Why a perfect dislocation splits in two
In the last guide you learned that a dislocation carries an elastic line energy that scales as the square of its Burgers vector — energy per length is roughly (1/2) G b^2. That single fact has a startling consequence: a crystal is always hunting for a way to shrink b. If one dislocation with Burgers vector b1 can split into two with b2 and b3, and if b2^2 + b3^2 comes out smaller than b1^2, the split is downhill in energy and it happens on its own. This energy test is Frank's rule, and it is the master accountant for every dislocation reaction in this guide.
Watch it play out in a face-centered cubic metal — copper, aluminium, gold. There the natural glide dislocation lives on a close-packed {111} plane with the shortest lattice vector, b = (a/2)[1 0 -1]. Rather than glide as one unit, it almost always dissociates into two partial dislocations called Shockley partials: (a/2)[1 0 -1] becomes (a/6)[2 -1 -1] plus (a/6)[1 1 -2]. Add those two back and you recover the original exactly, so nothing is lost — but now check the energies. The full vector gives b^2 = (1/2)^2 x (1+0+1) = 1/2, in units of a^2; the two partials give 2 x (1/6)^2 x (4+1+1) = 1/3. One-third sits comfortably below one-half, so Frank's rule says: split. The perfect dislocation is unstable against dissociating into two partials, and in FCC it essentially always does.
FCC dislocation dissociation on a {111} plane
---------------------------------------------
Perfect (full) dislocation, shortest vector on the
close-packed plane:
b = (a/2)[1 0 -1]
splits into TWO Shockley partials:
(a/2)[1 0 -1] -> (a/6)[2 -1 -1] + (a/6)[1 1 -2]
(full) (leading) (trailing)
check: (a/6)[2 -1 -1] + (a/6)[1 1 -2]
= (a/6)[3 0 -3] = (a/2)[1 0 -1] OK
Frank's rule -- compare b^2 (in units of a^2):
full : (1/2)^2 x (1+0+1) = 1/2 = 0.50
2 partials: 2 x (1/6)^2 x (4+1+1) = 1/3 = 0.33 <-- lower
So it splits, and a RIBBON OF STACKING FAULT opens between:
...======|########## fault ##########|======...
good fcc ^leading ^trailing good fcc
ABCABC partial wrong stacking partial ABCABC
|<------- width d ~ 1/(SFE) ------->|The stacking-fault ribbon
So what actually sits in the gap between the two partials? Recall how a close-packed crystal is built (from the close-packing rung): identical layers stacked ABCABC, each atom nestled in the hollows of the layer below. A Shockley partial does not carry a whole atom-to-atom step; it slides one close-packed layer out of its proper seat into the neighbouring hollow — an A-type layer landing on B-type sites. Between the leading and the trailing partial, therefore, the stacking is locally wrong: a thin slab where the sequence reads ...ABCBCABC... — a two-layer, hcp-like hiccup — instead of the perfect ...ABCABCABC... That wrong-stacking slab is a stacking fault, and the pair of partials plus the fault they enclose is the stacking-fault ribbon — an extended dislocation.
Now two forces argue over how far apart the partials sit. Elastically they repel: both carry Burgers vectors with a parallel edge component, so like strain fields push apart, and left alone the two would fly to infinity. But the fault between them costs energy — a surface tension gamma (joules per square metre) pulling the ribbon shut, because every extra square metre of wrong stacking adds gamma of energy. The ribbon settles at the width d where outward elastic repulsion exactly balances the inward pull of the fault, roughly d ≈ G b_p^2 / (2 pi gamma). The one qualitative law to carry away: ribbon width is inversely proportional to the stacking-fault energy. Cheap fault, wide ribbon; expensive fault, narrow ribbon.
Stacking-fault energy: one number that sets a metal's personality
Because the ribbon width follows 1/gamma, the stacking-fault energy quietly steers a whole crop of a metal's behaviours, and its value swings widely from metal to metal. Aluminium sits high, around 150 to 200 mJ/m^2, so its ribbons are so narrow they are barely resolvable — its partials huddle almost on top of each other. Copper is intermediate, roughly 45 mJ/m^2. Silver, brass, and austenitic stainless steels sit low, near 20 mJ/m^2 or less, so their ribbons spread out over several nanometres. These are honest ballpark figures — the exact number depends on alloying, temperature, and how it is measured — but the ordering is robust, and it matters enormously.
