Stress Fields, Line Energy, and Peierls Stress

The elastic knot around the line

So far this rung has treated a dislocation as a pure line: a place where the Burgers vector refuses to close, an edge with its extra half-plane or a screw with its spiral ramp. But a dislocation is not just a line drawn in empty space — it is a knot in a solid. To fit that extra half-plane in, the atoms around it have to shove aside, and that shoving means the whole neighbourhood is elastically strained. Strain means stress, so every dislocation trails a stress field that reaches far out into the crystal, long after the line itself is out of sight. Everything interesting in this guide follows from that field and the energy locked inside it.

The two pure types wear their fields differently. A screw dislocation is the tidier of the pair: its field is pure shear that circulates around the line like water swirling down a drain, with magnitude tau = G b / (2 pi r), where G is the shear modulus, b the size of the Burgers vector, and r your distance from the line. No squashing, no stretching — just twist. An edge dislocation is lumpier. Its extra half-plane crowds the atoms above the slip plane into COMPRESSION and leaves the atoms below in TENSION, so its field has a dilatational part (it changes volume) on top of the shear, and its peak stress carries an extra factor of 1/(1 - nu) from Poisson's ratio nu. Both fields die away only as 1/r — slowly. That slow falloff is the whole reason dislocations feel one another across the crystal.

EDGE DISLOCATION          | = extra half-plane pushed in from the top

   o  o  o  o  o  o  o        COMPRESSION  (atoms crowded together)
   o  o  o | o  o  o  o        above the extra half-plane
   o  o  o | o  o  o  o
  -o--o--(_|_)--o--o--o--   <- slip plane;  (_|_)  marks the edge line
   o   o    o    o    o       TENSION  (atoms pulled apart)
   o    o     o     o         below the half-plane

  screw field:  pure shear,  tau = G b / (2 pi r)   (no volume change)
  edge field :  shear + squeeze/stretch,  peak ~ G b / (2 pi (1-nu) r)

  both fall off slowly as 1/r ; inside r ~ 1-2 b lies the CORE,
  where the atoms are so displaced that linear elasticity gives up

The price of the field: line energy and the b-squared rule

Storing that strained field is not free. If you add up the elastic energy — one half of stress times strain — over all that surrounding volume, you get the line energy: the energy the crystal pays for every metre of dislocation it hosts. The integral runs from the core radius r0 out to the cutoff R, and because the field goes as 1/r the energy per unit length comes out as roughly (G b^2 / 4 pi) times ln(R / r0) for a screw, with the edge dividing by (1 - nu). The logarithm is mild and lands near 4 pi for a real crystal, so the whole thing collapses to a rule of thumb worth memorising: the energy per unit length is about G b^2 / 2, give or take a factor of two.

Push real numbers through it for copper. Copper's shear modulus is about G = 48 GPa, and its Burgers vector is the shortest lattice translation, b = a/2 along <110>, with |b| = a / sqrt(2) = 0.36 nm / 1.41 = 0.26 nm. So the line energy is about (1/2) times 48 x 10^9 Pa times (0.26 x 10^-9 m)^2, which works out to roughly 1.6 x 10^-9 J per metre — a nanojoule and a half per metre. Multiply by the length of one atom spacing and you get about 2 to 4 electron-volts for every atomic row of dislocation. A dislocation is genuinely expensive: that is why a perfect crystal contains none at equilibrium, and why the ones you find were forced in during growth or deformation.

Notice what the energy depends on: b squared, and almost nothing else. That single fact — the b^2 rule — runs a surprising amount of the whole subject. Because cost scales as b^2, a crystal always chooses the SHORTEST possible Burgers vector, which is exactly why real slip runs along close-packed directions where the repeat distance is smallest. It also means a dislocation, carrying so much energy per length, behaves like a stretched elastic string with a line tension of about G b^2 / 2: it resists being lengthened, tries to stay straight, and bows out under stress like a guitar string — the very bowing that guide 5 turns into the Frank-Read source that multiplies dislocations. And when two dislocations can merge into one with a smaller total b^2, they will; that energy accounting is Frank's rule, also waiting in guide 5.

Lattice friction: the Peierls-Nabarro stress

Guide 3 delivered the honest headline: gliding dislocations let a crystal shear one row at a time, so real metals are 10 to 100 times weaker than the theoretical shear strength of a perfect lattice, which is around G/10. But even a spotlessly clean crystal, with no other dislocations or impurities in the way, does not let a dislocation glide for absolutely free. As the core slides forward, it has to climb over the periodic hills and valleys of the lattice's own energy landscape, one atomic repeat at a time. The small stress needed to push it over each hill is the Peierls-Nabarro stress — the crystal's intrinsic lattice friction, the built-in resistance to glide that remains after every other obstacle is removed.

