%Attention Debbie Dietz

%I have re-formatted the manuscript file so that lines are < 72 chars
%long, hopefully this will avoid the truncation problem due to
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%Thankyou for your patience.
%                           Steve Chappell.
 

% All correspondence regarding this submission should be made to
%+--------------------------------------------------------------+
%| S.P.G.Chappell                      | Nuclear Structure Group|
%|                                     |  Department of Physics | 
%| :Phone (0904) 430000 ext. 2243/2244 |  University of York    | 
%| :Fax   (0904) 432214                |  Heslington            | 
%| :Email oef@yksc.york.ac.uk          |  York, YO1 5DD, 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 follows..

\documentstyle[preprint,eqsecnum,aps]{revtex}
\begin{document}
\draft
\preprint{Version 3.01}

%_______________________________________________________________________

\title{The $^{12}$C+$^{12}$C ``6-$\alpha$ Chain State'' Resonance}

%_______________________________________________________________________

\author{S.P.G. Chappell, D.L. Watson, S.P. Fox and C.D. Jones}
\address{Department of Physics, University of York, Heslington, York
YO1 5DD, United Kingdom.}
\author{W.D.M. Rae and P.M. Simmons}
\address{Department of Nuclear Physics, University of Oxford, Keble
Road, Oxford OX1 3RH,\\ United Kingdom.}
\author{M. Freer, B.R. Fulton, N.M. Clarke, N. Curtis, M.J. Leddy$^*$,
J.S. Pople, S.J. Hall,\\ R.P. Ward and G. Tungate}
\address{School of Physics and Space Research, University of
Birmingham, Edgbaston, Birmingham\\ B15 2TT, United Kingdom.}
\author{W.N. Catford, G.J. Gyapong and S.M. Singer}
\address{Department of Physics, University of Surrey, Guildford, Surrey
GU2 5XH, United Kingdom.}
\author{P.H. Regan$^{\dag}$}
\address{Department of Nuclear Physics, Research School of Physical
Sciences, The Australian National University, GPO Box 4, Canberra, ACT
2601, Australia.}
\author{}
\address{$^*$ Present address: Schuster Laboratory, University of
Manchester, Manchester M13 9PL,\\ United Kingdom.\\
$^{\dag}$ Present address: Department of Physics, University of Surrey,
Guildford, Surrey GU2 5XH, United Kingdom.}
%_______________________________________________________________________

\date{\today}
\maketitle 
\newpage

%_______________________________________________________________________

\begin{abstract}

The inelastic scattering of $^{12}$C+$^{12}$C to states above the decay
threshold into 6 $\alpha$-particles has been studied at center of mass
energies in the range 25 to 40 MeV. We observe strong resonant behavior
in the 0$_2^+$+3$_1^-$ and 0$_2^+$+0$_2^+$ channels around 32.5 MeV
with strikingly enhanced cross sections at 90$^o$ in the center of
mass. Further measurements suggest that the 0$_2^+$+(0$_3^+$) channel
also resonates in this energy region. These data are compared to recent
band crossing model calculations and the shape eigenstate
interpretation.


