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\begin{document}
\twocolumn[
\begin{center}
\mathversion{bold}
{\Large\bf Isoscalar and Isovector Spin-M1 Strengths in $^{11}$B}\\
\mathversion{normal}
\vspace{3mm}
{ \large
T.~Kawabata,
H.~Akimune$^{\mbox{a}}$,
H.~Fujimura$^{\mbox{b}}$,
H.~Fujita$^{\mbox{b}}$,
Y.~Fujita$^{\mbox{c}}$,
M.~Fujiwara$^{\mbox{b}}$,
K.~Hara$^{\mbox{b}}$,
K.~Y.~Hara$^{\mbox{a}}$,
K.~Hatanaka$^{\mbox{b}}$,
T.~Ishikawa$^{\mbox{d}}$,
M.~Itoh$^{\mbox{b}}$,
J.~Kamiya$^{\mbox{b}}$,
S.~Kishi$^{\mbox{d}}$,
M.~Nakamura$^{\mbox{d}}$,
K.~Nakanishi$^{\mbox{b}}$,
T.~Noro$^{\mbox{e}}$,
H.~Sakaguchi$^{\mbox{d}}$,
Y.~Shimbara$^{\mbox{b}}$,
H.~Takeda$^{\mbox{d}}$,
A.~Tamii$^{\mbox{b}}$,
H.~Toyokawa$^{\mbox{b}}$,
S.~Terashima$^{\mbox{d}}$,
M.~Uchida$^{\mbox{b}}$,
H.~Ueno$^{\mbox{f}}$,
T.~Wakasa$^{\mbox{e}}$,
Y.~Yasuda$^{\mbox{d}}$,
H.~P.~Yoshida$^{\mbox{b}}$
and M.~Yosoi$^{\mbox{d}}$\\
\vspace{2mm}
\it
\normalsize
Center for Nuclear Study, Graduate School of Science, University
of Tokyo \\
{$^{\mbox{a}}$}Department of Physics, Konan University\\
{$^{\mbox{b}}$}Research Center for Nuclear Physics, Osaka University\\
{$^{\mbox{c}}$}Department of Physics, Osaka University\\
{$^{\mbox{d}}$}Department of Physics, Kyoto University\\
{$^{\mbox{e}}$}Department of Physics, Kyushu University\\
{$^{\mbox{f}}$}RIKEN (The Institute of Physical and Chemical Research)\\
}
\end{center}
]
\vspace{3mm}
\noindent
%
The M1 transition strengths provide important information on the nuclear
structure because they could be a good measure to test theoretical
nuclear models. Recently, the M1 transition strengths are of interest
from a view of not only the nuclear physics but also neutrino astrophysics
because the spin part of the M1 operator is identical with the relevant
operators mediate neutrino induced reactions.
Raghavan {\it et al.} pointed out that the $^{11}$B nucleus
can be used as a possible neutrino detector to investigate stellar
processes \cite{a00_RAGH86}. High-energy neutrinos from the stellar
processes like the proton-proton fusion chain in the sun and the
supernova explosions excite
low-lying states in $^{11}$B and $^{11}$C by M1 and Gamow-Teller (GT)
transitions via the neutral-current (NC) and charged-current (CC)
processes, respectively.
Such neutrinos can be detected by measuring emitted electrons from the
CC reaction and $\gamma$ rays from the de-excitations of the low-lying
states. Since there is an isospin symmetrical relation between the
$^{11}$B and $^{11}$C and both the NC and CC reactions can be measured
simultaneously in one experimental setup, the systematic uncertainty in
measuring a ratio of the electron-neutrino flux to the entire neutrino
flux is expected to be small. Since the isospin of the ground state of
$^{11}$B is $T=1/2$, low-lying states in $^{11}$B are excited by both
the isovector and isoscalar transitions.
Therefore, both the isoscalar and isovector spin-M1
strengths are needed for estimating
the CC and NC cross sections.
The cross sections of hadronic reactions
provide a good measure for the weak interaction response since the
relevant operators in the hadronic reactions are identical with those in
$\beta$-decay and neutrino capture processes. Thus, we recently measured
cross sections for the $^{11}$B($^3$He,$t$) and $^{11}$B($d$,$d'$)
reactions to determine the isovector and isoscalar spin-M1 strengths in
$^{11}$B.
