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

6 QUADRUPOLE ROTATION INVARIANTS - PROGRAM SIGMAThe heavy-ion induced Coulomb excitation allows the measurement of essentially full sets of the E2 matrixelements for the low-lying states of the nuclei. It is interesting to ascertain to what extent these sets ofdata can be correlated using only a few collective degrees of freedom. The quadrupole rotational invariants[CLI72, CLI86] have been proven to be a powerful tool for extracting the collective parameters from thewealth of data produced by the Coulomb excitation. This procedure, outlined below, provides a completelymodel-independent way of determining the E2 properties in the rotating collective model frame of referencedirectly from the experimental E2 properties in the laboratory frame without recorse to models. Conversionof the measured E2 matrix elements to the quadrupole invariants is performed by the separate code, SIGMA,which uses the information stored by GOSIA on a permanent file during error calculation.6.1 FORMULATION OF THE E2 ROTATIONAL INVARIANT METHODElectromagnetic multipole operators are spherical tensors and thus zero-coupled products of such operatorscan be formed that are rotationally invariant, i.e., the rotational invariants are identical in any instantaneousintrinsic frame as well as the laboratory frame. Let us, for practical reasons, consider only the E2 operator.The instantaneous “principal“ frame can be definedinsuchaway,that:E(2, 0) =Qcosδ (6.1)E(2, 1) = E(2, −1) = 0E(2, 2) = E(2, −2) = (1/ √ 2) · Qsinδwhere Q and δ are the arbitrary parameters. This parameterization is general completely and modelindependent.It is analogous to expressing the radial shape of a quadrupole-deformed object in terms ofBohr’s shape parameters (β,γ). Using this parameterization the zero-coupled products of the E2 operatorscanbeformedintermsofQ and δ, e.g.:[E2 × E2] 0 = 1 √5Q 2 (6.2)no √0[E2 × E2] 2 2× E2 = √ Q 3 cos 3δ (6.3)35which are the two lowest order products. It is straightforward to form any order product, the limit beingset only by practical feasibility. On the other hand, one can evaluate the matrix elements of the E2 operatorsproducts by recursively using the basic intermediate state expansion:D¯s ¯(E2 × E2) J¯¯¯ EX½ ¾r = (−1)Is+Ir2 2 Jhs ||E2|| tiht ||E2|| ri(6.4)(2I s +1) 1/2 I s I r I ttwhich allows the rotational invariants built from Q and δ to be expressed as the sums of the products ofthe reduced E2 matrix elements using the experimental values of these matrix elements.As an example, an expectation value of Q 2 can be found directly using 6.2 and 6.4:√ √5Q 2 ­=s¯¯¯¯(E2 × E2)0¯¯¯¯(−1) 2I s 5 X½ ¾2 2 0s® = M(2I s +1) 1/2 (2I s +1) 1/2 sr M rs (6.5)I s I s I rwhere the abbreviation for the matrix elements:M sr = hs ||E2|| ri (6.6)½ ¾2 2 0is used. The Wigner’s 6-j symbolis equal to ([R0T59]):I s I t I rr119