%_______________________________________________________________________

%----------------------CUT HERE REVTEX FOLLOWS -------------------------
%_______________________________________________________________________


% All correspondence regarding this submission should be made to

% S.P.G. Chappell

% Nuclear Structure Group
% Department of Physics
% University of York
% Heslington
% York, YO1 5DD, UK
%  
% Phone (01904) 430000 ext. 2225/2244
% Fax   (01904) 432214
% Email oef@yksc.york.ac.uk

% Alternatively contact...

% W.D.M. Rae

% Department of Nuclear Physics
% University of Oxford
% Keble Road
% Oxford OX1 3RH, UK
% Phone (01865) 273368
% Fax  (01865) 273418
% Email B.Rae1@physics.oxford.ac.uk

%_______________________________________________________________________

%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

%_______________________________________________________________________

%REVTEX ...

\documentstyle[preprint,eqsecnum,aps]{revtex}
\begin{document}
\draft
\preprint{Accepted for Publication in PRC}

%_______________________________________________________________________

\title{Angular correlations for $\alpha$-particle decay in the 
reaction\\
$^{12}$C[$^{12}$C,$^{12}$C($3_1^-$)$\rightarrow$$^8$Be($0_1^+$)+$\alpha$
]$^{12}$C$^*$ at $E_{c.m.}=32.5$ MeV}

%_______________________________________________________________________

\author{S.P.G. Chappell}
\address{Department of Physics, University of York, Heslington, York
YO1 5DD, United Kingdom.}
\author{W.D.M. Rae}
\address{Department of Nuclear Physics, University of Oxford, Keble
Road, Oxford OX1 3RH,\\ United Kingdom.}
%_______________________________________________________________________

\date{\today}
\maketitle 
\newpage

\begin{abstract}

We present angular correlations in ``Basel'' co-ordinates for
$\alpha$-particle decay in measurements of $^{12}$C [ $^{12}$C,
$^{12}$C($3_1^-$)$\rightarrow$$^8$Be($0_1^+$)+$\alpha$ ] $^{12}$C$^*$
scattering at $E_{c.m.}=32.5$ MeV. A semi-classical model of the angular
correlation function has been used to describe the observed angular
correlations in the various exit channels. We demonstrate that the
semi-classical result can be used even in the case where there are two
non-zero spins in the exit channel if the decay products from both
ejectiles are detected. We apply this new method to the mutual
sequential breakup reaction $^{12}$C [$^{12}$C,
$^{12}$C($3_1^-$)$\rightarrow$$^8$Be($0_1^+$)+$\alpha$]
$^{12}$C($3_1^-$)$\rightarrow$$^8$Be($0_1^+$)+$\alpha$. We deduce that
the scattering occurs dominantly to a final state with fully aligned
spins and orbital angular momentum $L'=12$ or 14.

\end{abstract}

%_______________________________________________________________________


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

\newpage

%_______________________________________________________________________

\section{Introduction}

New experimental studies of scattering reactions to states high above
the 6$\alpha$ particle decay threshold in $^{24}$Mg have revealed new
resonant structure in the $^{12}$C+$^{12}$C system. Most notably, it
appears that the wide resonance first reported in the $^{12}$C [
$^{12}$C, $^{12}$C($0_2^+$) ] $^{12}$C($0_2^+$) reaction at
$E_{c.m.}=32.5$ MeV \cite{Wuo92}, with an unusually large strength of
the scattering cross section at $90^{\circ}$ (in the center-of-mass),
may also be common to a number of other channels. We have previously
reported on similar behavior for scattering to the
$^{12}$C($0_2^+$)+$^{12}$C($3_1^-$) and
$^{12}$C($0_2^+$)+$^{12}$C($0_3^+$) channels \cite{Cha95}, whilst
Freeman {\it et al.} \cite{Fre95} have found that the
$^{16}$O($0_1^+$)+$^8$Be($0_1^+$) channel also resonates at this energy
with the characteristic $90^{\circ}$ enhancement. This striking feature
in the angular distribution originally suggested that the 32.5 MeV
$0_2^+$+$0_2^+$ resonance be considered as an approximate shape
eigenstate \cite{Rae92,Mer93}, where many overlapping resonances with
different $L$ values are excited. It was thought that this deformed
structure was consistent with the formation of a 6$\alpha$ linear chain
configuration \cite{Mar86,Mer92} (LCC) at $E_x=46.4$ MeV in $^{24}$Mg.
The cranked $\alpha$-cluster model \cite{Mar86} predicts a band crossing
between the ground state and chain $\alpha$-cluster configurations in
$^{24}$Mg near $E_x=50$ MeV and $L=16$. This seemed to be confirmed by
the observation of dominant angular momenta $L=14$ and 16 \cite{Wuo92}
in the reaction.  However the resonant strength recently discovered in
other decay channels \cite{Cha95,Fre95,Ali94,Mir94} at this energy
presents a challenge to this interpretation. Since the dominant decay
mode of a 6$\alpha$ LCC is expected to be through symmetric fission to
two 3$\alpha$ LCC's or three $^8$Be nuclei \cite{Rae95}, the observation
of strong resonance for scattering to nuclei with no relation to chain
configurations is surprising. Moreover, the $0_2^+$ state in $^{12}$C is
widely accepted as having at best only a limited overlap with a
3$\alpha$ LCC structure \cite{Tak71,Tak72,Suz72}. It is thus difficult
to construct a 6$\alpha$ chain from two $^{12}$C($0_2^+$) states
\cite{Rae95}.

