the dislocation line energy
Making a dislocation costs energy. All those stretched and squeezed bonds in its surrounding stress field store elastic energy, and the crystal has to pay for them. The dislocation line energy is that stored energy counted per unit length of the line — how much it costs to have one metre of dislocation. Because the cost is proportional to length, a dislocation behaves like a stretched rubber band or a soap film: it has a line tension that pulls it taut and tries to make it as short as possible, which is why free dislocation segments bow and straighten rather than wander.
The size of this energy is set almost entirely by the Burgers vector, and it follows a rule worth memorising: the line energy is roughly E = alpha G b^2 per unit length, where G is the shear modulus, b the magnitude of the Burgers vector, and alpha a number of order 0.5 to 1 (about 1/2 for a screw and a bit more for an edge). The crucial point is the b SQUARED. Double the Burgers vector and you quadruple the energy. That single fact drives a lot of dislocation behaviour: crystals choose the shortest possible Burgers vectors, and a dislocation with a large b will often split into two or more with smaller b if the sum of their b^2 values is less than the original's — this is Frank's energy rule.
The energy also grows (slowly, as a logarithm) with the size of the crystal, because the 1/r stress field reaches so far; a full expression is E = (G b^2 / 4 pi K) ln(R/r0), with R an outer cutoff and r0 the core radius, and K a factor near 1 that depends on character. But the headline for a beginner is simpler: line energy is proportional to b^2, and it acts like a line tension. That tension is what makes a pinned dislocation bow out into a smooth arc under stress, sets the critical stress of a Frank-Read source, and, balanced against stacking-fault energy, fixes the width of a stacking-fault ribbon between partials.
Estimate for copper: E is about (1/2) G b^2 = 0.5 x 48e9 Pa x (0.256e-9 m)^2 = 1.6e-9 J/m, i.e. roughly 1.6 nanojoule per metre, or expressed per atom-length along the line about 4 eV per atomic spacing. Frank's rule uses only the b^2 part: since a/2[10-1] has b^2 = 0.5 a^2 while two a/6<112> partials total 0.33 a^2, the split lowers energy and occurs.
Line energy is proportional to b^2 and acts like a line tension pulling the dislocation short and straight.
The proportional-to-b^2 rule is what makes Frank's rule work, but it is an approximation: it ignores the interaction between segments and treats line tension as isotropic. The real line tension depends on character (edge vs screw), which is why dislocations are not perfectly flexible.