Hyperfine anomaly in Be isotopes and the neutron spatial distribution; a three-cluster model for $^9$Be.
Y. L. Parfenova
The study of the hyperfine (hfs) anomaly in neutron rich nuclei can give a very specific and unique way to study the neutron distribution and the clustering structure. We study the sensitivity of the hfs anomaly to the clustering effects in the $^9$Be isotope using two different nuclear wave functions obtained in the three-cluster ($\alpha+\alpha+n$) model. %of $^{9,11}$Be. The results are compared to those obtained for $^{9,11}$Be in a two-body core + neutron model to examine whether the hfs anomaly is sensitive to a halo structure in $^{11}$Be.
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\begin{document}

\preprint{APS/123-QED}

\title{Hyperfine anomaly in Be isotopes and the neutron
spatial distribution; a three-cluster model for $^9$Be.}% Force line breaks with \\

\author{Y. L. Parfenova}
\email{Yulia.Parfenova@ulb.ac.be}
\altaffiliation[Also at ]{ Skobeltsyn Institute of Nuclear Physics, Moscow State
University, 119992 Moscow, Russia.}%Lines break automatically or can be forced with \\
%\author{P. Descouvemont}%
%\email{pdesc@ulb.ac.be}
\affiliation{%
Physique Nucl\'eaire Th\'eorique et 
Physique Math\'ematique, CP229, Universit\'e Libre de Bruxelles
B 1050 Brussels, Belgium.}
%\author{L. V. Grigorenko}%
% \email{cwillain@ulb.ac.be}
%\affiliation{%
%Flerov Laboratory of Nuclear Reactions, JINR, RU-141980 Dubna, Russia.}
\author{Ch. Leclercq-Willain}
\email{cwillain@ulb.ac.be}
\affiliation{%
Physique Nucl\'eaire Th\'eorique et 
Physique Math\'ematique, CP229, Universit\'e Libre de Bruxelles
B 1050 Brussels, Belgium.}
%This line break forced with \textbackslash\textbackslash%

%\author{Charlie Author}
% \homepage{http://www.Second.institution.edu/~Charlie.Author}
%\affiliation{
%Second institution and/or address\\
%This line break forced% with \\
%}%

\date{\today}% It is always \today, today,
             %  but any date may be explicitly specified

\begin{abstract}
The study of the hyperfine (hfs) anomaly in neutron rich nuclei 
can give a very specific and unique way to study the neutron
distribution and the clustering structure. We study the 
sensitivity of the hfs anomaly to the clustering effects 
in the $^9$Be isotope using two
different nuclear wave functions obtained in the three-cluster 
($\alpha+\alpha+n$) model. 
%of $^{9,11}$Be.
The results are compared to those obtained for $^{9,11}$Be in 
a two-body 
core + neutron model to examine whether the hfs anomaly 
is sensitive to a halo structure in $^{11}$Be.
\end{abstract}


\pacs{32.10.Fn; 21.10.Gv; 21.60.Gx; 21.10.Ky }% PACS, the Physics and Astronomy
                             % Classification Scheme.
\keywords{hyperfine structure anomaly, neutron distribution, cluster model}%Use showkeys class option if keyword
                              %display desired
\maketitle



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction \label{sec:s1}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


The study of the hyperfine (hfs) anomaly in neutron rich nuclei,
in particular, those with loosely bound neutrons
can give a very specific 
and unique way to measure the neutron distribution. In a previous paper \cite{Par05} we have obtained the values of
the hfs anomaly calculated in a two-body core + neutron model 
for the $^{9,11}$Be isotopes.
The hfs anomaly $\epsilon$ is defined as the sum of the hfs anomaly 
related to the Bohr-Weisskopf effect ($\epsilon_{BW}$)\cite{Boh50}
%, in the $^{9,11}$Be isotopes in the core+neutron model.
and of the Breit-Rosenthal-Crawford-Shawlow (BRCS) correction
$\delta$ \cite{Ros72}, $\epsilon=\epsilon_{BW}+\delta$. 

It was found in Ref.\cite{Par05} that in  
Be isotopes, the 
value $\epsilon_{BW}$ is comparable to the BRCS correction 
$\delta$. The value $\epsilon_{BW}$ is very sensitive to
the weights of the partial states in the ground 
state wave function 
and might vary within 50\% depending on those weights. 
In \cite{Par05}, we found a difference of about 25\% for the total hfs 
anomaly value in the $^{9,11}$Be isotopes.

%In the core+neutron model \cite{Fuj99}, the weights of the 
%admixed states 
%are free parameters, which might be found in the shell 
%model or cluster model calculations. 
%A cluster model of the Be isotope is rather good for light 
%nuclei where 
%clusterization effects can be essential. 
In the present paper, we calculate the hfs anomaly in  
a three-cluster model of $^9$Be and study the sensitivity of 
the hfs anomaly to the clusterization effects. The calculations are performed with two different three-body wave 
functions \cite{Priv05,Gri02}. These wave functions differ from each other by the choice
of the cluster-cluster interaction potentials used in their calculations.
They are characterized by the same partial states but contributing with different weights.
They both reproduce the $^{9}$Be 
rms radius and the three-cluster ($\alpha$+$\alpha$+$n$) dissociation energy
(1.573 MeV).
We compare the results obtained with these two wave functions, and 
also compare the present results with the core + neutron model
calculations \cite{Par05,Fuj99}.

The accuracy of the description of the $^{9}$Be ground state wave function can be explored
by calculating the magnetic dipole and electric quadrupole moments which are
determined by the weights and quantum numbers of the states. 
The $^{9}$Be magnetic dipole moment is independent 
of the radial behavior of the wave function but this is not 
the case for the electric quadrupole  moment.
Calculation of both the magnetic dipole and electric quadrupole moments thus provides a 
precise test of the accuracy of all aspects of the ground state wave function (for the 
experimental data on the magnetic dipole moment in the Be isotopes, we
refer to Refs. \cite{Web76}, \cite{Dic49}; no experimental data exists on the hyperfine anomaly 
for these nuclei). We also compare the results for $^{9}$Be$^+$ to that for
$^{11}$Be$^+$, to investigate the sensitivity of the hfs anomaly to diffuse (halo)neutron structures.
 

% The value of the hfs anomaly $\epsilon $
%defined by the folding of the electronic and nuclear wave functions is
%rather sensitive to the spatial distribution of the valence neutron wave
%function and to the weights of the possible mixed states in the ground state
%description.   

%The relativistic electronic wave functions for the $2s$ electron 
%in Be$^+$ atom are obtained taking into account the screening of 
%the nuclear Coulomb potential by two $1s^2$ electrons in the closed shell. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Three-body wave function \label{sec:s3}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The wave 
function $\Psi _{JM}^{3b}$ of the fragment relative motion  
in the three-cluster model ($\alpha +\alpha +n$) of $^{9}$Be 
is described with the Jacobi coordinates in the 
method of hyperspherical harmonics \cite{Zhu93,Efr72}. 
%In the formalism based 
%on the hyperspherical harmonics, the three-body 
%wave function 
It is written as
\[
\Psi _{JM}^{3b}({\bf \xi_1},{\bf \xi_2})=
\sum \limits_{KLSS_x l_{1}l_{2}M_{L}} 
[\digamma _{LM_{L}}^{KSS_{x}l_{1}l_{2}} \otimes 
\chi_{SM-M_{L}}^{s_1(s_2s_3)S_x}]^{JM},
\]
where 
$\digamma _{LM_{L}}^{KSS_{x}l_{1}l_{2}}$ is the "active part" 
of the three-body wave function carrying the total orbital angular 
momentum $L$ with the projection $M_L$. $\chi _{SM_{S}}$ is 
the total spin function of the whole system with the total spin $S$ and 
projection $M_{S}$ (here restricted to the neutron spin function).
 The wave function 
$\digamma _{LM_{L}}^{KSS_{x}l_{1}l_{2}}(\xi_1 ,\xi_2 )$ depends 
on the relative coordinates (hyperradius and relative angles) 
and the neutron spin 
\[
\digamma _{LM_{L}}^{KSS_{x}l_{1}l_{2}}(\xi_1 ,\xi_2 )=\rho ^{-5/2} \Re
_{Kl_{1}l_{2}}^{LSS_x}(\rho )\Im _{KLM_{L}}^{l_{1}l_{2}}(\Omega _{5})  , 
\]
where $l_{1}$ and $l_{2}$ are the angular momenta of the relative 
motion corresponding to the $\xi_1$ and $\xi_2$ coordinates. 
% $m_{l_{1}}$ and $m_{l_{2}}$ are their projections, respectively. 
The sum $\vec{L}=\vec{l}_{1}+\vec{l}_{2}$ gives
the total angular momentum; $K$ is the hypermomentum. 

The hyperradius $\rho ^{2}=\xi_1^{2}+\xi_2^{2}$ is a collective 
rotationally and permutationally invariant variable, $\xi_1$ and 
$\xi_2$ are the translationally invariant normalized sets of 
Jacobi coordinates.
% which are conveniently introduced as
%\begin{eqnarray}
%\xi_1 &=& \rho \sin \theta, \label{coorch} \\
%\xi_2 &=&\rho \cos \theta, \nonumber
%\end{eqnarray}
We choose the Jacobi coordinates as follows:
%$y=\xi _{1}$ and $x=\xi _{2}$ defines a
%phase shift in the three-body wave function. Here and below the formulas
%are given for $x = \rho \sin \theta$ and $y = \rho \cos \theta$.
%The Jacobi coordinates are
\begin{eqnarray}
{\bf x} &=&(A_{23})^{1/2}{\bf{r_{23}}}, \label{jacxy} \\
{\bf y} &=&(A_{1(23)})^{1/2}{\bf{r_{1(23)}}}, \nonumber
\end{eqnarray}
where
${\bf{r_{23}}}$ is the relative coordinate of fragments 2 and 3, 
and ${\bf{r_{1(23)}}}$ is the coordinate of fragment 1 relative 
to the center of mass of the fragments 2 and 3. $A_{23}$ is the 
reduced mass number for the pair (2,3),
%$A_{23}=\frac{A_2A_3}{A_2+A_3}$. $S_x$ is the spin of the
similarly, $A_{1(23)}$ is the reduced mass of the fragment 1 with respect to the mass of the 
subsystem $(2,3)$.
%$A_{1(23)}=\frac{A_1(A_2+A_3)}{A}$ where $A=A_1+A_2+A_3$.
 