An alternative explanation of the $E_{c.m.}=32.5$ MeV resonances is
provided by the band crossing model (BCM) \cite{Abe78}.  This predicts
excitation function enhancements in the $0_2^+$+$0_2^+$,
$0_2^+$+$3_1^-$ and $0_2^+$+$0_3^+$ channels at this energy, when
realistic transition densities for $^{12}$C \cite{Kam81} are included
\cite{Hir94}. This model suggests that the structure of the
$0_2^+$+$0_2^+$ resonance can be understood in terms of a weakly
coupled 3$\alpha$ plus 3$\alpha$ configuration, where the 3$\alpha$
particles do not necessarily form a chain state. The BCM predicts a
dominant partial wave of $L=J=18$ for the $0_2^+$+$0_2^+$ channel,
somewhat larger than those in the experimental data ($L=14,16$
\cite{Wuo92}). $L$ is the orbital angular momentum and $J$ the total
spin. The $0_2^+$+$3_1^-$ resonance is predicted to be the result of a
similarly extended 3$\alpha$ structure coupled to an excited
$^{12}$C($3_1^-$) core with stretched angular momentum $L=J-3$. This
model has had some success in reproducing resonances at other energies
in the $^{12}$C+$^{12}$C system. The resonances discovered by Cormier
{\it et al.} \cite{Cor78} in the $0_1^+$+$2_1^+$ and $2_1^+$+$2_1^+$
channels and those by Fulton {\it et al.} \cite{Ful80} for scattering
to the $0_1^+$+$0_2^+$ and $0_1^+$+$3_1^-$ states are also observed in
BCM calculations \cite{Abe78,Hir94,Kon79}. Note that only the more
recent formulation of the BCM \cite{Hir94}, which incorporates a
distorted form factor for the $0_2^+$ level in $^{12}$C, can describe
the excitation function enhancement seen in the angular momentum
mismatched $0_1^+$+$0_2^+$ channel.

The physics underlying the BCM is rather simple, namely it predicts
that molecular resonances should be dominated by an aligned coupling
between the orbital angular momentum and intrinsic spin. One expects
this result quite naturally given the negative $Q$-value of these
reactions. Such alignments have been observed in studies of $^{12}$C on
$^{12}$C collisions for the single and mutual excitation of the
$^{12}$C(2$_1^+$) state using  particle-$\gamma$-ray angular
correlations \cite{Wil83,Tro84,Kon85}. Resonances in the cross-section
were found to be uniquely associated with a strong enhancement of the
spin alignment, particularly in the mutual 2$_1^+$ channel
\cite{Kon85}.  These data suggest that the resonant configuration is
characteristically different from that of the nonresonant weakly
aligned background and that the respective orientation of the deformed
$^{12}$C nuclei plays a decisive r\^{o}le in this resonance behaviour.

It would be interesting to make similar measurements of the alignment
in the case of the resonances observed in the $3_1^-$ channels in the
$^{12}$C+$^{12}$C system, especially in order to test the hypotheses
proposed to explain the 32.5 MeV resonances. Since the $3_1^-$ channels
involve excitations to nuclear states above the decay threshold for
particle emission, particle detection techniques for determining
angular correlations and spin alignments must be used. One such method
has been discussed previously and illustrated for the case of the
sequential breakup reaction $^{12}$C [ $^{12}$C,
$^{12}$C($3_1^-$)$\rightarrow$$^8$Be($0_1^+$)+$\alpha$ ]
$^{12}$C($0_1^+$) at $E_{c.m.}=27$ MeV \cite{Dav93}. The data presented
display a strong spin alignment at this energy which coincides with one
of the resonances observed by Fulton {\it et al.} \cite{Ful80} in this
channel.

