%Paper: 
%From: "Jean Cleymans" <CLEYMANS@physci.uct.ac.za>
%Date: 25 Oct 93 12:13:31 GMT+0200


% September 1993
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\twocolumn[
%\mbox{ }\hfill{\normalsize UCT-TP 201/93}\\
%\mbox{ }\hfill{\normalsize September 1993}\\

%\vspace{.5cm}

\begin{center}
{\Large \bf Hadron Production Ratios as Probes\\
\vspace{.3cm}
of Confinement and Freeze-Out}
\\
\vspace{.3cm}
{\bf E. Suhonen}$^{a}$, {\bf J. Cleymans}$^{b}$,
{\bf K. Redlich}$^{c}$, {\bf H. Satz}$^{d}$\\
\vspace{.5cm}
$^{a}$Department of Theoretical Physics,
University of Oulu, FIN-90570 Oulu, Finland.\\
$^{b}$Department of Physics, University of Cape Town,
Rondebosch 7700, South Africa.\\
$^{c}$Fakult\"{a}t f\"{u}r Physik, Universit\"{a}t
Bielefeld, D-33501 Bielefeld, Germany.\\
$^{d}$Theory Division, CERN, CH-1211 Geneva 23,
Switzerland.
\end{center}
\vspace{.3cm}
\begin{center}
%{\bf Abstract}\\
\noindent
Particle production in central S-A collisions at 200 GeV/A energy is
analysed within a thermal model. Present data imply that the strange
particles freeze out at a higher temperature than the non-strange
particles and that the strangeness saturation is incomplete.
\vspace{.3cm}
\end{center}
]

\underline{Introduction}: $\;$
A significant enhancement of strange particle production has been
observed in high energy heavy ion collisions in comparison to proton-
proton
collisions. This indicates a higher level of equilibration in
nuclear collisions where volume, life-time and energy density are
increased. Strongly interacting matter produced initially expands
rapidly, cools down and breaks up into observable hadrons. It is not
excluded that a quark-gluon plasma phase in which the colour degrees
of freedom are deconfined is achieved at an early stage of evolution.
Soft processes involved in the transition from quarks and gluons to
hadrons are, if at all, only poorly understood. It is possible that
the system expands too fast to retain equilibrium and a quark-gluon
plasma
suddenly disintegrates into the final hadrons. In this case the
observed
hadrons could carry information on parameters and properties of quark-
gluon
plasma and on confinement \cite{ref1,ref2}. We assume here a smooth
evolution in which a hadron gas thermalises before the final hadrons
stop interacting. Regardless of the exact nature of the produced
matter,
the observed hadrons will reflect the properties of the
last thermal state before the freeze-out, the equilibrium hadron gas.
In the present paper we shall study data on hadron production ratios
within the framework of a thermal model to determine the statistical
properties of their sources, i.e. of particle freeze-out states.

%\footnotesize{
%\makebox[5cm]{}   \\ \hline
%$^{*}$Presented by E. Suhonen at the EPS Conference
%1993, Marseille, France
%}

