  A crystallographic displacive model is proposed for the extension twins in
magnesium. It is based on a hard-sphere assumption previously used for
martensitic transformations. The atomic displacements are established, and the
homogeneous lattice distortion is analytically expressed as a continuous
angular-distortive matrix that takes the usual form of shear when the
distortion is complete. The calculations prove that a volume change of 3
percents occurs for the intermediate states and that the twinning plane, even
if untilted and restored when the distortion is complete, is not fully
invariant during the transient states. The crystallographic calculations also
show that the (90 deg, a) twins observed in magnesium nano-pillars and the (86
deg, a) twins observed in bulk samples come from the same mechanism, the only
difference being the existence of a slight obliquity angle (+/- 3.4 deg)
required to reduce the strains in the latter case. Continuous features in the
pole figures between the low-misoriented (86 deg, a) twin variants are
expected; they are confirmed by EBSD maps acquired on a deformed magnesium
single crystal. As the continuous mechanism of extension twinning is not a
simple shear, a "virtual work" criterion using the value of the intermediate
distortion matrix at the maximum volume change is proposed in place of the
usual Schmid's law. It allows predicting the formation of extension twins for
crystal orientations associated with negative Schmid factors.
