Other special microstructures are possible besides the wedge and these involve fitting multiple wedge microstructures together. Two such microstructures are the triangle and diamond microstructures, and they are of interest because both can grow from a point on a free surface of the specimen and can exist completely surrounded by the austenite phase. The triangle and diamond microstructures are very special microstructure in that it is only possible in materials with special lattice parameters. Not only must the material be able to form wedges, but the wedges must be able to be fitted together in a compatible manner.

**Triangle:** The triangle
microstructure is a triangular pyramid such that the
three habit planes separating the martensite regions
from the austenite intersect at a point and through this
point passes the line of intersection of the three midrib
planes between neighboring martensite regions. The triangles
considered have all martensite regions with either single
variants of martensite or twinned martensite. Further, the trace
of the habit and midrib planes for a triangle on a surface of
a specimen are depicted in Figure 1 below:

Figure 1: Triangle microstructure, where the straight lines are the intersection of the habit planes and midrib plane with the plane of the page.

From Figure 1, it can be appreciated that a triangle is possible
when a material can form wedges, and any deformation with
gradient say **A**, as in Figure 1, can form a wedge with
at least two other deformations with gradients **B**
and **C** as in Figure 1. In addition, the three habit
planes must be non-coplanar, while the three midrib planes
must be coplanar. Only materials with very special lattice
parameters can form the triangle, and the table
below gives the restrictions for several transformations.

**Diamond:** The diamond
microstructure is a diamond pyramid such that the
four habit planes separating the martensite regions
from the austenite intersect at a point and through this
point passes the line of intersection of the four midrib
planes between neighboring martensite regions. The diamonds
considered have all martensite regions with either single
variants of martensite or twinned martensite. The trace of
the habit and midrib planes for a diamond on a surface of
a specimen are depicted in Figure 2 below:

Figure 2: Diamond microstructure, where the straight lines are the intersection of the habit planes and midrib plane with the plane of the page.

Notice that the diamond can be formed by placing two wedges back-to-back, and the diamond is less restrictive than the triangle, because only two non-parallel midrib planes are needed to construct the diamond, but three non-parallel midrib planes are needed for the triangle. Furthermore, the same discussion as for the triangle above applies for the diamond, and the table below gives the restrictions for several transformations.

Various triangle microstructures are possible for different transitions as listed in the table below.

## Triangle Microstructures |
||
---|---|---|

Transition | Number | Observed |

Cubic-to-Trigonal | 0 | not possible |

Cubic-to-Tetragonal | 4 with = square root of 5/3 and = square root of 1/3 | not observed |

Cubic-to-Orthorhombic | 4 with =1
and =
f()
(*) |
not observed |

" | 12 with =1
and =
g()
(*) |
not observed |

" | 0 with twins | not possible |

Cubic-to-Monoclinic | 0 | not possible in a particular Ti-Ni alloy (**) |

In the table above, the first column is the transition; the second column lists the number of unique microstructures which can be formed; none of these microstructures have been observed in experiments.

(*) Note: for the cubic to orthorhombic
transition, the triangle microstructures is possible if and only
if one of the transformation stretches describing the transition
is equal to one, and the remaining two satisfy some algebraic
condition denoted by the functions *f* and *g*. These
are the same functions as for the single variant wedge.

(**) Note: for the cubic to monoclinic transition, the triangle is not possible for a particular Ti-Ni alloy, but restrictions as found for the orthorhombic transition may be found.

Various diamond microstructures are possible for different transitions as listed in the table below.

## Diamond Microstructures |
||
---|---|---|

Transition | Number | Observed |

Cubic-to-Trigonal | 0 | Not possible |

Cubic-to-Tetragonal | 0 | Not possible |

Cubic-to-Orthorhombic | 3 with =1
and =
g()
(***) |
not observed |

" | 18 on a curve with type I twins | not observed |

" | 18 on a curve with type II twins | not observed |

Cubic-to-Monoclinic | 0 | not possible in a particular Ti-Ni alloy (****) |

In the table above, the first column is the transition; the second column lists the number of unique microstructures which can be formed; none of these microstructures have been observed in experiments.

(***) Note: for the cubic to orthorhombic
transition, the diamond microstructures is possible if and only
if one of the transformation stretches describing the transition
is equal to one, and the remaining two satisfy some algebraic
condition denoted by the function *g*. This is the same
function as for the single variant wedge. Also, diamonds are
possible with twins.

(****) Note: for the cubic to monoclinic transition, there should be a number of surfaces defined by two parameters on which the diamond with twins is possible.

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