\newpage
\normalsize
\underline{Thermal Model}: $\;$
The state of an equilibrium hadron gas is specified by three
parameters, the temperature $T$, the baryon chemical potential
$\mu_{B}$ and the strangeness chemical potential $\mu_{s}$. The
requirement of vanishing overall strangeness fixes one of these
parameters, e.g. $\mu_{s}$, and the two remaining parameters,
$T$ and $\mu_{B}$, fully determine the local state in the thermal
model. For simplicity, we present the formulae in Boltzmann
statistics although in the actual calculations we have worked with
the quantum statistics. The partition function of the hadron gas is
given by \cite{ref1}
%%%EQ 1
\begin{equation}
\makebox{ln} \; Z(T,\mu_{B},\mu_{s}) = \sum_{i}
\left[ W_{i}^{m} + \left( \lambda_{B}^{B_{i}} \lambda_{S}^{-S_{i}}
+ \lambda_{B}^{-B_{i}} \lambda_{S}^{S_{i}} \right) W_{i} \right] .
\end{equation}
Here the first term refers to non-strange mesons and the second term
to particles which carry baryon numbers $B_{i}$ and strangeness
$S_{i}$. The fugacities related to the baryon number and strangeness
are $\lambda_{B} \equiv \makebox{exp} (\mu_{B}/T)$ and
$\lambda_{S} \equiv \makebox{exp} (\mu_{s}/T)$. The phase space
factor $W_{i}$ is of the form
%%EQ 2
\begin{equation}
W_{i} = \frac{d_{i} VT m_{i}^{2}}{2 \pi^{2}} \; K_{2} \;
        \left( \frac {m_{i}}{T} \right)
\end{equation}
where $d_{i}$ denotes the degeneracy and $m_{i}$ the mass of the
hadron state $i$, $V$ is the volume of the system and $K_{2}$ the
modified Bessel function of the second type. We have included in
eq.(1) all well established resonances up to mass of 2 GeV. The
thermal contribution of the particle multiplicity $N_{i}^{th} =
W_{i}$ calculated from eq.(1) has to be added by the resonance
contributions to get the particle multiplicity,
%%EQ 3
\begin{equation}
N_{i} = W_{i} + \sum_{j} \; \Gamma_{ij} \; W_{j} .
\end{equation}
Here $\Gamma_{ij}$ is the branching ratio of the decay of resonance
$j$ to particle $i$. If both thermal and chemical equilibrium were
established
and if there were a unique freeze-out for different hadron species,
then all hadron production ratios would be determined using the
values of $T$ and $\mu_{B}$ fixed by two measured ratios.

