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% |  Dr. S.P.G. Chappell              |  Nuclear Physics Laboratory|
% |                                   |  University of Oxford      |
% | :Phone (01865) 273312             |  Keble Road                |
% | :Fax   (01865) 273418             |  Oxford  OX1 3RH,  UK      |
% | :E-mail s.chappell1@physics.ox.ac.uk                           |
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%PACS numbers refer to..  
%25.70.Ef Resonances
%21.60.Gx Cluster models
%27.30.+t 20 <_ A <_ 38
%27.20.+n 6 <_ A <_ 19

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\renewcommand\baselinestretch{2}

\begin{document}


\begin{titlepage}

\begin{center}
{\Large New excitation function measurement for $^{12}$C+$^{12}$C
inelastic scattering to $\alpha$-unbound reaction channels}

\vspace{1cm}

{S.P.G. Chappell$^*$, R.L. Cowin, G.K. Dillon, D.L. Watson}\\
{\it Department of Physics, University of York, Heslington, York
YO1 5DD, U.K.}\\

\vspace{0.5cm}

{W.D.M. Rae}\\
{\it Nuclear Physics Laboratory, University of Oxford, Keble
Rd., Oxford OX1 3RH, U.K.}\\

\vspace{0.5cm}

{M. Freer}\\
{\it School of Physics and Space Research, University of
Birmingham, Edgbaston, Birmingham B15 2TT, U.K.}\\

\vspace{0.5cm}

{N. Curtis and M. Shawcross}\\
{\it Department of Physics, University of Surrey, Guildford, Surrey
GU2 5XH, U.K.}\\

\vspace{0.5cm}

{\it $^*$ Present address: Nuclear Physics Laboratory,
University of Oxford, Keble Rd.,\\
Oxford OX1 3RH, U.K.}\\

\vspace{1cm}

13$^{th}$ May 1997\\

\end{center}

\newpage

\begin{abstract}

Excitation functions have been measured for several
$^{12}$C($^{12}$C,$^{12}$C$^*$$\rightarrow 3\alpha$)$^{12}$C$^*$
reactions at centre-of-mass energies in the range $E_{c.m.}=31$ to 45
MeV. New resonant structures are observed near $E_{c.m.}$= 37.5 and
43.0 MeV in the $^{12}$C(3$_1^-$)+$^{12}$C(3$_1^-$) channel. These new
resonances correlate with pronounced structure in the angle integrated
cross-section of several other $^{12}$C+$^{12}$C inelastic channels
which do not lead to a $6\alpha$ final state. These results are
considered in the context of a simple $\alpha$-cluster model of
$^{12}$C.

\end{abstract} 


\vspace{2cm}
PACS number(s): 25.70.Ef, 21.60.Gx, 27.30.+t, 27.20.+n\\

Key words: NUCLEAR REACTIONS,
$^{12}$C($^{12}$C,$^{12}$C$^*$$\rightarrow 3\alpha$)$^{12}$C$^*$,
E=31-45 MeV, measured excitation functions, observed light heavy-ion
resonances. 