%For $^{9}$Be, the T-basis correspond to the choice neutron as
%the fragment 1. The spin of the fragments 2 and 3 we denote
%as ${\bf S}_x={\bf s}_2+{\bf s}_3$.

The five hyperspherical polar angles are 
$\Omega _{5}=\{\theta ,\hat{x},\hat{y}\}$ where 
$\theta$ is defined by the relations
\begin{eqnarray}
\xi_1 \equiv{x} &=&\rho \sin \theta, \label{coorch} \\
\xi_2 \equiv{y} &=&\rho \cos \theta. \nonumber
\end{eqnarray}
% =\;\arccos (y/\rho )$.
The choice of three different systems of Jacobi coordinates 
leads to three-body wave functions with different phase factor, 
the three different Jacobi coordinates systems being connected 
together by defined rotations. 
For $^{9}$Be, the T-basis correspond to choose the neutron as
fragment 1, the two $\alpha$-particles as fragments 2 and 3. 
The Y-basis associates the fragment 1 with one of the 
$\alpha$-particles.
%The spin of the fragments 2 and 3 we denote
%as ${\bf S}_x={\bf s}_2+{\bf s}_3$.


The values of the hypermomentum are $K=l_{x}+l_{y}+2n $ ($n=1,2,...$).
The hyperspherical harmonics have the form
\[
\Im _{KLM_{L}}^{l_{x}l_{y}}(\Omega _{5})=\psi _{K}^{l_{x}l_{y}}(\theta
)[Y_{l_{x}}(\hat{x}) \otimes Y_{l_{y}}(\hat{y})]_{LM_{L}}, 
\]
where 
\[
\psi _{K}^{l_{x}l_{y}}(\theta )=N_{K}^{l_{x}l_{y}}(\sin \theta
)^{l_{x}}(\cos \theta )^{l_{y}}P_{n}^{l_{x}+1/2,l_{y}+1/2}(\cos 2\theta ) 
\]
and $P_{n}^{\alpha ,\beta }$ is a Jacobi polynomial. 
$N_{K}^{l_{x}l_{y}}$ is the coefficient of normalization
\[
N_{K}^{l_{x}l_{y}}=\sqrt{\frac{2(n!)(K+2)(n+l_{x}+l_{y}+1)!}{\Gamma
(n+l_{x}+3/2)\Gamma (n+l_{y}+3/2)}} 
\]
and the normalization condition for the function 
$\psi _{K}^{l_{x}l_{y}}(\theta) $ is 
\[
\int\limits_{0}^{\pi /2}\psi _{K}^{l_{x}l_{y}}(\theta )\psi _{K^{\prime
}}^{l_{x}^{\prime }l_{y}^{\prime }}(\theta )\sin ^{2}\theta \cos ^{2}\theta
d\theta =\delta _{KK^{\prime }}\delta _{l_{x}l_{x}^{\prime }}\delta
_{l_{y}l_{y}^{\prime }} . 
\]

The charge density distribution of the three-body nucleus 
entering the calculation of the electronic wave functions 
can be obtained as 
\begin{equation}
\rho(\bf r)=\rho _{0}<\Psi _{JM}^{3b}\left| 
\sum \limits_{i} Z_{i}\rho _{i}({\bf r},{\bf x},{\bf y})\right| 
\Psi _{JM}^{3b}> , \label{rho}
\end{equation}
where $\rho_{0}$ is a normalization factor. In the case of
$^{9}$Be, the $\alpha $-particle density distribution
is approximated by a sum of 
Gaussians, with the parameters taken from \cite{deV87} and giving a charge radius
equal to 1.676 fm.
 

In the proceeding calculations two differing evaluations of $\Psi _{JM}^{3b}$ \cite{Priv05,Gri02} as noted in Section \ref{sec:s1} are used. 
%obtained in different approaches \cite{Priv05,Gri02}. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Magnetic hyperfine structure \label{HFSsec}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Here, we briefly mention the main points of the formalism
for the hfs anomaly calculations. 
For more details we refer to \cite{Par05} and references therein.

The magnetic hyperfine interaction Hamiltonian is defined by 
\begin{equation}
\mathcal{H}=-\int \bf{\mathcal{J}}(\mathbf{r})\cdot \mathbf{A}(\mathbf{r})\,d^{3}r,
\end{equation}
where $\bf{\mathcal{J}}$ is the nuclear current density and 
$\mathbf{A}$ is the
vector potential created by the atomic electrons.

The hyperfine interaction couples the electronic angular momentum 
$\mathbf{J}$ and the nuclear one $\mathbf{I}$ to a hyperfine 
momentum $\mathbf{F} = \mathbf{J} + \mathbf{I}$. The magnetic 
hyperfine splitting energy $W$ for a state $\mid I J F M_F=F >$ 
is defined as the matrix element of the Hamiltonian $\mathcal{H}$.

The functions $F^{\kappa J}$, $G^{\kappa J}$ entering the matrix
element are the radial parts
of the large and small components of the Dirac electronic wave 
function, with the quantum number ${\kappa}=\pm (J+\frac{1}{2})$ for
$J=l_e \mp \frac{1}{2}$ and the orbital angular momentum $l_e$.
%For details of the calculations of the electronic wave function in
%Be isotopes, we refer to \cite{Dav65,Ros61}.
The calculations are preformed with a realistic electronic
wave function (see \cite{Par05}). 

The magnetic dipole contribution to the hyperfine splitting
$W_{(I J)F F}$ has the form
\begin{eqnarray}
W_{(I J)F F } &=& <IJFF|\mathcal{H}|IJFF>    \\
&=& \frac{1}{2}[F(F+1)-I(I+1)-J(J+1)] a_{I}, \nonumber
\end{eqnarray}
where $a_I$ is defined as 
\begin{equation}
a_{I} = -\frac{2e \kappa \mu _{N}}{IJ(J+1)} < I I \mid \sum\limits_{i=1}^{A}
(M_{Z}^{l_i}(\mathbf{r}_i) + M_{Z}^{s_i}(\mathbf{r}_i))\mid I I >    \nonumber
\end{equation}
with the $Z$ components of the magnetic dipole moment 
$\mathbf{M}^{l_i(s_i)}(\mathbf{r}_i)$, related to the
angular momentum $l_i$ and spin $s_i$ of each nucleon; the summation runs over all the nucleons.

For an extended nuclear charge we have
\begin{eqnarray}
\mathbf{M}^{l}(\mathbf{r}_{i}) &=&g_{l}^{i}\mathbf{l}_{i} [\int \limits_{r_{i}}^{%
\infty }F^{\kappa J} G^{\kappa J}\;dr\;+\;\int \limits_{0}^{r_{i}}F^{\kappa
J} G^{\kappa J}\;(\frac{r}{r_{i}})^{3} dr] ,    \nonumber \\
\mathbf{M}^{s}(\mathbf{r}_{i}) &=&g_{s}^{i}\mathbf{s}_{i} \int \limits_{r_{i}}^{%
\infty }F^{\kappa J} G^{\kappa J}dr + \mathbf{D}%
_{i} \int \limits_{0}^{r_{i}}F^{\kappa J} G^{\kappa J}(\frac{r}{r_{i}}%
)^{3} dr ,     \nonumber
\end{eqnarray}
with 
$\mathbf{D}_{i}={-\sqrt{\frac{5}{2}}\;[\mathbf{s}^{1}\otimes C^2(%
\hat{r}_{i})]}^{1}$ and 
$C_q^{k}=\sqrt{\frac{4 \pi}{2k+1}}Y_{kq}(\hat{r}_{i})$.

The quantity $a_I$ can be expressed through the hfs 
constant for a point nucleus $a_I^{(0)}$ as
\begin{equation}
a_I \thickapprox a_I^{(0)} (1+\epsilon_{BW}+\delta)  , \label{eq10}
\end{equation}
%where $\epsilon_{BW}$ is defined as the hfs anomaly in the 
%Bohr-Weisskopf effect \cite{Boh50} and $\delta$ is the BRCS
%correction \cite{Ros72}. 
%$a_I^{(0)}$ is the hfs constant for the point nucleus
with 
\begin{equation}
a_I^{(0)}= -\frac{2 e \kappa \;\mu _{N} \mu}{IJ(J+1)} 
\int\limits_{0}^{\infty} F_0^{\kappa J}(r) G_0^{\kappa J}(r)dr .   
\end{equation}
Here, $\mu =< I I \mid \sum\limits%
_{i=1}^{A}(g_s^i s_i + g_l^i l_i) \mid I I >$ 
defines the magnetic dipole moment of the point nucleus in nuclear
magneton units $\mu _{N}$. The functions $F_0^{\kappa J}$, 
$G_0^{\kappa J}$ are the radial parts of the Dirac electronic
wave function in the point nucleus approximation.

The hfs anomaly in the Bohr-Weisskopf effect is 
\begin{eqnarray}
\epsilon_{BW} & = & - \frac{b}{\mu}
\sum\limits_{i=1}^3 \left[ \sum\limits_{j=1}^{n_i} 
\left\{ <II| (g_s^{(j)} s_j + g_l^{(j)} l_j) K^a(r_j)|II> \right.\right.\nonumber \\
&-&  \left.\left.<II| (g_l^{(j)} l_j + D_j) K^b(r_j) |II> \right\} \right] \label{EBW},      
\end{eqnarray}
where $b=[\int_0^{\infty} F_0^{\kappa J}G_0^{\kappa J}d r]^{-1}$,
and
\begin{eqnarray}
K^a(r_j) &=& \int\limits_{0}^{r_j}F^{\kappa J}G^{\kappa J}d r  ,      \label{elwfinta} \\
K^b(r_j) &=& \int\limits_{0}^{r_j}F^{\kappa J}G^{\kappa J} (\frac{r}{r_j})^3 d r. \label{elwfintb}
\end{eqnarray}

The index $j$ is related to the $A$ nucleons; $i(1-3)$ denotes one of the 
three-clusters of $n_i$ nucleons. $g_l^{(j)}$ and $g_s^{(j)}$ 
are the gyromagnetic ratios of the $j$-th nucleon orbital motion 
and spin, respectively.