\underline{Data Interpretation}: $\;$
Data we consider here are from sulphur collisions at 200A
GeV energy with tungsten- \cite{ref3,ref4}, silver- \cite{ref5} and
lead- \cite{ref2,ref6,ref7} targets. They are measured in the backward
hemisphere, 2.3 $<$ y $<$ 3.0, where the production ratios do not
depend much
on the target size. The WA85 experiment \cite{ref3,ref4} provides us
with
the following strange baryon and antibaryon ratios:
$\bar{\Lambda}/\Lambda = 0.2
\pm 0.01, \overline{\Xi^{-}}/\Xi^{-} = 0.45 \pm 0.05$, $\Xi^{-
}/\Lambda
= 0.095 \pm 0.006$, $\overline{\Xi^{-}}/\bar{\Lambda} = 0.21 \pm 0.02$
and $\overline{\Omega^{-}}/\Omega^{-} = 0.57 \pm 0.41$. The ratios for
$\Xi^{-}/\Lambda$ and $\overline{\Xi^{-}}/\bar{\Lambda}$ have been
corrected for the $p_{T}$ cut while the result for
$\overline{\Omega^{-}}/\Omega^{-}$
is preliminary and uncorrected for the acceptance. The first two
ratios
lead to the narrow bands in the $T-\mu_{B}$ plane which cross each
other in the small region of $T \simeq$ (190 $\pm$ 15) MeV and
$\mu_{B} \simeq$ (240 $\pm$ 40) MeV, as shown in Figure 1. The
crossing region corresponding to the ratios for $\Xi^{-}/\Lambda$
and $\overline{\Xi^{-}}/\bar{\Lambda}$ is quite different, as also
shown in
Figure 1. The thermal model in its original form thus turns out to be
%%%FIGURE 1
\begin{figure}[h]
\vspace{8.5cm}
\end{figure}
%%%
too idealized to explain WA85 data. A remedy we use is to leave out
the assumption from the complete chemical equilibrium between strange
and non-strange hadrons. In fact it has been suggested, on the basis
of small cross-sections of strange particle production, that the
strange particle phase space can reach only partial saturation
\cite{ref2,ref8,ref9}. The exchange processes among the strange
particle species are faster than strangeness producing processes and
therefore the strange particles are assumed to be in equilibrium
relative to each other but in relation to the non-strange particles
they may be suppressed by a phase space saturation factor $\gamma_{s}
< 1$
\cite{ref8}. This is achieved in our model by multiplying both
$\lambda_{S}$ and $\lambda_{S}^{-1}$ in eq.(1) by a parameter
$\gamma_{S}$
which then yields for the multiplicity of particle species $i$,
%%EQ 4
\begin{equation}
N_{i} = \gamma_{S}^{S_{i}} W_{i} + \sum_{j} \; \gamma_{S}^{S_{j}} \;
\Gamma_{ij} \; W_{j} .
\end{equation}
The ratios for $\bar{\Lambda}/\Lambda$ and $\overline{\Xi^{-}}/\Xi^{-
}$
are not changed by the modification while the results for
$\Xi^{-}/\Lambda$ and $\overline{\Xi^{-}}$ become multiplied by
factor
$\gamma_{S}$. As shown in the figure, the thermal model with
$\gamma_{S} = 0.7$ is compatible with the WA85 measurements on
$\Lambda$, $\bar{\Lambda}$, $\Xi^{-}$ and $\overline{\Xi^{-}}$
production. We notice, however, that the three independent data
points have been used to fix three statistical parameters
$T$, $\mu_{B}$ and $\gamma$ only. More data is needed to justify
the validity of the model. The model prediction for
$\overline{\Omega^{-}}/\Omega^{-}$ (1 $\pm$ 0.3) is higher than the
preliminary experimental ratio. On the other hand, the results for
$K_{s}^{0}/\Lambda$ and $K^{+}/K^{-}$, $K_{s}^{0}/\Lambda \cong$
1.2 $\pm$ 0.5 (2.3 $<$ y $<$ 2.8) and $K^{+}/K^{-} \simeq$ 1.5 $\pm$
0.5
(y $\simeq$ 2.3) obtained from NA35 measurements on rapidity
distributions in S-Ag collisions \cite{ref5} are well predicted by
the model with the above values of $T$ and $\mu_{B}$.
%%%FIGURE 2
\begin{figure}[h]
\vspace{8.5cm}
\end{figure}
%%%
Having extracted the freeze-out parameters for strange particles to
be $T \simeq$ 190 MeV, $\mu_{B} \simeq$ 240 MeV and
$\gamma_{S} \simeq$ 0.7 we turn our attention to the non-strange
hadron production. The non-strange particles dominate the charged
particle multiplicities which have been measured by EMU05 \cite{ref2}
and NA35 \cite{ref5} collaborations. From EMU05 we have a result for
the charge asymmetry ratio, $D_{Q} = (h^{+} - h^{-})/(h^{+} + h^{-})$
= 0.088 $\pm$ 0.007, measured in S-P collisions using the same
rapidity window as for WA85 results. The ratio $D_{Q}$ is closely
related to the entropy per baryon (S/B), $D_{Q}$(S/B) $\simeq$ 4.5,
from which one obtains S/B $\simeq$ 50 \cite{ref2}. In order to see
whether this value is consistent with the thermal model we show in
Fig.3 the entropy per baryon in the hadron gas for several different
%%%FIGURE 3
\begin{figure}[h]
\vspace{8.5cm}
\end{figure}
%%%
temperatures. It is seen that at $T$ and $\mu_{B}$ required by
strange particle ratios, S/B $\simeq$ 30, which is lower than the
measured value 50. The measured value would correspond to temperature
110 MeV $< T <$ 140 MeV as seen in Fig.3. Almost the same
temperature and chemical potential, 120 MeV $< T <$ 140 MeV and
200 MeV $< \mu_{B} <$ 270 MeV, can also be extracted from NA35
measurements on $h^{-}/(p - \bar{p})$ in S-Ag collisions at 2.3
$< y <$ 2.8 \cite{ref5}. Our interpretation for these results is that
there is no unique freeze-out for strange and non-strange particles.
The sequential freeze-out has immediate other experimental
consequences. It implies a different freeze-out radius for kaons
$R_{K}$ than for pions $R_{\pi}$. The interferometry studies of NA44
on S-Pb collisions indeed measure $R_{K} < R_{\pi}$ \cite{ref6,ref7}.
{}From free mean path arguments \cite{ref1}, assuming $\pi\pi$-cross-
section to be twice as big as $\pi K$-cross-section, one gets
$R_{K}/R_{\pi} \simeq$ 0.7 which is rather well in agreement with
NA44
results \cite{ref6,ref7}. For an isentropic expansion the freeze-out
radius is inversely proportional to the freeze-out temperature; hence
$T_{K} \simeq$ 190 MeV would imply $T_{\pi} \simeq$ 130 MeV. This is
consistent with the results of our analysis.

