= Solution
Let $c(t)$ be the fraction still crystalline. Under the <direct-impact amorphization model>, damage events occur randomly and independently; an event striking already amorphous material does not create that much additional amorphous volume. Only the crystalline fraction can be newly converted, so
$$
\frac{dc}{dt}=-Rvc,\qquad c(0)=1,\qquad f_a(t)=1-c(t)=1-e^{-Rvt}.
$$
The product of the given event rate density and damage volume has units of inverse time:
$$
k=Rv=(5.685\times10^{14})(1.706\times10^{-25})
=9.69861\times10^{-11} \mathrm{s^{-1}}.
$$
For $99\%$ <radiation-induced amorphization>, $c=0.01$, giving
$$
t_{99}=\frac{-\ln(0.01)}{k}=4.74828\times10^{10} \mathrm s
=\boxed{1504.64 \mathrm{yr}}.
$$
Using $365.25$ days per year, the corresponding calendar date is \b[approximately 3555 CE], measured from disposal preparation in 2050.
\Image[/past-exam-of-the-mphil-in-nuclear-energy-course-of-the-university-of-cambridge/2021/paper-ne4-amorphization.png]
{title=Random-overlap prediction for radiation-induced amorphization of the waste form}
A linear estimate $f_a=Rvt$ misses repeated hits to already damaged regions and therefore predicts too rapid an approach to $99\%$. The <exponential function> model never reaches exactly $100\%$ in finite time. Its assumptions include constant dose rate, fixed effective damage volume, no <annealing (materials science)> and no spatial correlation of damage. Actual <alpha decay> generates recoil damage as well as the <alpha particle>'s track; the stated damage volume must be understood as the effective volume per <radioactive decay> for the chosen model.
\b[Amorphous is not synonymous with dissolved or failed containment.] Radiation damage can change swelling, connected <porosity> and leach behaviour, so testing must compare structural evolution with element release. Accelerated self-irradiation and parallel dissolution measurements are useful precisely because crystallinity alone is not the performance endpoint.<JAERI ANSTO co-operative research on radiation damage and actinide behaviour in Synroc by S. Muraoka and others>[1994 primary conference abstract, accelerated self-irradiation and parallel aqueous-dissolution investigations]{ref}
Back to article page
