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\title{Spin-lattice relaxation of Nd$^{3+}$ and Tb$^{3+}$ in phosphate and sodium
borate silica glasses}
\author{{\it \ B.M. Epel, S.B. Orlinskii, 
R.M. Rakhmatullin and and Yu.K. Rosenzweig 
} \\
%EndAName
Kazan State University, Kremlevskaya str. 18, Kazan 420008, Russia \\
{\em E-mail: boris.epel@ksu.ru}}
\date{{\small Received April 20, 1999}\\
Accepted April 23, 1999}
\maketitle

\pagestyle{myheadings} 
\markright{{\sc Magnetic Resonance in Solids.  Electronic Journal  }
{\bf 3}, {\sc 1 (1999)}\hfill} \thispagestyle{empty}

\bigskip\bigskip

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\vbox{\parbox{2.51in}{\LARGE \bf {\it \ \ \ Volume} 3  \\  {\it No.} 1, 1999}
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\vspace*{1in}

\begin{center}
{\bf Abstract}
\end{center}

The experimental study of the electron spin-lattice relaxation (SLR) of Nd$%
^{3+}$ and Tb$^{3+}$ ions in phosphate and sodium borate silica glasses was
performed. Measurements were made with electron spin echo technique.
Temperature dependences of spin-lattice relaxation of the rare earth ions
reveal the contribution of different relaxation mechanisms. A stepped like
change of the SLR rate observed for  Tb$^{3+}$ and Nd$^{3+}$ ions was
explained by cross relaxation.

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

\item[Keywords]  : glass, EPR, ESE, spin-lattice relaxation
\end{description}

\newpage

\section{Introduction}

The study of the spin-lattice relaxation (SLR) of impurity ions in glasses
is of importance as it allows one to obtain information about their spatial
distribution and vibrational states in amorphous solids. The spectral
density of vibrational states in inorganic glasses at low energies deviates
from the Debye spectrum of crystalline solids. This is commonly supposed to
be caused by contributions of two-level systems and localized vibrational
excitations. Recently, a cross relaxation model was developed \cite{FTT}
that explained the observed SLR features of Yb$^{3+}$ ions in the phosphate
glass. It was supposed that the phase transition of tunneling two-level
systems (TLS) in the orientational pseudospin glass occurs at low
temperatures. The goal of this study was to find out if the similar features
in SLR can be detected with another impurity ions and on another glass
structure.

\section{Experimental}

The SLR of Tb$^{3+}$ and Nd$^{3+}$ ions in oxide glasses was studied.
Compositions of glasses are given in the Table 1. Samples were synthesized
in Vavilov Science Center in St. Petersburg.

Measurements of SLR times T1 were made on an electron-spin echo (ESE)
spectrometer with a working frequency of 9.4 GHz and a time resolution of 10$%
^{-7}$ s, in the temperature range of 1.6 - 7 K. The SLR time was measured
with a technique in which two pulses forming the echo signal followed by a
long microwave pulse saturating the spin system. The pulse sequence was long
saturating pulse (1 ms) - $t$ - 2$\pi $/3 - $\tau $ - 2$\pi $/3, where $\tau 
$ was kept constant, while $t$ was varied. Experiments showed that the
recovery profile of the ESE amplitude after the saturating pulse was
non-exponential so that relaxation times were measured from the exponential
tail. To avoid the sample heating some measurements were done in the liquid
helium medium.

\bigskip\ 

\begin{center}
Table 1. Compositions of studied glasses

\begin{tabular}{c|c|c|c}
Short & Concentration, mol\% & \multicolumn{2}{|c}{x Re$_2$O$_3$, mol\%} \\ 
\cline{3-4}
notation &  & Tb$_2$O$_3$ & Nd$_2$O$_3$ \\ \hline
PL & 75P$_2$O$_5$+(25-x)La$_2$O$_3$+x Re$_2$O$_3$ & 1 & 1 \\ 
&  & 0.3 & 0.3 \\ 
&  & 0.1 & 0.1 \\ 
&  &  & 0.03 \\ \hline
SNBL & 59SiO$_2$+20Na$_2$O+20B$_2$O$_3$+x Re$_2$O$_3$+ & 1 &  \\ 
& (1-x)La$_2$O$_3$ & 0.3 &  \\ 
&  & 0.1 &  \\ 
&  & 0.03 & 
\end{tabular}