\underline{Conclusions}: $\;$
The production rates of different hadrons provide tools for the study
of hadronisation and freeze-out stages in high energy heavy ion
collisions. The measured ratios of strange and non-strange hadrons
allowed us to determine the freeze-out parameters. Analysing data of
several CERN collaborations (WA85, NA35, EMU05 and NA44) within a
thermal model indicated that the strange particles freeze-out at
higher temperature ($T \cong$ 190 MeV) and at the same chemical
potential ($\mu_{B} \simeq$ 240 MeV) than the non-strange particles
($T \simeq$ 130 MeV). The sequential freeze-out is consistent with
the difference in the mean free paths of kaons and pions in the
medium. The saturation of the strangeness was found to be
incomplete.\\
%\newpage

\begin{thebibliography}{99}
%
\bibitem{ref1}
J. Cleymans, K. Redlich, H. Satz, E. Suhonen, {\it Z.~Phys.} {\bf C58}
(1993) 347.
%
\bibitem{ref2}
J. Letessier, J. Rafelski, A. Tounsi, V. Heinz, J.~Sollfrank,
{\it Phys. Rev. Lett.} {\bf 70} (1993) 3530 and references therein.
%
\bibitem{ref3}
D. Evans, New Results from WA85 on (anti)Hyperon Production,
proceedings of Quark Matter '93, Borl\~{a}nge.
%
\bibitem{ref4}
F. Antinori, Recent results from WA85 on Multistrange Hyperon
Production in S-W Interactions at 200 A GeV/c, these proceedings.
%
\bibitem{ref5}
D. R\~{o}hrich, Hadron Production in S-Ag, S-Au Collisions,
proceedings of Quark Matter '93, Borl\~{a}nge.
%
\bibitem{ref6}
T. Humarric, Particle Interferometry in NA44, proceedings of Quark
Matter '93, Borl\~{a}nge.
%
\bibitem{ref7}
C. Fabjan, Identified Particle Interferometry in Heavy-Ion
Collisions, results from NA44, these proceedings.
%
\bibitem{ref8}
J. Rafelski, {\it Phys. Lett.} {\bf B262} (1991) 333.
%
\bibitem{ref9}
U. Heinz, Strangeness Productions and Chemical Equilibration,
proceedings of Quark Matter '93, Borl\~{a}nge.

\end{thebibliography}

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%5 March 1993
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\begin{center}
{\Large \bf Differential cross section for n-p radiative capture at
E$_{\rm n}$ = 63.4 MeV}\\

\vspace{.5cm}

{\bf M.S.~Allie$^{1}$}, {\bf F.D.~Brooks$^{1}$},
{\bf D.G.~Aschman$^{1}$}, {\bf A.~Buffler$^{1}$},\\
{\bf W.A.~Cilliers$^{1}$}, {\bf R.W.~Fearick$^{1}$},
{\bf C.G.L.~Henderson$^{1}$}, {\bf M.J.~Oliver$^{1}$},\\
{\bf M.R.~Nchodu$^{1}$}, {\bf S.M.~Perez$^{1}$},
{\bf D.~Steyn$^{1}$}, {\bf W.R.~McMurray$^{2}$},\\
{\bf B.R.S.~Simpson$^{2}$}, {\bf F.D.~Smit$^{2}$},
{\bf H.G.~Miller$^{3}$}, {\bf K.~Bharuth-Ram$^{4}$}\\
and {\bf I.J.~van Heerden$^{5}$}