\end{titlepage}

\newpage

%_______________________________________________________________________

\section{Introduction}

The use of new segmented silicon strip detectors \cite{Dav90} has
enabled the study of reactions leading to multi-particle final
states. The $^{12}$C+$^{12}$C system was one of the first to be
studied using these detectors and new resonant behaviour has been discovered
near $E_{c.m.}=32-33$ MeV for scattering to
$^{12}$C($0_2^+$)+$^{12}$C($0_2^+$) and a 6$\alpha$-particle final
state
\cite{Wuo92}. In recent measurements in the same energy region, further
resonant structures have been reported in the
$^{12}$C($0_2^+$)+$^{12}$C($0_3^+$),
$^{12}$C($0_2^+$)+$^{12}$C($3_1^-$),
$^{12}$C($3_1^-$)+$^{12}$C($3_1^-$) and
$^{16}$O($0_1^+$)+$^{8}$Be($0_1^+$) exit channels
\cite{Cha95,Fre95,Ali94,Mir94,Mar96}. Several years ago Fulton
{\it et al.} \cite{Ful80} also measured enhancements in the cross
section for scattering to $^{12}$C($0_1^+$)+$^{12}$C($3_1^-$) near
$E_{c.m.}\sim 33$ MeV using more conventional techniques. These
observations represent just a few of the many resonances recorded
across a wide energy range in the $^{12}$C+$^{12}$C system over the
last three decades \cite{Erb84}.  However, the new resonance reported
by Wuosmaa {\it et al.} \cite{Wuo92} has proved particularly difficult
to interpret using conventional reaction models and has stimulated the
proposal of a number of controversial explanations. The first
observation of this resonance in scattering to
$^{12}$C($0_2^+$)+$^{12}$C($0_2^+$)
\cite{Wuo92} at $E_{c.m.}=32.5$ MeV suggested an association with a
6$\alpha$-chain state in $^{24}$Mg near $E_x=50$ MeV, as predicted by
the cranked $\alpha$-cluster model
\cite{Mar86}.  The strong increase in reaction cross-section observed
at 90$^{\circ}$ in the centre-of-mass was explained by the new concept
of an approximate-shape eigenstate \cite{Rae92,Mer93}. The subsequent
observations of resonances near the same energy in other channels
\cite{Cha95,Fre95,Ali94,Mir94,Mar96} are difficult to explain in terms
of these hypotheses. Indeed, the simple band crossing model of
Hirabayashi {\it et al.} \cite{Hir94} demonstrates that the resonances
in the $^{12}$C$^*$+$^{12}$C$^*$ channels could also be described
without requiring such an exotic intermediate state.

Rae, Fry and Merchant have recently suggested that the resonances
involving the $3_1^-$ state in $^{12}$C may result from the scattering
of $^{12}$C nuclei (which have threefold $\alpha$-cluster symmetry)
with aligned intrinsic spins \cite{Rae96}. The angular correlation
analyses of the $^{12}$C($3_1^-$)+$^{12}$C($0_2^+$,$3_1^-$) channels
at $E_{c.m.}=32.5$ MeV by Chappell and Rae \cite{Cha96} and the
$^{12}$C($0_1^+$)+$^{12}$C($3_1^-$) channel at $E_{c.m.}=27$ MeV by
Davis {\it et al.} \cite{Dav93} reveal spin alignments which are
consistent with this interpretation. In a semi-classical model
\cite{Rae96}, the cross-section enhancements are a result of resonances
dominated by a 3$\alpha$+3$\alpha$ molecular configuration in the
$^{12}$C($3_1^-$)+$^{12}$C($3_1^-$) channel where the spins of the two
$^{12}$C nuclei lie perpendicular to the plane of the $\alpha$'s and
are parallel to $L'$, the orbital angular momentum in the final
channel. This is analogous to the accepted description of the
resonances discovered by Cormier {\it et al.} \cite{Cor78} involving
the 4.43 MeV $2_1^+$ state in $^{12}$C \cite{Rae96,Rae88,Zha93}.  In
this latter case the resonances have been described in terms of a
$^{12}$C+$^{12}$C configuration where the spins of the two $^{12}$C nuclei
must lie in the plane of the three $\alpha$ particles in each $^{12}$C
\cite{Rae96}. If this interpretation of the $E_{c.m.}=32.5$ MeV
resonances in the $3_1^-$+$3_1^-$ channel is correct we may expect to
find a band of resonances in the mutually excited $3_1^-$ channel at
centre-of-mass energies corresponding to the excitation of different
values of the orbital angular momentum $L'$, analogous to the
observations of Cormier {\it et al.} \cite{Cor78}.

