J. P. Hirth et al. I Acta Materialia 54(2006)1917-1925 habit plane Fig. 5. Disconnections, terrace planes, and misfit strains at 8/a interface. 6. (a) Habit plane when disconnections remove misfit in the tion(b)Tilt wall produced by the b'2 component nections shown in Fig. 5 is b1 =D102. In order to calcu- late strains and the magnitude of the Burgers vectors, we =6.0 as given by Eq.(B 8)of Pond et al. [15].One half ise the lattice parameters cited by Zocco et al. [71, as of o partitions to each phase and so the habit plane in the 8 slightly modified by Lawson et al. [18]: as=0.4626 nm, phase is inclined to the terrace plane by o=0+o/2= aa=0.6199 nm, ba=0.4630 nm, ca=1070 nm, and 23.10. As shown in Fig. 6(a), the rotation 0 as produced B=101.820. For the fcc/hep disconnection, the step height by the step crystallographic orientation is changed by /2 is twice the spacing of the close-packed planes. Hence, the in each phase by the tilt wall corresponding quantities depicted in Fig. 5 are the ledge The lid defects remove the misfit E33=E33 and give added rotation of the a' lattice that changes the habit plane h=da1o=0.4630nm referred to o' but leaves the habit plane referred to 8 the Burgers vector normal to the terrace plane unchanged. Hence we compare the 8 habit plane angle of 23. 1 with the other results, since it is unchanged by the b2=2(a-di1)=-0.0712mm LID, which differs in the different approaches. The experi the normal strain component along the xz-direction ments of Zocco et al. [7]give 0= 22.2 and the IPS/PTMC analysis of Zocco et al. [7] gives 0= 12.8. As depicted in E2=b2/2am1=-0.1332 the stereographic projection in Fig. 7, the angular traces of the rotations of the habit planes from(111)s also differ and the normal strain component along the x3-direction The habit plane prediction of the TM is in much better B3=(100x-[10/1106=-0.053 agreement with experiment than that of the PTMC. The For the xr-direction, we define the Burgers vector and En in following analysis suggests that the agreement of the TM the median coherent reference lattice [13]. Thus the misfit may be even better. The normal to[10O],which would strain along xl, given by(bla-b18)/(b1a+b18),is coincide with the [1010 direction of the transformation partial in the hep pseudostructure, is inclined to the 1-)/2(02+n1) [102] direction by 5(see Fig. 4). Hence, in the comple tion of the transformation from hcp to a, in addition to =-0.1365nm/0.1820nm=-0.7499and 0.1820mm The choice of the angle is arbitrary in the following sense We have fixed the coincidence of the two structures along [100J(see Fig. 4), and have selected the normal to this 3. Results direction to define the angle. The shear then fixes an orthogonal box and the remainder of the atoms transform The procedure for predicting the habit plane is illus- by shuffles. One could pick another direction resulting in trated in Fig. 6(a). The strains and Burgers vectors are different shears and shuffles. Clearly, however, the mean transformed to the x, coordinates fixed on the habit plane. monoclinic structure is rotated with respect to the hcp exactly pacing is then selected so that the misfit pseudostructure, so any choice would lead to some shear strain G is exactly compensated by the average strain b1/L with consequences similar to those of the present selection of the disconnections, an analog of misfit removal in thin- Studies [19] of twinning in Pu-1.7 at Ga revealed new Im overlayers. This gives 0= 19.8 as determined from twinning systems in addition to those predicted by Crocker Eq(B 2)of Pond et al. [15]. The b, components of the dis- [5]. An example of the observed twinning is shown in connections produce a tilt wall and a corresponding misori- Fig 9: we note that the configuration is consistent with entation(rotation) from the original(1 11)//(0 1O)a of Fig. 8(c). These twins [7, 19], all of the(hOk) prismatic typenections shown in Fig. 5 is b1 ¼ 1 12 ½102 a. In order to calcu￾late strains and the magnitude of the Burgers vectors, we use the lattice parameters cited by Zocco et al. [7], as