%-----------------------------------------------------------------------*
% rcnp93a.tex : report for the proc. of RCNP work shop, in May 1993
% N.Tajima
% 24,30/June,1,2,5,6/July/93
%-----------------------------------------------------------------------*
\documentstyle[12pt]{article}
\topmargin=-1.0cm
\oddsidemargin=0.0cm
\evensidemargin=0.0cm
\textheight=24.0cm
\textwidth=16.0cm
%
%  General-use commands
%
\newcommand{\ket}[1]{\vert #1 \rangle}
%
\begin{document}
\title{
Signature dependence of the rotational states \\
of an odd-odd nucleus
}
\author{
N.~Tajima \\
Institute of Physics, College of General Education, \\
University of Tokyo, Komaba, Meguro, Tokyo 153, Japan.
}
\date{May 12, 1993}
\maketitle

\baselineskip=0.60cm

\begin{abstract}
{\normalsize

Rotational states of $^{124}$Cs are studied with a particle-rotor
model in which a proton and a neutron quasiparticles are coupled with
a triaxial rotor.  Both quasiparticles are excited in the Nilsson
orbitals deriving from the intruder h$_{11/2}$ orbital at sphericity.
A zero-range force including 
  $\sigma_{\rm p} \! \cdot \! \sigma_{\rmn}$
term is employed as the residual interaction between the
quasiparticles.

We confirmed that the spin-independent part of the interaction has to
be very strong, as was already reported by Ragnarsson and Semmes, in
order to reproduce the observed inversion of the sign of the signature
splitting for $I$ $\leq$ 16.  We also found that $\gamma$ deformation
with irrotational-flow moment of inertia plays an important role to
recover the normal-sign signature splitting for $I$ $>$ 16.  By
including both factors with opposite effects, the observed signature
splitting as a function of $I$ can be reproduced quantitatively well.

The band consisting of other two members of the signature quartette
is predicted to appear at an excitation energy of 0.6 MeV above the
yrast band.
}
\end{abstract}

\section{Signature in the cranking picture}

Before discussing about the {\em signature inversion}, it is worth
while reviewing the relation between the parity of the total angular
momentum $I$ (or $I-\frac{1}{2}$ for odd-$A$ systems) and the
signature quantum number widely used in nuclear structure assignments.
The former is what is observed in experiments, while the latter is
peculiar to the cranking picture, in which the nucleus is assumed to
perform a uniform rotation around a space-fixed axis.  There, the
signature quantum number is defined as the eigenvalue of an operator,
%
\begin{equation}
\hat{R_x} \equiv \exp ( -i \pi {\hat{I}}_x ),
\end{equation}
%
which rotates the intrinsic state around the $x$-axis.  The $x$-axis
is parallel to the cranking axis, which coincides with one of the
principal axes of the quadrupole deformation.  $R_x$ is a good quantum
number in the cranking model because both of the Coriolis field
($-\omega_{\rm rot} {\hat{I}}_{x}$) and the Nilsson potential commute
with ${\hat{R}}_x$.  Now, if one assumes that $I_x=I$, the signature
and the parity of $I$ (or $I - \frac{1}{2}$) are equivalent.

This assumption is not justified, however, when the fluctuations
$\langle \hat{I}_z^2 \rangle$ and/or $\langle \hat{I}_y^2 \rangle$ are
not negligible compared with $I(I+1)$, e.g., 1) when quasiparticles
are excited in orbitals having large $\Omega$ quantum number and 2)
when a triaxially deformed nucleus does three-dimensional rotations.
One should be very careful in applying the cranking model to these
cases.

Moreover, these deviations from cranking picture are larger at smaller
spins.  The signature inversion phenomena, which we are going to
discuss in this paper, take place at low spins.  This is why we use in
this paper the particle-rotor model, which can treat a wider variety
of rotational motions than the cranking model.  Another advantage of
the former model is its fully quantum mechanical nature, which enables
one to calculate transition amplitudes without any approximations.

