\documentstyle[psfig]{article} \newcommand{\text}{\rm} \begin{document} \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} \bigskip\bigskip \hbox{ \vbox{\psfig{figure=logo25a.eps,clip=}} \hspace{.5in} %\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} \vspace{.2in}}} \newpage \vspace*{1in} \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