Q$1$ a$.$ Li$-$based materials are actively used in new battery technology. The dielectric constant of the Li$-$Halide crystals is higher as compared to other Alkali$-$Halides

as shown in the figure below. Please explain why Li$-$based battery materials are effective as compared to other alkali metals; based on polarization mechanisms.

Also, provide a quantitative analysis based on the difference in the dielectric constant of the different alkali$-$halides based on the alkali atom.

Q$3.$ A crystal has a mm$2$ crystal symmetry as shown below by the reduced piezoelectric strain coefficient matrix:

$([0,0,0,0,$d$\_(15),0],[0,0,0,$d$\_(24),0,0],[$d$\_(31),$d$\_(32),$d$\_(33),0,0,0])$

a$.$ Calculate the induced strain under an electric field applied along the crystal a$=1$ axis.

b$.$ Calculate the induced strain under an electric field applied along the crystal c$=3$ axis.

Note: Explain all your assumptions in detail.

Q$4.$ What are the $5$ main takeaways from this course $($please cite the chapters and lectures wherever they are relevant$)?$ Please explain in detail. Please explain

your professional perspective based on the topics that were discussed in class $($you can discuss how this course might be useful in your future career plans$).$