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% All correspondence regarding this submission should be made to

%----
% +---------------------------------------------------+
%  Dr. S.P.G. Chappell  "s.chappell1@physics.ox.ac.uk"  
%
%  Tel: +44 (0)1865 273312
%  Fax: +44 (0)1865 273418 
%
%  Nuclear Physics Laboratory
%  University of Oxford
%  Keble Road                      
%  Oxford  OX1 3RH, U.K.         
% +---------------------------------------------------+

%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

%_______________________________________________________________________

\documentclass{elsart}
\usepackage[numbers,sort&compress]{natbib}
\begin{document}

\begin{frontmatter}
\title{High spin resonances in $^{12}$C+$^{12}$C scattering}
\author[a]{S.P.G. Chappell\thanksref{email}}
\author[a]{W.D.M. Rae}
\author[a]{C.A. Bremner}
\author[b]{G.K. Dillon}
\author[b]{D.L. Watson}
\author[b]{B. Greenhalgh}
\author[b]{R.L. Cowin}
\author[c]{M. Freer}
\author[c]{\and S.M. Singer}

\address[a]{Nuclear and Astrophysics Laboratory, University of Oxford, Keble
Rd., Oxford OX1 3RH, U.K.}
\address[b]{Department of Physics, University of York, Heslington, York
YO1 5DD, U.K.}
\address[c]{School of Physics and Space Research, University of
Birmingham, Edgbaston, Birmingham B15 2TT, U.K.}
\thanks[email]{E-mail: s.chappell1@physics.ox.ac.uk}

\begin{abstract}
We report on new experimental data which reveal high spin resonances
in $^{12}$C+$^{12}$C scattering. We have measured excitation functions
for the $^{12}$C [ $^{12}$C, $^{12}$C(3$_1^-$) + $^{12}$C(3$_1^-$) ]
reaction over the centre of mass energy range $E_{c.m.}=31$ to 45
MeV. Resonances have been observed at $E_{c.m.}=32.5$, 37.5 and 43
MeV. Detailed angular correlation measurements have been performed at
the energy corresponding to the peak of each resonance. From an
analysis of the angular correlations we suggest a dominant angular
momentum of $J=22$ at the highest energy. We interpret the results in
terms of an $\alpha$ cluster model of superdeformed states in
$^{24}$Mg.
\end{abstract}

\begin{keyword}
Nuclear reactions; $^{12}$C($^{12}$C,6$\alpha$); E=31-45 MeV;
excitation functions; spins; resonances; angular correlations.\\
\\
(PACS: 25.70.Ef, 21.60.Gx, 27.30.+t, 27.20.+n)\\
\end{keyword}
\end{frontmatter}

The $^{12}$C+$^{12}$C system displays a wide range of resonant
phenomena from the narrow resonances at energies near the Coulomb
barrier~\cite{Alm60,Erb84} to much wider structures in inelastic
scattering data~\cite{Cor78,Ful80,Rae87,Rae87b}. The origin of these
resonances has always been a matter of some
debate~\cite{Erb80}. However for a number of years it has been
suggested that there may be some connection between these resonances
and the superdeformed configurations in $^{24}$Mg predicted by various
different models~\cite{Erb84,Bet87,Mar86,Lea75,Flo84}. If we consider
the cranked Bloch-Brink $\alpha$ cluster model calculations of Marsh
and Rae~\cite{Mar86} then an interesting prediction of this model is
that only two configurations can support total angular momenta of
$J\geq16\hbar$. One of these configurations, labelled F1 by Flocard et
al.~\cite{Flo84}, has a well developed $^{12}$C+$^{12}$C structure. The
second structure is a 6$\alpha$-chain state. Thus at high spins, well
above the maximum that the compound nucleus $^{24}$Mg can support at
normal deformations, the system should show only resonances
corresponding to a maximum of two comparatively simple
configurations. In this paper we will not address the recent work on
the 6$\alpha$-chain state~\cite{Wuo92}.

