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\title{EPR and evidence for two structural nonequivalent states 
of Tl$^{2+}$(II)
paramagnetic centers near noncentral positions in Rb$_2$SO$_4$ 
and Cs$_2$SO$_4$ crystals}
\author{\it{\ G.V. Mamin and V.N. Efimov} \\
Kazan State University, Kremlevskaya, 18, Kazan 420008, Russia \\
{\em E-mail: george.mamin@ksu.ru, vladimir.efimov@ksu.ru}
}
\date{{\small Received September 10, 1999\\
Accepted October 5, 1999}}



\maketitle

\pagestyle{myheadings} 
%\markright{{\sc Magnetic Resonance in Solids.  Electronic Journal  }
%{\bf 3}, {\sc 2 (1999)}\hfill} \thispagestyle{empty}
\markright{{\sl Magnetic Resonance in Solids.  Electronic Journal.
 Vol. 3, No. 2 (1997)}\hfill}
%\markright{{ Magnetic Resonance in Solids.  Electronic Journal  }
%{ 3}, { 2 (1999)}\hfill} \thispagestyle{empty}

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%\vbox{\parbox{2.51in}{\LARGE \bf {\it \ \ \ Volume} 3  \\  {\it No.} 2, 1999} 
%\vbox{\parbox{2.51in}{ \bf { \ \ \ Volume} 3  \\  { No.} 2, 1999}
\vbox{\parbox{2.51in}{\LARGE\bf \ \  Volume 3  \\   No. 2, 1997}
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\begin{center}
{\bf Abstract}
\end{center}
The EPR spectrum temperature dependence of Tl$^{2+}$(II) paramagnetic
centers in Rb$_2$SO$_4$, Cs$_2$SO$_4$ crystals have been investigated. At $%
100K>T>40K$ the local symmetry of Tl$^{2+}$(II) paramagnetic centers is $C_1$%
. The broadening and doubling of EPR spectrum lines with decreasing of $T<40K
$ were observed. The angular dependence of lines positions indicate that two
structural nonequivalent positions of Tl$^{2+}$(II) paramagnetic center
exist at $T<20K$. The measurement of dielectric constant of crystals
displayed the absence of phase transition in this temperature range. The
transformation of EPR spectrum is explained by motion of Tl$^{2+}$(II)
paramagnetic center between two structural nonequivalent states arising near
noncentral positions. Parameters of the potential have been extracted by
fitting of the temperature dependence of EPR spectrum. Two nonequivalent
states are possible due to large difference between ionic radius of Tl$^{2+}$
and Rb$^{+}$, Cs$^{+}$.



\begin{description}
\item[PACS]  : 71.55; 76.30

\item[Keywords]: Electron paramagnetic resonance, noncentral ions,local dynamics
\end{description}

\newpage


The EPR method detects the noncentrality of the impurity paramagnetic Tl$%
^{2+}$(II) ions in K$_2$SO$_4$, Rb$_2$SO$_4$ and Cs$_2$SO$_4$ crystals \cite
{Efimov}. This defects perform the activation motion within the two
potential wells along the $c$ axis of the crystal. The activation motion
parameters ($E_a$=0.06 eV, $\tau _0$=5$\cdot $10$^{-13}$s) corresponds to
noncentral jumps \cite{Efimov,Mamin}. This defects motion leads to
broadening and doubling of EPR spectrum lines. The temperatures of the
broadening are different for each crystals ($T_{\mathrm{{K}_2{SO}_4}}=$30-50
K, $T_{\mathrm{{Rb}_2{SO}_4}}=$80-110 K, $T_{\mathrm{{Cs}_2{SO}_4}}=$70-100
K).

However, at low temperature ($T\sim $15-60 K) in Rb$_2$SO$_4$ and Cs$_2$SO$%
_4 $ crystals the additional doubling of EPR lines was observed \cite{Efimov}%
. This splitting testify the appearance of two structural non-equivalent Tl$%
^{2+}$(II) centers (Fig. \ref{figtempdep}, \ref{figAngle}). Since the phase
transitions exists in some crystals of K$_2$SO$_4$ group, we suppose that
the additional anomaly of EPR spectra can be associated with the structural
phase transition. This suggestion have been confirmed by the anomaly in
temperature dependence of $B_2^0$ spin-hamiltonian parameter of Mn$^{2+}$
ions in Cs$_2$SO$_4$ \cite{Mn}. In the present paper we report additional
studies of low temperature anomalies of Tl$^{2+}$(II) EPR spectra in Rb$_2$SO%
$_4$ and Cs$_2$SO$_4$ crystals.

