# 3.4: Problems

3.1. For each of the three crystalline materials below

• Find the crystal point group to which it belongs. (Hint: use http://www.mindat.org )
• Using Table 2.3.1, determine whether or not the material is piezoelectric.
• Using Table 2.3.1, determine whether or not the material is pyroelectric.
• Using Table 2.3.1, determine whether or not the material is Pockels electro-optic.

(a) ZnS (sphalerite)

(b) HgS (cinnabar)

(c) Diamond

3.2. Cane sugar, also called saccharose, has chemical composition C$$_12$$H$$_22$$O$$_11$$ and belongs to the crystal point group given by 2 in Hermann-Maguin notation [38]. Reference [38] lists values specified in cgse units for its piezoelectric constant as $$10.2 \cdot 10^{-8} \frac{\text{esu}}{\text{dyne}}$$ and its pyroelectric coefficient as $$0.53 \frac{\text{esu}}{cm^2 \cdot ^{\circ}\text{C}}$$. Convert these values to the SI units of $$\frac{m}{V}$$ and $$\frac{C}{m^2\cdot K}$$ respectively.

Hint: The electrostatic unit or statcoulomb is a measure of charge [7] where $1 \text{ esu} = 1 \text{ statC} = 3.335641 \cdot 10^{-10} C \nonumber$ and the dyne is a measure of force where $$1 \text{ dyne} = 10^{-5} N$$.

3.3. A material has relative permittivity $$\epsilon_{r\;x}$$ when no external electric field is applied. The coefficient $$\chi^{(2)}$$ is measured in the presence of an external electric field of strength $$|\overrightarrow{E}|$$. Assume that $$\chi^{(3)}$$ and all higher order coefficients are zero. Find the Pockels coefficient $$\gamma$$ as a function of the known quantities $$\epsilon_{r\;x}$$, $$\chi^{(2)}$$, and $$|\overrightarrow{E}|$$.

3.4. The first figure below shows the displacement flux density $$|\overrightarrow{D}|$$ as a function of the strength of an applied electric field intensity $$|\overrightarrow{E}|$$ in a non-electro-optic material. The second figure below shows the displacement flux density $$|\overrightarrow{D}|$$ as a function of the strength of an applied electric field intensity $$|\overrightarrow{E}|$$ in a ferroelectric electro-optic material. The solid line corresponds to an unpoled material. The dotted line corresponds to the material after it has been poled in the $$\hat{a}_z$$ direction, and the dashed line corresponds to the material after it has been poled in the $$-\hat{a}_z$$ direction.

(a) For the non-electro-optic material, find the relative permittivity, $$\epsilon_r$$. Also find the magnitude of the material polarization, $$\overrightarrow{P}$$.

(b) Assume the ferroelectric electro-optic material is poled by a strong external electric field, and then the field is removed. Find the magnitude of the material polarization $$|\overrightarrow{P}|$$ after the external field is removed.

(c) Assume the ferroelectric material is poled in the $$-\hat{a}_z$$ direction by a strong external field, and then the field is removed. A different external electric field given by $$\overrightarrow{E} = 100\hat{a}_z \frac{V}{m}$$ is applied. Find the approximate relative permittivity of the material.

3.5. A crystalline material is both piezoelectric and pyroelectric. When an external electric field of $$|\overrightarrow{E}| = 100 \frac{V}{m}$$ is applied, the material polarization is determined to be $$|\overrightarrow{P}| = 1500\epsilon_0 \frac{C}{m^2}$$. When both a stress of $$|\overrightarrow{\varsigma}| = 30 \frac{N}{m^2}$$ and an external electric field of $$|\overrightarrow{E}| = 100 \frac{V}{m}$$ are applied, the material polarization is determined to be $$|\overrightarrow{P}| = 6.0123 \cdot 10^{-6} \frac{C}{m^2}$$. When a temperature gradient of $$\Delta T = 50\; ^\circ C$$, a stress of $$|\overrightarrow{\varsigma}| = 30 \frac{N}{m^2}$$, and an external electric field of $$|\overrightarrow{E}| = 100 \frac{V}{m}$$ are applied, the material polarization is determined to be $$|\overrightarrow{P}| = 6.3 \cdot 10^{-6} \frac{C}{m^2}$$. Find:

• The relative permittivity of the material
• The piezoelectric strain constant
• The magnitude of the pyroelectric coefficient