The Small Angle Scattering of X-Rays from Cold-Worked Solids

The small angle scattering of x-rays or thermal neutrons from cold-worked crystals is calculated on the basis of two models, according to which the scattering arises predominantly from the density variation associated either with small cavities or with edge-type dislocations. The elastic distortions surrounding the cavities are unimportant, so that the scattering from cavities in a uniform medium of density $n$ is the same, to a good approximation, as that from partices (in vacuum) of density $n$ of the same size as the cavities. Thus the usual formulas of small angle scattering obtain, the scattered intensity has the familiar Gaussian-like dependence on scattering angle, and earlier results on multiple scattering may be applied. Around edge-type dislocations, on the other hand, the density variation is proportional to $\frac{sin\xi }{r}$, where $\xi$ is the angle measured from the slip direction in the plane perpendicular to the dislocation line and $r$ is the distance from the dislocation line. This angular variation results in a complete modification of the usual formulas, and, in fact, all of the terms ordinarily present disappear for this case, and conversely. The scattering shows a parabolic increase from zero at small angles, a maximum, and finally a monotonic decrease with increasing scattering angle. There is a large degree of anisotropy in the scattering, depending on the direction of the incident beam relative to the slip and dislocation axes. Multiple scattering from an array of dislocations in even a thick specimen is shown to be negligible. The theory is compared with Blin and Guinier's preliminary experimental results, and it is concluded that dislocations are incapable of explaining their data, although it is expected that under suitable conditions the measurement of the scattering from dislocations should be experimentally feasible.