In their original paper Marsh and Rae~\cite{Mar86} suggested that the
resonances first discovered by Cormier et al.~\cite{Cor78} should be
associated with this F1 configuration. The predicted moment of inertia
of the F1 band agrees with that of the Cormier resonances using the
measured spin values~\cite{Rae87,Rae87b,Sug85,Tan83}. These resonances
show up most clearly in the excitation function for the single and
mutual excitation of the 4.43 MeV 2$_1^{+}$ state in $^{12}$C.  We
will defer further detailed discussion of theoretical models to the
end of this letter. Here we just note that in the Bloch-Brink model of
$^{12}$C the ground state rotational band and the 3$_1^{-}$
state at 9.63 MeV share a common intrinsic shape, thus one might
search for resonances in the inelastic scattering of $^{12}$C+$^{12}$C
to final states which also include the 3$_1^{-}$ state. Together with the
recent development of an angular correlation method for mutual
excitation~\cite{Cha96} this was the motivation behind our search for
these new high spin resonances.

The 3$_1^{-}$ state in $^{12}$C decays predominantly via the ground state of
$^{8}$Be to three $\alpha$ particles. The necessary experimental
techniques for detecting these $\alpha$ particles and reconstructing
the $^{12}$C excitation spectrum have been developed by a number of
groups (e.g~\cite{Dav93,Wuo95,Cha95}). We use large 5 cm $\times$
5 cm Si wafers subdivided into sixteen 3 mm wide strips
which are independently instrumented. Signals are taken from
both ends of the strips and these provide both the total energy and
the position of the particle detected by any particular strip. The
details of the operation and calibration of these detectors can be
found elsewhere~\cite{Cur94,Cha97,Cow99}.

We have performed an excitation function measurement over the
laboratory energy range 62 to 90 MeV in 2 MeV steps using 4 detectors
in the 2 m scattering chamber at the Australian National University
Tandem Laboratory (ANU). Two detectors were placed at 25$^{\circ}$ and
53$^{\circ}$ either side of the beam and at a distance of 125 mm from
a 100 $\mu$gcm$^{-2}$ natural carbon target foil.  To reduce the
amount of electronics required the strips of each neighbouring pair of
detectors were connected electrically, so that signals were taken from
each end of the double detector and from the centre where the strips
were joined. Thus we required $3\times16\times2$ channels of
electronics.  This configuration has proved to be very
effective~\cite{Cha95} although the resolution of the detectors is
reduced if compared to results from independent operation. The signals
were processed conventionally using charge sensitive preamps followed
by shaping amplifiers and peak sensing ADC's~\cite{Cur94}.

The details of the data analysis are well described
elsewhere~\cite{Cha96,Cha95}. Briefly, a $^{12}$C spectrum is
reconstructed using energy and momentum conservation from the energies
and positions of $\alpha$ hits on each side of the beam. For the
coincident events most of our data comes from 5 fold hits, 3 on one
side of the beam and 2 on the other. The energy and momentum of the
sixth alpha particle can be deduced from the missing momentum and the
missing energy calculated using the known $Q$ value for the reaction.
Once a $^{12}$C excitation spectrum has been reconstructed for each
side of the beam gates were placed on the 3$_1^{-}$ peak in each
spectrum. The excitation function was then produced by plotting the
number of events detected simply normalised by the beam exposure at
each energy. The results are shown in figure 1. The angle integrated
yields cover the scattering angle range $45^{\circ}$ to $125^{\circ}$
in the centre of mass. No efficiency corrections have been applied to
these data. It is clear that the cross section for the mutual
excitation of the 3$_1^{-}$ states shows three broad resonant-like
structures in the energy region studied. For comparison in figure 1 we
also show a Monte-Carlo simulation of the reaction which gives an
estimate of the variation in detection efficiency assuming that all
decays are isotropic in the centre of mass.

Recently we presented triple angular correlation measurements for the
mutual 3$_1^{-}$+3$_1^{-}$ inelastic channel at $E_{c.m.}=32.5$
MeV~\cite{Cha96}. We suggested a dominant total angular momentum of
$J=18$ or 20 at this energy with a fully aligned configuration
i.e. the partial wave in the final channel was $L'=J-6$.  Since the
data in ref.~\cite{Cha96} were not sufficient to distinguish between a
dominant angular momentum of $J=18$ or 20 we have recently made
detailed angular correlation measurements at $E_{c.m.}=32.5$ MeV and
in addition at energies of $E_{c.m.}=37.5$ and 43 MeV using a new
detector array with new electronic instrumentation. The new detector
array is part of MEGHA recently installed at the ANU by the UK
Charissa collaboration~\cite{Cow99}. This new facility allows us to
mount and individually instrument 10 strip detectors with 320 channels
of electronics. The charge sensitive preamps and amplifiers are
basically of the same design as used previously but the ADC's have
been replaced with custom designed fast voltage to time converters
coupled to Lecroy 2277 TDC's~\cite{Hun99}. We used 8 detectors which
covered laboratory angles from 5$^{\circ}$ to 60$^{\circ}$ with an
azimuthal range of $\sim$100$^{\circ}$ on either side of the beam,
giving a large efficiency for the detection of high particle
multiplicities. This enabled us to use a thin target
(10$\mu$gcm$^{-2}$).  As a result the resolution obtained in these
measurements is significantly improved.