Here, we report on measurements of the scattering of $^{12}$C+$^{12}$C
to states above the $\alpha$-decay threshold across centre-of-mass
energies in the range $E_{c.m.}=31$ to 45 MeV. Two new resonances are
observed in the $^{12}$C($3_1^-$)+$^{12}$C($3_1^-$) channel. These are
correlated with pronounced structure in the angle integrated cross
section of several $^{12}$C$^*$+$^{12}$C$^*$ final states. We attempt
to make spin assignments but the present data do not provide
definitive results. However, the results are compatible with a
rotational band hypothesis.

\section{Experimental Details}

The data were recorded using $^{12}$C beams provided by the 14UD tandem
Van de Graaff accelerator at the Australian National University. The
beams were accelerated over selected laboratory energies in the range
62--90 MeV onto a self-supporting natural carbon target foil of aerial
density 100 $\mu$g cm $^{-2}$. Multiple particle exit channels were
measured using an array of four 25 cm$^2$ 500 $\mu$m thick position
sensitive strip detectors (PSSD's), each segmented into sixteen 5 cm
(long) by 3 mm (wide) strips. The PSSD's were arranged with centres 125
mm from the target at $+25^{\circ}$, $+53^{\circ}$, $-25^{\circ}$, and
$-53^{\circ}$ in the horizontal plane relative to the beam direction
and with the position sensitive strips oriented horizontally. PSSD's of
the type used generate 32 data channels, one for each end of the
sixteen strips. However, the maximum number of analogue-to-digital
converters (ADC's) available for the experiment was limited to 96. In
order to reduce the number of ADC channels required, the detectors were
operated in two pairs using a novel ``doubled'' configuration
\cite{Cha95} whereby the adjacent strip end contacts between each of
the two detector pairs were connected electrically.  Charge created at
either of these ends passed to the same preamplifier-amplifier-ADC
combination from which timing information was also generated. Particle
hits could be assigned to the correct detector by examining the ratio
of signals from the two uncoupled ends.  Thus, the detector system
consisted of 32 independent elements, 16 on either side of the beam
direction, and in principle allowed for the simultaneous measurement of
energy, position and arrival time for up to 32 particles, providing no
two particles struck the same or adjacent ``doubled'' strips. The
response of the detector array was calibrated using low energy elastic
scattering of a $^{12}$C beam ($E_{lab}=30$ MeV) from 5 $\mu$g
cm$^{-2}$ $^{197}$Au (on a 10 $\mu$g cm$^{-2}$ natural carbon backing)
and 100 $\mu$g cm$^{-2}$ $^{27}$Al targets. The method is described in
ref. \cite{Cha96b}.

No explicit particle identification was used in these measurements
since the overdetermination of reaction kinematics was sufficient to
allow reconstructions in terms of hit multiplicities with a low degree
of background contamination. This system allowed the scattering to
channels above the multiple $\alpha$-decay threshold to be studied.
Only those events were recorded during the experiment for which a
minimum of four coincidences were detected with at least three hits in
one of the PSSD pairs. These four, five and sixfold particle hit data
were then subjected to a detailed analysis. For example, each hit in a
sixfold event could be unambiguously assigned to an $\alpha$ particle
on measurement of a total reaction $Q$ value $\simeq -14.6$ MeV (Fig. 1).  
Similarly, the undetected $\alpha$ particle in fivefold
coincidences could be reconstructed from the missing momentum and the
6$\alpha$ final state studied.

The simultaneous detection of a ``jet'' of 3$\alpha$ particles is
characteristic of the scattering to $^{12}$C$^*$ above the $\alpha$
decay threshold (7.37 MeV). The direct breakup of the two states in
$^{12}$C of interest here, namely the 7.65 MeV 0$_2^+$ and the 9.63 MeV
3$_1^-$, to 3$\alpha$ particles is negligible \cite{Fre94}. Hence we
consider only those 3$\alpha$ events correlated as
$^8$Be($0_1^+$)+$\alpha$ to originate from a $^{12}$C nucleus. Typical
excitation energy spectra reconstructed from the momenta and energies
of the detected alpha particles in $^8$Be$\rightarrow 2\alpha$ and
$^{12}$C$^*$$\rightarrow$$^8$Be($0_1^+$)+$\alpha$ events are shown in
figure 2. The large angular coverage provided by the paired
PSSD's gives a greater efficiency for detecting
$^{12}$C(3$_1^-$)$\rightarrow 3\alpha$ decays than reported in previous
measurements \cite{Wuo92,Cha95}.