\bigskip\ 
\end{center}

\subsection{Nd$^{3+}$ in the PL glass}

Fig.1 shows the typical decay profile of SLR time of Nd$^{3+}$ in the semi -
logarithmic scale. The echo detected EPR (EDEPR) spectrum of Nd$^{3+}$ in

\begin{figure}[tbhp]
\centerline{\psfig{figure=pic1nnn.eps}} %, width=3.5in, clip=} }
\caption{The recovery profile of the ESE
amplitude after the saturating pulse. a - $T$ = 4K; b - $T$ = 1.6K. Nd$^{3+}$
in PL, $x=0.3mol\%$ $V(0)-V(t)=V(0)exp(-t/T1)$, where $V(t)$, $V(0)$ are
echo amplitudes at time t, zero; $T_1$ is the spin - lattice relaxation
time.}
\label{}
\end{figure}


\begin{figure}[tbhp]
\centerline{\psfig{figure=pic2nnn.eps}} %, width=3in, clip=} }
\caption{EDEPR spectra of Nd$^{3+}$ ions in the
PL glass. a - EDEPR spectra at time delays between pulses (from top) 5.14,
4.14, 3.19, 2.14, 0.9 mks. Oscillations are due to modulation effects. b -
The solid line is the EDEPR spectrum at 2$\tau $ = 8.5 mks, connected points
are the reconstructed spectrum $V(0)$. The concentration x = 0.03 mol \%, $T$%
=1.6 K.}
\label{}
\end{figure}

the PL glass presented in fig.2. The strong modulation due to anisotropic
hyperfine interaction of Nd$^{3+}$ ions with close $^{31}$P nuclei observed
in the EDEPR spectrum. To avoid distortions, caused by the modulation,
measurements were done at the increased time delay $\tau $. The temperature
dependences of SLR rates $T_1^{-1}$ of Nd$^{3+}$ ions are presented in
fig.3. SLR measurements were done at three magnetic field points 0.325 T,
0.434 T, 0.56 T. Because there is no any significant dependence of SLR times

\begin{figure}[tbhp]
\centerline{\psfig{figure=pic3nnn.eps,width=4.5in,clip=} }
\caption{ Temperature dependences of spin - lattice relaxation rates of Nd$%
^{3+} $ ions in the PL glass. Concentrations of Nd$^{3+}$ are denoted by x.
Straight lines correspond to the dependence $T_1^{-1}\sim T^n$. }
\label{}
\end{figure}

of Nd$^{3+}$ were found for the samples on magnetic field for all
concentrations, we present here the data measured at 0.56 T. At temperatures
3.5 - 5 K the dependence of $T_1^{-1}\sim T^9$ was observed. The dependence
of SLR on the concentration of Nd$^{3+}$ ions was found. We fitted
temperature and concentration dependences of SLR rates of Nd$^{3+}$ ions by
the formula 
\begin{equation}
T_1^{-1}=DT^n+CT^9,  \label{1}
\end{equation}

$^{}$where the first term is the power dependence typical for glasses, and
the second term is the Raman process. Parameters of $D$, $n$ for different
ion concentrations summarized in the table 2.

\bigskip\ 

\begin{center}
Table 2. Fitting parameters for temperature dependences of SLR rates of Nd$%
^{3+}$ ions in the PL glass.

\begin{tabular}{c|c|c|c}
Concentration, mol\% & $D$ & $n$ & $C$ \\ \hline
1 & 70.8$\pm $6.8 & 3.26$\pm $0.08 &  \\ 
0.3 & 29.5$\pm $1.6 & 2.32$\pm $0.15 & 0.0024$\pm $0.0001 \\ 
0.1 & 6.9$\pm $1.4 & 2.33$\pm $0.28 &  \\ 
0.03 & 3.3$\pm $0.4 & 1.9$\pm $0.15 & 
\end{tabular}

\bigskip\ 
\end{center}

\subsection{Tb$^{3+}$ in phosphate and sodium borate silica glasses.}

The low field part of the EDEPR spectrum of Tb$^{3+}$ ions in studied
glasses is shown in fig.4 \cite{Applied}. The shape of the low field EDEPR
spectrum of Tb$^{3+}$ ions was explained by the phase relaxation
influence.SLR times were measured at two maxima of the spectrum at magnetic
field points of 12 mT and 42 mT. Within an experimental error SLR rates were
independent on the magnetic fields. Temperature dependences of SLR rates of
Tb$^{3+}$ ions in SNBL glass are shown in fig.5. It can be seen that the SLR
rate $T_1^{-1}$ has the temperature independent part at a low concentration.