\vspace{.5cm}

\date{February 1993}

\end{center}

\begin{abstract}
\noindent
The angular distribution of photons from n-p
radiative capture of 63.4 MeV neutrons has been measured.
The results are transformed to deuteron
photodisintegration cross sections at the equivalent
energy, E$_{\gamma}$ = 33.9 MeV, and analysed in
combination with independent measurements of the
0$^{\circ}$ and 180$^{\circ}$ cross sections.  Legendre
polynomial coefficients obtained from the analysis are
consistent with values reported from global fits and
deviate marginally from predictions based on the Bonn and
Paris potentials.
\end{abstract}

\vspace{.5cm}

]

Searches now being made for evidence of sub-nucleonic
degrees of freedom in nuclei have led to renewed interest
[1,2] in deuteron photodisintegration and its inverse,
neutron-proton radiative capture.  These reactions have
been a rich source of information about the N-N
interaction for many years.  ``Conventional theories''
which include relativistic corrections and assume only
nucleon, meson and isobar degrees of freedom, have
achieved considerable success [1-4] in explaining data
from deuteron photodisintegration and n-p capture at photon energies
up
to about 40 MeV.  Experimental data [5,6] for the total
photodisintegration cross section are consistent with
values predicted by the theory in this energy range
[1,2].  The theory also appears to be in reasonable
agreement with most individual measurements of
differential cross sections, absolute or relative [1-4],
a notable exception being the data of Stephenson et al.
[7] for incident photon energies in the range 2-18 MeV.
Global fits, in which Legendre polynomials are used to
describe the cross section data over an extended range of
energy and angle [5-10], may be used to compare a larger
set of experimental data with theory.  Such comparisons
[1,2] have shown discrepancies between the Legendre
polynomial coefficients derived from experiment and theory
respectively, thus raising doubts as to whether
the conventional theory is entirely adequate, even at low
energies.


\newpage
\normalsize
The differential cross section data now available
at photon energies below 40 MeV have been
obtained mainly from measurements of the angular
distributions of protons or neutrons emitted from
deuteron photodisintegration.  For equivalent
incident neutron energies ($<$80 MeV) in the
n-p capture reaction, there have been a number
of measurements of differential cross sections at the
centre-of-mass angles 0$^{\circ}$ and 180$^{\circ}$, for
example [13-16], but angular distribution data at
intermediate angles are scarce.  For incident neutron
energies 19-50 MeV, Fink et al. [10] have reported
measurements at photon laboratory angles 55$^{\circ}$,
90$^{\circ}$ and 125$^{\circ}$.  Data from n-p capture
measurements are especially useful for comparison with
those obtained from photodisintegration because the
experimental techniques, and hence sources of error
too, are quite different.  We have therefore determined
the differential cross section for deuteron
photodisintegration at the incident photon energy of 33.9
MeV by measuring the angular distribution of photons from
n-p radiative capture of neutrons of laboratory energy
63.4 MeV.  Data measured at eight n-$\gamma$ laboratory
angles were transformed by means of detailed balance so
as to obtain differential cross sections for the
$^{2}$H($\gamma$,p)n reaction at eight $\gamma$-p centre-of-mass
angles
between 25$^{\circ}$ and 128$^{\circ}$.  The angular distribution
data, together with independently measured [16-18] values
of the 0$^{\circ}$ and 180$^{\circ}$ cross sections, were
then fitted to a Legendre polynomial expansion and the
coefficients determined in this way were compared with
those from other measurements and from theory.

%\section*{Experimental}

The experimental arrangement for the angular distribution
measurements is shown in schematic outline in fig. 1.  A
66 MeV pulsed proton beam from the k = 200
cyclotron of the South African National Accelerator
Centre, Faure, was directed through a natural lithium
metal target (2 mm thick) and then magnetically deflected
through an angle of 30$^{\circ}$ into a shielded beam
dump.  A collimated beam of neutrons from the (p,n)
reaction on lithium was formed by an aperture (50 x 50 mm) in the
iron and concrete shield (2.5 m thick), directly ahead the
proton target (fig. 1).  Pulsed neutrons were thus delivered to the
NE213 liquid scintillator (50 mm diam. x 50 mm), positioned 5.9 m from
the lithium target.  The neutron beam profile at this position was
measured and found to be uniform over a diameter of 60 mm, diminishing
sharply to 10\% of peak intensity at a radius of 43 mm.
Protons in the NE213 constituted the target for the n-p radiative
capture reaction and time-of-flight was used to select the strong
forward peak (E$_{\rm n}$ = 63.4 MeV) in the neutron spectrum,
corresponding to the transitions $^{7}$Li(p,n)$^{7}$Be (gs$+$0.43
MeV).  The timing resolution for the time-of-flight measurement
was 1.2 ns (FWHM).
%%%%%%%%%%%