Once the decay products of the two $^{12}$C nuclei have been
identified the angular correlation analysis proceeds using the Basal
coordinate system as described in ref.~\cite{Cha96}. The angular
correlation depends on five angles, namely the primary scattering
angle $\phi^*_B$ and the two decay angles $\chi_B$ and $\psi_B$ of the
$^{8}$Be-$\alpha$ relative velocity with respect to the beam for each
$^{12}$C nucleus. We demonstrated in ref.~\cite{Cha96} that the
semiclassical angular correlation function for mutual decay could be
approximated over the measured angular range by
\begin{eqnarray}\label{eq1}
W(\phi^*_B,\chi_B,\chi'_B,\psi_B,\psi'_B) & \propto & 
\frac{1}{\sin{\phi^*_B}}\left|
\sum_{L'} a_{L'} 
\sin\left[\left(L'+\frac{1}{2}\right)\phi^*_B + k\chi_B + k'\chi'_B
\right. \right. \nonumber \\
& & \left. \left. + \frac{\pi}{4}\right]\right| ^2 
d^I_{k0}(\psi_B)^2 d^{I'}_{k'0}(\psi'_B)^2  + \frac{A^2}{\sin{\phi^*_B}}
\end{eqnarray}
where $\psi_B$,$\psi'_B\simeq\pi/2$. Here $a_{L'}$ is the amplitude
for a single value of the partial wave in the final channel $L'$ and
$A^2$ is an incoherent background. The projections of the $^{12}$C
angular momentum on the $z$-axis, which is perpendicular to the
reaction plane, are $k$ and $k'$. In addition to resonant
contributions, we expect contributions to the
$^{12}$C(3$_1^-$)+$^{12}$C(3$_1^-$) channel from direct mutual
inelastic scattering. This direct process will lead to non-zero
amplitudes in a number of partial waves and some of these
contributions will add incoherently with the resonant amplitude. This
is partially accounted for by the ${A^2}/{\sin{\phi^*_B}}$ term in
equation \ref{eq1}.

In ref.~\cite{Cha96} we showed that by making assumptions about the
quantum numbers $k$ and $k'$ we could define a two-dimensional
correlation function $W(\phi^*_B,X)$ where $X=\chi_B\pm\chi_B'$ and
hence determine the dominant $L'$ and angular momentum $J=L'+k+k'$
from the projections of the ``ridge'' pattern onto a single axis
labelled by $\phi^*_P$. In this letter we take a slightly different
and more rigorous approach. We have performed a full Monte-Carlo
simulation of the reaction. The positions of all the detectors are fed
into this simulation along with minimum and maximum energy cuts for
each detector. The angular correlation function (Eq. \ref{eq1}) is
also fed into this simulation. We then pass these simulated data
through our sort code and treat them exactly as we treat the true
experimental data. So again we project the simulated angular
correlation function onto a two-dimensional plane $W(\phi^*_B,X)$ and
a single axis $\phi^*_P$.  But while in ref.~\cite{Cha96} we simply
looked for regular structure in the projections of the experimental
data to identify the correct assumptions for $L'$,$k$ and $k'$ in this
case we first determine the optimum projection angle $\theta_P$ which
maximises the peak to valley ratio in the one dimensional correlation
pattern. (The projection angle $\theta_P = 180^{\circ}-{\rm
atan}(\Delta X/\Delta \phi^*_B)$ is defined as the angle between the
$\phi^*_B$ axis and the line given by the gradient of the ridges
$\Delta X/\Delta \phi^*_B = (L'+ 0.5)/k$ in the
correlation~\cite{Cha96}.) We then directly compare the projected
correlation data with the Monte-Carlo predictions using the same
projection angle for the simulated correlation matrix.