Here, we report on similar measurements extended to the case of mutual
sequential breakup. In particular, we have examined the data of
Chappell {\it et al.} \cite{Cha95} and reconstructed $^{12}$C [
$^{12}$C, $^{12}$C($3_1^-$)$\rightarrow$$^8$Be($0_1^+$)+$\alpha$ ]
$^{12}$C$^*$ and $^{12}$C [ $^{12}$C,
$^{12}$C($3_1^-$)$\rightarrow$$^8$Be($0_1^+$)+$\alpha$ ] $^{12}$C
($3_1^-$)$\rightarrow$$^8$Be($0_1^+$)+$\alpha$ angular correlations at
$E_{c.m.}=32.5$ MeV. We choose to study these angular correlations
using the ``Basel'' co-ordinate system \cite{Dea84} as was used in the
particle-$\gamma$-ray correlation studies \cite{Tro84,Kon85}. The
theoretical angular correlation function reduces to a particularly
simple form using a semi-classical approximation in Basel co-ordinates
in which the physical interpretation of the data is transparent.  This
form of the correlation function also yields an efficient method of
projection by which to study the correlation ridge structures in one
dimension \cite{Rae85}. The ridge periodicity and projection gradient
are related to the spins and angular momenta in the final state. As
indicated above, we also present angular correlations for the mutual
inelastic scattering $^{12}$C [ $^{12}$C, $^{12}$C($3_1^-$) ]
$^{12}$C($3_1^-$).  The simultaneous measurement of decay angles for
the breakup of both recoil and ejectile nuclei provides a new technique
by which to examine the spin-orientation in sequential mutual breakup
reactions. Although the results from our existing data are ambiguous we
believe that important information can be extracted from both the
single and mutual
$^{12}$C($3_1^-$)$\rightarrow$$^8$Be($0_1^+$)+$\alpha$ correlation
functions in a simple way.  This paper is mainly concerned with the new
simple technique for extracting reaction information from the mutual
channel.

%_______________________________________________________________________

\section{Angular Correlations in Basel co-ordinates}

Particle decay angular correlation measurements have become
well-established in the study of heavy-ion sequential breakup reactions
using the techniques of resonant particle decay spectroscopy
\cite{Rae81,Rae84}. A sequential breakup reaction, $a(A,A^* \rightarrow
C+c)a$, consists of scattering a projectile $A$ from a target $a$ to
form an intermediate nucleus $A^*$ which then decays via particle
emission to leave a three-body final state $C+c+a$. Mutual sequential
breakup, $a(A,A^*\rightarrow C+c)a^*\rightarrow D+d$, is said to occur
when both the ejectile ($A^*$) and recoil ($a^*$) nuclei are excited
above the threshold for particle decay. The dependence of the direction
of particle decay on the initial scattering angle can reveal many
features of the reaction process. In particular, the gross structure of
the angular correlations may be used to determine the spins of the
nuclei, often in a model-independent manner, whilst the fine details may
in conjunction with reaction models reveal the $m$-substate populations
\cite{Pou79}.

The Basel co-ordinate system adopted in the present analysis is defined
in Fig. \ref{basel}. In this representation, the $z$-axis is defined on
an event-by-event basis to lie perpendicular to the reaction ($xy$)
plane and $x$ is taken as the beam direction. The trajectory of the
ejectile nucleus $A^*$ is then given by the angles $\theta^*_B$ and
$\phi_B^*$, where $\theta^*_B$ is fixed at $90^{\circ}$ by definition.
The angles $\psi_B$ and $\chi_B$ characterize the breakup by defining
the orientation of ${\bf v}_{rel}$ (the relative velocity vector
between the decay products $C$ and $c$) with respect to the
out-of-plane ($z$) and beam ($x$) axes. Most of the events considered
here are restricted by the detector geometry to be in-plane. We thus
take $\psi_B=\pi/2$.

The semi-classical angular correlation function in Basel
coordinates has been
obtained previously \cite{Hah67}.
In these co-ordinates, using the  decay
amplitude  
\begin{equation}
{
(-1)^k e^{-ik\chi_B}d^I_{k0}(\frac{\pi}{2})
}
\end{equation}
the semi-classical angular correlation function for single excitation
and a single incident ($L$) and final ($L'$) partial wave reduces to
\begin{equation}
{
W(\phi^*_B,\chi_B) \propto 
\frac{1}{\sin{\phi^*_B}}
\left| 
\sin[(L'+\frac{1}{2})\phi^*_B + k \chi_B + \frac{\pi}{4}] 
\right| ^2
d^I_{k0}(\frac{\pi}{2})^2
}
\label{W1}
\end{equation}
for a spin zero recoil. Here, $k$ is the projection of ejectile spin
($I$) on the $z$-axis.