In fourfold coincidences, grouped as 3$\alpha$ + 1 hits, the single hit
corresponds to either an $\alpha$-particle or a $^{12}$C nucleus below
the breakup threshold. We select the $^{12}$C($>$7.37
MeV)+$^{12}$C($<$7.37 MeV) events and reduce the $^{12}$C($>$7.37
MeV)+$\alpha$ background by requiring that the two body $Q$ values for
scattering to $^{12}$C$^*$ on either side of the beam direction agree
to within 500 keV. The range of $^{12}$C+$^{12}$C inelastic scattering
channels identified in the present data is shown in figure 3.

\section{Results}

Angle integrated cross sections have been extracted from data of the
type illustrated in figure 3 at each centre-of-mass energy
($E_{c.m.}$) sampled. The excitation functions are presented in
figures 4 and 5 for the 3$\alpha$+3$\alpha$ and 3$\alpha$+1 hit
events, respectively. The data have been normalized to beam exposure
($q_c$) including corrections for deadtime in the data acquisition
system such that ${\rm Yield=Counts}/q_c$ in units of 10$^4$
C$^{-1}$. The estimated laboratory cross-section is also shown in
figures 4 and 5. However, in these measurements, laboratory cross
sections are not of significant help in deciding what the true
reaction cross sections are to the different excited states. Since we
are dealing with a many particle final state it is not practical to
bin the data into a multi-dimensional differential
cross-section. Instead the variation of detector ``efficiency'' with
centre-of-mass energy and scattering channel has been modelled using a
simple Monte Carlo technique. The method assumes that the primary
scattering and subsequent decays in the reaction are isotropic in the
centre-of-mass. This is unrealistic but the only possible simple
assumption. The simulation incorporates the rejection of events where
the final state particle energies lie below the detection threshold
(which is position dependent for the PSSD's used). Multiple hits in
the same PSSD strip were also vetoed when performing the
calculations. These gross assumptions do not allow an absolute value
for the detection efficiency to be determined. Clearly, the
primary scattering distribution is dependent upon reaction mechanism
and the subsequent breakup for states with spin is not
isotropic. However, the percentage ``efficiencies'' calculated in this way
can be compared between different channels and centre-of-mass energies
in order to give some measure of the detection ``efficiency'' effects
in the data. The results of these relative ``efficiency'' simulations
are shown in figure 6. In all cases the detection ``efficiency''
varies smoothly with $E_{c.m.}$. It is clear that the reconstruction
of scattering channels from sixfold data is considerably less
efficient than for the fourfold data. Similarly it is clear that the
detection ``efficiency'' for the $3_1^-$ state in $^{12}$C is much
smaller than that for the 0$_2^+$ state.


The resonant behaviour of the observed cross sections near 32.5 MeV in
the 0$_2^+$+0$_2^+$ and 0$_2^+$+3$_1^-$ channels (fig. 4)
is consistent with previously reported measurements in this energy
region \cite{Wuo92,Cha95}. Interestingly, we detect more events in the
3$_1^-$+3$_1^-$ channel than in the 0$_2^+$+0$_2^+$ or 0$_2^+$+3$_1^-$
eventhough it is by far the most difficult channel to detect with the
lowest ``efficiency'' for 3$\alpha$+3$\alpha$ events. A large increase in
yield for scattering to this channel is also observed near
$E_{c.m.}=37.5$ and 43.0 MeV. The scattering to the 0$_2^+$+0$_2^+$ and
0$_2^+$+3$_1^-$ final states in this new energy range is rather weak by
comparison.