In figure 2 we show the two-dimensional correlation patterns for the
data at E$_{c.m.}=37.5$ MeV assuming that $|k|=|k'|$ for $k=k'$ and
$k=-k'$. As in ref.~\cite{Cha96} we see that ridges result only from
the assumption $k=k'$ and we can deduce that $k=3$; this strongly
suggests that the two $^{12}$C nuclei are aligned. The correlation
function for simulated data, assuming amplitudes $a_{14}=0.95$ and
$a_{16}=0.32$ for $L'=14$ and $L'=16$ respectively and $k=k'=3$, is
also shown in the figure. The simulation reproduces the ridge pattern
and data boundary. However, the data show a more complicated angular
correlation structure than can be reproduced using only one or two
partial waves in our simple model. Some of this is accounted for by
the incoherent term $A^2$, but this is a very crude approximation. One
dimensional projections of these data along the ridges are presented
in figure 3. The errors in the data and Monte-Carlo projections are
purely statistical and generally smaller than the size of the data
points. For clarity they are not shown. We also show a comparison of
Monte-Carlo calculations for three different consecutive partial waves
at each centre of mass energy (figure 3a to 3c). Note that the
different partial waves are out of phase away from the centre of the
$\phi^*_P$ range for each energy. In figures 3d to 3f we compare the
data with simulations for a coherent sum of two $L'$ values and an
incoherent background. The dominant $L'$ values indicated in the
figure are consistent with the optimum projection angle $\theta_P$ at
each energy. We suggest that the dominant total angular momentum at
32.5 MeV is $J=18$, which is consistent with our previous
measurement~\cite{Cha96}, while the dominant total angular momenta at
37.5 and 43 MeV are suggested to be $J=20$ and 22 respectively. We
have angular correlation data for many other channels and these will
be reported elsewhere ~\cite{Cha99}. Recently Wuosmaa et
al.~\cite{Wuo96} reported an assignment of $J=18$ at 33.5 MeV for the
0$_1^+$+3$_1^{-}$ channel. We also suggest a dominant total angular
momentum of $J=18$ and $L'=15$ at 32.5 MeV in this channel.  However,
for the 0$_2^+$+3$_1^{-}$ channel we suggest a dominant total angular
momentum of $J=16$ and final partial wave $L'=13$.

We now return to the interpretation of these results. We first assume
 that the spins of the resonances are equal to the dominant $J$ value
 suggested by the angular correlations. The $J$ values are then
 consistent with a rotational band with angular momenta up to
 $J=22$. We assume that the direct components may also include
 non-aligned $L'$ values although we have already noted that the angular
 correlations are dominated by the aligned partial wave.  This is also
 true of the Cormier resonances whose alignment has been measured
 using gamma-ray angular correlations~\cite{Kon85}. However, although
 this might indicate some alignment inside the angular momentum
 barrier, it may simply reflect the fact that the aligned barrier with
 $L'=J-6$ is always the lowest barrier for any $J$ value and thus we
 would expect the transmission factors to be much larger for this
 partial wave so that this may dominate any bias in the reduced widths
 themselves. Inside the barrier there are two limits of interest. The
 strong coupling limit applies to the Bloch-Brink $\alpha$ cluster
 model or the weak coupling limit typical of the Band Crossing
 Model(BCM)~\cite{Kon79}.

In the strong coupling limit, inside the barrier the F1 alpha cluster
model configuration is triaxial. Zhang~\cite{Zha95} has performed
angular momentum projected calculations for this configuration which
give rise to a number of bands with different $K$ values. For the
lowest values of $K$, these bands are nearly degenerate (see figure
4). Despite this Zhang estimates that $K$ mixing is quite
small. Outside the barrier $K$ is not a good quantum number and these
strongly coupled configurations will have non zero reduced widths for
several final state partial waves $L'$, but as we discussed above the
lowest partial wave will almost certainly dominate the total width. In
Zhang's calculations $K$ is the projection of the total spin along the
longest axis of the system and this is parallel to the line joining
the centres of the two $^{12}$C nuclei. In this picture $K$ must be
less than or equal to the sum of the spins of the two nuclei, thus for
the single excitation of one nucleus to the 2$_1^{+}$ state
$K\leq2$. The $K=5$ and 6 bands can only be observed in the
3$_1^{-}$+3$_1^{-}$ channel. In figure 4 we plot the centroids of the
experimental resonances using our suggested $J$ values where the error
bars represent an estimate of the widths of the experimental gross
structures.