The combined use of the Basel co-ordinate system and the semi-classical
approximation results in considerable physical insight to the nature of
the scattering reaction. The chosen $z$-axis, perpendicular to the
reaction plane, is also parallel to the orbital angular momentum vector
${\bf L'}$ semi-classically. Thus the angular momentum transfer in the
reaction is ${\bf L'}-{\bf L}$ and the projection of this on the
$z$-axis must be equal to the projection ($k$) of the ejectile spin
vector ${\bf I}$ on the $z$-axis in this approximation. If $k=I$ then
the vector ${\bf I}$ is fully aligned along ${\bf L}$ and ${\bf L}$ is
exactly parallel to ${\bf L'}$. For reactions with a very negative
$Q$-value the dominant $L'$ is generally much less than $L$ so that the
only important contribution to the cross-section comes from the maximum
possible value of $k$, namely $k=I$. However for $Q\simeq0$, there
would be a distribution of possible $k$ values although $k\simeq0$ may
dominate.  In such cases the alignment of ${\bf I}$ is not well defined
and for any incident partial wave $L$ there will be several exit
partial waves $L'$ contributing simultaneously.  The direction of the
vector ${\bf I}$ is then different for each possible partial wave $L'$.
(Note that the semiclassical approximation assumes that $I \ll L$ so
that the projection of ${\bf L'-L}$ on the $z$-axis is always assumed
to be approximately $L'-L$ although ${\bf L'}$ may not be exactly
parallel to ${\bf L}$.)

%_______________________________________________________________________

\section{Experimental Details}

The experimental method used in these measurements has been described in
detail elsewhere \cite{Cha95}. Briefly, the setup consisted of three 25
cm$^2$, 500 $\mu$m thick position sensitive strip detectors (PSSD's),
each segmented into sixteen strips. The PSSD's were arranged with their
centers 12 cm from the target at $+25^{\circ}$, $+52^{\circ}$ and
$-32^{\circ}$ in the horizontal plane relative to the beam direction. A
natural carbon target foil of aerial density 90 $\mu$g cm$^{-2}$ was
used. This system allowed the scattering to channels above the multiple
$\alpha$-decay threshold to be studied. For example, Fig.  \ref{q2c}
demonstrates the range of $^{12}$C$^*$+$^{12}$C$^*$ inelastic scattering
channels identified in the present data at $E_{c.m.}=32.5$ MeV. The
simultaneous detection of 3$\alpha$ + 3$\alpha$ hits on either side of
the beam is characteristic of the scattering to states in $^{12}$C$^*$
above the $\alpha$-decay threshold (7.37 MeV) (Fig.  \ref{q2c}a). For
the similar case of 3$\alpha$ + 1 hits (Fig.  \ref{q2c}b), the
individual hit may correspond to either an intact $^{12}$C nucleus below
the breakup threshold or an $\alpha$-particle.  In this spectrum, the
$^{12}$C$^*$($>$7.37 MeV) + $^{12}$C$^*$($<$7.37 MeV) case has been
selected by comparing the two-body $Q$ values for scattering to
$^{12}$C$^*$ on either side of the beam.  Events were discarded for
which the $Q$-values did not match to within 500 keV, thus reducing the
background. Note that, the over determination of reaction kinematics in
these measurements also allows the reconstruction of 6$\alpha$ final
states following the detection of only 5$\alpha$ coincidences. We
therefore consider four, five and sixfold particle-hit data in the
following analysis.

%_______________________________________________________________________

\section{Results \& Analysis}

\subsection{$^{12}$C[$^{12}$C,$^{12}$C($3_1^-$)$
\rightarrow$$^8$Be($0_1^+$)+$\alpha$]$^{12}$C($0_{1,2}^+$)
Angular Correlations}

Figure \ref{corr3g} shows the $^{12}$C($3_1^-$)$
\rightarrow$$^8$Be($0_1^+$)+$\alpha$ angular correlation pattern
observed in the $^{12}$C($3_1^-$)+$^{12}$C($0_1^+$) channel at
$E_{c.m.}=32.5$ MeV.  A series of prominent ridges can be seen in these
data within a segmented detection boundary. The limits of this boundary
agree well with the phase space coverage predicted by three-body
kinematics \cite{Rae84b} (Fig. \ref{corr3g}). The small differences
between the observed and calculated phase spaces in Fig. \ref{corr3g}
arise from deficiencies in the kinematics simulation. Most importantly,
the reaction proceeds to a four-body final state where $^8$Be($0_1^+$)
further decays into 2$\alpha$ particles. In addition, the
characteristic position-dependent energy thresholds of the PSSD's have
not been accurately reproduced.  The angular correlation pattern in
this channel is compared with the loci of maxima in the correlation
functions $W(\phi^*_B,\chi_B)$ (Eq.  \ref{W1}), for a number of $L'$
values and $k=3$, in Fig.  \ref{corr3gb}. 

