Past exam of the MPhil in nuclear energy course of the University of Cambridge / 2021 / Module NE3 / 3 / b / Solution

The resulting dislocation density reflects competing production, growth and recovery rates. The irradiation spectrum and displacement dose rate set the supply of recoil cascades and surviving point defects. Cascade structure controls the fraction born as isolated defects or clusters; equal nominal displacements per atom can therefore give different defect populations.

Temperature sets vacancy and interstitial mobilities and the rates of thermal emission and recovery. A simple radiation damage rate theory model illustrates the competition using number densities $c_i,c_v$:Here $G$ is the surviving mobile-pair production rate per volume, $K_{\mathrm{rec}}$ is the recombination coefficient, $D_i,D_v$ are diffusion coefficients, and $k_i^2,k_v^2$ are sink strengths. The model omits explicit cluster-size populations and thermal-equilibrium emission; those must be added for a quantitative description of a particular exposure.

The initial dislocation density, grain size, free surfaces, precipitates and existing cavities all change sink strengths. Dislocations can have an interstitial bias, so their capture rates for the two defect types differ. Solutes and transmutation gases alter defect binding and mobility. Crystal structure, elastic anisotropy and applied stress affect which loop orientations survive and grow.

For a number density $N_{\ell }$ of circular loops of radius $R$, their contribution to line length per volume isThus loop nucleation and loop growth both increase dislocation density, while shrinkage, annihilation and absorption at other sinks can decrease it. A change in loop count alone does not determine the change in total line density.

Schematically, ${\dot{\rho }}_d$ is a production-and-growth contribution minus recovery and annihilation. An initial rise followed by approximate saturation is possible when these balance, but the saturation value and approach rate depend on the evolving microstructure. There is no universal dislocation density fixed by dose alone.