\end{abstract}

%_______________________________________________________________________


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


\newpage

%_______________________________________________________________________

\section{Introduction}

The myriad of resonances observed in $^{12}$C+$^{12}$C scattering have
attracted intensive investigation for over three decades \cite{Erb64}.
Recently, observation \cite{Wu92} of a wide ($\Gamma_{cm}$=4.7 MeV),
Breit-Wigner shaped, resonance at $E_{cm}$=32.5 MeV in the
$^{12}$C($^{12}$C,$^{12}$C(0$_2^+$))$^{12}$C(0$_2^+$) inelastic
scattering reaction has caused some difficulty in interpretation with
conventional reaction models. The simple band-crossing model (BCM)
\cite{Ab78,Fu80} cannot account for such resonant behavior in the
strongly angular-momentum-mismatched mutual 0$_2^+$ channel, since in
general 0$^+$ bands do not cross. Furthermore, the angle-averaged
on-resonance cross section $\sigma_{av}$, for 0$_2^+$+0$_2^+$
($\sigma_{av}$=15 $\mu$b) and 0$_2^+$+0$_1^+$ ($\sigma_{av}$=100
$\mu$b) \cite{Fu80,Pa88} does not correlate with the factor of 100
difference in their Coulomb penetrabilities. Wuosmaa {\it et al }
\cite{Wu92} proposed an understanding of this structure through the
formation of a 6-$\alpha$ linear chain configuration (LCC) at
$E_x$=46.4 MeV in $^{24}$Mg. The cranked cluster model (CCM)
\cite{Ma86} predicts such a state some 15-20 MeV above the 6-$\alpha$
breakup threshold, with a band-head excitation of around 45.5 MeV
\cite{Me92}. This LCC is consistent with the large prolate deformations
found in both Hartree-Fock \cite{Flo84} and Nilsson-Strutinsky
\cite{Le75} potential energy surface calculations.  Taking the
rotational spacing of the 6-$\alpha$ LCC as $\hbar^2/2\cal{J}$=22 keV
(as predicted by the CCM) and that of two touching $^{12}$C nuclei in
the elastic band as 90 keV results in a band crossing at
$E_x$$\simeq$50 MeV for $L$=16 \cite{Ma86}, confirming observation of
the $L$=14,16 dominant angular momenta \cite{Wu92}.

An alternative explanation is offered by Satpathy and Sarangi, who have
applied a dynamic bonding-potential model \cite{Sa86} to the
0$_2^+$+0$_2^+$ channel \cite{Sa94}, as an extension of their
di-nucleus rotation-vibration picture of $^{12}$C+$^{12}$C scattering
resonances \cite{Sa90}. An increase in cross section at $E_{cm}$=33.09
MeV with $L$=14 is predicted, precluding the necessity for population
of any compound state in $^{24}$Mg. The apparent success of this
unified approach to molecular resonances is, however, criticized
\cite{Ka88} for inappropriate approximation of the centrifugal
potential which can produce spurious results.

Most importantly, the above theoretical descriptions fail to reproduce
the strikingly large enhancement in the angular distribution found at
90$^o$ (c.m.) on resonance. Experimentally, this has been found to be
the major source of resonant strength. An enhancement in such a
restricted angular range requires a coherent contribution from many
angular momentum waves in the scattering. Merchant and Rae \cite{Me93}
have, therefore, suggested a new type of mechanism whereby several
degenerate $l$-states are excited in a cluster band, characterized by a
fixed value of $2n+l$ (where $n$ and $l$ are the nodal and angular
momentum quantum numbers, respectively), to form an
approximate-shape-eigenstate \cite{Ra92} fixed in orientation with no
definite angular momentum. Here the degeneracy is accounted for by an
exotic structure with a very large moment of inertia (2R $\simeq$18
fm), consistent with a 6-$\alpha$ LCC, aligned with its long axis
perpendicular to the beam direction.

The dominant decay mode of a 6-$\alpha$ LCC is expected to be through
symmetric fission into two 3-$\alpha$ LCCs, where the $^{12}$C(0$_2^+$)
state at $E_x=7.65$ MeV is often associated with such a configuration
\cite{Br70,Ta71}. This identification, however, still remains a point
of contention. The 7.65 MeV 0$_2^+$ state lies only 280 keV above the
particle breakup threshold and in many generator coordinate model (GCM)
calculations the first excited 0$^+$ state does not turn out to be a
linear chain but a dominantly bent configuration \cite{Tak72}. Indeed a
structure in which the 3 $\alpha$-particles are loosely coupled, but
not necessarily as a chain, must be used to account for the large
$\alpha$-decay width of the 0$_2^+$ level \cite{Su71}. Furthermore an
artificial mixing of Brink-style equilateral triangle and other bent
3-$\alpha$ geometries is required to describe the strong
0$_2^+$$\rightarrow$0$_1^+$ monopole transition \cite{Ta71}. A true
linear chain might be expected to lie at higher excitation energies.
Using a recent model of $^{16}$O \cite{Rae94}, Rae and Merchant have
speculated that the LCC may be formed when the $^8$Be is in its 2$^+$
$M=0$ state. Thus the linear chain might be some 3 MeV (viz the
excitation of $^8$Be(2$^+$)) above the bent chain. Simple GCM
calculations \cite{Jie} support this speculation. Hence, one may
reasonably expect any symmetric 6-$\alpha$ LCC breakup to possibly
include higher excitations in $^{12}$C$^*$ with an LCC component. The
r\^{o}le of the 10.3 MeV (0$_3^+$) state, first proposed by Morinaga
\cite{Mo56} as the second level (2$_2^+$) of the 3-$\alpha$ rotational
band, may therefore be important.