The hfs anomaly can be approximated by
\begin{eqnarray}
\epsilon_{BW}&=&- \frac{b}{\mu} 
\sum\limits_{i=1}^{3}\left[\left\langle II| (g_s^{(i)} s_i + g_l^{(i)}l_i )
K^a (r_i ) |II\right\rangle \right.   \nonumber \\
&-& \left. \left\langle II |( g_{l}^{(i)}l_i  + D_i )K^b(r_i )|II \right\rangle
\right].    \label{anom1}
\end{eqnarray}

The index $i$ is related to the three-clusters of relative 
coordinate $r_i$ and angular momentum $l_i$ in respect to the 
center of mass system, and with the appropriated expressions 
$g_s^{(i)} , g_l^{(i)}$ and $D_i$, calculated for each cluster.
%where $c_i=\frac{A-A_i}{A}$ takes into account the motion
%of the center of mass, $r$ and $R$ are the electronic coordinate
%and the $i$-th fragment coordinate, respectively.
%$g_l^{(i)}$ and $g_s^{(i)}$ are the gyromagnetic ratios of 
%the $i$-th nucleon orbital motion and spin, respectively.
The BRCS correction is defined as $\delta=1-bK^a(\infty)$ 
\cite{Ros72}. This term is defined by the nuclear charge distribution and information on the neutron distribution is contained solely in $\epsilon_{BW}$.
%Thus, the sensitivity of the hfs anomaly $\epsilon$ to
%the neutron distribution depends 
%on the ratio of $\epsilon_{BW}$ and $\delta$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Hfs anomaly and nuclear moments of $^9$Be in the cluster 
model \label{sec:s4}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

To calculate the magnetic dipole moment and the hfs anomaly, 
we use three systems of coordinates related to each other by 
rotation: the T-basis with the neutron as cluster $i=1$, and the 
$Y(q)$ systems ($q=1$ or $2$) with one of the $\alpha$-particles 
as cluster $i=2$ or $3$.
The index $q=1$ and $q=2$ define respectively the rotations 
$1(23)\rightarrow 2(31)$ and $1(23)\rightarrow 3(12)$ between 
the T and Y-basis. 

%For calculations of the magnetic dipole  moment of the $^{9}$Be ground state we use 
The transformation of the hyperspherical 
harmonic function \cite{Efr72} from the T-basis to the 
Y-basis is defined by 
\begin{equation}
\Im _{KLM_{L}}^{l_{x}l_{y}}(\Omega _{5})
=\sum\limits_{l_{x}^{\prime }l_{y}^{\prime }}
\left\langle l_{x}^{\prime }l_{y}^{\prime }
|l_{x}l_{y}\right\rangle_{KL}^{q}\Im _{KLM_{L}}^{l_{x}^{\prime }l_{y}^{\prime }}
(\Omega _{5}^{\prime }) ,
\label{transL}
\end{equation}
where $\left\langle l_{x}^{\prime }l_{y}^{\prime }
|l_{x}l_{y}\right\rangle_{KL}^{q}$ are the
Raynal-Revai coefficients \cite{Efr72}. 
%The index $q$ defines the rotation between the T and Y(i)coordinates systems:$q$=1 corresponds to the rotation $1(23)\rightarrow 2(31)$ and 
%$q$ =2 to the rotation $1(23)\rightarrow 3(12)$.

The transformation of the spin part of the wave function 
is written as
\begin{equation}
\chi _{SM_{S}}^{s_1(s_2s_3)S_x}
=\sum\limits_{S_{x}^{\prime }}\left\langle S_{x}^{\prime
}|S_{x}\right\rangle _{S}^{q}
\chi _{SM_{S}}^{s_1^{\prime} (s_2^{\prime}s_3^{\prime})S_x^{\prime}}
\label{transS}
\end{equation}
with $s_1^{\prime}=s_{2(3)}$, $s_2^{\prime}=s_{3(1)}$, and 
$s_3^{\prime}=s_{1(2)}$ for $q=1(2)$,
\begin{eqnarray}
\left\langle S_{x}^{\prime
}|S_{x}\right\rangle _{S}^{q} &=& (-)^{s_{1(2)}+2s_{2(1)}+ s_3 + S_{x}^{\prime }
(S_{x})} \hat{S}_{x}\hat{S}_{x}^{\prime } \nonumber \\
 &\times& \left\{ 
\begin{array}{ccc}
s_{2(3)} & s_{3(2)} & S_{x} \\ 
s_{1} & S & S_{x}^{\prime } 
\end{array} 
\right\} ,
\end{eqnarray}
where ${\bf S=s_2+s_3+s_1}$, ${\bf S_{x}=s_2+s_3}$, and 
$ {\bf S_{x}^{\prime }=s_{3(1)}+s_{1(2)}}$.

%The transformation of coordinates is as follows:
%\begin{eqnarray*}
%x^{\prime } &=&\cos \phi x+\sin \phi y=\sin (\theta +\phi ) \\
%y^{\prime } &=&-\sin \phi x+\cos \phi y=\cos (\theta  +\phi )
%\end{eqnarray*}

To calculate the magnetic dipole moment, we consider the
contribution of each fragment with respect to the rest system. 
In $^9$Be, the neutron spin and the $\alpha $-particle 
orbital motion contribute to the magnetic dipole  moment as 
%The mean value of the spin projection 
%$\left\langle s_i^z \right\rangle$ 
%of the $i$-th fragment ($i=2,3$) is
%\begin{equation}
%\left\langle S_{x}|s_i^z|S_{x} \right\rangle_{S} 
% = \sum\limits_{S_{x}^{\prime }S_{x}^{\prime \prime}}
%
%\left\langle S_{x} |S_{x}^{\prime} \right\rangle_{S}^{q}
%\left\langle S_{x}^{\prime} | s_i^z | S_{x}^{\prime \prime} \right\rangle ^q  
%\left\langle S_{x}^{\prime \prime}
%|S_{x}\right\rangle_{S}^{q}  \label{eq13a} 
%\end{equation}
%
%In the case of $^9$Be, 
%$\left\langle s_i^z \right\rangle 
%=\left\langle s_i^z \right\rangle ^q = m_s $.\\
%
%The mean value of the angular momentum projection
%$\left\langle l_{y(q)}^z \right\rangle$ of the $\alpha $-th fragment 
%($q=2,3$) is 
%\begin{eqnarray}
%\left\langle  l_{y(q)}^z  \right\rangle 
%&=& \sum \limits_{l_{y}l_{y}^{\prime }}
%%
%\left\langle l_{x}l_{y} |l_{x}l_{y}^{\prime} \right\rangle_{KL}^{q} \label{eq13b} \\
%
%& \times & 
%\sum\limits_{l_{x}^{\prime \prime}l_{y}^{\prime \prime}
%l_{y}^{\prime \prime \prime}}
%
%\left\langle l_{x}l_{y}^{\prime} 
%| l_{y(q)}^z | l_{x}l_{y}^{\prime \prime} \right\rangle  
%\left\langle l_{x}l_{y}^{\prime \prime }
%|l_{x}l_{y} \right\rangle_{KL}^{q}  \nonumber
%\end{eqnarray}
%where
%\begin{eqnarray}
%\left\langle l_{x}l_{y}^{\prime}| l_{y(q)}^z | l_{x}l_{y}^{\prime \prime} \right\rangle 
%&=& m_y \delta(l_{y}^{\prime},l_{y}^{\prime \prime})
%\end{eqnarray}
%Whence the magnetic dipole  moment is 
\begin{equation}
\mu =  \sum\limits_{\varsigma}\omega _{\varsigma}^{2}
\left\langle m_{s} \right\rangle g_{s}^{(n)} \label{magmom}
+ \frac{5}{9}\sum\limits_{q=1,2}\sum\limits_{\varsigma^{\prime}} \beta
_{\varsigma^{\prime}}^{2}
\left\langle m_{y}\right\rangle_q g_{l_y^{\prime}}^{(\alpha )}. 
\end{equation}
$\left\langle m_{s} \right\rangle ,\left\langle m_{y}\right\rangle_q$ 
are the expectation values of the spin and angular momentum 
projections of the fragments. $\left\langle m_{s} \right\rangle$ 
is found in the T-basis and associated with the neutron spin; 
$\left\langle m_{y}\right\rangle_q$ is found in the Y-basis 
obtained by the rotation $q=1(2)$ for $i=2(3)$ (see (\ref{transL}) 
and (\ref{transS})) and associated with the $\alpha$-particle 
orbital momentum.
In Eq. (\ref{magmom}) $\omega _{\varsigma}^{2}$  is 
the weight of the partial state in the T-basis for the channel 
with quantum numbers $\varsigma=Ll_{x}l_{y}$; and 
$\beta _{\varsigma \prime}^{2}$ is the weight of the partial 
state obtained with (\ref{transL}) and (\ref{transS})  
%of the $\alpha -(\alpha- n)$ system obtained with (\ref{eq13a}),(\ref{eq13b}) 
for each channel with quantum numbers 
$\varsigma^{\prime}=Ll_{x}^{\prime}l_{y}^{\prime}$ in the 
Y-basis. Here, we mean by channel the set $\varsigma$ of quantum 
numbers characterizing the partial waves contributing to the 
ground state wave function. 
In the case of $^9$Be represented by the system 
$\alpha + \alpha + n $, this set of quantum numbers is $L$, $l_x$, 
and $l_y$ defined in Section \ref{sec:s3}. The nuclear wave function is summed over 
the hypermomentum $K$ which is not included explicitly as the matrix 
element does not depend on it.
 
In Eq. (\ref{magmom}), 
%the mean values $<m_{s}> \, <m_{y}>$ 
%are defined by
%\begin{eqnarray}
%<m_{s}>&=&\sum_{M_{L_{\alpha }},m_{l},m_x,m_y, M_{S}}\left(
%C_{l_{x}m_{x}l_{y}m_{y}}^{L(m_{x}+m_{y})}C_{SM_{S}L(m_{x}+m_{y})}^{\frac{3}{2}\frac{3}{2}}\right) ^{2} \nonumber \\
%<m_{\alpha }>&=&\sum_{ M_{L_{\alpha }},m_{l},m_x, m_y, M_{S}}\left( C_{l_{x}m_{x}l_{y}m_{y}}^{L(m_{x}+m_{y})}C_{SM_{S}L(m_{x}+m_{y})}^{\frac{3}{2}\frac{3}{2}}\right) ^{2}
%\end{eqnarray}
%$l_{y}$, $m_{y}$ ($l_{x}$, $m_{x}$) 
%are the expectation values of the spin and orbital angular momentum 
%projections, $S$, $M_{S}$ are the total spin and its projection. 
%In (\ref{magmom}), 
%$\mu _{n}$
$\mu _{n}=\frac{1}{2}g_{s}^{(n)}$ denotes the magnetic dipole moment of 
the neutron and $\mu _{l^{\prime}_y}^\alpha = 
g_{l^{\prime}_y}^{(\alpha)} m_y = 
2 \frac{A_1+A_2}{AA_3} g_{l_y}^{(p)} m_y $ is the
magnetic dipole moment of the $\alpha $-particle in the state with 
$l_{y}$ as angular momentum in the Y-basis. Note that the 
magnetic dipole moments are obtained using for the $g$ factors 
$g_{s}^{(n)}=-3.8260854(90)$ and $g_{l_y}^{(p)}=1$.
 