%---------------------------------------------------------------------------
\section{Band structure of a signature quartette}

We discuss about the possible configurations of two qp's in pure-$j$
orbitals.  It is important to distinguish between rotationally aligned
(RA) orbitals and deformation aligned (DA) ones.  RA orbitals are
labelled by the magnetic quantum number along the $x$ axis.  They are
signature eigenstates:
%
\begin{equation}
  \hat{R_x}|j,m_x\rangle=e^{-i \pi m_x}|j,m_x\rangle.
\end{equation}
%
Orbitals
  $\lbrace \ket{j,j}, \ket{j,j-2}, \cdots , \ket{j,-j+1} \rbrace$
have a signature $e^{-i \pi j}$, while orbitals 
  $\lbrace \ket{j,j-1}, \ket{j,j-3}, \cdots , \ket{j,-j} \rbrace$
have the opposite signature $e^{i \pi j}$.
The former set of orbitals are called the {\em favored} signature
states because they include the orbital $\ket{j,j}$, which has the
smallest expectation value of the Coriolis field.  The latter ones are
called the {\em unfavored} signature states.

For DA orbitals, the signature eigenstates are obtained by taking
linear combinations of a pair of degenerated Nilsson orbitals:
%
\begin{equation}
  \ket{j,|m_z|,R_x=e^{\pm i \pi j}} = \frac{\ket{j,m_z} \pm \ket{j,-m_z}}{\sqrt{2}}.
\end{equation}
%
It can be shown through perturbative calculation that orbitals with
$R_x$ = $e^{- i \pi j}$ ($e^{i \pi j}$) are favored (unfavored) by the
Coriolis field $-\omega_{\rm rot} \hat{j_z}$, in agreement with the
nomenclature of favored/unfavored of the RA orbitals.

With two qp's (a proton and a neutron in the following), there are
four states in a pair of angular momenta $I$ and $I+1$, i.e., 
  $\lbrace ff, fu, uf, uu \rbrace$ 
where $fu$ means a state in which the proton qp is in the favored
signature orbital and the neutron qp is in the unfavored one and so
on.  They are called the {\em signature quartette}.

In fig.\ \ref{f_q}, we show how the quartette are grouped into
rotational bands. Both the proton and the neutron qp's are assumed to
be in the h$_{11/2}$ orbital.  The first portion shows the case in
which both qp's are in RA orbitals. States with $f$ and $u$,
i.e. $\ket{j,m_x=j}$ and $\ket{j,m_x=j-1}$ respectively, are regarded
to construct different intrinsic structures and form different
rotational bands.

The second portion treats a combination that proton (neutron) qp is in
RA (DA) orbital.  The third portion is obtained by exchanging the
orbitals of the proton and the neutron.  In both cases, the difference
of the signature of the RA orbital makes different bands while that of
the DA orbital is responsible for the signature staggering of energy.
(The states with a qp in $f$- and $u$-signature DA orbitals can be
regarded as having the same internal structure but differing only in
the way of collective rotation.)  The phase of the signature
staggering of the higher-lying band is opposite to that of the
lower-lying one.  This fact may be useful to tell which excited band
is the signature partner of a band in odd-odd nuclei.

The last portion treats a case in which both qp's are in DA orbitals.
In this case the independent qp picture employed by the cranking model
is no more correct.  The signature quartette should be mixed among
them to construct states having $K$ = $| \Omega_{\rm p}$ +
$\Omega_{\rm n} |$ as a good quantum number rather than the signature
of each qp, because the rotational energy depends on $K$ as $E_I$
$\propto$ $I(I+1)-K^2$.  This is another shortcoming of the cranking
model.  In this respect, too, it is worth while using the
particle-rotor model to examine the applicability of the cranking
picture to signature inversion phenomena.

% \newpage      % here comes figure 1.

