$5($i$).$ An electron probe of $10nm$ diameter with a convergence semi$-$angle $$ of $5{10}^{-3}$ radians is

formed in a $35$keV SEM. Ignoring any lens aberration effects, calculate how many electrons pass

through the probe per second, if the electron source brightness $={10}^{10}amps{m}^{-2}s{r}^{-1}.$

$($ii$).$ Calculate the theoretical size $($resolution$),d(min),$ of the spot onto which a scanning electron

microscope beam can be focused if the objective lens has a spherical aberration, $C_S,$ coefficient of

$1cm,$ and a chromatic aberration coefficient, $C_c,$ of the same magnitude for an effective probe

semi$-$angle, $,$ of $0\mathrm{.}003$ radians. The desired incident beam current, $I_0.$ is $2{10}^{-12}$amp for an

accelerating potential, $E,$ of $20kV,$ and a tungsten cathode temperature, $T,$ of $2300C.($Assume a

voltage stability, $\frac{E}{E},$ of $1{10}^{-5}.)$

$d(min)$ is given, in general for a thermionic emitter, by a combination of four terms as shown

in the equation below. The first term represents a space charge limitation on the spot size, that is$,$

as the current $I_o$ in the spot is increased the electrons come closer together and so repel each other

more. The second and third terms are due to the spherical and chromatic lens aberrations. The

final term is the diffraction$-$limited spot size, where $r_d$ is the radius of the disc of confusion based

on a Rayleigh criterion.

$d(min)=\sqrt[\phantom{2}]{\frac{6I_{0}kT}{J_{o}eE{}^2}+(\frac{C_5{}^3}{2}{)}^2+(C_c\frac{E}{E}{)}^2+4r_d^2}$

where $k$ is Boltzmann's constant and $J_0.$ the current density at the cathode corresponding to

temperature $T,$ is given by the Richardson$-$Dushman equation:$=$

$J_o=\frac{4m{k}^{2}e{T}^2}{h^3}$exp$[\frac{E_w}{kT}]$

Explain every single step.