The experiment was performed at Research Center for Nuclear Physics,
Osaka University using 450-MeV $^3$He and 200-MeV deuteron beams. The
measured cross sections were shown in Figs.~\ref{a00_fig:het} and
\ref{a00_fig:b11dd}.
Since the ground state of $^{11}$B has non-zero spin, the cross sections
for the $^{11}$B($^3$He,$t$) and $^{11}$B($d$,$d'$) reactions are
described by an incoherent sum over the cross section of the different
multipole contributions,
$$
\frac{d\sigma}{d\Omega}=\sum_{{\Delta}J}\frac{d\sigma}{d\Omega}({\Delta}J).
$$
In order to determine the spin-M1 strengths, the cross
section for each
${\Delta}J$ transition must be given to extract the ${\Delta}J=1$
contribution.
\begin{figure}[ht]
\centering
\resizebox{6.0cm}{!}{\includegraphics{./a00_het.eps}}
\vspace{-3mm}
\caption{\label{a00_fig:het}
Cross sections for the $^{11}$B($^3$He,$t$) reactions compared with
the DWIA calculation.
The dash-dotted, dashed, and dotted curves show
${\Delta}J=0$, ${\Delta}J=1$ and ${\Delta}J\ge2$ contributions,
respectively. The solid curves are sums of the all multipole
contributions.
}
\end{figure}
\begin{table}
{\tabcolsep=1.5mm
\begin{tabular}{cccc}
\hline\hline
$E_x$ & $J^\pi$ & \multicolumn{2}{c}{$B({\rm GT})$} \\
(MeV) & & Present & ($p$,$n$) \\
\hline
0.00 & $3/2^-$ & \multicolumn{2}{c}{$0.345\pm0.008$} \\
2.00 & $1/2^-$ & $0.402\pm0.031$ & $0.399\pm0.032$ \\
\begin{tabular}{c} 4.32 \\ 4.80 \\ \end{tabular}&
\begin{tabular}{c} $5/2^-$ \\ $3/2^-$ \\ \end{tabular}&
\begin{tabular}{c} $0.454\pm0.026$ \\ $0.480\pm0.031$ \\ \end{tabular}&
$\Bigr\}$\hspace{4pt} $0.961\pm0.060$ \hspace{6pt}\mbox{ } \\
\begin{tabular}{c} 8.10 \\ 8.42 \\ \end{tabular}&
\begin{tabular}{c} $3/2^-$ \\ $5/2^-$ \\ \end{tabular}&
\begin{tabular}{c} $\le 0.003$ \\ $0.406\pm0.038$ \\ \end{tabular}&
$\Bigr\}$\hspace{4pt} $0.444\pm0.010$ \hspace{6pt}\mbox{ }\\
\hline\hline
\end{tabular}
\caption{\label{a00_tab:bgt}
Measured $B({\rm GT})$ values compared with the ($p$,$n$) result \cite{a00_TADD90}.}
}
\end{table}
For the $^{11}$B($^{3}$He,$t$) analysis, the cross
section for the each ${\Delta}J$ transition was calculated by the
distorted wave impulse approximation (DWIA) as seen in
Fig.~\ref{a00_fig:het}. Since the GT strength $B({\rm GT})$ for the ground-state
transition
is known to be $0.345\pm0.008$ from the $\beta$-decay strength, the
cross sections for the ${\Delta}J=1$ transitions to the excited states
in $^{11}$C can be related to the $B({\rm GT})$ values by assuming the
linear proportional relation.
The obtained $B({\rm GT})$ values are compared with
the previous ($p$,$n$) result \cite{a00_TADD90} in Table~\ref{a00_tab:bgt}.
The present results are consistent with the ($p$,$n$) result although
several states are not separately resolved due to the poor energy
resolution in the ($p$,$n$) measurement.
Assuming isospin symmetry is conserved,
the GT strengths are easily related to the isovector spin-M1 strength
$B(\sigma\tau_z)$,
$$
\frac{B({\rm GT})}{B(\sigma\tau_z)}=\frac{8\pi}{3}
\frac{{\langle}T_i,T_{iz},1,\pm1|T_f,T_{fz}\rangle^2}
{{\langle}T_i,T_{iz},1,0|T_f,T_{fz}\rangle^2}.
$$
Although the isospin-symmetry breaking changes this ratio, the
variation is usually small. Therefore, the GT strengths obtained from
the charge exchange reaction are still useful to study the isovector
spin-M1 strengths.
\begin{figure}
\centering
\resizebox{6.0cm}{!}{\includegraphics{./a00_b11dd.eps}}
\vspace{-3mm}
\caption{
Cross sections for the $^{11}$B($d$,$d'$) reactions.