%----------------------------------------------------------------------------
\section{The effect of the spin-spin interaction between
a proton and a neutron quasiparticle}

It is known that the proton-neutron interaction prefers spin-triplet
state to singlet one.  It is consistent with an empirical law called
the GM rule \cite{GM58}, i.e,.  a state with
  $K$ = $K_{>}$ $\equiv$ $\Bigl| |\Omega_{\rm p}| + |\Omega_{\rm n}| \Bigr|$
usually has lower energy than a state with
  $K$ = $K_{<}$ $\equiv$ $\Bigl| |\Omega_{\rm p}| - |\Omega_{\rm n}| \Bigr|$
when $j$ = $l$ + $\frac{1}{2}$.
Because the $K$ quantum number and single-particle signatures are
incompatible for more than one qp configurations, the spin-spin
interaction may affect the signature dependence of some quantities.
Let us estimate the size of the effects for a non-correlated state of
a proton and a neutron:
%
\begin{equation}
  |(l \mbox{\small $\frac{1}{2}$} ) j \Omega_{\rm p} \rangle_{\rm p}
  |(l \mbox{\small $\frac{1}{2}$} ) j \Omega_{\rm n} \rangle_{\rm n}.
\end{equation}
%
The probabilities of spin singlet and triplet components of this state
are, respectively,
%
\begin{eqnarray}
  {\rm Prob}. (S=0) & =  \frac{1}{4} & -
  \frac{\Omega_{\rm p} \Omega_{\rm n}}{4j^2}, \\
  {\rm Prob}. (S=1) & =  \frac{3}{4} & +
  \frac{\Omega_{\rm p} \Omega_{\rm n}}{4j^2}.
\end{eqnarray}
%
In our case of $^{124}$Cs, by substituting
  $j$ = $\frac{11}{2}$, $\Omega_{\rm p}$ = $\frac{1}{2}$,
and $\Omega_{\rm n}$ = $\frac{7}{2}$,
one obtains the difference of the probabilities of only 3 \%.
Thus the difference in the diagonal matrix elements (the GM splitting)
seems very small.  On the other hand, the non-diagonal effects are not
negligible: the spin-spin interaction enhances normal-sign signature
splitting according to our numerical study.

%-----------------------------------------------------------------------------
\section{The Particle-Triaxial-Rotor Model for odd-odd nuclei}

The hamiltonian is
%
\begin{equation}
  \hat{H} = \sum_{\kappa=1}^{3} \frac{\hbar^2}{2 {\cal J}_\kappa}
  \hat{R}_{\kappa}^2 + \hat{h}_{\rm Nilsson} + V_{\rm pn},
\end{equation}
%
where
\footnote{
  There may be difference by a factor of $(1/2)^{1/2}$ in the definition
  of the strengths of $V_{\rm pn}$ between the numerical
  calculations\cite{BP76,SR90,SR92} and the equation
  (\ref{e_vpn})\cite{BP76,SR90,SR92}.
}
%
\begin{eqnarray}
 {\cal J}_{\kappa} & = & \frac{4}{3}{\cal J}_0
 \sin^2(\gamma + \frac{2 \pi}{3} \kappa ) \ \ \ \ \ \ \;
 \mbox{(irrotational flow)} \label{e_moi} \\
 \hat{R}_{\kappa} & = & \hat{I}_{\kappa} - \hat{j}_{\kappa}^{\rm p}
 - \hat{j}_{\kappa}^{\rm n},\\
 V_{\rm pn} & = & 4 \pi \sqrt{\frac{\pi b^3}{\sqrt{2}}} \delta (
 \vec{r}_{\rm p} - \vec{r}_{\rm n} ) ( u_0 + u_1 \vec{\sigma}_{\rm p} \cdot
 \vec{\sigma}_{\rm n} ). \label{e_vpn}
 \end{eqnarray}
%
The subspace in which we diagonalize $\hat{H}$ is
%
\begin{equation}
  \mbox{
    (1qp in $\pi$h$_{11/2}$) $\times$ (1qp in $\nu$h$_{11/2}$) $\times$
    ($|R_3|$ = 0, 2, 4, 6)
  }
\end{equation}
%
The parameters of the model take on the following values:
%
\begin{enumerate}
\item
$\epsilon_2=0.22$ $\leftarrow$ Experiment, 
a Nilsson-Strutinski calculation\cite{Sh93}
\item
$\gamma=23^{\circ}$ $\leftarrow$ fitting
$E(2_{\gamma}^{+})/E(2_{\rm gr}^{+})$
in neighboring even-even nuclei.\footnote{
Assuming {\em harmonic} $\gamma$ vibration, one obtains $\gamma=43^{\circ}$.
(Too large for harmonic vibration.)
}
\item
$u_0=-7.2$MeV, $u_1=-0.80$MeV
$\leftarrow$
$u_1$ was obtained by fitting GM shifts\cite{BP76}.
$u_0$ was recommended by Semmes and Ragnarsson for $^{120}$Cs to
reproduce the signature inversion\cite{CLJ92}.
\item
$3\hbar^2/{\cal J}_0=E(2_{\rm gr}^{+})=200$keV $\leftarrow$
fitting $\partial E / \partial I$ at
$I \cong 16$ where the signature inversion occurs.
\item
$G_{\rm p}$ and $G_{\rm n}$ are multiplied by 0.95 to take into
account the blocking effect.  $G_{\rm p}$ $\rightarrow$ 18.24 MeV,
$G_{\rm n}$ $\rightarrow$ 7.03 MeV.  $(15Z)^{1/2}$ proton and
$(15N)^{1/2}$ neutron orbital pairs below and above each fermi level
are taken into account.
\end{enumerate}
%
More details of the model are explained in refs.\cite{SR90,SR92}.

