Show $(111)$ plane and $[110]$ direction in a cubic unit cell. $(20.$P$)$

Determine the planar atomic density of $(100)$ plane of a face centered cubic $($FCC$)$ crystal lattice using an appropriate diagram and a formula. $(20$ P$)$

Determine the linear atomic density along $[111]$ direction of a body centered cubic $($BCC$)$ unit cell using an appropriate diagram and a formula. $(20$ P$)$

Write the equations used to show the effect of $($a$)$ grain size and $($b$)$ dislocation density on the strength of metallic materials and give the names of their parameters. $(20$ P$)$

Show the effect of percent cold deformation on the hardness, strength, ductility, electrical conductivity, and grain size of metallic materials by drawing appropriate curves. $(20$ P$)$