$[25$ points$]$: In a solid$-$state diffusion experiment to measure diffusion coefficients, a layer of

radioactive gold atoms $($denoted $A{u}^{**})$ is deposited onto a block of non$-$radioactive fcc gold.

$($a$)$ Explain briefly how the experiment would be conducted to determine the diffusion coefficient

$($b$)$ During the experiment the activation energy $($migration plus formation$)$ for diffusion in this

material is determined to be $Q=1\mathrm{.}67eV.$ Given that diffusion occurs by a vacancy mechanism,

calculate the diffusion coefficient in $Au$ at $1070K.$ The lattice parameter for fcc gold is $a_o=0\mathrm{.}408$

$nm$ and the vibration frequency is $5{10}^{12}Hz.$

$($c$)$ Using the diffusion coefficient above, calculate the concentration of radioactive gold at $x=20$

microns after one week. How long would one have to wait to obtain a concentration of $50\%$

radioactive gold at that distance? Consider that the concentration of radioactive gold at $x=0$ is

$100\%$ throughout the experiment.