The observed ridges are clearly 60$^{\circ}$ apart in $\chi_B$; from
Eq. \ref{W1} the spacing $\Delta\chi_B$ must be $\pi/k$ and this yields
$k=3$. The gradient of the ridges is
$\Delta\chi_B/\Delta\phi^*_B=(L'+\frac{1}{2})/k$. Note that
the semi-classical correlation function is independent of
$L'$ at $\phi^*_B=90^{\circ}$. Since the present data span only a
narrow range about $\phi^*_B=90^{\circ}$, an unambiguous assignment of
the dominant $L'$ value in this channel is not possible.  However, we
may examine the structure in this correlation more closely by assuming
possible ridge gradients and projecting the data onto the $\chi_B=0$,
$\phi^*_B$-axis.  This is equivalent to averaging the data along lines
of constant $\phi^*_B+\chi_Bk/(L'+\frac{1}{2})$, as in Fig.
\ref{corr3gp}. This averaging process is advantageous since from Eq.
\ref{W1} the angular correlation depends only on a linear combination
of the angles $\phi^*_B$ and $\chi_B$. If the data are dominated by a
single combination of one $L'$ and one $k$ then the averaging procedure
enhances the statistics without any loss of information since the
correlation is the square of a pure sine wave.  However, even if the
latter condition of a single combination of $L'$ and $k$ is true, we do
not necessarily know {\it a priori} which values of $L'$ and $k$
dominate. These must be deduced from the experimental data. A
convenient way to do this is to  evaluate
$\phi^*_B+\chi_Bk/(L'+\frac{1}{2})$ on an event by event basis and
histogram the data as a function of this variable. This can be done for
several values of $k$ and $L'$. When the correct choice of $k$ and $L'$
is made (which is equivalent to choosing the projection gradient) a
regular sequence of maxima and minima should be seen in the
one-dimensional projection of the correlation spectrum. If the wrong
choice of $k$ or $L'$ is made and the data span a sufficiently large
phase space in the $\phi^*_B$ and $\chi_B$ variables then the maxima
and minima will be smeared out. As we noted our data are restricted in
$\phi^*_B$-space and do not determine the gradient at all well. Since
the correlation function is nearly independent of $L'$ all projections
show similar structure.

Figure \ref{corr30} shows the $^{12}$C($3_1^-$)$
\rightarrow$$^8$Be($0_1^+$)+$\alpha$ angular correlation pattern
observed in the $^{12}$C($3_1^-$)+$^{12}$C($0_2^+$) channel.
One-dimensional projections of this correlation are given in Fig.
\ref{corr30p}. As was noted above the data do not determine $L'$ well.
However note that if, in both of the above examples, the measured range
of $\phi^*_B$ was extended the correlation analysis would determine
$L'$. This is illustrated below for the mutual case where the range of
angles obtained from our existing data set is larger.

\subsection{Spin-orientation Alignment in
$^{12}$C[$^{12}$C,$^{12}$C($3_1^-$)]$^{12}$C($3_1^-$) Scattering}

We will now consider the particle decay angular correlations for the
mutual inelastic scattering $^{12}$C [ $^{12}$C, $^{12}$C($3_1^-$) ]
$^{12}$C($3_1^-$).  In this case, the variation over the possible $m$
substates for recoil nuclei with spin destroys any ridge pattern in the
simple ($\phi_B^*,\chi_B$) plot (Fig. \ref{corr33}). However the
present measurements allow us to investigate both the correlation angles
$\chi_B$ and $\chi_B'$ for the breakup of both the ejectile and recoil
nuclei simultaneously. The semi-classical angular correlation function
for mutual excitation and decay can be obtained in a similar way to
that used by Hahne and Rae \cite{Hah67}. The decay amplitude for a
specific $k$ and $k'$ for the case of two separate nuclei is
\begin{equation}
{
(-1)^k e^{-ik\chi_B}d^I_{k0}(\frac{\pi}{2})
(-1)^{k'} e^{-ik'\chi_B'}d^{I'}_{k'0}(\frac{\pi}{2})
}
\end{equation}
Here $k$ and $k'$ are the projections of the two $^{12}$C spins (${\bf
I}$ and ${\bf I'}$) on the $z$-axis and $k+k'$ is now identified with
$L-L'$. Using the formalism of Hahn and Rae, this amplitude gives the
final angular correlation function
\begin{equation}
{
W(\phi^*_B,\chi_B,\chi_B') \propto 
\frac{1}{\sin{\phi^*_B}}
\left| 
\sin[(L'+\frac{1}{2})\phi^*_B + k \chi_B + k' \chi_B ' + \frac{\pi}{4}] 
\right| ^2
d^I_{k0}(\frac{\pi}{2})^2
d^{I'}_{k'0}(\frac{\pi}{2})^2
}
\end{equation}
where again $L'$ is the final state orbital angular momentum.  If we
assume that $k$ and $k'$ have equal magnitudes then we can plot the
correlation as a function of $X=\chi_B \pm \chi_B'$.  We find that
ridge structure is observed in plots of $\phi^*_B$ versus $X$ when
$X=\chi_B+\chi_B'$ (Fig. \ref{corr33x}a) but not when
$X=\chi_B-\chi_B'$ (Fig. \ref{corr33x}b).  Projections of these
correlations are shown in Fig. \ref{corr33xp}. A regular sequence of
peaks can be seen in the $X=\chi_B+\chi_B'$ projections for $L'=12$ and
14. No structure is observed for the $X=\chi_B-\chi_B'$ case. Note that
again the ridges have a spacing of 60 degrees in the $X=\chi_B+\chi_B'$
direction suggesting that $k$ and $k'$ are predominantly both equal to
3.  Also note that in this case the projected correlations do indeed
provide some information on $L'$. Structure in the projected
correlations is only visible if we assume $L'$ to be either 12 or 14 in
our event by event histogramming. Again a wider angular coverage would
make this assignment unambiguous.  These results imply that the
$^{12}$C($3_1^-$) nuclei scatter dominantly with fully aligned spins,
i.e. $K=k+k'=6$ and $L'=12$ or 14 in this channel. This is consistent
with the very negative $Q$-value for the reaction so that for each
incident $L$ only $L'=L-6$ contributes. This suggests a resonant spin
of $J=L'+K=18$ or 20. Hence considerable information can be gleaned from
these mutual angular correlations by the simple technique of projection
of the three-dimensional correlation $W(\phi^*_B,\chi_B,\chi_B')$ onto
a single axis using various assumptions for $k$, $k'$ and $L'$.

