1918 J. P. Hirth et al. cta Materialia 54(2006 )1917-192 (IPS) solutions by Choudry and Crocker [8 and Adler et al. [9]. These solutions were applied to the Pu-1.7 at Ga alloy and correlated with transmission electron micros- copy observations by Zocco et al. [7]. The lattice corre- spondence in the latter analysis were consistent with the plane and direction parallelism mentioned in the paragraph These IPS solutions followed the procedures of the phe nomenological theory [10, 1l] of martensite transforma- tions (PTMc) for cases where the lattice invariant deformation(LID)was associated with twinning in the a phase. The PTMC, given a lattice correspondence and a LID, determines an IPS solution that gives a strain-free habit plane and a transformation shear displacement on that plane. However, while the transformation is implicitly related to possible transformation defects, the defects are usually not explicitly determined An alternative topological model(TM) has been devel- oped by Pond and Hirth [12, 13]. This approach first con- siders the transformation defects and then predicts a consistent habit plane and lattice orientation relationship directly from the geometry of the defects. The defects are identified as transformation disconnections. also called transformation dislocations. As shown in Fig. 2, the dis- connections have both step and dislocation character [14]. Fig. 1. Four monoclinic unit cells(dotted lines)of a-Pu viewed along In the propagation of the martensite transformation, the 010 The eight different equivalent sites are labeled. The blue aton disconnections move a diffusionless m (layer A)are all at y=0.75 while the orange atoms (layer B) are at coherent terrace planes that separate the two phases. The =0.25. The A and B layers are identical except for a rotation of 180. Lid is treated in the final step of the analysis as the defor- The larger unit cell shown with dashed lines has a monoclinic p angle of mation required to relieve misfit strain in the terrace plane 116. 80 (close to 120. for the hcp cll) instead of 101. 82 for the smaller with displacement perpendicular to that of the transforma tion shear In the TM, one can define a stress-free habit plane that pseudostructure are indicated in Table I by parentheses. In involves only transformation disconnections and LID all cases there are six near neighbors in the(010)plane, as defects(slip or twinning dislocations )in the habit plane there are in the(000 1) plane of the hcp structure. Hence, The PTmc solutions only yield such a solution if there is this structure can be mapped into a hcp structure without no misfit between the matrix and product phases normal changing atom neighbors. The neighbors in parentheses to the terrace plane. Otherwise the TM and PTMC solu in Table I become shuffled into 12-fold coordination with tions would coincide only if the PTMC structure acquired other atoms. After analyzing transformations from 8 to additional dislocations with a Burgers vector normal to the the hcp pseudostructure, and the formation of twins terrace plane [14]. Pond et al. [15] have compared the TM herein, the pseudostructure can be transformed to a' by and the PTmC predictions for the p-transformation means of shuffles and one shear. The shear does not corre- in a Ti-Mo alloy and for the orthorhombic to monoclinic spond to a difference between the 120 angle of the hcp unit transformation in ZrO2[16]. In both cases, the lattice misfit cell and the 101.82 B angle of the monoclinic unit cell; is small and the difference between the two models is small rather, the shear corresponds to the difference in the but significant. High-resolution transmission electron 116.80 angle of the monoclinic angle of the double size microscopy has been applied to both systems [16, 17]. unit cell outlined by dashed lines in Fig. I. The a and c axes revealing coherent terrace/disconnection structures at the of the latter unit cell are parallel to pseudo-close-packed interface, as predicted by the TM directions in the(010)plane. Additional details are given The lattice misfit is large in the case of the Pu-Ga alloy Fig. 4 and in Section 2. Another important feature, where there is a 20% volume contraction associated with found by Olsen [6], is the near parallelism of the (11 1)s the 8a transformation. Thus, a comparison of the TM and (010)a planes as well as of the [1 10] and [100] and PTMC models for this case may yield significant directions There have been several studies of the transformation The a' martensite plates have a large aspect ratio in the crystallography, as reviewed by Zocco et al. [7], but perti- Pu-1.7 at Ga alloy as revealed in Fig 3. Here, we ana- nent to the present work are the invariant plane strain lyze the ideal stress-free habit plane and orientation rela-pseudostructure are indicated in Table 1 by parentheses. In all cases there are six near neighbors in the (0 1 0) plane, as there are in the (0 0 0 1) plane of the hcp structure. Hence, this structure can be mapped into a hcp structure without changing atom neighbors. The neighbors in parentheses in Table 1 become shuffled into 12-fold coordination with other atoms. After analyzing transformations from d to the hcp pseudostructure, and the formation of twins therein, the pseudostructure can be transformed to a0 by means of shuffles and one shear. The shear does not corre￾spond to a difference between the 120 angle of the hcp unit cell and the 101.82 b angle of the monoclinic unit cell; rather, the shear corresponds to the difference in the 116.80 angle of the monoclinic angle of the double size unit cell outlined by dashed lines in Fig. 1. The a and c axes of the latter unit cell are parallel to pseudo-close-packed directions in the (0 1 0) plane. Additional details are given in Fig. 4 and in Section 2. Another important feature, found by Olsen [6], is the near parallelism of the (1 1 1)d and (0 1 0)a planes as well as of the ½110 d and [1 0 0]a directions. There have been several studies of the transformation crystallography, as reviewed by Zocco et al. [7], but perti￾nent to the present work are the invariant plane strain (IPS) solutions by Choudry and Crocker [8] and Adler et al. [9]. These solutions were applied to the Pu–1.7 at.% Ga alloy and correlated with transmission electron micros￾copy observations by Zocco et al. [7]. The lattice corre￾spondences in the latter analysis were consistent with the plane and direction parallelism mentioned in the paragraph above. These IPS solutions followed the procedures of the phe￾nomenological theory [10,11] of martensite transforma￾tions (PTMC) for cases where the lattice invariant deformation (LID) was associated with twinning in the a0 phase. The PTMC, given a lattice correspondence and a LID, determines an IPS solution that gives a strain-free habit plane and a transformation shear displacement on that plane. However, while the transformation is implicitly related to possible transformation defects, the defects are usually not explicitly determined. An alternative topological model (TM) has been devel￾oped by Pond and Hirth [12,13]. This approach first con￾siders the transformation defects and then predicts a consistent habit plane and lattice orientation relationship directly from the geometry of the defects. The defects are identified as transformation disconnections, also called transformation dislocations. As shown in Fig. 2, the dis￾connections have both step and dislocation character [14]. In the propagation of the martensite transformation, the disconnections move in a diffusionless manner across coherent terrace planes that separate the two phases. The LID is treated in the final step of the analysis as the defor￾mation required to relieve misfit strain in the terrace plane with displacement perpendicular to that of the transforma￾tion shear. In the TM, one can define a stress-free habit plane that involves only transformation disconnections and LID defects (slip or twinning dislocations) in the habit plane. The PTMC solutions only yield such a solution if there is no misfit between the matrix and product phases normal to the terrace plane. Otherwise the TM and PTMC solu￾tions would coincide only if the PTMC structure acquired additional dislocations with a Burgers vector normal to the terrace plane [14]. Pond et al. [15] have compared the TM and the PTMC predictions for the b ! a0 transformation in a Ti–Mo alloy and for the orthorhombic to monoclinic transformation in ZrO2 [16]. In both cases, the lattice misfit is small and the difference between the two models is small but significant. High-resolution transmission electron microscopy has been applied to both systems [16,17], revealing coherent terrace/disconnection structures at the interface, as predicted by the TM. The lattice misfit is large in the case of the Pu–Ga alloy, where there is a 20% volume contraction associated with the d ! a0 transformation. Thus, a comparison of the TM and PTMC models for this case may yield significant differences. The a0 martensite plates have a large aspect ratio in the Pu–1.7 at.% Ga alloy as revealed in Fig. 3. Here, we ana￾lyze the ideal stress-free habit plane and orientation rela- 8 8 8 8 8 8 8 8 8 1 1 1 1 1 1 7 7 7 7 7 7 8 8 8 8 8 8 8 7 7 7 7 7 7 2 2 2 2 1 1 1 1 1 1 3 3 3 3 3 3 5 5 5 5 5 5 6 6 6 6 4 4 4 4 4 4 4 4 5 5 5 5 5 5 3 3 3 3 3 3 2 2 2 2 6 6 6 6 8 8 c a Fig. 1. Four monoclinic unit cells (dotted lines) of a-Pu viewed along [0 1 0]. The eight different equivalent sites are labeled. The blue atoms (layer A) are all at y = 0.75 while the orange atoms (layer B) are at y = 0.25. The A and B layers are identical except for a rotation of 180. The larger unit cell shown with dashed lines has a monoclinic b angle of 116.80 (close to 120 for the hcp cell) instead of 101.82 for the smaller unit cell. 1918 J.P. Hirth et al. / Acta Materialia 54 (2006) 1917–1925