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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 estimateHence the nominal damage is about $55$ displacements per atom under the assumed composition. In the thin-sheet approximation the density and thickness cancel: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.

An energetic neutron can collide with a lattice atom and create a primary knock-on atom. That recoil transfers energy to other atoms in a displacement cascade. Some atoms leave their normal lattice sites, creating Frenkel pairs: a vacancy and an extra atom at an interstitial position.

Many nearby defects recombine during the cascade and its thermal relaxation. The surviving point defects are therefore fewer than a simple count of all displacement events. Some defects are already clustered within the cascade; others migrate and aggregate later.

A collection of vacancies can collapse into a platelet bounded by a vacancy-type dislocation loop. An aggregation of self-interstitial atoms can create an extra atomic platelet bounded by an interstitial-type dislocation loop. The boundary is a line defect because a circuit around it has a nonzero Burgers vector. The allowed Burgers vectors and habit planes depend on the material's crystal structure and fault energies.

Loops can grow by absorbing mobile point defects, shrink by emitting or absorbing the opposite defect type, and interact with existing dislocations. Some remain faulted, while suitable loops can unfault or become part of a larger dislocation network. Repeated cascades and subsequent defect transport therefore create a population of extended defects even in material with a low initial dislocation density.

This describes the main displacement-damage mechanism. Ionizing energy deposited in a material does not automatically create the same defect population as a neutron recoil cascade; recoil spectrum, displacement thresholds and material response matter.