The Burgers Vector and Its Circuit

A single vector that fingerprints a dislocation

In guide 1 you met the three faces of the line defect as pictures: the edge with its extra half-plane wedged in, the screw with its spiral car-park ramp, and the mixed blend of the two. Pictures are a fine start, but to reason and calculate you want a number. That number is the Burgers vector, written b. Think of it as the exact step — how far, and in which direction — the crystal on one side of the defect has been shifted relative to the other. It is the slip the dislocation has already carried out, frozen and packed into the width of a single line.

The vector b sits in a beautifully simple relationship with the direction the dislocation line runs, which we call the line sense and write with the unit vector t (the tangent along the line). For a pure edge, b is perpendicular to t — the extra half-plane pushes sideways across the line. For a pure screw, b is parallel to t — the ramp winds up along the line. A mixed dislocation sits at any angle in between, and its 'character' at each point is just the angle between b and t. So b and t together are the full genetic code of a dislocation: give me those two vectors and I can tell you edge from screw, which way it will glide, and how much energy it stores.

One vital fact you will lean on repeatedly: b is constant everywhere along a given dislocation line. The line can bend, curl, and swap character — edge here, screw there, mixed in between — yet its Burgers vector stays exactly the same. It is t that changes as the line curves, not b. A single closed dislocation loop is edge on two facing arcs and screw on the other two, all sharing one unchanging b. Hold on to this; nearly every rule that follows is really this fact in disguise.

The Burgers circuit: measuring b by walking a loop

How do you actually pin b down for a real dislocation? You walk a loop around it, counting atomic steps — this is the Burgers circuit, and it is delightfully concrete. Start on some atom near the line and hop from atom to atom, taking, say, five steps right, five up, five left, and five down. Around a defect-free patch of crystal this loop closes perfectly: you land back on the atom you started from. But thread that same loop around a dislocation and, remarkably, it still closes in the real crystal — you can always trace atom-to-atom and get home. The trick is what happens when you replay the identical sequence of steps in a perfect reference lattice.

In the perfect reference crystal that replayed loop does NOT close. It finishes one lattice step short of where it began, and the little vector you would need to travel to get from the Finish point back to the Start point is exactly b. That gap is the closure failure, and it is the whole measurement: no gap means no dislocation was enclosed; a gap of one lattice translation means you looped a single dislocation whose Burgers vector is that translation. This is why b is always a genuine crystal lattice vector for a perfect (or 'full') dislocation — the atoms far from the line must knit back into flawless registry once the defect has passed, and only a whole lattice translation vector leaves the surroundings looking untouched.

THE BURGERS CIRCUIT  (RH / FS convention)

STEP A -- in the REAL crystal, loop atom-to-atom around the line,
          equal step-count each way.  It CLOSES -- you get home.

              . . . . . . .
              . +->->->+ . .       right-handed loop:
              . ^     v . .          thumb along line sense t,
              . ^  _|_ v . .          fingers curl this way
              . ^   T  v . .        _|_ , T  = extra half-plane
              . +<-<-<-+ . .                  of an EDGE dislocation
              . . . . . . .                  (line points out of page)

STEP B -- replay the SAME step sequence in a PERFECT crystal.
          Now it does NOT close: a one-lattice-step gap is left.

              S = Start     F = Finish
              F <---gap---- S
                  |____|  =  b        ( Finish  ->  Start  =  b )

For an FCC metal:   b = (a/2)[110]  ,   |b| = a / sqrt(2)
                    Cu: a ~ 3.6 A  ->  |b| ~ 2.55 A

Why b is the shortest lattice vector

Not every lattice vector makes an equally happy Burgers vector. A dislocation stores elastic energy in the strained lattice around it, and to a very good approximation that stored line energy is proportional to the square of the Burgers vector: energy per unit length is roughly (1/2) G b^2, where G is the shear modulus. The square is the key. It means a dislocation with a short b is cheap, and one with a long b is expensive out of all proportion. Nature, ever thrifty, overwhelmingly prefers the shortest lattice translation available — which in a close-packed metal is a vector along a close-packed direction, where the atoms sit nearest together.

