A computer simulation of vacancy diffusion is performed on a hexagonal lattice of

$10,000$ screen pixels with a spacing of $0\mathrm{.}1mm.$ The computer performs the

calculations so that the vacancy jumps at a constant rate of $100Hz.$ There is just

one vacancy in the simulation cell, and it moves by nearest$-$neighbor jumps; it

remains within the cell by using periodic boundary conditions at the borders.

a$.(5$ points$)$ Estimate the vacancy diffusion coefficient in this simulation,

assuming that the vacancy moves by a random walk.

b$.(5$ points$)$

One tracer atom, represented by a specially marked pixel, is

initially located at the center of the simulation cell. The

vacancy is introduced at a random location and then moves

by a random walk. Estimate the tracer diffusion coefficient.

c$.(5$ points$)$

In real materials, it is likely that a tracer atom will encounter

not only single vacancies, but also divacancies $($or vacancy

pairs$),$ which refer to two vacancies on neighboring sites. Do

you expect the population of divacancies to increase or

decrease with increasing temperature? Explain.