Integral X-ray diffuse scattering for studying irradiation-induced dislocation loops in single crystals

Numerical diffuse scattering cross-section calculations are used to establish a rigorous basis for determining the concentration and size distribution of dislocation loops in irradiated single crystals from integral X-ray diffuse scattering (XRDS) measurements. Differential XRDS intensities for prismatic {111} type dislocation loops are numerically calculated as a function of loop radius R and wavevectors q relative to Bragg reflections in tungsten. The results show the well known 1/q² Huang scattering form at small q that transitions to a ∼1/q⁴ dependence associated with the Stokes–Wilson approximation for q ≳ 1/R. More importantly, they show further that the 1/q⁴ falloff is not the asymptotic large-q form of the diffuse scattering for small loops (R < 200 Å) as has often been assumed. Rather, for loop sizes as small as R ≃ 5 Å with strong curvature, the calculations show definitively that the scattering transitions to a robust 1/q⁵ falloff at larger q that arises due to the local strains near the dislocation core defining the circumference of the dislocation loops. The presence of this 1/q⁵ asymptotic form for both small and large loops is experimentally confirmed using an integral XRDS measurement around the 110 reflection on self-ion-irradiated tungsten combined with numerically calculated integral XRDS cross-sections. Accordingly, the historical two-region theoretical treatment of the cross-sections for integral XRDS is extended to a three-region model that has direct sensitivity to the (first-moment) dislocation line lengths of dislocation loops. These developments enable the use of both numerical and analytically modeled cross-sections to make accurate integral XRDS determinations of dislocation loop sizes and concentrations using modest-intensity laboratory X-ray sources.