In the weak coupling picture each gross peak is a combination of
resonances in two or more channels~\cite{Kon79}. Recently we have
attempted coupled channels calculations with deep potentials derived
from a folding model~\cite{Fry98,Rae97}. Although we can obtain good
agreement between theory and data for parameters such as the energies
of the resonances and their widths the coupling potentials are very
strong and without modification the results do not agree with the BCM
calculations of ref.~\cite{Kon79}.

Unfortunately there are no detailed predictions of the energies and
widths of the resonances in the $^{12}$C(3$_1^-$)+$^{12}$C(3$_1^-$)
channel for either the BCM or the Bloch-Brink model. Both models can
accommodate the possibility of multiple resonances within each broad
structure and we believe that both models will predict similar
alignments. A challenge to the BCM is the observation of resonances in
the $^{16}$O+$^8$Be channel~\cite{Rae97,Cha99} at the same energies as
those in the $^{12}$C(3$_1^-$)+$^{12}$C(3$_1^-$) channel. These
resonances would fit naturally into any detailed alpha cluster model
predictions.

In conclusion we have observed high spin resonances in
$^{12}$C+$^{12}$C scattering in the mutual 3$_1^{-}$ channel. Assuming
these resonances dominate the angular correlations at the centre of
each peak in the excitation function we suggest that they have angular
momenta up to $J=22$. We believe that these resonances extend the
sequence of broad resonances first observed by Cormier et
al.~\cite{Cor78}. A comparison of resonance energies and the suggested
spins with theoretical calculations support this view. Above $J=16$ we
argue that the $^{12}$C+$^{12}$C system should be much simpler to
interpret since these high angular momenta cannot be supported by a
$^{24}$Mg nucleus of normal deformation. As a result the level density
is much lower and the absorptive potential in this region should be
weak aiding the observation of these resonances, the only significant
absorption being due to other direct reaction channels.

The authors wish to thank the staff of the Department of Nuclear
Physics at the Australian National University for assistance in
running the experiments. We acknowledge the financial support of the
U.K. Engineering and Physical Sciences Research Council (EPSRC). The
experimental work was performed under a formal agreement between the
EPSRC and ANU.

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\newpage
\section*{Figure Captions}

Fig. 1. Experimental excitation functions for the
$^{12}$C$^*$(3$_1^-$)+$^{12}$C$^*$(3$_1^-$) channel. Also shown is a
Monte-Carlo calculation of the detection efficiency as a function of
$E_{c.m.}$.\\
\\
Fig. 2. Experimental angular correlations $W(\phi_B^*,X)$ at
E$_{c.m.}=37.5$ MeV assuming $|k|=|k'|$ and (a) $k=-k'$ or (b)
$k=k'$.  (c) Monte-Carlo simulation for $a_{14}=0.95$, $a_{16}=0.32$
and $k=k'=3$ (without the incoherent background).\\
\\
Fig. 3. Comparisons of measured and simulated angular correlations
projected onto a single axis using the optimum projection angle
$\theta_P$ indicated at each of the three centre of mass
energies. (a--c) Simulation for three consecutive $L'$ values and no
incoherent background. (d--f) Simulation including a coherent
contribution of two partial waves and an incoherent background. (The
background is shown as the dotted curves in the figures scaled by a
factor of 0.5 for clarity). (d) $E_{c.m.}=32.5$ MeV with
$a_{12}=0.92$, $a_{14}=-0.39$ (e) $E_{c.m.}=37.5$ MeV with
$a_{14}=0.95$, $a_{16}=0.32$ (f) $E_{c.m.}=43.0$ MeV with
$a_{16}=0.98$, $a_{18}=-0.22$. The projection angles
$\theta_P=103^{\circ}, 101.5^{\circ}$ and $100.5^{\circ}$ yield
$L'=12.5, 14.2$ and 15.7 respectively which is consistent with our
assignments.\\ 
\\
Fig. 4. Angular momentum projected $\alpha$-cluster model calculations 
for the F1 configuration in $^{24}$Mg~\cite{Zha95} and the 
resonances in the present data and those of
Cormier~\cite{Cor78,Rae87,Rae87b,Sug85,Tan83} assuming our suggested
$J$ values.\\


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