%-----------------------------------------------------------------------------
\section{The results of calculations}

For experimental information, see refs. \cite{KFH90,KFH92}.  In
fig. \ref{f_ss1}, we show signature splitting calculated without
$V_{\rm pn}$.  The splitting is defined by interpolating the energy
spectrum for odd-$I$ sequence of states to even-$I$ spins (using
6-point Lagrange polynomial interpolation formula).  Experimental
values are denoted by dots connected by a solid curve.  Below $I$ =
16, its sign is negative (i.e., the signature inversion occurs).
Calculated splittings with $\gamma$ = 0$^{\circ}$ and 23$^{\circ}$
rotor are also shown.  In the figure, ``$\gamma$ = $-23^{\circ}$''
means that irrotational-flow moment of inertia (eq.\ (\ref{e_moi})) is
used while ``$\gamma$ = $+23^{\circ}$'' designates that ${\cal J}_x$
and ${\cal J}_y$ are exchanged\cite{HM83}.  With ``$\gamma$ =
$-23^{\circ}$'', the signature splitting is reproduced quite well.

In fig. \ref{f_ss2}, calculations without $V_{\rm pn}$ are shown.
The agreements with the experiment are very poor.

The ratio B(M1;$I \rightarrow I-1)$/B(E2;$I \rightarrow I-2$) is shown
in fig.\ \ref{f_em}. The employed parameters are those used in 
fig.\ \ref{f_ss1} with ``$\gamma$ = $-23^{\circ}$''.  One sees that
the signature dependence of transition amplitudes are reproduced well
with the same parameter set.

One of the disagreement of our calculation with experiment is that the
spectrum near the bandhead is too compressed.  To improve the
spectrum, we tried introducing the Coriolis attenuation factor.
Attenuation factor of 1 means no artificial attenuation while that of
0 means that the Coriolis coupling is not taken into account.  In
fig.\ \ref{f_ca}, the effect of Coriolis attenuation factor on the
spectrum is shown.  One does not see any remarkable improvements.  In
fact, this compression of the spectrum is a long-standing problem
haunting probably any particle-rotor-type models. More elaborate
investigations will be necessary to solve the problem.  Anyway, the
transition energy (i.e., the rotational frequency in the cranking
picture) is in good agreement with the experiment around 
$I$ $\sim$ 16 where the inversion occurs.  Therefore, our arguments
concerning the reproduction of the inversion do not suffer much from
this compression of the spectrum near the bandhead.


