Tetragonal-1o-monoclinie phase transformation in ceria-zirconia 541 Table 1. The lattice parameters of the monoclinic and tetragonal phases in ZrO2-12 mole% CeO2 [15]. b(nm) B(deg) Monoclinic 0.5203 0.5217 0.5128 0.5224 evaluating T and factoring it into R, B and P. These methods differ mathematically and are commonly referred to as different theories, but they rest on the same physical assum ptions and so, of course, yield identical results The lattice deformation can be expressed in terms of the lattice parameters of the parent and product phases by an appropriate choice of the lattice correspondence In general, the lattice deformation alone does not give rise to the invariant-plane condition and both the rigid-body rotation and the lattice-invariant deformation (such as slip or twinning)are required to take place in the product phase, so as to minimize the elastic strain energy On applying the crystallographic theory to the t-m transformation in ceria- zirconia, (101) [101] shear is considered as the lattice-invariant strain and the lattice deformation is chosen from the lattice correspondence(LCB) between unit cells of the tetragonal and monoclinic lattices. The lattice deformation matrix, B, for LCB-I can be written as Cm am sin(90-B)0 0=cos(90-6)0 where at, Ct, and am, bm, Cm are the lattice parameters of the tetragonal and monoclinic phases, respectively, and B is the monoclinic angle. The lattice param eters of the parent and product phases for ZrO2-12 mole% CeO2 [15] are given in table 1. The procedure for obtaining crystallographic parameters has been thoroughly described by WLR. Exactly following WLR, all crystallographic param- eters such as the habit plane orientation, the amount of Lis, the direction of the total shape deformation, etc. have been calculated for the t-m transformation in ZrO-12 mole% CeO,. Table 2 summarizes the results of numerical calculation of crystallographic parameters for LCB-l and LCB-2 In the ID theory, all solutions can be expressed in simple and analytical forms Moreover, since the ID approach assumes that the absolute magnitude of each distortion component is much smaller than unity, the successive occurrence of defor mations can be expressed by the addition of matrices with no attention paid to the order of occurrence of the deformations. The id theory has been applied successfully to the discussion of the crystallography of various martensites [20-22] Equation (I)in the ID analysis becomes T=R+B+Pevaluating T and factoring it into R, B and P. These methods differ mathematically and are commonly referred to as different theories, but they rest on the same physical assumptions and so, of course, yield identical results. The lattice deformation can be expressed in terms of the lattice parameters of the parent and product phases by an appropriate choice of the lattice correspondence. In general, the lattice deformation alone does not give rise to the invariant-plane condition and both the rigid-body rotation and the lattice-invariant deformation (such as slip or twinning) are required to take place in the product phase, so as to minimize the elastic strain energy. On applying the crystallographic theory to the t!m transformation in ceria￾zirconia, (101)t [101]t shear is considered as the lattice-invariant strain and the lattice deformation is chosen from the lattice correspondence (LCB) between unit cells of the tetragonal and monoclinic lattices. The lattice deformation matrix, B, for LCB-1 can be written as B ¼ cm at am at sinð90
Þ 0 0 am at cosð90
Þ 0 0 0 bm ct 0 B BBB BB@ 1 C CCC CCA ð2Þ where at, ct, and am, bm, cm are the lattice parameters of the tetragonal and monoclinic phases, respectively, and
is the monoclinic angle. The lattice param￾eters of the parent and product phases for ZrO2-12 mole% CeO2 [15] are given in table 1. The procedure for obtaining crystallographic parameters has been thoroughly described by WLR. Exactly following WLR, all crystallographic param￾eters such as the habit plane orientation, the amount of LIS, the direction of the total shape deformation, etc. have been calculated for the t!m transformation in ZrO2-12 mole% CeO2. Table 2 summarizes the results of numerical calculation of crystallographic parameters for LCB-1 and LCB-2. In the ID theory, all solutions can be expressed in simple and analytical forms. Moreover, since the ID approach assumes that the absolute magnitude of each distortion component is much smaller than unity, the successive occurrence of defor￾mations can be expressed by the addition of matrices with no attention paid to the order of occurrence of the deformations. The ID theory has been applied successfully to the discussion of the crystallography of various martensites [20–22]. Equation (1) in the ID analysis becomes T ¼ R þ B þ P: ð3Þ Table 1. The lattice parameters of the monoclinic and tetragonal phases in ZrO2-12 mole% CeO2 [15]. Phase a (nm) b (nm) c (nm)
(deg) Monoclinic 0.5203 0.5217 0.5388 81.09 Tetragonal 0.5128 0.5224 Tetragonal-to-monoclinic phase transformation in ceria-zirconia 541