the Peierls-Nabarro stress

Even a perfectly clean dislocation, with nothing in its way, does not glide utterly for free. As it moves forward by one atomic spacing, it has to climb out of the comfortable valley it sits in, over a small bump, and down into the next valley — because the atomic landscape it travels through repeats period by period. The minimum shear stress needed to push a straight dislocation over that periodic bump, in an otherwise perfect crystal at absolute zero, is the Peierls-Nabarro stress. It is the crystal lattice's own built-in friction against dislocation glide.

The size of this friction depends sensitively on how spread-out the dislocation's core is, and the model captures it with an exponential: tau_PN is roughly (2G / (1 - nu)) exp(-2 pi w / b), where G is the shear modulus, nu Poisson's ratio, b the Burgers vector, and w the core width (how many atoms the disregistry is smeared over). The exponential is the whole story: a WIDE, spread-out core (large w) gives a tiny Peierls stress, while a NARROW, concentrated core gives a large one. Wide cores occur on close-packed planes with large interplanar spacing, so slip on those planes is easy — this is a deeper reason why close-packed planes are the slip planes.

This one quantity explains a striking divide in the strength of materials. Face-centred-cubic metals have wide dislocation cores and a Peierls stress so low it is almost negligible — they are soft and stay ductile even in the cold. Covalent crystals like silicon and diamond have narrow, stiff, directional-bond cores and an enormous Peierls stress — they are hard and brittle at room temperature, their dislocations frozen in place until heated. Body-centred-cubic metals sit in between, with a Peierls stress high enough that their screw dislocations move sluggishly at low temperature, which is exactly why steels can turn brittle in the cold (the ductile-to-brittle transition).