%_______________________________________________________________________

\section{Conclusions}

We have compared angular correlation measurements for
$^{12}$C[$^{12}$C,$^{12}$C($3_1^-$)$\rightarrow$$^8$Be($0_1^+$)+$\alpha$
]$^{12}$C$^*$ scattering at $E_{c.m.}=32.5$ MeV to a semi-classical
description of the correlation function in Basel co-ordinates.  The
present analysis suggests that the dominant partial wave in
$^{12}$C(3$_1^-$) + $^{12}$C(3$_1^-$) scattering at $E_{c.m.}=32.5$ MeV
is $L'=12$ or 14 and that the dominant partial wave in the incident
channel is $L=18$ or 20 (The incident partial wave is equal to the
total spin $J$). This result cannot be compared with the band crossing
model of this reaction \cite{Hir94} since the mutual 3$_1^-$ channel
was not considered in those calculations. Our experimental results may
be consistent with a new model of the 32.5 MeV resonance presented
recently by Rae, Fry and Merchant \cite{Rae95b} but more extensive
angular correlation data are required to make the dominant partial wave
assignments less ambiguous, especially in the 0$^+_{1,2}$+3$_1^-$
channels. It will also be necessary to establish that the
$3_1^-$+$3_1^-$ channel is resonant at this energy. In the new model
proposed by Rae, Fry and Merchant, the $E_{c.m.}=32.5$ MeV resonance is
thought to be the result of the scattering of $^{12}$C nuclei with
3-fold $\alpha$-cluster symmetry and aligned spins. The present data
indicate that a correlated full spin alignment between the deformed
nuclei in the $^{12}$C($3_1^-$)+$^{12}$C($3_1^-$) channel has been
observed at $E_{c.m.}=32.5$ MeV.

%_______________________________________________________________________

\acknowledgements

We would like to thank our Charissa and A.N.U. collaborators for their
work on the original experiment. The experiment was carried out under
an agreement between the E.P.S.R.C. and the A.N.U. .

%_______________________________________________________________________

\begin{references}

\bibitem{Wuo92} 
 A.H. Wuosmaa {\it et al.},
 Phys. Rev. Lett. {\bf 68}, 1295 (1992);
 Phys. Rev. C {\bf 50}, 2909 (1994).

\bibitem{Cha95}
 S.P.G. Chappell {\it et al.},
 Phys. Rev. C {\bf 51}, 695 (1995).

\bibitem{Fre95} R.M. Freeman, F. Haas, A. Elanique, A. Morsad 
 and C. Beck, Phys. Rev. C {\bf 51}, 3504 (1995).

\bibitem{Rae92}
 W.D.M. Rae, A.C. Merchant and B. Buck,
 Phys. Rev. Lett. {\bf 69}, 3709 (1992).

\bibitem{Mer93}
 A.C. Merchant and W.D.M. Rae,
 J. Phys. G {\bf 19}, L89 (1993).

\bibitem{Mar86}
 S. Marsh and W.D.M. Rae,
 Phys. Lett.  {\bf 180B}, 185 (1986).

\bibitem{Mer92} A.C. Merchant and W.D.M. Rae,
 Nucl. Phys. {\bf A549}, 431 (1992);
 Phys. Rev. C {\bf 46}, 2096 (1992).

