= Solution
Use the same <atomic fractions> and collision assumptions as in part(c). Let $A$ be the irradiated area. The number of atoms in the sheet is $N_t=ntA$, while the number of incident <neutrons> is $\Phi A$, where $\Phi$ is the <neutron fluence>.
Counting one <primary knock-on atom> for each neutron that scatters and using $\nu=430$ displacements per recoil gives the first-collision estimate
$$
\begin{aligned}
d&=\frac{\nu\Phi A P_s}{ntA}\\
&=\frac{430(4\times10^{26})(0.0273697)}
{(8.6452\times10^{28})(10^{-3})}\\
&=54.45.
\end{aligned}
$$
Hence \b[the nominal damage is about $55$ <displacements per atom>] under the assumed composition. In the thin-sheet approximation the density and thickness cancel:
$$
\boxed{d\simeq\nu\Phi\bar\sigma_s
=430(4\times10^{26})(3.21\times10^{-28})=55.21.}
$$
The $1.4\%$ difference comes from using first-collision attenuation rather than the linear approximation. Treating repeated collisions after deflection would require angular and energy information not provided; this correction is already smaller than the precision warranted by the composition assumption.
There is also a counting convention in “other atoms”: $430$ counts secondary displacements, while including the displaced <primary knock-on atom> itself gives $\nu=431$. The corresponding first-collision result is $54.58$ rather than $54.45$ dpa, an increase of only $0.23\%$. Both conventions round to about $55$ dpa. The numerical estimate is explicitly conditional on assigning the stated displacement yield to each scattering event.
A value greater than one is physically possible. <Displacements per atom> counts displacement events relative to the initial atom inventory: atoms can be displaced repeatedly, and many <vacancies> and <self-interstitials> recombine. It does not mean $55$ surviving vacant sites per original lattice site, or that every atom has moved exactly $55$ times. This is a large nominal exposure for which evolving <dislocation loops>, cavities and <irradiation hardening> matter; the surviving <microstructure> additionally depends on <irradiation temperature>, <displacement dose rate>, recombination and <defect sinks>.
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