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void swelling has a temperature window, rather than increasing monotonically with irradiation temperature. At sufficiently low temperature, vacancy migration and cavity growth are slow. At intermediate temperature, mobile defects, stable nuclei and unequal sink capture can support rapid cavity growth. At higher temperature, increased vacancy emission, recombination and changing sink populations can reduce the vacancy supersaturation and net void swelling. The precise peak depends on alloy, spectrum, dose rate and starting microstructure; lower void swelling at high temperature does not mean that creep or helium embrittlement is harmless there.

Distinguish void swelling $S=\Delta V/V_0$ from its slope. With displacement dose $D$, a simple schematic description iswhere $D_0$ represents incubation and $k(T)$ an approximately constant post-transient slope over a limited range. Real nucleation and sink evolution make the transition smooth; neither $D_0$ nor $k$ is a universal material constant. At larger dose, microstructural or compositional evolution may change the slope again.

neutron fluence is an integrated exposure, $\Phi =\int \varphi dt$. Under an unchanged spectrum it can track displacement dose, but equal fluence does not necessarily imply equal damage or helium production. Even equal dpa can give different void swelling at different dose rates because recombination and transport compete in real time. Thus compare specimens at specified temperature, spectrum, dose rate and metallurgical condition; do not transfer one void swelling curve using fluence alone.

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.

The first equation is a prescribed first-order removal law. For a constant absorption coefficient,Integrating from the initial condition givesand henceThe neutron fluence $\Phi$ is the time integral of scalar neutron flux; it has inverse-area units, making $\sigma \Phi$ dimensionless. Constant flux is not needed for this integration, but a constant effective coefficient is.

Use neutron fluence $\Phi$ as the independent variable and abbreviate $a={\sigma }_{a\mathrm{Th}}$, $b={\sigma }_{a\mathrm{U}}$ and $c={\sigma }_{c\mathrm{Th}}$. Substituting the parent solution into the second linear differential equation givesThe integrating factor is $e^{b\Phi }$. Integrating from the initial state yieldsThe first term is survival of the initial population; the second accumulates production at earlier exposures, weighted by subsequent survival. For $b=a$, the continuous limit is $(N_{U0}+c{N}_{Th0}\Phi )e^{-a\Phi }$. The same formula applies to a time-varying prescribed flux through $\Phi (t)$; intervals of zero flux leave both populations unchanged.