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\begin{document}

Simplest example is 2-d hexagonal ionix crystal. Charges alternate
on hexagonal grid. Stepping on grid creates electric dipole.

For square matrix there is a center of symmetry which disallows dipole
creation.

magnitude of hexagonal dipole:

\( \frac{qa_{unitcell}}{3} \) so dipole size of \( p=\frac{1}{3}qa\epsilon  \). The density of dipoles is \( P=\frac{1}{a^{3}}p=\frac{\rho }{m_{A}}p=\frac{1}{3}\frac{q\epsilon }{a^{2}} \). Electric field
induced is \( 4\pi p*dipoledensity\sim 4\pi P \).

Then the electric charge density is \( \Phi =4\pi Pl\sim \frac{4\pi }{3}q(\frac{\rho }{m_{A}})^{\frac{2}{3}}\epsilon l \) \( \sim 4*4.8*10^{-10}(\frac{3}{10^{2}*1.6*10^{-24}})^{\frac{2}{3}}\frac{\delta l}{cm}\sim (10^{9}V/cm=d^{-1})*\delta l \). For quartz, \( d=3*10^{-10}cm/V \)

For Rochelle Salt \( (kNaC_{4}H_{4}O_{6}*4H_{2}0) \) and Barium Titamate \( BaTi0_{3} \), it is \( d=3*10^{-8}cm/V \). These are ferromagnetic
with the curie temperature just below room temperature, so small electric
fields can produce large domains.

uses:

\begin{itemize}

\item cameras

\item oscillating quartz in watches

\item microphones

\item spark toy. strain, \( \epsilon  \)=\( \frac{\sigma }{\mu }=\frac{210^{6}dyne/cm^{2}}{10^{12}}=2*10^{-6} \) . This means \( \phi \sim d^{-1}\epsilon \sim 6000V \).

\end{itemize}

\end{document}
