For each of the following characterization problems, use a few sentences $(3-5)$ to briefly explain the most appropriate way of using a scanning probe microscope or an electron microscope to:

i$)$ Identify the surface reconstruction of a GaN single crystal surface.

ii$)$ Examine the $3-$dimensional shape and surface faceting of a $ZnQ$ nanorod.

iii$)$ Determine the orientation relationship between a metal catalyst nanoparticle and its oxide crystal support $($e$.$g$.,$ platinum on titania$).$

iv$)$ Determine the local hardness of a nanocomposite film comprising $10$nm diameter fullerene particles embedded in an amorphous carbon matrix.

v$)$ Determine if a silicon nanowire contains any substitutional gold atoms at a doping level of less than $0\mathrm{.}01$ atomic $\%.$

vi$)$ Map the magnetic domain structure of an ordered self$-$assembled array of $40nm$ CoPt nanoparticles.

vii$)$ Faithfully map the surface topography of an etched silicon surface that has a series of very deep vertical sided grooves.

viii$)$ Determine the structure of an inorganic crystalline compound grown within the central cavity of a carbon nanotube.

ix$)$ Determine the chemical composition of an inorganic crystalline compound contained within the central cavity of a carbon nanotube.

x$)$ A colleague claims to have substituted a significant fraction of the carbon atoms making up the wall of a carbon nanotube with $B$ and $N$ atoms. How would you either confirm or refute this claim?