the Thompson tetrahedron
/ TOMP-son /
Face-centred-cubic metals have four different {111} slip planes, and on each of them dislocations can take several Burgers vectors and split into partials — a bookkeeping nightmare of directions and signs. The Thompson tetrahedron is an elegant map that tidies the whole mess onto a single little four-sided pyramid. Build a tetrahedron whose four triangular faces are the four {111} planes of the crystal, and suddenly every important dislocation vector in FCC is an edge, a corner-to-centre line, or a face of this one shape. It is the crystallographer's cheat-sheet for FCC slip.
Here is how to read it. Label the four corners A, B, C, D, and the centres of the four faces (each opposite a corner) with the Greek letters alpha, beta, gamma, delta. Then: the six EDGES of the tetrahedron, like AB or BC, are the six perfect Burgers vectors a/2<110> — the full slips. The lines from a corner to a face-centre, like A-delta (written A-delta) or delta-B, are the a/6<112> Shockley partials — the half-steps. And a line from a face-centre to the tetrahedron's own centre is an a/3<111> Frank partial. A perfect dislocation dissociating into two Shockley partials then reads as a single geometric statement, for example AB -> A-delta + delta-B, with the triangular face it crosses being the stacking fault.
The payoff is that dislocation reactions in FCC become almost visual. Frank's energy rule (does a reaction lower total b^2?) turns into simple geometry on the tetrahedron; whether two dislocations on different {111} planes can react to form a lock is read off from whether their vectors share an edge; and the notorious Lomer-Cottrell lock — a sessile barrier that helps FCC metals work-harden — appears naturally as a stair-rod dislocation along an edge where two faces meet. Anyone doing serious FCC dislocation analysis keeps a Thompson tetrahedron close, because it converts algebra with indices into pictures with letters.
On the tetrahedron, the perfect dislocation AB = a/2[10-1] dissociates on face delta (a {111} plane) as AB -> A-delta + delta-B, i.e. a/2[10-1] -> a/6[2-1-1] + a/6[11-2]. When two such dissociated dislocations meet along the edge where faces delta and gamma join, two of their partials react into a stair-rod a/6<110> lying on that edge — a sessile Lomer-Cottrell lock that blocks further glide.
The four {111} planes as tetrahedron faces: edges are perfect a/2<110>, corner-to-face lines are a/6<112> partials.
The Thompson tetrahedron is a notation and mnemonic, not a physical object inside the crystal — it just encodes FCC geometry compactly. It applies specifically to FCC (and by extension the FCC-like glide of some other structures); it is not a general tool for BCC or HCP.