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9.3. Nuclear Transparency Calculations 117
12 C(e, e ′ p) data was performed in order to extract the correct values for the 1s and 1p
spectroscopic factors from experiment. It was shown there that such an extraction of
the spectroscopic factors at low Q 2 [Q 2 < 0.6 (GeV/c) 2 ] lead to a summed (average)
spectroscopic factor S12 C = 0.575 ± 0.02 for 12 C. These calculations were performed
with an optical potential model to treat the final state interactions. On the other
hand, the authors of Ref. [120] also performed Glauber calculations in the higher Q 2
range [Q 2 > 0.8 (GeV/c) 2 ] with the result that they found a summed spectroscopic
factor of 0.84 ± 0.03 at Q 2 = 1.08 (GeV/c) 2 and slightly increasing values with
increasing energy. The results of this analysis are shown in Fig. 9.6. We note that
the authors of Ref. [120] adopt the convention in which the spectroscopic factor
for a particular angular level is normalized according to the occupation number
in the independent-particle model (i.e. 2 and 4 for the 1s and 1p proton levels
in 12 C, respectively). It is evident from the results for the spectroscopic factors
in 12 C that the calculated transparencies will differ enormously from each other
depending on which values for the spectroscopic factors are adopted. When using
the high Q 2 spectroscopic factors of Fig. 9.6 to calculate the nuclear transparencies
in the 12 C(e, e ′ p) reaction, the authors found values T12 C ∼ 0.5 − 0.6 in the Q 2 range
between 1 and 8 (GeV/c) 2 . On the other hand, when using the low Q 2 spectroscopic
factors, transparencies T12 C ∼ 0.8 − 0.9 were found.
As already pointed out above, one can also perform the inverse procedure, where
one combines the experimental transparency and the theoretical transparency to
obtain an average spectroscopic factor S12 C. If the theoretically calculated transparency
is scaled with the correct averaged spectroscopic factor, it should equal the
experimentally found transparency. In other words the summed or average spectroscopic
factor is simply found by considering the ratio T exp /T theo . This simple relationship
gives the results of Fig. 9.5 a whole new dimension. For one, this means that
we should only consider the calculations that overshoot the data points as realistic.
The uncorrelated Glauber calculation using the CC2 current operator for example,
largely undershoots the data points (T theo < T exp ), assuming a spectroscopic factor
larger than unity. Second, the closer the theoretical curves are to the experimental
data points, the higher the average spectroscopic factor. Complete agreement with
the experimental results would assume a full occupancy of the nuclear shell levels.
In Fig. 9.5 the correlated RMSGA calculations predict an average spectroscopic
factor that approaches unity in the Q 2 > 0.6 (GeV/c) 2 regime. This observation concurs
with the findings of Fig. 9.6. These spectroscopic factors are large compared to
the ones typically found in low Q 2 analysis. From Fig. 9.5 we can also infer a rather
modest Q 2 dependent rise of the spectroscopic factors in the correlated RMSGA calculations
using the CC1 operator. This also concurs with the findings of Ref. [120],
as can be seen from Fig. 9.6. For the calculations using the CC2 current operator,
there’s no real indication for a Q 2 dependence of the spectroscopic strength, when
experimental error margins are kept in mind. However, the most striking feature
of Fig. 9.6 is the abrupt rise of the spectroscopic strength when going from the low