A martensitic twin involves fitting
only two martensite variants together. Other microstructures are
possible which use more than two variants, and one such is the
parallelogram martensite microstructure. A *parallelogram*
microstructure is formed when twins cross. Macroscopically,
this microstructure appears as depicted in Figure 1 below.

Figure 1: Local parallelogram microstructure

Each of the deformation gradients, **A**, **B**, **C**, and
**D** takes values from one of the martensite energy wells. They
can be taken as **A**=,
**B**=, **C**=
, and **D**=, where the **R**'s are rotations
and the **U**'s are variants of the martensite phase. Consequently,
the compatibility equation for the parallelogram martensite microstructure
can be written as

Each of these compatibility equations is a twinning equation, and therefore the vectors

Geometrically, the four planes with normals **m**'s must intersect
along a line; thus, they must be coplanar. Further, it can be shown that
the twin shears **b**'s must also be coplanar in order for all of
the compatibility equations to be satisfied.

If the local compatibility equations are all satisfied, then the microstructure as depicted in Figure 1 can be repeated in a tiling pattern to fill all of the specimen as shown in Figure 2.

Figure 2: Global parallelogram microstructure

Some of the parallelogram microstructures which are possible for the different transitions as listed in the table below.

## Parallelogram Martensite Microstructures |
||||
---|---|---|---|---|

Transition | Twin Type | Number | Observed | Comment |

Cubic-to-Trigonal | Compound | 6 | Au-Cd, Ti-Ni, Ti-Ni-Al, Ti-Ni-Fe | possible for all trigonal angles |

Cubic-to-Tetragonal | Compound | 0 | No | not possible for any lattice parameters |

Cubic-to-Orthorhombic | Compound and Type I | 6 | Cu-Al-Ni | possible for all lattice parameters |

Cubic-to-Monoclinic | Compound | 6 | Ti-Ni | possible for all lattice parameters |

In the table above, the first column is the transition; the second gives the twins used in the construction; the third column lists the number of unique parallelogram martensite microstructures which can be formed using all of the variants of the martensite phase; and the fourth column gives alloys for which some of the twin crossings are observed. Moreover, for the cubic-to-trigonal, -orthorhombic and -monoclinic transitions, there may be other parallelogram microstructures.

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