These unresolved issues have prompted us to perform detailed angular
distribution measurements, over the energy region of this resonance,
for $^{12}$C($^{12}$C,6$\alpha$) reactions and study the formation of
$^{12}$C$^*$ above the 3-$\alpha$ breakup threshold. Our results are
compared with recent BCM calculations and the shape eigenstate
interpretation.


%_______________________________________________________________________

\section{Experimental Method}

The data were taken at the Australian National University's 14UD tandem
Van de Graaff facility using $^{12}$C beams, accelerated over selected
laboratory energies in the range 50 to 80 MeV. The target consisted of
a self-supporting 90 $\mu$g cm$^{-2}$ natural carbon foil. Three 25
cm$^{2}$, 500 $\mu$m thick position sensitive strip detectors (PSSDs)
were arranged with their centers 12 cm from the target at +25$^o$,
+52$^o$ and -32$^o$ in the horizontal plane relative to the beam
direction. The PSSDs were fabricated from a single wafer of n-type
silicon implanted with boron ions to form a resistive p-type layer,
that serves as the front anode of the detector and forms the p-n
junction. Division of this electrode into 16 horizontally arranged 3 mm
strips, with contacts made to either end of each strip, allows for the
determination of position, energy and timing of up to 16 simultaneous
hits in the detector, providing no two particles enter via the same
strip. Thus the PSSDs were particularly suited to the study of the
$^{12}$C($^{12}$C,6$\alpha$) reaction, in particular to the recording
of all 6 $\alpha$-particles in coincidence from the assumed sequential
breakup of resonant states in $^{24}$Mg$^*$ through $^{12}$C$^*$ then
$^{8}$Be$_{gs}$. The data acquisition system used, however, limited the
maximum number of available ADC channels to 80. Hence, the single
(-32$^o$) detector was operated in normal mode and the neighboring
(+25$^o$ \& +52$^o$) PSSDs were coupled into a novel ``doubled''
configuration. This was achieved by the electrical connection of
adjacent strip ends between the PSSDs, so that charge collected at
either of these ends passed to the same preamplifier-amplifier-ADC
combination. These signals could then be determined as originating from
one detector or the other by the application of suitable logic. For
example, the true strip of any particular hit must have both signals
from the coupled and uncoupled ends present above threshold. No
explicit particle identification was made during the experiment, since
over determination of the reaction kinematics allowed reconstruction of
the data in terms of multiplicity with a low degree of background
contamination. In this case, each hit in a 6-fold event could be
unambiguously assigned to an $\alpha$-particle on measurement of a
total reaction Q-value $\simeq$-14.6 MeV. Only those events in which at
least 5 PSSD strips fired above threshold, including a minimum of 2 in
the double-detector, were recorded for detailed analysis.

For 6-$\alpha$ coincidences, occurring as 3-$\alpha$ hits either side
of the beam, we calculate the excitation $E_x$ and scattering angle of
both decaying $^{12}$C$^*$ nuclei from the momenta of the
$\alpha$-particles, viz

\begin{equation}
{
E_x = \sum_{i=1}^3 E_{\alpha_i} - 
\frac{P^2}{ 2M } + Q_{thresh}
}
\label{eq1}
\end{equation}
where the $^{12}$C momentum, 
\begin{equation}
{
P = \sqrt{ 
(\sum_{i=1}^3 P_{\alpha_i x})^2+
(\sum_{i=1}^3 P_{\alpha_i y})^2+
(\sum_{i=1}^3 P_{\alpha_i z})^2
 }
}
\label{eq2}
\end{equation}
$Q_{thresh}$ is the $^{12}$C$\rightarrow$3$\alpha$ breakup threshold
(7.27 MeV) and $M$ is the mass of $^{12}$C. The $E_{\alpha_i}$ and
$P_{\alpha_i}$ are the measured $\alpha$-particle energies and
components of momentum respectively. This method of reconstruction is
independent of the process by which the $\alpha$-particles are formed,
although here we consider only those 3-$\alpha$ events correlated as
$^{8}$Be$_{gs}$+$\alpha$ to originate from a $^{12}$C nucleus. The
accuracy of this reconstruction is dominated by the position response
of the detection system, where the PSSDs localize incident events to
within 200 $\mu$m and are linear to within a few percent.

