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

where ē is an arbitratry vector. Denoting a given function of matrix elements by S( ¯M) we have to locate thepoints ¯M on the contour yielding the extremum values of S( ¯M). Let us consider the function S( ¯M) whichcan locally be appoximated by the linear expansion in ¯M:∆ s ( ¯M) =∇ s • ( ¯M − ¯M s ) (6.33)where we expand S( ¯M) in a vicinity of ¯M 2 .Tofind the extrema of 6.33 on the coutour 6.30 we have tofind the vectors ē satisfying:µd 2δ • ( ¯∇ s • ē)=0 (6.34)dē ¯∇ o ē +[(¯∇ o ē) 2 +2δēJē] 1/2which yields:¯∇ s (2δ − ¯∇ o ē) − ¯∇ o ( ¯∇ s ē) − ( ¯∇ s ē) • ¯Je =0 (6.35)From 6.35 it is clear that the vector ē must be a linear combination of J −1 ¯∇s and J −1 ¯∇o .Insertingto 6.35 we get, using the identityē = αJ −1 ¯∇s + βJ −1 ¯∇o (6.36)¯∇ s J −1 ¯∇o = ¯∇ o J −1 ¯∇s (6.37)resulting from the symmetry of J :µ 2δ + ¯∇o J −1 1/2¯∇0α = ±¯∇ s J −1 ¯∇ ; β = −1 (6.38)swhich is valid for any arbitrarily chosen origin ¯M o . However, in our case we can assume that ¯M o ,thevector of fitted matrix elements, is a close approximation of the minimum, thus ¯∇ o ≈ 0. Neglecting theterms containing ¯∇ o we finally get:µ 1/2¯M = ¯M 2δo ±¯∇ s J −1 ¯∇ J −1 ¯∇s (6.39)swhere a positive sign corresponds to the maximum of 6.33 on the contour 6.30 and a negative signcorresponds to its minimum. This formula gives an exact solution for any linear function of ¯Mo . Fornon-linear functions 6.39 can be used iteratively following the scheme:¯∇ (0)s = ¯∇ s ( ¯M o )=⇒¯M(1)¯∇ (1)s = ¯∇ s ( ¯M (1) )=⇒ ¯M (2) (6.40)until the convergence is achieved. This procedure is used in SIGMA to estimate the errors of Q 2 , cos3δand their statistical moments. Because of the fact that the implementation of the rotational-invariant sumrules is only possible for the cases in which virtually all the E2 matrix elements for low lying states areknown, implying that the underlying Coulomb excitation problem is well overdetermined, it is reasonable toassume a least-squares statistic increase δ= 1.TheJ matrix is estimated using the gradients computed byGOSIA during the calculation of correlated errors (see 4.6) and stored on a permanent file. Applying thequadratic approximation 6.1 and assuming ¯M o = minimum (i.e. ∇ o =0)onecanwrite:¯∇( ¯M) =Ĵ( ¯M − ¯M o ) (6.41)During the correlated errors calculation GOSIA evaluates the gradients in points -M for which only onematrix element at the time is perturbed from its central value. This means that ¯M − ¯Mo has only onenon-zero component, thus 5.41 defines k-thcolumnofJ if M k has been perturbed. Two estimates of J ki125