%Their values are $\mu _{n}=\frac{1}{2}g_{s}^{(n)}$, 
%and $\mu _{l_y}= \frac{l_{y}}{2}g_{l}^{(p)}$, 
%where $g_{s}$ is the gyromagnetic 
%ratio of the spin of the nucleon
 
%for a neutron and $g_{s}^{(p)}=5.585694772(126)$for a proton), 
%and  are the gyromagnetic factors of a neutron and of an orbiting 
%proton with angular momentum $l$). 
%(for a neutron $g_{l}^{(n)}=0$).
% The gyromagnetic ratio for the $\alpha$-particle is 


The factor $\frac{5}{9}$ in (\ref{magmom}) is
 the center of mass factor $\frac{A-A_{2(3)}}{A}$ in the 
 Y-basis corresponding to the $\alpha +(\alpha + n)$ system.
 
We assume that the $\alpha$-particle angular momentum is defined by two orbiting protons, neglecting 
the spin contribution of the nucleons.
%Using the three-body wave function described in �II, the expression of the hfs anomaly is given by:
%\begin{eqnarray}
%\epsilon_{BW}&=&- \frac{b}{\mu_I} 
%\left[\left\langle II| [(g_s^{(1)} s_1 + g_l^{(1)} c_1 l_y )K^a ( c_1 y ) \right.\right. \nonumber\\
%&-& \left.\left.( g_{l}^{(1)} c_1 l_y  + D_1 )K^b( c_1 y )] |II \right\rangle \right]_T \nonumber \\ 
%&+& \sum\limits_{i=2,3}\left[\sum\limits_{y(i)}\left\langle II_T |II_{y(i)}\right\rangle \right.\nonumber\\
%&\times & \left\langle II_{y(i)}|[( g_s^{(i)} s_i + g_l^{(i)} c_i l_{y(i)} )K^a (c_i y(i) )\right.\nonumber\\ 
%&-& \left.( g_{l}^{(i)} c_i l_{y(i)}  + D_i)K^b(c_i y(i) )]|II_{y(i)} \right\rangle \nonumber\\
%&\times & \left.\left\langle II_{y(i)} |II_T\right\rangle \right] \label{anom2}
%\end{eqnarray}
%where $c_i = \frac{A-A_i}{A}$ takes into account the motion of the center of mass in each basis system.\\
%
%In the matrix element (\ref{anom2}) the first term is associated with the contribution of the neutron (cluster $i=1$) and the hyperspherical harmonic wave function is defined in the T-system. The two other terms are related to the contribution from the $\alpha$
%particles (clusters $i= 2$ and $3$); the matrix element is calculated in the appropriated $Y(i),i= 2,3$ system and the transformation of the wave function from the $Y(i)$ to the T basis system is used.

The electric quadrupole  moment is 
\begin{equation}
Q = 2 \sum \limits_{i=1}^3 \left\langle II|r_i^2 C_0^{2}(\hat{r}_{i})| II \right\rangle ,
\end{equation}
where the summation runs over the fragments.

In the three-cluster model of $^9$Be, the hfs anomaly in the
Bohr-Weisskopf effect is 
found as the weighted sum
\begin{eqnarray}
\epsilon_{BW} &=& - \frac {b}{\mu_I}\left\{ 
\sum\limits_{\varsigma} \omega
_{\varsigma}^{2} \left\langle m_{s} \right\rangle g_{s}^{(n)}  
\right. \nonumber \\
& \times & \left[
K_{\varsigma}^{a}-K_{\varsigma}^{b}\left( 1\mp \frac{6}{4}
\frac{2I+1}{2(I+1)}\right) \right] \label{EpsBW} \\
&+&\left. \frac{5}{9}\sum\limits_{q = 1,2}\sum\limits_{\varsigma^{\prime}} \beta
_{\varsigma^{\prime}}^{2} \left\langle m_{y} \right\rangle_q
g_{l_y^{\prime}}^{\alpha}\left[ K_{\varsigma^{\prime}}^{a}
-K_{\varsigma^{\prime}}^{b}\right]
\right\} , \nonumber 
\end{eqnarray}
where the sign $\mp $ corresponds to $I=l_y\pm \frac{1}{2}$, and where $\mu$ is replaced by the
experimental value ($\mu_I=-1.1778(9)\mu_N$ \cite{Web76}) of the $^9$Be magnetic dipole moment. 


Here we denote 
\begin{eqnarray}
K_{\varsigma}^{a} 
&=&\int\limits_{0}^{\infty }|\Phi_{\varsigma}(R)|^{2} K^a(\frac{A-A_i}{A}R)R^{2}dR \label{KA} \\
K_{\varsigma}^{b}
&=&\int\limits_{0}^{\infty }|\Phi_{\varsigma}(R)|^{2} K^b(\frac{A-A_i}{A}R)R^{2}dR  \label{KB} 
\end{eqnarray}
where the ratio $\frac{A-A_i}{A}$ takes into account the center 
of mass motion, $A_i$ is the valence fragment mass (the neutron 
mass $(i=1)$ in the T-basis, and the $\alpha $-particle 
mass $(i=2 ,3)$ in the Y-basis).

So, we can write 
\begin{equation}
\epsilon_{BW}=\sum\limits_{\varsigma} 
\omega_{\varsigma}^{2} \epsilon_{BW}^{\varsigma} \label{epschan}
\end{equation}
where 
$\epsilon_{BW}^{\varsigma}$ is obtained for each channel 
$|\varsigma >$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results and discussion \label{sec:s5}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Three-body model of $^9$Be} 
In the calculations we use two ground
state wave functions of the $^9$Be described as the three body
$\alpha+\alpha+n$ system. These wave functions are
obtained with different $\alpha-\alpha$ and $\alpha-n$ interaction 
potentials. 

The first wave function, WF1 \cite{Priv05}, is obtained with supersymmetric 
equivalent potential \cite{Des03}. With the $\alpha $-particle
charge radius $1.676$ fm \cite{deV87} this wave function 
gives the value $2.564$ fm for the $^{9}$Be charge rms radius. This 
value agrees with the experimental ones, 
$2.519(12)$ and $2.50(9)$ fm 
(see \cite{deV87} and references therein). 
The $^{9}$Be magnetic dipole moment
$\mu _{^{9}Be}=-1.0531 \mu_N $ is less by $10\%$ compared to the
experimental values, $-1.177432(3) \mu_N $ 
\cite{Ita83} and $-1.1778(9) \mu_N $ \cite{Web76}. 
The calculated electric quadrupole  moment is 53.39 mb which is in good
agreement with the experimental value 52.88(38) mb \cite{Sun91}.

The second wave function, WF2 \cite{Gri02}, is obtained 
with the Ali-Bodmer potential \cite{Fed96} and the 
$\alpha $-neutron interaction potential \cite{Cob98}. 
The three-body interaction potential is adjusted to 
fit the three-cluster dissociation energy $1.573$ MeV.
%Supersymmetric transformation of the attractive
%s-wave Vna potential gives a purely repulsive partner (see,
%e.g.),  This partner shows the same s-wave
%phase shift, and has a ��Pauli core�� which excludes from the
%an potential the 0s state occupied by the a-core neutrons.
%PC: The repulsive potential may be approximated @4# by a
%Gaussian potential of size b52.3 fm and Vna(l50)
%5150 MeV, as this also reproduces the experimental s1/2
%phase shifts @83#.
%
%A-A   Ali-Bodmer  a0
%A-n   Gaussov potencial, kak vo vseh rabotah Zhukov-Danilin
The value of the $^{9}$Be charge radius, $2.707$, fm is 
overestimated compared to the experimental one. The $^{9}$Be 
magnetic dipole moment %(\ref{magmom})
$\mu _{^{9}Be}= -1.316 \mu_N $ is $10\%$ larger than 
the experimental value. The electric quadrupole  moment is $Q=65.42$ mb.

\begin{table}[tbp]
\caption{\label{Table1} The weights of the partial waves 
$\protect\omega _{Ll_{x}l_{y}}^{2}$, the mean 
$^8$Be-neutron distance $r_n$, the neutron contribution 
$\epsilon_{BW}^{\varsigma (n)}$ to the values of the hfs 
anomaly obtained with WF1. 
%with rms$_\alpha$=1.676 fm
}
\begin{ruledtabular} 
\begin{tabular}{cccc}
$\varsigma$ & $\omega_{\varsigma}^{2}$ & 
                    $r_n$  &  $\epsilon_{BW}^{\varsigma (n)}$ \\ 
$Ll_{x}l_{y}$  &   & fm    &  \%      \\ \hline
101    &  0.498355 & 3.274 & -0.0332 \\ 
121    &  0.315975 & 3.159 & -0.0229 \\ 
221    &  0.152790 & 2.976 &  0.0123 \\ 
123    &  0.019661 & 3.758 & -0.0312 \\ 
223    &  0.003674 & 3.585 & -0.0286 \\ 
143    &  0.006608 & 3.689 & -0.0303 \\ 
243    &  0.001595 & 4.397 &  0.0245 \\ 
\end{tabular}
\end{ruledtabular}
\end{table} 
 
\begin{table}[tbp]
\caption{\label{Table2} The same as in Table \ref{Table1}
calculated with the wave function WF2.}
\begin{ruledtabular} 
\begin{tabular}{cccc}
$\varsigma$ & $\omega _{\varsigma}^{2}$ & 
                    $r_n$  & $\epsilon_{BW}^{\varsigma (n)}$ \\ 
$Ll_{x}l_{y}$  &   & fm    &  \%     \\ \hline
101    &  0.592607 & 3.778 & -0.0402 \\ 
121    &  0.286695 & 3.381 & -0.0248 \\ 
221    &  0.090244 & 3.284 &  0.0141 \\ 
123    &  0.020740 & 3.870 & -0.0313 \\ 
223    &  0.002710 & 3.753 & -0.0296 \\ 
143    &  0.005362 & 3.770 & -0.0299 \\ 
243    &  0.001154 & 3.760 &  0.0179 \\ 
\end{tabular}
\end{ruledtabular}
\end{table}

The difference between these wave functions is in the radial $y$
dependence of the valence neutron wave function (obtained by 
integration over the $x$ coordinate and the angles), 
the core charge radius and the weights of the partial states 
$L$, $l_x$, $l_y$ for the ground state wave function 
(see Tables \ref{Table1} and \ref{Table2}).