The model gives a striking formula: tau_PN is about (2G / (1 - nu)) times exp(-2 pi w / b), where w is the WIDTH of the dislocation core, roughly the interplanar spacing d of the slip plane divided by (1 - nu). The exponential is the whole story. A wide, spread-out core — one whose misfit is smeared over many atoms — makes 2 pi w / b large, so the exponential is tiny and the friction is minuscule. Wide cores come from slip planes that are widely spaced, and the most widely spaced planes are the close-packed ones. This is the deep reason face-centred-cubic metals like copper, aluminium, and gold are so soft: their close-packed slip gives a fat core and a Peierls stress down around 10^-4 to 10^-5 of G, small enough that dislocations glide freely even in the cold.

Now flip it. In covalently bonded crystals like silicon or diamond, and in the body-centred-cubic transition metals, the bonds are stiff and directional and the core is narrow and compact. That drives the exponent down and the Peierls stress up, to something like 10^-2 to 10^-3 of G — hundreds of times higher. A narrow core is also strongly temperature-sensitive, because thermal vibration can help the line hop over the Peierls hills by nucleating little kink pairs; cool it down and that help vanishes. This is exactly why body-centred-cubic iron has a ductile-to-brittle transition (soft when warm, glassy when cold) while silicon's dislocations are nearly frozen at room temperature, making it brittle. Be honest, though: the Peierls-Nabarro formula is a one-dimensional cartoon, and its exact numbers need atomistic calculation. What it gets right, and robustly, is the trend — spread the core wide and glide is easy; pinch it narrow and the crystal turns strong and temperature-fussy.

Climb: the move that eats vacancies

Everything above concerns glide — a dislocation sliding within its slip plane, which conserves matter (no atoms are created or destroyed, so it can happen at any temperature). But an edge dislocation has a second, sneakier way to move: perpendicular to the slip plane, by making its extra half-plane grow or shrink. This is climb, and it is non-conservative — it can only happen by moving atoms in and out, which means it needs a supply of point defects. To retreat the half-plane by one atomic row, the atoms along its bottom edge must peel off and vanish into passing vacancies; to advance it, the edge must capture atoms (equivalently, spit vacancies out). A pure screw has no half-plane to grow, so it cannot climb at all — climb is an edge-only trick.

1. A vacancy, wandering by thermal diffusion, arrives at the bottom edge of the extra half-plane.
2. An atom at that edge hops into the vacancy, so the half-plane loses its lowest atom and the vacancy is annihilated.
3. The edge of the half-plane — the dislocation line — has now moved UP by one atomic plane, out of its old slip plane. That is one unit of climb.
4. Because vacancies arrive at different points along the line at different times, climb usually happens unevenly, leaving little steps called jogs along the dislocation.

The catch is the vacancies. A crystal only carries an appreciable population of them near its melting point (their concentration climbs as exp(-E/kT)), and they only travel by vacancy-mediated diffusion, which is itself sluggish and thermally activated. So climb is a HOT process: negligible cold, important only above roughly 0.4 of the melting temperature. That single restriction sets the boundary between two whole regimes of behaviour. Down in the cold, dislocations can only glide, so an obstacle stops them dead and they pile up. Up in the heat, they can climb around obstacles, and opposite-signed edges can climb together and annihilate — which is why hot metals creep slowly under load, and why annealing softens a cold-worked metal by letting its tangled dislocations climb, meet, and cancel.

Putting the field to work

Step back and the four ideas lock together into one picture of how a real crystal behaves. The long-range 1/r stress field means dislocations are never truly alone: two of the same sign on the same plane repel, opposite signs attract and annihilate, and as deformation packs more and more line into the crystal — the stress fields overlapping ever more densely — their mutual repulsion makes each one harder to push. That is work hardening in a nutshell, and its bookkeeping is the dislocation density, the total line length per unit volume, which climbs from about 10^10 per square metre in an annealed metal to 10^15 or more in a heavily worked one. Line energy sets the currency (b^2) for which slip systems and reactions win; Peierls stress sets the intrinsic floor of strength; climb decides what can heal, and when.

The b^2 rule also plants the seed for the next guide. If splitting one dislocation into two partials lowers the total b^2, energy says it should split — and it does, spreading into a ribbon of stacking fault whose width is set by the stacking-fault energy. Meanwhile the line tension we met here, bowing a pinned segment out under stress, is precisely the Frank-Read source that lets a single dislocation spawn thousands more. Partials, stacking-fault ribbons, and Frank-Read multiplication are exactly where guide 5 goes next — every one of them an accounting move in the b^2 energy ledger you now hold.