The angular distribution of gammas from n-p radiative
capture in the NE213 scintillator was determined by
observing coincidences between this scintillator and (any
one of) the eight identical sodium iodide crystals
surrounding it (fig. 1).  The crystals (75 mm diam.  $\times$ 125 mm)
were mounted in identical lead shields, including a 5 mm thickness of
lead over the flat surface facing the NE213 cell to reduce unwanted
coincidences from recoil protons emerging from the cell.
The gamma flight time over the 175 mm gap between the cell and the
crystals
(fig. 1) was used to reject backgrounds due to non-relativistic
particles
detected in the latter.

Signals from the NE213 detector were fed to a modified
Link Systems Model 5010 Pulse Shape Discriminator, which
provided two outputs from which the NE213 pulse height L
and pulse shape S were derived [19] in the offline
analysis.  These two outputs were recorded event by event
on magnetic tape, together with the pulse height from the
sodium iodide, the incident neutron time-of-flight and
the coincidence time delay between NE213 and sodium
iodide.  A pattern register was also used to identify the
gamma detector active in the event and to veto events in
which more than one sodium iodide detector registered.
The design of the electronic system ensured that the same
dead time was effective for all the sodium iodide
detectors.

Fig. 2a shows a perspective view of number of events
against NE213 pulse height L and pulse shape S, for
coincidences in which the sodium iodide crystal, located
at $\theta_{{\rm n}\gamma}$(lab) = 90$^{\circ}$, detected a gamma of
energy greater than 12 MeV.  Events corresponding to
protons, deuterons and alphas detected in the NE213
produce the ridges labelled p, d and $\alpha$
respectively in the L-S plane (fig. 2a) and n-p
radiativecaptures form a sharp peak which can be clearly
identified on the deuteron ridge.  Events on this
deuteron ridge (d-$\gamma$ coincidences) were selected by
a cut in the L-S plane and projected onto the L-axis to give
the deuteron pulse height spectra shown in fig. 2b, for
the different gamma angles.  The peak corresponding to n-p
radiative capture events is seen to move to higher
pulse height as the gamma angle increases, as expected
from the kinematics, and a good peak-to-background ratio
is observed at all the angles studied.
%%%%%%%%%%%


The peaks (fig. 2b) were integrated and corrected for
background to determine the angular distribution of
gammas from n-p radiative capture.  Backgrounds, which are
probably due mainly to the $^{12}C(n,d \gamma)^{11}B$ reaction,
were estimated by interpolating between the spectra at pulse
heights below and above the peaks.  The raw data obtained
in this way were first converted to relative differential
cross sections for the n-p radiative capture reaction by
correcting for the variation of gamma detection
efficiency and solid angle with gamma angle, using the
code ESTE [20].  Detailed balance was then used to
convert to differential cross sections for proton
emission in the inverse reaction, deuteron
photodisintegration at the corresponding laboratory
photon energy of 33.9 MeV.  The results, normalised as
outlined below to an absolute differential cross section
scale, are listed in Table 1 and plotted in fig. 3a.
%%%%%%%%%
%\section*{Results and analysis}