To determine the nature of anomalies we have measured the temperature
dependence of dielectric constant $\varepsilon $ in Rb$_2$SO$_4$ and Cs$_2$SO%
$_4$ crystals with 0.1-0.5\% thallium impurity in the temperature range
which corresponds to the transformation of EPR spectra (Fig. \ref{figepsilon}%
). However, no anomalies of the temperature dependence of $\varepsilon $
have been detected. This confirms the absence of ferroelectric phase
transitions in Cs$_2$SO$_4$ crystals.

\begin{figure}[tbph]
\centerline{\psfig{figure=fig1a.eps,width=4.75 in,clip=} }
\caption{ The temperature dependence of dielectric constant in a) Rb$_2$SO$%
_4 $ and b) Cs$_2$SO$_4$ crystals }
\label{figepsilon}
\end{figure}

The temperature dependence of Tl$^{2+}$(II) ions EPR spectrum in Cs$_2$SO$_4$
crystals is shown in Fig. \ref{figtempdep}a. The similar transformation of
line shape is appeared when the EPR spectrum averaged by the motion of
defects between two states with not equal occupation probabilities \cite
{Bloch}. This transformation can be described by Bloch equations, which were
modified for defects motion processes \cite{Bloch}. The magnetic moment,
whose imaginary part is proportional to EPR absorption, can be written as

\begin{equation}
\widehat{M}=K\frac{f_a\left( H-\widehat{H}_b\right) +f_b\left( H-\widehat{H}%
_a\right) }{\left[ \left( H-\widehat{H}_a\right) +iAP_{ab}\right] \left[
\left( H-\widehat{H}_b\right) +iAP_{ba}\right] +A^2P_{ab}P_{ba}},
\label{bloch}
\end{equation}
where $P_{ab}$ - the probability of defect jump from $a$ state to $b$ state, 
$P_{ba}$ - the jump probability from $b$ state to $a$ state, $f_a=\frac{%
P_{ba}}{P_{ab}+P_{ba}}$ and $f_b=\frac{P_{ab}}{P_{ab}+P_{ba}}$ are the
populations of potential minima,$\widehat{H}_{a,b}=H_{a,b}+i\frac 1{AT_2}$ , 
$H_a$, $H_b$ - magnetic field corresponding to the EPR of defect in $a$ or $%
b $ state, $T_2$ -spin-spin relaxation time, $A=\frac{4\pi \nu _{EPR}}{%
H_a+H_b} $. The intensities of line for defects in $a$ and $b$ states are
different at low temperatures indicating minima of multi-well potential with
various depths. The probabilities $P_{ab}$ and $P_{ba}$ can be written from
Arenius activation law

\begin{equation}
P_{ab}=\frac 1{\tau _0}\exp \left( -\frac{E_a}{kT}\right) ,P_{ba}=\frac
1{\tau _0}\exp \left( -\frac{E_a-E_0}{kT}\right) ,  \label{P}
\end{equation}
where $E_a$- the activation energy, $E_0$ - the difference between depths of
potential minima. We substituted \ref{P} in the equation \ref{bloch} and
calculated EPR line shape. The result of fitting is shown on Fig. \ref
{figtempdep}a by dotted line. One can see that experimental EPR spectra can
be approximated by the model of Tl$^{2+}$(II) ions motion in non symmetric
two well potential with parameters shown in table 1.

\begin{figure}[tbph]
\centerline{\psfig{figure=fig2a.eps,width=4.5in,clip=} }
\caption{ The temperature dependence of EPR spectra transformation in case of
a) low-field and b) high-field transitions at 
${\mathbf H}\vert\vert{\mathbf a}$. 
The solid line - experimental data, dotted line - approximation
by equation \protect\ref{bloch}. }
\label{figtempdep}
\end{figure}
\begin{center}
Table 1. Parameters of thallium motion in two-well energy
potential.
\begin{tabular}{cccc}
\hline
Crystals & $E_a$ (eV) & $E_0$ (eV) & $\tau _0$ (10$^{-11}$s) \\ \hline
Rb$_2$SO$_4$ & 0.026$\pm $0.001 & 0.004$\pm $0.0001 & 1$\pm 0.3$ \\
Cs$_2$SO$_4$ & 0.019$\pm $0.001 & 0.002$\pm $0.0001 & 1.1$\pm 0.3$ \\ \hline
\end{tabular}
\end{center}