Put real numbers on it in an FCC metal like copper or aluminium. The shortest lattice translation is (a/2)[110], running face-diagonally between touching atoms; its length squared is b^2 = (a/2)^2 times (1^2 + 1^2 + 0^2) = a^2/2. Compare a would-be dislocation carrying the full cube-edge vector a[100], with b^2 = a^2. The edge vector costs twice the energy of the face-diagonal one, so FCC crystals simply never bother with a[100] dislocations — the (a/2)[110] type wins, and it is exactly the vector that lets one close-packed plane glide over another. This is not a coincidence; it is why the slip system of FCC metals is {111} planes gliding in <110> directions, the subject of the next guide.

The same b^2 accounting drives a subtler move you will meet in guide 5. Sometimes a dislocation lowers its energy by splitting into two partial dislocations with shorter Burgers vectors, dragging a ribbon of stacking fault between them. In FCC, (a/2)[110] can dissociate into two partials of type (a/6)<112>; check the arithmetic and (a^2/2) on the left exceeds 2 times (a^2/6) = a^2/3 on the right, so the split genuinely pays for itself. This energy-versus-b^2 bookkeeping, formalised as Frank's rule (a reaction is favourable when the summed b^2 goes down), is the single most useful tool for predicting what dislocations will do.

A dislocation line cannot end inside a crystal

Here is one of the deepest and most useful rules about the line defect, and it falls straight out of the Burgers circuit. A dislocation line cannot simply stop at a point in the middle of an otherwise perfect crystal. Picture why: imagine the line just ended somewhere. Draw a small Burgers circuit around the line on one side of that supposed end-point — you enclose the line, so you measure a closure failure b. Now slide the very same circuit past the end-point to the other side, where by assumption there is no line — you should enclose nothing and measure zero. But you cannot smoothly change the closure failure from b to zero without the loop crossing the line somewhere; the crystal would have to tear. So a free-floating loose end is forbidden.

If a line cannot end in the bulk, then every dislocation must do one of just three things: close on itself into a loop, run all the way out to a free surface or a grain boundary, or meet other dislocations at a node where several lines join. At such a node the Burgers vectors obey a conservation law exactly like Kirchhoff's current rule for wires: taking all lines to point into the node, the Burgers vectors sum to zero. This is again just 'b is conserved along a line', now applied where lines branch — a big circuit around the whole node encloses everything and must read the same as the sum of small circuits around each branch.

Reading b and t together, and where this leads

1. Fix a line sense t along the dislocation and hold it for the whole analysis.
2. Trace a right-handed Burgers circuit around the line in the real crystal so it closes.
3. Replay the identical steps in a perfect reference lattice; the Finish-to-Start gap is b.
4. Compare b with t: b perpendicular to t is pure edge, b parallel to t is pure screw, any angle between is mixed.
5. The magnitude b^2 gives the line energy and drives Frank's rule for reactions; the plane containing both b and t is the slip plane the line can glide on.

That last step hides a lovely subtlety worth naming. For an edge dislocation b and t are perpendicular, so they define a single unique plane — the edge has exactly one plane it can glide on, and it is stuck with it. For a screw dislocation b and t are parallel, so they do NOT define a unique plane at all; any plane containing the line is fair game, and a screw can therefore cross-slip from one plane to another, a freedom that matters enormously for how metals work-harden. The very same b that fixes the energy also, through its angle to t, decides how footloose the dislocation is.

With b in hand you now hold the quantitative key to the whole rung. In guide 3 the same vector explains why gliding dislocations let a crystal shear one row of bonds at a time, so real metals turn out 10 to 100 times weaker than a flawless lattice. In guide 4 it sets the elastic stress field and the Peierls lattice friction the line must overcome to move. And in guide 5 it governs how b splits into partials and how Frank-Read sources multiply line. Every one of those stories is, at heart, a story about one small vector and the loop you draw to measure it.