An essay on $d_0$ in neutron-based strain measurement techniques: equilibrium-based methods for non-destructive $d_0$ estimation

Determination of the stress-free reference lattice spacing, $d_0$, is a central and often underestimated difficulty in diffraction-based strain measurement. Although commonly treated as a material constant, $d_0$ often varies with position and measurement direction as a result of composition, phase, texture and residual stress at the sub-bulk scale. While direct measurement of $d_0$ from coupons is usually preferable, this is sometimes impractical or even impossible. This paper reviews the $d_0$ problem in general and develops a hierarchy of non-destructive alternatives based on mechanical equilibrium. This ranges from established techniques based on force-balance, to the application of boundary conditions and finally an approach based on point-wise equilibrium within a sample, implemented through eigenstrain analysis. In all cases, examples are provided in the form of real samples including an additively manufactured Inconel cube, ancient Roman bronze medical probes and an ancient bronze dagger of purported Persian origin. These latter examples demonstrate that equilibrium can provide a practical physical constraint for estimating $d_0$ when destructive reference measurements are entirely inappropriate.