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\begin{document}
\title{Barium Titanate}
\author{ Saroj Aryal\\
	\\
	Department of Physics \\
	University of Cincinnati \\
	Cincinnati, Ohio 45221\\[1em]}
\date{Dec 8, 95 }
\maketitle


\begin{abstract}
Barium Titanate is a ferroelectric material. The study of Barium Titanate enables us to better understand ferroelectric phenomenon. Ceramics of Barium titanate can be polarized in a particular direction and used for various purposes. Various physical properties of single crystal of Barium Titanate are described in this article.


\end{abstract}

\newpage
 

\section{Introduction:}



                 Barium Titanate(BT) is a  dielectric crystelline material which
 exhibits the anamalous properties typical for ferroelectricity. The
 ferroelectric activity of BT was reported in 1945-46 by Von Hippel and
 coworkers. BT is interesting from the view point of practical
 applications,firstly because it is chemicallly and mechanically very stable,
 secondly because it exhibits ferroelectric properties at and above room 
 temperature and it can be easily prepared and used in the form of ceramic
 polycrystalline samples. The first single crystals of BT were produced in 
 Switzerland in 1947.



\section{Structure}



                The structure of BT belongs to the hexagonal system and
 perovskite-type.The perovskite-type crystal is stable below $1460^o$C and
 becomes ferroelectric below $130^o$C (Curie point). It belongs to the cubic 
system below $130^o$C and its unit cell contains one molecular unit of BT
Fig 1(i).


\begin{figure}
\vspace{0.50in}
\epsfxsize=5in
\epsfysize=2.50in
\centerline{\epsffile{Fig1}}
\vspace{0.25in}
\caption{(i)Cubic Structure of Barium titanate($BaTiO_3$) with general formula $ABO_3$ (ii){Unit cells of four phases of Barium Titanate(a)cubic above $120^o$C (b)Tetragonal (c) Orthorhombic (d) Rhombodedral[1]}
}
\label{}
\end{figure}



  The symmetry of the polar phase,i.e; the phase below $130^o$C, is tetragonal, with point group 4mm. In this state the crystal spontaneously acquires
an electric polarization(a permanent electric dipole moment). As the 
polarization can in general point in any one of the originally cubic crystal
axes, the crystal consists, in the ferroelectric state, of regions(domains) which differ in the direction of the spontaneous polarization. An applied
electric field can align the polarization in the whole crystal and can reverse
the polarization whereby electric hysteresis occurs. At lower temperatures,
$5^o$C and $-90^oC$, additional phase transitions take place.The spontaneous 
polatization successively changes its direction relative to the crystellographic
 axes and increases in absolute value. The direction of polarization,from the
high temperature end, are (100), (110) and (111) in terms of crystal system.the crystal belongs, in above order, to tetragonal, orthorhombic and rhombohedral
 systems[2]


\section{Dielectric Properties:}




                In cubic phase there is only one principle dielectric constant,
 while in the tetragonal phase there are two[2]. BT has large dielectric
 constant in the vicinity of Curie point



$\epsilon = \epsilon_0+ C /(T - T_0)$



where $\epsilon_0$ is the part of the dielectric constant independent
 of temperature, C Curie-Weiss constant, T temperature and To Curie point.  

\begin{figure}
\vspace{0.50in}
\epsfxsize=3.50in
\epsfysize=4.50in
\centerline{\epsffile{Fig2}}
\vspace{0.25in}
\caption{(a)Dielectric connstants of Barium Titanate as a function of temp.(b) Spontaneous polarization of BT,measured along the pseudo-cubic edge as a function of temp.[4]}
\label{}
\end{figure}

The field dependence of dielectric constant of BT can be found in [3].
 The refractive index of the cubic phase is unusually large n = 2.4, it is
 temperature dependent. The refractive index of tetragonal phase is practically
 independent of temperature and shows no anomaly at the Curie point. In the
 cubic phase, the refractive index increases with decreasing temperature
 indicating that the electric polatizability increases as the  Curie point is
 approached from above.
 \section{The Spontaneous Polarization:}


 The temperature dependence of spontaneous polarization $P_s$ of BT
 single crystal is shown in Fig2(b)


The experimental points were obtained from measurements of the spontaneous
polarization along pseudo-cubic (100) direction, the magnitude of this vector
is obtained by multiplying the corresponding values of fig-2 with Sqrt[2].
Similarly in the rhombohedral phase, the experimental results reported in
fig-2 must be multiplied with $Sqrt{3}$ in order to obtain the magnitude on $P_s$. The spontaneous polarization at Curie point is discontinuous.The value of
spontaneous polarization at room temperature is $26*10^{-}6 C/cm^2$.