If this transformation corresponds to the defect motion, but not phase
transition, the splitting temperature $T^{*}$ must depend on $\Delta
H=H_a-H_b$. The $\Delta H$ value was measured at low temperature, when jumps
between states of defects are frozen. The Tl$^{2+}$ ions have $%
[Xe]f^{14}5d^{10}6s^1$ electron shell configuration. Because of the strong
contact interaction of $6s$ electron with ion's nucleus, the hyperfine
structure constant is big 
($A_{hfs} \approx$ 110 GHz ${>>}g\beta H$). 
The big $A_{hfs}$ value leads to two EPR hyperfine
transitions with different $\Delta H$ values. The temperature dependencies
of EPR line form for two hyperfine transitions with $\Delta H$= 162 Oe and $%
\Delta H$= 221 Oe are shown in Fig. \ref{figtempdep} a, b. However, as one
can see, the temperature range of spectrum transformation is wider, than
possible shift of $T^{*}$. The $E_a$, $E_0$ and $\tau _0$ parameters were
obtained from approximation of the line shape transformation of high-field
hyperfine EPR transition by equations \ref{bloch}, \ref{P}. The values of
this parameters are nearly equal to those obtained for the low-field
hyperfine transition (Fig. \ref{figtempdep} a). This indicates, that Tl$%
^{2+} $(II) ions in Rb$_2$SO$_4$, Cs$_2$SO$_4$ crystals are involved in
additional motion. This type of motion is absent in K$_2$SO$_4$ crystals but
appears in Rb$_2$SO$_4$, Cs$_2$SO$_4$ crystals, when the radius of thallium
ions is less than the radius of the substituted cation ($R_{Tl^{2+}}$=1.36
\AA , $R_{K^{+}}$=1.33 \AA , $R_{Rb^{+}}$=1.49 \AA , $R_{Cs^{+}}$=1.65 \AA\ 
\cite{Rions}). Therefore we can assume, that this motion can appear due to
of Coulomb interaction of thallium ion with surrounding ions. The calculated
potential of Tl$^{2+}$(II) ions for K$_2$SO$_4$, Rb$_2$SO$_4$, Cs$_2$SO$_4$
crystals is shown in Fig. \ref{figPotencial}

\begin{figure}[tbph]
\centerline{\psfig{figure=fig3a.eps,width=4.5in,clip=} }
\caption{ The energy potential of Tl$^{2+}$(II) ions in the cation position
of a) K$_2$SO$_4$, b) Rb$_2$SO$_4$, c) Cs$_2$SO$_4$ crystals. The energy
unit is eV. The arrows show the direction of z axis projection of $\widehat{%
g^2}^{*}$-tensor on ($ab$) plane of crystal. }
\label{figPotencial}
\end{figure}

In our calculations we used the following assumptions:

1. The calculation of the electrostatic field was implemented in point
charge model.

2. The ion's position was fixed.

3. The electron shell of oxygen ions was assumed to be a sphere.

4. The energy repulsion between oxygen and thallium ions was taken as

\begin{equation}
E=E_o\exp \left( -\frac{R_{Tl-O}}{\rho _o}\right) ,
\end{equation}
where $E_O$=$E_{Cl}$=1.2$\cdot $10$^4$ eV, $\rho _O$=$\rho _{Cl}$-$R_{Cl}$+$%
R_O$=0.23 \AA . The $E_{Cl}$ and $\rho _{Cl}$ values for KCl crystals were
taken from ref.\cite{KClpar}.

5. The existence of high temperature non central motion was out of our
consideration.

One can see from Fig. \ref{figPotencial} that the energy potential in Rb$_2$%
SO$_4$, Cs$_2$SO$_4$ crystals has two minima in ($ab$) plane with various
depths. The big values of $E_a$ and $E_0$ parameters can be explained by
assumed simplifications. The values of of $E_a$ and $E_0$ can decrease if
one will take account the high temperature non central motion of thallium
ions.

