Tetragonal-10-monoclinic phase transformation in ceria-zirconia 543 In order for T to describe the invariant-plane deformation, the following form T=00T3 The physical meaning of equation(7)is that no distortion exists on the xf and x2 planes. The resultant matrix T has six unknown parameters, i.e., g, w1, W2, W3, 0 and These six parameters can be solved using the condition of the invariant-plane deformation in equation(7). In other words, there are six simultaneous equations to solve for the six unknowns: Tll=T12=T21=T22=T31=T3=0. The expli forms of these six equations can be obtained from equations(4)and(5)as TH1=(1-g/2)os2φ-8 sin cosφ+e2sin2p=0 T12=(E1-g/2)sin cos cos 0+& cos A cos -w3 cos6-(w2-g/2)sin A cos o E2Sinφcosφcos6-wsinφsinb=0 21=(E1-g/2)in中cos中cos-8cos6sin2φ+m3cosb-E2 sin p coscos 6 +w1sinφsin6=0 T22=(E1-g/2)sin2cos26+8cos20 sin o cos o+E? cos2 cos20 T31=(s1-g/2)sinφcosφsin6-8 sin e sin2φ+m3sine-s2sinφcosφsin +(-w2+g/2)cos B cos o-w, sin o cos 8=0 T32=(E1-g/2)sin* cos e sin 0 +8 cos 0 sin 0 sin o cos o +(-w2 +g/2)sin o + E2 cos"o sin e cos 8+ w cos -(E3+g/2)sin e cos 6=0 These six unknown parameters can thus be found. Therefore, the rotation matrix on the p coordinate system is obtained as Etan中-E E? tan o 0 where tanp=(8-[82-4E2(E1 +E3)) 2)/2E2. Using the numerical values for EI, E2, E3 and 8 given previously, the normal to the habit plane is found to be [0.3009, 0.9536, 0] All unknown parameters have been solved, and the direction of the total shape deformation, d, and its magnitude, m, can be obtained from the following equations d"=[13, T23, T331 (T3)2+(y2+(T3)2] Thus, the infinitesimal deformation approach has also been applied successfully to the t-m transformation in ZrO2-12 mole% CeO, for LCB-1. Exactly the same procedure has been repeated to obtain the crystallographic solutions for LCB-2 Two sets of solutions are listed in table 4In order for T to describe the invariant-plane deformation, the following form must be satisfied: Tn ¼ 0 0 Tn 13 0 0 Tn 23 0 0 Tn 33 0 B @ 1 C A n ð7Þ The physical meaning of equation (7) is that no distortion exists on the xn 1 and xn 2 planes. The resultant matrix Tn has six unknown parameters, i.e., g, w1, w2 , w3, and . These six parameters can be solved using the condition of the invariant-plane deformation in equation (7). In other words, there are six simultaneous equations to solve for the six unknowns: Tn 11 ¼ Tn 12 ¼ T n 21 ¼ T n 22 ¼ T n 31 ¼ T n 32 ¼ 0. The explicit forms of these six equations can be obtained from equations (4) and (5) as T n 11 ¼ ð"1 g=2Þcos2  sin  cos  þ "2 sin2  ¼ 0 T n 12 ¼ ð"1 g=2Þsin  cos  cos þ cos cos2  w3 cos ðw2 g=2Þsin cos  "2 sin  cos  cos w1 sin  sin ¼ 0 T n 21 ¼ ð"1 g=2Þsin  cos  cos cos sin2  þ w3 cos "2 sin  cos  cos þ w1 sin  sin ¼ 0 ð8Þ T n 22 ¼ ð"1 g=2Þsin2  cos2 þ cos2 sin  cos  þ "2 cos2  cos2 þ ð"3 þ g=2Þsin2 ¼ 0 T n 31 ¼ ð"1 g=2Þsin  cos  sin sin sin2  þ w3 sin "2 sin  cos  sin þð w2 þ g=2Þcos cos  w1 sin  cos ¼ 0 T n 32 ¼ ð"1 g=2Þsin2  cos sin þ cos sin sin  cos  þðw2 þ g=2Þsin  þ "2 cos2  sin cos þ w1 cos  ð"3 þ g=2Þsin cos ¼ 0: These six unknown parameters can thus be found. Therefore, the rotation matrix on the p coordinate system is obtained as Rp ¼ 0 "2 tan  "3 "2 tan  0 0 "3 0 0 0 @ 1 A p ð9Þ where tan  ¼ { [ 2 4"2("1 þ"3)]1/2}/2"2. Using the numerical values for "1, "2, "3 and given previously, the normal to the habit plane is found to be [0.3009, 0.9536, 0]. All unknown parameters have been solved, and the direction of the total shape deformation, d, and its magnitude, m, can be obtained from the following equations dn ¼ ½Tn 13, T n 23, T n 33n m ¼ ðTn 13Þ 2 þ ðTn 23Þ 2 þ ðT n 33Þ 2
1=2 : ð10Þ Thus, the infinitesimal deformation approach has also been applied successfully to the t!m transformation in ZrO2-12 mole% CeO2 for LCB-1. Exactly the same procedure has been repeated to obtain the crystallographic solutions for LCB-2. Two sets of solutions are listed in table 4. Tetragonal-to-monoclinic phase transformation in ceria-zirconia 543