\section{The Coercive Field:}




 			The hysteresis loops of good single crystals of BT
 exhibit rather sharp corners and a marked rectangular appearence. The value of
 corcive field, mearured at room temp, varies from a minimum of 500 V/cm to a
 maximum of about 2000 V/cm[M2]. These experimental values are much smaller than
 the value expected on the basis of the thermodynamic theory. The theoretical
 value is of the order of $10^5$ V/cm. THe relatively small expt. values of the
 coercive field tell us that the crystal actually leaves a metastable state long
 before this state becomes unstable.




\section{Specific heat and lattice distortion}



The Curie trnaition in BT is of the first order therefore it should
 characterized by a latent heat. However , the specific heat measurements do not
 indicate abrupt changes. THis is due to the fact that internal strains, surface
 effects, temp inhomogeneities and other causes cooperate in spreading the
 anamalous heat changes over a finite temp range. The three phase changes occuring in BT are characterize by an expansion of the
 original cubic lattice in the direction of spontaneous polarization and a
 contraction in the direction perpendicular to it. BT is not piezoelectric in the non-polar phase compared to similar materials.
  When electric field is applied to the crystal, it produces strains
 proportional to the square of the field. 



\section{Lattice Vibration:}

	Information about the character of lattice vibrations and their change
with temperature can be obtained from infrared and optical absorption studies.
The most complete infrared absorption studiy is due to Last[5] who investigated
 BT and a no. of perovskite-type compounds.
	As there are five atoms in the cubic unit cell of BT, there are fifteen
 degrees of freedom, of which three are related to translational and three to
 torsional motion of the unit cell. The remaining nine modes aue associated with
 vibrational degrees of freedom.

		The optical absorption edge of BT lies in the vecinity of 3800A 
 and is only slightly affected by Curie transition at $120^oC$. This result might
 indicate that the electronic structure of BT does not undergo tadial changes at
 the Curie point.

	Information about the effect of internal field on the degeneracy of the spin level can be obtained from the study of paramagnetic resonance in BT single crystals. Pure crystals, however, do not contain paramagnetic ions and therefore the effect was studied on BT crystals containing smallamount of paramagnetic impurities.


\section{Uses:}



             Since BT has a high dielectric constant, typically 2000 for a
 ceramic sample at room temperature, capacitors of very small dimensions can be
 produced for electronic applications. 

  BT ceramics, when suitably doped with a trivalent oxide such as Lanthanul
 Oxide, for example, exhibit a variation of the electrical conductivity by a
 factor upto $10^6$ in the vicinity of Curie-point .This effect can be used for
 sensitive temperature-control devices. Ceramics of BT and, in particular, of a
  mixed lead titanate zirconate can be poled by an electric field and retain a
 substantial remanent polarization. These materials then have a strong
 piezoelectric effect.In addition, they are mechanically rugged and rather
 insensitive to temperature and humidity. Therefore, these ceramics find wide
 applications in devices such as microphones, ultrasonic and underwater
 transducers, and spark generators.

\begin{thebibliography}{99}

\bibitem{refA} Jona and Shirane, Ferroelectric Crystals
\bibitem{refB} Jona and Papinsky, Phys. Rev. 105 861 (1957)
\bibitem{refC} Triebwasser,S., Phys. Rev. 118,110 (1960)
\bibitem{refD} Merz,W. Phys. Rev. 76,1221 (1949)
\bibitem{refE} Last,J. Phys. Rev. 105, 1740 (1957) 
\bibitem{refF} Merz,W.J.,Phys Rev.,91,513,(1953)
\bibitem{refF} Mitsui et al., An introduction to the physics of
               ferroelectrics (1976)


\end{thebibliography}

\end{document}  