Crude estimates of the relative partial decay probabilities have been
calculated for the three resonant energies of interest. The probability
for decay to channel $A$ relative to channel $B$ is given approximately
by
\begin{equation}
{
\frac{N_A}{N_B}\simeq\frac{\rm Yield_A}{\rm Yield_B}
\frac{\rm Efficiency_B}{\rm Efficiency_A}
}
\end{equation}
where the yield in each scattering channel is normalized by the
detection ``efficiency''. The uncertainty here is due to the assumptions
made in the ``efficiency'' calculation for the primary and breakup angular
distributions. The crude partial decay probabilities are shown relative
to the 0$_2^+$+0$_2^+$ channel in Table 1, from which all
other combinations can be extracted. Although the 0$_2^+$+0$_2^+$ and
0$_2^+$+3$_1^-$ channels (fig. 4) do not show much resonant
behaviour except at $E_{c.m.}\simeq32.5$ MeV, the $3\alpha$+1 events
(fig. 5) do show correlated enhancements with the
resonances in the 3$_1^-$+3$_1^-$ channel. Note that the estimated
relative yield in the 3$_1^-$+3$_1^-$ channel is approximately equal to
that in the $0_1^+$+3$_1^-$ channel at $E_{c.m.}=37.5$ and 43.0 MeV.

We have performed a simple angular correlation analysis in Basel
co-ordinates on the 3$_1^-$+3$_1^-$ channel at $E_{c.m.}=37.5$ MeV. The
co-ordinate system and method are described in ref. \cite{Cha96}.
Briefly, the semiclassical angular correlation function for mutual
excitation and decay may be written as
\begin{equation}
{
W(\phi^*_B,\chi_B,\chi_B') \propto 
\left| 
\sin\left[\left(L'+\frac{1}{2}\right)\phi^*_B + k \chi_B + k' \chi_B '
+ \frac{\pi}{4}\right]
\right| ^2
}
\end{equation}
where $L'$ is the final state orbital angular momentum. $k$ and $k'$
are the projections of ejectile and recoil spin ($I$) on the axis
perpendicular to the plane of the primary scattering. We can examine
projections of the three-dimensional correlation
$W(\phi^*_B,\chi_B,\chi_B')$ between the primary scattering angle
$\phi^*_B$ and the subsequent decay angles (in the scattering plane)
$\chi_B$ and $\chi_B'$ using various assumptions for $k$, $k'$ and
$L'$. In particular, if $k$ and $k'$ have equal magnitudes we can write
$X=\chi_B \pm \chi_B'$. This gives either one of two possible
two-dimensional correlation functions $W(\phi^*_B,X)$ which can be
projected onto a single axis. Such projections are shown in figure 7. 
Periodic structure is only visible in the projected
correlations for $X=\chi_B + \chi_B'$ ($k=k'$) and $L'=14$ or 16. The
peak structure apparent in the $L'=12$ spectrum is not regular and
typical when using the incorrect assumption for $k$, $k'$ or $L'$ (see
ref. \cite{Cha96}). This suggests that the scattering at
$E_{c.m.}=37.5$ MeV may occur dominantly to nuclei with fully aligned
spins, i.e., $K=k+k'=6$ and orbital angular momentum $L'=14$ or 16.
This is consistent with the very negative $Q$ value for the reaction,
so that for each incident partial wave $L$ only $L'=L-6$ contributes.
This suggests a possible resonant spin of $J=L=20$ or 22. Previously
\cite{Cha96} we reported $L'=12$ or 14 for the 32.5 MeV resonance and
we deduced a $J$ value of 18 or 20. This is consistent with the
assignment $J=18$ in a recent measurement of scattering to the
$^{12}$C(3$_1^-$)+$^{12}$C(0$_1^-$) channel at the nearby energy
$E_{c.m.}=33.5$ MeV \cite{Wuo96}. We note that the level of
background in the projected correlation spectra at $E_{c.m.}=37.5$ MeV
(Fig. 7) is higher than that measured previously at
$E_{c.m.}=32.5$ MeV (Fig. 10 in ref. \cite{Cha96}). We believe that the
increased background is due to experimental problems associated with
non-linear detection response and not the significant contribution of
additional partial waves or non aligned channels in the scattering at
$E_{c.m.}=37.5$ MeV. This is illustrated in figure 8 where we
cannot resolve the previously observed correlation structure (Fig. 10
in ref. \cite{Cha96}) in the present data at $E_{c.m.}=32.5$ MeV.
However even with the poorer detector performance it does appear that
the new measurement may be consistent with a rotational band of
resonances with spins differing by two units of angular momentum.

