a coherent interface

When two crystals meet at an interface, the friendliest possible arrangement is for their atomic rows to line up and continue straight across the seam, one crystal's planes flowing into the other's without a break. A coherent interface is exactly this: the lattice planes are continuous across the boundary, every atom on one side has a partner directly across from it, and bonds carry on almost undisturbed. It is the interface equivalent of two neighbours whose floorboards line up perfectly at the doorway.

For a truly coherent interface the two lattices must have matching atomic spacings along the interface plane. If they match perfectly there is essentially no misfit and the boundary energy comes only from the change in chemistry across it, which is very low. Usually the two phases have slightly different natural spacings, so to stay coherent one or both lattices must STRETCH to fit; this stores elastic coherency strain in the surrounding material. The misfit is measured by delta = (a2 - a1)/a1, the fractional difference in lattice parameter. Coherency can be maintained as long as that strain energy stays cheaper than the alternative (introducing dislocations), which is only while the particle is small.

Coherent particles are the secret of many strong alloys: the tiny gamma-prime Ni3Al precipitates in nickel superalloys are coherent with the matrix, giving low-energy, stable, dislocation-pinning particles that survive at high temperature. Because a coherent interface is low-energy, coherent precipitates form easily and resist coarsening. Honest note: coherence is not free. It is bought with elastic strain, and once a growing particle gets big enough that storing that strain costs more than making dislocations, the interface loses coherency and becomes semicoherent.