Tables \ref{Table1} and 
\ref{Table2} show  the neutron contribution
$\epsilon_{BW}^{\varsigma(n)}$ 
to the hfs anomaly (first term in (\ref{EpsBW}))
%and $\epsilon_{BW}^{\varsigma(\alpha)}$, in 
%the Bohr-Weisskopf effect in $^{9}$Be$^{+}$ 
and the root mean square distance of
the valence neutron $r_n$ from the $^8$Be center of mass
in each channel $\varsigma$. 

The contributions to the hfs anomaly in the Bohr-Weisskopf effect, 
$\epsilon_{BW}^{(n)}$ and $\epsilon_{BW}^{(\alpha)}$,
and the BRCS correction $\delta$ calculated with
WF1 and WF2 are listed in Table \ref{Table3}.
%Here, $\epsilon_{BW}^{(\alpha)}$ is the contribution 
%to the hfs anomaly from the  
%$\alpha $-orbital motion (second term in (\ref{EpsBW})).

%\begin{eqnarray}
%\epsilon_{BW}^{(n)} &=& \sum\limits_{\varsigma} \epsilon_{BW}^{\varsigma(n)} \\
%&=& \frac {b}{\mu_I} 
%\sum\limits_{\varsigma} \omega
%_{\varsigma}^{2} \left\langle m_{s} \right\rangle g_{s}^{(n)} 
%\left[
%K_{\varsigma}^{a}-K_{\varsigma}^{b}\left( 1\mp \frac{6}{4}\frac{2I+1}{%
%2(I+1)}\right) \right].     \nonumber
%\end{eqnarray}

%The contribution of the hfs anomaly from the $\alpha$-particle
%orbital motion, $\epsilon_{BW}^{(\alpha)}$, 
%calculated in the rotated Y-basis is 
%\begin{eqnarray}
%\epsilon_{BW}^{(\alpha)} &=& \sum\limits_{\varsigma^{\prime}} %\epsilon_{BW}^{\varsigma^{\prime}(\alpha)} \\
%&=& \frac {b}{\mu_I} \frac{10}{9}
%\sum\limits_{\varsigma^{\prime}} \beta
%_{\varsigma^{\prime}}^{2} \left\langle m_{y} \right\rangle
%g_{l_y^{\prime}}^{\alpha}\left[ K_{\varsigma^{\prime}}^{a}
%-K_{\varsigma^{\prime}}^{b}\right] .      \nonumber 
%\end{eqnarray}


The contribution of the hfs anomaly from the $\alpha$-particle
orbital motion, $\epsilon_{BW}^{(\alpha)}$ (second term in (\ref{EpsBW}))
is small compared to that from the neutron, $\epsilon_{BW}^{(n)}$; 
and its variation is small also (see Table \ref{Table3}).
% compared to the variations of $\delta$ and $\epsilon_{BW}^{(n)}$.
%As the result,
%So, the hfs anomaly $\epsilon_{BW}$ is mostly
%determined by the neutron contribution. 

\begin{table}[tbp]
\caption{\label{Table3} The $^{9}$Be charge rms radius, the neutron radial distance $r_n$, the contributions to the hfs anomaly from the
neutron spin ($\epsilon_{BW}^{(n)}$) and from the
$\alpha$-particle orbital motion ($\epsilon_{BW}^{(\alpha)}$), 
the values of $\epsilon_{BW}$ and of 
$\delta$ calculated for the two wave functions $WF1,2$ ($\alpha$-particle radius: $1.676$ fm)
are compared to the (core+neutron) results. WF2$^*$ refers to another choice of the $\alpha$-particle radius ($1.46$ fm).}

\begin{ruledtabular} 
\begin{tabular}{lcccc}
Value                           &core+neutron & WF1       & WF2      & WF2$^*$    \\ \hline
rms (fm)                        & 2.519       & 2.564     & 2.707    & 2.533   \\ 
$r_n$ (fm)                      &             & 3.207     & 3.621    &         \\ 
$\epsilon_{BW}^{(n)}$ (\%)      &             & -0.02281  & -0.03059 &         \\ 
$\epsilon_{BW}^{(\alpha)}$ (\%) &             & 0.00085   &  0.00088 &          \\ 
$\epsilon_{BW}$ (\%)            &  -0.0236    & -0.02112  & -0.02882 & -0.03032  \\ 
$\delta$                        &  -0.0451    & -0.04644  & -0.04926 & -0.04634  \\ 
$\epsilon$                      &  -0.0687    & -0.06756  & -0.07809 & -0.07666 \\ 
\end{tabular}
\end{ruledtabular}
\end{table}


To explore the sensitivity of the results to the charge radius of 
the $\alpha$-clusters in $^9$Be, we vary the value of 
the $\alpha$-particle charge radius from $1.676$ to $1.636$ fm.
The last value of the charge radius is obtained with regard to
the negative contribution of the neutron charge distribution, $(0.34)^2$ fm$^2$
(see \cite{Sic05,Ana99}).

Correspondingly, the $^9$Be charge radius changes from 2.564 to 
2.534 fm for WF1 and from 2.707 to 2.678 fm for WF2.
Owing to the radial behavior of the electronic wave functions
(see Fig. \ref{fig1}) entering the expressions (\ref{elwfinta}),
(\ref{elwfintb}), and (\ref{EpsBW})-(\ref{KB}), with a smaller 
charge radius (rms and rms$_C$) we get a larger value of the 
hfs anomaly in the Bohr-Weisskopf effect. Thus, in our case, 
we get an increase in the hfs anomaly value 
$\epsilon_{BW}$ by 1.6\% for WF1 and 0.5\% for WF2.

The $\alpha$-$\alpha$ distance is smaller when calculated with WF1, so
correspondingly, the $^9$Be charge radius is
smaller and the value of the BRCS correction $\delta$ is 
less (see Table \ref{Table3}).
$\delta$ varies within 6\% depending on the description of the nuclear wave function.

%The BRCS correction is approximately the same (in general
%it has a tendency to increase with charge radius). 

\begin{figure}
\includegraphics{figure1}% Here is how to import EPS art
\caption{\label{fig1} The electronic wave functions
obtained for the distributed nuclear charge, $G$ (dashed line),
and for the point nuclear charge, $G_0$ (solid line).}
\end{figure}

%Nevertheless, as seen from Table \ref{Table3}, the smaller 
%values of the hfs anomaly $\epsilon_{BW}$  
%obtained with WF1 
%correspond to the smaller values of rms$_C$ and $r_n$.
$\epsilon_{BW}$ and the total hfs anomaly 
$\epsilon$ which is the 
sum ($\epsilon_{BW}^{(n)}+\epsilon_{BW}^{(\alpha)}+\delta$), 
are mainly determined by the neutron distribution. 
There is no direct correspondence of the hfs anomaly
value to the nuclear charge radius.
For example, if we put the radius of the $\alpha$-particle at
1.46 fm, the $^9$Be charge radius obtained with WF2 is 
2.533 fm and the hfs anomaly $\epsilon$=-0.07666\% (see Table \ref{Table3} WF2$^*$).
These values are larger than those obtained with WF1 with the
charge radius 2.564 fm. 
Thus, even with smaller charge radius values one can
get larger values of $\epsilon_{BW}$ and $\epsilon$.

Therefore, we can conclude that the hfs anomaly is more sensitive 
to the neutron spatial distribution than to the charge 
distribution of the whole nucleus, and
the value $r_n$ is a crucial parameter  
for the hfs value in the Bohr-Weisskopf effect. 

With the different nuclear wave functions, the $r_n$ value varies of about
$10\%$ and the hfs anomaly $\epsilon_{BW}$ of about 30\%. 
As the value of the hfs anomaly $\epsilon_{BW}$ is 
about two times smaller than $\delta$,the total
hfs anomaly $\epsilon$ varies within 14\% with the choice of the wave function.


%We also perform the calculations using two values
%of the $\alpha$-particle rms radius, $1.46$ and $1.676$ fm.
%The last value correspond to the $\alpha$-particle charge
%radius from Ref. \cite{deV87}. This will give the measure of the 
%sensitivity of the results to the rms radii of the fragments.


%In our case, the smaller value of the $^{9}$Be 
%charge rms radius obtained with WF1 correspond to the
%smaller value of the hfs anomaly. We should mention here,that 
We should also mention that
the value of the hfs anomaly is very sensitive to the
contribution of the different partial states in 
the $^{9}$Be ground state wave function. 
In particular, the hfs anomaly $\epsilon_{BW}^{(n)}$ in the channels
$\varsigma=121$ and $\varsigma=221$ (that is of 50\% of the
nuclear wave function) is twice as small as $\epsilon_{BW}^{(n)}$
in the channel $\varsigma=101$. Thus, the relative weights of 
these states are of major importance. As found in \cite{Vor95} the magnetic dipole moment in the
$\alpha+\alpha+n$ cluster model is also rather sensitive to
these weights. In the case of $^{9}$Be, three channels mostly contribute 
to the magnetic dipole moment and the hfs anomaly, $\varsigma=101$, 
$\varsigma=121$, and $\varsigma=221$. The magnetic dipole moment 
in this case is \cite{Vor95}
\begin{equation}
\mu \thickapprox -1.857 \omega_{101}^2- 1.191 \omega_{121}^2
+ 1.914 \omega_{221}^2 + \mu _{res}.
\end{equation}

According to \cite{Vor95} the experimental magnetic dipole 
moment value can be reproduced under the condition
$\omega_{221}^2 < 16$ \%. Otherwise the 
calculated value is underestimated.

Let us analyze the correlation between the
weights and the values of the $^{9}$Be hfs anomaly,  
magnetic dipole and electric quadrupole moments, noting that the analysis is model dependent.