The dashed, dotted, and dash-dotted curves show ${\Delta}J=0$,
1 and 2 contributions, respectively. The
solid curves are sums of the all multipole contributions.
}
\label{a00_fig:b11dd}
\end{figure}
\begin{figure}[ht]
\begin{center}
\centering
\resizebox{7.0cm}{!}{\includegraphics{./a00_smexp.eps}}
\vspace{-3mm}
\caption{Measured $B({\rm GT})$ ($B(\sigma\tau_z)$) and $B(\sigma)$ values
are compared with the shell model predictions using the Cohen-Kurath
wave functions \cite{a00_COHE65}. The open bar in the right-upper panel
shows the $B(\sigma)$ value for the 4.44-MeV state estimated from
$B({\rm GT})$ (see text).}
\label{a00_fig:smexp}
\end{center}
\end{figure}
For the $^{11}$B($d$,$d'$) analysis, the cross
section for each ${\Delta}J$ transition was determined from the
$^{12}$C($d$,$d'$) reaction. Since the
ground state of $^{12}$C has a zero spin, transitions to the discrete
states in $^{12}$C are expected to be good references for the angular
dependence of the cross sections for certain ${\Delta}J$ transitions.
As shown in Fig.~\ref{a00_fig:b11dd}, the cross section for the
$^{11}$B($d$,$d'$) reaction was successfully decomposed into the each
${\Delta}J$ contributions.
Although the 4.44-MeV ($5/2^-_1$) state can be excited by both the
${\Delta}J^\pi=1^+$ and $2^+$ transitions, the main part of the
transition is due to ${\Delta}J^\pi=2^+$.
This result is explained by the fact that the strong coupling between
the ground and 4.44-MeV states is expected since the 4.44-MeV state is
considered to be a member of the ground-state rotational band.
Since the observed ${\Delta}J^\pi=2^+$ transition strength is
much larger than the expected ${\Delta}J^\pi=1^+$ strength, the
${\Delta}J^\pi=1^+$ component of the transition strength can not be
reliably extracted for the 4.44-MeV state. The transition strength for
the 6.74-MeV ($7/2^-_1$) state is also dominated by the
${\Delta}J^\pi=2^+$ component, but the ${\Delta}J^\pi=1$ transition
to this state is not allowed.
The isoscalar spin-M1 strength $B(\sigma)$ for the
transition to the 2.12-MeV ($1/2^-_1$) state is deduced to be
$0.037\pm0.008$ from the $\gamma$-decay widths of the
mirror states and the
$B({\rm GT})$ value \cite{a00_BERN92}. Using this value, the cross section for
the ${\Delta}J=1$ transitions to the other excited states can be related
to the $B(\sigma)$ values.
Since the ${\Delta}J=1$ cross section for the 4.44-MeV state was
not reliably obtained in the ($d$,$d'$) analysis, the isoscalar
spin-M1 strength was determined from the measured $B({\rm GT})$ value
and the relative strength of the isoscalar
transition to the isovector transition calculated by using the
Cohen-Kurath wave functions (CKWF) \cite{a00_COHE65}.
The obtained $B({\rm GT})$ ($B(\sigma\tau_z)$) and $B(\sigma)$ values are
compared with the shell model predictions using the CKWFs in
Fig.~\ref{a00_fig:smexp}. The CKWFs reasonably explain the
experimental result except the quenching by a factor of 0.5-0.7.
The present result will be useful in the measurement of the
stellar neutrinos using the NC and CC reactions on $^{11}$B.
\bigskip
\bigskip
\begin{thebibliography}{99}
\bibitem{a00_RAGH86}R.S.~Raghavan, Sandip Pakvasa and B.A.~Brown,
Phys. Rev. Lett. {\bf 57} (1986) 1801.
\bibitem{a00_TADD90}T.N.~Taddeucci {\it et al.},
Phys. Rev. C {\bf 42} (1990) 935.
\bibitem{a00_BERN92}
J.~Bernab\'{e}u, T.~E.~O.~Ericson, E.~Hern\'{a}ndez and J. Ros
Nucl. Phys. B {\bf 378} (1992) 131.
\bibitem{a00_COHE65}S.~Cohen and D.~Kurath, Nucl. Phys. {\bf 73} (1965) 1.
\end{thebibliography}
\end{document}
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