In order to make a meaningful Legendre polynomial fit to
the differential cross section for photodisintegration it
is important to have data covering a wide range of angles
and thus, in the present case, to include independent
measurements for the angles 0$^{\circ}$ and
180$^{\circ}$.  Groups at Mainz [17,18] and at Louvain
[16] have recently reported new absolute measurements at
these angles, for photon energies close to 34 MeV.  These
are summarised in Table 2.  The Mainz data are also
plotted, together with the present measurements, in fig. 3a.
A weighted least-squares Legendre polynomial fit, in
which the present data were normalised as follows, was
made to the combined data set shown in fig. 3a.  A
scaling factor was introduced for the present data only,
and was adjusted in the fitting procedure so as to equate
the Legendre polynomial coefficient A$_{0}$ to
the value 22.9 $\mu$b/sr [11,21], which
corresponds to a total photodisintegration cross section
$\sigma_{\rm T}$ = 4$\pi{\rm A}_{0}$ =  287 $\mu$b.  The
present data are normalised, in other words, to a total
photodisintegration cross section of 287 $\mu$b.  This
value, based on the Bonn r-space potential [11,21], is
consistent, within 2\%, with the cross section calculated
from the Paris potential [12,21] and with the value
obtained from the global fit of Rossi et al. [5], which is
based on an extensive set of experimental data.  It
should also be noted that the absolute values assumed for
the 0$^{\circ}$ and 180$^{\circ}$ cross sections, which
were not scaled during the fitting, have negligible
influence on the normalisation procedure used.  This is
due to the effect of the $\sin \theta$ factor introduced
in the integration of the differential cross section to
obtain the total cross section.  Fitting was carried out
using the matrix inversion technique [22], weighting
each datum in proportion to the inverse square of its
standard deviation.  The Legendre expansion was limited
to third order because fitting to order 4 led to an
insignificant value for A$_{4}$ (0.2 $\pm$ 1.1) and an
increase in the $\chi^{2}$ per degree of freedom from 2.8
to 3.3.  The F-test, which was used to measure how much
each additional term improved the fit, also indicated
that the A$_{4}$ term should be rejected with high
probability (94\%).
%%%%%%%%%%

%%%%%%%%%%
The Legendre coefficients obtained from the fitting,
together with the uncertainties given by the error
matrix, are listed as set (c) in Table 3.  The
differential cross section calculated from these
coefficients is shown by the dashed curve in fig. 3a.
The solid curve was calculated using coefficients (set
(a) in Table 3) predicted [11,21] by the Bonn potential.
Other coefficients listed in Table 3 are: set (b),
predicted [12,21] by the Paris potential including
meson exchange currents (MEC), isobar configurations
(IC) and relativistic corrections (RC); set (d),
obtained by fitting the present data in combination with
the Louvain data instead of the Mainz data (Table 2); and
sets (e) and (f), calculated from global fits made by
Rossi et al. [5] and De Pascale et al. [8]
respectively, to extensive sets of experimental data. The Legendre
coefficients A$_{1}$--A$_{3}$ from Table 3 are also displayed in
fig. 3b, to facilitate comparisons between them.
%%%%%%%%%%%%

%%%%%%%%%%%%
%\section*{Discussion}

As has been noted before [1,2], for photon energies in
the range of this experiment, the Legendre polynomial
coefficients calculated from the Bonn potential [11,21]
agree well with those calculated using
the Paris potential [12,21].   This is illustrated by the
agreement between sets (a) and (b) in Table 3.  The good
agreement between the A$_{0}$ coefficients (Table 3) of
these sets with those of the global fits (sets (e) and (f))
illustrates the success of the conventional theory in
predicting the total cross section for deuteron
photodisintegration at this energy.  The values of
A$_{0}$ in sets (c) and (d) are fixed to that of set (a) by the
normalisation, as already noted.  The present relative
measurements of d$\sigma$/d$\Omega$ may be compared with
other experimental data taken at the same or nearby
photon energy by comparing ratios of the n $\geq$ 1
Legendre coefficients to A$_{0}$.  Such a comparison shows that
the present data are in good agreement with those of
Weissman and Schultz [23,24] at E$_{\gamma}$ = 35 MeV,
and also with other measurements [25,26] at nearby
energies.  These measurements form part of the data sets
upon which the global fits [5,8] are based and from which
coefficient sets (e) and (f) in Table 3 are derived.
Comparison of then $\geq$ 1 coefficients derived from the present
measurements,
the global fits and theory respectively, show that the present values
(sets (c) and (d)) of A$_{1}$ and A$_{3}$ are consistent
with those obtained from the global fits and are, like these,
slightly and systematically larger in magnitude than the
theoretical predictions. The A$_{2}$ coefficient obtained
from the present analysis (sets (c) and (d)) is sensitive
to the set of 0$^{\circ}$ and 180$^{\circ}$ cross sections used
(Mainz or Louvain). Note that these cross sections are in strong
disagreement with one-another for the value at 0$^{\circ}$ (Table 2).
When the Mainz data are used in the present analysis the value
obtained for
A$_{2}$ (set (c)) is reasonably consistent with the values
obtained from the global fits.