To estimate how the hfs anomaly $\epsilon_{BW}^{(n)}$
varies with the weights of the different states, we consider 
the contributions of these three channels only
(so that $\omega_{101}^2+\omega_{121}^2 + \omega_{221}^2 + \omega_{res}^2=1$)
and find the weights satisfying the experimental value of the
$^9$Be magnetic dipole moment ($\mu_I=-1.177432(3)\mu_N$). Under
this condition, we get the hfs anomaly value plotted in 
Fig. \ref{fig2} as a function of $\omega_{101}^2$ for the wave 
functions WF1 and WF2 (solid and dashed lines, respectively).

\begin{figure}
\includegraphics{figure2}% Here is how to import EPS art
\caption{\label{fig2} The variation of the Bohr-Weisskopf hfs anomaly value 
from the neutrons with the weight $\omega_{101}^{2}$ for WF1 (solid line) and WF2 
(dashed line)($\alpha$-particle radius: $1.676$ fm). The dots on the lines are the values in agreement 
with the electric quadrupole  moment(see text). The weight $\omega_{221}^{2}$ is indicated for each point.} 
\end{figure}

Similarly one can 
express the electric quadrupole  moment through the weights of 
these dominant states. For WF1 we get
\begin{eqnarray}
Q & \thickapprox & 3.157 \omega_{101}^2 - 4.553 \omega_{121}^2 - 
4.527 \omega_{221}^2 \nonumber \\ 
 &+&  2 \omega_{101} \omega_{121} 32.859  
+ 2 \omega_{101} \omega_{221} 32.039 + \label{quad1} \\
 &+&  2 \omega_{121} \omega_{221} 7.314 + Q_{res},  \nonumber
\end{eqnarray}
 where $Q_{res}=11.97$ mb.
 
For WF2 this relation is
\begin{eqnarray}
Q & \thickapprox & 4.204 \omega_{101}^2 - 6.635 \omega_{121}^2 - 
6.662  \omega_{221}^2 \nonumber \\ 
&+& 2 \omega_{101} \omega_{121} 41.908 
+ 2 \omega_{101} \omega_{221} 41.610 \label{quad2} \\
&+& 2 \omega_{121} \omega_{221} 9.918 +Q_{res}, \nonumber
\end{eqnarray}
where $Q_{res}=13.11$ mb.

$\omega_{res}^2$, $\mu _{res}$ and $Q_{res}$ are the contributions of the neglected channels respectively
to the weights, the magnetic dipole and electric quadrupole moments. 
The square dots on the lines in Fig. \ref{fig2} point 
the hfs values obtained with the weights which also
satisfy the experimental value of the $^9$Be electric quadrupole  
moment 52.88(38) mb.
For each wave function we get a few
points - two for WF1 or even four for WF2. 
%This is explained 
%by very curved line for the values of the hfs, corresponding 
%to the experimental value of the 
%electric quadrupole  moment. 
%Here, the dots denote the cross of these lines with
%those lines corresponding to the description of the $^9$Be 
%magnetic dipole moment.
On this figure we also report the corresponding values of the weight 
$\omega_{221}^2$ ($\omega_{221}^2<16\%$).
One can see the ranges of the hfs anomaly values  
obtained with WF1 and WF2 respectively 
%$0.0255< |\epsilon_{BW}| <0.0260$ and
%$0.0275<|\epsilon_{BW}|<0.0305$. 
Therefore, an experimental estimation 
of the hfs anomaly in the Bohr-Weisskopf effect could give
values for the weights of the partial states in the
ground state wave function.

%We should mention here, that in the calculations of the 
%hfs anomaly with (\ref{EpsBW}) we use the experimental 
%value of the nuclear magnetic dipole moment. The use of the 
%calculated value leads to the factors 1.118 and 
%0.895  for the values of the hfs anomaly obtained
%with WF1 and WF2, respectively.  In any case, our
%conclusion remains the same.

The figure shows that the hfs anomaly is a very critical 
quantity to test the nuclear wave function, all other
parameters being equivalently well described, 
in particular, the electronic part. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Comparison with the core+neutron model \label{sec:s6}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


In Ref. \cite{Par05}, the hfs anomaly for $^{9}$Be was calculated 
within the core+neutron model of $^{9}$Be. 
%In this model 
%$^{9}$Be is considered as a system composed by the $p_{3/2}$-wave 
%neutron and the $^{8}$Be core in the ground (0$^{+}$) and excited 
%(2$^{+}$) states. 
The $^9$Be ground state wave function was given by the 
superposition of states 
\begin{eqnarray}
\left| ^{9}{\rm Be} \left( 3/2^{-}\right) \right\rangle 
&=&\omega_{0^+} \left|
[^{8}Be \left( 0^{+}\right) \otimes n_{p_{3/2}}]_{3/2^{-}}\right\rangle  \label{WF9}  \\
&+&\omega_{2^+} \left| [^{8}Be \left( 2^{+}\right) \otimes
n_{p_{3/2}}]_{3/2^{-}}\right\rangle  .    \nonumber   
\end{eqnarray}
corresponding to the $^8$Be core 
in the ground (0$^{+}$)  and excited (2$^{+}$) states with
the neutron separation energies 1.665 and 4.705 MeV, respectively. 

With the weights $\omega_{0^+}^2= 0.535$ and $\omega_{2^+}^2=0.465$ 
obtained with the spectroscopic factors from Ref. \cite{Coh67} the 
magnetic dipole moment is $\mu=-1.0687$ $\mu_N$ and $\epsilon_{BW}$, $\delta$ and $\epsilon$ have the values reported in Table \ref{Table3}. 
%With the change of the weights to $\omega_{0^+}^2=  0.579$
%and $\omega_{2^+}^2=0.421$ we reproduce the experimental value of
%the $^{9}$Be magnetic dipole moment -1.1447 $\mu_N$ and get the value 
%$\epsilon_{BW}=-0.0256$ \%.
%The value of the hfs anomaly in the Bohr-Weisskopf effect, 
%obtained in 
%the core+neutron model and with 
The $\epsilon_{BW}$ is close to the values $\epsilon_{BW}=-0.0249$\% \cite{Fuj99} (obtained with the
weights $\omega_{0^+}^2=\omega_{2^+}^2=0.5$) and 
$\epsilon_{BW} =-0.0243$\% from \cite{Yam00}. 

The $\epsilon_{BW}$ values obtained for the $0^+$ state in the 
(core+neutron) model ($\epsilon_{BW}= - 0.0440$\%) and for the 
$l_x = 0$ state of $WF1$ or $WF2$ in the ($\alpha + \alpha + n$) 
model ($\epsilon_{BW}= -0.0332\%$ or  $-0.0402$\%) are relatively 
close to each other. On the contrary, the $\epsilon_{BW}$ 
values for the different partial states $l_x = 2$ in the 
three cluster model exceed by a few times 
(see Tables \ref{Table1} and 
\ref{Table2}) the value obtained for the $2^+$ state
in the (core + neutron) model ($\epsilon_{BW}= -0.0063$\%).

%The value obtained with the wave functions WF1 and WF2 are 
%close to the theoretically calculated values $\epsilon_{BW} =0.0249$\% 
%from \cite{Fuj99} and $\epsilon_{BW} =0.0243$\% from \cite{Yam00}. 

The BRCS correction obtained 
with the two-body wave function
is $-0.0451$\%, close to that obtained in the three-body 
calculations. Thus in the core+neutron model we get 
$\epsilon=-0.0687$\%, to be compared with 
the values $\epsilon=-0.06756$\% and $\epsilon=-0.07809$\% 
obtained with the three clusters 
$ \alpha + \alpha + n $ wave functions.

%The BRCS correction is related to the 
%charge distribution and is independent of the neutron distribution. 
%So the value of the hfs anomaly $\epsilon$ has a lower limit for
%light nuclei. For the Be isotopes it is about
%$-0.044$ \%, that contradicts the hfs value 
%$\epsilon =-0.0243$\%, extracted from experimental data \cite{Yam00}.

Thus the clustering effect, revealing itself in the set 
of states contributing to the ground state wave function,
lead to a variation of the hfs value $\epsilon$ of less than 2\% for WF1 and of about 
14\% for WF2. Compared to the  $^{11}$Be nucleus the difference in the
 value of the hfs anomaly in the Be isotopes is about 25\%. 
This value gives us the accuracy of the measurements
of the hfs anomaly needed to study clustering effects
in light nuclei.


%This result gives a precision of the hfs anomaly measurements
%needed for the neutron distribution studies.

This results corroborates the 
conclusion in Ref. \cite{Fuj99}, that the
value of the hfs anomaly reflects the extended neutron 
distribution in $^{11}$Be and might indicate a neutron halo, 
but the difference for the different isotopes is not so pronounced 
as was found in Ref. \cite{Fuj99}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion \label{sec:s7}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In the present paper, we have calculated the hfs 
anomaly in the $^{9}$Be$^+$ ion with the nucleus described in a 
three-cluster model. The $1s^22s$ electronic wave 
function is obtained taking into account the charge distribution of the clustered ($\alpha+\alpha +n$)nucleus and the shielding effect of two electrons in the $1s^2$ configuration. 

%It is found that, in the case of Be isotopes the hfs anomaly 
%is mostly sensitive to the neutron spatial distribution.
%Thus the hfs anomaly can be used for study of the valence
%neutron distribution in the neutron-rich light nuclei.

The result of the calculations strongly depends 
on the weights of the partial waves contributing to the ground state
wave function. Together with the magnetic dipole and electric quadrupole moments 
the value of the hfs anomaly can be used to study the
clustering effects in neutron rich light nuclei.

The total hfs anomaly is the sum of $\delta$ and $\epsilon_{BW}$.
The BRCS correction $\delta$ is only determined
by the nuclear charge distribution and slightly varies from
isotope to isotope. The value of the BRCS correction is comparable 
or larger than the value of $\epsilon_{BW}$. 
The hfs anomaly in $^{11}$Be differs from that in $^{9}$Be by 25\%. 
The clustering effect leads to variations of the hfs value
within 15\%. The calculated magnitude and differential change in the value of the hfs 
anomaly is indicative of the experimental precision that must be achieved to study 
the clustering effect and the neutron distribution in neutron rich light nuclei.

%The difference of the hfs values in the Bohr-Weisskopf 
%effect obtained with distinct wave 
%functions in the two- and three-cluster models is about 
%31\%. The BRCS correction $\delta$ is only determined
%by the nuclear charge distribution and varies slightly from
%isotope to isotope. This correction is comparable 
%or even larger than the value of $\epsilon_{BW}$. Thus,
%the corresponding change in the value $\epsilon=
%\delta+\epsilon_{BW}$ is 14\%.
%This value gives the accuracy of the measurements
%of the hfs anomaly needed for study of the clustering effects
%in the light nuclei.