Extensive Monte-Carlo simulations of the detection process have been
performed for each of the 6-body final states discussed herein. The
simulation assumes an isotropic center of mass angular distribution for
the primary scattering with subsequent decays of the $^{12}$C$^*$
nuclei having either isotropic or $1/sin(\theta)$ distributions for
zero and non-zero spin states respectively. Detection of the
$\alpha$-particles is then modelled in detail, including energy
threshold effects and rejection for multiple hits in a single strip.
The resulting profiles then allow relative normalization of the data
for variations in efficiency with respect to energy, angle and
excitation of the reconstructed nuclei.


%_______________________________________________________________________

\section{Results}

The summed excitation energy for detected $^{12}$C$^*$+$^{12}$C$^*$
events at 32.5 MeV (c.m.) is given in figure \ref{exsum}, where the
combinations of 0$_2^+$ and 3$_1^-$ mutual excitations have been
identified. Selection of $^{12}$C(0$_2^+$) events in the single-PSSD
enables states in $^{12}$C$^*$ up to $\sim$12 MeV excitation to be
studied in the larger area double-PSSD combination. Since the origin of
5 out of the 6 $\alpha$-particles has been determined (i.e.
$^{12}$C+$^{12}$C$\rightarrow$$^{12}$C(0$_2^+$)+$^8$Be$_{gs}$+$\alpha$),
any background contamination is expected to be small. Figure \ref{12C}
is an example of this measurement, again at 32.5 MeV.  Note that the
relative detection efficiency decreases by approximately 80\% between
$E_x=$7.65 and 12 MeV as the average breakup cone of
$^{12}$C$^*$$\rightarrow$3$\alpha$ trajectories increases, from
$\sim$13$^o$ to $\sim$52$^o$, and considerably reduces any overlap with
the double-PSSD's angular range. The 0$_2^+$ and 3$_1^-$ states are
clearly resolved in the efficiency corrected data, above an apparently
structured background. Subtraction of these peaks via Gaussian fitting
reveals two major structures comprising the remaining background: a
wide peak at $E_x$$\simeq$9.9$\pm$0.5 MeV and a smaller contribution at
$E_x$$\simeq$11 MeV. These are consistent with the (0$_3^+$) and 1$^-$
cluster states in $^{12}$C$^*$, respectively. Integration of this
background, between the limits given in Fig.\ref{12C}, then provides an
estimate of the yield in the 0$_2^+$+(0$_3^+$) and 0$_2^+$+1$^-$
channels. The relevant excitation functions, as extracted from Figs.
\ref{exsum} and \ref{12C} at each of the c.m. energies sampled, are
presented in Fig.\ref{exfunc}. The large errors are representative of
uncertainties in the correction for low efficiency. The 0$_2^+$+0$_2^+$
and 0$_2^+$+3$_1^-$ channels are observed to resonate in the region of
32.5 MeV, along with the 0$_2^+$+(0$_3^+$) background contribution.
Efficiency corrected angular distributions, at selected energies, for
the 0$_2^+$+0$_2^+$, 0$_2^+$+3$_1^-$ and 3$_1^-$+3$_1^-$ channels are
shown in Figs. \ref{oo} to \ref{tt}. A dramatic increase in cross
section at 90$^o$, compared to the neighboring maxima, is a
characteristic of resonance in the 0$_2^+$+0$_2^+$ and
0$_2^+$+3$_1^-$channels. Note that, the very small detection
probability ($<$ $\sim$0.1 \%) and difficulty in accounting for the
position dependent energy thresholds of the PSSDs makes the Monte-Carlo
simulation for normalization of the low 3$_1^-$+3$_1^-$ yield rather
more inaccurate. This is reflected in the angular distributions
presented  for this channel, where symmetry about 90$^o$ is not
restored. This demonstrates that any quantitative comparison to theory
must accurately account for the phase space sampled in these
multi-particle reactions and not rely on the simplified assumptions of
Monte-Carlo codes for correction of the projected data.