Insufficient data and poor detector performance in the present
measurements similarly prevented the extraction of angular momenta at
the other resonant energy ($E_{c.m.}=43$ MeV) in the
$^{12}$C(3$_1^-$)+$^{12}$C(3$_1^-$) channel. The unambiguous
determination of the angular momenta in other channels was also not
possible due to the poor statistics and the fact that our angular
coverage in the centre-of-mass frame around 90$^{\circ}$ is not
sufficient to distinguish different $L'$ values (see ref.
\cite{Cha96}).

It should be noted that the only way to increase the efficiency for
these measurements is to extend the experimental sensitivity over a
much larger solid angle. Placing detectors closer to the beam would
increase the solid angle but reduce the angular resolution and make
the identification of reaction channels ambiguous. Hence the only
practical solution is to use many more detectors. The ``array'' of
four 16-strip PSSD's used here generated 96 signals which was the
maximum number that could be accommodated by our data-readout
system. Thus these investigations are fundamentally very challenging
and await the development of even larger highly segmented detector
arrays and associated data acquisition.

\section{Conclusion}

We have measured excitation functions for several
$^{12}$C($^{12}$C,$^{12}$C$^*$$\rightarrow 3\alpha$)$^{12}$C$^*$
reactions at centre-of-mass energies in the range $E_{c.m.}=31$ to 45
MeV. New resonances have been observed near $E_{c.m.}$= 37.5 and 43.0
MeV in the $^{12}$C(3$_1^-$)+$^{12}$C(3$_1^-$) channel. These
resonances appear to be correlated with enhancements in the
cross-section of several other $^{12}$C+$^{12}$C inelastic channels
which do not lead to a 6$\alpha$ final state. The resonance spin at
32.5 MeV is most likely different from the dominant partial waves in
the $0_2^++0_2^+$ channel ($l=14,16$ \cite{Wuo92}). The mutual $0_2^+$
channel also shows little structure in the excitation function as
compared to the scattering to $^{12}$C(3$_1^-$)+$^{12}$C(3$_1^-$)
across the measured energy range $E_{c.m.}=31$ to 45 MeV, except at
32.5 MeV. These differences suggest that these two reaction channels
are populated by different mechanisms.

The unified description of resonance structures observed in the
$^{12}$C+$^{12}$C system remains one of the longstanding problems of
nuclear physics. These data in $\alpha$-unbound reaction channels
extend the systematics of these phenomena to higher energies and will
serve as a test of theoretical models at higher excitation.

Our data suggest that the resonances at $E_{c.m.}=32.5$ and 37.5 MeV
are two units of angular momentum apart. This is consistent with
the model presented by Rae, Fry and Merchant \cite{Rae96}, where the
resonances are thought to be the result of the scattering of $^{12}$C
nuclei with threefold $\alpha$-cluster symmetry and aligned spin.  It
will be necessary to establish the spin alignment systematics of the
resonances before any detailed comparison can be made with competing
models of the reaction mechanics. Such experiments are planned using
the new ``MEGHA'' array
\cite{Meg96} being installed by the ``Charissa''\footnote{A group of
researchers from the Universities of Oxford, Birmingham, York, Surrey
and Stafford.} collaboration at the Australian National University,
Canberra.