\section{Acknowledgments}

The authors are grateful to Dr P. Descouvemont and Dr. L.V. Grigorenko 
for helpful discussions and for having provided the numerical values
of the wave functions they have calculated for $^9$Be in a three-body model.

This paper has been supported by the Belgian Program P5-07 of Inter-university Attraction Poles initiated by the Belgian-state Federal Services for Scientific Politics. Y.L.P. has received an one year postdoctoral research grant from the Cooperation of the Belgian Scientific Politics with Central and Oriental European countries.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage %Just because of unusual number of tables stacked at end
%\bibliography{apssamp}% Produces the bibliography via BibTeX.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{99}

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APS-123-QED, Feb 2005. 9pp. 
e-Print Archive: nucl-th/0502032 
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\bibitem{Boh50}  A. Bohr and V.F. Weisskopf, Phys. Rev. \textbf{77},
94 (1950).

\bibitem{Ros72}  H.J. Rosenberg and H.H. Stroke, Phys.~Rev.~\textbf{A5},
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\bibitem{Priv05} Private Communications
\textit{Three clusters approach of the $^9$Be ground state wave function}
\textbf{WF1}: P. Descouvemont,
Physique nucl\'eaire th\'eorique et Physique Math\'ematique,
U.L.Bruxelles, Bruxelles, Belgium. pdesc@ulb.ac.be

\bibitem{Gri02} L.V. Grigorenko, R.C. Johnson, I.G. Mukha, I.J. Thompson, M.V. Zhukov,
Eur. Phys. J. \textbf{A15}, 125 (2002) 

\bibitem{Fuj99}  T. Fujita, K. Ito, and T. Suzuki, Phys. Rev. 
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\bibitem{Zhu93}  M. V. Zhukov, B. V. Danilin, D. V. Fedorov, J. M. Bang, I. J. Thompson, J. S. Vaagen, 
Phys. Rep. \textbf{231}, 151 (1993).

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\bibitem{deV87}  H. de Vries, C. W. de Jager, and C. de Vries, At. Data and
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\bibitem{Dav65}  A.S.Davydov, "\textit{Quantum Mechanics}", Pergamon Press,
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\bibitem{Ros61}  M.E. Rose "\textit{Relativistic Electron Theory}", John
Wiley and Sons, New York, 1961. 


\bibitem{Des03}  P. Descouvemont and C. Daniel, 
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\bibitem{Ita83}  W. M. Itano, Phys. Rev. \textbf{B27}, 1906 (1983).
 
\bibitem{Sun91}  D. Sundholm and J. Olsen 
Chem. Phys. Lett. \textbf{177}, 91 (1991). 

\bibitem{Fed96}  D.V. Fedorov, A.A. Jensen, Phys. Lett. \textbf{B389}, 631
(1996).

\bibitem{Cob98}  A. Cobis, D.V. Fedorov, A.S. Jensen, 
Phys. Rev. \textbf{C58}, 1403 (1998).

\bibitem{Sic05}  I. Sick, Eur. J. Phys. A \textbf{24}, 65 (1999).

\bibitem{Ana99}  G.S. Anagnostatos, A.N. Antonov, P. Ginis, 
J. Giapitzakis, M.K. Gaidarov, J. Phys. G: Nucl. Part Phys. 
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\bibitem{Vor95}  V.T. Voronchev, V.I. Kukulin, V.N. Ponomarev,
and G.G. Ryzhikh. Few Body Systems {\bf 18}, 191 (1995).

\bibitem{Coh67}  S. Cohen and D. Kurath, 
Nucl. Phys. {\bf A101}, 1 (1967).

\bibitem{Yam00}  N. Yamanaka, Hyperfine interactions \textbf{127}, 129
(2000). 

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%
%\bibitem{Bel63}  M. Le Bellac, Nucl. Phys. \textbf{40}, 645 (1963).

%\bibitem{Dan04}  B.V. Danilin

%\bibitem{Des02}  P. Descouvemont, Nucl. Phys. \textbf{A699}, 463 (2002).

%\bibitem{Wad97} M. Wada, K. Okada, H. Wang, K. Enders, F. Kurth, 
%T. Nakamura, S. Fujitaka, J. Tanaka, H. Kawakami, S. Ohtani, and 
%I. Katayama, Nucl.Phys. \textbf{A626}, 365c (1997).

%\bibitem{Iso93}  ISOLDE Collaboration, 
%H. T. Duong, C. Ekstrom, M. Gustafsson, T. T. Inamura, P. Juncar, 
%P. Lievens, I. Lindgren, S. Matsuki, T. Murayama, R. Neugart,
%T. Nilsson, T. Nomura, M. Pellarin, S. Penselin, J. Persson, 
%J. Pinard, I. Ragnarsson, O. Redi, H. H. Stroke, and J. L. Vialle,
%Nucl. Instr. and Methods in Phys. Research, Sect. A, 325, 465 (1993).

%\bibitem{Gei99}  W. Geithner et al., Phys. Rev. Lett. \textbf{83}, 3792
%(1999) .

%\bibitem{Suz95}  T. Suzuki, T. Otsuka, and A. Muta, Phys. Lett. \textbf{B364},
%69 (1995) .

%\bibitem{For99}  S. Fortier, S. Pita, J.S. Winfield, W. N. Catford,
%N. A. Orr, J. Van de Wiele, Y. Blumenfeld, R. Chapman, S. Chappell, 
%N. M. Clarke, N. Curtis, M. Freer, S. Gales, K. L. Jones,
%H. Langevin-Joliot, H. Laurent, I. Lhenry, J. M. Maison, P. Roussel-Chomaz,
%M. Shawcross, M. Smith, K. Spohr, T. Suomijarvi, and A. de Vismes, 
%Phys. Lett. \textbf{B461}, 22 (1999).

%\bibitem{War92}  E. K. Warburton and B.A. Brown, Phys. Rev. C {\bf 46},%
%923 (1992).

%\bibitem{Nun96}  F. M. Nunes, J. A. Christley, I. J. Thompson, R. C.
%Johnson, and V. D. Efros, Nucl. Phys. {\bf A609}, 43 (1996).

%\bibitem{Aut70}  D. L. Auton, R. G. H. Robertson, Nucl. Phys.{\bf A157}%
%~(1970) 305.

%\bibitem{Aum00}  T. Aumann, A. Navin, D. P. Balamuth, D. Bazin, B. Blank,
%B. A. Brown, J. E. Bush, J. A. Caggiano, B. Davids, T. Glasmacher, 
%V. Guimaraes, P. G. Hansen, R. W. Ibbotson, D. Karnes, J. J. Kolata,
%V. Maddalena, B. Pritychenko, H. Scheit, B. M. Sherrill, and J. A. Tostevin, 
%Phys. Rev. Lett. \textbf{84}, 35 (2000).

%\bibitem{Par00}  Yu. L. Parfenova, M. V. Zhukov, and J. S. Vaagen, Phys. Rev.  
%\textbf{C62}, 044602 (2000).

%\bibitem{Fed93}  D.V. Fedorov, A.S. Jensen, and K. Riisager, Phys. Lett. \textbf{%
%B312}, 1 (1993).

%\bibitem{Ros78}  F. R\"{o}sel, H. M. Fries, K. Adler, and H. C. Pauli, At.
%Data. Nucl. Data Tables \textbf{21}, 92 (1978).

%\bibitem{Ots93} T. Otsuka, N.Fukunishi, and H. Sagawa. Phys. Rev. Lett. \textbf{70},
%1385 (1993) .

%\bibitem{Oza01}  A. Ozawa \textit{et al.},
%, O. Bochkarev, L. Chulkov, D. Cortina, H. Geissel,
%M. Hellstr\"{o}m, M. Ivanov, R. Janik, K. Kimura, T. Kobayashi, \textit{et al,} 
%Nucl. Phys.~\textbf{A691,}~599~(2001).

%\bibitem{Oza01}  A.Ozawa, T. Suzuki, and I. Tanihata, Nucl. Phys. {\bf A693} 32 (2001).

%\bibitem{Tan88}  I. Tanihata \textit{et al.},
%, T. Kobayashi, O Yamakawa, S. Shimoura, K. Ekuni, K. Sugimoto, N. Takahashi, T. Shimoda, 
%Phys. Lett. \textbf{B206,} 592
%(1988).

%\bibitem{Lia90}  E. Liatard \textit{et al.}, Europhys. Lett.\textbf{13}, 401 (1990) . 
%\bibitem{XX76}  PL A56 (1976) 446.

%\bibitem{End95}  K. Enders, O.Becker, L. Brand, J. Dembcszynski, G. Marx, G.
%Revalde, P. B. Rao, and G. Werth, Phys.~Rev.~A~{\bf 52}~(1995)~4434.

%\bibitem{Neu00}  R. Neugart. Hyperfine Interactions {\bf 127} (2000) 101.

%\bibitem{Aum00}  T. Aumann, A. Navin, D. P. Balamuth {\it et al.}, 
%D. Bazin, B. Blank B,
%B. A. Brown, J. E. Bush, J. A. Caggiano, B. Davids, T. Glasmacher, 
%V. Guimaraes, P. G. Hansen, R. W. Ibbotson, D. Karnes, J. J. Kolata,
%V. Maddalena, B. Pritychenko, H. Scheit, B. M. Sherrill, J. A. Tostevin, 
%Phys. Rev. Lett. {\bf 84} 35 (2000).

%\bibitem{Boh94}  A. Bohr and V.F. Weisskopf, Phys.~Rev.~{\bf 77}%
%~(1950)~94.

%\bibitem{Til02}  D.R. Tilley, et al., Nucl.~Phys.~{\bf A708}~(2002)~3.

%\bibitem{XX49}  PR 75 (1949) 1769.

%\bibitem{Zwi79}  B. Zwieglinski, W. Beneson, R. G. H. Robertson, Nucl.
%Phys.~{\bf A315}~(1979) 124.

%\bibitem{Str61}  H.H. Stroke, R.J. Blyn-Stoyle, V. Jaccarino, Phys.~Rev.~%
%{\bf 123}~(1961)~1326.

