The figure below shows a transmission electron micrograph of a self$-$assembled twodimensional raft of Au nanoparticles having a bimodal size distribution. The particles have no preferred crystallographic orientation relationship relative to each other and are supported on an amorphous carbon film.

$50nm$

$($i$)$ Draw the smallest repeat unit cell of this lattice and use this to deduce the number ratio of large$-$to$-$small particles.

$($ii$)$ Estimate the approximate number of atoms in one of the smaller and one of the larger particles. $[$Assume Au has an f$.$c$.$c$.$ unit cell with a lattice parameter of $0\mathrm{.}407nm].$

$($iii$)$ Sketch the low angle electron diffraction pattern you would expect to see from this ordered bimodal latticle of nanoparticles. Try to relate this diffraction pattern to the observed superlattice planes.

$($iv$)$ Describe the behavior you might expect to see if this raft of Au nanoparticles were slowly heated from room temperature up to $800C.$