\bibitem{Ali94} M. Aliotta {\it et al.},
 in {\it Proceedings of the V International Conference on Nucleus
 Nucleus Collisions}, Taormina, Italy, May 1994, edited by M. Di Toro,
 E. Mignelo and P. Piatelli, Nucl. Phys. {\bf A583}, 281 (1995).

\bibitem{Mir94} E.T. Mirgule {\it et al.},
 in Ref. \cite{Ali94} p. 287.

\bibitem{Tak71} N. Takigawa and A. Arima,
 Nucl. Phys. {\bf A168}, 593 (1971).

\bibitem{Tak72}
 N. De Takacsy,
 Nucl. Phys. {\bf A178}, 469 (1972); 
 C. Bargholtz, 
 {\it ibid.} {\bf A243}, 449 (1975); 
 H. Freidrich, L. Sapathy, A. Weiguny, 
 Phys. Lett. B {\bf 36}, 189 (1971).

\bibitem{Suz72}
 Y. Suzuki, H. Horiuchi, and K. Ikeda,
 Prog. Theor. Phys. {\bf 47}, 1517 (1972).

\bibitem{Rae95}
 W.D.M. Rae and A.C. Merchant,
 Phys. Rev. Lett. {\bf 74}, 4145 (1995).

\bibitem{Abe78} Y. Abe,
 {\it Nuclear Molecular Phenomena}, edited by N. Cindro, (North-Holland,
 Amsterdam, 1978) p. 211; Y. Kondo, T. Matsuse and Y. Abe, Prog. Theor.
 Phys. {\bf 59}, 465 (1978); T. Matsuse, Y. Abe and Y. Kondo, {\it
 ibid.} {\bf 59}, 1009, 1037 and 1904 (1978); Y. Kondo, Y. Abe and T.
 Matsuse, Phys. Rev. C {\bf 19}, 1356 (1979); Prog. Theor. Phys. {\bf
 63}, 722 (1980); Phys. Rev. C {\bf 22}, 1068 (1980); Y. Abe, Y. Kondo,
 and T.  Matsuse, Prog. Theor. Phys. Suppl. {\bf 68} Chapt. IV, 303
 (1980).

\bibitem{Kam81} M. Kamimura,
 Nucl. Phys. {\bf A351}, 456 (1981).

\bibitem{Hir94} Y. Hirabayashi, Y. Sakuragi and Y. Abe,
 Yukawa Institute Kyoto, Report No. YITP/K-1093, 1994 (unpublished); 
 Phys. Rev. Lett. {\bf 74}, 4141 (1995).

\bibitem{Cor78} T.M. Cormier {\it et al.}, 
 Phys. Rev. Lett. {\bf 40}, 924 (1978) and
 {\bf 38}, 940 (1977).

\bibitem{Ful80} B.R. Fulton, T.M. Cormier and B.J. Herman,
 Phys. Rev. C {\bf 21}, 198 (1980).

\bibitem{Kon79} Y. Kondo, Y. Abe and T. Matsuse,
 Phys. Rev. C {\bf 19}, 1356 (1979).

\bibitem{Wil83} S.J. Willett, S.K. Korotky, R.L. Phillips
 D.A. Bromley and K.A. Erb,
 Phys. Rev. C {\bf 28}, 1986 (1983).
 
\bibitem{Tro84} W. Trombik, W. Trautmann, F. Krug, W. D\"{u}nnweber,
 D. Konnerth, W. Hering, R. Singh and D. Zeppenfeld,
 Phys. Lett. {\bf B135}, 271 (1984).

\bibitem{Kon85} D. Konnerth {\it et al.},
 Phys. Rev. Lett. {\bf 55}, 588 (1985).

\bibitem{Dav93} N.J. Davis, A.C. Shotter, E.W. Macdonald, T. Davinson,
 K. Livingston, P.J. Sellin, D. Branford, R.D. Page and P.J. Woods,
 Nucl. Instr. and Meth. {\bf A330}, 165 (1993).

\bibitem{Dea84} 
 D.R. Dean and N. Rowley, 
 J.Phys. G {\bf 10} 493 (1984).

\bibitem{Rae85}
 W.D.M. Rae {\it et al.},
 Phys. Lett. {\bf 156B}, 167 (1985).

\bibitem{Rae81} 
 W.D.M. Rae  {\it et al.}, 
 Phys. Lett. {\bf 105B}, 417 (1981); 
 W.D.M. Rae, A.J. Cole, B.G. Harvey and R.G. Stokstad, Phys.
 Rev. C {\bf 30}, 158 (1984).