Here is why a materials engineer cares. To let a screw dislocation change slip planes — to cross-slip, the escape route that lets metals flow around obstacles — the two partials must first be squeezed back into a point, a constriction. A wide ribbon (low gamma) makes that costly, so cross-slip is suppressed: slip stays confined to single planes (planar slip), the metal work-hardens steeply, and deformation twinning becomes easy — the physics behind tough, high-work-hardening TWIP steels and brass. A narrow ribbon (high gamma, like aluminium) lets screws cross-slip freely, giving wavy slip lines and easy dynamic recovery. One number — the stacking-fault energy — and you have already predicted whether a metal will twin, how it hardens, and how it creeps.
Bookkeeping the reactions: Frank's rule and the Thompson tetrahedron
Frank's rule is not just for splitting one dislocation in two — it is the referee for every dislocation reaction. When two dislocations meet and can react, b1 + b2 -> b3, the reaction runs if it lowers the total b^2; when it does, the two lines fuse along their intersection into a new segment. Sometimes the product is a dislocation that cannot glide on either original plane — a sessile lock, like the Lomer-Cottrell lock in FCC — and it sits there as an immovable obstacle that other dislocations pile up against. So the very same energy bookkeeping both splits perfect dislocations into mobile partials AND welds them into barriers; whether a reaction helps a metal flow or helps it harden is just a question of which b^2 comes out smaller.
Keeping track of all these {111} planes and <110> vectors by hand is dizzying, so crystallographers use a beautiful map: the Thompson tetrahedron. Take the four close-packed {111} slip planes of an FCC crystal and they turn out to be the four faces of a regular tetrahedron; its six edges are exactly the six (a/2)<110> perfect Burgers vectors, and the lines from each corner to the centre of the opposite face are the (a/6)<112> Shockley partials. Reading off a dissociation, or checking whether two dislocations can react, becomes a matter of tracing edges to face-centres on this little solid. Be clear about what it is, though: the Thompson tetrahedron is a bookkeeping device and a naming scheme, not a new piece of physics — it just makes the geometry that Frank's rule scores impossible to get lost in.
Where dislocations multiply: the Frank-Read source
One deep puzzle is left. When a crystal deforms, dislocations glide to the surface and slip out — so their number ought to fall. Yet measurements show the opposite: the dislocation density rockets up a thousandfold or more as you work a metal, from perhaps 10^10 lines threading each square metre in an annealed crystal to 10^15 in a heavily cold-worked one. Dislocations must be manufactured during deformation. The classic factory is the Frank-Read source, and its mechanism is wonderfully simple to picture.
- Start with a dislocation segment pinned at both ends — held fast by two precipitates, junctions, or nodes a distance L apart — free to bow only in between.
- Apply a shear stress. The segment bows outward on its slip plane like a soap film pushed across a wire loop, while its own line tension pulls back.
- The bow is hardest to push when it reaches a semicircle of radius L/2; that peak needs a critical stress of about tau ≈ G b / L. Push past it and the loop turns unstable and keeps growing.
- The two arms swing around behind the pinning points, meet, and — being of opposite sign there — annihilate, pinching off a complete closed dislocation loop that expands away.
- The original segment snaps back to its starting shape and repeats — an assembly line spitting out loop after loop, as long as the stress holds.
Now let a thousand such sources run at once and the consequences cascade. The loops they pump out glide on many intersecting planes, cut across one another, drag jogs, and pile into dense tangles — a 'forest' of dislocations that every new dislocation must fight its way through. Each line now moves in the stress field of all the others, and the flow stress climbs with the square root of the density, tau ≈ alpha G b sqrt(rho) (the Taylor relation). That is work hardening in one line: pump up rho and you raise the strength. It is the deepest reason a paperclip stiffens, then finally snaps, as you bend it back and forth — you are running its Frank-Read sources and tangling the crystal solid.