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

of the a kχ tensor to the RN system is done using rotation matrices Dχχ k definedinequation2.38. Operators0U and V are defined as:U = T +2cos(θ) (2.60)where:V = T 2 +4cos(θ) · T +6cos 2 (θ) (2.61)T = 1 2 [sin(θ)e-iφ L + − sin(θ)e iφ L − ] (2.62)L + and L − being the raising and lowering operators for spherical harmonics:L + Y kχ =[(k − χ)(k + χ +1)] 1/2 Y k+1χ (2.63)L − Y kχ =[(k + χ)(k − χ +1)] 1/2 Y k−1χ (2.64)The transformation 2.58 changes the even-order spherical harmonics to linear combinations of sphericalharmonics. Therefore, the effect of the modification 2.58 on the angular distribution of γ-rays can berepresented as a modification of ρ kχ tensors. Ordering the relativistic correction by the powers of β gives:dW LAB (θ, φ)= dW RN (θ, φ)+ β X b kχ Y kχ (θ, φ)+β X 2 b kχ Y kχ (θ, φ) (2.65)dΩ γ dΩ γkkodd evenwhere the b kχ tensors result from equation 2.58 and are given by linear combinations of the a kχ components.The result of equation 2.65 can be represented as modified statistical tensors and rotated back to thelaboratory frame of coordinates. It should be noted that due to the relativistic correction, terms involvingodd-k spherical harmonics are introduced and the maximum k value is increased by 2. The relativisticcorrection formalism [LES71] presented above results in an elegant modification of the decay statisticaltensors and is used to the full extent in GOSIA despite the minor importance of some of the correction termsinvolved.2.2.3 Gamma Detector Solid Angle Attenuation FactorsDiscussing the angular distribution of the γ-rays we have been so far concerned only with a well-defined directionof observation, given by the pair of angles (θ γ ,φ γ ). To reproduce an experiment one must, however,take into account the finite size of the γ detectors as well as the γ-ray absorption mechanism. A practicalmethod to account for these effects has been described by Krane ( [KRA72] and references therein). Thismethod is applicable for the coaxial germanium detectors, almost exclusively used in the type of the experimentswe are concerned with. The method of Krane introduces γ energy-dependent attenuation factors,Q k (E γ ), multiplying the final decay statistical tensors, defined as:where:Q k (E γ )= J k(E γ )J O (E γ )(2.66)J k (E γ )=Z αmaxOP k (cos α)K(α)(1 − exp(−τ(E γ )x(α)) sin(α)dα (2.67)and α is the angle between the detector symmetry axis and the γ-ray direction, τ is an energy-dependentabsorption coefficient of the active germanium layer and x stands for the angle-dependent path length in thedetector. P k are standard Legendre polynomials, while the function K(α) can be introduced to account forthe effect of the γ-ray passing through the inactive, p-type core of the detector as well as the absorbers infront of a detector, commonly used to attenuate X-rays. The function K(α) can be written as:24