Criteria: net vectors <20 Å; lengths of corresponding vectors agree within 15 percent; corresponding angles in networks agree within 10°. Contact planes are given in Miller index (intercept) notation. Net vectors r, and r given in terms of unit cell translations; a, is angle between ve and 2a is similarly defined in net B. The sort key used in these calculations was ||T|-|FR|| /[(|ra|+|r|)/2] + ||TA2|-|82|| /[(гA2+TB2)/2] + |a, -a /[(a,+a)/2] + 0.15[maximum (area a, area b)-25]/400. BI B2 "Disloc" is an estimate of the additional number of dislocations due to misfit introduced at the interface assuming no elastic strain in either compuner i.e., b This match included because of statement in Frondel et al. [9]; length of 301 vector is 29.07 Å (outside usual range). Other merit figures for this ta' were 5.37, 5.31, 5.23, and 5.22, corresponding to different sets of parallel planes. complete criterion for ranking twinning possibilities. though we considered fits in only two dimensions, the ht cases of possible merohedry (first 8 lines in table 1) ve the highest merit figures (≥6.27), consistent with the ventional twinning view of coincident lattices. It is teworthy that identity matches do not require the misfit locations usually necessary [10] to accommodate nensional misfit between different substrates. It is also teworthy that in some twins the thickness of the undary layer may reflect a gradual change over many it cells (see reference 11 for a specific example) and may ow the effect of elastic strain. It is not obvious in such es that the relationship between individuals of the twin 1 be described in terms of a coincident lattice. 7. Epitaxy Between Ca,(PO4)3OH and Other Calcium Ortho-Phosphates The calcium phosphates we have considered here as ssibilities for forming epitactical relationships with ¿(PO4)3OH are Ca¡H2(PO)·5H2O [12,13], CaHPO, 2H2O ·], CaHPO [15], Ca(H2PO)1⁄2·H2O [16,17], Ca1O(PO4)2 }], B-Ca3(PO4)2 [19] and Ca5(PO4)2SiO4 [20] (see table 2 common names). These materials comprise the more ble and common calcium phosphates. Table 2 contains the matched nets obtained using only tric criteria with the constraints described earlier. The ge number of acceptable matches is partly a result of the erous constraints, but also results from ignoring ctural aspects. Table 3 contains details of the first five matching nets each case, ranked by sort key [4]. An estimate of the location density due to misfit in the absence of all elastic ain is provided. An estimate of the corresponding misfit location density is given by 2 inclusion of dislocations at the interface for films >200 Å. At this stage, we may say that the most favorable case of epitaxy in which all the misfit is taken up by dislocations will require an additional 108 dislocations/cm2 at the interface, to be compared with dislocation densities of 102 to 102 dislocations/cm2 in homogeneous solids. A dislocation density of 1012 dislocations/cm2 corresponds to an inter-dislocation spacing of ~100 Å, i.e., ~30 ions for close-packed Ca and PO, ions. For each of the seven pairs of compounds, the structurematching procedure was applied to the 10 matching nets. determined from the sort key to be the most probable candidates for epitaxy. The Ca and P positions in the crystal structures were used to specify their gross structural details. The matching nets with the highest merit figures are given in table 4. Table 4 also shows the contact planes, the sort order, and the atoms found to be involved in the structural matches. The number of each type of atom is given for equivalent areas of the contact planes. In all cases, a 2 Å thick slice was used and all subsets and redundancies were removed, e.g., 2Ca(2),2P, being a subset TATA2 TB (82) of a parallel slice 2Ca(1),2Ca(2), 2P in Cag(PO4)3OH, was not considered in the (010) vs. (100) matching of Cas(PO4)3OH with CagH2(PO4)6.5H2O. B2 ere a, is the angle between vectors Fal Reasons for considering only Ca and P positions in the crystal structure include (i) reducing the computational expense to manageable proportions, (ii) allowing the program to fit in the computer (>65K of storage may otherwise be required although an overlay scheme has now reduced the importance of this second consideration), and (iii) specifying only the major features of the structure because dislocations and elastic strain at the interface may change the local details somewhat. One assumption when only Ca and P atoms are included is that these atoms are sufficient to specify the major features of the structure. CaHPO4+2H20 [102,100] [001,100] Ca(H2PO4)2-H2O B-Ca3(PO4)2 Cas(PO4)2SiO4 (010) (311) [102,100] [01ī,112] 0.129 2.23 X 108 [011,112] 0.129 2.23 × 108 [010,001] 0.130 2.31 X 1010 [īlī,11ī] 0.132 9.08 × 1010 0.132 0.053 [111,111] 0.103 TABLE 4. Details of matches for best inter-plane merit figures (>4.5) between Ca (PO4)3OH and several other calcium phosphates. Results obtained with program MATCH2. Thickness of slice through structure = 2 Å total (1 Å out from central plane). Motif radius = 6:0 Å. Input was results of MATCHI Contact planes and sort key are as given in table 3. The ordinal number of the match in the output from MATCH1 (sorted on the sort key) is given in the sort order column. 'Merit figure includes allowing for only half occupancy of Ca(4) site. NO. OF OCCURENCES 20 15 10 This will not be true if for example a structure contains an 8. Feasible Cases of Epitaxy Figure 3 shows the frequency of occurrence vs. interplane merit figure for the 364 inter-plane merit figures obtained in this study. Few cases of epitaxy have interplane merit figures greater than our proposed threshold value of 6.0. Those that do involve epitaxy between Cas (PO4)3OH and Cas H2(PO4)6.5H2O, Cas (PO4)3OH and Ca4O(PO4)2, and Ca¿(PO4)3OH and CaŚ(PO4)2SiO4. 0.0 FIGURE 3. Distribution of number of occurrences versus interplane figureof-merit for calculations described in this paper. Table 5 shows the detailed atom:atom matching for the highest inter-plane merit figures for the various cases of epitaxy considered. For the Cas(PO4)3OH/CaH2(PO4)∙5H.0 case with an inter-plane merit figure of 8.32, the detailed atom-atom matching is reasonable both chemically (a indicated by the magnitudes of the individual atom-atom matches and by visual inspection of the structures) and |