\begin{figure}[tbhp]
\centerline{\psfig{figure=pic4nn.eps,width=4.5in,clip=} }
\caption{ EDEPR spectra of Tb$^{3+}$ ions in glasses. a - PL + $x$Tb$_2$O$_3$%
, b - SNBL + $x$Tb$_2$O$_3$. Spectra was recorded at temperature $T$ = 1.6K,
at the microwave magnetic field b parralel to the extenal magnetic field B, 2%
$\tau $ = 0.62 mks, the working frequency 9.35 GHz. }
\label{}
\end{figure}

\begin{figure}[tbhp]
\centerline{\psfig{figure=pic5nn.eps,width=4.5in,clip=} }
\caption{ Temperature dependences of spin - lattice relaxation rates of Tb$%
^{3+} $ ions in the SNBL glass at different concentrations x and at
different magnetic fields B. $\nu $ = 9.35 GHz. The straight line is the
dependence$T_1^{-1}$ = 250 $T^2$. }
\label{}
\end{figure}

\begin{figure}[tbhp]
\centerline{\psfig{figure=pic6nn.eps,width=4.5in,clip=} }
\caption{ Temperature dependences of spin - lattice relaxation rates of Tb$%
^{3+} $ ions in the PL glass. $\nu $ = 9.35 GHz. 1 - the dependence $%
T_1^{-1} $ = 1122 $T^{1.87}$ , 2 - the dependence $T_1^{-1}$ = 269 $T^{1.38}$%
. }
\label{}
\end{figure}

Temperature dependences of SLR rates of Tb$^{3+}$ ions in the PL glass
presented in fig.6 were fitted with the formula:

\begin{equation}
T_1^{-1}=DT^n.  \label{2}
\end{equation}
Parameters of $D$, n for Tb$^{3+}$ ions in the PL glass are summarized in
Table 3.

\bigskip\ 

\begin{center}
Table 3. Fitting parameters of the temperature dependences of Tb$^{3+}$ ions
in the PL glass.

\begin{tabular}{c}
Concentration, mol\% \\ \hline
0.3 \\ 
0.1
\end{tabular}
\begin{tabular}{|c|}
$D$ \\ \hline
1122$\pm $108 \\ 
269$\pm $19
\end{tabular}
\begin{tabular}{c}
$n$ \\ \hline
1.87$\pm $0.08 \\ 
1.38$\pm $0.055
\end{tabular}

\bigskip\ 
\end{center}

\section{Discussion}

Electron SLR models of paramagnetic centers in glasses were considered in
several works. Orbach with coworkers \cite{Orbacha}, \cite{Orbachb}
developed the theory of SLR taking into account a localization of vibrations
in amorphous media. They used the concept of localized phonons and fractons
and obtained temperature dependences of SLR rates for Raman and direct
processes. In case of localized fractons authors \cite{Orbachb} deduced for
the Raman like process

\begin{center}
\begin{equation}
\begin{tabular}{l}
$\left( T_1^{ave}\right) ^{-1}\propto T^{4.63}$ for Kramers ions \\ 
$\left( T_1^{ave}\right) ^{-1}\propto T^{2.63}$ for non Kramers ions
\end{tabular}
\label{3}
\end{equation}
\end{center}

In case of localized phonons the temperature dependences of SLR have the
same function as in crystalline solids for the Raman process

\begin{center}
\begin{equation}
\begin{tabular}{l}
$\left( T_1^{ave}\right) ^{-1}\propto T^9$ for Kramers ions \\ 
$\left( T_1^{ave}\right) ^{-1}\propto T^7$ for non Kramers ions
\end{tabular}
\label{4}
\end{equation}
\end{center}

Neither of the predicted temperature dependences observed in our
experiments. Another theory of SLR in glasses was proposed independently by
Askew \cite{Askew} and Stutzman \cite{Stutzman}. They supposed that TLS are
in strong interaction with the phonon system and therefore a nearby spin can
relax through it. Supposing that energies of TLS considerably larger that
the energies of the spin system and that the density of the energy
distribution of TLS follows to $P(E)\sim E^\lambda $ in some energy range $%
Emin\leq E\leq Emax$, they give the expression for the average SLR time

\begin{center}
\begin{equation}
\left( T_1^{ave}\right) ^{-1}\propto T^{2+\lambda }\int \limits_{x_{\min
}}^{x_{\max }}\frac{x^{1+\lambda }}{\sinh (x)}dx.  \label{5}
\end{equation}
\end{center}