\section*{Acknowledgments}

The financial support of the U.K. Engineering and Physical Sciences
Research Council is acknowledged. This work was performed under an
agreement between the E.P.S.R.C. and the A.N.U..

%_______________________________________________________________________

\newpage
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\end{thebibliography}

%_______________________________________________________________________

\newpage
\section*{Figure Captions}

Fig. 1. Total Q value spectra for five and sixfold coincidences 
at $E_{c.m.}=32.5$ MeV.\\
\\
Fig. 2. Excitation energy in (a) $^8$Be and (b) $^{12}$C$^*$ deduced
from 2$\alpha$ and $^8$Be($0_1^+$)+$\alpha$ $\rightarrow 3\alpha$ hits
respectively at $E_{c.m.}=32.5$ MeV.\\
\\
Fig. 3. $^{12}$C$^*$+$^{12}$C$^*$ scattering channels identified at
$E_{c.m.}=32.5$ MeV. from (a) 3$\alpha$+3$\alpha$ events and (b)
3$\alpha$+1 hits on either side of the beam direction.\\
\\
Fig. 4. Experimental excitation functions for
$^{12}$C($^{12}$C,$^{12}$C$^*$$\rightarrow
3\alpha$)$^{12}$C$^*$$\rightarrow 3\alpha$ channels.\\
\\
Fig. 5. Experimental excitation functions for
$^{12}$C($^{12}$C,$^{12}$C$^*$$\rightarrow
3\alpha$)$^{12}$C$^*$($<$7.37 MeV) channels.\\
\\
Fig. 6. Monte Carlo relative ``efficiency'' profiles for the detection of
3$\alpha$+3$\alpha$ (filled symbols) and 3$\alpha$+1 hits (open
symbols) for the channels indicated.\\
\\
Fig. 7. Projections of the $W(\phi_B^*,X)$ angular correlations at
$E_{c.m.}=37.5$ MeV in the $^{12}$C($3_1^+$)+$^{12}$C($3_1^+$) channel
for $k=k'=3$ and the $L'$ values indicated.\\
\\
Fig. 8. Poor resolution in projections of the $W(\phi_B^*,X)$ angular
correlations in the $^{12}$C($3_1^+$)+$^{12}$C($3_1^+$) channel at
$E_{c.m.}=32.5$ MeV for $k=k'=3$ and the $L'$ values indicated.\\

%_______________________________________________________________________

\newpage
\section*{Tables}


Table 1. Crude estimates of the partial decay probability relative to the
$0_2^++0_2^+$ channel measured at each of the three resonant energies.\\


%_______________________________________________________________________

\newpage

Table 1\\
\\
\begin{tabular}{c||c|c|c}
\hline\hline
$A$ & \multicolumn{3}{c}{$N_A/N_{0_2^++0_2^+}$}\\
\hline
e&  \multicolumn{3}{c}{$E_{c.m.}$ (MeV)}\\
& 32.5 & 37.5 & 43.0\\
\hline\hline
$0_2^++3_1^-$ & 11.3 $\pm$ 1.5 &  12.4 $\pm$  1.7 &  5.3 $\pm$ 0.8\\
$3_1^-+3_1^-$ & 141.0 $\pm$ 37.1 & 394.9 $\pm$ 79.1 & 164.7 $\pm$ 27.9\\
\\
$0_2^++0_1^+$ & 23.1 $\pm$ 1.9 & 35.1 $\pm$ 3.7 & 2.9 $\pm$ 0.3\\
$0_2^++2_1^+$ & 6.3 $\pm$ 0.5 & 14.0 $\pm$ 1.5 & 5.7 $\pm$ 0.6\\
$3_1^-+0_1^+$ & 304.8 $\pm$ 31.3 & 332.4 $\pm$ 33.7 & 145.6 $\pm$ 16.1\\
$3_1^-+2_1^+$ & 65.3 $\pm$ 8.0 & 240.4 $\pm$ 29.3 & 44.6 $\pm$ 5.0\\
\hline\hline
\end{tabular}

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\end{document}