In order to define the changes in thallium local environment it is necessary
to know the change of crystal field gradient. The gradient variation can
changes the $\widehat{g}$- and $\widehat{A}$-tensor axes directions. In the
assumption, that the axes of $\widehat{g}$ and $\widehat{A}$ tensor coincide 
\cite{gtensor}, from the angular dependence of line positions for one of
hyperfine transitions one can get the axes directions of $\widehat{g}^{*}$%
-tensor with effective spin $S^{*}$=1/2. The angular dependence of line
position is shown in Fig. \ref{figAngle}. At temperatures higher than $T^{*}$
the point symmetry of Tl$^{2+}$(II) center is $C_1$. With cooling, below $%
T^{*}$ the new thallium centers appears. One of the Tl$^{2+}$(II) centers ($%
a $) have higher line intensity than the other ($b$ center). However the
point symmetry of both centers remains unchanged. Main values and axes
directions of effective $\widehat{g^2}^{*}$-tensors are different for Rb$_2$%
SO$_4$, Cs$_2$SO$_4$ crystals. The projection of $z$-axis $\widehat{g^2}^{*}$%
-tensors of ($ab$) plane are shown in Fig. \ref{figPotencial}. One can see
that $z$-axis projection direction corresponds to direction of maximal
gradient of crystals field.

\begin{figure}[tbph]
\centerline{\psfig{figure=fig4a.eps,width=4.5in,clip=} }
\caption{ The angular dependence of EPR spectrum line positions for
low-field hyperfine transitions of Tl$^{2+}$(II) ions in a) Rb$_2$SO$_4$ and
b) Cs$_2$SO$_4$ crystals. }
\label{figAngle}
\end{figure}

At $T>T^{*}$ the anomaly of the EPR linewidth angular
dependence of Tl$^{2+}$(II) was observed (Fig. \ref{figAngleWidth}). In the
case of averaging motion , EPR linewidth can be written as \cite{Bloch}

\begin{equation}
\delta H=\frac{\left( H_a-H_b\right) ^2}{8AP_{act}}+\delta H_0,
\label{width}
\end{equation}
where $\delta H_0$ - homogeneous linewidth (<2 Oe.), $H_a$ and $%
H_b$ magnetic fields, which corresponds to the EPR line position of centers $%
a $ and $b$. Neglecting the temperature dependence of spin hamiltonian
parameters, one can define $H_a$ and $H_b$ from the angular dependence of
line positions at $T<T^{*}$ (Fig. \ref{figAngle}). We can calculated the
angular dependence of EPR linewidth (Fig. \ref{figAngleWidth} solid line).
One can see that the simple model describes the change of EPR linewidth
fairly well.

\begin{figure}[tbph]
\centerline{\psfig{figure=fig5a.eps,width=4.5in,clip=} }
\caption{ The angular dependence of EPR linewidth of Tl$^{2+}$(II) ions in
a) Rb$_2$SO$_4$ and b) Cs$_2$SO$_4$ crystals. }
\label{figAngleWidth}
\end{figure}

So the low temperature anomaly of Tl$^{2+}$(II) ions EPR spectrum in Rb$_2$SO%
$_4$, Cs$_2$SO$_4$ crystals can be explained by motion of Tl$^{2+}$(II)
paramagnetic centers between two structural nonequivalent states arising
near noncentral positions. These two structural nonequivalent states can be
due to Tl$^{2+}$(II) ions displacements under the influence of electrostatic
force.

The work was in part supported by the Russian State Foundation for Basic
Research, Grant No 98-02-18037.

\begin{thebibliography}{9}
\bibitem{Efimov}   Efimov V.N., Mamin G.V.  {\it Mod. 
Phys. Lett. B} {\bf 11}, 579 (1997)

\bibitem{Mamin}   Mamin G.V., Efimov V.N.
{\it Mod. Phys. Lett. B} {\bf 12}, 929 (1998)

\bibitem{Mn}  Abdulsabirov R.Ju., Stepanov V.G., Gresnev Ju.S.,
Zaripov M.A. {\it Fizika Tverdogo Tela} {\bf 14} 1816 (1972)
(in Russian)

\bibitem{Bloch} Carrington A., McLachlan A.D. {\it Introduction to
magnetic resonance with application to chemistry and chemical physics.}
Harper and Row publisher, 1967

\bibitem{Rions}  Bokij G.B. {\it Kristallohimija.} Moskow, Nauka, 
1978 (in Russian)

\bibitem{KClpar} Kittel C. {\it Introduction to solid state physics.}
Fourth edition, John Wiley \& Sons, Inc., NY, 1976 


\bibitem{gtensor} Eremin M.V., Silkin  N.I. 
{\it Phys. Stat. Sol.(b)} {\bf 84}, 803  (1977)
\end{thebibliography}

\end{document}