%_______________________________________________________________________

\section{Discussion}

Returning to the theoretical interpretation of these data, the band
crossing model of Abe {\em et al} \cite{Ab78} has recently been
extended in calculations by Hirabayashi \cite{Hir94} {\em et al} to
include realistic transition densities for $^{12}$C. Since the 0$_2^+$
state in $^{12}$C is highly deformed, in these new calculations the
0$_2^+$ bands have higher moments of inertia and thus cross the elastic
band. These calculations appear to reproduce the angle integrated cross
sections for both the 0$_2^+$+0$_1^+$ and 0$_2^+$+0$_2^+$ bands and
also predict correlated enhancements in the 0$_2^+$+3$_1^-$ and
0$_2^+$+0$_3^+$ channels roughly in line with our measurements.
Reproduction of the 0$_2^+$+0$_2^+$ angular distribution behavior about
90$^o$ has also been achieved by Hirabayashi {\em et al}, but with a
dominant partial wave ($L$=18) somewhat larger than those in the
experimental data ($L$=14,16 \cite{Wu92}).

The observation of the 90$^o$ enhanced resonance in the 0$_2^+$+3$_1^-$
channel appears to contradict the 6-$\alpha$ chain state
interpretation, although not the shape eigenstate hypothesis which only
requires a very large moment of inertia. In addition, there have been
reports of this resonance being observed in other channels near 32.5
MeV with similar enhancements at 90$^o$ in the angular distributions.
Aliotta {\it et al} \cite{Al94} and Freeman {\it et al} \cite{Fre94}
have observed decay to $^{16}$O$_{gs}$+$^8$Be, while Mirgule {\it et
al} \cite{Mir94} have measured resonant cross sections for decay to the
13.9 MeV 6$^+$ and 20.5 MeV 8$^+$ states in $^{20}$Ne. These data also
appear to rule out the population of a LCC in $^{24}$Mg, not least due
to the fact that $^{16}$O$_{gs}$ is spherical. However, Rae and
Merchant \cite{Rae94,Rae94b} have proposed a mechanism which is
consistent with both the 6-$\alpha$ chain state and shape eigenstate
hypotheses and can explain the observed decay into these other
channels.

Rae and Merchant propose that there is considerable mixing of the chain
state with other configurations in $^{24}$Mg at this excitation
energy.  They suggest that the spreading width of this chain
configuration is probably much larger than the escape (or fission)
width. The mechanism of this mixing can be visualized in terms of an
inelastic reorientation process. To describe a linear chain state
composed of two $^{12}$C $\alpha$-chains the $^{12}$C nuclei cannot
have zero spin since then no orientation would be preferred and the
structure would not be linear.  Thus the 6-$\alpha$ LCC must involve
$J=2,4$ and possibly higher $^{12}$C spins. Also in order to have a
linear structure with axial symmetry the $^{12}$C nuclei must have
$M=0$ along the long axis of the chain state (both to preserve this as
a symmetry axis and provide the alignment).  Rae and Merchant propose
that this wavefunction can decay in two ways:  1) by fission into two
$^{12}$C nuclei and 2) by reorientation of the $^{12}$C spins from
$M=0$ to some other $M\neq0$. This is essentially an inelastic
scattering reorientation mechanism. After this reorientation the system
no longer has the symmetry of a chain state and the $^{12}$C nuclei are
no longer aligned end to end. The state can then collapse inwards (this
is essentially fusion) and change shape as it does so.

In this mechanism, since the 6-$\alpha$ chain state resonance is at a
higher excitation energy than almost all other resonances in
$^{12}$C+$^{12}$C scattering, these shape changes could result in an
excited configuration in any of the other possible local total energy
minima. Thus we should expect to see all $\alpha$-cluster model
channels populated - possibly with an essentially statistical
probability. This decay mechanism then automatically explains the
formation of the chain state from the $^{12}$C$_{gs}$+$^{12}$C$_{gs}$
entrance channel, which is just the inverse of the decay process, viz
that inelastic processes produce the alignment which results in the
formation of the LCC.

