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The number of alpha decay events per material volume accumulated over time. For constant alpha-decay event rate density $R$, $D=Rt$; otherwise $D(t)={\int }_0^{t}R(s)ds$. Multiplying by a damaged volume per event gives a dimensionless overlap parameter.

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 new amorphous volume is thereforeDividing by $\mathcal{V}dt$ givesSeparate variables and integrate:Hence, with cumulative alpha-decay event density $D=Rt$,The product $Rt{V}_{\alpha }$ is dimensionless: events per cubic metre accumulated over time multiply cubic metres damaged per event. Here “dose” is an 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 $D{V}_{\alpha }$. Its probability of remaining crystalline is the zero-hit probability $e^{-D{V}_{\alpha }}$; the complementary fraction has been hit at least once. Overlapping events do not create a second copy of already amorphous material.

At small dose, $f_a\simeq D{V}_{\alpha }$; at larger dose, overlap bends the curve towards one. For example, $D{V}_{\alpha }=1$ gives $f_a=0.6321$, not full radiation-induced amorphization, and $f_a=0.99$ requires $D{V}_{\alpha }=\ln 100=4.6052$.

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 ${\int }_0^{t}R(s)ds$. If annealing, material evolution or spatially correlated actinide distributions matter, this simple law need not hold. The subscript on $V_{\alpha }$ indexes an alpha decay event: it does not require the emitted alpha particle itself to produce the damaged volume.