= Solution
Consider a representative bulk volume $\mathcal V$ and let $f_a(t)$ be its <amorphous fraction>. With <alpha-decay event rate density> $R$, during $dt$ the expected number of independent <alpha decay> events is $R\mathcal V\,dt$. Each produces <effective amorphization volume per decay> $V_\alpha$, but only the fraction $1-f_a$ of a randomly placed damage region was previously crystalline. The expected \i[new] amorphous volume is therefore
$$
d(f_a\mathcal V)=R\mathcal V\,dt\;V_\alpha(1-f_a).
$$
Dividing by $\mathcal V\,dt$ gives
$$
\frac{df_a}{dt}=RV_\alpha(1-f_a),\qquad f_a(0)=0.
$$
Separate variables and integrate:
$$
\int_0^{f_a}\frac{df}{1-f}=V_\alpha\int_0^tR\,ds,
\qquad -\ln(1-f_a)=RV_\alpha t.
$$
Hence, with <cumulative alpha-decay event density> $D=Rt$,
$$
\boxed{f_a(D)=1-e^{-DV_\alpha}=1-e^{-RtV_\alpha}}.
$$
The product $RtV_\alpha$ is dimensionless: events per cubic metre accumulated over time multiply cubic metres damaged per event. Here “dose” is an \b[event density], not an absorbed energy per kilogram.
An independent <probability> interpretation reaches the same result. A particular point is covered by a number of random damage events with a <Poisson distribution> of mean $DV_\alpha$. Its <probability> of remaining crystalline is the zero-hit <probability> $e^{-DV_\alpha}$; the complementary fraction has been hit at least once. Overlapping events do not create a second copy of already amorphous material.
\Image[/past-exam-of-the-mphil-in-nuclear-energy-course-of-the-university-of-cambridge/2020/paper-ne4-overlap-model.png]
{title=Direct-impact amorphization as a function of dimensionless accumulated alpha-decay dose}
At small dose, $f_a\simeq DV_\alpha$; at larger dose, overlap bends the curve towards one. For example, $DV_\alpha=1$ gives $f_a=0.6321$, not full <radiation-induced amorphization>, and $f_a=0.99$ requires $DV_\alpha=\ln100=4.6052$.
The derivation assumes a homogeneous, initially crystalline material; independent random event locations; one-event <radiation-induced amorphization> of a fixed effective volume; and no recovery of that volume. It is the direct-impact model, not a model requiring several prior hits to cross a damage threshold. If $R$ varies while $V_\alpha$ stays fixed, replace $Rt$ by $\int_0^tR(s)\,ds$. If <annealing (materials science)>, material evolution or spatially correlated <actinide> distributions matter, this simple law need not hold. The subscript on $V_\alpha$ indexes an \b[<alpha decay> event]: it does not require the emitted <alpha particle> itself to produce the damaged volume.
Back to article page