\vspace{\baselineskip}

\noindent
{\large {\bf Acknowledgements}}

The author thanks I.~Ragnarsson and P.~Semmes for providing the
particle-triaxial-rotor model code for odd-odd nucleus.  He is also
grateful to Drs. K.~Furuno and T.~Komatsubara for useful information
and discussion on the subject.  Most of the numerical calculation for
this talk was done on the computer VAX 6440 of the Meson Science
Laboratory, Faculty of Science, University of Tokyo.  The author was
supported in part by the Grant-in-Aid for Scientific Research
(no. 05740167) from the Ministry of Education, Science and Culture of
Japan.

%--------------------- references ------------------------------------%

{\small     % \footnotesize
\begin{thebibliography}{99}
\bibitem{GM58} % GM rule
         C.J.~Gallagher and S.A.~Moszkowski, Phys.Rev. 111 (1958) 1282.
\bibitem{Sh93} % equilibrium deformation of 124Cs
         Y.R.~Shimizu, private communication.
\bibitem{BP76} % The effective neutron-proton interaction
         J.P.~Boisson and R.~Piepenbring, Phys.Rep. {\bf 26} (1976) 99.
\bibitem{CLJ92} % experiment of 120Cs
         B.~Cederwall, F.~Lid\'en, A.~Johnson, L.~Hildingsson, R.~Wyss,
         B.~Fant, S.~Juutinen, P.~Ahonen, S.~Mitarai, J.~Mukai, J.~Nyberg,
         I.~Ragnarsson, and P.~Semmes, Nucl. Phys. {\bf A542} (1992) 454.
\bibitem{SR90} % Strasbourg conference
         P.B.~Semmes and I.~Ragnarsson, Conf. high spin physics and gamma-soft
         nuclei, Pittsburg, September 1990
         (World Scientific, Singapore, 1991) p.500.
\bibitem{SR92} % Strasbourg conference
         P.B.~Semmes and I.~Ragnarsson, Conf. future directions in nuclear
         physics with 4 $\pi$ detection systems of the new generation,
         Strasbourg, March 1991 (AIP, New York, 1992) p.566.
\bibitem{KFH90} % experiment of 124Cs (short note)
         T.~Komatsubara, K.~Furuno, T.~Hosoda, J.~Espino, J.~Gascon,
         G.B.~Hagemann, Y.~Iwata, D.~Jerrestam, N.~Kato, T.~Morikawa,
         J.~Nyberg, G.~Sletten, and P.O.~Tj\/om,
         Z. Phys. {\bf A335} (1990) 113.
\bibitem{KFH92} % experiment of 124,126Cs and 124La
         T.~Komatsubara, K.~Furuno, T.~Hosoda, J.~Mukai, T.~Hayakawa,
         T.~Morikawa, Y.~Iwata, N.~Kato, J.~Espino, J.~Gascon, N.~Gj\/orup,
         G.B.~Hagemann, H.J.~Jensen, D.~Jerrestam, J.~Nyberg, G.~Sletten,
         B.~Cederwall, and P.O.~Tj\/om, Conf.  rapidly rotating nuclear
         structure, Tokyo, October 1992.
\bibitem{HM83} % gamma trick
         I.\ Hamamoto and B.R.\ Mottelson, Phys.\ Lett.\ {\bf B132} (1983) 7.
\end{thebibliography}
}

% \newpage

\noindent {\bf FIGURE CAPTIONS}

\newcounter{figno}
\begin{list}
{FIGURE \arabic{figno}. }{\usecounter{figno}
   \setlength{\labelwidth}{2cm}
   \setlength{\labelsep}{0.5mm}
   \setlength{\leftmargin}{8.5mm}
   \setlength{\rightmargin}{0mm}
   \setlength{\listparindent}{0mm}
   \setlength{\parsep}{0mm}
   \setlength{\itemsep}{0.5cm}
   \setlength{\topsep}{0.5cm}
}
\baselineskip=0.45cm
\item \label{f_q}
    Possible band structure of a signature quartette.
\item \label{f_ss1}
    The signature splitting with $V_{\rm pn}$.
\item \label{f_ss2}
    The signature splitting without $V_{\rm pn}$.
\item \label{f_em}
    M1/E2 mixing ratio of intra band transitions.
\item \label{f_ca}
    The effect of Coriolis attenuation factor on energy.
\end{list}

\end{document}