\end{thebibliography}

 
\end{document}
%
% ****** End of file apssamp.tex ******


nucl-th0506007.extracted_bib
\begin{thebibliography}{99}

\bibitem{Par05} Y.L. Parfenova and C. Leclercq-Willain,
APS-123-QED, Feb 2005. 9pp. 
e-Print Archive: nucl-th/0502032 
Accepted in Phys. Rev. \textbf{C}, June (2005).  

\bibitem{Boh50}  A. Bohr and V.F. Weisskopf, Phys. Rev. \textbf{77},
94 (1950).

\bibitem{Ros72}  H.J. Rosenberg and H.H. Stroke, Phys.~Rev.~\textbf{A5},
 1992 (1972).


\bibitem{Priv05} Private Communications
\textit{Three clusters approach of the $^9$Be ground state wave function}
\textbf{WF1}: P. Descouvemont,
Physique nucl\'eaire th\'eorique et Physique Math\'ematique,
U.L.Bruxelles, Bruxelles, Belgium. pdesc@ulb.ac.be

\bibitem{Gri02} L.V. Grigorenko, R.C. Johnson, I.G. Mukha, I.J. Thompson, M.V. Zhukov,
Eur. Phys. J. \textbf{A15}, 125 (2002) 

\bibitem{Fuj99}  T. Fujita, K. Ito, and T. Suzuki, Phys. Rev. 
\textbf{C59}, 210 (1999). 
 
\bibitem{Web76}  E. W. Weber and J. Vetter, Phys. Lett. \textbf{A56}, 446
(1976).

\bibitem{Dic49} W. C. Dickinson and T. F. Wimett, Phys. Rev. 
\textbf{75}, 1769 (1949).

\bibitem{Zhu93}  M. V. Zhukov, B. V. Danilin, D. V. Fedorov, J. M. Bang, I. J. Thompson, J. S. Vaagen, 
Phys. Rep. \textbf{231}, 151 (1993).

\bibitem{Efr72} Ya. A. Smorodinskii and V. D. \'Efros, 
Yad. Fiz. {\bf 17}, 210 (1973) 
[Sov. J. Nucl. Phys. {\bf 17}, 107 (1973)].

\bibitem{deV87}  H. de Vries, C. W. de Jager, and C. de Vries, At. Data and
Nucl. Data Tables \textbf{36}, 495 (1987).

\bibitem{Dav65}  A.S.Davydov, "\textit{Quantum Mechanics}", Pergamon Press,
Oxford, London, Edinburg, New York, Paris, Frankfurt, 1965.

\bibitem{Ros61}  M.E. Rose "\textit{Relativistic Electron Theory}", John
Wiley and Sons, New York, 1961. 


\bibitem{Des03}  P. Descouvemont and C. Daniel, 
Phys. Rev. \textbf{C67}, 044309  (2003).

\bibitem{Ita83}  W. M. Itano, Phys. Rev. \textbf{B27}, 1906 (1983).
 
\bibitem{Sun91}  D. Sundholm and J. Olsen 
Chem. Phys. Lett. \textbf{177}, 91 (1991). 

\bibitem{Fed96}  D.V. Fedorov, A.A. Jensen, Phys. Lett. \textbf{B389}, 631
(1996).

\bibitem{Cob98}  A. Cobis, D.V. Fedorov, A.S. Jensen, 
Phys. Rev. \textbf{C58}, 1403 (1998).

\bibitem{Sic05}  I. Sick, Eur. J. Phys. A \textbf{24}, 65 (1999).

\bibitem{Ana99}  G.S. Anagnostatos, A.N. Antonov, P. Ginis, 
J. Giapitzakis, M.K. Gaidarov, J. Phys. G: Nucl. Part Phys. 
A \textbf{24}, 69 (1999). 

\bibitem{Vor95}  V.T. Voronchev, V.I. Kukulin, V.N. Ponomarev,
and G.G. Ryzhikh. Few Body Systems {\bf 18}, 191 (1995).

\bibitem{Coh67}  S. Cohen and D. Kurath, 
Nucl. Phys. {\bf A101}, 1 (1967).

\bibitem{Yam00}  N. Yamanaka, Hyperfine interactions \textbf{127}, 129
(2000). 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%\bibitem{Bel63}  M. Le Bellac, Nucl. Phys. \textbf{40}, 645 (1963).

%\bibitem{Dan04}  B.V. Danilin

%\bibitem{Des02}  P. Descouvemont, Nucl. Phys. \textbf{A699}, 463 (2002).

%\bibitem{Wad97} M. Wada, K. Okada, H. Wang, K. Enders, F. Kurth, 
%T. Nakamura, S. Fujitaka, J. Tanaka, H. Kawakami, S. Ohtani, and 
%I. Katayama, Nucl.Phys. \textbf{A626}, 365c (1997).

%\bibitem{Iso93}  ISOLDE Collaboration, 
%H. T. Duong, C. Ekstrom, M. Gustafsson, T. T. Inamura, P. Juncar, 
%P. Lievens, I. Lindgren, S. Matsuki, T. Murayama, R. Neugart,
%T. Nilsson, T. Nomura, M. Pellarin, S. Penselin, J. Persson, 
%J. Pinard, I. Ragnarsson, O. Redi, H. H. Stroke, and J. L. Vialle,
%Nucl. Instr. and Methods in Phys. Research, Sect. A, 325, 465 (1993).

%\bibitem{Gei99}  W. Geithner et al., Phys. Rev. Lett. \textbf{83}, 3792
%(1999) .

%\bibitem{Suz95}  T. Suzuki, T. Otsuka, and A. Muta, Phys. Lett. \textbf{B364},
%69 (1995) .

%\bibitem{For99}  S. Fortier, S. Pita, J.S. Winfield, W. N. Catford,
%N. A. Orr, J. Van de Wiele, Y. Blumenfeld, R. Chapman, S. Chappell, 
%N. M. Clarke, N. Curtis, M. Freer, S. Gales, K. L. Jones,
%H. Langevin-Joliot, H. Laurent, I. Lhenry, J. M. Maison, P. Roussel-Chomaz,
%M. Shawcross, M. Smith, K. Spohr, T. Suomijarvi, and A. de Vismes, 
%Phys. Lett. \textbf{B461}, 22 (1999).

%\bibitem{War92}  E. K. Warburton and B.A. Brown, Phys. Rev. C {\bf 46},%
%923 (1992).

%\bibitem{Nun96}  F. M. Nunes, J. A. Christley, I. J. Thompson, R. C.
%Johnson, and V. D. Efros, Nucl. Phys. {\bf A609}, 43 (1996).

%\bibitem{Aut70}  D. L. Auton, R. G. H. Robertson, Nucl. Phys.{\bf A157}%
%~(1970) 305.

%\bibitem{Aum00}  T. Aumann, A. Navin, D. P. Balamuth, D. Bazin, B. Blank,
%B. A. Brown, J. E. Bush, J. A. Caggiano, B. Davids, T. Glasmacher, 
%V. Guimaraes, P. G. Hansen, R. W. Ibbotson, D. Karnes, J. J. Kolata,
%V. Maddalena, B. Pritychenko, H. Scheit, B. M. Sherrill, and J. A. Tostevin, 
%Phys. Rev. Lett. \textbf{84}, 35 (2000).

%\bibitem{Par00}  Yu. L. Parfenova, M. V. Zhukov, and J. S. Vaagen, Phys. Rev.  
%\textbf{C62}, 044602 (2000).

%\bibitem{Fed93}  D.V. Fedorov, A.S. Jensen, and K. Riisager, Phys. Lett. \textbf{%
%B312}, 1 (1993).

%\bibitem{Ros78}  F. R\"{o}sel, H. M. Fries, K. Adler, and H. C. Pauli, At.
%Data. Nucl. Data Tables \textbf{21}, 92 (1978).

%\bibitem{Ots93} T. Otsuka, N.Fukunishi, and H. Sagawa. Phys. Rev. Lett. \textbf{70},
%1385 (1993) .

%\bibitem{Oza01}  A. Ozawa \textit{et al.},
%, O. Bochkarev, L. Chulkov, D. Cortina, H. Geissel,
%M. Hellstr\"{o}m, M. Ivanov, R. Janik, K. Kimura, T. Kobayashi, \textit{et al,} 
%Nucl. Phys.~\textbf{A691,}~599~(2001).

%\bibitem{Oza01}  A.Ozawa, T. Suzuki, and I. Tanihata, Nucl. Phys. {\bf A693} 32 (2001).

%\bibitem{Tan88}  I. Tanihata \textit{et al.},
%, T. Kobayashi, O Yamakawa, S. Shimoura, K. Ekuni, K. Sugimoto, N. Takahashi, T. Shimoda, 
%Phys. Lett. \textbf{B206,} 592
%(1988).

%\bibitem{Lia90}  E. Liatard \textit{et al.}, Europhys. Lett.\textbf{13}, 401 (1990) . 
%\bibitem{XX76}  PL A56 (1976) 446.

%\bibitem{End95}  K. Enders, O.Becker, L. Brand, J. Dembcszynski, G. Marx, G.
%Revalde, P. B. Rao, and G. Werth, Phys.~Rev.~A~{\bf 52}~(1995)~4434.

%\bibitem{Neu00}  R. Neugart. Hyperfine Interactions {\bf 127} (2000) 101.

%\bibitem{Aum00}  T. Aumann, A. Navin, D. P. Balamuth {\it et al.}, 
%D. Bazin, B. Blank B,
%B. A. Brown, J. E. Bush, J. A. Caggiano, B. Davids, T. Glasmacher, 
%V. Guimaraes, P. G. Hansen, R. W. Ibbotson, D. Karnes, J. J. Kolata,
%V. Maddalena, B. Pritychenko, H. Scheit, B. M. Sherrill, J. A. Tostevin, 
%Phys. Rev. Lett. {\bf 84} 35 (2000).

%\bibitem{Boh94}  A. Bohr and V.F. Weisskopf, Phys.~Rev.~{\bf 77}%
%~(1950)~94.

%\bibitem{Til02}  D.R. Tilley, et al., Nucl.~Phys.~{\bf A708}~(2002)~3.

%\bibitem{XX49}  PR 75 (1949) 1769.

%\bibitem{Zwi79}  B. Zwieglinski, W. Beneson, R. G. H. Robertson, Nucl.
%Phys.~{\bf A315}~(1979) 124.

%\bibitem{Str61}  H.H. Stroke, R.J. Blyn-Stoyle, V. Jaccarino, Phys.~Rev.~%
%{\bf 123}~(1961)~1326.

\end{thebibliography}