the dislocation stress field
A dislocation is not just a flaw right at its core — it disturbs the crystal for a long way around. Because the extra half-plane squeezes some atoms together and pulls others apart, the whole neighbourhood is left under strain, like the wrinkles that radiate far out across a bedsheet from a single tuck. This surrounding region of stretched and compressed bonds is the dislocation's stress field, and even though the atoms far away are displaced only slightly, that gentle long-range field is what lets one dislocation feel another across many atomic spacings.
The shape of the field differs for the two characters. Around an EDGE dislocation the field has both shear and a squeeze/stretch part: the atoms above the slip plane (on the extra-half-plane side) are in compression and those below are in tension — a dilatational field that, crucially, attracts undersized atoms below the line and oversized atoms above. Around a SCREW dislocation the field is pure shear with NO compression or tension at all; the atoms just twist around the line. In both cases the stress falls off gently as 1/r, where r is the distance from the line: for a screw the shear stress is tau = G b / (2 pi r), with G the shear modulus and b the Burgers vector magnitude. Close to the core (within a nanometre or so) the 1/r formula blows up and linear elasticity fails, so we simply excise a small core region and treat it separately.
This 1/r field is the reason dislocations interact at a distance, and those interactions run the show. Two like dislocations on the same plane repel; opposite ones attract and can annihilate. The tension-and-compression lobes of an edge dislocation pull solute atoms toward it to form a Cottrell atmosphere that pins it (the origin of a sharp yield point). The field also stores the dislocation's energy — integrate the elastic energy density around the line and you get the line energy, which is why that energy scales as b^2 and diverges only logarithmically with crystal size. Almost every strengthening mechanism is, at bottom, a manipulation of these overlapping stress fields.
For a screw dislocation in copper (G = 48 GPa, b = 0.256 nm), the shear stress at r = 10 nm from the core is tau = G b / (2 pi r) = (48e9 x 0.256e-9) / (2 pi x 10e-9) = 0.20 GPa. At r = 100 nm it is ten times smaller, 0.020 GPa — the 1/r decay makes the field long-range but ever weaker with distance.
Stress falls as 1/r from the line; a screw's field is pure shear, an edge's also has compression above and tension below.
The 1/r elastic field is unphysical right at the core, where strains are too large for linear elasticity; the standard fix is to cut out a core of radius ~b and account for it separately. Only the edge field has a hydrostatic (dilatational) part — a screw attracts no solute by size misfit.