\bibitem{Rae84} 
 W.D.M. Rae and R.K. Bhowmik, Nucl. Phys. {\bf A420}, 320 (1984);
 R.K. Bhowmik and W.D.M. Rae, Phys. Lett. {\bf 136B}, 149 (1984);
 S. Marsh and W.D.M. Rae, {\it ibid.} {\bf 153B}, 21 (1985).

\bibitem{Pou79}
 F. Pougheon {\it et al.},
 Nucl. Phys. {\bf A325}, 481 (1979).

\bibitem{Hah67}
 F.J.W. Hahne, 
 Nucl. Phys. {\bf A104}, 545 (1967);
 W.D.M. Rae, 
 in {\it Proceedings of the IV International conference on Clustering
 Aspects of Nuclear Structure}, Chester, United Kingdom, 1984, edited by
 J. S. Lilley and M. A. Nagarajan, (D. Reidel, Lancaster, 1985), p. 261.

\bibitem{Rae84b}
 W.D.M. Rae {\it et al.}, Phys. Rev. C {\bf 30}, 158 (1984).

\bibitem{Rae95b} 
 W.D.M. Rae {\it et al.},
 Oxford University Report, OUNP-95-12 (1995).

\end{references}

%_______________________________________________________________________

\begin{figure}
\caption{Velocity vectors in the center-of-mass for the sequential
breakup reaction $(A, A^* \rightarrow C+c)$ defining the correlation
variables ($\theta^*_B=90^{\circ},\phi^*_B,\psi_B,\chi_B$) in the
``Basel'' co-ordinate scheme.}
\label{basel}
\end{figure}

\begin{figure}
\caption{Two-body $Q$-value for $^{12}$C$^*$+$^{12}$C$^*$ events at
$E_{c.m.}=32.5$ MeV reconstructed from (a) 3$\alpha$+3$\alpha$ and (b)
3$\alpha$+1 hits identified on either side of the beam direction.}
\label{q2c}
\end{figure}

\begin{figure}
\caption{The $^{12}$C[ $^{12}$C, $^{12}$C($3_1^-$) $\rightarrow$
$^8$Be($0_1^+$)+$\alpha$ ]$^{12}$C($0_1^+$) angular correlation compared
to the ($\phi^*_B,\chi_B$) phase space (shown as the grid of dots)
defined by the detector geometry.}
\label{corr3g}
\end{figure}

\begin{figure}
\caption{The $^{12}$C[ $^{12}$C, $^{12}$C($3_1^-$)$ \rightarrow$
$^8$Be($0_1^+$)+$\alpha$ ]$^{12}$C($0_1^+$) angular correlation
overlayed with the calculated loci of maxima in the semi-classical
correlation functions $W(\phi^*_B,\chi_B)$ for $k=3$ and the $L'$ values
indicated.}
\label{corr3gb}
\end{figure}

\begin{figure}
\caption{One-dimensional projections of the $k=3$ angular correlation in
the $^{12}$C($0_1^-$)+$^{12}$C($3_1^-$) channel for a number of $L'$
values.}
\label{corr3gp}
\end{figure}

\begin{figure} 
\caption{The $^{12}$C[ $^{12}$C, $^{12}$C($3_1^-$)$ \rightarrow$
$^8$Be($0_1^+$)+$\alpha$ ]$^{12}$C($0_2^+$) angular correlation and loci
of maxima in the correlation functions $W(\phi^*_B,\chi_B)$ for $k=3$
and the $L'$ values indicated.}
\label{corr30} 
\end{figure}

\begin{figure}
\caption{One-dimensional projections of the $k=3$ angular correlation in
the $^{12}$C($0_2^-$)+$^{12}$C($3_1^-$) channel for a number of $L'$
values.}
\label{corr30p}
\end{figure}

\begin{figure}
\caption{The $^{12}$C[ $^{12}$C, $^{12}$C($3_1^-$)$ \rightarrow$
$^8$Be($0_1^+$)+$\alpha$ ]$^{12}$C($3_1^-$) angular correlation.}
\label{corr33}
\end{figure}

\begin{figure}
\caption{$^{12}$C[ $^{12}$C, $^{12}$C($3_1^-$)$ \rightarrow$
$^8$Be($0_1^+$)+$\alpha$ ]$^{12}$C($3_1^-$)$ \rightarrow$
$^8$Be($0_1^+$)+$\alpha$ angular correlations including consideration of
the recoil breakup angle: (a) $X=\chi_B+\chi_B'$ and (b)
$X=\chi_B-\chi_B'$.}
\label{corr33x}
\end{figure}

\begin{figure}
\caption{Projections of the $W(\phi^*_B, X)$ angular correlations in
the $^{12}$C($3_1^-$)+$^{12}$C($3_1^-$) channel for $k=k'=3$ and the
$L'$ values indicated.}
\label{corr33xp}
\end{figure}

%_______________________________________________________________________

\end{document}
