Model dependent analysis

The much improved sensitivity provided by modern
<math>4\pi</math> high-resolution
<math>\gamma</math>-ray
detector facilities, such as Gammasphere when coupled to
<math>4\pi</math> recoil-ion detectors like [[CHICO | CHICO]], has greatly expanded the number of collective
bands and levels observed in heavy-ion induced Coulomb excitation
measurements. Since the late
1990s this improved sensitivity has led to an explosive increase in the number of
<math>E\lambda</math>
matrix elements involved in least-squares fits to Coulomb excitation data. For
example, current heavy-ion induced Coulomb excitation measurements can
populate <math>\approx 100</math> levels in <math>\approx 10</math>
collective bands coupled by <math>\approx 3000</math> <math>E\lambda</math>
matrix elements. This rapid increase in the number of unknowns, and
concomitant increase in the dimension of the least-squares search problem,
coupled with a reduction in available beam time due to closure of a
substantial fraction of the arsenal of heavy-ion accelerator facilities, has
made it no longer viable to obtain sufficiently complete sets of Coulomb
excitation data for a full model-independent Gosia analysis. Fortunately the
collective correlations are strong for low-lying spectra of most nuclei making
it viable to exploit these collective correlations to greatly reduce the
number of fitted parameters to accommodate the smaller data sets. For example,
for the axially-symmetric rigid rotor the fit to the diagonal and transition
E2 matrix elements can be reduced to fitting the one intrinsic quadrupole moment
of the band, which, in principle, only requires measurement of one
E2 matrix element. Model-dependent analyses of Coulomb excitation data can be
used to extract the relevant physics even when the data set is insufficient to
overdetermine the many unknown matrix elements model independently.

A model-dependent approach that has been highly successfully involves
factoring the system into subgroups of levels that have similar collective
correlations, and assuming that a model can adequately relate the
<math>E\lambda</math> matrix elements for each of these localized subgroups of levels. That is, the
model is used to relate the <math>E\lambda</math> matrix elements in a localized region by coupling all of the "dependents" to
one "master" member of the subgroup. The least-squares fit then is made to
find the best values of the master matrix elements with the dependents varying
in proportion to the masters. This approach can reduce the number of
parameters being fit by an order of magnitude. The best-fit values for these
master matrix elements then can be used iteratively to refine the
model-dependent coupling employed for correlating the matrix elements in the
localized subgroups. As an example, in strongly-deformed nuclei the ground
rotational band levels below the first band crossing can be broken into one or
two subgroups each of which are coupled to common intrinsic
E2 moments, the levels above the band crossing also can be broken into similar
subgroups, while the individual matrix elements around the band crossing can
be treated as independent parameters. An iterative procedure then can be
employed where models are assumed to be sufficient accurate locally to
extrapolate the measured sensitive matrix elements to model-dependently
predict the less sensitive matrix elements.

The model-dependent approach must be treated with considerable care because
the strong cross correlations can lead to erroneous conclusions. The fact that
one model fits the experimental Coulomb excitation data is not a proof that
this solution is unique. For example, the Davydov-Chaban rigid-triaxial rotor
model reproduced well the measured Coulomb excitation yields for an early
multiple Coulomb excitation studies of
<math>^{192,194,196}</math>Pt<ref>I.Y. Lee, D. Cline, P.A. Butler, R.M. Diamond, J.O. Newton, R.S. Simon and F.S. Stephens, Phys. Rev. Lett. 39:684 (1977).</ref> implying the existence of rigid triaxial
deformation. A subsequent more extensive Coulomb excitation
study,<ref>C.Y. Wu, Ph.D. Thesis, University of Rochester (1983)</ref>,<ref>C.Y. Wu, D. Cline, T. Czosnyka, et al., Nucl. Phys. A 607:178 (1996).</ref> that was analysed using Gosia,
determined that these nuclei are very soft to
<math>\beta</math>- and
<math>\gamma</math>-vibrational degrees of freedom. The earlier erroneous conclusion that this
nucleus behaved like a rigid triaxial rotor was fortuitous because the early
measurements were sensitive only to the centroids of the
<math>\beta</math> and
<math>\gamma</math> shape degrees of freedom. Another example is that the axially-symmetric model
often can give an almost equally acceptable fits to Coulomb excitation data of
triaxially-deformed nuclei where erroneous
<math>B(E2)</math>
strengths compensate for incorrectly assumed static electric quadrupole
moments. Thus it is important to compare the quality of model-dependent fits
using various competing collective models to derive reliable conclusions as to
the relative efficacy of different collective models.

==Notes==

<references/>