slightly modified by Lawson et al. [18]: ad = 0.4626 nm, aa = 0.6199 nm, ba = 0.4630 nm, ca = 1.070 nm, and b = 101.82. For the fcc/hcp disconnection, the step height is twice the spacing of the close-packed planes. Hence, the corresponding quantities depicted in Fig. 5 are the ledge height h ¼ da 010 ¼ 0:4630 nm the Burgers vector normal to the terrace plane b2 ¼ 2ðda 020 dd 111Þ¼0:0712 nm the normal strain component along the x2-direction e22 ¼ b2=2dd 111 ¼ 0:1332 and the normal strain component along the x3-direction e33 ¼ ð½100 a ½110 dÞ=½110 d ¼ 0:0537 For the x1-direction, we define the Burgers vector and e11 in the median coherent reference lattice [13]. Thus the misfit strain along x1, given by ðb1a b1dÞ= 1 2 ðb1a þ b1dÞ, is e11 ¼ 1 12 ½102 a 1 6 ½1 12 d 1 2 1 12 ½102 a þ 1 6 ½1 12 d ¼ 0:1365 nm=0:1820 nm ¼ 0:7499 and b1 ¼ 0:1820 nm. 3. Results The procedure for predicting the habit plane is illus￾trated in Fig. 6(a). The strains and Burgers vectors are transformed to the x0 i coordinates fixed on the habit plane. The disconnection spacing is then selected so that the misfit strain e0 11 is exactly compensated by the average strain b0 1=L of the disconnections, an analog of misfit removal in thin- film overlayers. This gives h = 19.8 as determined from Eq. (B.2) of Pond et al. [15]. The b0 2 components of the dis￾connections produce a tilt wall and a corresponding misori￾entation (rotation) from the original (1 1 1)d//(0 1 0)a of u = 6.6 as given by Eq. (B.8) of Pond et al. [15]. One half of u partitions to each phase and so the habit plane in the d phase is inclined to the terrace plane by x = h + u/2 = 23.1. As shown in Fig. 6(a), the rotation h as produced by the step crystallographic orientation is changed by u/2 in each phase by the tilt wall. The LID defects remove the misfit e0 33 ¼ e33 and give added rotation of the a0 lattice that changes the habit plane referred to a0 but leaves the habit plane referred to d unchanged. Hence we compare the d habit plane angle of 23.1 with the other results, since it is unchanged by the LID, which differs in the different approaches. The experi￾ments of Zocco et al. [7] give h = 22.2 and the IPS/PTMC analysis of Zocco et al. [7] gives h = 12.8. As depicted in the stereographic projection in Fig. 7, the angular traces of the rotations of the habit planes from (1 1 1)d also differ. The habit plane prediction of the TM is in much better agreement with experiment than that of the PTMC. The following analysis suggests that the agreement of the TM may be even better. The normal to [1 0 0]a, which would coincide with the ½1 01 0 direction of the transformation partial in the hcp pseudostructure, is inclined to the [1 0 2]a direction by 5 (see Fig. 4). Hence, in the comple￾tion of the transformation from hcp to a0 , in addition to shuffles there is a shear e13 that produces this rotation. The choice of the angle is arbitrary in the following sense. We have fixed the coincidence of the two structures along [1 0 0]a (see Fig. 4), and have selected the normal to this direction to define the angle. The shear then fixes an orthogonal box and the remainder of the atoms transform by shuffles. One could pick another direction resulting in different shears and shuffles. Clearly, however, the mean monoclinic structure is rotated with respect to the hcp pseudostructure, so any choice would lead to some shear with consequences similar to those of the present selection. Studies [19] of twinning in Pu–1.7 at.% Ga revealed new twinning systems in addition to those predicted by Crocker [5]. An example of the observed twinning is shown in Fig. 9: we note that the configuration is consistent with Fig. 8(c). These twins [7,19], all of the (h0k) prismatic type T T x1 x x ′ 2 2 x ′1 h λ L terrace plane habit plane T T T T T T T T x ′ φ 1 α′ δ θ b a Fig. 6. (a) Habit plane when disconnections remove misfit in the x0 1- direction. (b) Tilt wall produced by the b0 2 component. Fig. 5. Disconnections, terrace planes, and misfit strains at d/a interface. J.P. Hirth et al. / Acta Materialia 54 (2006) 1917–1925 1921