Some of the temperature dependences of SLR can be fitted with this formula,
and we suppose that this indicates interaction with TLS. But the
concentration dependence as well as temperature dependence $T_1^{-1}\sim T^9$
should be explained by the other or additional mechanism. The presence of
temperature independent regions in SLR strongly indicates cross relaxation
processes in our experiments. To explain the concentration dependence in our
experiments we used the model that was applied to interpret results of the
study of SLR of Yb$^{3+}$ in the PL glass \cite{FTT}. This cross relaxation
model \cite{FTT} takes into account the fast relaxing center (FC) which is
the local paramagnetic center that tunnels between states of different wells
of the many-well potential. The case of the states localized in separate
wells which refers to the small value of a tunneling matrix element as
compared with the natural width 1/$\tau _0$ of the level or with the
splitting between single-well states $\Delta U_{ij}$ within a random
(defect) field was considered in \cite{Vihnin}. The FC arises in this case 
\cite{Vihnin} when the external magnetic field is applied. Namely, when the
average Zeeman splitting $\beta B_0(g_1+g_2)/2$ is equal (with an accuracy
of 1/$\tau _0$) to the corresponding value of $\Delta U_{ij}$, where $g_1$, $%
g_2$ are $g$-factors of spin states in potential wells ''1'' and ''2'' with
a tunneling transition between them. In this case the states with a
different spin projection being in different wells are in resonance. TLS
efficiently interact with each other through the acoustic phonon field. As
the temperature decreases a percolate phase transition can occur which
results in the tunneling pseudospin systems in the orientational glass
phase. Calculations showed that the transition temperature $T_g$ is of the
order of some Kelvin. The distribution function $g(\Delta U)$ has a minimum
in the orientational pseudospin glass phase. This is related with the
appearance of the near order in the pseudospin system. As a result the
number of FC sharply diminishes, this leads to the sharp decrease of the SLR
rate. Earlier the sharp decrease of the SLR rate of Yb$^{3+}$ in PL was
reported at $T_g$ $\approx $ 2 K. A stepped like change of the SLR rate is
observed for the Tb$^{3+}$ ion in PL. For some concentration of Nd$^{3+}$
ions in PL the small stepped like change in SLR also was observed. The
position of the transition temperature $T_g$ is independent on the type of
ion. Similar dependences were observed for the SNBL glass, where $T_g\approx 
$ 2.5 K. Such a small difference of Tg between two glasses is consistent
with theory \cite{Parshin}.

\subsection{ Conclusions}

Temperature dependences of SLR rates for Nd$^{3+}$ and Tb$^{3+}$ ions in
phosphate and sodium borate silica glasses were investigated. A stepped like
change of the SLR rate observed for the Tb$^{3+}$ and Nd$^{3+}$ ions in some
concentrations. This experimental result was explained by cross relaxation
theory developed in \cite{FTT}. While some of the features of SLR of rare
earth ions in studied glasses can be explain by separate theories, they are
failed to explain all SLR features from the unified approach. This indicates
the contribution of different mechanisms to the SLR of paramagnetic centers
in glasses and the necessity to develop a new theory of SLR.

\begin{thebibliography}{9}
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Rozentsvaig Y.K. Phys. Solid State, V 37, p.818 - 824. (1995).

\bibitem{Applied}  Gromov I.A., Orlinskii S.B., Rakhmatullin R.M. Applied
Mag. Res., N3, p.1147-1158. (1992).

\bibitem{Orbacha}  Alexander S., Entin-Wohlman Ora and Orbach R. Phys. Rev.
B., V 32, N 10, p.6447- 6455. (1985).

\bibitem{Orbachb}  Alexander S., Entin-Wohlman Ora and Orbach R. Phys. Rev.
B., V 33, N 6, p.3935- 3955. (1986).

\bibitem{Askew}  Askew T.R., Stapleton H.J. and Brower K.L. Phys.Rev. B., V
33, N 7, p.4455-4463. (1986)

\bibitem{Stutzman}  Stutzman M., Biegelsen D.K., and Stuke J., Bull. Am.
Phys Soc., V 28, p. 532. (1983).

\bibitem{Vihnin}  Vikhnin V.S. Sov.Phys.Solid State, V 20, 771. (1978).

\bibitem{Parshin}  Parshin D.A. Phys. Solid State, V 36, p. 991-1024. (1994).
\end{thebibliography}

\end{document}