattering angle, θ p ,and the bombarding energy, E b , are constant for a given experiment. Using the definition of the “point“yields (4.16) the integrated yields, Y i (IV I f ), are given by:41