The combined present and Mainz [17,18] data (fig. 3a)
are therefore seen to be in fair agreement with the
differential cross sections predicted [11] from theory.
The experimental data, as represented by the dashed curve
in fig. 3a, display an asymmetry about 90$^{\circ}$
which is slightly greater than that predicted by the theory
(solid curve).  The same effect is manifested by the
systematic deviations between experimental and theoretical values for
the
A$_{1}$ and A$_{3}$ Legendre coefficients (Table 3 and fig. 3b).  The
present
analysis therefore provides some confirmation of this previously
noted [1,2] trend.  However, more accurate measurements
are needed in order to test whether these discrepancies
in A$_{1}$ and A$_{3}$ are as large (several standard deviations) as
indicated
by comparing the global fit of Rossi et al. [5] with the theoretical
predictions (Table 3 and fig. 3b). Calculations based on the Paris
potential
[12] show that, at this energy, inclusion of MEC/IC/RC
increases the magnitude of the coefficient A$_{2}$ by $<$6\% and has
an even
smaller effect ($<$1\%) on A$_{1}$ and A$_{3}$. The theoretical
predictions
are thus very insensitive to the MEC/IC/RC corrections, especially
for the
coefficients A$_{1}$ and A$_{3}$, hence it seems very unlikely that
these
corrections will prove to be the source of any discrepancies suggested
(fig. 3b) by the data now available.


We wish to thank: the Foundation for Research Development
for financial support; Dr D.Reitmann and staff of the NAC
for their cooperation in carrying out the experiments; Mr
P.A.Back and Mr D.Momsen for constructing equipment
used; and Drs H.Arenh\"{o}vel, P.Leleux and A.Zuchiatti
for valuable discussions and for providing us with data and
computer codes.
%
\vspace{.5cm}

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\subsubsection*{Figure Captions}
\begin{description}
\item[Fig. 1]  Experimental arrangement, schematic and not to scale.

\item[Fig. 2]  (a) Counts (vertical) versus NE213 pulse
height L and pulse shape S, for coincidences between the NE213 and
the NaI(Tl) crystal positioned at $\theta_{{\rm n}\gamma}$(lab) =
90$^{\circ}$,
showing ridges due to protons (p), deuterons (d) and alphas
($\alpha$) detected
in the NE213.  The sharp peak on the deuteron ridge is the signature
for n-p
radiative capture. (b) Projected NE213 pulse height spectra for
coincidences
identified as d in NE213 and $\gamma$ in NaI(Tl).  The spectra
associated with different n-$\gamma$ laboratory angles are offset by
200 counts.

\item[Fig. 3]  (a) Differential cross section and
               (b) Legendre polynomial coefficients, for protons
emitted from
    deuteron photodisintegration at E$_{\gamma}$(lab) = 33.9 MeV.
    In (a), data shown are from the present work (open circles)
    and from Mainz [17,18] (solid circles), the dashed curve
    shows a weighted least squares Legendre polynomial fit
    (see text) to the data and the solid curve shows the
    differential cross section predicted by the Bonn potential
    [11]. Panel (b) displays the Legendre polynomial coefficients
    A$_{1}$--A$_{3}$ listed in Table 3 showing: set (a), Bonn
potential
    (solid lines); set (b), Paris potential (dashed lines); set (c),
    present and Mainz data (solid circles); set (d), present and
    Louvain data (open circles); and values obtained from global fits,
    set (e), Rossi et al. [5] (squares) and set (f), De Pascale et al.
    [8] (triangles).  The uncertainties for set (e) (squares)
    are smaller than the size of the points.
\end{description}
\end{document}