This explanation suggests that the 32.5 MeV resonance should show
structure if studied in finer energy steps. The widths of this finer
structure should be comparable to the intermediate structure resonances
observed in $^{12}$C+$^{12}$C scattering, since to first order the
chain state is mixing with other alpha cluster configurations which are
thought to provide a consistent interpretation of the $^{12}$C+$^{12}$C
scattering resonances \cite{Rae94c}. Thus it would be interesting to
see if this prediction is borne out by experiment. Such a detailed
excitation function measurement would be able to distinguish the band
crossing model from the $\alpha$-cluster model interpretation.


%_______________________________________________________________________

\section{Conclusion}

We have presented data which suggest that the resonance observed in the
$^{12}$C(0$_2^+$)+$^{12}$C(0$_2^+$) channel by Wuosmaa {\it et al} can
also be observed in the $^{12}$C(0$_2^+$)+$^{12}$C(0$_3^+$) channel, as
predicted by Zhang and Rae, and in the
$^{12}$C(0$_2^+$)+$^{12}$C(3$_1^-$) channel. The latter observation
presents a challenge to the original structural interpretation of the
32.5 MeV resonance as a 6-$\alpha$ LCC in $^{24}$Mg. We have discussed
two theoretical models which attempt to describe these data - a BCM
calculation by Hirabayashi {\it et al} and a cluster model calculation
by Rae and Merchant. The latter suggests that the chain state should be
strongly mixed with other cluster configurations in $^{24}$Mg at this
excitation energy and thus should be observable in many decay
channels.  A more detailed analysis of the angular correlation data for
the $^{12}$C(0$_2^+$)+$^{12}$C(3$_1^-$) channel is underway. This
should provide further information on the reaction mechanism and allow
a more stringent test of the model calculations.

%_______________________________________________________________________

\acknowledgements

The authors are grateful for the invaluable support and continued
hospitality of the ANU technical and research staff. SPGC would like to
express his sincere thanks to Professor Y. Abe and Dr. Y. Hirabayashi
for many encouraging discussions and the provision of their BCM
calculations before publication. The financial support of the U.K.
Engineering and Physical Sciences Research Council (formerly Science
and Engineering Research Council) through the award of studentships to
SPGC, SPF, PMS, NC, MJL (CASE), JSP and SMS is acknowledged.

%_______________________________________________________________________

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\end{references}


%_______________________________________________________________________

\begin{figure}
\caption{Summed excitation energy for $^{12}$C$^*$+$^{12}$C$^*$ events
at $E_{cm}$=32.5 MeV. The excitation energies, $E_x$ are reconstructed
in both (s)ingle and (d)ouble PSSDs.  The data has been normalized to
beam exposure only.}
\label{exsum}
\end{figure}

\begin{figure}
\caption{Excitation energy in $^{12}$C$^*$ for the
$^{12}$C($^{12}$C,$^{12}$C$^*$)$^{12}$C(0$_2^+$) reaction at
$E_{cm}$=32.5 MeV. The data has been normalized to beam exposure (Raw)
and detection efficiency (Corrected). The integration limits used for
estimation of the (0$_3^+$) and (1$^-$) contributions are shown.}
\label{12C}
\end{figure}

\begin{figure}
\caption{Experimental excitation functions for selected channels in
$^{12}$C($^{12}$C,$^{12}$C$^*$)$^{12}$C$^*$.}
\label{exfunc}
\end{figure}

\begin{figure}
\caption{0$_2^+$+0$_2^+$ angular distributions at the energies (c.m.)
indicated in each frame.}
\label{oo}
\end{figure}

\begin{figure}
\caption{0$_2^+$+3$_1^-$ angular distributions at the energies (c.m.)
indicated in each frame.}
\label{ot}
\end{figure}

\begin{figure}
\caption{3$_1^-$+3$_1^-$ angular distributions at the energies (c.m.)
indicated in each frame.}
\label{tt}
\end{figure}

%_______________________________________________________________________


\end{document}
