Reconstructive phase transitions have a strong first-order thermodynamic character, and they are sharply discontinuous from the structural point of view. The latter feature is stressed by the fact that neither of the space groups of the two phases is subgroup of the other one. This contrasts with displacive or order-disorder transformations, whose continuous or semi-continuous nature is coupled with a group-subgroup relation between the two structural symmetries. However, a model was proposed for reconstructive transitions, where the continuous character lost on thermodynamic grounds is recovered from the kinetic point of view. Indeed, the crystalline solid is assumed to undergo a continuous transformation along a non-equilibrium 'transition state', which is intermediate between the two end structures and which keeps a common subset of their symmetry operators. The 'transition coordinate' has thus a kinetic rather than thermodynamic meaning.

Let H₁ and H₂ be isomorphous subgroups (with the same space group type H) of the symmetry groups G₁ and G₂ of the two end phases, respectively: H₁⊂G₁ and H₂⊂G₂. A number of conditions must be strictly fulfilled, in order that an intermediate transition state with H symmetry may be accepted. The first condition is that the number of formula units per primitive unit-cell is the same in H₁ and H₂: Z(H₁) = Z(H₂). This leads to the relation i_k,1Z(G₁) = i_k,2Z(G₂), and then to i_k,1/i_k,2 = Z(G₂)/Z(G₁). The klassen-gleich indexes i_k of the H₁ and H₂ subgroups must be inversely proportional to the corresponding Z values in G₁ and G₂. Within this constraint, all the pairs H₁^m,H₂ⁿ should be considered, with m and n running over all the classes of conjugate subgroups H₁^m⊂G₁ and H₂ⁿ⊂G₂. Each pair may in principle generate a distinct transition state with space group type H, and then a corresponding physically independent transformation pathway. Then every pair should be checked with respect to the second important condition to be fulfilled. The Wyckoff positions of the parent G₁ and G₂ space groups must split into the same final set of Wyckoff positions of the H subgroup type, according to the G₁→H₁ and G₂→H₂ symmetry reductions. This is clearly necessary to make the atomic arrangement of the intermediate state compatible with both parent structures G₁ and G₂. Further, the transformation path coordinate(s) should be selected within the free coordinates of the Wyckoff positions in H.

A transition path characterized by a common subgroup of G₁ and G₂ of type H is said to be of maximal symmetry, if there is no other acceptable transition path corresponding to a common subgroup type H', such that H⊂H'⊂G₁, and H⊂H'⊂G₂. The study of the maximal symmetry transition states should precede the analysis of those with non-maximal symmetry. However, the latter ones may prove to be preferred by energy considerations.

A rough estimate of the energetic factor in evaluating different transition paths may be provided by a consideration of the lattice strain and atomic shifts which relate the two parent structures represented within the common H reference frame. This can give an approximate hierarchy criterion to order the acceptable transformation pathways.

The code TRANPATH calculates all possible transition paths between two phases of symmetries G1 and G2 with no group-subgroup relations, on the basis of an intermediate state with a common subgroup type H. The only constraints are: a maximum value of the klassen-gleich ik index: i_k,1≤i_k, i_k,2≤i_k; a maximum value of the lattice strain and a maximum value of the atomic shifts relating the two parent structures in the H reference frame. At first the COMMONSUBS module is run, which determines all the subgroup types shared by the G₁ and G₂ space groups. Then every subgroup type is analyzed in terms of its conjugacy classes, and the different pairs of classes of H₁ and H₂ are considered and analyzed. By means of the WYCKSPLIT module, the Wyckoff positions of G₁ and G₂ are split according to the G₁→H₁ and G₂→H₂ branches; then a consistency check is performed between the obtained H₁ and H₂ Wyckoff positions, in order to accept or to reject the given pair of H1,H2 classes as a possible transition path. If the path is accepted, then the transformation matrices relating the unit-cells of the parent phases to the unit-cell of the intermediate state are determined. Finally, the lattice strain between starting and end points of the transformation is computed, together with the corresponding atomic displacements: such values are compared with the thresholds given for them in input.

A number of examples are worked out, showing how reconstructive phase transitions can be analyzed by means of the TRANPATH program in order to determine transformations pathways. The first case concerns the zincblende- to rocksalt-type transition, which is observed for several semiconductors like SiC and ZnS at high pressure. The G₁ and G₂ space groups are F-43m (zincblende) and Fm-3m (rocksalt); Z₁ = Z₂ = 4 for the corresponding conventional unit-cells.

The second example refers to the zincblende-(F-43m) to antilitharge-type (P4/nmm) transformation, shown by AgI at moderate pressures. In this case the numbers of formula units for conventional unit-cell are Z₁ = 4 and Z₂ = 2.

The third example is the phase transition from the antilitharge- (P4/nmm) to the rocksalt-type (Fm-3m) structure of AgI, observed at higher pressures than the previous case. The corresponding contents of the conventional unit-cells are Z₁ = 2, Z₂ = 4.

The fourth example is the transformation from the rocksalt- (Fm-3m) to CsCl-type (Pm-3m) phase, which is observed at high pressure for several systems like NaCl, CaO, and others. We have now Z₁ = 4, Z₂ = 1.

For every case, at least one of the possible transition pathways is determined and worked out in detail by means of the TRANPATH code.

Supplementary Material